Physics Reports vol.435

Physics Reports 435 (2006) 1 – 31 www.elsevier.com/locate/physrep

Complementarity and uncertainty in Mach–Zehnder interferometry and beyond Paul Buscha, b,∗ , Christopher Shilladayc a Department of Mathematics, University of York, UK b Perimeter Institute for Theoretical Physics, Waterloo, Canada c Department of Mathematics, University of Hull, UK

Accepted 14 September 2006 Available online 24 October 2006 editor: J. Eichler

Abstract A coherent account of the connections and contrasts between the principles of complementarity and uncertainty is developed starting from a survey of the various formalizations of these principles. The conceptual analysis is illustrated by means of a set of experimental schemes based on Mach–Zehnder interferometry. In particular, path detection via entanglement with a probe system and (quantitative) quantum erasure are exhibited to constitute instances of joint unsharp measurements of complementary pairs of physical quantities, path and interference observables. The analysis uses the representation of observables as positive-operator-valued measures (POVMs). The reconciliation of complementary experimental options in the sense of simultaneous unsharp preparations and measurements is expressed in terms of uncertainty relations of different kinds. The feature of complementarity, manifest in the present examples in the mutual exclusivity of path detection and interference observation, is recovered as a limit case from the appropriate uncertainty relation. It is noted that the complementarity and uncertainty principles are neither completely logically independent nor logical consequences of one another. Since entanglement is an instance of the uncertainty of quantum properties (of compound systems), it is moot to play out uncertainty and entanglement against each other as possible mechanisms enforcing complementarity. © 2006 Published by Elsevier B.V. PACS: 03.65.Ta; 03.75.Dg; 03.65.Ca Keywords: Complementarity principle; Uncertainty principle; Joint measurement; Mach–Zehnder interferometer; Path marking; Erasure

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. The notions of complementarity and uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Bohr, Heisenberg, and the consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. The notion of complementarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1. Complementarity in Bohr’s writings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2. Complementarity and wave–particle duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∗ Corresponding author.

E-mail addresses: [email protected] (P. Busch), [email protected] (C. Shilladay). 0370-1573/$ - see front matter © 2006 Published by Elsevier B.V. doi:10.1016/j.physrep.2006.09.001

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2.2.3. Modern formulations of complementarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4. Preparation complementarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.5. Measurement complementarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.6. Complementarity principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. The notion of uncertainty and the uncertainty principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1. Origin of the uncertainty principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2. Three varieties of uncertainty relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3. Uncertainty principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4. Uncertainty or indeterminacy? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Complementarity vs. uncertainty? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Interlude: measurement theoretic concepts and tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Effects and POVMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Joint measurability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Measurement implementation of a POVM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Path marking and erasure in an atom and Mach–Zehnder interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. The atom-interferometric experiment of SEW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Mach–Zehnder interferometer: basic setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Path detection in outputs D1 , D2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Interference detection at D1 , D2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5. The path-marking setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6. Quantum erasure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7. Quantitative erasure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Complementarity and uncertainty in Mach–Zehnder interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Manifestations of complementarity in Mach–Zehnder interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Value complementarity from preparation uncertainty relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Quantitative duality relations are uncertainty relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Measurement complementarity from measurement inaccuracy relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5. Disturbance versus accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6. Uncertainty, disturbance, or entanglement? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1. Introduction Soon after the inception of quantum mechanics, the notions of complementarity and uncertainty were introduced to highlight features of the new theory, unknown to classical physics, which amounted to limitations of the preparation and measurement of atomic systems. Complementarity and uncertainty continue to attract the attention of researchers, inspiring novel experimental tests and demonstrations. One such experiment was proposed by Scully, Englert, and Walther (SEW) [1]. These authors described an atom interferometer in which entanglement is utilized to store information about the path of an atom: we will use the term “path marking” to refer to this process of path information storage in a probe system by way of establishing correlations between the atom and the probe. SEW’s claim is that in their scheme, the standard explanation for the loss of interference upon path marking in terms of classical random momentum kicks, and hence the uncertainty relation, is not applicable. The fact that in the SEW experiment the interference pattern was wiped out not by classical kicks, which supposedly could be associated with an indeterminate momentum, but by entanglement led to the suggestion that the principle of uncertainty is less significant than complementarity: “The principle of complementarity is manifest although the position–momentum uncertainty relation plays no role” [1, p. 111]; and in reply to a critique [2], SEW go further, stating that “the principle of complementarity is much deeper than the uncertainty relation although it is frequently enforced by
x
p
h¯ /2” [3]. A path-marking experiment similar to that proposed by SEW was experimentally realized in 1998 by Dürr et al. [4] and confirmed the conclusion of SEW, that neither mechanical disturbance nor the position–momentum uncertainty relation could explain the loss of interference. This, in turn, caused a controversy and led to grossly misleading articles in popular science journals, announcing “An end to uncertainty—Wave goodbye to the Uncertainty Principle, you don’t need it anymore.” [5]. Numerous papers appeared with conflicting conclusions as to whether some forms of

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random disturbance or uncertainty relations could always be invoked to explain the loss of interference when path marking was effected via entanglement, or whether entanglement was the more fundamental mechanism for this form of complementarity (e.g. [6–12]). The main aim of this paper is to study the roles and relative significance of the uncertainty principle (in an appropriate understanding), “disturbance” in measurement, and entanglement in the explanation of complementary quantum phenomena, such as the mutual exclusivity of path marking and interference detection. It seems to us that the continuing lack of consensus over these questions is largely due to persistent differences of general outlook on the principles of complementarity and uncertainty that are manifest in the development just reviewed, and due to the fact that these differences are usually not reflected upon. The two notions are often considered as logically related, and the uncertainty relation is frequently presented as a quantitative expression of complementarity. By contrast, the work of SEW raised controversy over the question whether the uncertainty relation is at all relevant to complementarity. In effect, the considerations of SEW and the conclusions drawn by Dürr et al. from their experiment suggest that complementarity and uncertainty refer to aspects of quantum physics that can be discriminated experimentally. We therefore find it necessary to develop a coherent account of the notions and principles of complementarity and uncertainty before we illustrate the manifestations of these principles in the analysis of a set of simple interferometric experiments. To this end, we will briefly analyze the existing contrasting views on complementarity and uncertainty, and review the development of various formalizations of these so-called principles (Section 2). Separating the broader “ideological” issues from the technical aspects will help to elucidate the interesting specific physical and conceptual questions indicated above, which can then be effectively addressed in terms of suitably defined notions of complementarity and uncertainty. The analysis of the experimental illustrations of these concepts will be carried out using the general descriptions of observables in terms of positive-operator-valued measures (POVM). Projection-valued measures (PVMs) are special cases of POVM and are called sharp observables. POVMs that are not PVMs are called unsharp observables. Some of the experimental schemes presented here are instances of joint measurements of an unsharp path and an unsharp interference observable. The measurement theoretic concepts and tools required for our analysis will be briefly reviewed in Section 3. In Section 4, we will discuss a range of atom and Mach–Zehnder interferometric experiments. We will briefly review the type of which-path and erasure experiments proposed by SEW and carried out by Dürr et al. (Section 4.1). This will prepare the definition and study of analogous Mach–Zehnder interferometric setups in Sections 4.2–4.7. The following experiments are presented within one common setting and analyzed in detail: use of the interferometer for path detection; an interference detection setup; introduction of path marking via entanglement with a probe system, which precedes the interferometer; quantum erasure; and quantitative quantum erasure. The language of POVMs proves useful in Section 5 for the analysis of the relationships between specific versions of complementarity and uncertainty as they manifest themselves in the present group of experiments. General conclusions are drawn in Section 6. In the presentation of complementarity and uncertainty developed here, these features are seen to be logically related within quantum mechanics, although they have their separate identities and roles. In a nutshell, uncertainty is a direct consequence of the linear structure of the Hilbert space formalism in that every vector state can be expressed as a linear combination of eigenvectors of some observable, whose values are therefore uncertain (or, more precisely, indefinite) in that state. This fact may be considered as the broadest formulation of the uncertainty principle, which is then usually expressed as a trade-off relation between pairs of noncommuting observables. The complementarity principle highlights an extreme form of that relation, stating that there are pairs of quantities that are such that complete determination of one quantity implies maximal uncertainty about the other (and vice versa). Thus, the formulation of complementarity presupposes the notion of uncertainty, but the principles of complementarity and uncertainty cannot be reduced to one another. Similarly, it is pointed out that entanglement is an instance of uncertainty. It is therefore pointless to attempt to separate entanglement as a “mechanism” enforcing complementarity in addition to, and independently of, the uncertainty principle. We hope that this study helps not only to settle some long-standing foundational issues of quantum mechanics, but also to promote the idea that the time has come to introduce the concept and application of POVMs in the undergraduate teaching of quantum mechanics. POVMs are already being taught as a standard tool in courses on quantum

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mechanics, quantum computation and quantum information in a small but growing number of institutions around the world. The concepts of the underlying modern quantum theory of measurements are slowly being brought to the attention of teachers of undergraduate quantum physics. For example, simple implementations of POVMs for quantum optical experiments were presented by H. Brandt in the American Journal of Physics [13]. There are a number of books giving elementary demonstrations of the use of POVMs in the analysis of conceptual problems and quantum experiments, the first being the fine text monograph by the late Asher Peres [14]. We dedicate this paper to the memory of Asher Peres, friend and mentor to one of us (P.B.), and mediator between quantum foundations and quantum applications. 2. The notions of complementarity and uncertainty 2.1. Bohr, Heisenberg, and the consequences The principles of complementarity and uncertainty were introduced by Niels Bohr [15] and Werner Heisenberg [16] some 80 years ago in their efforts to develop an intuitive understanding of quantum physics. Their concern was to explain the dramatic deviations of this new theory from classical physics, which manifested themselves in wave–particle duality and the impossibility of defining and observing sharp particle trajectories. From the beginning there was disagreement over the relative significance and merits of the notions of complementarity and uncertainty [17]. Bohr considered complementarity as fundamental for his interpretation of quantum mechanics, which was to be based on classical concepts and intuitions, rather than mathematical formalism. He thought of the uncertainty relation (primarily) as a formal, quantitative expression of complementarity. By contrast, Heisenberg seemed to have had little use for the notion of complementarity; he sought to develop an intuitive understanding of quantum mechanics that derived from the formalism itself. In his reminiscences, Heisenberg summarized the key that led him to the discovery of the uncertainty relation in the famous sentence that he ascribed to Einstein: “it is the theory which decides what can be observed” [18]. It made him realize that there was room in quantum mechanics for simultaneous approximate values of position and momentum, and thus for unsharply defined trajectories as they are observed in cloud chambers. Bohr and Heisenberg eventually reached a compromise on their divergent views in the terminology of what came to be known as the Copenhagen interpretation of quantum mechanics. However, the fundamental differences in their philosophical outlooks on quantum mechanics were never resolved, as is manifest from their writings and recorded interviews and discussed, for example, in the book of Jammer [17]. The appearance of unity maintained nevertheless by the Copenhagen pioneers has made it difficult to obtain a coherent account of the so-called Copenhagen interpretation, and there is a vast body of historico-philosophical literature on this subject with rather divergent conclusions. Here we are concerned with the reception of the notion of complementarity by the physics community. A survey of the textbook literature exhibits that three points of view have evolved concerning the relationship and interplay between the principles of complementarity and uncertainty. However, it seems to us that a fairly unambiguous, systematic account of the possible formalizations of the notion of complementarity within quantum mechanics has been obtained. We briefly recall the three distinct interpretational stances on the status of the principle of complementarity, and then review the possible rigorous formalizations of the notion of complementarity. There is one group of authors who reiterate the account that has become part and parcel of the so-called Copenhagen interpretation: according to this view, the uncertainty relation constitutes the quantitative expression of the principle of complementarity (e.g., [19]), and quantum mechanics has been said to be best understood as the theory of complementarity [15,19–21]. Within this group, some describe the uncertainty principle as enforcing complementarity, while others deny the uncertainty relation the status of principle, playing down its significance in favor of complementarity [20]. (Asher Peres had his own, characteristically humorous way of doing this: in his textbook [14], the only occurrence of the term “uncertainty principle” is in the index, with the page reference indicating this very index page. On the other hand, the principle of complementarity is also described only briefly, while there are numerous occurrences of variants of uncertainty relations.) A second group of authors (including, e.g., Dirac, von Neumann, Feynman) avoids carefully any mention of the term complementarity, apparently in accordance with a widespread perception that Bohr’s presentations of this concept had remained rather obscure. These authors invoke the uncertainty principle as the reason for the impossibility of

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making simultaneous path and interference observations with arbitrary precision. This approach is consistent with Heisenberg’s point of view, according to which it is possible but not necessary to refer to complementarity for an intuitive interpretation of quantum mechanics. Finally, today there is a widespread sense that complementarity and uncertainty are best regarded as consequences of quantum mechanics which highlight characteristic features of that theory but need not be held up as independent principles. We will reassess these positions after the following brief review of the evolution of current formulations of the complementarity and uncertainty principles. 2.2. The notion of complementarity 2.2.1. Complementarity in Bohr’s writings Bohr introduced the term complementarity in his 1927 lecture in Como: “The very nature of the quantum theory thus forces us to regard the space–time co-ordination and the claim of causality, the union of which characterizes the classical theories, as complementary but exclusive features of the description, symbolizing the idealization of observation and definition, respectively.” [15, p. 580]. Complementarity was thus originally conceived as a relationship between pairs of descriptions, or phenomena, which are mutually exclusive but nevertheless both required for a complete account of the physical system under consideration. Bohr considered complementarity as a “rational generalization” of the classical notion of a causal space–time description of physical phenomena. He argued that in quantum physics, causality (represented by conservation laws and deterministic equations of motion) and space–time description fall apart, as a consequence of what he called the quantum postulate, according to which every observation (i.e., measurement) induces an uncontrollable, unavoidable, and nonnegligible change of the phenomenon (i.e., the observed system). In fact, as Bohr put it, space–time coordination requires observation and hence uncontrollable state changes, whereas a causal account of phenomena requires the possibility of defining a state and hence precludes interaction, and thus measurement. For Bohr, the definition of a state and its observation constituted idealizations which were simultaneously applicable to any desired degree of accuracy in classical physics. According to the quantum postulate, the finite magnitude of Planck’s quantum of action introduces a limitation to the simultaneous applicability of these idealizations: the influence of interactions required for an observation is no longer negligible for atomic systems where the characteristic quantities of the dimension of an action are comparable in magnitude to Planck’s constant. According to Bohr, such measurement interactions leave the object and measuring apparatus in a situation that does not allow an independent description of either system: “Now, the quantum postulate implies that any observation of atomic phenomena will involve an interaction with the agency of observation not to be neglected. Accordingly, an independent reality in the ordinary physical sense can neither be ascribed to the phenomena nor to the agencies of observation” [15]. We agree with Howard [22] that this passage should be read as expressing the entanglement of object and apparatus after the measurement interaction. Following Bohr’s own practice, it has become customary to interpret his informal descriptions of complementarity in terms of position and momentum observables as follows: these observables are complementary in that for a deterministic description of the trajectory of a particle, the values of both observables are required, but according to the quantum mechanical limitations of preparing and measuring the values of these observables, sharp values cannot be assigned to them simultaneously. This impossibility of defining sharp trajectories then gives room for the existence and explanation of interference phenomena. A lucid account of complementarity along these lines was given by Pauli as early as 1933 [19]. In his reply of 1935 to the famous challenge of Einstein et al. [23], Bohr defines the complementarity of position and momentum [24] by reference to “the mutual exclusion of any two experimental procedures, permitting the unambiguous definition of complementary physical quantities.” Complementarity thus describes, according to Bohr, the limited way in which classical concepts can be applied in the description of quantum experiments. Such a limitation is imposed by quantum mechanics (the quantum postulate), in accordance with the failure of classical physics to explain atomic phenomena. The importance of complementarity derives, for Bohr, from the necessity of expressing all physical experience in classical physical language. 2.2.2. Complementarity and wave–particle duality It should be noted that in contrast to the classical exclusivity of wave and particle theories that gave rise to the quantum puzzle of wave–particle duality, the concepts of position and momentum are simultaneously applicable in

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classical physics but not (without qualification) in quantum mechanics. This implies that wave–particle duality cannot simply be considered as an instance of complementarity, as is often suggested. There are pairs of phenomena (and of related observables) that the same type of system (e.g., electrons) can display and that are associated with intuitive ideas relating to either particles or waves. But there are also instances of particle and wave behaviors which do manifest themselves in the same experimental setup, such as the Compton scattering of a photon with an electron observed through a
-ray microscope, followed by the subsequent wave-like propagation of the light through the microscope, which accounts for the finite optical resolution. In this and similar examples, Bohr freely used wave and particle aspects in the analysis of one and the same phenomenon. Another simple example is the interference observation in a double slit experiment where the interference is explained by wave superposition while at the same time the pattern is built up by the successive recordings of the photons or electrons through their inelastic collisions with the molecules in the photographic plate. According to Bohr, the apparent contradiction that lies in the wave–particle duality is resolved through the realization of the limited simultaneous applicability of classical concepts that is complementarity. A dissolution, within quantum theory, of the problem of wave–particle duality was carried out somewhat differently and more formally by Heisenberg in his 1929 Chicago lectures [25]. He identifies particle and wave pictures with the quantized theories of particles and fields, and sees the consistency of the two pictures established by the equivalence of the quantum field theory, restricted to the subspace of N quanta, with the quantum mechanics of N particles. On this account, there is no need for a simultaneous application of wave and particle ideas since both formalisms can be used to give equivalent accounts of the experiments in question. These different assessments of the significance of wave–particle duality and of its resolution in quantum mechanics are one example of the persistent discrepancies between Bohr and Heisenberg. A detailed analysis of Heisenberg’s account of wave–particle duality, with similar conclusions to ours, has recently been given by Camilleri [26]. 2.2.3. Modern formulations of complementarity The complementarity idea was thus gradually transformed into the notion of complementarity of pairs of physical quantities, with an emphasis on the negative aspect of mutual exclusiveness of value assignments or experimental setups (such as path or interference detection). As will become evident from the discussion of the uncertainty principle below, the positive aspect of mutual completion, or complementation, that comprises the original meaning of the term complementarity, has been delegated to the uncertainty principle. The statement of mutual exclusiveness of experimental setups can be interpreted as constituting three distinct forms of complementarity; one that refers to the possibilities of state preparation, and two that refer to the possibilities of joint and sequential measurements, respectively. This distinction allows one to take into account the freedom in placing the Heisenberg cut between preparation (object system) and measurement (probe system, measuring apparatus), as well as the fact that often the state-preparing effects of measurements are utilized. Preparation complementarity is the impossibility of preparing states in which the two observables in question are simultaneously assigned sharp values. Measurement complementarity is the impossibility of performing simultaneous sharp measurements of these observables, or the impossibility of performing their measurements in sequence without mutual disturbance. In these general forms, complementarity applies to practically all pairs of noncommuting quantities (exceptions are pairs with some common eigenstates). Complementarity is usually understood to be a more specific relationship that singles out certain important pairs of observables, including, but not restricted to, canonically conjugate pairs such as position and momentum, or components of angular momentum and their associated angles. A comprehensive discussion of definitions of preparation and measurement complementarity together with examples of complementary pairs of observables are given in the monograph [27, (Section IV.2)] and the review [28]. We will restrict ourselves here to a brief summary with sufficient detail for the purposes of our subsequent analyses. 2.2.4. Preparation complementarity The most widely adoped form of preparation complementarity (e.g., [20,29,1,12]) is probably the following one that we will refer to as value complementarity (following [27]): two observables are value complementary if whenever one has a definite value, the values of the other are maximally uncertain. A value of observable A is definite if it occurs with probability equal to one in a measurement of A, and the values of observable B are maximally uncertain if they occur with equal probabilities in a measurement of B.

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Formally, the value complementarity of two observables A, B with discrete nondegenerate spectra and associated eigenbasis systems {k : k = 1, . . . , n}, {
:
= 1, . . . , n} amounts to the statement that any two eigenstates have constant overlap, that is, the numbers |k |
| are independent of k,
. Pairs of orthonormal basis systems with this property are called mutually unbiased. It is well known that any observable A with nondegenerate eigenstates 1 , . . . , n has at least one partner B with which it forms a value complementary pair; in fact
there are infinitely many such partners: first one can define any B with orthonormal system of eigenstates 
= √1n k e2ik
/n k ,
= 1, . . . , n. It is easily verified that A, B are value complementary. Note that the eigenvalues of the observables do not enter the definition of value complementarity. Further value-complementary partners BU of A are obtained by taking BU := U BU ∗ , where U is any unitary operator that commutes with A. One must note that the notion of value complementarity does not easily extend to continuous quantities or those with unbounded spectra. For example, in order to consider position and momentum as value complementary, one must use their improper eigenstates; one could try to capture the idea of nearly sharp position values by means of sequences of normalizable states whose position distributions approach a Dirac distribution. In that sequence, the momentum distributions become arbitrarily widely spread out, but it is not clear how to express the idea that these momentum distributions approach a uniform distribution, which does not exist on the basis of normalizable states. Similarly, in the case of number and conjugate phase, it is true that for a number eigenstate, the phase is distributed uniformly, but there are no states with definite, sharp values of the phase, nor are there states with uniform number distributions [30]. To maintain the idea of value complementarity, one would have to allow it to become a nonsymmetric relation. These limitations do not arise if one adopts a slightly weaker form of preparation complementarity, known as probabilistic complementarity. Observables A, B are probabilistically complementary if and only if for their spectral projections E A (X), E B (Y ) associated with bounded intervals X, Y (such that the projections E A (X), E B (Y ) are different from the null operator O and the unit operator I ) it is true that whenever the probability tr[E A (X)] = 1 then 0  = tr[E B (Y )]  = 1, and vice versa. This is to say that if the value of A is definitely in interval X, the value of B is never certain to be in any interval Y (or its complement). This notion applies without difficulty to the position–momentum and number-phase pairs; but it also allows any two spin components of a spin- 21 system to be complementary. Probabilistic complementarity is known to be equivalent to the statement that for all bounded intervals X, Y (such that E A (X), E B (Y )  = O, I ), the associated spectral projections of A and B satisfy E A (X) ∧ E B (Y ) = O,

(1)

E A (X) ∧ E B (R\Y ) = O,

(2)

E A (R\X) ∧ E B (Y ) = O.

(3)

Here, P ∧ Q denotes the intersection of the projections P and Q, that is, the projection onto the intersection of their ranges; R\X denotes the complement of the set X. The equation P ∧ Q = O is equivalent to the statement that P and Q have no common eigenvectors associated with their eigenvalue 1. 2.2.5. Measurement complementarity Measurement complementarity can be specified in a similar vein to refer to a pair of observables A, B for which a sharp measurement of one of them makes any attempt at measuring the other one simultaneously or in immediate succession completely obsolete. The first form of measurement complementarity, the impossibility of joint measurements, is a special instance of von Neumann’s theorem [31], according to which two observables are jointly measurable if, and only if, they commute. In a more specific, stronger form, the measurement complementarity of two observables A, B can be expressed as an exclusion relation for the quantum operations describing the state changes due to the measurements of A and B. This characterization was again found to be equivalent to the relation (1) (see [27, Section IV.2.3]), thus demonstrating the match, stipulated already by Bohr, between the possibilities of preparation and the possibilities of measurement. Measurement complementarity in the case of sequential measurements will be taken to mean that due to the effect of the A measurement, the B measurement will not recover any information whatsoever about B in the (input) state immediately prior to the A measurement. In extreme cases it may happen that, although the second setup is devised to measure observable B, the statistics obtained in the presence of the A measurement is independent of the input state; the

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observable effectively measured in the second measurement provides no information about the system state prior to the A measurement. This is again a special instance of a general theorem in the quantum theory of measurement (discussed, e.g., in [27, pp. 42–44]), according to which a sequence of a measurement of observable A followed immediately by a measurement set up to measure observable B constitutes a joint measurement of A and some unsharp observable B  (represented by a POVM) that commutes with A. If A and B are value complementary observables with mutually unbiased eigenbases and if A is measured first by a von Neumann measurement, then the observable B  actually measured by a subsequent B measurement scheme is in fact trivial, in that its statistics carries no information about the system state prior to the A measurement. Measurement complementarity is thus seen to reflect the fact that in quantum mechanics, every nontrivial measurement must alter the system’s state, at least in the case of some input states. There can thus be no information gain without “disturbance” in quantum mechanics. As will be discussed in the next subsection, the related issue of a measurement of one variable disturbing the distributions of other, noncommuting variables has been highlighted by Heisenberg by means of his famous thought experiments illustrating the uncertainty relation. This feature of quantum mechanics—the fact that measurements necessarily alter the system under investigation, hence the impossibility to “see” the system in its undisturbed form—has impressed the scientifically interested public as being of such fundamental importance that it is now widely known under the name “Heisenberg effect” or also “observer effect” (as is quickly confirmed by an internet search). In fact, these terms are used to denote loosely analogous phenomena in a variety of disciplines, ranging from sociology, political science or market research over anthropology and population ecology to computer science (where certain forms of programming errors are called Heisenbugs). We will see value complementarity and the two versions of measurement complementarity (for joint or sequential measurements) at work in the discussion of the Mach–Zehnder interferometry experiments; if the path observable is represented by z , any of its associated interference observables, represented by cos x + sin y , is a complementary partner. 2.2.6. Complementarity principle The complementarity principle is the statement that there are pairs of observables which stand in the relationship of complementarity. As we saw above, this is satisfied in quantum mechanics for observables the eigenvector basis systems of which are mutually unbiased. We conclude that the “principle” of complementarity, as formalized here, is a consequence of the quantum mechanical formalism. There seems to be no need to speak of a complementarity principle, unless one sets out to use such a principle in a more general framework to deduce quantum mechanics. Here we follow the common practice of speaking of the complementarity principle as a description of a remarkable feature of quantum mechanics. 2.3. The notion of uncertainty and the uncertainty principle 2.3.1. Origin of the uncertainty principle The uncertainty principle was introduced in Heisenberg’s seminal paper of 1927 [16]; although he did not speak of a principle, he made it very clear that he considered the uncertainty relation as fundamental for an intuitive understanding of quantum mechanics. He saw it as an expression of the physical content of the canonical commutation relations for conjugate pairs of quantities, and considered that these algebraic relations should form the basis for a derivation of the formalism of quantum mechanics. Heisenberg introduced quantum mechanical uncertainty with the following words: “It is shown that canonically conjugate quantities can be determined simultaneously only with a characteristic inaccuracy,” and “…the more precisely the position is determined the less precisely the momentum is known and conversely.” Often the uncertainty principle is reduced to the idea, referred to above as the Heisenberg effect, that any measurement “disturbs” the system in question. In the case of the canonically conjugate position and momentum observables of a particle, this “disturbance” is often identified with a momentum kick imparted on the particle during the measurement. If taken as generally valid, the explanation of the uncertainty principle in terms of momentum kicks is an incorrect conflation whose origin must be seen in Heisenberg’s discussion of a position measurement with a
-ray microscope. Heisenberg pointed out that the higher the accuracy of the position determination was, the shorter the wavelength of the photon, and therefore the larger the momentum exchange with the observed particle. In a “note added” at the end of his 1927 paper, he credits Bohr for the correction that the magnitude of the momentum transfer did not constitute a cause

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for the particle’s momentum to become indeterminate. Bohr’s explanation of the necessary momentum uncertainty was based on the dual nature of the light used in the observation: the particle aspect of the photon accounts for the momentum exchange resulting from Compton scattering, while the wave aspect gives rise to an uncertainty in the momentum inference due to the finite aperture of the microscope. Thus one could say that according to Bohr it is the quantum nature of the probe system utilized in a measurement that enforces the necessary uncertainty trade-off. 2.3.2. Three varieties of uncertainty relations A careful reading of Heisenberg’s 1927 paper and his 1930 Chicago lecture notes [25] shows that he has in fact distinguished three variants of uncertainty relations. It is evident that Heisenberg considered measurements to produce (approximate) eigenstates of the measured observable, corresponding to the measured value. Thus he describes the outcome of an attempted joint measurement of position and momentum in terms of the standard deviations of position and momentum observables in a Gaussian wave function centered at the measured values. The position and momentum uncertainties in this conditional final state are then taken to represent the inaccuracies of the joint measurement. Here Heisenberg brings together two versions of uncertainty relations: the uncertainty relation (Q, ) · (P , )


h¯ 2

(4)

for state preparations, according to which separate measurements of position and momentum in a (vector) state  have distributions with widths (standard deviations) satisfying this uncertainty relation; and a trade-off relation
q ·
p


h¯ 2

(5)

for the inaccuracies of joint measurements of these noncommuting observables. Heisenberg did not have at his disposal a precise quantum-mechanical notion of joint measurement of noncommuting observables. Such a notion was developed only several decades later, after POVMs had been introduced and made available to describe unsharp or approximate measurements. Heisenberg does grapple with the notion of joint unsharp measurement and comes close to a solution by considering sequences of measurements. For example, he considers the diffraction of a matter wave at a slit and shows that if the particle’s momentum was initially sharp, this precision in the definition of momentum becomes degraded during the passage through the slit which effects an approximate localization of the particle. Considered as a sequence of a sharp momentum measurement followed immediately by an approximate position measurement, the outcome of the sharp momentum determination is thus seen to be modified into an unsharp momentum determination, due to the “disturbing” influence of the approximate position determination. The resulting inaccuracies in the definitions of position and momentum are shown to satisfy an uncertainty relation. Indeed, the variance (P , ) of momentum after passage through the slit can be taken to represent both the accuracy
p of the momentum determination in the combined measurement scheme and a measure of the disturbance Dp of the (initially sharp) momentum through the unsharp position determination: (P , ) =
p = Dp. Likewise, the position uncertainty (Q, ) upon passage through the slit reflects the accuracy
q of the position measurement. We thus have h¯
q · Dp
. 2

(6)

This version of uncertainty relation constitutes an accuracy–disturbance trade-off relation for sequences of measurements. It has been carefully discussed in the context of interference experiments by Pauli in his 1933 review [19]. In the form described here, the disturbance of the distribution of an observable B through a measurement of A is measured in terms of the variance of B in the state immediately after the selective A measurement operation, which is to be compared to the (near) zero variance of B in an initial (near) eigenstate. In this formulation, the accuracy–disturbance relation follows from the preparation uncertainty relation. In a more general approach, the disturbance of the distribution of B during a measurement of A should be described by some measure of the difference between the distributions of B before and after the nonselective A measurement, and it should depend on the accuracy of the A measurement. Interestingly, rigorous and general formulations of such disturbance uncertainty relations have been investigated only rather recently (e.g., [29,32,33]). A review of rigorous formulations of all three types of uncertainty relations for position and momentum is given in [34].

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2.3.3. Uncertainty principle We propose that the term uncertainty principle refers to the broad statement that there are pairs of observables for which there is a trade-off relationship in the degrees of sharpness of the preparation or measurement of their values, such that a simultaneous or sequential determination of the values requires a nonzero amount of unsharpness (latitude, inaccuracy, and disturbance). This comprises the above three versions of uncertainty relations. There are a variety of measures of uncertainty, inaccuracy, and disturbance with which such trade-off relations can be formulated, usually in the form of inequalities. The term “principle” refers here to the fact that the uncertainty relations highlight an important nonclassical feature of quantum mechanics. They are a formal consequence of the noncommutativity of the observables in question. Being inequalities, the uncertainty relations can hardly be considered adequate as postulates from which to derive quantum mechanics; however, they have been used to rule out the field of real numbers in favor of the complex numbers as the underlying field for the Hilbert space formulation of quantum mechanics [35]. 2.3.4. Uncertainty or indeterminacy? A discussion of the uncertainty principle would not be complete without a comment on the nature of quantum mechanical uncertainty. The fact that many observables do not have definite values even in pure states, which represent maximal available information about the physical system under consideration, stands in strong contrast to the situation in classical physics. If in a pure state  the probability for a property represented by a projection P is neither one nor zero, the value of any physical quantity represented by that projection must be considered to be objectively indeterminate or indefinite, rather than only subjectively unknown or uncertain. According to the Kochen–Specker theorem (discovered by Kochen and Specker [36] and independently by Bell [37]), any attempt at hypothetically assigning definite values to sets of non-commuting observables leads to contradictions even in the case of rather small, finite sets of such observables. (Asher Peres was one of the champions in finding smaller and smaller sets of observables with Kochen–Specker contradictions; see his book [14].) The assignment of definite values to the position and momentum of a particle in the Bohm interpretation is no exception to this conclusion since these value attributions are contextual, that is, they are different for different measurement setups. The conclusion is that the term indeterminacy principle appears to be more appropriate. However, “uncertainty principle” has become the standard name and shall be used here, with the proviso that one should beware the misleading connotation of subjective ignorance that it carries. 2.4. Complementarity vs. uncertainty? The above review shows that the concepts of complementarity and uncertainty highlight two aspects of one and the same feature of quantum mechanics: (I) the impossibility of assigning simultaneously sharp values to certain pairs of noncommuting observables, be it by preparation or measurement and (P) the possibility of simultaneously assigning unsharp values to such observables by preparation of measurement. One may say that in contrast to Bohr, who emphasized the negative aspect of complementarity in the sense of (I), Heisenberg moved further to make a positive statement in the form of uncertainty relations which, if satisfied, enabled the option (P). We believe that this account is endorsed by Bohr who occasionally refers to the uncertainty relation as expressing complementarity, in the sense of (I), as well as (P). This seems evident from the following passage in the published version of the Como lecture which we quote in full length: In the language of the relativity theory, the content of relations (2) [the uncertainty relations] may be summarized in the statement that according to the quantum theory a general reciprocal relation exists between the maximum sharpness of definition of the space–time and energy–momentum vectors associated with the individuals. This circumstance may be regarded as a simple symbolical expression for the complementary nature of the space–time description and claims of causality. At the same time, however, the general character of this relation makes it possible to a certain extent to reconcile the conservation laws with the space–time co-ordination of observations, the idea of a coincidence of well-defined events in a space–time point being replaced by that of unsharply defined individuals within finite space–time regions [15, Section 2].

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For Bohr, the “mutual completion” part of complementarity refers to the necessity of making use of the exclusive descriptions (or observables) in different experimental contexts that cannot be created simultaneously. Incidentally, the above quote seems to constitute the first occurrence of the term “unsharp” in connection with the question of simultaneous preparation or measurement of position and momentum, which today is formalized in terms of unsharp observables, that is, POVMs. It seems largely a matter of terminological or interpretational preference which aspects of the two statements (I) and (P) are to be subsumed under the complementarity principle or the uncertainty principle. However, the formalizations of the features (I) and (P) reviewed in the preceding subsections, which are those most commonly used in the recent research literature, have clearly identified (I) as an expression of the idea of complementarity and (P) as the essence of the uncertainty principle. This constitutes a break with two older traditions which gave preference either to the complementarity principle or the uncertainty principle. It appears to us that with this terminological shift, a more balanced assessment has been achieved: compared to the view that emphasized complementarity over uncertainty, the positive role of the uncertainty relations as enabling joint determinations and joint measurements is now highlighted more prominently; and even though it is true (as we show later) that the uncertainty statement (P) entails (I) in a suitable limit sense, it is still appropriate to point out the strict mutual exclusivity of sharp value assignments which, after all, is the reason for the quest for an approximate reconciliation in the form of simultaneous but unsharp value assignments. Irrespective of the particular terminological or interpretational preference, formalizing the respective statements (I) and (P) has opened up new and interesting questions: (I) and (P) have become claims that can or cannot be proven as consequences of the theory, and it becomes possible to study the logical relationships between these statements. Questions like these will be studied in the remaining part of this paper with respect to Mach–Zehnder interferometric or, more generally, qubit observables. 3. Interlude: measurement theoretic concepts and tools In this section we review the representation of observables as POVMs, their use in the definition of joint unsharp measurement of noncommuting observables, and the measurement theoretic implementation of a POVM. We restrict ourselves to observables with finitely many values and some examples in the context of qubit observables, as they will be used in the following sections. 3.1. Effects and POVMs The standard representation of observables in quantum mechanics uses self-adjoint operators, or equivalently, the associated spectral measures, acting in the Hilbert space H associated with the physical system under consideration. For example, a standard observable A with discrete spectrum {a1 , a2 , . . . , an } has a spectral representation of the form
A = nk=1 ak Pk . The eigenvalues ak and the associated mutually orthogonal spectral projections Pk define the spectral measure P : ak  → Pk , which satisfies the normalization condition  Pk = I .

(7)

(8)

k

A projection-valued map of form (7) with property (8) is also called a PVM. The probabilities for the outcomes (eigenvalues) am in a measurement of A on state  are given by |Pm | =: Pm  = Pm .

(9)

The normalization of probabilities for all unit vectors  is ensured by the normalization condition (8). Standard observable A and PVM P : ak  → Pk will also be referred to as sharp observable. In generalizing the formalism to include POVMs as representations of imperfect or inaccurate measurements, it is noted that probabilities can generally be represented by expectation values of positive operators that are not necessarily

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projection operators nor necessarily commuting. Such operators are called effects. Hence, a measurement with outcomes { 1 , 2 , . . . , m } is represented by a POVM E :
 → E
,

(10)

with a unique collection of effects E1 , E2 , . . . , Em . The normalization condition  E
= I

(11)




is required to ensure the normalization of the probability distributions
 → |E
| for each vector state . Often we will simply denote a POVM E in terms of the set of its effects {E1 , . . . , Em }. A POVM which is not a PVM will be called unsharp observable. represent a coarse grained, or A POVM
 → E
is taken to

smeared version of a sharp observable (7) if there is a stochastic matrix (w
k ) (w
k
0,
w
k =1) such that E
:= k w
k Pk . Such a POVM, which is always commutative, represents an approximate measurement of the sharp observable. For example, the spectral representation of the Pauli spin-1/2 operator x in H = C2 has the form x = a1 P1 + a2 P2 = 21 (I + x ) − 21 (I − x )

(12)

with eigenvalues a1 = 1, a2 = −1 and spectral projections P1 = 21 (I + x ), P2 = 21 (I − x ). A smeared version of this spectral measure is obtained by applying a stochastic matrix   1 1+f 1−f (fj k ) := , −1f 1. 2 1−f 1+f The smeared version of the spectral measure of x is F = {F1 , F2 } where F
:= F1 = 21 (I + f x ),

F2 = 21 (I − f x ).


2

k=1 f
k Pk .

(13) This gives (14)

Similarly, a smeared version of z , G = {G1 , G2 } can be defined as, G1 = 21 (I + gz ),

G2 = 21 (I − gz ),

(15)

where −1 g 1. In this example, we see that the class of POVMs is wide enough to include trivial POVMs: these are composed of effects which are all multiples of the unit operator I. The above POVM F becomes trivial if we put f = 0. Trivial POVMs give probabilities that do not depend on the state; they provide no information at all. We have already found it convenient to make reference to observables represented as trivial POVMs in the preceding section. 3.2. Joint measurability In a joint measurement of two observables F and G, one sets out to infer the values of these observables from the output readings. Thus for every pair of values of F and G there has to be a pointer value and the statistics of these pointer values should reproduce the probabilities for the values of F and G in the object’s input state. Thus, there should be a POVM, E, whose probabilities are the joint probabilities for the outcomes of F and G. This means that the probability distributions of F and G should be obtained as marginal distributions of the probability distributions of E. Such a POVM, E, is called a joint observable of F and G; observables F and G are the marginals of E. According to a theorem of von Neumann [31, Section III.3] two sharp observables have a (sharp) joint observable exactly when they commute. Thus two non-commuting observables cannot be sharply measured together. However, it has been found that smeared versions of two noncommuting sharp observables may have a joint observable. The pair F = { 21 (I ± f z )}, G = { 21 (I ± gz )} are known [38] to have a joint observable exactly when, f 2 + g 2 1.

(16)

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Thus for two POVMs to be jointly measurable their degrees of unsharpness |f |, |g| must be limited by this tradeoff inequality. In this case it is straightforward to give an example of a joint observable E, assuming for simplicity 0 f, g 1, E11 = 41 (I + f x + gz ),

E21 = 41 (I − f x + gz ),

E12 = 41 (I + f x − gz ),

E22 = 41 (I − f x − gz ).

(17) (18) 

Each operator Ek
is positive because the eigenvalues are 41 (1 ± |(f, g)|) = 41 (1 ± f 2 + g 2 )
0 due to (16). Moreover, the marginality relations are fulfilled: E11 + E12 = F1 ,

E21 + E22 = F2 ,

(19)

E11 + E21 = G1 ,

E12 + E22 = G2 .

(20)

In Sections 4.5–4.7 we will give measurement implementations of similar joint observables. 3.3. Measurement implementation of a POVM Next, we give a brief explanation of how a POVM can be implemented in a suitable measurement scheme. In a measurement a probe is coupled with the object system by a unitary evolution after which the probe pointer is read. In addition, if the object system is still available after the measurement interaction has ceased, one may also perform another measurement on it. The following situation is encountered in Sections 4.5–4.7: a photon gets entangled with a probe system and then passes through a Mach–Zehnder interferometer. Finally, a joint measurement is made of a probe observable and a detector observable. The purpose of this joint measurement is to obtain information about the photon state immediately prior to the interaction with the probe and subsequent passage through the interferometer. Such information is available in the form of the output probabilities if these can be expressed in terms of the photon’s input state. Given that the initial state of the probe and the interferometer settings are fixed in each run, it follows that the output probabilities are indeed the expectation values of a POVM for the photon input state. Let i = |1 + |2 (| |2 + |2 = 1) denote the input state of the photon, |p0  the initial probe state, U the unitary evolution operator representing the probe and the passage through the interferometer. Then the final state of the combined system is f = U i ⊗ |p0 . On this the sharp output observable with projections Mk
= |kk| ⊗ |r
r
| is measured. Here |k, k = 1, 2, are the eigenstates of an object observable measured after the interaction with the probe, and |r
,
= 1, 2, are eigenstates of a pointer or output observable of the probe. (In Sections 4.6 and 4.7, different choices will be made for the |r
.) The output probabilities are then  f |Mk
| f  = p0 |i |U ∗ Mk
U |i |p0 

= | |2 p0 |1|U ∗ Mk
U |1|p0  + ∗ p0 |1|U ∗ Mk
U |2|p0  + ∗ p0 |2|U ∗ Mk
U |1|p0  + ||2 p0 |2|U ∗ Mk
U |2|p0 .

(21)

These can be written as 11 12 21 22  f |Mk
| f  = | |2 Ek
+ ∗ Ek
+ ∗ Ek
+ ||2 Ek
.

(22)

Being probabilities, these numbers are nonnegative and hence the expression (22) is a quadratic form for the variables ij

and . That is to say, for each k,
, the matrix (Ek
)i,j =1,2 is positive semi-definite; thus. it represents a positive operator Ek
defined in the Hilbert space of the photon. Hence,  f |Mk
| f  = i |Ek
|i 

(23)


for all i . Normalization of the output probability entails k
Ek
=1. Thus (k,
)  → Ek
is a POVM, which represents the (input) observable of the object system measured by the measurement scheme using the output observable (k,
)  → Mk
. Formula (23) is the basis for the analysis of all the measurement schemes discussed in the coming sections. The above consideration illustrates a general theorem that states that every measurement scheme defines a POVM for the object system. The converse statement is also true: every POVM admits an implementation in terms of a measurement

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scheme. For a general introduction to these results and for original references, the reader may wish to consult [39]. To our knowledge, the first proposals of realistic models of joint measurements of unsharp qubit observables were developed in [40]. Further detailed examples of measurement implementations of POVMs are given, for instance, in the monograph [27] and in the book of de Muynck [41]. 4. Path marking and erasure in an atom and Mach–Zehnder interferometry 4.1. The atom-interferometric experiment of SEW In a two slit atom interferometer [1] each atom is prepared in a superposition 0 (r) =

√1 [1 (r) + 2 (r)] 2

(24)

of path states 1 (r) and 2 (r) which represent the passage through the two slits. On the far capture screen (position coordinate R) an interference pattern will be observed according to P0 (R) = |0 (R)|2 = 21 [|1 |2 + |2 |2 + ∗1 2 + ∗2 1 ].

(25)

SEW consider the situation in which atoms, prepared in an excited internal state |a and propagating in a superposition of states corresponding to two collimated beam paths, arrive singly at micro-maser cavities preceding each of the double slits [1, Fig. 3]. Once in the cavity, the atoms will make a transition |a → |b, spontaneously emitting a microwave photon. The state of atom plus field changes from 0 (r) =

√1 [1 (r) + 2 (r)]|a|01 02  2

(26)

to the entangled state (r) =

√1 [1 (r) |b |11 02  + 2 (r)|b |01 12 ], 2

(27)

where |11 02  and |01 12  represent the field states corresponding to one photon in cavity 1 and none in cavity 2 and vice versa. Thus, the micro-maser cavities act as which-way detectors only if a photon left in the cavity changes the electromagnetic field in a detectable way. The probability density on the capture screen is given by P (R) = 21 [|1 |2 + |2 |2 + ∗1 2 11 02 |01 12  + ∗2 1 01 12 |11 02 ] = 21 [|1 |2 + |2 |2 ].

(28)

The interference (cross) terms vanish because the field states |11 02  and |01 12  are orthogonal. SEW also consider the possibility of recovering coherence and thus the interference pattern by deleting or “erasing” the path information left in the microwave cavity detectors [1, 26–31]. To model this they consider a new arrangement whereby the two cavities are separated by a shutter–detector combination [1, Fig. 5a]. This allows for the radiation either to be confined to the upper or lower cavity when the shutters are closed or for the radiation to be absorbed by a detector behind each shutter when it is opened. In the latter case the path marking information can be said to be erased. This will be explained presently. In the erasure experiment one makes use of the fact that the state (27) has the equivalent form (r) =

√1 [+ (r) |b |+ + − (r) |b |−], 2

(29)

where ± =

√1 [1 (r) ± 2 (r)], 2

|± =

√1 [|11 02  ± |01 12 ]. 2

(30)

To display the effects of the erasure, the experimental procedure is described as follows: after an atom arrives on the far screen, the shutters are opened and the state of the detector behind the shutters, which may have changed from its

P. Busch, C. Shilladay / Physics Reports 435 (2006) 1 – 31

15

initial state |d to a new state |e orthogonal to |d, is recorded. The possible transitions are as follows, reflecting the sensitivity of the detector to the field state |+ rather than |−: |+ |d → |00 |e,

|− |d → |− |d.

(31)

In half of the observations, the detector was found in an excited state indicating that there had been a photon in one of the cavities which has been absorbed. In the remaining cases there is no detection. Thus, the total state makes the following transition: √1 [+ |b |+ + − |b |−]|d 2

→

√1 [+ |b |01 02  |e + − |b |− |d]. 2

(32)

As seen in Eq. (29) the symmetric atom state + is coupled with the symmetric cavity field state; thus the state of the atom arriving at the screen selected if the detector is found in state |e is + . The probability density of those atoms will show the maxima and minima (fringes) of an interference pattern, P+ (R) = |+ (R)|2 = P0 (R) (Eq. (25)). Atoms arriving at the screen for which there is no corresponding signal from the erasure detector (i.e., the detector is found in |d) will display “anti-fringes”, P− (R) = |− (R)|2 , corresponding to the selected state − . If all the events are counted, irrespective of the erasure detector state, the distribution is, 1 2

P+ (R) +

1 2

P− (R) = 21 [|1 |2 + |2 |2 ] = P (R).

(33)

The maxima of one pattern coincide with the minima of the other one, washing out the fringes. This consideration shows that in the entangled state, (r) (Eq. (27)), the information about path as well as interference is fully available. Choosing to measure the path marking basis states, |11 02 , |01 12 , of the probe yields which way information. Measuring instead the field states |+, |− allows the recovery of interference fringes or anti-fringes, respectively. The two options are mutually exclusive; in the first case it is the interference information which is lost whereas in the second case it is path information which is lost. It should be noted that this situation is related to the Einstein, Podolsky, Rosen (EPR) experiment (in Bohm’s version for spin 1/2 pairs) which also makes use of an entangled state with more than one biorthogonal decomposition as in Eqs. (27) and (29). 4.2. Mach–Zehnder interferometer: basic setup Consider a special case of the Mach–Zehnder interferometer in Fig. 1 with no path marking and no phaseshifter. The two possible input states from I1 , I2 will be represented by orthogonal unit vectors |1, |2, of a two dimensional Hilbert space, H = C2 . When a photon entering via I1 (represented by a “path” state |1) arrives at the beam splitter BS1 its state is changed to an equally weighted superposition, with appropriate phase shift by /2 upon reflection; and similarly for an input state |2: |1 →

√1 [|1 + i|2], 2

|2 →

√1 [i|1 + |2]. 2

(34)

Arriving at detector D1 will be a component via the path I1 BS1 M1 reflected by BS2 carrying a total phase shift of /2 from M1 plus /2 from BS2 , and a component via the path I1 BS1 M2 transmitted by BS2 also carrying a total phase shift of /2 from BS1 plus /2 from M2 . Hence, detector D1 will register an output as these two components are in phase and interfere, constructively. Arriving at detector D2 will be a component via the path I1 BS1 M1 transmitted by BS2 carrying a total phase shift of /2 from M1 , and a component via the path I1 BS1 M2 reflected by BS2 carrying a total phase shift of /2 from BS1 plus /2 from M2 plus /2 from BS2 . Hence, detector D2 will register no output as these two components are out of phase by  and interfere destructively. So, if I1 BS1 M1 BS2 D1 is the path represented by |1, any measurement of the output of D1 has associated with it projector |11| representing one of the eigenstates of the measured input observable according to Eq. (23) (which defines the measured POVM). We identify this with the spectral projection of the Pauli operator z associated with the eigenvalue 1, |11| = 21 (I + z ). Similarly, I2 BS1 M2 BS2 D2 corresponds to |2 and any measurement of the output of D2 has associated with it the input projector |22| = 21 (I − z ).

16

P. Busch, C. Shilladay / Physics Reports 435 (2006) 1 – 31

Fig. 1. A Mach–Zehnder interferometer with path marking and phase shifter.

We are now in a position to consider a Mach–Zehnder interferometer with path marking before the beam splitter BS1 . This will be implemented by introducing a probe system which interacts with the photon. The probe is represented by a two dimensional Hilbert space, H = C2 , with path marking (“pointer”) states |p1  and |p2 . The interaction between the photon and the probe is to be arranged in such a way that |p1  becomes correlated with |1 and |p2  with |2, so that registration of these pointer states allows one to infer that the photon was in the corresponding path eigenstate (see Eq. (37) below). A phase shifter,
in one path after BS1 completes the analogy with the SEW experiment. A general input from I1 , I2 without the path marking interaction switched on (object in input state i and probe remaining in a neutral state |p0 ) can be represented by i ⊗ |p0  = ( |1 + |2) ⊗ |p0 .

(35)

Taking into account the phase shift in path 1 (see Fig. 1), the photon input states |1, |2 undergo the following evolution upon passage through the interferometer and before entering one of the detectors D1 , D2 : |1 → 21 [(−ei
− 1)|1 + i(ei
− 1)|2],

|2 → 21 [i(−ei
+ 1)|1 − (1 + ei
)|2].

(36)

When the path marking is turned on the total state after the path marking interaction has ceased and before the photon enters BS1 is e = |1 ⊗ |p1  + |2 ⊗ |p2 .

(37)

The photon states |1, |2 evolve according to (36); this leads to the total final (output) state after the photon passes through beam splitter BS2 as
f = 21 [(−ei
− 1)|1 + i(ei
− 1)|2] ⊗ |p1  + 21 [i(−ei
+ 1)|1 − (1 + ei
)|2] ⊗ |p2 .

(38)

We are now ready to discuss a variety of possible experiments. 4.3. Path detection in outputs D1 , D2 The simplest case of this Mach–Zehnder interferometer is with no path marking, |p1  = |p2  = |p0  and no phase shift,
= 0, analogous to a double slit interferometer (SEW) with no path marking and the far field detector placed

P. Busch, C. Shilladay / Physics Reports 435 (2006) 1 – 31

17

centrally; the output state is, of = −( |1 + |2) ⊗ |p0 .

(39)

Observing the output of detectors D1 , D2 with no path marking is represented by the projections M1 = |11| ⊗ I , M2 = |22| ⊗ I . The probabilities for an output at D1 and D2 are:  of |M1 | of  = i |11|i  = | |2 ,  of |M2 | of  = i |22|i  = ||2 .

(40)

The input observable measured by this experiment is the POVM E 0 = {E10 , E20 } defined by  of |Mk | of  = i |Eko |i 

(41)

for all i and k = 1, 2. It follows that E 0 is a PVM with projections E10 = 21 (I + z ),

E20 = 21 (I − z ).

(42)

This reproduces the discussion of path detection connected with Fig. 1: If i = |1, then the probabilities of a detection at D1 and D2 are 1|E10 |1 = 1 and 1|E20 |1 = 0, respectively. A similar consideration applies to an input state i = |2. Thus, the measured observable is the path observable z . 4.4. Interference detection at D1 , D2 We now consider the use of the Mach–Zehnder interferometer for an interference measurement. In a double slit interferometer both slits would be open and a detector placed at the first minimum. Here we choose
= −/2; then the output state is,   −(1 − i) 1 1 −/2 f = √ (43)

√ (|1 − |2) +  √ (|1 + |2) ⊗ |p0 . 2 2 2 If a measurement of Mk =|kk|⊗I, k =1, 2 is now applied by observing the outputs of Dk the associated probabilities are −/2

 f

−/2

|M1 | f

−/2

|i , −/2 −/2 −/2 2 1  f |M2 | f  = 2 (| − | ) = i |E2 |i , −/2

from which E1 −/2

E1

 = 21 (| + |2 ) = i |E1 −/2

and E1

= 21 (I + x ),

(44)

are extracted: −/2

E2

= 21 (I − x ).

(45)

Following customary practice we consider an interference observable one with the form cos
x + sin
y , 0 
< , given that the path is represented by z . Interference observables are singled out by the condition that the interference contrast can assume its maximum possible value. In this case their eigenstates give equal probabilities of 1/2 to the path projections |11|, |22|. In the present experiment, the measured input observable is defined by the projections of Eq. (45); these are the spectral projections of the operator x , which is indeed an interference observable. 4.5. The path-marking setup Now consider the case where |p1  and |p2  are mutually orthogonal, p1 |p2  = 0. This is an analog of SEW’s path-marking experiment. We can find the influence of the path marking on the outputs of the detectors using each of  = |kk| ⊗ |p p |, k,
= 1, 2, e.g. the four measurement projections of path |k jointly with marker |p
, Mk


 |
f  = | 21 (ei
+ 1)|2 = 41 | |2 (1 + cos
). 
f |M11

(46)

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P. Busch, C. Shilladay / Physics Reports 435 (2006) 1 – 31

The input POVM, the measured observable is again defined by Eq. (23),   
f |Mk
|
f  = i |Ek
|i ,

(47)

and we obtain

1 1   E11 = (I + z ) cos2 , E21 = (I + z ) sin2 , 2 2 2 2

1 1   E12 = (I − z ) sin2 , E22 = (I − z ) cos2 . 2 2 2 2 These effects are all fractions of a path projection. We now give the marginal input POVM associated with the detectors D1 , D2 ,

(48)

  F1 = E11 + E12 = 21 (I + cos
z ),   F2 = E22 + E21 = 21 (I − cos
z ).

(49)

This POVM represents a path observable smeared by cos
. The unsharpness inherent in the detector marginal is reflected in the non-zero probability of the marker indicating path 1 but detector D2 firing even if the input state is a path eigenstate |1. Here we see the effect of the perfect path-marking interaction: irrespective of the phase parameter value, the Mach–Zehnder interferometer does not detect any interference effects. When
= 0, the POVM {F1 , F2 } becomes a sharp path observable and when the interferometer is set to observe interference,
= /2, this POVM is reduced to being trivial, F1 = 21 I = F2 , giving no path nor interference information. This is in line with the prediction of SEW: path marking results in the interference pattern being washed out. After the path marking interaction, all the detectors are able to “see” is a “shadow” of the path information provided by the path marker: indeed, the marginal POVM measured by the path marker is given by the effects   G1 = E11 + E21 = 21 (I + z ),   + E12 = 21 (I − z ). G2 = E22

(50)

These represent a sharp path observable irrespective of the value of
. It is possible to define a third ‘marginal’,
  H1 = E11 + E22 = cos2 I , 2
  H2 = E12 + E21 = sin2 I , 2 which in the present experiment also turns out to be trivial.

(51)

4.6. Quantum erasure In the previous experiment, each path was correlated with one of two orthogonal marker states. We can now consider a new set of pointer states, which are superpositions of the two orthogonal path-marker states, |q1  =

√1 (|p1  + ei |p2 ), 2

|q2  =

√1 (|p1  − ei |p2 ). 2

(52)

Observing these symmetric states involves outputs for which both |p1  and |p2  are equally likely, so no information about the path is recorded. The final state (38) in terms of the new pointer states is
,

f =

1 √ [(− (1 + ei
) + ie−i (1 − ei
))|1 + (−i (1 − ei
) − e−i (1 + ei
))|2] ⊗ |q1  2 2 1 + √ [(− (1 + ei
) − ie−i (1 − ei
))|1 + (−i (1 − ei
) + e−i (1 + ei
))|2] ⊗ |q2 . 2 2

(53)

As before we can find the four joint probabilities for the marker and detector outputs, defined as the expectations of  = |kk| ⊗ |q q |, k,
= 1, 2. the projections Mk




P. Busch, C. Shilladay / Physics Reports 435 (2006) 1 – 31

19

The associated input POVM E  is determined via the relation
,


,

   f |Mk
| f  = i |Ek
|i .

(54)

We obtain  E11 = 41 (I − sin
cos x − sin
sin y + cos
z ) = 41 (I − n · ),  E21 = 41 (I + sin
cos x + sin
sin y − cos
z ) = 41 (I + n · ),  E12 = 41 (I + sin
cos x + sin
sin y + cos
z ) = 41 (I + m · ),  E22 = 41 (I − sin
cos x − sin
sin y − cos
z ) = 41 (I − m · ).

(55)

Here we have introduced the unit vectors n = (sin
cos , sin
sin , − cos
),

m = (sin
cos , sin
sin , cos
).

(56)

The marginal POVM associated with the detector outputs is obtained by summing over both probe outputs:   F1 = E11 + E12 = 21 (I + cos
z ),

  F2 = E21 + E22 = 21 (I − cos
z ).

(57)

This is a smeared path observable. The marginal POVM associated with the probe outputs is obtained by summing over both detection outputs:   G1 = E11 + E21 = 21 I,

  G2 = E12 + E22 = 21 I .

(58)

This is a trivial observable, it provides no information about the input state i . The fact that the detector POVM is a smeared path observable and the probe POVM is trivial can be understood as follows. The entanglement between probe and photon is devised to establish a strict correlation between the path states |1, |2 and the pointer states |p1 , |p2 , for any photon input state i . This correlation information is not accessible by measuring a probe observable with the eigenstates |q1 , |q2  because these are equal weight superpositions of the path marker states. Further, the reduced state of the photon after the coupling has been established is a mixture of the path states, so that any phase relation between these states has been erased. Accordingly, the detector outputs cannot detect any interference indicative of coherence between the path input states, and the only information left about the input is path information. A third ‘marginal’ input POVM is defined as follows:   H1 = E11 + E22 = 21 (I − sin
(cos x − sin y )),   H2 = E12 + E21 = 21 (I + sin
(cos x + sin y )).

(59)

This is a smeared interference observable, the unsharpness being determined by the term sin
and the direction of the associated Poincaré sphere vector being given by ±(cos , sin , 0). By varying from 0 to 2, all possible interference observables can be realized. The erasure scheme presented here constitutes a joint unsharp measurement of path and interference observables as represented by the marginal POVMs {F1 , F2 } and {H1 , H2 }. We also see that for
= −/2, all four effects Ek
are fractions of spectral projections of a sharp interference observable; the marginal {H1 , H2 } becomes a sharp interference observable and the marginal {F1 , F2 } becomes a trivial observable. Here we have recovered the observation of SEW, that the detector statistics conditional on the probe output readings display perfect interference with perfect visibility. In fact, we have found somewhat more: independently of the photon input state, the conditional probabilities for detections at D1 , D2 given a probe recording of |q1  (say) are:

    1 i |E11 i prob(D1 |q1 ) = = i (I + cos x + sin y )i , i |G1 i  2

  1  |E i  prob(D2 |q1 ) = i 21 (60) (I − cos

 − sin

 ) =  x y i . i i |G1 i  2

20

P. Busch, C. Shilladay / Physics Reports 435 (2006) 1 – 31

For = 0 and the input state i = √1 (|1 + |2), this gives prob(D1 |q1 ) = 1 and prob(D2 |q1 ) = 0. This corresponds to 2 the case of perfect interference fringes. Similarly, for the detector probabilities conditional on |q2  and the above input eigenstate of x , we obtain probabilities 0 and 1 for D1 and D2 , respectively, which are characteristic of interference antifringes. This situation is a consequence of the fact that for the above input and
= −/2, = 0, the state e and also the −/2 is an EPR state, analogous to the state described in the SEW erasure setup of Section 4.1: total output state f   −(1 − i) 1 1 −/2 f = √ √ (|1 − |2) ⊗ |p1  + √ (|1 + |2) ⊗ |p2  2 2 2 −(1 − i) 1 (61) = √ √ [|1 ⊗ |q1  + |2 ⊗ |q2 ]. 2 2 4.7. Quantitative erasure The two possible experimental options discussed in the preceding subsections, of path marking and erasure are mutually exclusive in that they require settings and operations that cannot be performed at the same time: for path determination, the probe eigenstates |p1 , |p2  must be read out, while for the recovery of interference it is necessary to record the detector outputs conditional on the probe output states |q1 , |q2 . Erasure was achieved by choosing
= /2,  } being constituted of (fractions of) spectral projections of an interference observable. which led to the POVM {Ek
Accordingly, the only non-trivial marginal is the sharp interference observable {H1 , H2 }.  } is a joint observable for If, however, the interferometric parameter
is varied between 0 and /2, then the POVM {Ek
an unsharp path and an unsharp interference observable. In this case, the experiment provides simultaneous information about these noncommuting quantities. In the limit of
= 0, the interference marginal {H1 , H2 } becomes trivial and the path marginal {F1 , F2 } becomes sharp. The possibility of obtaining some joint information about both observables, path and interference, can also be achieved by modifying the path marking coupling in such a way that the correlation between the paths and the probe indicator observable is not perfect. This has been described as quantitative erasure (e.g., [42]). We show here that quantitative erasure is again an instance of a joint unsharp measurement. We take the path-marking interaction to be of the same form as before, Eq. (38), but now we specify the marker states |p1 , |p2  to be nonorthogonal. Their associated Poincaré sphere vectors will be chosen to be tilted by an angle  away from the ±z axis towards the positive x axis. As pointer states we choose |q1 , |q2  equal to the up and down eigenstates of z . Thus we define   |p1  = cos |q1  + sin |q2 , 2 2

  |p2  = sin |q1  + cos |q2 . 2 2

The final state after the path-marking interaction is  

  
, i
i
f = − cos (1 + e ) + i sin (1 − e ) |1 2 2 2 2   

   + −i cos (1 − ei
) − sin (1 + ei
) |2 ⊗ |q1  2 2 2 2   

  + − sin (1 + ei
) + i cos (1 − ei
) |1 2 2 2 2   

   i
i
+ −i sin (1 − e ) − cos (1 + e ) |2 ⊗ |q2 . 2 2 2 2

(62)

(63)

 associated with the output projections M  = |kk| ⊗ |q q | via Now we determine the input effects Ek


k

,
,    f |Mk
| f  = i |Ek
|i ,  E11 = 41 [I (1 + cos  cos
) − sin
sin x + (cos
+ cos )z ]

= 41 [I (1 + cos  cos
) + m · ],

P. Busch, C. Shilladay / Physics Reports 435 (2006) 1 – 31

21

 E21 = 41 [I (1 − cos  cos
) + sin
sin x − (cos
− cos )z ]

= 41 [I (1 − cos  cos
) − n · ],

 E12 = 41 [I (1 − cos  cos
) − sin
sin x + (cos
− cos ) z ]

= 41 [I (− cos  cos
) + n · ],

 E22 = 41 [I (1 + cos  cos
) + sin
sin x − (cos
+ cos )z ]

= 41 [I (1 + cos  cos
) − m · ],

(64)

where m = (− sin
sin , 0, (cos
+ cos )), n = (− sin
sin , 0, (cos
− cos )).

(65)

The three marginal POVMs are determined as before:   F1 = E11 + E12 = 21 (I − sin
sin x + cos
z ),   F2 = E22 + E21 = 21 (I + sin
sin x − cos
z ),   1 G 1 = E11 + E21 = 2 (I + cos z ),   1 G 2 = E22 + E12 = 2 (I − cos z ),   H1 = E11 + E22 = 21 I (1 + cos  cos
),   H2 = E12 + E21 = 21 I (1 − cos  cos
).

(66)

For
= −/2, the first marginal POVM (corresponding to the detector statistics) becomes an unsharp interference observable, while the second marginal POVM (corresponding to the probe output statistics) is an unsharp path observable. In both cases the unsharpness is determined by the parameter . We note for later reference that instead of the choice of pointer states |q1 , |q2 , one could have measured any pair of orthogonal probe states |r1 , |r2 . It is straightforward to show that the marker marginal is always an unsharp path observable. 5. Complementarity and uncertainty in Mach–Zehnder interferometry 5.1. Manifestations of complementarity in Mach–Zehnder interferometry The sequence of experiments in Section 4 is a demonstration of complementarity in different guises. In the first two experiments path detection (4.3) and interference detection (4.4) are mutually exclusive because this requires settings of the parameter
which cannot be realized in the same experiment, namely,
= 0 for path (z ) measurement and
= −/2 for interference (x ) measurement, respectively. Here we have an instance of the complementarity of measurement setups or measurement complementarity: these two noncommuting sharp observables do not admit any joint measurement. These experiments can also be used to confirm preparation complementarity. We recall that sending a path eigenstate −/2 |1 or |2 into the Mach–Zehnder interferometric setup to observe interference leads to the probability 1|E1 |1 = −/2 1|E2 |1 = 1/2, interference is completely uncertain, and, if we feed an interference eigenstate, | ± x, into the interferometer to measure path (
= 0), the path observable is uncertain ±x|E10 | ± x = ±x|E20 | ± x = 1/2. Value complementarity can indeed be used to explain the disappearance of interference fringes resulting from path marking. Once perfect correlation between the path states and the marker states is established in the entangled state (38), the reduced state of the photon is a mixture of the path eigenstates |1 and |2. In each of these, the path is definite, and therefore, in accordance with value complementarity, the outcomes of a subsequent interference measurement are equally probable. No interference fringes show up. Indeed this remains true for any mixture of path eigenstates.

22

P. Busch, C. Shilladay / Physics Reports 435 (2006) 1 – 31

This account in terms of preparation complementarity views the path marking interaction as part of a preparation process. An alternative explanation is possible in terms of measurement complementarity as follows. In the experiment of Section 4.5, where sharp path marking is followed by the interference setup with
= −/2, it was found that the path measurement interaction leads to a complete loss of interference information detectable in D1 , D2 . All that the detectors can “see” is path information, irrespective of the value of
(Eq. (49)). If the sharp path marking is relaxed into unsharp path marking, Section 4.7, setting the interferometer with
= −/2 defines an unsharp interference observable, which is jointly measured with the path that can be recorded at the path marker. 1 It is found that the less accurate the path marking is set by making cos  in G 1,2 = 2 (I ± cos z ) smaller, the sharper  = 1 (I ∓ sin  ) becomes larger. will be the interference measurement as sin  in F1,2 x 2 We see here that measurement complementarity follows in the limits of making the path marginal or the interference marginal perfectly sharp, rendering the other trivial. A similar analysis applies to the erasure setup (Section 4.6): if 0 <
< /2, this setup realizes a joint measurement  , H  } which are unsharp path and interference observables. of the POVMs {F1, , F2 } and {H1, 2  = 1 I and H  = 1 (I ∓ cos  ∓ sin  ). Here we have a sharp Consider the case of
= /2, where F1,2 x y 1,2 2 2  = 1 (I ±  ) and H  = 1 I , so that we have a interference measurement and no path measurement. With
= 0, F1,2 z 1 2 2 sharp path measurement and no interference. These two limit cases of a joint measurement scheme illustrate once more measurement complementarity. 5.2. Value complementarity from preparation uncertainty relations We recall that a pair of value complementary observables A, B is characterized by the condition that in each of the eigenstates of A, all eigenvalues of B are equally likely to occur as outcomes of a B measurement; and vice versa. We now show that for qubit observables such as those occurring in the Mach–Zehnder interferometry measurements discussed here, the value complementarity property can always be obtained as a consequence of some suitable uncertainty relation for the observables in question. In what follows we allow general states represented as density operators  = 21 (I + r · ), |r| 1. For the observables represented by x , y and z , we then have k  = rk2 ,

k = 1, 2, 3,

(67)

and the variances, defined as Var(A) ≡ Var(A, ) = operation), are Var(k ) = 1 − k 2 = 1 − rk2 ,

A2 

−

A2

(where A = tr[A], etc., tr[·] denoting the trace

k = 1, 2, 3.

(68)

With these expressions it is straightforward to confirm the general uncertainty relation for variances, with the commutator and covariance terms contributing the lower bound: Var(x )Var(z )


2 1 4 |[x , z ]|

+ 41 [x z + z x  − 2x z ]2 .

(69)

It is easy to verify that the pair x , z is value complementary: in an eigenstate of z , we have r12 = 1 and so r2 = 0. So, probability 1 for a value of z goes along with probability 1/2 for the values of x ; and vice versa. However, when it comes to deciding whether the statement of value complementarity can be inferred from the uncertainty principle, one should look at the uncertainty relation (69) alone, rather than using the explicit values of its terms. But using solely the above variance inequality one cannot recover value complementarity without further information on the terms of the left hand side. Still it suffices to use the algebraic and spectral properties of the Pauli operators (we imagine that we are given only this information but not the explicit expression of the probabilities or expectations): then one finds the right hand side of Eq. (69) to be equal to y 2 + x 2 z 2 , and one can argue as follows: if the path is definite, that is, if =|| with  an eigenstate of z , the left hand side of the uncertainty relation (69) is zero, and therefore the terms on the right hand side must vanish, too. Thus as z  = 1, then x  = y  = 0. Since the eigenvalues of these quantities are ±1, it follows that these observables are uniformly distributed in . By symmetry, z is uniformly distributed if x has a definite value. (Note that we did not have to use the full explicit expressions for the expectation values, which would of course make this consideration entirely trivial.)

P. Busch, C. Shilladay / Physics Reports 435 (2006) 1 – 31

23

There are other forms of uncertainty relations which yield the statement of value complementarity without recourse to specific properties of the Pauli operators. Here we

only mention the entropic uncertainty for two observables A, B with spectral representations A = 2i=1 ai Pi , B = 2k=1 bk Qk . The (Shannon) entropy of A in a state  is defined as H (A, ) = −

2 

|Pi | log2 |Pi |.

(70)

i=1

This quantity is a measure of uncertainty concerning the value of A as encoded in the probability distribution of A in the given state: note that 0 H (A, )log2 2 = 1, where the lower bound is assumed for any eigenstate of A and the upper bound arises for any state which assigns equal probability 1/2 to all eigenvalues. The following additive trade-off relation then holds [43]:   ||Pi Qk || . (71) H (A, ) + H (B, )
− 2 log2 max i,k Pi  Qk  If A, B are a pair of value √complementary observables, so that any pair of eigenstates i of A and k of B have overlap given by |i |k | = 1/ 2, it follows that the lower bound on the right hand side is log2 2. Thus, for A = z , B = x , one obtains [44]: H (z , ) + H (x , )
1.

(72)

The combined lack of information about z and x is never less than one bit. Now, it is easily seen from this inequality alone that if one observable has a definite value, e.g., H (z , ) = 1 (which happens in the eigenstates of z ), then the other observable is maximally uncertain, it is uniformly distributed since H (x , ) = 1. We note that the explicit expressions for the probabilities have gone into the derivation of this entropic inequality. But all that is needed to recover the property of value complementarity is contained in this inequality. The inequality (72) is indeed easily verified by using the expressions 21 (1 ± rz ), 21 (1 ± rx ) for the probabilities and applying calculus to determine the minima. It is similarly straightforward to verify the following triple uncertainty relation: H (x , ) + H (y , ) + H (z , )
2.

(73)

5.3. Quantitative duality relations are uncertainty relations In the debates of the 1990s over complementarity in the context of interferometry and which-path experiments, the meaning of the term wave particle duality has gradually shifted away from a relation of strict exclusion of path determination and interference observation (in the same setup) to the broader idea of a continuous trade-off between approximate path determination and approximate interference determination. These discussions were eventually linked with earlier theoretical and experimental work of the 1980s on simultaneous but imperfect path determination and interference observation (e.g., [40,45–47]), as reviewed in [11]. The original intuitive ideas of the pioneers on an approximate reconciliation of complementary operational options (cf. Section 2) have thus been turned into precise trade-off relations which are being tested experimentally. Trade-off relations of the form P 2 +V 2 1 were derived as characterizations of the duality between path predictability and interference visibility [48–50]. (In [50], a stronger result, P 2 + V 2 = 1, was shown to hold for certain experimental situation.) Soon afterwards, similar relations were formulated for quantum erasure (for a lucid and comprehensive review, see [42]). Vivid debates took place over the question whether the associated quantities are connected with uncertainties, and it has been shown that the respective trade-off relations are related in various ways to some forms of uncertainty relations. In the context of the Mach–Zehnder interferometer, we refer, in particular, to the work of Björk et al. [9], Dürr and Rempe [11], and Luis [12]. Using familiar measures of uncertainty, we give a simple demonstration that in the context of Mach–Zehnder interferometry experiments, quantitative duality relations are indeed equivalent to the uncertainty relation for an appropriate pair of associated observables.

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Any state  can be represented by a matrix of the following form in the basis of eigenvectors of z :    w±
0, w+ + w− = 1, w+ re−i √ where = 0 r  w+ w− , 0  < 2. rei w−

(74)

We define the path contrast of  as CP = CP () := |prob(z = +1, ) − prob(z = −1, )| = |w+ − w− |.

(75)

This is identical to the predictability P entering the duality relation P 2 + V 2 1. Similarly we define the interference contrast of  as CI = CI () = |prob(x = +1, ) − prob(x = −1, )| = 2r cos .

(76)

With the specification  = 0, or with an alternative choice of interference observable, this reduces to the visibility V := (Imax − Imin )/(Imax + Imin ) = 2r (where Imax , Imin denote the maximal and minimal intensities of the measured interference pattern, obtained by variation of the interference observables). Using r 2 w+ w− , the following duality relation is easily verified: 2 2 + w− − 2w+ w− + 4r 2 cos2  1. CP2 + CI2 = w+

(77)

Now we observe that CP2 = z 2 = 1 − Var(z ),

CI2 = x 2 = 1 − Var(x ).

(78)

Therefore, the above duality inequality can be equivalently expressed as Var(z ) + Var(x ) = 2 − (CP2 + CI2 )
1.

(79)

Thus, our duality relation is equivalent to a form of uncertainty trade-off relation. As before, value complementarity is again entailed as a limit case. We now show that this last inequality is actually a direct consequence of the uncertainty relation (69). Using Eq. (67), that relation is readily found to be equivalent to x 2 + y 2 + z 2 1,

(80)

which expresses the positivity of the state . Using Eq. (68), we thus see that the uncertainty relation (69) is indeed equivalent to the following inequality: Var(x ) + Var(y ) + Var(z )
2.

(81)

We note that besides x , the operator y also constitutes an interference observable with respect to the path z . Thus, 2 , and a similar termVar( ) = 1 − C 2 , we obtain a substituting Var(z ) = 1 − CP2 , Var(x ) = 1 − CI2 ≡ 1 − CI,x y I,y generalized and sharpened duality relation which involves one path and two complementary interference observables: 2 2 + CI,y 1. CP2 + CI,x

(82)

Thus the full uncertainty relation for x , z , including the commutator and covariance terms, is equivalent to the additive triple trade-off relation for the variances of x , y , z as well as this new trade-off relation for three mutually complementary observables. 5.4. Measurement complementarity from measurement inaccuracy relations The measurement schemes of Sections 4.6 and 4.7 were found to constitute joint measurements of unsharp path and interference observables of the form F = {F1,2 = 21 (I ± f x )} and G = {G1,2 = 21 (I ± gz )}. For instance, in Eq. (66), setting
= −/2, we have f = sin  and g = cos , so that we have f 2 + g 2 = 1. This is an instance of the criterion (16) which ensures the joint measurability of the POVMs F, G.

P. Busch, C. Shilladay / Physics Reports 435 (2006) 1 – 31

25

For a general state  = 21 (I + r · ), |r| 1, we define the contrasts of the distributions of F and G, CF () = |tr[F1 ] − tr[F2 ])| = |f r 1 |, CG () = |tr[G1 ] − tr[G2 ]| = |gr 3 |.

(83)

The contrasts of the POVMs F, G are the respective maximal contrasts over all states : CF = |f |,

CG = |g|.

(84)

These quantities measure the degrees of unsharpness, UF := 1 − CF2 = 1 − f 2 ,

2 UG := 1 − CG = 1 − g2 ,

(85)

in the POVMs F, G. The unsharpness of F can also be defined as the minimum variance of the distribution of F for all states . Indeed, it is not hard to verify that Var  (F ) = 1 − f 2 r12 = 1 − f 2 + f 2 (1 − r12 ) = UF + (1 − UF )Var  (x )
UF .

(86)

Taking the minimum over all  gives Var min (F ) = UF .

(87)

The above joint measurability criterion can be written in terms of the degrees of unsharpness: UF + UG
1.

(88)

This inequality is an uncertainty trade-off relation which must be satisfied if the two noncommuting unsharp path and interference observables F and G are to be jointly measurable. We have here an instance of Heisenberg’s uncertainty principle for the inaccuracies which are necessarily present in joint measurements. As far as we are aware, this is one [38] of two cases in which an inaccuracy relation has been proven as a necessary condition for joint measurability. The other example is the case of position and momentum [51,33], and the corresponding uncertainty relation for joint measurements is reviewed in [34]. Finally we note that the variances of the marginals F, G in a joint measurement satisfy the uncertainty relation Var  (F ) + Var  (G)
UF + UG
1.

(89)

Measurement complementarity is obtained as a limiting case for a pair F, G which are jointly measurable: if it is stipulated that one marginal, say F, becomes sharp, UF = 0, or |f | = 1, then the other marginal, G, becomes a trivial POVM, g = 0, G1,2 = 21 I . Thus, if the path F is measured sharply, any attempt at obtaining information on interference will fail as the only unsharp interference observable G that can be measured jointly with F is trivial. 5.5. Disturbance versus accuracy The setup discussed in Section 4.7 corresponds to a sequence of measurements where first path marking and registration can be achieved and then an interferometric measurement is carried out. As was observed in Section 5.1, if the path marking correlation is perfect, all the interferometric detectors can “see” is unsharp path information. If the setting is
= /2, then the input observable indicated by the detectors is trivial: no interference information whatsoever about the input state is observed. If the path marking is unsharp, due to imperfect correlations or nonorthogonal path marker states |p1 , |p2 , then some interference information about the input state can pass through the path marking interaction. Here we will give a quantitative expression of this trade-off between the “disturbance” of the interference information through (imperfect) path marking and the accuracy of this path-marking process. The path-marking interaction transforms the initial state ( |1 + |2)|p0  into e := |1|p1  + |2|p2 .

(90)

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This state has reduced photon-state operator e = | |2 |11| + ||2 |22| + ∗ p2 |p1 |12| + ∗ p1 |p2 |21|.

(91)

In what follows we want to express the necessary “disturbance” of the interference detection due to the path-marking entanglement. We start with a situation where |p1 , |p2  are not necessarily orthogonal, but arranged as described in Section 4.7. We also allow a general set of orthogonal pointer states |r1 , |r2 . The quality of the path marking can be determined by following the entangling interaction with a path detection at D1 , D2 , where
= 0. If a reading r1 (r2 ) is taken to infer the path to be path 1 (path 2), then the subsequent path detection in the interferometer can be used to verify this inference. Thus, a joint measurement is made of the PVM with projections Mk
= |kk| ⊗ |

|, with probabilities pk
=  0f |Mk
| 0f  = |k|
| e |2 .

(92)

The probability prob(corr) of a correct inference is given as the sum of the probabilities of the corresponding coincident outputs: prob(corr) = p11 + p22 = i |H10 |i  = | |2 |r1 |p1 |2 + ||2 |r2 |p2 |2 .

(93)

Similarly, the probability prob(err) of error is given by the non-coincident combinations: prob(err) = p12 + p21 = i |H20 |i  = | |2 |r2 |p1 |2 + ||2 |r1 |p2 |2 .

(94)

Here we have introduced the input marginal H 0 = {H10 , H20 } which represents the coincidence events in this joint measurement: H10 = |11||r1 |p1 |2 + |22||r2 |p2 |2 , H20 = |11||r2 |p1 |2 + |22||r1 |p2 |2 .

(95)

On writing |rm rm | = 21 (I + rm · ), r1 = r, r2 = −r, and |p
p
| = 21 (I + p
· ), we obtain the following forms for H10 , H20 : H10 = 21 (I (1 + 21 r · (p1 − p2 )) + 21 r · (p1 + p2 )z ),

(96)

H20 = 21 (I (1 − 21 r · (p1 − p2 )) − 21 r · (p1 + p2 )z )

(97)

in this notation the probability prob(corr) becomes prob(corr) = 21 (1 + r · (| |2 p1 − ||2 p2 )).

(98)

In order to determine the maximum quality path determination available by a suitable choice of the path marker output observable (with eigenstates |r1 , |r2 ), we maximize the probability prob(corr) of correct inferences on the path from the registrations of r1 , r2 . The maximum probmax (corr) is obtained at r = r0 =

| |2 p1 − ||2 p2 , || |2 p1 − ||2 p2 |

(99)

and its value is probmax (corr) = 21 (1 + || |2 p1 − ||2 p2 |) =: L. The (path) distinguishability is defined as D = 2L − 1 [49]. We obtain D = || |2 p1 − ||2 p2 | = 1 − 4| |2 ||2 |p1 |p2 |2 .

(100)

(101)

We note that the probabilities for correct and wrong inferences can be expressed in terms of the coincidence POVM H 0 = {H10 , H20 }: noting that Probmin (err) = 1 − probmax (corr) = 1 − L, we have D = [i |H10 |i  − i |H20 |i ]r=r0 .

(102)

P. Busch, C. Shilladay / Physics Reports 435 (2006) 1 – 31

We calculate the variance of the distribution of H 0 in the state i , using

t¯ = t di |Ht0 |i  = i |H10 |i  − i |H20 |i , we obtain Var(H, i ) =

(t − t¯)2 di |Ht0 |i  = 1 − [i |H10 |i  − i |H20 |i ]2 .

27

(103)

(104)

Thus we obtain that D 2 = 1 − Var(H 0 , i )|r=r0 .

(105)

The distinguishability is thus found to be directly related to the uncertainty of the coincidence observable. It gives a measure of the accuracy of the path determination, and it is dependent on the degree of entanglement between the photon and the marker system, which may be quantified by the overlap quantity |p1 |p2 |. We now determine the visibility Ve of an interference observation available in the reduced photon state e after the entangling interaction. Thus we consider a joint measurement of the kind studied in Section 4.7. The usual definition of visibility in terms of the difference between maximal and minimal probability of an outcome at D1 (or D2 ) reduces to (106) Ve = V (e ) = tr(e |+, n+, n|) − tr(e |−, n−, n|) , where the interference observable n = |+, n+, n| − |−, n−, n| = n ·  =



0 e i


e−i
0

 ,

(107)

(with n = (cos
, sin
, 0)) is chosen such that the above difference becomes maximal: Ve = 2|Re( ∗ p2 |p1  ei
)|max = 2| ||||p2 |p1 |.

(108)

Since Ve = |n |, we have Ve2 = V (e ) = 1 − Var(n , e ),

(109)

and finally [50] Ve2 + D 2 = 1.

(110)

This is a limiting case of a general inequality Ve2 + D 2 1, reviewed in [42], where it is shown that equality arises whenever the total system of photon plus marker is in a pure state as is the case in the present context. The above quantitative erasure duality relation can be written in terms of the associated uncertainties: Var(H 0 , i ) + Var(n , e ) = 1.

(111)

This relation shows the trade-off that happens as a result of the path-marking interaction: the better the path marking is set, the more the interference term becomes attenuated. This is to be understood as a relation between the accuracy of path determination and the “disturbance” of the interference capability of the quantum state that passes through the path marking interaction. This can be seen particularly clearly if the input state i is chosen to be an interference √ eigenstate, | |=||=1/ 2. In this case the distinguishability becomes minimal among all input states, and the visibility becomes maximal, and both quantities depend solely on the overlap between the marker states and hence the degree of entanglement between photon and marker: D = |p1 − p2 | = 1 − |p1 |p2 |2 , Ve = |p1 |p2 |. (112) The deviation of Ve from 1, its original value for the input interference eigenstate, represents the minimal degradation of the interference capability required by the gain in path distinguishability. If the path marking is made perfect, by requiring p1 |p2  = 0, then D = 1 and Ve = 0, that is, the disturbance of the interference observation becomes maximal, no interference can be detected.

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5.6. Uncertainty, disturbance, or entanglement? We finally turn to the question of whether entanglement provides a more fundamental or more general explanation of the loss of interference in path-marking experiments than uncertainty. We think that this question arose from the erroneous conflation of the uncertainty principle with the idea of classical random disturbance. The discussion of that conflation has led to some very interesting clarifications and distinctions between classical, random momentum transfers (or phase kicks) and quantum-mechanical momentum transfers (or phase shifts) [6,8], with the conclusion that often, if not always, one and the same experiment may admit explanations of the loss of interference both in terms of classical random disturbances and in terms of quantum disturbances. It seems to be a desire for causal explanation that induced the search for mechanical causes (classical or quantum) enforcing the uncertainty principle. In the preceding subsection we have seen that the reverse perspective leads to a satisfactory account of the disturbance of interference through path marking: this disturbance is expressed by means of an uncertainty relation. We have used the term “disturbance” in the operational sense of a change in the distribution of the values of the interference observable. There was no question of a (random) mechanical kick. Quite the contrary, the coupling for the path marking process was arranged so as to constitute a nondemolition measurement: if the input is a path eigenstate, say |1, the total state after path marking is |1|p1 ; that is, the system has remained undisturbed, it is still in the original path eigenstate. The magnitude of the loss of interference capability or coherence that arises in the transition from an interference eigenstate as input to the reduced photon state e after the path marking interaction is determined solely by the overlap c := |p1 |p2 |, Eq. (112). The quantity c describes the degree of correlation between the path eigenstates and the marker states, as well as the degree of entanglement in the total state e . It is evident that in the present quantitative erasure experiment, the explanations of the degradation of coherence either in terms of uncertainties for the photon system alone (Eq. (111)) or in terms of entanglement draw on the same crucial entity—the quantity c. The deeper reason for this may be seen in the fact that entanglement can itself be explained as an instance of uncertainty. This was made manifest in the papers of Kim and Mahler [10] and Björk et al. [9], who used uncertainty relations for observables of the entangled object and probe system to explain the loss of interference. A pure state of a compound system is entangled if it cannot be represented as a product state. Product states are also called separable. The (normalized) state e = |1|p1  + |2|p2  is entangled exactly √ 1 |p2 | < 1 and √ when c = |p 0 < | |, || < 1. As an entangled state, e possesses a biorthogonal decomposition e = w1 ⊗1 + 1 − w2 ⊗2 , where 1 |2  = 0 = 1 |2 . The only values of the (nonnegative) parameter w for which e is separable are w = 1 or w=0. Then the state e is separable exactly when the adapted observable S =|1 1 |⊗|1 1 |−|2 2 |⊗|2 2 | has a definite value. Conversely, the state e is entangled if and only if the outcomes of an S measurement are uncertain. Seen in this way, entanglement is quite generally an instance of uncertainty, and like uncertainty, it is rooted in the existence of states which are superpositions of orthogonal families of states. It is therefore to be expected that whenever there is an explanation for a loss of interference due to entanglement, there is also an associated explanation in terms of an uncertainty relation. Our consideration of the previous subsection shows that such uncertainty relations can even be formulated without recourse to the probe system. It has been argued that in the case of the experiments of SEW and Dürr et al., the position–momentum uncertainty relation does not provide an explanation of the loss of interference, and that no explanation based on another uncertainty relation has been found [11]. However, the loss of interference capability in the SEW double-slit experiment, discussed in Section 4.1 above, is due to the loss of coherence in the transition from 0 = √1 [1 + 2 ] to the associated 2 mixed state, |0 0 | −→ 21 |1 1 | + 21 |2 2 |,

(113)

which is completely analogous to the transition |i i | → e of the photon state in the Mach–Zehnder context. All that matters for the explanation of the loss of interference is the presence or absence of a definite phase relation between the path states 1 and 2 (or |1, |2). This can be fully described by analogs of Mach–Zehnder interferometric observables in the two dimensional subspace spanned by the path states 1 , 2 . Thus if the path observable is represented as z := |1 1 | − |2 2 |,

(114)

P. Busch, C. Shilladay / Physics Reports 435 (2006) 1 – 31

29

then the phase relation between the path states can be tested by a phase sensitive observable such as x := 21 |1 + 2 1 + 2 | − 21 |1 − 2 1 − 2 |.

(115)

The explanations and formulations of loss of interference in terms of uncertainty relations given in the context of Mach–Zehnder interferometry in the present section can then be literally transferred to the case of double-slit interference and path-marking experiments. The same conclusion has been drawn by Luis [12] who based his argument on a continuous phase POVM conjugate to the path observable (z ), instead of the phase-sensitive interference observables used in the present paper. Similarly, Björk et al. [9] and de Muynck and Hendrikx [52] have demonstrated that analogous atom-interferometric experiments can be interpreted in terms of joint unsharp measurements of path and interference observables, and that complementarity emerges as a limiting case of an entropic uncertainty relation for the accuracies of these joint measurements. 6. Conclusion We have reviewed the evolution of the understanding and current formalizations of the concepts of complementarity and uncertainty. In particular, we exhibited three distinct versions of complementarity and uncertainty relations, respectively, referring to the limitations and possibilities of preparing and measuring simultaneously sharp or unsharp values, and to a necessary trade-off between accuracy and state disturbance. It was shown that these formulations can be usefully applied in the context of Mach–Zehnder interferometry, to explain the loss or degradation of interference due to path marking. In particular, it was found that value complementarity and measurement complementarity can be recovered from appropriate uncertainty relations for preparations, joint, and sequential measurements, respectively. Furthermore, duality relations for the trade-off between partial path determinations and reduced-visibility interference observations were shown to be expressible as uncertainty relations. Next, we have noted that entanglement is properly understood as an instance of uncertainty in the context of the description of compound systems, rather than a separate feature. It is therefore to be expected that whenever an explanation of the loss of interference can be given in terms of entanglement, this can be accompanied with an explanation in terms of a form of uncertainty relation. We have pointed out that “disturbance” in quantum measurements cannot be reduced to the naive notion of classical (random) kicks; the generic concept of disturbance concerns the necessary state change induced by measurements, which inevitably goes along with the build-up of entanglement between object system and probe. This applies, in particular, to the nondemolition measurement couplings employed here for path marking, which are exactly of the same kind as the coupling used in the SEW and Dürr et al. experiments. Such (approximately) repeatable measurements were also at the heart of Heisenberg’s thought experiments illustrating the various types of uncertainty relations. The necessary state change required for the extraction of path information leads to a disturbance of the distribution of an associated interference observable. All these notions—uncertainty, complementarity, entanglement, and disturbance—are ultimately rooted in the linear structure of quantum mechanics, with the concomitant noncommutativity of observables. Taken all this together, it seems indeed moot to try and establish a hierarchy of principles of uncertainty, complementarity, or entanglement within quantum mechanics. As seen from within this theory, these features are linked with each other but cannot be claimed to be reducible to one another. They are not logically independent, nor simply consequences of each other. Such logical relations can only be analyzed within a framework more general than quantum mechanics, in which each principle can be introduced as a separate postulate. In fact, it has been shown that there are theories in which there is an uncertainty principle without a complementarity principle (suitably specified), and theories with complementarity but without a Heisenberg uncertainty relation [53]. In quantum mechanics, uncertainty and complementarity are consequences of the formalism which are neither reducible to each other nor entirely unrelated. There is thus no need in quantum mechanics to speak of a complementarity or uncertainty “principle”, unless one uses this term informally as a way of highlighting these “principal” implications of the theory, fundamental for its intuitive understanding. If, however, one sets out to use complementarity or uncertainty as postulates, from which to deduce quantum mechanics in Hilbert space starting from a more general framework, then it is indeed appropriate to refer to them as principles. In this spirit, Bohr [15] and Pauli [19] have referred to quantum mechanics as “the theory of complementarity”, and the programme implied by this slogan has been carried out in quite different ways by Lahti together with the late Bugajski [54] (who use the convexity, or operational approach) and by Schwinger [20]

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(based on his idea of measurement algebra). Similarly compelling derivations of quantum mechanics from the uncertainty principle, as envisaged by Heisenberg [16], are still outstanding. This task seems impossible as long as the uncertainty principle is expressed only in terms of an inequality; however, if the uncertainty principle is reinterpreted in the stronger formal sense of the canonical commutation relations, which represent the characteristic shift covariance properties of localization observables, then a derivation of quantum mechanics from such an enhanced postulate becomes possible starting, for example, within a quantum logical framework [55]. Acknowledgment We are indebted to James Brooke (Saskatoon), Joy Christian (Oxford and PI), Gregg Jaeger (Boston), Pekka Lahti (Turku), Abner Shimony (Boston) and Stefan Weigert (York) for valuable comments and suggestions on various draft versions of this work. Part of this work was carried out during P. Busch’s stay at Perimeter Institute. C. Shilladay gratefully acknowledges support and hospitality extended to him during his short visit at PI. References [1] [2] [3] [4]

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Physics Reports 435 (2006) 33 – 156 www.elsevier.com/locate/physrep

Dynamics of Loschmidt echoes and fidelity decay Thomas Gorina , Tomaž Prosenb,∗ , Thomas H. Seligmanc, d , Marko Žnidariˇcb a Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Street 38, D-01187 Dresden, Germany b Physics Department, Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia c Centro Internacional de Ciencias, Apartado postal 6-101, C.P.62132 Cuernavaca, Morelos, Mexico d Instituto de Ciencias Físicas, University of Mexico (UNAM), C.P.62132 Cuernavaca, Morelos, Mexico

Accepted 13 September 2006 editor: C.W.J. Beenakker

Abstract Fidelity serves as a benchmark for the reliability in quantum information processes, and has recently attracted much interest as a measure of the susceptibility of dynamics to perturbations. A rich variety of regimes for fidelity decay have emerged. The purpose of the present review is to describe these regimes, to give the theory that supports them, and to show some important applications and experiments. While we mention several approaches we use time correlation functions as a backbone for the discussion. Vanicek’s uniform approach to semiclassics and random matrix theory provides important alternatives or complementary aspects. Other methods will be mentioned as we go along. Recent experiments in micro-wave cavities and in elastodynamic systems as well as suggestions for experiments in quantum optics shall be discussed. © 2006 Elsevier B.V. All rights reserved. PACS: 03.65.Nk; 03.65.Sq; 03.65.Ud; 03.65.Yz; 03.67.Lx; 03.67.Pb; 05.45.Mt; 05.45.Pq; 43.20.Gp; 76.60.Lz Keywords: Loschmidt echo; Fidelity

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Historical overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Outline of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. General theoretical framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Fidelity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1. Quantum Zeno regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2. Temporally stochastic perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3. Effect of conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4. Baker–Campbell–Hausdorff expansion of the echo operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Perturbation with vanishing time average . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Average fidelity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1. Time-averaged fidelity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35 36 38 39 39 41 41 42 43 43 45 45

∗ Corresponding author.

E-mail addresses: [email protected] (T. Gorin), [email protected] (T. Prosen), seligman@fis.unam.mx (T.H. Seligman), [email protected] (M. Žnidariˇc). 0370-1573/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physrep.2006.09.003

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T. Gorin et al. / Physics Reports 435 (2006) 33 – 156 2.3.2. State averaged fidelity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Estimating fidelity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Errors in approximate quantization schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6. Echo measures beyond fidelity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1. Scattering fidelity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2. Polarization echo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3. Composite systems, entanglement and purity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.4. Inequality between fidelity, reduced fidelity and echo purity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.5. Uncoupled unperturbed dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.6. Linear response expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum echo-dynamics: non-integrable (chaotic) case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Fidelity and dynamical correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1. Long time behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Vanishing time-averaged perturbation and fidelity freeze . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Composite systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Semiclassical theories in terms of classical orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Random matrix theory of echo dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Linear response theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Supersymmetry calculation for the fidelity amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Quantum freeze . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1. Linear response approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2. Beyond fidelity freeze . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3. Commutator perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4. RMT freeze in dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Echo purity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5. Purity decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum echo-dynamics: integrable case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Action-angle operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Perturbations with non-zero time average . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1. Coherent initial states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2. Random initial states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3. Average fidelity for coherent states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Perturbations with zero time average . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1. Coherent initial states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2. Freeze in a harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3. Rotational orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4. Random initial states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Composite systems and entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5. Mixed phase space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classical echo-dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Short time decay of classical fidelity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1. Linear response regime for Lipschitz continuous initial density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2. Linear response regime for discontinuous initial density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.3. General (multi-)Lyapunov decay for chaotic few body systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Asymptotic long time decay for chaotic dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. Asymptotic decay for regular dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4. Classical fidelity in many-body systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5. Universal decay in dynamically mixing systems which lack exponential sensitivity to initial conditions . . . . . . . . . . . . . . . . . . . . . . . Time scales and the transition from regular to chaotic behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. Chaotic dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Regular dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3. Comparison, chaotic vs. regular . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4. Increasing chaoticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5. Echo measures in terms of Wigner functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1. Illustration with Wigner functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.2. Wigner function approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Application to quantum information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1. Fidelity studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2. Decreasing the errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1. Modifying the algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2. Dynamical decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A report on experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46 47 48 48 48 50 51 52 53 53 54 54 57 60 62 64 68 70 72 75 75 76 80 82 83 85 87 88 89 90 91 93 93 94 97 98 98 100 103 104 105 106 107 107 110 111 112 112 113 113 114 115 117 118 118 120 121 122 123 123 125 127

T. Gorin et al. / Physics Reports 435 (2006) 33 – 156 9.1. Echo experiments with nuclear magnetic resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2. Measuring fidelity of classical waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3. Fidelity decay and decay of coherences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1. The Gardiner–Cirac–Zoller approach and the kicked rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. Time-averaged fidelity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix B. Models for numerics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1. The kicked top . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1.1. Two coupled kicked tops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2. The kicked Ising-chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix C. Scattering fidelity in different approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.1. Limit of weak coupling, diagonal S-matrix elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.2. Rescaled Breit–Wigner approximation and the perturbative regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.3. Rescaled Breit–Wigner approximation and linear response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix D. Echo purity in RMT: technicalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.1. The decoupled case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix E. Higher order terms in the Born series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35 127 128 131 132 134 135 135 137 137 140 141 142 142 142 143 144 147 149 150

1. Introduction Irreversibility of macroscopic behavior has attracted attention ever since Boltzmann introduced his H-theorem in a seminal paper in 1872 [1] (for English translation of some of Boltzmann’s papers see book [2]), together with the equation bearing his name. The problem of irreversibility is how to reconcile the apparent “arrow of time” with the underlying reversible microscopic laws. How come that macroscopic systems always seem to develop in one direction whether the underlying dynamics is symmetric in time? This so called reversibility paradox is usually attributed to Josef Loschmidt. He mentioned it briefly at the end of a paper published in 1876 [3] discussing the thermal equilibrium of a gas subjected to a gravitational field, in an attempt to refute Maxwell’s distribution of velocities for a gas at constant temperature. He also questioned Boltzmann’s monotonic approach towards the equilibrium. Discussing that, if one would reverse all the velocities, one would go from equilibrium towards the initial non-equilibrium state, he concludes with: “. . . Das berühmte Problem, Geschehenes ungeschehen zu machen, hat damit zwar keine Lösung . . .”. Bolzmann was quick to answer Loschmidt’s objections in a paper from 1877 [4], pointing out the crucial importance of the initial conditions and of the probabilistic interpretation of the second law. The story goes that Boltzmann’s reply to Loschmidt’s question regarding velocity reversal was: “Then try to do it!”. It is obvious that such velocity reversal is for all practical purposes impossible due to large number of particles involved and also due to high sensitivity to small errors made in the reversal. The first written account of the reversibility paradox is actually not due to Loschmidt but due to William Thompson (later Lord Kelvin), although it is possible that Loschmidt mentioned the paradox privately to Boltzmann before. Boltzmann and Loschmidt became good friends while working at the Institute of Physics in Vienna around 1867, for a detailed biography of Boltzmann see book [5]. In a vivid paper [6] from 1874 Thompson gave a very modern account of irreversibility. When discussing the heat conduction and the equalization of temperature he says: “. . . If we allowed this equalization to proceed for a certain time, and then reversed the motions of all the molecules, we would observe a disequalization. However, if the number of molecules is very large, as it is in a gas, any slight deviation from absolute precision in the reversal will greatly shorten the time during which disequalization occurs. . . Furthermore, if we take account of the fact that no physical system can be completely isolated from its surroundings but is in principle interacting with all other molecules in the universe, and if we believe that the number of these latter molecules is infinite, then we may conclude that it is impossible for temperature-differences to arise spontaneously . . .”. The interesting question is then, how short is this short disequalization time? An analysis of the disequalization time is very near to the modern concept of echo dynamics. From discussions of Boltzmann, Loschmidt and Thompson it is clear that this disequalization time will depend on the imperfections made in the reversal, i.e. the change in the initial condition for the backward evolution, and on the perturbations to this backward evolution, i.e. on how much the backward evolution after the reversal differs from the forward evolution. The

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quantity measuring sensitivity to perturbations of the backward evolution is nowadays known as the Loschmidt echo after Ref. [7], or the fidelity (we shall interchangingly use both terms, Loschmidt echo and fidelity). It is the overlap of the initial state with the state obtained after the forward unperturbed evolution followed by the backward perturbed evolution (Loschmidt echo) or equivalently the overlap of states obtained after the forward unperturbed evolution and the state after the forward perturbed evolution (fidelity). It can be considered in classical mechanics as well as in quantum mechanics. If unperturbed and perturbed evolutions are the same the Loschmidt echo is obviously equal to one while for unequal evolutions it generally decreases with time. The decay time of the Loschmidt echo will then be the time of disequalization in question. Surprisingly, despite its importance for statistical mechanics and irreversibility it was not considered until some years ago, even though various “problems” regarding irreversibility appear again and again. Echo dynamics and with it connected Loschmidt echo is the subject of the present report. Echo dynamics has its applications beyond statistical mechanics. Performing echo evolution is a standard experimental technique in NMR where the earliest measurements of the Loschmidt echo were performed [8–11]. It can also be related to other situations such as the seismic response in volcanos [12]. The quantum information community has adopted the autocorrelation function of echo dynamics, known as fidelity, as a standard benchmark for the quality of any implementation of a quantum information device [13]. Quantum information theory enables one to do things not possible by classical means, e.g., perform quantum computation. The main obstacle in producing quantum devices that manipulate individual quanta are errors in the evolution, either due to unwanted coupling with the environment or due to internal imperfections. Therefore, the goal is to build a device that is resistant to such perturbations. For this one ought to understand the behavior of fidelity in different situations to know how to maximize it. The theoretical framework for decreasing the unwanted evolution in quantum information devices, called dynamical decoupling [14,15], is intimately connected to fidelity theory, especially to the interesting phenomenon of quantum freeze [16–18]. But the ultimate motivation for theoretical study of fidelity neither came from quantum information theory nor from statistical mechanics but from the field of quantum chaos [19,20]. The exponential instability of classical systems is a well known and much studied subject. As the underlying laws of nature are quantum mechanical the obvious question arises how this “chaoticity” manifests itself in quantum systems whose classical limit is chaotic. The field of quantum chaos mainly dealt with stationary properties of classically chaotic systems, like spectral and eigenvector statistics. Despite classical chaos being defined in a dynamical way it was easier to pinpoint the “signatures” of classical chaos in stationary properties. There were not that many studies of the dynamical aspects of quantum evolution in chaotic systems, some examples being studies of the reversibility of quantum evolution [21,22], the dynamical localization [23,24], energy spreading [25,26] or wave-packet evolution [27,28]. Classical instability is usually defined as an exponential separation of two nearby trajectories in time. In quantum mechanics one could be tempted to look at the sensitivity of quantum mechanics to variations of the initial wave function. But quantum evolution is unitary and therefore preserves the dot product (i.e. the distance) between two states and so there is no exponential sensitivity with respect to the variation of the initial state. The sensitivity to perturbations as measured by the Loschmidt echo on the other hand naturally allows for the comparison between quantum and classical situation. For classical systems Loschmidt echo gives the same exponential sensitivity to perturbations of the evolution as to perturbations of initial conditions. The quantum fidelity though can behave in a very different way, displaying a rich variety of regimes. Some other theoretical questions like quantization ambiguity, i.e. the difference between different quantizations having the same classical limit, can also be connected to fidelity decay [29]. Considering that fidelity lies at the crossroad of three very basic areas of physics: statistical mechanics, quantum information theory and quantum chaos, and that its understanding is crucial for building successful quantum devices it is not surprising that it received a lot of attention in recent years. Let us give a brief historical overview, listing the most important discoveries. The list of references is by no means complete, for detailed references see the later sections of the paper. 1.1. Historical overview Fidelity has been first used as a measure of stability by Peres in 1984 [30], see also his book [31]. Peres reached the conclusion that the decay of fidelity is faster for chaotic than for regular classical dynamics. As we shall see the general situation can be exactly the opposite. Non-decay of fidelity for regular dynamics in Peres’s work was due to a very special choice of the initial condition, namely that a coherent wave packet was placed in the center of a stable island. Such a choice is special in two ways, first the center of an island is a stationary point and second the number of constituent eigenstates of the initial state is very small. After Peres’s work the subject lay untouched for about

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37

a decade. In 1996 Ballentine and Zibin [32] numerically studied a quantity similar to fidelity. Instead of perturbing the backward evolution, they took the same backward evolution but instead perturbed the state after forward evolution by some instantaneous perturbation, like shifting the whole state by some
x. They also looked at the corresponding classical quantity. The conclusion they reached was that for chaotic dynamics quantum stability was much higher than the classical one, while for regular dynamics the two agreed. All these results were left mainly unexplained. Gardiner et al. [33,34] proposed an experimental scheme for measuring the fidelity in an ion trap. In a related work, Schack and Caves [35–37] studied how much information about the environment is needed to prevent the entropy of the system to increase. Fidelity studies received fresh impetus from a series of NMR experiments carried out by the group of Levstein and Pastawski. In NMR echo experiments are a standard tool. The so called spin echo experiment of Hahn [8] refocuses free induction decay in liquids due to dephasing of the individual spins caused by slightly different Larmor frequencies experienced due to magnetic field inhomogeneities. By an appropriate electromagnetic pulse the Zeeman term is reversed and thus the dynamics of non-interacting spins is reversed. The first interacting many-body echo experiment was done in solids by Rhim et al. [9]. Time reversal, i.e. changing the sign of the interaction, is achieved for a dipolar interaction whose angular dependence can change sign for a certain “magic” angle, that causes the method to be called magic echo. Still, the magic echo showed strong irreversibility. Much later a sequence of pulses has been devised enabling a local detection of polarization [10] (i.e. magnetic moment). They used a molecular crystal, ferrocene Fe(C5 H5 )2 , in which the naturally abundant isotope 13 C is used as an “injection” point and a probe, while a ring of protons 1 H constitutes a many-body spin system interacting by dipole forces. The experiment proceeds in several steps: first the 13 C is magnetized, then this magnetization is transferred to the neighboring 1 H. We thus have a single polarized spin, while others are in “equilibrium”. The system of spins then evolves freely, i.e. spin diffusion takes place, until at time t the dipolar interaction is reversed and by this also spin diffusion. After time 2t the echo is formed and we transfer the magnetization back to our probe 13 C enabling the detection of the polarization echo. Note that in the polarization j j +1 echo experiments the total polarization is conserved as the dipole interaction has only “flip-flop” terms like S+ S− , which conserve the total spin. To detect the spin diffusion one therefore needs a local probe. With the increase of the reversal time t the polarization echo—the fidelity—decreases in approximately exponential way. The nature of this decay has been further investigated by Pastawski et al. [38]. The group of Levstein and Pastawski performed a series of NMR experiments where they studied in more detail the dependence of the polarization echo on various parameters [11,39,40]. They were able to control the strength of the residual part of the Hamiltonian, which was not reversed in the experiment and is assumed to be responsible for the polarization echo decay. By diminishing the residual interactions they obtained a Gaussian decay with a decay rate, saturated and independent of the perturbation strength, i.e. the strength of the residual interaction. That would imply the existence of a perturbation independent regime. While there is still no complete consensus on the interpretation of these experimental results they triggered a number of theoretical and even more numerical investigations. We shall brieflylist just the most important ones, first regarding quantum fidelity. Using the semiclassical expansion of the quantum propagator Jalabert and Pastawski [7] derived a perturbation independent quantum fidelity decay for localized initial states and chaotic dynamics, also called “Lyapunov decay” due to its dependence on the Lyapunov exponent. This regime has been numerically observed for the first time in Ref. [41]. Studying fidelity turned out to be particularly fruitful in terms of the correlation function [42], see also Ref. [43], which is also the approach we use in the present review. Perturbation dependent exponential decay of quantum fidelity observed in Refs. [42,44–46], also called the Fermi golden rule decay, has been derived in Ref. [44] using random matrix theory and in Ref. [45] using semiclassical methods, and independently in Refs. [42,46] using Born expansion in terms of correlation functions. Perhaps the most interesting result of this approach is that one can, by increasing chaoticity of the corresponding classical system, increase quantum fidelity, i.e. improve the stability of the quantum dynamics. Note that classical correlators already appeared in Ref. [7], and also later in Ref. [45]. For sufficiently small perturbations fidelity decays as a Gaussian, which is the so-called perturbative decay [44–46]. Refs. [44,45] were the first which presented a unified theoretical treatment of all main regimes of fidelity decay, and identified the scales of perturbation strength which controlled the transitions among them. Decoherence, as characterized by the decay of off-diagonal matrix elements of the reduced density matrix, has been connected to the fidelity in Refs. [47,48]. Echo purity, which is a generalization of purity to echo dynamics, has been introduced in Ref. [49] and the so-called reduced fidelity, measuring the stability on a subsystem, in Ref. [50]. A rigorous inequality between fidelity, reduced fidelity and echo purity has been proved in Refs. [50,51]. It is well known that the

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quantization of classical system is not unique. Kaplan [29] compared the quantization ambiguity in chaotic and regular systems, reaching the conclusion that in chaotic systems the quantization ambiguity is suppressed as compared with regular ones. A detailed random matrix (RMT) formulation of fidelity has been presented in Ref. [52], although random matrix theory for describing fidelity decay has been used before [44,53,54]. A supersymmetric method has been used to derive an analytic expression for fidelity decay in RMT models [55,56]. Quantum fidelity decay in regular systems has been discussed in Ref. [46]. An interesting phenomenon of prolonged stability for certain type of perturbations, called quantum freeze, has been discovered in Refs. [16–18]. Using a semiclassical propagator fidelity has been expressed as a Wigner function average of phases due to action differences [57,58], the so-called dephasing representation, which is particularly handy for numerical evaluation of quantum fidelity in terms of classical orbits. Classical fidelity as compared to the quantum one received much less attention. It has been first defined and linear response derived in Ref. [46]. Numerical results on the classical fidelity and its correspondence with the quantum fidelity in chaotic systems and in systems exhibiting diffusion have been presented in Ref. [59]. Classical fidelity in regular and chaotic systems has also been theoretically discussed [60]. A detailed explanation of the asymptotic decay in chaotic systems has been given in Ref. [61]. Short time Lyapunov decay of classical fidelity has been obtained in Ref. [62], showing that the Lyapunov decay of quantum fidelity is a consequence of quantum-classical correspondence. Classically regular systems on the other hand have been worked out in Ref. [63]. 1.2. Outline of the paper A wide variety of methods have been used to treat echo dynamics, and shall be discussed in this report, yet a linear response treatment, which relates fidelity decay to the decay of the correlation function of the perturbation in the interaction picture, is sufficient to understand most of the results. The method is based on writing the so-called echo propagator in the interaction picture, thereby focusing attention only on the perturbation that actually causes fidelity to decay. By employing the Baker–Campbell–Hausdorff formula or a linear response expansion, one is then able to obtain approximations in a perfectly controllable way. In this review we shall therefore use this method as skeleton and discuss other methods as they are needed or have been used in the literature. All the theoretical work has been accompanied by extensive numerical experiments. Finally we shall see, that a random matrix model represents real and numerical experiments under chaotic conditions quite well. Throughout this review we shall pay special attention to the role of the initial state (localized packets versus random states) and the type of dynamics (integrable versus chaotic dynamics). Furthermore, we shall discuss random matrix models. We shall now describe the organization of this review. In the process we shall point out a hierarchical structure, that will allow the reader interested in some particular aspect, to read only the next section and a few subsections to obtain the information he needs. Thus Section 2 contains the basic definitions of echo dynamics and fidelity, which will be indispensable throughout the paper. Also we shall see, that the interaction picture provides the ideal representation to understand echo dynamics, because it isolates the effect of the perturbation on the wave function from that of the unperturbed dynamics. To get a first understanding we shall in Section 2.1 see how dynamical correlations of the perturbation control the basic behavior of fidelity decay, in a linear response analysis. This subsection is essential for the entire paper, because we shall use it as a skeleton for most considerations, and all new results will be based on its contents. The section will give definitions of other measures corresponding to different experimental and theoretical situations, such as scattering, polarization echo, or the evolution of entanglement as measured by purity under echo dynamics. The analysis of the precision of different quantization schemes is also connected to fidelity decay. Section 3 deals with dynamical chaotic systems, and we shall explicitly see the effects of correlations of the perturbation on the fidelity decay. Special attention will be given to perturbations with zero diagonal, which result in the so-called quantum freeze. We shall discuss echo measures for composite systems. In Section 4 random matrix theory of echo dynamics will be developed, and we shall discuss when we expect the results found by RMT to represent a chaotic system well. The quantum freeze will be analyzed in the context of RMT. New results on quantum freeze and decoherence will be presented. Section 5 will deal with integrable systems and has similar internal structure as the one on chaotic dynamics. Though, due to the lack of universality the behavior is richer. In this context the difference between coherent states and radom states will play a significant role. Some previously unpublished results are presented in this section. These include

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39

Section 5.2.3 describing decay of fidelity averaged over position of initial packets and Section 5.3.2 describing freeze in a harmonic oscillator. The following Section 6 deals with the decay of classical fidelity. First linear response is derived already showing marked difference between classical and quantum fidelity. Then long time Lyapunov decay is discussed as well as the decay in regular systems. The section concludes with the discussion of classical fidelity decay in many-body systems. Section 7 on time scales is an overview of various decays derived for chaotic and regular systems. It can serve as a brief summary of results obtained in Sections 3 and 5. The scaling of decay time scales on different parameters is given and the decay is compared between chaotic and regular dynamics. In certain situations quantum regular systems can be more sensitive to perturbations than chaotic ones and additionally, increasing the chaoticity can improve the stability. The decay of fidelity is also illustrated in terms of Wigner functions. Unpublished material consists of some figures illustrating and comparing decay time scales in chaotic and regular systems. Application of fidelity to quantum information is described in Section 8. First we give an incomplete list of various studies of the influence of errors on quantum computation and point out how different results obtained in previous sections can be employed to understand results obtained in the literature. We give an illustrative example of how more randomness can improve stability of quantum computation. A connection between dynamical decoupling and fidelity freeze is also pointed out. Finally in Section 9 we shall describe experiments measuring fidelity. We shall discuss NMR experiments, microwave experiments and elasticity experiments, for which fidelity measurements have already been performed as well as possible atom optics experiment. Although our exposition is intended as a unified and self-contained review of the results which have mainly been published before in the quoted literature, there is however a substantial body of new results which have never been published. Let us here pin down the main original results presented in this report: (i) the results about quantum freeze of fidelity in the random matrix framework (Section 4.3); (ii) the extension of the results on scattering fidelity to the weak coupling regime (Appendix C); (iii) random matrix treatment of purity decay in composite systems, both in the framework of echo dynamics and of forward dynamics of a central system weakly coupled to an environment; and (iv) treatment of the quantum freeze of fidelity for quadratic Hamiltonians (Section 5.3.2). 2. General theoretical framework 2.1. Fidelity We consider a general quantum Hamilton operator Hε (t) which may depend explicitly on time t, as well as on some external parameter ε, like magnetic field, interaction strength, shape of potential well, etc. Without essential loss of generality we shall assume that the parameter dependence is linear, namely Hε (t) = H0 (t) + εV (t). Let Uε (t) be the propagator
  i t  Uε (t) = Tˆ exp − dt Hε (t  ) , 2 0

(1)

(2)

where Tˆ designates the time ordering. If | is some arbitrary initial state, then the time evolution, depending on the time variable t as well as on the perturbation parameter ε of the Hamiltonian, is given by |ε (t) = Uε (t)|.

(3)

Having the tools defined above, the fidelity amplitude, with respect to the unperturbed evolution U0 (ε), is defined as the overlap of the perturbed and unperturbed time-evolving states as fε (t) = 0 (t)|ε (t) = |U0 (−t)Uε (t)|.

(4)

The square of its modulus is fidelity Fε (t) = |fε (t)|2 .

(5)

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At this point we wish to stress the dual interpretation of fidelity, which is the central characteristic of echo dynamics often called ‘Loschmidt echo’: on one hand Fε (t) is the probability that the states of unperturbed and perturbed time evolution are the same, but on the other hand, due to unitarity of time evolutions, Fε (t) is also the probability that after an echo—composition of forward perturbed and backward unperturbed dynamics—we arrive back to the initial state. In the following, we shall write the expectation value in a given initial state simply as A = |A|. The fidelity amplitude may now be compactly written in terms of the expectation value of the so-called echo-operator Mε (t) = U0 (−t)Uε (t),

(6)

fε (t) = Mε (t).

(7)

as

One mayalso use a non-pure initial state, namely a statistical mixture described by a density matrix , with A = tr A = k pk k |A|k , where pk and |k  are, respectively, the eigenvalues and eigenvectors of the density matrix . One might be lead to an intuitive belief that incoherent superpositions of fidelity amplitude, such as the above for non-pure initial states, would not have a clear physical significance. Nevertheless, it turns out that this is not the case and that such a generalized object may be experimentally observable [64] and exhibit certain interesting theoretical features [65]. Writing the perturbation operator in the interaction picture V˜ (t) = U0 (−t)V (t)U0 (t),

(8)

one can straightforwardly check by differentiating Eq. (6) that the echo-operator solves the evolution equation i d Mε (t) = − ε V˜ (t)Mε (t) dt 2

(9)

with the (effective) Hamiltonian ε V˜ . The echo-operator is nothing but the propagator in the interaction picture, which can be written in terms of a formal solution of Eq. (9)
  t i  ˜  ˆ Mε (t) = T exp − ε dt V (t ) . (10) 2 0 Again, a time-ordered product has to be used since V˜ (t) for different times does not in general commute. The computation of fidelity becomes rather straightforward if one plugs the expression for the echo-operator (10) into definition (7). In particular, this approach is ideally suited for a perturbative treatment when the perturbation strength ε can be considered as a small parameter. One can write explicitly the Born series for the echo operator, which has an infinite radius of convergence provided that the perturbation V (t) is an operator, uniformly bounded for all t:  ∞  (−iε)m t Mε (t) = 1 + dt1 dt2 · · · dtm Tˆ V˜ (t1 )V˜ (t2 ) · · · V˜ (tm ). (11) 2m m! 0 m=1

If we truncate the above series at the second order, m = 2, and insert the expression into Eq. (7), we obtain    iε t  ˜  ε 2 t  t  ˜  ˜  fε (t) = 1 − dt V (t ) − 2 dt dt V (t )V (t ) + O(ε 3 ). 2 0 2 0 t Taking the square modulus we obtain the fidelity   ε 2 t  t  Fε (t) = 1 − 2 dt dt C(t  , t  ) + O(ε 4 ), 2 0 0

(12)

(13)

where C(t  , t  ) = V˜ (t  )V˜ (t  ) − V˜ (t  )V˜ (t  ),

(14)

is the 2-point time-correlation function of the perturbation. This high-fidelity approximation (13) shall be called linear response expression for fidelity. We stress that the validity of the linear response formula (13) is by no means restricted

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41

to short times. The only condition is to have high-fidelity, namely  := 1 − Fε (t) has to be small. The error of the linear approximation is typically of order O(ε 4 ). However, using the same technique one can go beyond the linear response approximation for certain particular cases of dynamics, for example we shall derive Fermi golden rule decay for strongly mixing dynamics (Section 3.1) and various non-universal decays for integrable dynamics (e.g. Section 5.2). Note that the linear-response formula (13) already establishes a very important physical property of quantum echodynamics. One observes an intimate connection between fidelity decay and temporal-correlation decay, namely faster decay of the correlation function C(t  , t  ), as |t  −t  | grows, implies slower decay of fidelity and vice versa. For example, for quantum systems in the semiclassical regime, whose classical limit is chaotic and fast mixing, such that C(t  , t  ) decays faster than const /|t  − t  |, fidelity is expected to decay as a linear function of time Fε (t) = 1 − const ε 2 t, whereas for regular systems, with integrable classical limit, C(t  , t  ) is expected to have oscillating behavior with generally nonvanishing time average, hence fidelity is expected to decay as a quadratic function of time Fε (t) = 1 − const ε2 t 2 . This implies the seemingly paradoxical conclusion that the fidelity of regular dynamics may decay faster than the fidelity of chaotic ones. Detailed discussion and comparsion between chaotic and regular dynamics is given in Section 7. The linear response formula (13) can be cast into another intuitively useful form. Let us define an integrated perturbation operator (t)  t (t) = dt  V˜ (t  ). (15) 0

Then, the RHS of Eq. (13) rewrites in terms of an uncertainty of operator (t): Fε (t) = 1 −

ε2

{2 (t) − (t)2 } + O(ε 4 ). (16) 22 Now, another interpretation of the quantum fidelity decay in the quantum chaotic—or stochastic—versus regular regime can be given. For quantum chaotic dynamics, one expects to have diffusive1 behavior of typical observables, like V, hence 2  − 2 ∝ t, whereas for regular dynamics one expects ballistic behavior, 2  − 2 ∝ t 2 . Another point has to be stressed: The fidelity amplitude is in general a c-number whose phase can be shifted by a perturbation which is a multiple of an identity operator. So, adding a constant to the perturbation, e.g. forcing it to have a vanishing expectation value V  = V − V 1, rotates the fidelity amplitude by a unimodular factor fε (t) = exp(iεtV /2)fε (t) while it leaves the fidelity Fε (t) unchanged. This follows trivially from the representation of fidelity amplitude (7) in terms of the echo operator (10). 2.1.1. Quantum Zeno regime Note, however, that for very short times, below a certain time scale tZ , namely before the correlation function starts to decay, |t  |, |t  | < tZ , C(t  , t  ) ≈ C(0, 0) = V 2 , the fidelity always exhibits (universal) quadratic decay 1/2 1/2

C(0, 0) V 2  ε2 2 2 for |t| < tZ = =2 . (17) F (t) = 1 − 2 V t d2 C(0, t)/dt 2 [H0 , V ]2  2 This short-time regime (also discussed in [30]) may be identified with the quantum Zeno effect [66,67], and the time scale tZ referred to as the Zeno time. 2.1.2. Temporally stochastic perturbations It is instructive to explain what happens in the case when the perturbation V (t) is explicitly time dependent and stochastic, i.e. being a Gaussian delta correlated white noise, with variance V (t  )V (t  ) = v 2
(t  − t  )1.

(18)

• here denotes an average over an ensemble of stochastic perturbation operators. The dynamical correlation C(t  , t  ) is obviously insensitive to the nature of unperturbed dynamics; hence the correlation function is again a delta function 1 Of course, for a finite quantum system, diffusive behavior can only be observed on finite time scales, typically below the Heisenberg time. The issue of time scales is discussed extensively in later sections, in particular in Section 7.

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T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

C(t  , t  ) = v 2
(t  − t  ). Furthermore, due to the Gaussian nature of the noise, higher-order correlation functions, say of order 2n, in the expression for the echo operator (11) factorize with the multiplicity (2n − 1)!! due to the Wick theorem, yielding a simple exponential decay
2  ε 2 F (t) = exp − 2 v t . (19) 2 For stochastic uncorrelated perturbations fidelity thus decays exponentially with the rate which depends on the magnitude of perturbation only and not on dynamics of the unperturbed system. 2.1.3. Effect of conservation laws One should note that the correlation integral  t  t  Ci (t) = dt dt  C(t  , t  ) = 2 (t) − (t)2 0

(20)

0

always increases quadratically, even in the case of chaotic quantum dynamics, if the perturbation has a nonvanishing component in the direction of some (trivial) conserved quantities such as the energy, the Hamilton operator H0 (if time-independent) or the angular momentum, etc. For example, let {Qn , n = 1, 2 . . . M} be an orthonormalized set of conserved quantities with respect to the initial state |, such that Qn Qm  =
nm .2 Then any time-independent perturbation can be decomposed uniquely as V =

M 

cm Q m + V 

(21)

m=1

with coefficients cm =V Qm  and V  being the remaining non-trivial part of the perturbation, by construction orthogonal to all trivial conservation laws, Qm V   = 0

for all m.

In such a case the correlation integral will always grow asymptotically as a quadratic function   M  2 Ci (t) → cm t 2 ;

(22)

(23)

m=1

hence fidelity will decay quadratically. Therefore it is desirable to subtract this trivial effect by always choosing perturbations which are orthogonal to all known trivial conservation laws. For example, if H0 is time-independent, then it is a trivial invariant. The fact that a general perturbation V may not be orthogonal to H0 , H0 V  = 0 is equivalent to the fact that V will change the density of states (in the first order in ε) in the region of eigenstates of H0 populated by the initial state . This means that the Heisenberg time will be different for forward (perturbed) and backward (unperturbed) dynamics, so fidelity will decay quadratically due to this trivial effect. This effect is simply removed by replacing the perturbation by V  = V − H0 V H0 /H02 , or equivalently by measuring forward and backward time evolution in units of their respective Heisenberg times. We note that this effect has actually been discussed and taken care of in beautiful experimental studies of fidelity by Lobkis and Weaver [68] and Stöckmann et al. [69,70] (see Section 9.2). Nevertheless it is important to add that such “trivial” perturbations are non-trivial in the framework of quantum information applications, and indeed this shows, that the control of such perturbations is particularly important. There is another consequence of conservation laws, namely they divide the Hilbert space of unperturbed evolution U0 (t) into blocks specified by eigenvalues of the conserved operators, or quantum numbers. As a result the effective Heisenberg time—which will be one of the key time scales in later discussions—is reduced due to a reduced average density of states with fixed quantum numbers. 2 Any set of physical conservation laws Q can be orthogonalized using a standard Gram–Schmidt procedure. In order to prevent degeneracy of m the scalar product we only have to assume independence of conservation laws in the sense that all the M states Qm | should be linearly independent.

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2.1.4. Baker–Campbell–Hausdorff expansion of the echo operator For the purposes of semi-classical analysis in subsequent sections it will be useful to perform another formal manipulation on the echo operator (10), namely we may want to rewrite it approximately in terms of a single exponential. For this aim, we apply the well known Baker–Campbell–Hausdorff expansion eA eB = exp(A + B + (1/2)[A, B] + · · ·), to the infinite product (10) yielding   t  t  ε t  ˜  ε2   ˜   ˜ Mε (t) = exp −i dt V (t ) + 2 dt dt [V (t ), V (t )] + · · · 2 0 22 0  t  
i 1 = exp − (24) (t)ε + (t)ε 2 + · · · , 2 2 where (t) =

i 2



t

dt 

0



t t

dt  [V˜ (t  ), V˜ (t  )].

(25)

Similar algebraic manipulations are well known in quantum field theory and are sometimes known as the Magnus expansions, see for example Ref. [71]. Two general remarks concerning the above asymptotic BCH expansion (24) are in order. First, the double integral of the commutator, which defines the operator (t), does in general grow only linearly in time, namely one can show that for arbitrary pair of states | and |, ||(t)|| < C t,

(26)

where C is some constant, which may depend on the choice of the states | and |. Second, the matrix elements of the third and fourth order terms of the BCH expansion (24) can be estimated by const ε 3 t and const ε 4 t 2 , respectively. Therefore, Eq. (24) provides in general a good approximation of the echo operator up to times t < const ε−1 , which can be made arbitrary long for sufficiently weak perturbations. 2.2. Perturbation with vanishing time average Let us define the time average of the perturbation operator  (t) 1 t  ˜  V¯ = lim dt V (t ). = lim t →∞ t t →∞ t 0

(27)

Generally, an arbitrary perturbation V can be decomposed into its time average V¯ and the rest, denoted by Vres and called the residual part, V = V¯ + Vres .

(28)

Let us, for the rest of this discussion, assume for simplicity that the unperturbed Hamiltonian H0 and the perturbation V are time-independent operators. Let Ek and |Ek  denote, respectively, eigenenergies and eigenvectors of the unperturbed Hamiltonian H0 and assume that the spectrum {Ek } is non-degenerate, which is typical for non-integrable and often true even for integrable systems. In this case the time average is the diagonal part of the perturbation V¯ = diagV , and the residual part Vres is the off-diagonal part. The operator (t) can be asymptotically written as (t) = V¯ t + O(t 0 ), where the second term has off-diagonal matrix elements only, if expressed in the eigenbasis of H0 . A further important condition is that Vres does not grow asymptotically with time. Therefore, the second order correlation integral Ci (t) when expressed as (20) does not grow with time either and remains bounded for all times if V¯ = 0 or, more precisely, if and only if V¯ 2  − V¯ 2 = 0. Even though one may argue that the case of vanishing time-averaged perturbation V¯ = 0 may be rather special and non-generic, it deserves to be studied separately as it exhibits very unusual and perhaps surprisingly stable behavior of echo-dynamics. Furthermore, it can be realized in several important physical situations: • When the perturbation V can be written as time-derivative of another observable or equivalently as a commutator with the Hamiltonian then obviously, by construction, the diagonal part V¯ vanishes.

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T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

• When the unperturbed system is invariant under a certain unitary symmetry operation P, say parity, P H 0 = H0 P , whereas the symmetry changes sign of the perturbation P V = −V P . This means that the perturbation breaks the unitary symmetry in a maximal way and matrix elements of V are nonvanishing only between the states of opposite parity. • When the unperturbed system is invariant under a certain anti-unitary symmetry operation T, say time-reversal, T H 0 =H0 T , whereas the symmetry changes sign of the perturbation T V =−V T . This means that the perturbation breaks the antiunitary symmetry in an optimal way and the matrix of V in the eigenbasis of H0 can be generally written as Vmn = iAmn where A is a real antisymmetric matrix. • Sometimes diagonal elements of the perturbation can be taken out by hand and treated as a trivial first order perturbation of the unperturbed part. In few-body and many-body physics this approach is commonly known as the mean-field approximation. One may be generally interested in the effect of residual perturbations and analyze it through echo-dynamics. In the present subsection we discuss some general properties of echo dynamics for perturbations with V¯ = 0, also called residual perturbations because V = Vres . As we shall see, the short time fidelity will still be given by the operator (t). Because its norm does not grow with time, the fidelity will freeze at a constant value for the time of validity of the linear response approximation. After a sufficiently long time it will again start to decay due to the operator (t). We claim that this behavior is rather insensitive to the nature of unperturbed dynamics, for example whether it is regular or chaotic. Provided that the spectrum of H0 is non-degenerate, any residual perturbation can be defined in terms of another operator W by the following prescription3 i V = (d/dt)W (t) = [H0 , W ], W (t) = U0 (−t)W U 0 (t). 2 Indeed, given a residual perturbation one easily determines the matrix elements of W as Wj k := −i2

Vj k , Ej − E k

(29)

(30)

where Ei are eigenenergies of H0 . For the corresponding definition in a discrete-time case, e.g. for kicked systems, see [16,17]. From the definition of W in Eq. (30) we can see that the matrix elements Wj k are large for near degeneracies. For W to be well behaved the matrix elements Vj k of the perturbation must decrease smoothly approaching the diagonal. Setting diagonal elements of V to zero will typically [e.g. in classically chaotic systems (Section 3.2) or in models of random matrices (Section 4.3)] produce singular behavior and reduce the effect of fidelity freeze. However, for regular unperturbed systems the effect of quantum freeze is even more robust due to abundance of selection rules, namely the matrix Vj k is typically sparse so singularities due to near degeneracies may not occur even if we take out the diagonal part by hand. With this newly defined operator W, the expression for the integrated perturbation (t) is extremely simple, (t) = W (t) − W (0).

(31)

Similarly, the expression for (t) (25) is considerably simplified to i (t) = R (t) − [W (0), W (t)], 2 where



t

R (t) =

R(t  ) dt  ,

i R = [W, (d/dt)W (t)], 2

R(t) = U0 (−t)RU 0 (t).

(32)

(33)

0

Apart from the “boundary term” i/2[W, W (t)], the operator (t) has, similarly to (t), the structure of a time integral of another operator R. From this representation it follows that any matrix element or the norm of operator (t) grows at most linearly with time. The aforementioned boundary term can be used to factorize the expression for the echo 3 Note that in the classical limit one can replace the commutator with a Poisson bracket, (1/i2)[A, B] → {A, B}.

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operator which will be the form most useful for applications in the following sections. Namely, to order O(tε 3 ) it can be written as


 i ε2 i i Mε (t) = exp − W (t)ε exp − R (t) exp W (0)ε . (34) 2 2 2 2 From expression (34) we see that, since W (t) = W (0) , a time scale t2 = O(1/ε) should exist such that for times t < t2 , the term in the second exponential R (t)ε 2 can be neglected as compared to the one in the first and the third, W (t)ε, W (0)ε. Therefore, for t < t2 we can write



ε

 2 ε



(35) F (t) = exp −i W (t) exp i W (0) . 2 2 As we shall see later, the RHS of the above equation will typically not depend on time, beyond some timescale t1 , much shorter than t2 , namely for t1 < t < t2 , but we can claim quite generally that RHS of Eq. (35) has a strict lower bound. Expanding the formula (35) to the second order in ε one can show [17] that the fidelity has a lower bound irrespective on the nature of dynamics, given by F (t)
1 − 4

ε2 22

r 2,

t < t2 ,

(36)

where r 2 = sup[W (t)2  − W (t)2 ].

(37)

t

For a residual perturbation the fidelity therefore stays high up to a classically long time t2 ∼ 1/ε and only then starts to decay again. We call this phenomena fidelity freeze. The freeze happens for arbitrary quantum dynamics, irrespective of the existence and the nature of the classical limit. After t2 the term (t) ∼ t R¯ in the second exponential of (34) will become important and eventually this will cause the fidelity to decay. In the regime t > t2 the expression of the echo operator is formally similar to the original form (10) where the perturbation εV has to be replaced by an effective perturbation operator 21 ε 2 R. This point shall be discussed in more detail later. We have to stress that the freeze of fidelity is of purely quantum origin. The classical fidelity discussed in Section 6 does not exhibit a freeze. 2.3. Average fidelity 2.3.1. Time-averaged fidelity In a finite Hilbert space the fidelity will not decay to zero but fluctuate around some small plateau value, which is equal to the time-averaged fidelity. For ergodic systems this time-averaged value is in turn equal to the phase space averaged one. For a finite Hilbert space of size N, fidelity will start to fluctuate for long times due to the discreteness of the spectrum of the evolution operator. The size of this fluctuations can be calculated by evaluating time-averaged fidelity F¯  1 t  F¯ = lim dt F (t  ). (38) t →∞ t 0 We expand the initial state in the eigenstates |En  of the unperturbed Hamiltonian H0 , write the eigenenergies and eigenstates of the perturbed Hamiltonian as Enε and |Enε , and denote the matrix elements between unperturbed and perturbed eigenstates by Okl = Ek |Elε .

(39)

The transition matrix O is unitary, and if both eigenvectors can be chosen real it is orthogonal. This happens if H0 and Hε commute with an antiunitary operator T whose square is identity. The fidelity amplitude can now be written as  f (t) = (O † )lm Oml exp(−i(Elε − Em )t/2), (40) lm

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T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

with lm = El |(0)|Em  being the matrix elements of the initial density matrix in the unperturbed eigenbasis. To calculate the average fidelity F¯ we have to take the absolute value square of f (t). Averaging over time t we shall assume that the phases are non-degenerate and find exp(i(Elε − Elε + Em − Em )t/2) =
mm
ll  .

(41)

This results in the average fidelity F¯ =



|(O)ml |2 |Oml |2 .

(42)

ml

The time-averaged fidelity therefore understandably depends on the initial state  as well as on the “overlap” matrix O. For small perturbation strengths, say ε smaller than some critical εrm , the unitary matrix O will be close to identity. Using O → 1 for ε>εrm (42) gives us F¯weak =



(ll )2 .

(43)

l

One should keep in mind that for the above result F¯weak we needed the eigenenergies to be non-degenerate, Elε = Em (41), and at the same time O → 1. This approximation   is justified in lowest order in ε, if the offdiagonal matrix elements are |Oml |2 ∝ ε 2 . For pure initial states l 2ll = l |El ||4 is just the inverse participation ratio of the initial state expressed in terms of the unperturbed eigenstates. Note that these expressions should be symmetric with respect to the interchange of the unperturbed and perturbed basis. On the other hand, for sufficiently large perturbation strength ε and complex perturbations V, such that the two bases—the perturbed one and the unperturbed one—become practically unrelated, one might assume O to be close to a random matrix, unitary or orthogonal. In the limit N → ∞, the matrix elements Oml can be treated as independent complex or real Gaussian random numbers. Then we can average the expression (42) over a Gaussian distribution ∝ exp(− N |Oml |2 /2) of matrix elements Oml , where we have = 1 for orthogonal O and = 2 for unitary O. This averaging gives |Oml |4  = (4 − )/N 2 and |Oml |2  = 1/N for the variance of Oml (brackets • denote here averaging over the distribution of matrix elements and not over the initial state). The average fidelity for strong perturbation can therefore be expressed as l =m 4−  2 1  ¯ Fstrong = ll + |lm |2 . N N l

(44)

l,n

More details on various cases of initial states as well as on crossover of the average fidelity from the case of weak to the case of strong perturbations can be found in Appendix A. 2.3.2. State averaged fidelity Sometimes the average fidelity is of interest, i.e. the fidelity averaged over some ensemble of initial states. Such an average fidelity is also more amenable to theoretical treatment. Easier to calculate is the average fidelity amplitude f (t) which is of second order in the initial state | while fidelity F (t) is of fourth order in |. We shall show that the difference between the absolute value squared of the average fidelity amplitude and the average fidelity is small for large Hilbert space dimensions N. Let us look only at the simplest case of averaging over random initial states, denoted by •. In the asymptotic limit of large Hilbert space size N → ∞ the averaging is simplified by the fact that the expansion coefficients cm of a random initial state in an arbitrary basis become independent Gaussian variables, with variance 1/N . Quantities bilinear in the initial state, like the fidelity amplitude or the correlation function, result in the following expression 

 |A|= : A =

 ml

∗ cm Aml cl

=

1 tr A, N

(45)

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where A is an arbitrary operator. The averaging is done simply by means of a trace over the whole Hilbert space. For the fidelity F (t) which is of fourth order in | we get after some analysis F (t) =



∗ cm [Mε (t)]ml cl cp [Mε (t)]∗pr cr∗  =

mlpr

|f (t)|2 + 1/N . 1 + 1/N

(46)

The difference between the average fidelity and the average fidelity amplitude is therefore of order 1/N [72]. Note that the random state average (46) is in fact exact for any N. There are two reasons why averaging over random initial states is of interest. First, in the field of quantum information processing these are the most interesting states as they have the least structure, i.e. can accommodate the largest amount of information. Second, for ergodic dynamics and sufficiently long times, one can replace expectation values in a specific generic state | by an ergodic average. Assuming ergodicity for sufficiently large Hilbert spaces there should be no difference between averaging the fidelity amplitude or the fidelity or taking a typical single random initial state. For mixing dynamics the long time fidelity decay is independent of the initial state even if it is non-random, whereas in the regular regime it is state dependent. For instance, the long time Gaussian decay (Section 5) depends on the position of the initial coherent state. The fidelity averaged over this position of the initial coherent state might be of interest and will not be equal to the fidelity averaged over random initial states. In the special case of very strong perturbation, ε > 2, the average fidelity depends on the way we average even for mixing dynamics. This happens due to large fluctuations of fidelity for different initial coherent states, see Ref. [73] and also [74]. Systematic study of fluctuation of fidelity with respect to an ensemble of initial coherent states has been performed in Ref. [75] and showed that variance of fidelity is a non-monotonous function of time with a well defined maximum, where the standard deviation of fidelity can dominate its average value. 2.4. Estimating fidelity For relatively short times or sufficiently weak perturbations, there exists an inequality giving a lower bound on fidelity in terms of the quantum uncertainty of the time-evolving perturbation operator. This is a time dependent version of Mandelstam–Tamm inequality [76] (sometimes also called Fleming’s bound [77]) which is usually used to derive bounds on the time necessary for a given state to evolve into an orthogonal state [78]. In the context of fidelity it was used by Peres [30], see also Ref. [47]. In order to derive the inequality one starts with an observation that the time derivative of fidelity can be written as d iε F (t) = − 0 (t)|[Pε , V ]|0 (t), dt 2

(47)

where Pε =|ε (t)ε (t)| is the projector onto the time-evolving state of the perturbed evolution. Using the Heisenberg uncertainty relation for the operators Pε and V,
V (t)
Pε (t) 21 |0 (t)|[Pε , V ]|0 (t)|

(48)

with the time-evolving quantum uncertainty of operator A defined as
A(t) = (0 (t)|A2 |0 (t) − 0 (t)|A|0 (t)2 )1/2 , we can estimate the time-derivative of fidelity as



d

2ε 2ε d

− F (t)  F (t)


V (t)
Pε (t) =
V (t)F (t)(1 − F (t)). dt dt 2 2 Separating the variables and integrating we arrive at the final inequality  ε t  dt
V (t  ). F (t)
cos2 ((t)), (t) = 2 0 We should note, however, that inequality (51) can be used only for ||  /2.

(49)

(50)

(51)

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2.5. Errors in approximate quantization schemes Let us sidetrack our discussion for a moment by mentioning a closely related subject. An early and very fundamental application of fidelity decay was given by Kaplan, who analyzed extensively the discrepancy between semi-classical quantization and the exact one [29,79–81]. While most research concentrates directly on the behavior of the spectra, here the temporal deviation between the exact and the approximate solution are discussed. Considering the deviation at Heisenberg time will then yield information about the correctness of the level sequence and also shed light on the question whether localization can be explained in the framework of semi-classics. Iterating Bogomolny’s semi-classical solution for a quantum Poincaré map [82] provides a crude approximation. The fidelity taken between this approximation and the real quantum time evolution will show a significant decay long before the Heisenberg time. On the other hand, it is by no means practical to pursue semi-classical evolution up to Heisenberg time, because the exponential growth in the number of classical orbits, which we have to consider, precludes such a task. What Kaplan shows, is that semi-classics can be performed for times long compared to the return time T of a simple Poincaré map, but short compared to Heisenberg time TH i.e. for a time TK such that T >TK >TH . Iterating this map we can then reach Heisenberg time with a good semi-classical approximation. With other words this iterated dynamics is an arbitrarily good approximation to the true semiclassics, the difference falling off exponentially as exp(−const TK /T ). Therefore the fidelity decay between true quantum evolution and exact semiclassics is essentially the same as between quantum evolution and iterated semiclassics as long as TK ?T . The upshot of these investigations is in agreement with our general findings. Fidelity decay of even the best semiclassical approximation will go as 1—const t 2 for integrable systems and as 1—const t for chaotic ones. Thus semiclassical approximations are found to be more stable for chaotic systems than for integrable ones. In particular the level sequence becomes marginally correct in two dimensions for integrable systems, while for chaotic ones three dimensions seem to be just about the limit. Note that this does not imply that semi-classics cannot be usefully applied for some purposes in higher dimensional cases. Yet if we use a correct level sequence as the criterion, then this limitation exists for sure. 2.6. Echo measures beyond fidelity The quantum state overlap—the fidelity amplitude—is only one of the measures of quality of the echo, or of the deviation between two quantum time evolutions. For example in real experiments, there is often different information at our disposal to make the comparison between two quantum states. The information about the system’s state is based on the measurements of certain observables whose outcomes, in turn, only partially determine the state in question. It is thus relevant to define other measures of echo-dynamics based on the partial informations between two quantum states. 2.6.1. Scattering fidelity Fidelity, as it is usually defined, also applies to scattering systems. A wave packet can be evolved with two slightly different scattering Hamiltonians. This would be the standard fidelity of a scattering system. In contrast, “scattering fidelity” stands for a quantity which can be obtained from simple scattering data, though under certain conditions it agrees with the standard fidelity [69,70,83]. Typically scattering theory is developed around the scattering or S-matrix, and in this context it is only logical to inquire on the stability of S-matrix elements under perturbations. If we take into account that the S-matrix is due to some Hamiltonian that now describes an open system, we can again consider an unperturbed Hamiltonian and its perturbation, which define the S-matrices S and S  . Usually the S-matrix is given in the energy domain, i.e. an S-matrix element is written as Sab (E), where a and b denote the scattering channels. By taking the Fourier transform  (52) Sˆab (t) = dE e−2 iEt Sab (E), of any S-matrix element, we obtain the S-matrix Sˆ in the time domain. It now seems natural to consider the correlation function in the time domain ∗   ˆ ab , Sab ](t) ∝ Sˆab (t)∗ Sˆab (t) C[S

(53)

T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

49

as an appropriate measure of scattering fidelity. Yet this is not the case, because this quantity is dominated by the behavior of the autocorrelation functions. This can be amended by normalizing with the autocorrelation functions to obtain the scattering fidelity amplitude s ∗  ∗  ∗  ˆ ab ˆ ab ˆ ab fab (t) = C[S , Sab ](t)[C[S , Sab ](t)C[S , Sab ](t)]−1/2 .

(54)

While we formally started out in the energy domain, to conform with the usual language of scattering theory, it must be noted, that Lobkis and Weaver [68] reach essentially the same definition when analyzing scattering data taken in the time domain. The connection with fidelity was established in [83]. We may ask how the scattering fidelity amplitude is related to the usual concept of fidelity. In principle, we could do this by considering the scattering process in terms of wave packet dynamics, where fidelity can be defined in the usual sense, i.e. as the overlap between two forward evolutions. Yet, the relation between the scattering matrix and the scattering of wave packets is somewhat involved and not very practical, even if it is standard textbook knowledge (see [84]). We shall therefore ask a question which looks more practical: Under what circumstances will scattering fidelity be equivalent to fidelity or maybe state averaged fidelity in the bound system used to describe the interaction region? For such a concept to be well defined we need a fairly weak coupling to the asymptotic channels. We shall analyze this question below, but it is important to note, that the interest of scattering fidelity is by no means limited to such a situation, because a perturbation of ideal couplings to the continuum is very relevant, and then no simple analogy to standard fidelity exists. Scattering theory usually distinguishes between short time signals usually referred to as direct reactions and long time signals which are termed very fittingly coda in elasticity. It is the latter, that tends to be universal and associated to some form of equilibration in the system. It is thus for this part, that we expect a close analogy with standard fidelity decay. This is fortunate, because the theory of fidelity decay that is available is also restricted to this part. This becomes clear, in the formulation of correlations of the perturbation in the interaction picture. The particular shape of this decay is usually not considered, and will give very specific properties that may be strongly system dependent. It will also depend on the initial state even if we consider chaotic or mixing systems. On the other hand we expect the signal of scattering fidelity to become independent of the channels for the coda. To investigate the analogy of scattering fidelity with standard fidelity we shall use the simplest and most common model for scattering with a resonant part, which we present in the following insertion exclusively for the discussion of this analogy [69,70]. The effective Hamiltonian approach. In the effective Hamiltonian approach to scattering [85,86], the S-matrix reads Sab (E) =
ab − V (a)

†

1 V (b) , E − Heff

Heff = H0 − i/2,

 = V V † + W ,

(55)

where V (a) is the column vector of V corresponding to the scattering channel a. For later use, we introduce the absorption width W , which is simply a scalar in the present context. This is equivalent to model absorption with infinitely many very weakly coupled channels, whose partial widths add up to W [87]. The Fourier transform of Sab (E) reads (for t = 0)  † ˆ Sab (t) = dE e−2 iEt Sab (E) = −2 i (t)V (a) e−2 iHeff t V (b) . (56) Assume, H0 and H0 = H0 +
W are two slightly different Hamiltonians. According to Eq. (55) these define two  , as well as two scattering matrices S (E) and S  (E). We consider the crosseffective Hamiltonians Heff and Heff ab ab correlation function  † †  Sˆab (t)∗ Sˆab (t) = 4 2 (t)V (b) e2 iHeff t V (a) V (a) e−2 iHeff t V (b) . †

(57)

The singular value decomposition of V provides an orthogonal (unitary) transformation of the S(E) and S  (E) where the channel vectors are orthogonal to each other (Engelbrecht–Weidenmüller transformation [88]). Thus, we may restrict ourselves to the case of orthogonal channel vectors; this implies that all direct or fast processes will happen in the elastic channels. We then have   Sˆab (t)∗ Sˆab (t) = 4 2 wa wb (t)vb |e2 iHeff t |va va |e−2 iHeff t |vb . †

(58)

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T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

√ V (a) = wa |va  with normalized vectors |va . The general idea of what follows is to assume the state vectors |va  and |vb  to be random and independent. This equation looks very suggestive to obtain the desired relation with the standard fidelity for a random state |vb  f (t) = vb |e2 iH0 t e−2 i(H0 +
W )t |vb .

(59)

In the eigenbasis of H0 we have f (t) =



|vj b |2 e2 iEj t ,

j

where Ujj (t) =

Ujj (t) =

1  2 iEj t e , N

Ujj (t) =

j





j |ok e−2 iEk t ,

ok |j  =



k

1  2 2 i(Ej −E  )t k , Oj k e N

(60)

jk



Oj2k e−2 iEk t .

(61)

k

Comparing with Eq. (58) we see that the only remaining problem resides in the fact, that we have the effective Hamiltonian in the exponent, which includes the coupling term. This can be handled for weak and intermediate coupling. While the former will be presented in what follows, more involved cases of intermediate coupling are presented in appendix Appendix C, still leading to the same result. We assume that all coupling parameters wa go to zero. Then the anti-Hermitian part of Heff becomes a scalar and we are essentially left with transition amplitudes of Hermitian Hamiltonians.  Sˆab (t)∗ Sˆab (t) ∼ 4 2 wa wb (t)e−W t vb |e2 iH0 t |va va |e−2 i(H0 +
W )t |vb    ∗ vj∗b e2 iEj t vj a vka k|ol e−2 iEl t ol |mvmb . = 4 2 wa wb (t)e−W t

(62)

j,klm

For a = b, we find by averaging over the initial states:    Sˆab (t)∗ Sˆab (t) ∼ 4 2 wa wb (t)e−W t |vj b |2 |vj a |2 e2 iEj t j |ol e−2 iEl t ol |j  j,l

w a wb 1  2 2 i(Ej −E  )t wa wb l = 4 2 Oj l e = 4 2 (t)e−W t (t)e−W t f (t). N N N

(63)

jl

The prefactor, which decays exponentially in time, is exactly the normalization by the two autocorrelation functions, which we postulated at the beginning, as the two agree in this approximation up to the factor wa or wb , respectively. An average over initial states will often be impractical. However, for chaotic (ergodic) systems it may not be necessary. We may use a spectral average and/or average over different samples, instead. Indeed the results obtained for stronger coupling in the appendix assume chaotic behavior of the long time dynamics in the interaction region more explicitly, and probably the relation to standard fidelity is of practical importance only in such a situation. Yet it must be emphasized, that scattering fidelity is a relevant concept in its own right, if we take into account the fundamental role played by the S-matrix in many approaches to quantum systems. 2.6.2. Polarization echo In NMR experiments the quantity that can naturally be measured is the echo in the polarization of the local nuclear spin. One prepares the initial state |0  of a many-spin system in an eigenstate of a projection of the spin (1/2) of a specific nucleus, say s0z s0z |0  = m0 |0 ,

(64)

where m0 = ±1/2. Next one performs the echo experiment, i.e. we apply the echo operator and then measure local polarization of the same nucleus. It has the expectation value mε (t) = Mε† (t)s0z Mε (t).

(65)

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51

The polarization echo Pε (t) is defined as the probability that the local polarization of the spin is restored after the echo dynamics Pε (t) =

1 2

+ 2m0 mε (t) =

1 2

+ 2s0z Mε† (t)s0z Mε (t).

(66)

This quantity has been extensively measured in a series of NMR experiments performed by the group of Levstein and Pastawski [38,40]. One can try to generalize this quantity to an echo with respect to some arbitrary quantum observable A. Again, in order that the quantity makes sense as the echo of a observable A, the system has to be prepared in the initial state which is an eigenstate |, or eventually a statistical mixture  of eigenstates, of observable A. Due to this property, the averages have the property AB = BA for any other observable B. Then we define the echo of an observable A as its correlation function with respect to echo-dynamics PεA (t) =

AM †ε (t)AM ε (t) . A2 

(67)

One can write a general linear response expression for this quantity. Inserting into the previous expression the echo operator to second order ε2 ε Mε (t) = 1 − i (t) − 2 Tˆ 2 (t) + O(ε 3 ) 2 22

(68)

and retaining terms to second order, we obtain a simple expression to order O(ε 4 ) PεA (t) = 1 −

ε 2 A2 2 (t) − A(t)A(t) . A2  22

(69)

This can for example be used for the polarization echo. 2.6.3. Composite systems, entanglement and purity In various studies of decoherence and effects of external degrees of freedom to the system’s dynamics one studies composite systems. Coupling with the environment is usually unavoidable so that the evolution of our system is no longer Hamiltonian. To preserve the Hamiltonian formulation we have to include the environment in our description. We therefore have a “central system”, denoted by a subscript “c”, and an environment, denoted by subscript “e”. The names central system and environment will be used just to denote two pieces of a composite system, without any connotation on their properties, dimensionality etc. The central system will be that part which is of actual interest. The Hilbert space is a tensor product H = Hc ⊗ He and the evolution of a whole system is determined by a Hamiltonian or a unitary propagator on the whole Hilbert space H of dimension N = Nc Ne . The unperturbed state |(t) and the perturbed one |ε (t) are obtained with propagators U (t) and Uε (t). Fidelity would in this case be the overlap of two wave functions on the whole space H. But if we are not interested in the environment, this is clearly not the relevant quantity. Namely, the fidelity will be low even if the two wave functions are the same on the subspace of the central system and differ only on the environment. We can define a quantity analogous to the fidelity, but which will measure the overlap just on the subspace of interest i.e. on the subspace of the central system. Let us define the reduced density matrix of the central subsystem c (t) := tr e [(t)],

M M c (t) := tr e [ (t)],

(70)

where tr e [•] denotes a trace over the environment and M (t) = Mε (t)(0)Mε (t)† is the so-called echo density matrix. Throughout this chapter we shall assume that the initial state is a pure product state i.e. a direct product, |(0) = |c (0) ⊗ |e (0) =: |c (0); e (0),

(71)

where we also introduced a short notation |c ; e  for pure product states. The resulting initial density matrix (0) = |(0)(0)| is of course also pure. Fidelity can be written as F (t) = tr[(0)M (t)] and in a similar fashion we shall define a reduced fidelity [50] denoted by FR (t), FR (t) := tr c [c (0)M c (t)].

(72)

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T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

The reduced fidelity measures the distance between the initial reduced density matrix and the reduced density matrix after the echo. Note that our definition of the reduced fidelity agrees with the information-theoretic fidelity [89,90,13] on a central subspace Hc only if the initial state is a pure product state, so that c (0) is also a pure state. One of the most distinctive features of quantum mechanics is entanglement. Due to the coupling between the central system and the environment the initial product state will evolve after an echo into the pure entangled state Mε (t)|(0) and therefore the reduced density matrix M c (t) will be a mixed one. For a pure state |(t) there is a simple criterion for entanglement. It is quantified by purity I (t), defined as I (t) := tr c [2c (t)],

c (t) := tr e [|(t)(t)|].

(73)

Purity, or equivalently von Neumann entropy tr(c ln c ), is a standard quantity used in decoherence studies [91]. If the purity is less than one, I < 1, then the state | is entangled (between the environment and the central system), otherwise it is a product state. Similarly, one can define a purity after an echo, called purity fidelity in [49] by 2 FP (t) := tr c [{M c (t)} ].

(74)

We shall rename this quantity more appropriately echo purity. All three quantities, the fidelity F (t), the reduced fidelity FR (t) and the echo purity FP (t) measure stability with respect to perturbations of the dynamics. If the perturbed evolution is the same as the unperturbed one, they are all equal to one, otherwise they are less than one. Fidelity F (t) measures the stability of a whole state, the reduced fidelity gives the stability on the subspace Hc and the echo purity measures separability of M (t). One expects that fidelity is the most restrictive quantity of the three—(0) and M (t) must be similar for F (t) to be high. For FR (t) to be high, only the reduced density matrices c (0) and M c (t) must be similar, and finally, the echo purity FP (t) is high if only M (t) factorizes. Fidelity is the strongest criterion for stability. 2.6.4. Inequality between fidelity, reduced fidelity and echo purity Actually, one can prove the following inequality for an arbitrary pure state | and an arbitrary pure product state |c ; e  [50,51],





c ; e | 4  c |c |c  2 tr c [2 ], c

(75)

where c := tr e [||]. Proof. Uhlmann’s theorem [89], i.e. noncontractivity of the fidelity, states that tracing over an arbitrary subsystem cannot decrease the fidelity, tr[|c ; e c ; e |||]tr[|c c |c ].

(76)

Then, squaring and applying the Cauchy–Schwartz inequality |tr[A† B]|2 tr[AA† ]tr[BB † ] we immediately obtain the wanted inequality (75). The rightmost quantity in the inequality I = tr[2c ] is nothing but the purity of state | and so does not depend on |c ; e . One can think of inequality (75) as giving us a lower bound on purity. An interesting question for instance is, which state |c ; e  optimizes this bound for a given |, i.e. what is the maximal attainable overlap |c ; e ||4 (fidelity) for a given purity. The rightmost inequality is optimized if we choose |c  to be the eigenstate of the reduced density matrix c corresponding to its largest eigenvalue 1 , c |c  = 1 |c . To optimize the left part of the inequality, we have to choose |e  to be the eigenstate of e := tr c [] corresponding to the same largest eigenvalue

1 , e |e  = 1 |e . The two reduced matrices e and c have the same eigenvalues [92], 1
2
· · ·
Nc . For such choice of |c ; e  the left inequality is actually an equality, |c ; e ||4 = |c |c |c |2 = 21 and the right inequality is

21 tr[2c ] =

Nc  j =1

2j ,

(77)

T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

53

with equality iff 1 =1. If the largest eigenvalue is close to one, 1 =1−ε, the purity will be I =(1−ε)2 +O(ε 2 ) ∼ 1−2ε and the difference between the purity and the overlap will be of the second order in ε, I −|c ; e ||4 ∼ ε2 . Therefore, for high purity the optimal choice of |c ; s  gives a sharp lower bound, i.e. its deviation from I is of second order in the deviation of I from unity. For our purpose of studying stability to perturbations, a special case of the general inequality (75) is particularly interesting. Namely, taking for | the state after the echo evolution Mε (t)|(0) and for a product state |c ; e  the initial state |(0) (71), we obtain F 2 (t)FR2 (t)FP (t).

(78)

As a consequence of this inequality we find: If fidelity is high, reduced fidelity and the echo purity are also high. In the case of perturbations with zero time average, the fidelity freezes and from the inequality we can conclude that the reduced fidelity and the echo purity will display a similar behavior. 2.6.5. Uncoupled unperturbed dynamics A special, but very important, case arises if the unperturbed dynamics U0 represents two uncoupled systems, so we have U0 = Uc ⊗ Ue .

(79)

This is a frequent situation if the coupling with the environment is “unwanted”, so that our ideal evolution U0 is uncoupled. Under these circumstances the reduced fidelity FR (t) and the echo purity FP (t) have especially nice forms. The reduced fidelity (72) can be rewritten as ε FR (t) = tr c [c (0)M c (t)] = tr c [c (t)c (t)],

(80)

where c (t) is the unperturbed state of the central system and εc (t) := tr e [Uε (t)(0)Uε† (t)] the corresponding state obtained by perturbed evolution. Whereas for a general unperturbed evolution the reduced fidelity was an overlap of the initial state with an echo state, for a factorized unperturbed evolution it can also be interpreted as the overlap of the (reduced) unperturbed state at time t with a perturbed state at time t, similarly as for fidelity. Echo purity can also be simplified for uncoupled unperturbed evolution. As U0 factorizes we can bring it out of the innermost trace in the definition of echo purity and use the cyclic property of the trace, finally arriving at 2 ε 2 FP (t) = tr c [{M c (t)} ] = tr c [{c (t)} ] = I (t).

(81)

Echo purity is therefore equal to the purity of the forward evolution. The general inequality gives in this case F 2 (t)FR2 (t)I (t),

(82)

and so the fidelity and the reduced fidelity give a lower bound on the decay of purity. Because the purity is frequently used in studies of decoherence this connection is especially appealing. In most of our theoretical derivations regarding the echo purity we shall assume a general unperturbed evolution, but one should keep in mind that the results immediately carry over to purity in the case of uncoupled unperturbed dynamics. Also a large part of our numerical demonstration in next two sections will be done on systems with an uncoupled unperturbed dynamics as this is usually the more interesting case. 2.6.6. Linear response expansion We proceed with the linear response expansion of reduced fidelity (72) and echo purity (74). We use the notation c := |c (0)c (0)| for a pure initial density matrix for the central system and e := |e (0)e (0)| for the environment. For explicit calculations it is convenient to use an orthogonal basis |j ; , j = 1, . . . , Nc ,  = 1, . . . , Ne , with the convention that the first basis state |1; 1 := |c ; e  is the initial state. Then, inserting the second order approximation to the echo operator (68) into the definitions, and keeping only quantities to second order in ε, we have up to O(ε 4 )

54

T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

(since third orders exactly vanish) ε 2 {1; 1|2 (t)|1; 1 − 1; 1|(t)|1; 12 }, 1 − F (t) = 2 ⎧ ⎫ Ne ⎬ ε 2 ⎨  |1; |(t)|1; 1|2 , 1; 1|2 (t)|1; 1 − 1 − FR (t) = ⎭ 2 ⎩ =1 ⎧ ⎫ Ne Nc ⎬ ε 2 ⎨   |1; |(t)|1; 1|2 − |j ; 1|(t)|1; 1|2 . 1 − FP (t) = 2 1; 1|2 (t)|1; 1 − ⎭ 2 ⎩ =1

(83)

j =2

Writing the expectation value in the initial product state as usual, •=tr[(c ⊗e )•], we can rewrite the linear response result in basis free form ε 2 (t)(1 ⊗ 1 − c ⊗ e )(t), 1 − F (t) = 2 ε 2 (t)(1 − c ) ⊗ 1(t), 1 − FR (t) = 2 ε 2 (t)(1 − c ) ⊗ (1 − e )(t). (84) 1 − FP (t) = 2 2 The linear response expansion of course also satisfies the general inequality (78). The difference between FR (t) and F (t) as well as between FP (t) and F (t) results from off-diagonal matrix elements of operator (t). One may compare these results to time-independent perturbative expansions valid up to quantum Zeno time tZ [93,94]. Depending on the growth of the linear response terms with time we shall again have two general categories, that of mixing dynamics and that of regular dynamics. In the general linear response expressions above we have not assumed anything on the particular form of perturbation V and unperturbed system H0 . It may be however interesting to study separately the special case where the unperturbed system is uncoupled and the perturbation is the coupling. This we shall do for different situations in later sections. 3. Quantum echo-dynamics: non-integrable (chaotic) case 3.1. Fidelity and dynamical correlations In the previous section we have elaborated in detail on different general properties of echo dynamics. At this point we specialize our interest to the case of systems which produce maximal possible dynamical disorder. For systems which possess a well defined classical limit this means that the latter should be non-integrable and fully chaotic. For other systems, we assume that the system’s stationary properties are well described by Gaussian or circular ensembles of random matrices and hence fall into the general category of quantum chaos. Yet, we postpone the discussion of the random matrix formulation of echo dynamics to the next section and here concentrate on individual physical dynamical systems which possess the property of dynamical mixing, either in the semiclassical or the thermodynamic limit. Again, we use the Born expansion (11) and express fidelity in terms of dynamical correlations. At first we assume that a well defined classical limit exists and that the classical system has the strong ergodic property of mixing such that the correlation function of the perturbation V decays sufficiently fast; this typically (but not necessarily) corresponds to globally chaotic classical motion. However, reference to underlying classical dynamics is not strictly necessary as the corresponding quantum mixing can sometimes be established in the thermodynamic limit without taking the classical limit. For the application of echo-dynamics in such a situation see Refs. [42,49]. Due to ergodicity and mixing we can assume that averages of any dynamical observables over a specific initial state can be replaced by a full Hilbert space average, • = tr (•)/N , at least after times longer than a certain relaxation time-scale tE 0 (t)|A|0 (t) ≈ A

for |t|
tE .

√

(85)

For minimal-uncertainty wave packet initial state with typical phase space diameter 2 and exponentially unstable classical dynamics with characteristic exponent , the time scale tE can be estimated as tE ∼ log(1/2)/ and is sometimes

T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

55

known as the Ehrenfest time. This is the time needed for an initially localized wavepacket to spread effectively over the accessible phase space [95]. For other types of initial states this time is in general even shorter. For example, for a random initial state, Eq. (85) is satisfied for any t, by definition of a random state (45). Therefore, for any initial state (e.g. in the worst case for a minimal-uncertainty wave packet) one obtains identical results for F (t) for sufficiently long times,4 i.e. longer than tE . The state averaged quantum correlation function is homogeneous in time, i.e. C(t, t  ) = C(t − t  ), so we simplify the second order linear response formula (13) for fidelity  ε2 t  dt (t − t  )C(t  ) + O(ε 4 ). (86) F (t) = 1 − 2 2 2 0 If the decay of the correlation function C(t) is sufficiently fast, namely if its integral converges on a certain characteristic mixing time scale tmix , meaning that C(t) should in general decay faster than O(t −2 ), then the above formula can be further simplified. For times t?tmix we can neglect the second term under the summation in (86) and obtain a linear fidelity decay in time t (in the linear response) F (t) = 1 − 2(ε/2)2 t,

(87)

with the transport coefficient  being  ∞ 2 (t) − (t)2 = dt C(t) = lim . t →∞ 2t 0

(88)

Note that  has a well defined classical limit obtained from the classical correlation function and in the semiclassical limit this classical cl will agree with the quantum one. We can make a stronger statement in a non-linear-response regime if we make an additional assumption on the factorization of higher order time-correlations, namely assuming n-point mixing [96]. This implies that 2m-point correlation V (t1 ) · · · V (t2m ) is appreciably different from zero for t2m − t1 → ∞ only if all (ordered) time indices {tk , k = 1 . . . 2m} are paired with the time differences within each pair, t2k − t2k−1 , being of the order or less than tmix . Then we can make a further reduction, namely if t?mt mix the terms in the expansion of the fidelity amplitude f (t) (7,11) are  t dt1 dt2 · · · dt2m V (t1 )V (t2 ) · · · V (t2m ) Tˆ 0  t (2m)! → Tˆ dt1 dt2 · · · dt2m V (t1 )V (t2 ) · · · V (t2m−1 )V (t2m ) → (2t)m . (89) m!2m 0 The fidelity amplitude is therefore f (t) = exp(−ε 2 t/22 ) and the fidelity is F (t) = exp(−t/m ),

m =

22 , 2ε 2 cl

(90)

where m = O(ε −2 ) is the time scale for decay and the subscript “m” stands for mixing dynamics. If the system has a classical limit, one may take also a classical limit of the transport coefficient cl , so the decay time-scale m can be computed from classical mechanics. We should stress again that the above result (90) has been derived under the assumption of true quantum mixing which can be justified only in the limit N → ∞, e.g. either in the semiclassical or the thermodynamic limit. Thus for the true quantum-mixing dynamics the fidelity will decay exponentially. The same result can also be derived using the standard Fermi golden rule interpreting fidelity as the transition probability and estimating the square of perturbation matrix elements in terms of classical correlation functions [44,45]. This regime of exponential fidelity decay is often referred to as a Fermi golden rule regime. It is consistent with the effective treatment of fidelity as a Fourier transform of the local density of states of the eigenstates of Hε expressed in the eigenbasis of H0 , or vice versa. Within the standard random matrix assumptions corresponding to classical chaos, the local density of states has a Lorentzian form and this corresponds to an exponential decay (90). 4 The exception might be systems with non-ergodic quantum behavior, for example exhibiting dynamical localization.

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T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

0.03

σcl(t)

0.025 0.02

Ccl(t)

0.015 0.01

10-1 10-2 10-3 10-4 10-5 10-6 0

0.005

5

10 t

15

20

0 -0.005 -0.01 -0.015 0

5

10

15

20 t

25

30

35

40

Fig. 1. The classical correlation function of perturbation V (91) for chaotic kicked top and  = /6 (top solid curve) and  = /2 (bottom broken curve). The finite time integrated correlation function is shown in the inset.

100

γ =π/2

10-1 10-2

F(t)

10-3 10-4 10-5 10-6 10-7 10-8 0

500

1000

1500 (εS)2

2000

2500

3000

t

Fig. 2. Quantum fidelity decay in the chaotic regime for  = /2 and three different perturbation strengths ε = 5 × 10−4 , 1 × 10−3 and 3 × 10−3 (solid, dashed and dotted curves, respectively) is shown. The chain line gives theoretical decay (90) with the classically calculated  seen in Fig. 1.

To numerically check the above exponential decay, we shall use the kicked top (B.1) with parameter  = 30, giving totally chaotic classical dynamics. As argued before, one can calculate the transport coefficient  (88) by using the classical correlation function of the perturbation (B.3), Vcl = v = 21 z2 .

(91)

We consider two different values of kicked top parameter , namely  = /2 and /6. The classical correlation functions is shown in Fig. 1. The correlation function (obtained by averaging over 105 initial conditions on a sphere) is shown in the main frame. The correlation functions have qualitatively different decay towards zero for the two chosen ’s. In the inset the convergence of the classical  (88) is shown. One can see that the mixing time is tmix ≈ 5. The values of cl are cl = 0.00385 for  = /2 and cl = 0.0515 for  = /6. These values are used to calculate the theoretical decay of fidelity F (t) = exp(−ε 2 S 2 2cl t) according to Eq. (90) which is compared with numerical simulations in Figs. 2 and 3. We used averaging over the whole Hilbert space, and checked that due to ergodicity there was no difference for large S if we choose a fixed initial state, say a coherent state. As fidelity will decay only until it reaches its finite size fluctuating

T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

100

57

γ =π/6

10-1

F(t)

10-2 10-3 10-4 10-5 10-6 0

50

100

150

200

(εS)2 t Fig. 3. Similar figure as 2, only for  = /6 and perturbation strengths ε = 1 × 10−4 , 2 × 10−4 and 3 × 10−4 (solid, dashed and dotted curves, respectively).

value F¯ (A.3) we choose a large S = 4000 in order to be able to check the exponential decay over as many orders of magnitude as possible. In Fig. 2 the decay of quantum fidelity is shown for  = /2. The agreement with theory is excellent. Note that the largest ε shown corresponds to m ≈ 1 so that the condition for n-point mixing t?tmix is no longer strictly satisfied, hence we observe some oscillations around the theoretical curve. However, overall agreement with the theory is still good due to the oscillatory nature of the correlation decay (see Fig. 1) fulfilling the factorization assumption (89) on average. In Fig. 3 for  = /6 a similar decay can be seen. In both cases fidelity starts to fluctuate around F¯ calculated in Section 2.3.1 for sufficiently long times t∞ (see Appendix A). 3.1.1. Long time behavior So far, we have assumed that the quantum correlation function C(t) decays to zero and its integral (or sum, for discrete-time dynamical systems) converges to . However, for a system with a Hilbert space of finite dimension N, the ¯ similar to the finite asymptotic correlation function asymptotically does not decay but has a non-vanishing plateau C, ¯ value of fidelity F . This will cause the double correlation integral to grow, asymptotically, quadratically with time. Because this plateau C¯ is small, the quadratic growth will overtake linear growth 2cl t only for large times. The time-averaged correlation function C(t, t  ) (14) can be calculated assuming a nondegenerate unperturbed spectrum Ek as 2   t  t   1     2 C¯ = lim 2 dt dt C(t , t ) = kk (Vkk ) − kk Vkk , (92) t →∞ t 0 0 k

k

where kk are diagonal matrix elements of the initial density matrix (0) and Vkk are diagonal matrix elements of the perturbation V in the eigenbasis of the unperturbed propagator U0 . One can see that C¯ depends only on the diagonal matrix elements, in fact it is equal to the variance of the diagonal matrix elements. Since the classical system is ergodic and mixing, we shall use a version of the quantum chaos conjecture [97–101] saying that Vmn are independent Gaussian random variables with a variance given by the Fourier transformation S() (divided by N) of the corresponding classical correlation function Ccl (t) at frequency  = (Em − En )/2. On the diagonal wehave  = 0 and an additional factor t t of 2 due to the random matrix measure of the diagonal elements. Using 2cl t = 0 dt  0 dt  C(t  , t  ) = S(0)t we can write 2S(0) 4cl C¯ = = . N N

(93)

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T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

10 tH 0.103t /S

ΣC(j,k)/(2t)

1

γ =π / 6

0.1 0.0077t /S γ = π/2

0.01

0.001 1

10

100

1000

10000

10000

t  Fig. 4. The finite time quantum correlation sum (t) = t−1 j,k=0 C(j, k)/2t (solid curves) together with the corresponding classical sum  cl (t) = t−1 C (j, k)/2t (dashed curves saturating at  and ending at t ∼ 1000) is shown for the chaotic kicked top. Quantum data are cl cl j,k=0 for a full trace  = 1/N with S = 1500. Upper curves are for  = /6 while lower curves are for  = /2. Chain lines are best fits for asymptotic ¯ linear functions corresponding to Ct/2 = 0.0077t/S for  = /2 and 0.103t/S for  = /6.

Because of ergodicity, for large N, C¯ does not depend on the statistical operator  used in the definition of the correlation function, provided we do not take some non-generic state like a single eigenstate |Ek  for instance. If U0 has symmetries, so that its Hilbert space is split into s components of sizes Nj , the average C¯ will be different on different subspaces, C¯ j = 4cl /Nj . Averaging over all invariant subspaces then gives 4scl , C¯ = N

(94)

so that C¯ is increased by a factor s compared to the situation with only a single desymmetrized subspace. The fidelity ¯ 2 ≈ decay will start to be dominated by the average plateau (93) at time tH when the quadratic growth takes over, Ct H 2cl tH , tH = 21 N ∝ 2−d ,

(95)

which is nothing but the (dimensionless) Heisenberg time associated with the inverse (quasi)energy density. Note that for autonomous systems where energy is a conserved quantity, Heisenberg time scales differently tH ∝ 21−d . Again, if one has s invariant subspaces, the Heisenberg time is reduced tH = N/(2s). In Fig. 4 we show numerical calculation of the correlation integral (sum) for the chaotic kicked top at  = 30. We compare the classical correlation sum (the same data as in Fig. 1) and quantum correlation sum. One can nicely see the crossover from linear growth of quantum correlation sum 2cl t for small times t < tH , to the asymptotic quadratic ¯ In addition, numerically fitted asymptotic growth 0.103t/S and 0.0077t/S nicely growth due to correlation plateau C. ¯ agrees with formula for C, using N = S and classical values of transport coefficients cl = 0.0515 and 0.00385 for  = /6 and /2, respectively. For times t > tH , and provided ε is sufficiently small, the correlation sum will grow quadratically and the linear response fidelity reads F (t) = 1 −

ε 2 4cl 2 t . 22 N

(96)

To derive the decay of fidelity beyond the linear response regime one needs higher order moments of diagonal elements of perturbation V. If we use the leadingorder BCH expression of the echo operator (24), neglecting the term (t)ε 2 , we have the fidelity amplitude f (t) = k exp(−iVkk εt/2)/N , where we choose an ergodic average  = 1/N . In the

T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

100

γ = π/2

59

γ = π/6

p(Vkk)

p(Vkk)

100

10

10

1 1 0.16

0.162 0.164 0.166 0.168 Vkk

0.17

0.172 0.174

0.145 0.15 0.155 0.16 0.165 0.17 0.175 0.18 0.185 0.19 Vkk

Fig. 5. Histogram of the normalized distribution of the diagonal matrix elements Vkk for the chaotic kicked top and S = 4000 on OE subspace (B.4). √ The dotted line is the theoretical Gaussian distribution with the second moment C¯ and the two chain lines are expected Ni statistical deviations if there are Ni elements in the ith bin. Note the different x-ranges in two figures due to different cl for the two chosen .

limit N → ∞ we can replace the sum with an integral over the probability distribution of diagonal matrix elements p(Vkk ) = p(V ),  f (t) =

dVp(V ) exp(−iV εt/2).

(97)

For long times the fidelity amplitude is therefore a Fourier transformation of the distribution of diagonal matrix elements. For classically mixing systems the distribution is conjectured to be Gaussian with the second moment equal to C¯ (93). This is confirmed by numerical data in Fig. 5. The mean  value of diagonal matrix elements is perturbation specific and is for our choice of the perturbation (B.3) equal to k Vkk /(2S + 1) = (2S + 1)(S + 1)/12S 2 . From the figure we can see that the distribution is indeed Gaussian with the variance agreeing with the theoretically predicted C¯ = 4cl /S. The Fourier transformation of a Gaussian is readily calculated and we get a Gaussian fidelity decay  F (t) = exp(−(t/p ) ), 2

p =

N 2 . 4cl ε

(98)

In order to see a Gaussian fidelity decay for mixing systems the perturbation strength ε must be sufficiently small. Otherwise, the fidelity will decay exponentially (90) to its fluctuating plateau F¯ before time tH when the Gaussian decay starts. Requiring that the time-scale of exponential decay m is smaller than tH = N/2 gives the critical perturbation strength εp , εp = √

2 . cl N

(99)

For ε < εp we will have a Gaussian decay (98), otherwise the decay starts as an exponential (90) and goes over to a Gaussian at the Heisenberg time tH (see Section 4 for a uniform theory within a random matrix model). If the decay reaches the plateau F before or around tH , the decay remains purely exponential (for details see Section 7). Again we illustrate the predicted Gaussian decay for the chaotic kicked top with  = 30 and S = 1500 and a full trace average over the Hilbert space. The results of numerical simulation, together with the theory are shown in Fig. 6. The regime of Gaussian decay is sometimes referred to as the perturbative regime [45,44] because it can be derived using perturbation theory in lowest order. Writing the eigenenergies in first order Ekε = Ek + Vkk ε and the overlap matrix in zeroth order Okl =
kl + O(ε) (40), one arrives at the Fourier transform formula (97).

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T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

1

0.1 γ = π/2 γ = π/6

F(t)

0.01

0.001

0.0001

1

10

100 S(ε

1000

t)2

Fig. 6. Quantum fidelity decay for ε < εp in the chaotic regime. For  = /2 data for ε = 1 × 10−6 (solid curve) and 5 × 10−6 (dotted curve) are shown. For  = /6, ε = 3 × 10−7 (solid) and 1 × 10−6 (dotted) are shown. Note that for both  the curves for both ε practically overlap. The chain curves are theoretical predictions (98) with classically computed cl .

3.2. Vanishing time-averaged perturbation and fidelity freeze Here we shall assume that the perturbation is of particular form, namely that it can be written as a time derivative of some observable W (29). We shall show that a simple semi-classical theory can be developed which describes anomalously slow fidelity decay (fidelity freeze) in such a case. Other types of perturbations with vanishing timeaverage will be considered within the random matrix framework in Section 4.3. The presentation here follows Ref. [17]. Based on the general discussion of Section 2.2, we write a semiclassical expression for the fidelity below the time scale t2 = O(ε −1 ), which shall be specified later, and above the Ehrenfest time scale tE , tE < t < t2 , as


   2




iε iε

exp W

. (100) F (t) ≈ Fplat = exp − w 2 2 cl This equation is derived from the general Equation (35) in three steps: (i) the higher order middle factor in Eq. (35) can be neglected for t < t2 ; (ii) we assume that due to mixing dynamics, the correlation function of exp(−iW (t)ε/2) can be factorized for t > tmix as exp(−iW (t)ε/2) exp(iW (0)ε/2) ≈ exp(−iW (t)ε/2)exp(iW (0)ε/2); and (iii) for t > tE the average W (t) = W  is approximated by the classical average  d xw(x) wcl =   ,  d

(101)

where w(x) is the classical observable corresponding to the operator W and  is the classical invariant ergodic component related to the initial state , e.g. the energy surface for E = |H0 |. The plateau of fidelity can be written in a compact form for the two simplest extreme cases of initial states: (a) for a coherent initial state (CIS), where the initial state average can be approximated by the classical observable evaluated at the center x∗ of the wave packet W  ≈ w(x∗ ), hence |exp(−iW ε/2)| ≈ 1; and (b) for a random initial state (RIS), where the initial state average is equivalent to an ergodic average, exp(−iW ε/2) = exp(−iW ε/2) ≈ exp(−iwε/2)cl . Defining a classical generating function as G(z) = exp(−izw)cl ,

(102)

T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

61

one can compactly write CIS ≈ |G(ε/2)|2 , Fplat

RIS Fplat ≈ |G(ε/2)|4 .

(103)

RIS ≈ (F CIS )2 . Curiously, the same relation is Note that the two plateau levels are satisfying universal relation Fplat plat satisfied for the case of regular dynamics (see Section 5.3). If the argument z = ε/2 is large, the analytic function G(z) can be calculated generally by the method of stationary phase. In the simplest case of a single isolated stationary point x∗ in N dimensions

N/2



|G(z)| 



| det jxj jxk W ( x ∗ )|−1/2 . (104) 2z

This expression gives an asymptotic power law decay of the plateau height independent of the perturbation details. Note that for  a finite phase space we will have oscillatory diffraction corrections to Eq. (104) due to a finite range of integration d which in turn causes an interesting situation for specific values of z, namely that by increasing the perturbation strength ε we can actually increase the value of the plateau. Next we shall consider the regime of long times t > t2 . Then the second term in the exponential of (24) dominates the first one, however even the first term may not be negligible. Up to terms of order O(tε 3 ) we can factorize Eq. (34) as Mε (t) ≈ exp(−iε/2(W (t) − W (0))) exp(−iε 2 /22R (t)). When computing the expectation value f (t) = Mε (t) we again use the fact that in the leading semiclassical order the operator ordering is irrelevant and that any time-correlation can be factorized, so also the second term of (t) (25) vanishes. Thus we have


 2



ε2

(105) F (t) ≈ Fplat exp −i R (t)

, t > t2 . 22 This result is quite intriguing. It tells us that apart from a prefactor Fplat , the decay of fidelity with residual perturbation is formally the same as fidelity decay with a generic (non-residual) perturbation, Eqs. (7) and (10), when one substitutes the operator V with R and the perturbation strength ε with εR = ε 2 /2. The fact that time-ordering is absent in Eq. (105) as compared with (10) is semiclassically irrelevant. Thus we can directly apply the general semiclassical theory of fidelity decay described in previous subsection, using a renormalized perturbation R of renormalized strength εR . Here we simply rewrite the key results in the ‘non-Lyapunov’ (perturbation-dependent regime), namely for εR < 2. Using a classical transport rate  := limt → ∞ 2t1 (2R (t)cl − R (t)2cl ) where R (t) is a classical observable corresponding to R (t). We have either an exponential decay
 ε4 F (t) ≈ Fplat exp − 2 t , t < tH (106) 22 or a (perturbative) Gaussian decay 
ε4 t 2 , F (t) ≈ Fplat exp − 2  22 tH

t > tH ,

(107)

and the crossover again happens at the Heisenberg time tH . This is just the time when the integrated correlation function of R(t) becomes dominated by quantum fluctuation. Comparing the two factors in (106,107), i.e. the fluctuations of two terms in (24), we obtain a semiclassical estimate of t2   tH cl 2cl , (108) t2 ≈ min ,  ε ε 2 where cl is dispersion of classical observable corresponding to W, 2cl = w 2 cl − w2cl .

(109)

Interestingly, the exponential regime (106) can only take place if t2 < tH . If one wants to keep Fplat ∼ 1, or have exponential decay in the full range until F ∼ 1/N, this implies a condition on dimensionality: d
2. The quantum fidelity and its plateau values have been expressed entirely (in the leading order in 2) in terms of classical quantities.

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T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

1

(a)

0.99

F

0.98 0.97 0.96 0.95 0.94 1 (b)

F

0.1

0.01

0.001 100

101

102

103

104

105

t Fig. 7. F (t) for the kicked top, with ε = 10−3 (a), and ε = 10−2 (b), upper curves (dashed) for CIS and the lower curves (full) for RIS. Horizontal lines are theoretical plateau values (103), vertical lines are theoretical values of t2 (108). Points represent calculation of the corresponding classical fidelity for CIS which follows quantum fidelity up to the Ehrenfest (log 2) barrier and exhibits no freezing.

While the prefactor Fplat depends on the details of the initial state, the exponential factors of (106,107) do not. Yet, the freezing of fidelity is a purely quantum phenomenon. The corresponding classical fidelity (discussed in Section 6) does not exhibit freezing. Let us illustrate our theoretical findings by numerical examples. First we consider a quantized kicked top as an example of a one-dimensional system (d = 1) with spin quantum number S = 1000. The model is described in Appendix B.1, Eqs. (B.6), (B.7), and (B.8). In Fig. 7 we show that the analytical expressions in Eq. (103) for the plateau values agree very well with numerical results, not only for weak perturbation ε = 10−3 shown in Fig. 7(a) where linearized (linear response) expressions for the plateau values could be used, strong perturbation ε = 10−2 as shown in Fig. 7(b). Integration over the sphere yields  but also for √

i

/4 G(εJ ) = 2εJ erf i(e εJ /2). Comparing with the asymptotic general formula (104) for G(z) we now also find a diffractive contribution due to the oscillatory behavior of the complex Error function. Small (quantum) fluctuations around the theoretical plateau values in Fig. 7 lie beyond the leading order semiclassical description. In Fig. 7 we also demonstrate that the semiclassical formula (108) for t2 works very well. To demonstrate the Gaussian and exponential long-time decay of fidelity (106) with renormalized perturbation strength we look at a system of two (d = 2) coupled tops described in Appendix B.1, Eqs. (B.15)–(B.17). Here we work with the spin quantum number S = 100. The results of the numerical simulations are shown in Fig. 8. We show only the long-time decay, since at short times, the behavior is qualitatively the same as for d = 1. If the perturbation is sufficiently strong, one obtains an exponential decay as shown in Fig. 8(a), while for a smaller perturbation we obtain a Gaussian decay as shown in Fig. 8(b). Numerical data have been successfully compared with the theory (106), (107) using classically calculated  = 9.2 × 10−3 , and with the “renormalized” numerics using the operator R. A similar phenomenon as quantum freeze has been observed in Ref. [102] for small perturbations consisting of a phase space displacement. 3.3. Composite systems As for other measures of echo-dynamics for mixing and ergodic dynamical system we again assume a bipartite decomposition of the Hilbert space H = Hc ⊗ He . For mixing dynamics the correlations decay and the linear response term will grow linearly with time. For large times one can argue that (t) should look like a random matrix

T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

63

100

F

10-1 10-2 10-3 10-4 10-5 0

1000

2000

3000

4000

5000

(a) 100

F

10-1 10-2 10-3 10-4 10-5 0 (b)

0.2·105 0.4·105 0.6·105 0.8·105 t

1.0·105 1.2·105

Fig. 8. Long-time fidelity decay in two coupled kicked tops. For strong perturbation ε = 7.5 × 10−2 (a) we obtain an exponential decay, and for smaller ε = 2 × 10−2 (b) we have a Gaussian decay. Chain curves give semiclassical expressions (106,107) using classical inputs, full curves give direct numerical simulations and open circles give “renormalized” numerics in terms of renormalized perturbation operator R and perturbation strength εR = ε 2 /2.

and the terms giving the difference between the fidelity and the echo purity and the reduced fidelity can be estimated as Nc  2 2 1 j =2 |j ; 1|(t)|1; 1| j |[(t)](j ;1),(1;1) |  ∼ ∼ , (110) 2 2 Ne 1; 1| (t)|1; 1 j, |[(t)](1;1),(j,) | because there are more terms in the sum for fidelity. Therefore we can estimate the difference FP (t) − F 2 (t) ∼ 1/Nc + 1/Ne and FR (t) − F (t) ∼ 1/Nc . Provided both dimensions Nc,e are large and for sufficiently long times, so the “memory” of the initial state is lost, we can expect the decay of all three quantities to be the same. Discussing linear response results (Section 2.6.6) in the case of mixing dynamics we have shown that the linear decay is the same for all three quantities. Similar random matrix arguments as for the linear response can be used also for higher order terms and therefore one expects that in the semiclassical limit of small 1/Nc + 1/Ne we will have the same exponential decay (90) FP (t) ≈ FR2 (t) ≈ F 2 (t) = exp(−2t/m ),

(111)

with the decay time m = 22 /2ε 2 cl (90) independent of the initial state. This result is expected to hold when (t) can be approximated with a random matrix for large times (110) and V does not contain terms acting on only one subspace. Such terms could cause fidelity to decay while having no influence on purity. For a numerical demonstration of this result we chose for the system of coupled kicked tops such parameters that the corresponding classical dynamics is practically completely chaotic and mixing. The exact form of the perturbation and the parameter values are given in Appendix B.1, Eq. (B.20). The unperturbed system consists of two uncoupled kicked tops such that the coupling is due to the perturbation. In this case, echo purity FP (t) is the same as purity I (t) of the forward evolution of the perturbed system. The perturbation strength and the spin size were chosen as ε = 8 × 10−4 and S = 200, respectively, such that Nc,e √ = 2S + 1 =√401. The initial state was chosen as a product of two coherent states placed at the positions ϑ∗c,e = / 3, ∗c,e = / 2 on the canonical sphere, for both tops. We show in Fig. 9 the decay of the fidelity F (t), the reduced fidelity FR (t), and the purity I (t). Clean exponential decay is observed in all three cases, on a time scale m (111) given by the classical transport coefficient cl . We numerically calculated the

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T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

100

I

10-2

F2,FR2,I

10-4

FR2

10-6

10-8 F2 10-10 0

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 t

Fig. 9. Decay of F 2 (t), FR2 (t) and I (t) (dotted curves) in the mixing regime of the double kicked top. The solid line gives the theoretical exponential decay (111) with m calculated from the classical cl = 0.056. Horizontal chain lines give the saturation values of the purity and the reduced fidelity, 1/200 and 1/4002 , respectively.

classical correlation function Ccl (t) = [zc (t)zc (0)cl ]2 ,

(112)

where we took into account that the unperturbed dynamics is uncoupled and is the same for both subsystems and that zcl = 0. Taking only the first term Ccl (0) = 1/9 would give cl = 1/18 (88) while the full sum of Ccl (t) gives a slightly larger value cl = 0.056. Exponential fidelity decay persists up to the saturation value determined by the dimension of the Hilbert space. We also wish to illustrate what happens if dimension of one of the subspaces, say of the central system Nc is not large, but we only let the dimension of the environment Ne to become large. For this purpose we choose a model of a kicked Ising spin chain for parameter values for which the model is non-integrable and operates in the regime of quantum chaos (see Appendix B.2). We note that in the linear response expression for purity decay, we have a reduced slope of purity echo increase with a factor 1 − (Ne + Nc )/(Ne Nc ) with respect to [F (t)]2 , whereas for long times, the decay of purity, with the appropriate plateau value subtracted and rescaled, is determined by the asymptotic decay of the echo operator, squared, i.e. FP (t) − FP∗ ≈ [F (t)]2 . 1 − FP∗

(113)

This is illustrated in Fig. 10 for different cases of division of the spin chain into the central system and the environment, and discussed in detail in Ref. [51]. For a related work on entanglement growth in chaotic composite systems see [103,104]. Some other aspects of entanglement in chaotic systems have been studied in Refs. [47,105–114]. 3.4. Semiclassical theories in terms of classical orbits In the semiclassical limit (small 2) quantum fidelity can be calculated using the semiclassical expression for the quantum propagator in terms of classical orbits. Historically this was the method used by Jalabert and Pastawski [7] to derive the perturbation independent Lyapunov decay of quantum fidelity. They used the perturbation consisting of a uniform distribution of scattering potentials with a Gaussian spatial dependence. It is noted that in the absence of the perturbation F (t) ≡ 1 is recovered exclusively if only diagonal terms are retained, i.e. in the notation we use later this

T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

65

1 Lc = L / 2,alt.,L= 20 Lc = L / 2,alt.,L= 16 Lc = L / 2,alt.,L= 12 Lc = L / 2,con.,L= 20 Lc = 1,L= 20 theory

|FP(t)-F∗|/(1-F∗)

0.1

0.01

0.001

0

100

200

300

400

500

600

700

t Fig. 10. Decay of echo purity in kicked Ising chain in the mixing regime hz = 1.4 for different types of division [full curve, broken dashed curves = central system is composed of every other spin, dashed curve = central system is a connected half of the chain, dotted curve = central system is a single spin] and different sizes L (indicated in the legend) and ε  = 0.01. Theoretical decay (for L → ∞) is given by sampled symbols.

means orbits for which jε = j0 , resulting in a single sum over orbits (instead of double sum). For non-zero perturbation averaging over impurities is argued to “select” only diagonal terms. Later it was shown that the same Lyapunov decay is found also for the classical fidelity [62]. This suggests that quantum Lyapunov decay is a consequence of the quantum-classical correspondence; see Section 6 for more details. The correspondence will hold until the Ehrenfest time when a wave packet is spread over a sufficiently large portion of Hilbert space and quantum interferences become important. We shall discuss in detail parameter ranges and time scales when various regimes of fidelity decay occur in Section 7, and we shall see that a Wigner function representation proposed in Ref. [115] illustrates this nicely. For a detailed discussion of the range of the Lyapunov decay in a Lorentz gas with disorder see Ref. [116]. Fidelity decay for disordered system with diffractive scatterers has been studied using √ a diagrammatic expansion in Ref. [117]. It is worth mentioning that for sufficiently strong perturbations (2 < ε < 2) there are large fluctuations [73,75] and as a consequence the average fidelity decay depends on the way we average. For typical initial packets the evolution is “hypersensitive” to perturbations, resulting in a double exponential fidelity decay [73], F ∝ exp(−const × e2 t ). Semiclassics has been used in Refs. [45,44] to derive the so-called Fermi golden rule decay (90), see also [53,54]. The Fermi golden rule decay has been derived independently using correlation function formalism in Refs. [42,46]. Fermi golden rule type expressions involving the correlation function have also appeared as an intermediate step in Ref. [44]. Lyapunov decay in a Lorentz gas was the subject of a numerical study in Ref. [41]. Lyapunov as well as Fermi golden rule regimes in a stadium billiard with disorder have been studied in Ref. [118]. The transition between Fermi golden rule and Lyapunov decay in a Bunimovich stadium has been numerically considered in Ref. [119] while the short time decay of fidelity in the same system has been discussed using local density of states in Ref. [120]. The importance of the delicate interplay between classical perturbation theory and the structural stability of manifolds in chaotic systems has been stressed in selecting the diagonal terms. The argumentation has been further elaborated in Ref. [121], resulting in the so-called dephasing representation [57,58]. Using the dephasing representation it is possible to calculate quantum fidelity decay numerically ranging from the Fermi golden rule to the Lyapunov regime. For strong perturbations one can use dephasing representation to investigate deviations from a simple Lyapunov decay for chaotic systems for which the hyperbolic stretching rate is not uniform across the phase space [122], or perhaps even more interestingly, to investigate new types of perturbation dependent fidelity decay in ‘weakly’ chaotic systems with classically diffusive behavior [123]. Semiclassics can also be used to calculate purity. For chaotic systems the exponential decay of purity has been derived in [124], confirming earlier predictions in Refs. [91,125,50]. A chaotic time dependent oscillator is treated in Ref. [126]. Instead of using classical orbits in semiclassical derivations, e.g. in the Van Vleck–Gutzwiller propagator, one can directly use the formalism of Weyl quantization to obtain a semiclassical expression of fidelity. Such an approach

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T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

has been used in Ref. [127] where the propagation of Gaussian wave packets is studied. Two terms are identified, the position of the center of the packet accounts for its translation while the metaplectic operator takes care of the dispersion. The metaplectic operator represents the symplectic transformation (i.e. canonical transformation) of the linearized motion around the central orbit. Independently, a similar approach has been used in Ref. [128]. In the present section we give a brief derivation of the semiclassical expression for fidelity in terms of the Wigner function of the initial state, the so-called dephasing representation derived in Refs. [57,58]. With r, r denoting points in configuration space, the fidelity amplitude can be written as  f (t) = 0 (t)|ε (t) = dr ∗0 (r ; t)ε (r ; t). (114) If (r, 0) is the initial state, the wave function at time t is given by  ε (r ; t) = dr r |Uε (t)|r(r; 0).

(115)

The step to semiclassics consists in replacing the quantum propagator r |Uε (t)|r by its semiclassical approximation Kεsc (r , r, t), i.e. r |Uε (t)|r → Kεsc (r , r, t). The first argument of Kεsc will denote the end point after time t (quantity with prime) and the second argument the initial point (without prime). The semiclassical Van Vleck–Gutzwiller propagator is [129,130], 
  1 i

sc   Kε (r , r, t) = (116) Sj (r , r, t) − i mjε , Cjε exp 2 ε 2 (2 i2)d/2 jε

where the sum extends over all orbits jε that connect points r and r in time t, i.e. sum over all initial momenta pjε , such t

that the evolution for time t results in (r, pjε ) → (r , pjε ). For chaotic systems the number of contributing orbits jε will be very large already for small times and will grow exponentially with time, because the sum goes over all momenta, i.e. over all energies. The action is given by  t  Sjε (r , r, t) = dt  Lε (˜r(t  ), r˙˜(t  ), t  ), (117) 0

  and Cjε jε /jr jr)| is the absolute value of the Van Vleck determinant. Taking into account that jSjε (r , r, t)/ jr = −pjε , where pjε is the initial momentum, we see that Cjε is the Jacobian of the transformation between the initial

= | det(j2 S

jp

momentum and the final position, Cjε = | det jrjε |. The integer mjε is a Maslov index. Writing the fidelity amplitude (114) in terms of Kεsc and K0sc and taking the expectation value leads to an expression involving a three-fold integral over the positions: r (the position argument of the propagated wave functions at time t and r, r˜ (the position arguments of the initial states to be propagated). In addition we have a double sum over orbits jε of the perturbed dynamics, t t connecting (r, pjε ) → (r , pjε ), and orbits j0 of the unperturbed one, connecting(˜r, p˜ j0 ) → (r , p˜ j0 ),   1 f (t) = dr d˜r ∗ (˜r; 0)(r; 0) d (2 i2) jε ,j0
   i

× dr Cj0 Cjε exp {Sjε (r , r, t) − Sj0 (r , r˜, t)} − i {mjε − mj0 } . (118) 2 2 In the expression for fidelity F (t) we have twice as many terms! The last expression is still rather involved but for our case of echo dynamics several simplifications are possible. One is to use the shadowing theorem (see references in [57]) to identify those orbits that give the dominating contribution to the phase factor and the other is to transform the integration from the final position to the initial momentum. Note that the semiclassical propagator (116) does not satisfy a semigroup condition, i.e. a product of two propagators K sc (t) for time t is only approximately equal to a propagator for twice the time, K sc (2t). To preserve the semigroup property, i.e. preserving normalization (unitarity), a product of semiclassical propagators must be supplemented by the “matching” of momenta of two orbits occurring in two propagators. Formally this condition is implemented by making

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67

Fig. 11. Orbits involved in the semiclassical calculation of the fidelity amplitude (118) after taking into account matching condition (119). Left figure denotes situation in the classical limit of exact matching and the right one for finite 2 where shadowing orbit pairs give major contribution.

a stationary phase approximation in the integral over r , for details see e.g. [131]. If the semiclassical limit 2 → 0 is taken prior to integration, the stationary phase condition jSjε /jr − jSj0 /jr has to be obeyed exactly. Exact matching indeed gives the decay of classical fidelity (see Section 6). The matching condition at time t, p˜ j0 = pjε ,

(119)

forces the initial condition for the unperturbed orbit (˜r, p˜ j0 ) to be implicitly determined trough the matching condition for a forward perturbed evolution from (r, pjε ) followed by a backward unperturbed evolution. This suppresses one space integration over d˜r in (118). The orbits involved in calculating this classical fidelity are schematically shown in the left panel of Fig. 11. The action difference
S(r, p, t) after forward perturbed evolution followed by backward unperturbed evolution is given by  t  0     
S(r, p, t) = Sjε (r , r, t) − Sj0 (r , r˜, t) = dt Lε (r(t ), r˙(t )) + dt  L0 (r(t  ), r˙(t  )), (120) 0

t

where the initial condition for the second integral with L0 is the final orbit position of the first integral. This difference will, in general, be large for chaotic systems, because perturbed and unperturbed orbits will explore vastly different regions of phase space. For sufficiently large time the action difference will depend only on the chaotic properties of the system and not on the perturbation strength. This results in the perturbation independent decay of classical fidelity. For finite 2, i.e. first evaluating the integral and then taking the semiclassical limit, the above matching condition (119) needs to hold only approximately. Approximate continuity of momenta for finite 2 means that we have to consider also almost continuous orbits for which discontinuity p˜ j0 − pjε is sufficiently small. This is shown in the right panel of Fig. 11. Of course, the approximate continuity is automatically taken care of if we perform the stationary phase integration properly. Yet here we would like to identify the dominating contribution by physical arguments, namely using the shadowing theorem. It states for chaotic systems that, while two orbits starting from the same initial condition will generally exponentially diverge, if evolved with two slightly different evolutions, we can find exponentially close initial condition for one evolution such that its orbit will closely follow (shadow) the other one. Because the two initial conditions5 are exponentially close they should in fact be summed over (for finite 2) in the formula for the fidelity amplitude f (t) (118). As the orbit and its shadow, a ‘shadowing pair’, closely follow each other, their action difference
S will be small, i.e.
S ∼ O(ε) whereas one has
S ∼ O(1) for non-shadowing pairs. Shadowing pairs therefore cause the largest contribution to f (t). It is thus sufficient to take into account shadowing pairs only. To arrive at the final expression we change the integration variables dr and d˜r in (118) to their difference x = r − r˜ and their mean q¯ = (r + r˜)/2. As the major contribution to f (t) results from shadowing pairs, due to their small
S the difference x will be small and we can expand
S around q¯ to linear order in x, Sjε (r , q¯ + x/2, t) − Sj0 (r , q¯ − x/2, t) = Sjε (r , q¯ , t) − Sj0 (r , q¯ , t) − x¯p + · · · .

(121)

where we denoted the mean initial momentum by p¯ = (pjε + p˜ j0 )/2. This expansion does not depend on the initial state being localized. Next, we transform the integration over the final positions r to an integration over initial mean 5 In our case there are actually two exponentially close “final” conditions at time t.

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momenta p¯ . Together with the original Jacobian prefactor





jp

jp˜

| det jp¯jε || det jpj¯0 | which though is close to one for shadowing orbits. For shadowing orbits Maslov indices are equal, mjε = mj0 . We thus obtain  x x 1 ∗ ¯ ¯ ¯ ¯ d q d p dx  ; 0  q + ;0 q − f (t) = 2 2 (2 i2)d
 
! i i   × exp Sjε (r , q¯ , t) − Sj0 (r , q¯ , t) exp − p¯ x . (122) 2 2 Cj0 Cjε this gives

The action difference Sjε (r , q¯ , t) − Sj0 (r , q¯ , t) is calculated by using forward perturbed orbit jε followed by an unperturbed backward evolution along shadowing orbit j0 , where the initial condition is such that (¯q, p¯ ) = (r + r˜, pjε + p˜ j0 )/2, the so-called center chord in Weyl formalism [132]. By transforming to initial momenta p¯ we have eliminated one summation over momenta in Eq. (118). Therefore, we should sum over all initial conditions with the same center chord (¯q, p¯ ). As shadowing orbits are very close to each other there will be typically only one orbit having a given center chord and furthermore, to lowest order, we can assume that both orbits are actually the same for the purpose of calculating the action difference. For this single orbit we can take the initial condition to be the center chord (¯q, p¯ ). Combining all these simplifications for  tshadowing orbits and noting that Lε −L0 =−εV , if V is the perturbing potential, we find Sjε (r , q¯ , t) − Sj0 (r , q¯ , t) ≈ 0 dt  V (¯q(t  ), p¯ (t  ), t  ). We recognize the Wigner function for the state (r; 0) in Eq. (122),
  x x i 1 ∗ dx  q¯ − ; 0  q¯ + ; 0 exp − p¯ x , (123) W (¯q, p¯ ) = 2 2 2 (2 2)d and we finally get
  t  i     f (t) = dq¯ dp¯ W (¯q, p¯ ) exp − ε dt V (¯q(t ), p¯ (t ), t ) . 2 0

(124)

The final result (124) is the so-called dephasing representation [57,58]. It has a very appealing form: fidelity is given as the Wigner function average of the phases due to action differences. The importance of this expression is that it is valid from the Lyapunov to the Fermi golden rule regime. It can also in general describe fidelity decay in regular systems, e.g., Gaussian decay or even decay after plateau in quantum freeze [133]. In particular it is very handy for numerical evaluation of quantum fidelity decay in various non-generic situations or for complicated initial states. It should be also useful for systems with many degrees of freedom due to favorable scaling of its running time as compared to exact quantum calculations. The statistics of action differences has been discussed in Ref. [134]. While in practice the numerical evaluation of the dephasing representation (124) turns out to reproduce the exact quantum fidelity decay in chaotic, regular as well as in mixed systems very well [57,58], there are nevertheless some limitations. For instance, it is unable to describe the perturbative Gaussian decay for small perturbations which occurs due to finite size effects [121]. The exponential term in the expression for f (t) (124) is very reminiscent of our approximate quantum echo operator Mε (t) ≈ exp(−iε(t)/2) (24), where (t) is the integral of the perturbation (15). Indeed, the semiclassical fidelity amplitude could be obtained directly, using the Weyl–Wigner quantization [132], i.e. replacing Mε (t) by its Weyl symbol and the density matrix  by its Wigner function W we recover the dephasing representation. 4. Random matrix theory of echo dynamics From the previous section we have a surprising dichotomy. We have two slightly different approaches which yield for weak and very weak perturbation respectively the Fermi Golden rule and the perturbative regime. Both results are based on perturbation expansions, and there should be a unified theory to describe these regimes. As random matrix models have been extremely successful in describing a wide field of phenomena in physics ranging from elasticity to particle physics [135,86] associated in some sense with chaos, it seems natural to attempt a formulation of echo dynamics in this framework. A number of papers have appeared on this subject [53,54,136]. We shall basically follow the lines of [52], which uses the linear response approximation expressed in terms of correlation integrals, very similar to our reasoning in Section 2.1. This method provides a uniform approximation encompassing both regimes.

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69

It proved successful in explaining two independent experiments (see Section 9). Some exact results for the random matrix theory (RMT) model used are also available [56,55,18,137]. We give the outlay of the model and derive a linear response formula for fidelity decay extending its validity to long times in a form which is strictly valid in the weak perturbation limit. We compare this result to new more detailed calculations for the kicked top and to the exact RMT solution. We then present a considerable body of new material extending this model to purity decay, and with modified perturbations we discuss various situations where quantum freeze can occur [18,137,111]. The RMT model we present, allows to describe fidelity decay in a chaotic system, under a static global perturbation; that model could be extended to treat also noisy perturbations, but at least in the case of uncorrelated noise, a direct statistical treatment is more adequate (see Section 2.1.2). Chaoticity is meant to justify choosing the unperturbed system from one of the Gaussian invariant ensembles [138–140]. With the word “global” we mean that in the eigenbasis of the unperturbed Hamiltonian, the perturbation matrix is not sparse. Banded matrices may occur, as long as their bandwidth is so large, that a full matrix would yield similar results. We shall mainly concentrate on this case, because it is most amenable to analytic treatment. Where other Hamiltonians are used we shall point this out and explain why it is necessary. We consider a perturbed Hamiltonian in a form, typical for RMT (see e.g. [141,142]): Hε = cos(ε  )H0 + sin(ε  )H1 ,

(125)

where H0 and H1 are chosen from one of Cartan’s classical ensembles [143,86]. This scheme has the advantage that the perturbation does not change the level density of the Hamiltonian, and thus avoids the need to normalize time in the echo dynamics, as discussed at the end of Section 2.1.3. We will generally be interested in situations where ε  √ scales as 1/ N , where N denotes the dimension of the Hamiltonian matrices. In this case, the matrix elements of the perturbation couple a finite number of neighboring eigenstates of the unperturbed system, largely independent of N. This assumption allows to describe all regimes, from ε  = 0 up to the Fermi golden rule regime, and beyond. For large N, we may therefore linearize the trigonometric functions in Eq. (125). It is then convenient, to fix the average level spacing √ of H0 to be one in the center of the spectrum, and to require that the off-diagonal matrix elements of V = H1 / N have unit variance such that Hε = H0 + εV , (126) √  where ε = N ε . It is easy to check that corrections to the Heisenberg time are of order O(1/N ). The matrix H0 can have special properties (for example in Section 4.4 on echo purity), but typically it will be taken from the classical ensembles. We may use different ensembles for H0 and V. In many cases, the ensemble of perturbations is invariant under the transformations that diagonalize H0 . We can then choose H0 to be diagonal with a spectrum {Ej0 } with given spectral statistics. In this situation we can unfold the spectrum that defines H0 , to have average level density one along the entire spectrum, or we can restrict our considerations to the center of the spectrum. This restricts us to situations, where the spectral density may be assumed constant over the energy spread of the initial state. Other cases could be important, but have, to our knowledge, so far not been considered in RMT. The one parameter family Hε defines echo dynamics as discussed in Section 2. In what follows, the eigenbasis of H0 will be the only preferred basis except in Section 4.4, where we deal with entangled subsystems. Unless stated otherwise, we consider initial states to be random, but of finite span in the spectrum of H0 . Eigenstates of H0 are one limiting case and random states with maximal spectral span the other. The spectral span of the initial state and the spreading width of the H0 -eigenstates in the eigenbasis of Hε , determine the only relevant time scales. They should be compared to the Heisenberg time, which has been fixed to tH = 1, by unfolding the spectrum of H0 . In the limit N → ∞, the Zeno time (of order tH /N ; see Section 2.1.1) plays no role. Here, we are interested in the decay of fidelity or some other measure of echo dynamics. The main results cover essentially the range from the perturbative up to the Fermi golden rule regime [44,45]. The analysis of the quantum freeze and an exact analytical result for the random matrix model will provide additional and/or different regimes. The Lyapunov regime [7,44] as well as the particular behavior of coherent states are certainly not within the scope of RMT. We shall find that in many situations the fidelity amplitude is self averaging; see Eq. (131). Therefore we mainly concentrate on the fidelity amplitude, and do not bother with the more complicated averages for fidelity itself. This section is organized as follows: The linear response approximation and the exact analytic treatment of the fidelity amplitude are discussed in Subsections 4.1, and 4.2, respectively. The quantum freeze case is studied in Section 4.3. Composite systems (two coupled subsystems) are considered in Sections 4.4 and 4.5.

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4.1. Linear response theory Recall that the echo operator Mε (t) can be written to second order in ε as [46,52] 

t

Mε (t) = 1 − i2 ε

dt V˜ (t  ) − (2 ε)2 



t

dt

0





t

dt  V˜ (t  )V˜ (t  ) + O(ε 3 ).

(127)

0

0

To obtain the fidelity amplitude we have to compute the average of Mε (t) with respect to both, the ensemble defining the random perturbation V, and the one defining the spectrum {E0 } of H0 . Provided that H0 and V are statistically independent, the linear term in ε averages to zero. The quadratic term is determined by the two-point time correlation function   [V˜ ()V˜ ( )],  = V, V, e2 i[(E −E )+(E −E ) ]  

=
,



2 +
( −  ) − b2 ( −  ) . V

(128)

The middle expression contains two separate averages, the first is taken over the ensemble of V and the second over the spectral ensemble of the unperturbed Hamiltonian. The average over the matrix elements of V yields delta functions that allow to rewrite the spectral average in terms of the two-point correlation function of the spectrum; as the spectral ensemble is invariant under rearrangements of the eigenenergies, the result is independent of the index . The spectral two-point correlation function is then translational invariant and the spectral form factor b2 [144] is conveniently introduced. The constant V depends on the classical ensemble from which the perturbation was taken. It can take the values 1,2, and 4 for the orthogonal, unitary, and symplectic ensembles, respectively. Inserting this result into Eq. (7) for the fidelity amplitude we get [52] " #    fε (t) = 1 − (2 ε)2 t 2 / V + t/2 −

t



d

d b2 () + O(ε 4 ).

(129)

0

0

Any stationary ensemble from which H0 may be chosen, yields a particular two-point function b2 . The correlation integrals over b2 for the GOE and the GUE are discussed in [52]. To demonstrate the effect of spectral correlations by contrast, we occasionally use a random level sequence. In this case b2 (t) = 0 and the last term in (129) vanishes. Typically (at least in the case of the classical ensembles), spectral correlations lead to a positive b2 , such that the fidelity decay will be slowed down. The result (129) shows two remarkable features: The first is that the linear and the quadratic term in t scale both with ε2 The second is about the two possibly different ensembles used for the perturbation and for H0 . The characteristics of V affect only the prefactor of the t 2 -term, while the characteristics of H0 affect only the two-point form factor b2 . In experiments or numerical simulations averaging over H0 may be unnecessary, due to the self-averaging properties of fidelity. For this to be effective, we need an initial state which covers a sufficiently large number of eigenstates of H0 . The two-point form factor b2 can then be obtained by averaging over energy or frequency intervals (spectral average). As an expansion in time, Eq. (129) contains the leading terms for the perturbative as well as the Fermi golden rule results [30,44,45]. Both are exponentials of the corresponding terms. It is then tempting to simply exponentiate the entire ε 2 -term to obtain  " #    fε (t) = exp −(2 ε)2 t 2 / V + t/2 −

t

0



d

d b2 ()

.

(130)

0

This expression will prove to be extremely accurate for perturbation strengths up to the Fermi golden rule regime. Some justification for the exponentiation is given in [72]. While exponentiation in the perturbative regime is trivially justified, our result shows that for times t>tH , we always need the linear term in t to obtain the correct answer. In experiments the interplay of both terms has been proven to be important [69,70,83]; see also Section 9.2. On the other hand, for stronger perturbations fidelity has decayed before Heisenberg time to fluctuation levels or to levels where our

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71

〈F(t)〉

1

0.1 0

0.5

1 t /tH

1.5

2

Fig. 12. Simulation of fidelity decay for a dynamical model (the desymmetrized kicked rotor) in the cross-over regime. The three thin solid lines show the result of the simulation (for three different initial conditions), the dashed line the corresponding exponentiated linear response approximation. An additional dashed line shows the exponential part, another one the Gaussian part of that approximation (for details see text). The Heisenberg time is tH = 1000.

approximation fails. Comparison with the exact result [56,55] (Section 4.2) will show that the exponentiation allows to extend the linear response result from a validity of f (t) ≈ 1 to a validity range of f (t)0.1 Note that the pure linear response result (129) is probably all we need for quantum information purposes, as processes with fidelity less than 1 − , where  ∼ 10−4 , are not amenable to quantum error correction schemes [13]. For considerations about stability in echo dynamics, on the other hand, the exponentiated formula (130) gives a clear and simple expression. The exact treatment will show where to expect additional effects, but experiments at this time are still limited to f (t)0.1 [69,70,83]. Fidelity can be calculated in the linear response approximation along the same lines as above. One obtains [52]: Fε (t) = |fε (t)|2  = fε (t)2 + (2 ε)2 (2/ V )iprt 2 + O(ε 4 ). (131)  Here ipr =  |E ||4 indicates the inverse participation ratio (ipr) of the initial state expanded in the eigenbasis of H0 . This equation displays two extreme effects: on the one hand it shows the self-averaging properties of this system. For states with a large spectral span in H0 the correction term that marks the difference between Fε (t) and |fε (t)|2 goes to zero as the inverse participation ratio becomes small (∼ 1/N ), see also Eq. (46) for the difference between the average fidelity amplitude and the average fidelity. On the other hand, for an eigenstate of H0 , ipr = 1, and hence the quadratic term in Eq. (131) disappears. Moreover, the correlations cancel the linear term after the Heisenberg time. Thus, we find that after Heisenberg time the decay stops for an H0 taken from a GUE and continues only logarithmically for a GOE [52]. That situation is very similar to the quantum freeze considered in Section 4.3. Situations involving states with a small spectral span have been analyzed in Ref. [145]. There fidelity decay has been computed for two perturbed Hamiltonians Hε and H−ε , with perturbations of opposite sign. Under these circumstances, it has been found that the evolving states have maximal overlap for different evolution times. Note that by construction, the Heisenberg times are the same for both evolutions, in distinction to the cases we discuss in the Sections 2.1.3 and 9.2. We illustrate the above results with the help of dynamical models, the kicked rotor and the kicked top (see Appendix B.1). We show cases, where the time-reversal symmetry is conserved (by both the unperturbed Hamiltonian H0 and the perturbation V), and other cases, where the time-reversal symmetry is broken (again by both, H0 and V). The former corresponds to a GOE spectrum, perturbed by a GOE matrix, the latter to a GUE spectrum, perturbed by a GUE matrix. In Fig. 12 we show fidelity decay in the cross-over regime. The numerical calculations have been carried out by Martinez [146]. The quantum propagator for the kicked rotor is constructed in position space, such that the Floquet matrix is symmetric [147]. However, we changed the kick potential from cos  to cos  − sin 2 in order to break the reflection symmetry. That assures that the time reversal symmetry is the only remaining symmetry in the system. The perturbation is implemented by a small increment of the kicking strength K, which has been chosen in the regime

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1014 GOE theory 10-6

1012 10-5

-lnF(t)/ε2KT

1010 10-4

108 10-3 106 εKT = 10-2

104 102 100

101

102

103

104

105

106

t Fig. 13. The scaled logarithm of the fidelity as a function of time for a time-reversal invariant kicked top. The quantity is chosen such that different perturbation strengths ε give the same theoretical curve (solid line), based on the Eq. (130). The crossover between the exponential (Fermi golden rule regime) and Gaussian decay (perturbative regime) occurs around the Heisenberg time tH = 400. The numerical data for different perturbation strengths are shown by dashed curves.

of complete chaos K ≈ 10. The perturbation strength 2 ε ≈ 0.554 has been computed from the integrated classical correlation function, cl as explained in Section 3.1. Thus, in the comparison of theory and numerical experiment, there is no free parameter. We plot three curves, for different initial states. Those were coherent (Gaussian) wave packets located at arbitrarily chosen positions on the p = 1/2 axis in phase space. In this way, the initial states have real coefficients in the position basis. The additional dashed lines are just to guide the eye, as to the validity of the Fermi golden rule and the perturbative expressions, respectively. To check explicitly the scaling in the perturbation strength we finally perform simulations with the kicked top (Appendix B.1). Numerical results for time-reversal invariant kicked top with S = 400 (propagator (B.1)) and perturbations ranging over a whole set of kicked top perturbation strengths εKT = 10−6 to εKT = 10−2 are shown in Fig. 13. The RMT perturbation parameter ε is given in terms √ of the “physical” kicked top perturbation εKT (written simply as ε in propagators in Appendix B.1) as 2 ε = SεKT 2cl N, resulting in 2 ε ranging from 7.9 to 7.9 × 10−4 for the −2 shown εKT , see also Section 4.3.4 for further details on obtaining ε from εKT . Plotting −εKT ln Fε (t) versus t on a double-log scale has the consequence that all curves should follow a single line given by Eq. (130). However, the numerical results deviate at the very end and the very beginning. Differences for long times are due to the scaling, since the saturation value of fidelity does actually not depend on ε. At short times the individual properties of the dynamical system show up, and short periodic orbits give the expected non-universal contributions. Fig. 13 shows that for most perturbations considered, the exponentiated linear response expression fits very well and explains the transition from linear to quadratic behavior. The strongest perturbation is the exception. Here saturation sets in before non-generic effects have died out, without leaving any space for generic RMT behavior. To complete the picture, we show in Fig. 14 a similar calculation for a kicked top which breaks the time-reversal symmetry (Appendix B.1, Eq. (B.14)) and spin size S = 200. It corresponds to the random matrix model (126), where both parts, H0 and V are chosen from a GUE. Again we see essentially the same features as in Fig. 13. The comparison with the exponentiated linear response formula shows similar agreement as for the GOE case. 4.2. Supersymmetry calculation for the fidelity amplitude The exponentiated linear response formula (130) agrees very well with dynamical models as we have seen above, and as it has been reported in the literature [52,148]. It also agrees with experiments [69,70,83] on which we shall report in Section 9.2. Nevertheless it is quite clear, that this approach is not justified for perturbations stronger than one, even if we omit the fact, that the exponentiation is only heuristically justified. In the last twenty years many problems

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1014

73

10-6

GUE theory

1012 -lnF(t)/ε2KT

10-5 1010 10-4 108 10-3 3•10-3

106

εKT = 10-2

104 102 100

101

102

103

104

105

106

107

t Fig. 14. The scaled logarithm of the fidelity as a function of time for a kicked top, with broken time-reversal invariance (similar graph, as in Fig. 13). The theoretical curve (solid line) is based on Eq. (130), with V = 2 and b2 (t) describing GUE level fluctuations. The Heisenberg time is tH = 400. The numerical data for different perturbation strengths are shown by dashed curves.

in RMT have been solved exactly, and indeed recently Stöckmann and Schäfer [56,55] have solved the model given in Eq. (126) exactly, for GOE or GUE matrices, in the limit of infinite dimensions. More specifically, they choose H0 and V independently but both either from the GOE or the GUE, and compute the fidelity amplitude fε (t) with the help of supersymmetry techniques. A detailed discussion would not be adequate in this review, and we refer to [149] for an introduction to the techniques used and to the original paper [56] for details. They obtain fε (t) =

1 t



min(t,1)

du(1 + t − 2u)e−(2 ε)

2 (1+t−2u)t/2

(132)

0

for the GUE case and  fε (t) = 2



t

u

du max(0,t−1)

×e

dv 0

(t − u)(1 − t + u)v((2u + 1)t − t 2 + v 2 )  (t 2 − v 2 )2 (u2 − v 2 )((u + 1)2 − v 2 )

−(2 ε)2 [(2u+1)t−t 2 +v 2 ]/2

(133)

for the GOE case. These solutions are valid for arbitrary but fixed perturbation strength, in the limit N → ∞. In Fig. 15, we reproduce two graphs from [56]. The left one, compares the exact and the exponentiated linear response result for fε (t) for the GOE case. For large perturbations we see a qualitative difference in the shape of fidelity decay as a shoulder is forming in the exact results. For even stronger perturbations depicted on the right hand side, this becomes notorious as a revival appears at Heisenberg time. Yet, the revival is noticeable only for very small fidelities of the order of 10−4 for the GUE and 10−6 for the GOE. In all cases agreement with the exponentiated linear response formula is limited to f (t)0.1. It is thus adequate for most applications, and indeed it was difficult to come up with a dynamical model which can show the revival. Naturally the random matrix model does not capture the Lyapunov regime, which depends on semi-classical properties. Yet it is not obvious that such a regime always exists. For example for a kicked spin chain we have never discovered this regime, and it may well be that there is no semi-classical regime for this system. This dynamical model has been used in Ref. [150], to illustrate the partial revival, shown in Fig. 15. The authors used a multiply kicked Ising spin chain in a Hilbert space spanned by 20 spins, and averaged only over a few initial conditions. The aim was to obtain the partial revival with as little averaging as possible, relying on the self-averaging properties of the fidelity amplitude.

74

T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

100

1.0000 GOE

= 100 10−2

0.1000 f ( )

f ( )

10−4 0.0100

10−6 10−8

0.0010 10−10 10−12

0.0001 0

1

2

3

4

5

(a)

0.0

0.5

1.0

1.5

(b)

Fig. 15. (Color online) Fidelity amplitude decay for the random matrix model, defined in Eq. (126) (taken from [56]). Part (a) shows fε (t) for the GOE case, as obtained from the exact expression, Eq. (133), (solid lines), together with the same quantity, as obtained from the exponentiated linear response result, Eq. (130) (dashed lines). The perturbation strength has been set to the following values: (2 ε)2 = 0.2, 1, 2, 4 and 10. Part (b) shows fε (t) with (2 ε)2 = 100 for the GUE case (solid line) and the GOE case (dashed line), as obtained from the exact expressions, Eqs. (132) and (133), respectively.

0 -1

f

-2 -3 -4

0.2

0.4

0.6

0.8

1

1.2

1.4

t Fig. 16. The decay of the fidelity amplitude in a dynamical system, in the case of broken time-reversal symmetry, in a regime where the partial revival is observable (taken from Ref. [150]). More precisely, log10 f (t) is plotted. The triangles show √ the real part of the fidelity amplitude as obtained from the numerical simulation; the circles show the imaginary part, which goes to zero as 1/ N due to state averaging. The perturbation strength is (2 ε)2 = 31.78. The solid line shows the exact theoretical result, Eq. (132), the dashed line shows the exponentiated linear response result, Eq. (130).

It has been checked that the model shows random matrix behavior as far as its spectral statistics is concerned. The result for the decay of the fidelity amplitude is reproduced in Fig. 16. Yet it will probably be difficult to see this effect in an experiment, and we do not know how it may appear in a dynamical system which does display Lyapunov decay at times short compared to the Heisenberg time. The partial revival of the fidelity amplitude occurs not only in the case of a full GOE (GUE) perturbation as discussed here, but also in the case of a perturbation with vanishing diagonal elements (see Section 4.3, below). In [55] the partial revival has been explained with an analogy to the Debye–Waller factor. A direct, semiclassical and hence dynamical explanation could be obtained by periodic orbit expansions similar to [151,152] (for general chaotic systems) or [153] (for a quantum graph model).

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75

4.3. Quantum freeze In Section 4.1, we found that after the Heisenberg time, fidelity decay is essentially Gaussian [cf. discussion below Eq. (130)]. It is determined by the diagonal elements, i.e. the time average of the perturbation (in the interaction picture). If this term is zero or very small, we should see a considerable slowing down of fidelity decay. That such a possibility exists in principle was first noted in [16] for integrable systems (see Section 5.3) and in [17] for chaotic systems, as described in Section 3.2. However, in both cases the perturbation was chosen to depend in a very particular way on the unperturbed Hamiltonian, with the effect that fidelity became largely independent of the dynamics in the unperturbed system. We shall discuss that particular choice in the context of RMT at the end of this subsection. Here we shall mainly consider situations where the perturbation is some RMT matrix with zero diagonal. In the notation of Section 2.2, it means that the off diagonal elements will form the residual interaction Vres = V . The different options to implement such a scenario have been discussed in Section 2.2. We concentrate here on the case, where H0 is chosen from a GOE and V is an antisymmetric hermitian random Gaussian matrix. This case is of interest, because it is the only random matrix model for which an exact analytical result has been obtained, so far [18,137]. We shall show these below, yet we shall mainly concentrate on an extension of the previous linear response treatment, as well as an approach based on second order (time independent) perturbation theory. These approaches give more insight into the mechanism, and they are more easily generalized to other interesting cases. Indeed we shall again find a plateau in fidelity decay, which will be flat if H0 is chosen from a GUE and almost flat (up to logarithmic corrections) if it is chosen from the GOE. This plateau begins at Heisenberg time, and we are able to describe the decay to the plateau and the plateau itself in terms of the linear response results. The finally ensuing decay is obtained from second order perturbation theory. A brief description of some of these results can be found in [18]. 4.3.1. Linear response approximation In linear response, we have seen that the expectation value Vij Vkl  is the essential ingredient for the calculation of fidelity. We can compute this average with equal ease for the following three ensembles of perturbations: (a) If V is taken from a GOE with deleted diagonal, i.e. Vij Vkl  = (
ik
j l +
il
j k )(1 −
ij ), then we find Vik Vkj  = (
ik
kj +
ij )(1 −
ik ) =
ij (1 −
ik ).

(134)

(b) If V is taken from a GUE with deleted diagonal, i.e. Vij Vkl  =
il
j k (1 −
ij ) then we just obtain the same result for Vik Vkj . Thus, in both cases, we obtain:     e2 i(Ei −Ek ) e2 i(Ek −Ej )
ij (1 −
ik ) =
ij e2 i(Ei −Ek )(− ) . (135) V˜ ()V˜ ( )ij = k =i

k

(c) If V is taken from an ensemble of imaginary antisymmetric matrices we may write V = iA,

Aij Akl  =
ik
j l −
il
kj .

(136)

This yields V˜ ()V˜ ( )ij = −





e2 i(Ei −Ek ) e2 i(Ek −Ej )

(
ik
kj −
ij ) =
ij





e2 i(Ei −Ek )(− ) .

(137)

k =i

k

In all three cases, we obtain the same result, which is in fact quite similar to the cases without deleted diagonal considered in Section 4.1. If we write the sum of exponentials in terms of the spectral two-point correlation function (as it has been done in Eq. (129)), we obtain " #    t 2  fε (t) = 1 − (2 ε) t/2 − d d b2 () + O(ε 4 ). (138) 0

0

Comparing this expression to Eq. (129), we note that the t 2 -term is missing. This has the peculiar consequence that the characteristics of V [i.e. whether we consider case (a), (b) or (c)] have no effect on the linear response result. It solely depends on the spectral statistics of H0 , encoded in the two-point form factor b2 . If we deal with an H0 taken from a GUE then the decay will stop at Heisenberg time, while for one selected from a GOE we will have a very slow

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T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

1 0.998

〈 f(t)〉

0.996 0.994 0.992 0.99 0.988 0

1

2

3

4

5

t /tH Fig. 17. Numerical simulation of fidelity amplitude decay for a GOE spectrum perturbed with a GOE matrix with deleted diagonal. The perturbation strength was ε = 0.025 (for further details, see text). The thick solid line shows the result of the simulation, the short dashed line shows the linear response result, Eq. (138). The long dashed line shows the linear response result for an ordinary GOE perturbation.

decay, determined by the logarithmic behavior of b2 . We can get explicit formulae for the freeze after Heisenberg time. Evaluating the integral in Eq. (138) for GOE and GUE cases, we get for times t > tH , log(2t) + 2 + O(ε 4 ) : GOE 12 fε (t) = 1 − (2 ε)2 61 + O(ε 4 ) : GUE. fε (t) = 1 − (2 ε)2

(139)

We can see the time independence for systems without time-reversal symmetry, while for time-reversal invariant systems freezing is not perfect as the plateau value decreases logarithmically, i.e. very slowly with time. These predictions are illustrated and compared to simulations for a dynamical model in Figs. 22 and 23 in Section 4.3.4. In Appendix E we show that the Born series can be computed to fourth order and estimates of higher orders can be given in the limit of large N. This is done both for case (a) and case (c), and again the results agree. By consequence, any difference between the two kinds of perturbations must be of strictly non-perturbative type. That is also confirmed by numerics and thus in Fig. 17, we only show case (a), i.e. we give, for a GOE Hamiltonian perturbed by a GOE with deleted diagonal, a comparison of the above formula with a Monte Carlo simulation of the random matrix model. Here we used just a linear rescaling of the unperturbed spectrum to obtain unit average level spacing in the center. Therefore we used rather large matrices of dimension N = 400, and random initial states with small spectral span (over the 20 central eigenstates of H0 ). With an ensemble of nrun =2000, we obtained for the fidelity amplitude the thick solid curve. The linear response approximation, Eq. (138) is plotted by a thin short-dashed line. The agreement is rather good up to 2tH . From then on, we find increasing deviations. This point of departure from the linear response approximation can be moved towards larger times, by decreasing the perturbation strength further. A strict plateau cannot be observed due to the remaining logarithmic decay of the linear response result. The great gain in fidelity can be appreciated by comparing to the decay of the fidelity amplitude for an ordinary GOE perturbation (long-dashed line). Note that this derivation does not depend in any way on the time-averaged perturbation being identically zero. Rather we use the fact that diagonal and off diagonal elements enter independently, the former influencing the t 2 -term in Eq. (129) only. We can therefore equally well deal with diagonal elements which are simply suppressed by a constant factor. That factor would lead to a corresponding suppression of the above mentioned quadratic term. This opens a new range of applications which will have to be considered in the future. 4.3.2. Beyond fidelity freeze Higher order terms in ε will cause an end of the freeze, and we shall use time-independent perturbation theory to describe this phenomenon. Recall that in the case of a standard perturbation such considerations lead to quadratic decay setting in around Heisenberg time [44,45]. It is precisely the absence of this term in the present situation that is

T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

77

responsible for the freeze. Here, we consider the case, where the second order term is needed, e.g. due to the unusually small or vanishing diagonal elements in the perturbation matrix. In a formally exact manner, the fidelity amplitude can be written in terms of the eigenvalues of the perturbed and the unperturbed system, and the orthogonal transformation between their respective eigenbases (40):  ˜ f (t) = ∗  e2 iE t O e−2 iE t O . (140) 

We assume random initial states  with well defined local density of states (in the eigenbasis of H0 ). The spectrum of H0 is denoted by {E }, while the spectrum of Hε is denoted by {E˜  }. For times which are long compared to the Heisenberg time, we may neglect phases from energies of different states due to phase randomization. This amounts to approximate the above sum by its diagonal contribution  = as  2 −2 i t f (t) = | |2 O e ,  = E − E˜  . (141) 

2 may be set equal to one.6 This leads to Finally, we may assume a sufficiently small perturbation, such that O  | |2 e−2 i t . (142) f (t) ≈



In second order perturbation theory, the level shifts  are given by  = εV + ε 2

 |V |2 + O(ε 3 ). E − E 

(143)

 =

The first order term, containing the diagonal elements of V, is statistically independent from the second order term. As a consequence, the ensemble average inside the sum factorizes into a purely Gaussian decay and a factor which corresponds to the quantum freeze situation

  2

| |2 e−2 iεV t S (E ), S (E ) = e−i >1 |V1 | /(E1 −E )

, (144) f (t) ≈ E1 =E



where  = 2 ε2 t. The above construction allows to treat the energy argument in S (E) as a free variable. For the following, we discard the exponential which contains the diagonal matrix elements of V (it usually factors out, anyway). In addition we assume that in the eigenbasis of H0 , the initial state has non-zero coefficients only in a narrow region in the center of the spectrum. That allows to remove the sum, and consider S (E) at the energy E = 0. More general cases can be considered, but require to replace the sum over  by a convolution-type integral. For E = 0, S (E) can be reduced to S (0), by employing a principal value integral to express the smooth part of the sum in the exponent. We thus consider initial states, such that fε (t) ≈ S (0). Then, the decay of the fidelity only depends on amplitude 2 the spectral correlations of the ensembles considered. It involves the level curvature  = N >1 |V1 | /E − E1 , which has been studied in some detail in the literature (see Refs. [154–156] and references therein). Using a closed form of the distribution of  it is a simple exercise to compute fε (t):  C cos  P() = d : S (0) = C ,  = 2 ε 2 t, (145)  1+ /2 (2 + 2 ) (2 + 2 )1+ /2 where C assures proper normalization of P(). The cosine integral in Eq. (145) gives  fε (t) ≈ S (0) =

e−  (1 + ) : = 2, : = 1,

K1 ( ) e−  : = 0,

(146)

6 A more phenomenological assumption of statistically independent eigenvectors and eigenvalues, which would allow to pull the average O 2   out of the sum, does not prove very useful, if compared to the exact analytical result [18].

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T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

1 1

0.8 〈 f(t)〉

0.99

〈 f(t)〉

0.6

0.98 0.97

0.4

0.96 0

2

4

0.2

6

8

10

12

14

t/tH

0 0

0.5

1

1.5

2

= 2π2t /tH Fig. 18. Numerical simulation of the fidelity amplitude decay for a GOE spectrum perturbed with a GOE matrix with deleted diagonal. The perturbation strength and all other parameters were chosen as in Fig. 17. The thick solid line shows the result of the simulation, the short dashed line shows the result from perturbation theory, Eq. (146) with = 1. In the inset, we show the transition from linear response (long dashed line) to time independent perturbation theory (short dashed line). The numerical data are plotted with diamonds.

where K1 (z) is the modified Bessel function [157]. The resulting fε (t) gives the decay of fidelity (within the second order perturbative approximation) after the freeze plateau. Note that the decay time scales as 1/ε2 as opposed to 1/ε in the absence of the freeze. In all cases, we assume H0 to be diagonal. The case = 2 describes a GUE spectrum perturbed by a GUE matrix V, with deleted diagonal. It means that the moduli squared |V1 |2 in Eq. (144) have an exponential distribution. The case = 1 describes a GOE spectrum perturbed by a GOE matrix, with deleted diagonal. In that case, the moduli squared |V1 |2 have a Porter-Thomas distribution [158]. This perturbation is equivalent to an antisymmetric imaginary random Gaussian matrix. This perturbation has been treated exactly using supersymmetry techniques [18,137]. The last case = 0 describes a spectrum with uncorrelated eigenvalues, while V is chosen as in the = 1 case. That case has not been considered in [156]. It fits rather accidentally to the formula (145) for P(), when setting = 0. We include that case, in order to get a more complete overview on the effects of level repulsion on the decay of the fidelity amplitude in the quantum freeze situation. The fact that a spectrum without correlations gives indeed an exponential decay, can be easily checked by direct evaluation of the separable average [159]. Furthermore, one finds that this result only depends on the average of the matrix element |V1 |2 , and not on its distribution. In Fig. 18 we show the decay of the fidelity amplitude for a GOE spectrum, perturbed by a GOE matrix with deleted diagonal. The perturbation strength ε = 0.025 and the other parameters are exactly the same as in Fig. 17; but here we show the long time behavior (main graph). The figure shows that our perturbation theory provides a very accurate description of the data. In the inset we focus on smaller times, where the transition from the linear response regime to the perturbative regime can be observed. For the GUE case ( = 2) and the Poisson case ( = 0) we obtain similar agreement. The effect of spectral correlations (level repulsion). In view of Eq. (143), it is clear that level repulsion plays a crucial role  for the behavior of the fidelity amplitude. The degree of level repulsion determines whether the level curvature 2 = N >1 |V1 | /E , has a finite second moment, or not. If it has, we may expand S (0) as defined in Eq. (144) up to the quadratic term, and average term by term. That would give f0 (E, t) ∼ 1 −

2 2  . 2

(147)

It is not difficult to see that 2  is finite for the GUE spectrum, only. For a GOE spectrum, or a Poisson spectrum (without correlations) that average diverges. Eq. (147) will be used below to consider the case of a GUE spectrum perturbed by a GOE matrix with deleted diagonal. The small time behavior of the fidelity amplitude can be obtained from Eq. (146). Here, small time really means small ; the time t should still be much larger than the Heisenberg time. The corresponding asymptotic expansions

T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

79

1 0.8

〈 f(t)〉

0.6 0.4 0.2 0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

= 2π2t /tH Fig. 19. Numerical simulation of fidelity amplitude decay for different random spectra, perturbed with a GOE matrix with deleted diagonal. For the GOE (long-dashed line) and the Poisson spectrum (solid line) we show the analytical results of Eq. (146). For the GUE spectrum we show a numerical simulation (short-dashed line) and the leading order quadratic behavior (dash-dotted line). For additional details, see text.

yield to lowest order  fε (t) ∼

1 − 2 2 /2 1 − 21 (− ln( ) + 1 − 

1 2

+ ln 2 − ) 2 2

: GUE, : GOE, : Poisson,

(148)

where  is Euler’s gamma constant [157]. These expansions demonstrate that in the GOE and the Poisson case, the second moment of the level curvature  do not exist. In Fig. 19, we illustrate the effect of level correlations on the decay of the fidelity amplitude, in the presence of a GOE perturbation (with deleted diagonal). For the GOE and the uncorrelated spectrum, the curves shown are just the analytical results given in Eq. (146). For the GUE spectrum, there is no complete analytical result available (note that it is a rather artificial situation), and we therefore show an actually quite accurate7 numerical simulation. The Poisson spectrum has no correlations at all, and by consequence the decay of the fidelity amplitude is the fastest. From  = 0 it starts of linearly. Next comes the GOE spectrum, with linear level repulsion, and therefore much slower decay f (t). The GUE spectrum leads to the slowest decay of the fidelity amplitude. It is the only case, where the f (t) behaves quadratically for small . The corresponding coefficient will be calculated below; the resulting quadratic decay is plotted with a dash-dotted line. As mentioned above, a GUE spectrum perturbed by a GOE matrix (with deleted diagonal), is a rather artificial construction. Even though, there is no complete analytical result available (in terms of the perturbation theory, developed above), we may still obtain the leading order behavior at small times. This is due to the fact that for the GUE spectrum, the level curvature has a finite second moment 

N 

2  = |V1 |4  (E1 − E )−2

, (149)

=2 E1 =0

where |V1 |4  = 2(3) for a GUE (GOE) perturbation. By comparison of Eq. (147) with the = 2 case in Eq. (146) we find that fε (t) ∼ 1 − 3 2 2 /4 (dash–dotted line in Fig. 19).

7 We expect this simulation to reproduce the result of the perturbation theory exactly, within the finite linewidth in the figure. In particular, we checked that further decrease of ε or increase of N (the dimension of the Hamiltonian matrix) gives no noticeable change on the scales of the figure.

T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

1

2b +1 〈 f(t)〉

〈Σk e2 i ( Ei − Ek ) t〉

80

0

1/2b

0 tH

(a)

1 − 2b 2

(b)

tH

Fig. 20. Schematic behavior of the fidelity amplitude [panel (b)] and the two-point correlations [panel (a)], which are the determining quantities for that behavior (see text for details). For the purpose of illustration, we assume a small band width. The solid lines show the results for a GOE spectrum, the dashed lines that for an uncorrelated (Poisson) spectrum.

4.3.3. Commutator perturbation Let us take a perturbation which is proportional to the commutator of a random GOE matrix W with H0 (in order to remain Hermitean, the perturbation must be imaginary). H = H0 + εV ,

Vkl = iWkl (Ek − El ).

(150)

In the interaction picture, the perturbation V is equivalent to the time-derivative of W, as detailed in Section 2.2. Here, we use a banded perturbation matrix Wkl , and consequently also banded Vkl , because otherwise the limit of infinite matrix dimension N → ∞ will be ill defined. It is then convenient to assume that the basis states are arranged in such a way that Ei < Ej for i < j (for technical convenience we also assume a non-degenerate spectrum). Linear response. The crucial quantity is the average    V˜ik ()V˜kj ( ) = e2 i(Ei −Ek ) e2 i(Ek −Ej ) Vik Vkj . (151) k

k

Since W is hermitian, we obtain under the above assumptions:   V˜ ()V˜ ( )ij =
ij (Ei − Ek )2 e2 i(Ei −Ek )(− ) |Wij |2 ,

|Wij |2  = (b − |i − j |),

(152)

k

where (j ) = 1 for j
0 and (j ) = 0 otherwise. That is the simplest case of a banded matrix, but more realistic band profiles can be treated along the same lines. Inserting this into Eq. (127) for the echo operator in the linear response approximation, we get ⎡ ⎤ min(N,i+b)  Mε (t)ij =
ij ⎣1 + ε 2 (e2 i(Ei −Ek )t − 2 i(Ei − Ek )t − 1)⎦ k=max(1,i−b)

=
ij [1 − ε C(t)]. 2

(153)

For an initial state which stays away from the border of the spectrum, the linear term averages to zero (as a spectral average). In the limit of large band-width, the remaining sum of exponential phases tend to an expression involving the two-point form factor, similar to the case of Eq. (128): min(N,i+b) 

e2 i(Ei −Ek )t →
(t) − b2 (t) + 1.

(154)

k=max(1,i−b)

With that information in mind, we plot in Fig. 20 the schematic behavior of that sum for finite band width b; panel (a). The solid line shows the result for a GOE spectrum, the dashed line for the case b2 (t) = 0 (uncorrelated levels). The time t = tH /(2b) may be considered as the Ehrenfest time of the system. So we may say that the correlation integral C(t) increases from zero to some value between 2b and 2b + 1, within a time which is of the oder of the Ehrenfest time. Panel (b) of Fig. 20 shows the schematic behavior of the fidelity amplitude; its very fast decay to the practically

T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

81

1

5

0.8 (1 − 〈 f(t)〉) × 10

〈 f(t)〉

0.9

0.7 0.6

0 5 10 15 20 25

0

0.5

0.5 0

1 t/tH

0.005

1.5

0.01

2

0.015

0.02

0.025

= 2π2t /tH Fig. 21. The fidelity amplitude in the case of a banded commutator perturbation. The case of a GOE spectrum (solid lines) and a Poisson spectrum (long dashed lines). The theoretical expectations based on linear response (fplateau ; dotted line) and second order perturbation theory (short dashed lines). The main figure shows the long time behavior, where we scaled time to coincide with , as used in Eq. (157). The insert shows the short time behavior for times of the order of the Heisenberg time. For details on the parameters used in the simulation, see text.

constant value of fplateau = 1 − 2bε2 . This shows that a commutator perturbation yields a behavior of the fidelity amplitude which is to a large extent insensitive to the particular dynamics of the unperturbed system. This is in line with the “surprising” similarities of quantum freeze between integrable and chaotic systems [16,17]. For perturbations which have a vanishing diagonal due to other reasons, the situation is different, as we have seen above. Time independent perturbation theory. As the diagonal of the perturbation V is zero, we must evaluate the eigenenergies of H in second order perturbation theory, just as in Eq. (143). For a perturbation of the form (150), the level shifts are given by   = E˜  − E = ε 2 |W |2 (E − E ). (155)  =

For this expression to be valid, the shifts in the eigenenergies must be small compared to the average level distance (which is set equal to one). For definiteness, let us consider the banded random matrix model, Eq. (153), as used in the linear response calculation. Then we obtain, from Eq. (144):   fε (t) ≈ | |2 ei ,  = |W |2 (E − E ), (156) 

1  |−|  b

where  = 2 ε2 t. In this equation, we have assumed that the initial state stays away from the matrix borders. Then, as long as the level density remains constant, the phase average ei  is translational invariant, and we obtain fε (t) ≈ ei  = 1 −

2 2   + O(4 ). 2 

(157)

In distinction to the perturbations with deleted diagonal, here the second moment of the level shifts usually exist. We find:
 b 
3  b b2 2 4 2 2 4   ≈ 2|W |  dxx + b /2. = 2|W |  + + O(b) . (158) 3 2 0 The sum over  is here approximated with an integral. Here, we added the next leading term of the Euler-Maclaurin expansion (it amounts to the inversion of the familiar trapezoidal integration rule) [157]. Note that the b2 -term also depends on the spectral correlations. Thus, the expansion above is correct only in the case of an uncorrelated (Poisson) spectrum. For a GOE spectrum, the b2 -term must be modified. In Fig. 21 we show the behavior of the fidelity amplitude for a banded commutator perturbation. We performed two simulations, one with a GOE spectrum and another one with a Poisson (uncorrelated) spectrum. We use a small

82

T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

100 ε = 0.35 ε = 0.035

10-1

1-F

10-2 10-3

ε = 3.5•10-3

10-4

ε = 3.5•10-4

10-5 10-6 10-7 10-3

10-2

10-1

100

101

102

103

104

105

106

τ =t / tH Fig. 22. Log-freeze in a kicked top, GOE case, the same data as in [18]. The solid line shows the exact result obtained by supersymmetry [18], the chain line shows the long-time approximation Eq. (161), the thick dotted lines show numerical simulations with the kicked top. At small times, 1 − F increases linearly as given by the standard Fermi golden rule decay (159). Around the Heisenberg time tH = 400 this regime ends and the freeze begins. The logarithmic behavior ends where the asymptotic decay of the freeze begins. This is well described by the long-time approximation.

band-width b = 10 to observe finite size effects in the band-width. The perturbation strength was ε = 0.003, and the dimension of the matrices N = 100. We performed an ensemble average over nrun = 40000 realizations, and an average over ntr = 50 initial states, which were eigenstates of the unperturbed system, located in the center of the spectrum. The main figure shows the long time behavior for both simulations, (GOE spectrum: solid line, Poisson spectrum: long-dashed line) together with the theoretical expectation, Eq. (157). We find good agreement in the Poisson case, while the correlations in the GOE spectrum lead to a slower decay. As explained above, we expect the difference between correlated and uncorrelated spectra to disappear in the limit of large band-width. The insert shows the short time behavior for times of the order of the Heisenberg time. We compare the simulations with the plateau value fplateau found from the linear response approximation. We find good agreement, and we recognize a weak influence of the two-point form factor, as explained in Fig. 20. 4.3.4. RMT freeze in dynamical systems We shall demonstrate the theory for fidelity freeze in RMT by a dynamical model. We again choose the kicked top. As the numerical results for a commutator type perturbations have already been presented (Figs. 7 and 8), we shall focus on perturbations with vanishing diagonal elements, either due to symmetries or due to setting them to zero by hand, see also general discussion in Section 2.2. First we are going to discuss an unperturbed system with time reversal symmetry, therefore corresponding to GOE sym theory (COE for a kicked system). For this we take a symmetrized chaotic kicked top Uε given in Appendix B, sym sym Eq. (B.12). Because the perturbation for Uε is complex antisymmetric (in the eigenbasis of unperturbed U0 ) we sym expect to find quantum freeze. The symmetrization, i.e., splitting operator P in Uε into two parts, is essential. Without symmetrization there is no freeze and fidelity decays on a much shorter time scale. For instance, for parameters used in Fig. 22 the fidelity would decay on a time scale which is approximately 1000 times shorter! In order to compare RMT with the numerics we have to re-introduce physical units. The Heisenberg time is N = 2S, if S is the spin size, and the second moment of the matrix elements of the perturbation is |Vj k → j |2  = 2cl /N, where cl is an integral of the classical correlation function (see Section 3.1). For times smaller than the Heisenberg time fidelity will decrease linearly, according to Eq. (87), 1 − F ≈ (εKT S)2 2cl t,

(159)

where we temporarily use εKT for the kicked top perturbation strength (denoted simply by ε in Appendix B.1) to  distinguish it from the RMT perturbation strength ε, which is given as 2 ε = εKT 4cl S 3 . For intermediate times,

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83

when we have a log-freeze (139), we have instead 2 1 − F ≈ εKT S 3 4cl

ln(t/2S) + 2 + ln 2 . 6

(160)

For large times we can use the approximate theoretical prediction (146) noting that in physical units we have 2 S 2  t/ and thus  = εKT cl 4 2 S 4 2cl t 2 [K1 (εKT S 2 cl t)]2 . F = εKT

(161)

In Fig. 22 we compare numerical results with the theoretical prediction (161) and with the exact result from Ref. [18], all for S = 200 and different εKT = 10−6 , . . . , 10−3 . We have averaged over 400 independent kicked top realizations, taking one random initial state each time. Independent kicked top realizations are obtained by drawing the parameter  in Eq. (B.12) at random from a Gaussian probability distribution with the center at  = 30 and unit variance. To ensure statistical independence the standard deviation divided √ by the number of realizations must be large as compared to the so-called level collision time which scales as ∼ 1/ N and is determined by the level velocity and the mean level spacing. The averaging over independent spectra is here absolutely essential to reproduce the log-freeze. The agreement with √ the log-freeze (160) holds only up to the time determined by the smallest level spacing and therefore scales as ∼ m, if m is the number of ensembles used in the averaging. The asymptotic result (161) is similarly sensitive to averaging. Next, we shall give an example for a system without time-reversal symmetry, belonging to the CUE. The one step propagator is given by Eq. (B.14). For the numerical simulations, we use exactly the same parameters as in the GOE case; this refers to the spin magnitude S = 200, the random distribution for , the initial states, and the perturbation strengths. In order to obtain freeze we set the diagonal elements of the perturbation to zero by hand. The short time decay does not depend on the symmetry class and it is the same as for the GOE case (159). The plateau during quantum freeze is for the GUE case independent of time, 2 1 − F = εKT S 3 4cl 31 .

(162)

On the other hand, Eq. (146) yields for the long time decay 2 2 F = (1 + εKT S 2 cl t)2 exp(−2εKT S 2 cl t).

(163)

In Fig. 23 we again find nice agreement between numerical simulations and theory. 4.4. Echo purity The random matrix model developed in [52] and described above can be generalized to cover the evolution of entanglement following the lines of Refs. [160,161]. The new essential ingredient is to write the Hilbert space H as a tensor product of two Hilbert spaces H = Hc ⊗ He , which we will for convenience call “central system” and “environment”. For the dimension N of H, we get: N = nc ne , where nc and ne are the dimensions of Hc and He , respectively. We first consider an unperturbed Hamiltonian H0 ∈ GOE, which couples the two subspaces randomly and strongly, while the perturbation V will be drawn independently from the same ensemble. The entire argument that follows will go through analogously for H0 ∈ GUE. Echo purity, i.e. the purity of an initial state evolved with the echo operator, is defined in Section 2.6, Eq. (74). To calculate that quantity we need a representation of the echo operator M˜ ε (t) in a product basis with respect to the two factor spaces. Some orthogonal transformation will yield this as M˜ ε (t) = OM ε (t)O T .

(164)

As H0 and V are chosen from orthogonally invariant ensembles, the orthogonal transformation O will be random and independent of the product spaces Hc and He . We shall average over it using the invariant measure of the orthogonal group as in [160,161]. Since purity is of second order in the density matrix the expressions become involved and we have to digress to some formal considerations before actually working out the linear response approximation. Purity form. Since purity is defined as a trace of the square of a density operator, its treatment in RMT usually leads to a rather messy jungle of indices [160,161]. To avoid that, we define a linear form p[.] on the space of operators

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100 ε = 0.35 ε = 0.035

10-1

1-F

10-2 10-3

ε = 3.5•10-3

10-4

ε = 3.5•10-4

10-5 10-6 10-7 10-3

10-2

10-1

100

101

102

103

104

105

106

τ =t / tH Fig. 23. Freeze in a kicked top, for the GUE case. The diagonal elements are put to zero by hand. The thick dashed lines show the numerical simulations, the solid lines correspond to the exact RMT result from [18], and the chain lines show the asymptotic result (163). The plateau during freeze is constant in the GUE case, as described by Eq. (162).

in H⊗2 . To this end we require that for any tensor product of operators 1 , 2 acting on H the purity functional evaluates to p[1 ⊗ 2 ] = tr c (tr e 1 tr e 2 ).

(165)

Due to linearity this suffices for a complete definition of the purity form. The purity itself, reads: I (t)=p[⊗]. For that expression to be physically meaningful,  should be a density matrix. However, the following algebraic manipulations will eventually involve operators, which are not density operators. For H = H0 + εV and M (t) denoting the initial state (0) propagated with the echo operator M(t) = U0† (t)U (t), the echo purity may be written as FP (t) = p[M (t) ⊗ M (t)].

(166)

In order to compute the echo purity in linear response approximation, we use the corresponding expression for the echo operator, Eq. (127) in the eigenbasis of H0 . Due to the bilinearity of the tensor product, we obtain for the linear response expression of the echo operator in the product basis ˜ M(t) ≈ 1 − 2 iε I˜ − (2 ε)2 J˜,  t T ˜ d e2 iE  V e−2 iE  , I = OI O , I =

(167) (168)

0

J˜ = OJ O T ,

J =

 

 

t

d 0

0





d e2 iE  V e−2 iE (− ) V e−2 iE  ,

(169)

where O T H0 O = diag(E ). As mentioned above, V will be drawn from a GOE, while the orthogonal matrix O is taken from the orthogonal group provided with the invariant Haar measure. We shall see that in linear response approximation, echo purity depends on the spectrum of H0 only via its two-point correlations, which are conveniently expressed in terms of the two-point form factor b2 . All three components are assumed to be statistically independent.8 8 This assumption is crucial and should be carefully checked, in practice.

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85

The starting point is thus more general than what we have considered in Section 2.6, inasmuch as we allow for non-separable (with respect to Hc and He ) and even non-pure (in H) initial states. While we treat the technical details in Appendix D, we state here, that the echo purity is of the general form FP (t) ≈ p[(0) ⊗ (0)] − 2(2 ε)2 CP (t),

(170)

where the correlation integral CP (t) is independent of ε. In the case of a pure and separable initial state (0) = |c (0)c (0)| ⊗ |e (0)e (0)| we may use the orthogonal invariance of H0 and V under arbitrary orthogonal transformations, and the invariance of purity under orthogonal transformations in the factor spaces, to obtain the following compact expression: 
    t n c + ne − 2 2  d db2 (E, ) . (171) t + t/2 − CP (t) = 2 1 − N +2 0 0 Note that the average over the random orthogonal transformation, implies that the decay is influenced by the spectral correlations from everywhere in the spectrum. For that reason we use the notation b2 (E, ) which means a global spectral average of the two-point form factor over the whole spectrum. The decay of the echo purity is of the same form as the fidelity amplitude decay, but faster by a factor of 4[1 − (nc + ne − 2)/(N + 2)]. This fact is consistent with the inequality (75) which states that the square of fidelity must be smaller or equal to purity; for our RMT model in linear response approximation the equality holds exactly in the limit nc , ne → ∞. These results, in particular Eqs. (170) and (171), apply to the quantum simulation of a dynamical model (the kicked Ising spin chain, Appendix B.2) considered in Section 3.3. While those considerations were restricted to the Fermi golden rule regime (exponential decay), the present results cover the whole range from the Fermi golden rule to the perturbative regime. Being obtained from linear response theory, their validity is naturally restricted to time regions where the echo operator is sufficiently close to the identity. However, we expect that this defect can be cured by phenomenological exponentiation of the linear response result, similar to the fidelity amplitude case. If in addition the possibly quite large limit values of the echo purity are taken into account (as described in Section 3.3), we expect to obtain an accurate general theoretical description of echo purity decay. Numerical simulations to check these ideas are considered, but have not been performed, so far. See however, Fig. 10, which presents some cases for relatively small Hilbert space dimensions, where the echo-purity deviates very clearly from an exponential decay. These deviations indicate the presence of an additional term, quadratic in time, which would be in line with Eq. (171). Unfortunately, the presence of a number of discrete symmetries in the dynamical system used, makes a quantitative comparison exceedingly difficult. Some special cases could be considered. In many dynamical models, such as the Jaynes–Cummings model with counter-rotating terms [162] the perturbation may only act in one of the two subsystems. As on the other hand the eigenstates of H0 (when transformed to a product basis) imply a complicated orthogonal transformation, we still expect the above result to provide a reasonable description of the decay of echo purity. One special case of fundamental importance will be treated in the following Section 4.5. There we shall assume H0 to be separable, the perturbation providing the only coupling. 4.5. Purity decay The results we developed in the previous subsection can readily be modified to obtain an excellent model for the decay of purity in a pure forward time evolution. To that end, consider a Hamiltonian H0 which does not establish any interaction between the two subsystems, such that it can be written as H0 = hc ⊗ 1 + 1 ⊗ henv , where hc and henv act on Hc and He , respectively. Clearly the purity of any density matrix for the central system will not be affected by a time evolution with this Hamiltonian. Thus entanglement and purity decay will be entirely due to the forward time evolution. A simulation with a corresponding dynamical model is discussed in Section 3.3, Fig. 9. While the focus there is on the semiclassical behavior which involves quantum mechanically strong perturbations to reach the Fermi golden rule regime, we consider decay times of the order of the Heisenberg time. As before, the results here will therefore cover the whole range from the Fermi golden rule to the perturbative regime. Assuming that the perturbation V as well as the Hamiltonians hc and henv are taken from appropriate random matrix ensembles, we can still apply our results obtained for echo purity. However, at the level of echo dynamics, we make

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1 0.8

〈I(t)〉

0.6 0.4 0.2 0 0

2

4

6

8

10

t /tH Fig. 24. Purity decay in a random matrix simulation. The initial state is a tensor product of two random states in the respective factor spaces. The simulations are shown by diamonds (nc = ne = 10) and by circles (nc = 2, ne = 50). The corresponding theoretical curves show that bare linear response result, Eq. (173) together with its exponentiated version for the case nc = ne = 10 (solid lines) and the case nc = 2, ne = 50 (dashed line). The exponentiated version contains a phenomenological treatment of the asymptotic value (for details, see text).

an additional approximation. Since the additivity of the eigenenergies of hc and henv in the unperturbed system tend to destroy spectral correlations, anyway, we replace the spectrum of H0 by a sequence of uncorrelated random levels. We approximate the purity decay I (t) of a separable H0 with the decay of the echo purity FP (t) of a simpler H0 , which is no longer separable. In this way, we get for the growth of entanglement as measured by the purity of an evolving state with initial density matrix 0 I (t) ≈ FP (t) = I (0) − 2(2 ε)2 CP (t) + O(ε 4 ),

(172)

with a simplified correlation integral due to the absence of correlations in the spectrum of H0 (see Appendix D.1). For the case of pure initial states we thus get the following linear response result:  ( ) t nc + ne − 1 − Ipr 2 0 (173) 1− + O(ε4 ), I (t) ≈ 1 − 4(2 ε)2 t 2 [1 − Ipr 2 0 ] + 2 N where Ipr 2 = Ipr(tr e 0 ) + Ipr(tr c 0 ) − Ipr0 . Here, we have generalized the inverse participation ratio to density matrices by defining: Ipr = tr(diag )2 , where diag  is the diagonal part of the density matrix . In the case of pure states,  = ||, this definition reduces to the familiar definition of the inverse participation ratio. Note that even if 0 is assumed pure in this equation, the partial traces of 0 are generally not. Eq. (173) can be tested by random matrix simulations. To this end we assume the initial state to be pure and separable, I (0) = 1. We choose rather small dimensions of the Hilbert spaces involved, and compare in Figs. 24 and 25 the cases nc = ne = 10 (diamonds) and nc = 2, ne = 50 (circles). In those figures we show the average purity I (t) (it equals the average echo purity FP (t) in the present model), as a function of time, restricting ourselves to rather small perturbations ε = 0.025. In Fig. 24 we use initial states which are products of random states in the respective subsystems. In that case, Ipr 2 0 = 3/(nc + 2) + 3/(ne + 2) − 9/((nc + 2)(ne + 2)). Here, the perturbation strength is actually strong enough to observe the cross-over from linear to quadratic decay. For each case, nc = ne = 10 (solid lines) and nc = 2, ne = 50 (dashed lines), we plot two theoretical curves, the behavior according to Eq. (173) and its exponentiated version. For the latter we also took into account the finite limit values of the purity at large times, in the way as described in Section 3.3, Eq. (113) I (t) ≈ FP∗ + (1 − FP∗ ) exp[−2CP (t)(1 − FP∗ )−1 ].

(174)

In the absence of a theoretical prediction for the limit value FP∗ , we fitted it to the numerical simulation. There are noticeable differences between the numerical simulations and the analytical predictions. We believe these differences to be due to the particular form of the level density for H0 and the fact that we use random initial states with maximal

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87

1 0.99 0.98 〈I(t)〉

0.97 0.96 0.95 0.94 0.93 0.92 0

0.5

1

1.5

2 t /tH

2.5

3

3.5

4

Fig. 25. Purity decay in a random matrix simulation. The initial state is a product state from the product basis of eigenststes of the uncoupled Hamiltonian with an energy located at the center of the spectrum. The simulations are shown by diamonds (nc =ne =10) and by circles (nc =2, ne =50). The corresponding theoretical curves show the plain linear response result, Eq. (175), by a solid line (nc = ne = 10) and a dashed line (nc = 2, ne = 50), respectively.

spectral span. Since H0 is the sum of two random spectra with constant level density, the level density of H0 has a triangular shape, whereas in the analytical work the level density is assumed to be constant. Fig. 25 shows the purity decay for initial states which are eigenstates of H0 . In that case, the quadratic term in Eq. (173) cancels, and one obtains
 nc + ne − 2 2 (175) + O(ε 4 ). I (t) = 1 − 2(2 ε) t 1 − N We find that the linear response result is indeed valid, also for that case. At small times the theoretical curves converge to the numerical results. However, even when the saturation value of purity is taken into account (as done in Fig. 24), the exponentiated linear response formula (not plotted) does not correctly describe the numerical simulations. Note that due to the absence of the t 2 -term the saturation values are much higher. In [111] a similar result is obtained for an even more involved situation relevant for quantum information. 5. Quantum echo-dynamics: integrable case We shall now turn our attention to integrable dynamics. The system can actually also have a mixed phase space, provided the initial state is located in the integrable part of phase space so that we can locally define action-angle variables used in the following theoretical derivations. We shall call such systems regular. Throughout this section we shall assume for simplicity that the perturbation is time independent9 in the Schrödinger picture, i.e. V (t) = V . The action-angle variables used in the theoretical derivations are introduced in Section 5.1. Afterwards, we discuss fidelity decay for perturbations with non-zero time average (Section 5.2) and then for perturbations with zero time average, resulting in the so-called fidelity freeze (Section 5.3). In Section 5.4 measures for composite systems are discussed, in particular purity. Section 5.5 discusses fidelity decay in systems with mixed phase space, for cases where the purely regular theory exposed in the previous subsections is not sufficient. If the dynamics is integrable it is useful to define a time-averaged perturbation V¯ (27),  1 t  ˜  (t) V¯ = lim dt V (t ) = lim . (176) t →∞ t 0 t →∞ t For sufficiently small 2 the above integral converges for some classical averaging time tave . Observe that V¯ is by construction a constant of motion, [V¯ , U0 ] = 0. For chaotic dynamics V¯ goes to zero in the semiclassical limit. For 9 This of course does not mean that the perturbation V˜ (t) in the interaction picture is time independent.

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T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

regular dynamics on the other hand there can be two different situations depending on the operator V¯ : (i) the typical case where V¯ = 0 and (ii) the special case where V¯ = 0, which gives rise to an entirely different behavior known as quantum freeze [16], for a general discussion of freeze see Section 2.2. If V¯ is nonzero the double integral of the correlation function C(t  , t  ) (14) will also grow quadratically with time and so one can also define an time-averaged ¯ correlation function C, 1 C¯ = lim 2 t →∞ t

 t 0

t

dt  dt  C(t  , t  ).

(177)

0

C¯ is the plateau value of the correlation function for t → ∞ (or the value around which the correlation function oscillates). Correlation functions in regular systems will typically have a plateau due to the existence of conserved quantities. Equivalently C¯ can be written in terms of the average perturbation operator V¯ , C¯ = V¯ 2  − V¯ 2 ,

(178)

i.e., C¯ is the variance of the time-averaged perturbation. For sufficiently small 2, the quantum correlation function can be replaced by its classical counterpart, i.e., the quantum operator V can be replaced by the classical observable v, such that C¯ is given by classical dynamics. The time scale on which C¯ converges is again tave . Physically, this is the time in which the classical correlation functions decay to their asymptotics. The linear response expression for fidelity (13) can now be written as F (t) = 1 −

ε2 ¯ 2 Ct + · · · , 22

(179)

valid for t > tave . The time scale for fidelity decay scales as ∼ 1/ε. Here, the quadratic decay is not a consequence of trivial conservation laws, as discussed in Section 2.1.3, but of the regular dynamics with non-decaying correlations. Also it must not be confused with the so-called Zeno decay, Section 2.1.1, which is also quadratic but happens on a very short time-scale on which the correlation function does not yet decay, regardless of the dynamics [66,67]. For a numerical comparison of Zeno time-scale and our quadratic decay; see Fig. 39. Of course one should keep in mind that the behavior of the correlation function (14) depends on the dynamics as well as on the perturbation V itself. For instance, even for integrable dynamics the correlation function can decay to zero for a sufficiently “random” perturbation. In such a case the fidelity decay in the linear response regime will be linear even for regular dynamics. Indeed, exponential decay of fidelity has been found in Ref. [163] also in a regular regime if a random transformation of the basis is made before applying the perturbation (making the perturbation effectively random). Note that such a random perturbation has no classical limit, nevertheless it could be important for quantum computing where typical errors acting on individual qubits may not have a classical limit [164] either. The application of perturbations which have no classical limit can result in a situation where the presence of symmetries can cause the fidelity decay to be exponential and slower than in the absence of symmetries [165]. A more detailed account of fidelity studies in the context of quantum computing is given in Section 8. Passing to action-angle operators will turn out to make derivations easier and will furthermore enable us to use classical action-angle variables in the leading semiclassical order, thereby expressing quantum fidelity in terms of classical quantities, even though in some cases quantum and classical fidelity may behave quite differently. Therefore, before proceeding with the evaluation of F (t) beyond the linear response let us have a look at the quantum action-angle operator formalism. 5.1. Action-angle operators Since we assume the classical system to be integrable (in the case of near-integrable systems at least locally, by KAM theorem) we can employ action-angle variables, {jk , k , k = 1 . . . d}, in a system with d degrees of freedom. In the present section, dealing with the regular regime we shall use lowercase letters to denote classical variables and capital letters to denote the corresponding quantum operators. For instance, the quantum Hamiltonian will be given as H (J,
) whereas its classical limit will be written as h(j,
).

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89

As our unperturbed Hamiltonian is integrable, it is a function of actions only, i.e. h0 = h0 (j). The solution of the classical equations of motion is simply j(t) = j,


(t) =
+ (j)t (mod 2 ),

(180)

with a dimensionless frequency vector (j) = jh0 (j)/jj. In Section 5.3 it will come handy to expand the classical limit v(j, ) of a perturbation operator V into a Fourier series,  vm (j)eim·
, (181) v(j,
) = m∈Zd

where the multi-index m has d components. The classical limit of the time-averaged perturbation V¯ is v¯ = v0 (j), i.e. only the zeroth Fourier mode of the perturbation survives time averaging. In quantum mechanics, one quantizes the action-angle variables using the EBK procedure (see e.g. Ref. [166]) where one defines the action (momentum) operators J and angle operators exp(im ·
) satisfying the canonical commutation relations, [Jk , exp(im ·
)] = 2mk exp(im ·
), k = 1, . . . , d. As the action operators are mutually commuting they have a common eigenbasis |n labeled by the d-tuple of quantum numbers n = (n1 , . . . , nd ), J|n = Jn |n.

(182)

Here Jn = 2{n + } is an eigenvalue of operator J in an eigenstate |n and 0 k 1 are the Maslov indices which are irrelevant for the leading order semiclassical approximation we shall use. It follows that the angle operators act as shifts, exp(im ·
)|n = |n + m. The Heisenberg equations of motion can be solved in the leading semiclassical order by using classical equations of motion, i.e. replacing quantum H0 (2n) − H0 (2(n − m)) with its classical limit m · (2n) and disregarding the operator ordering, J(t) = eiH0 t/2 Je−iH0 t/2 = J, eim·
(t) = eiH0 t/2 eim·
e−iH0 t/2 eim·(J)t eim·
,

(183)

in terms of the frequency operator (J). Throughout this paper we use the symbol for ‘semiclassically equal’, i.e. asymptotically equal to leading order in 2. Similarly, time evolution of any other observable is obtained to leading order by substitution of classical with quantal action-angle variables. For instance the perturbation V˜ (t) (181) is  V˜ (t) = eiH0 t/2 V e−iH0 t/2

vm (J)eim·(J)t eim·
. (184) m

5.2. Perturbations with non-zero time average In this section we assume V¯ = 0. We have seen (24) that the echo operator for sufficiently small ε can be written as 
 1 i , (185) (t)ε + (t)ε 2 Mε (t) ≈ exp − 2 2 where (t) (25) grows at most linearly with t. If V¯ = 0, the first term in the exponential will grow as (t) = V¯ t for times longer than the classical averaging time tave (176). Therefore, provided ε < 1, the term involving (t) will be 1/ε times smaller than the first one and can be neglected. Replacing (t) with V¯ t we write the fidelity amplitude as ¯

f (t) = e−iεt V /2 ,

t?tave .

(186)

As the average perturbation operator V¯ is diagonal in the eigenbasis of actions, V¯ |n = V¯n |n, the above expectation value is especially simple in the action eigenbasis |n,  f (t) = exp(−iεt V¯n /2)D (2n), D (2n) = n||n. (187) n

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T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

Now we make a leading order semiclassical approximation by replacing quantum operator V¯ by its classical limit v(j) ¯ and replacing the sum over quantum numbers n with the integral over classical actions j (note small-cap letter). By denoting with d (j) the classical limit of D (2n), we arrive at the fidelity amplitude, −d



f (t) 2

ε dd j exp −i t v(j) ¯ d (j). 2

(188)

This equation will serve us as a starting point for all calculations of fidelity decay in regular systems for non-residual perturbations. The replacement of the sum with the action space integral (ASI) is valid up to such classically long times ta , that the variation of the argument in the exponential across one Planck cell is small, ta =

1 ∼ 20 /ε. |jj v|ε ¯

(189)

Subsequently we shall see that the fidelity decays in times shorter than this, and thus the ASI approximation is justified. The ASI representation (188) will be now used to evaluate the fidelity for different initial states d (j). 5.2.1. Coherent initial states First we take the initial state to be a coherent state |j∗ ,
∗  centered at action and angle position (j∗ ,
∗ ). Expansion coefficients of a general coherent state (a localized wave packet) can be written as
d/4  2 1 1/4 ∗ ∗ ∗ |det | exp − (Jn − j ) · (Jn − j ) − in ·
, n|j ,
 =

22 ∗

∗

(190)

where  is a positive symmetric d × d matrix of squeezing parameters. The classical limit of D (187) is therefore d (j) = (2/ )d/2 |det |1/2 exp(−(j − j∗ ) · (j − j∗ )/2).

(191)

The ASI (188) for the fidelity amplitude can now be evaluated by the stationary phase method with the result [46], * + F (t) = exp −(t/r )2 ,

 1 r = ε

22 v¯  · −1 v¯ 

,

(192)

where the derivative of the average perturbation is v¯  =

jv(j ¯ ∗) , jj

(193)

and is evaluated at the position of the initial packet j∗ . Comparing the fidelity (192) with the linear response formula (179) we see that the average correlation function for a coherent initial state, C¯ = 21 2(¯v · −1 v¯  ), is proportional to 2 because the size of the packet scales with 2. We thus find a Gaussian decay of fidelity for coherent initial states in regular systems. This has very simple physical interpretation. As the fidelity is an overlap of the initial state with the state obtained after an echo, i.e. after propagation with Mε (t) = exp(−iV¯ (J)εt/2), we can see that the effective Hamiltonian for this evolution is V¯ (J). It depends only on action variables and therefore its classical limit v(j) ¯ generates a very simple classical evolution. Only the frequencies of tori are changed by the amount
 = ε¯v , while the shapes of tori do not change. This change in frequency causes the “echo” packet Mε (t)|(0) to move ballistically away from its initial position and as a consequence fidelity decays. The functional form of this decay is directly connected with the shape of the initial packet. For coherent initial states it is Gaussian while for other forms of localized initial packets it will be correspondingly different but with the same dependence of the regular decay time r on the ballistic separation “speed” v¯  and perturbation strength ε. Note that the decay of classical fidelity is more complicated, see Section 6.

T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

91

100

10-2

F(t)

10-4

tb

10-6

10-8

10-10 0

50

100

150

200 t

250

300

350

400

Fig. 26. Quantum fidelity for a coherent initial state of the regular kicked top (B.1) [from [46]]. The solid line shows the predicted Gaussian decay (195), the point symbols “+” show the numerical results. The vertical line indicates the theoretical beating time (194) which is particular to one degree of freedom systems.

In one dimensional systems (d = 1) another phenomenon will be observable. After long times the echo packet will make a whole revolution around the torus causing the fidelity to be large again. This will happen after the so-called beating time tb determined by the condition10 ε v¯  (j ∗ )tb = 2 , tb =

2

. v¯  ε

(194)

This beating phenomena is particular to one dimensional systems as in general the incommensurability of frequencies will suppress the revivals of fidelity in more than one degree of freedom (DOF) systems (i.e. tb for derivative of v¯ in each action direction would have to be the same), for a numerical example of a two DOF system see Ref. [46]. Strong revivals of fidelity have been discussed also in Ref. [167]. In case of commensurate frequencies one can have revivals also in many-dimensional systems, for an example of d = 2 dimensional Jaynes–Cummings model see Refs. [168,51]. In Ref. [169] quantum fidelity has been exactly calculated for a system with the Hamiltonian Hε = p2 /2 + g(t)q 2 /2 + ε/q 2 , with an arbitrary periodic function g(t), and it has been shown that there are perfect periodic recurrences of fidelity. To illustrate the above theory for fidelity decay we compare it in Fig. 26 with the results of numerical simulation. The model we choose is a kicked top with a unit-step propagator given in Eq. (B.1), using  = /2 and small  = 0.1 giving regular dynamics. Evaluation of the theoretical formula (192) in the limit of vanishing  gives for this particular perturbation the theoretical fidelity decay F (t) = exp(−ε2 St 2 y 2 (1 − y 2 )/8),

(195)

if y is the position of the initial coherent state on a sphere. Nice agreement of numerics (S = 100 and ε = 0.0025) with the theory can be seen as well as fidelity recurrence at the predicted time tb . 5.2.2. Random initial states In this section we shall consider random initial states, that is pure states whose expansion coefficients are independent Gaussian random numbers. In the semiclassical limit, i.e. for large Hilbert space dimension N, the expectation value of an echo operator is self averaging, meaning that the fidelity for one particular random state is equal to the fidelity averaged over the whole Hilbert space. Therefore, we calculate f (t) via Eq. (7) choosing the initial state as the density 10 In the presence of symmetries the beating time can be shorter than the one given by Eq. (194).

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1

F(t)

0.1

0.01

0.001 1

10

100

1000

Sεt Fig. 27. Power law decay of fidelity for the regular kicked top (B.1) [from [46]] and random initial state. The solid curve gives the results of a numerical simulation for  = 1/N, the dotted line gives a predicted asymptotic decay ∝ t −1 , and the wiggling dashed curve represents the numerics for a single random state, N = 100.

matrix  = 1/N. The expansion coefficients take the form D (J) = 1/N (187) and the corresponding classical quantity is d (j) =

(2 2)d , V

(196)

where we calculated the dimension of the Hilbert space N using the Thomas-Fermi rule with V being the volume of classical phase space. For short times the decay of fidelity is again quadratic (179), albeit with a different C¯ than for coherent states. For longer times the ASI (188) giving f (t) can be calculated using a stationary phase method if the phase ε vt/2 ¯ of the echo operator changes fast enough. This is valid for t > 2/ε. Note on the other hand, that there is also an upper limit for ASI determined by ta ∼ 1/ε (189). In contrast to the coherent initial state, we can now have more than one stationary point. If we have p points, j ,  = 1, . . . , p, where the phase is stationary, jv(j ¯  )/jj = 0, the ASI gives [46]



p ¯  )ε/2 − i } (2 )3d/2

2

d/2  exp{−it v(j f (t) = , (197)



¯  |1/2 V tε | det V =1 ¯  }kl = j2 v(j where {V ¯  )/jjk jjl is a matrix of second derivatives at the stationary point j , and  = (m+ − m− )/4 where m± are the numbers of positive/negative eigenvalues of the matrix V¯  . In above derivation we also assumed that phase space is infinite. In a finite phase space we shall have diffractive oscillatory corrections in the stationary phase formula, see numerical results below or Ref. [16]. It is interesting to note the power-law dependence on time and perturbation strength. In the simple case of one stationary point, fidelity will asymptotically decay as , -d (198) F (t)  2/(tε) , where the sign  will denote “in the asymptotic limit” throughout this section. An interesting, though still largely unexplored, question is what happens in the thermodynamic limit d → ∞? From the above formula we see that with increasing dimensionality d of a system the decay gets faster. This allows for a possible crossover to a Gaussian decay when approaching the thermodynamic limit. Such behavior has been observed in a class of kicked spin chains [42]. Agreement with Gaussian or exponetial decay beyond linear response is frequently observed also for finite d, e.g. in a spin model of quantum computation [43]. In Fig. 27 we show results of numerical simulation for the same regular kicked top used for coherent initial states (B.1). Note the oscillating decay due to finite phase space.

T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

5.2.3. Average fidelity for coherent states One might be also interested in fidelity averaged over the position of the initial coherent state,  (2 )d dd jF (t, j). F (t)j = V

93

(199)

The asymptotic decay of average fidelity will be dominated by regions in the action space where the decay F (t, j) is slowest. Denoting all j dependent terms by a non-negative scalar function g(j), the fidelity decay for a single coherent state can be written as F (t, j) = exp(−ε 2 t 2 g(j)/2) (192). For large ε 2 t 2 /2 the main contribution to the average will come from regions around zeros of g(j), where the fidelity decay is slow. In general there can be many zeros due to divergences in r (192), but for simplicity let us assume there is a single zero at j∗ of order . The asymptotic decay can then be calculated and scales as [170] 
2 d/ F (t)j  . (200) ε2 t 2 The asymptotic decay is therefore algebraic with the power depending on the number of degrees of freedom d and the order  of the zero. For infinite phase space  can only be an even number, whereas for a finite space  can also be odd. Note that for regular systems the fidelity averaged over the position of coherent states F (t)j is not necessarily equal to the fidelity for a random initial state (198).11 Whereas for fidelity decay of a random initial state zeros of v(j) ¯ matter, for F (t)j singularities of r (192) determine asymptotic decay (e.g. zeros of v¯  ). To illustrate the above points we calculated the average fidelity decay for a kicked top and two different perturbations. In the first case we take the system already used for coherent states (B.1), for which g(j ) = j 2 (1 − j 2 )/8 (195). We have three zeros, with the asymptotic decay given by the one of highest order,  = 2, √ 2

F (t)j  √ . (201) εt S As a second example we take the same regular kicked top system and parameters, only the perturbation is now in rotation angle (B.5), whereas before it was in the twist parameter . The function g is in this case g(j ) = (1 − j 2 )/2. As opposed to the previous case, we now have only two zeros of order  = 1 at j∗ = ±1. The asymptotic decay is therefore F (t)j 

1 . ε2 t 2 S

(202)

In Fig. 28 the results of a numerical simulation are compared with theory. Good agreement is observed with the asymptotic predictions (201) and (202). Strong revival of fidelity for the case of perturbation in the rotation angle (B.5) seen in the figure is particular to this perturbation because the beating time tb (194) does not depend on j. Generally such revivals are absent from F (t)j . We can therefore see that the asymptotic decay of average fidelity F (t)j is algebraic, the power being system dependent. 5.3. Perturbations with zero time average Above we considered general perturbations, for which the double correlation integral (13) in regular systems was growing quadratically with time, i.e. time-averaged perturbation V¯ is non-zero. In the present section we discuss the situation when this correlation integral does not grow with time (asymptotically for large times), that is the timeaveraged perturbation is zero, V¯ ≡ 0. This are the so called residual perturbations because V = Vres . Most of the results have been published in Ref. [16]. We shall heavily rely on general derivations presented in Section 2.2. As the short time fidelity is given by the operator (t) (31) and its norm does not grow with time, the fidelity will freeze at a constant value called the plateau (35). After a sufficiently long time it will again start to decay due to the operator (t) (32). The main tool for theoretical derivations will be similar as for non-residual perturbations, that is using action-angle variables and evaluating resulting semiclassical ASI using stationary phase method. 11 For chaotic systems the two averages of course agree.

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T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

1



0.1

0.01

0.001 10

100

1000

10000

t Fig. 28. Average fidelity decay F (t)j for a regular kicked top [from [170]]. For two different perturbations, Eq. (B.1) and Eq. (B.5), different asymptotic power law is obtained. Top set of curves is for a perturbation in  (B.1) and bottom is for a perturbation in the rotation angle (B.5). Full curves in both cases represent numerics while dotted lines are theoretical predictions (201) and (202) without free fitting parameters.

5.3.1. Coherent initial states Similarly as we expanded the perturbation V (181) into a Fourier series we can also expand the operator W (t) which is needed to calculate the freeze plateau (35). We write it in terms of action-angle operators, W = W (J,
) and then write the corresponding classical quantity w(j,
) as,  w(j,
) = wm (j)eim·
. (203) m =0

The expansion coefficients wm are easily expressed in terms of vm , wm = −i

vm . m·

The time dependent value of the plateau (35) is now in the leading semiclassical order . / . / fplat (t) exp −iεw(J,
+ t)/2 exp iεw(J,
)/2 .

(204)

(205)

For sufficiently large time, say t > t1 , the phase (J)t in the argument of the first exponential function will change over the initial state by more than 2 . In this case the averaging over initial states will yield the same result as averaging over time. Therefore, for t > t1 we can replace the time dependent fplat (t) with its time average,  1 t  dt f (t  ). (206) fplat lim t →∞ t 0 But averaging over time is equal to averaging over angle, so we can write the expression for fidelity plateau as  ε ε

 dd x

 fplat exp i w(J,
) exp −i w(J, x) . (207) 2 2 (2 )d This is a general expression for fplat valid for any initial state. Now we shall assume the initial state to be a coherent state. Let us first estimate the time t1 after which the above averaging is justified. We demand m
t1 > 2 , where
 is a frequency change over the packet and m is the mode number of certain Fourier mode (203) wm . The frequency change ∗ packet. √ For coherent over the initial packet is 
 ≈ 
j, where j k = jj (j )/jjk and
j is the size of the initial √ states (191) it is
j ≈ 2/ so that we get an estimate of t1 for coherent initial states, t1 ≈ /(m 2). A more

T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

95

rigorous derivation [16] shows that the Fourier mode wm decays with a Gaussian envelope on the time scale t1 , t1 = 

1 / 2. m · −1 T m 4

∝ 2−1/2 .

(208)

Therefore, all time dependent terms in fplat (t) decay with a Gaussian envelope on the above time scale t1 . After t1 , but before t2 when (t) becomes important, the fidelity is constant and we have a fidelity freeze. We proceed to evaluate the plateau value of this freeze fplat (207) for coherent initial states. We shall use the fact that the expectation value of an arbitrary quantity is to leading order equal to this quantity evaluated at the position of the packet, exp(−(iε/2)g(J,
)) exp(−(iε/2)g(j∗ ,
∗ )), for sufficiently smooth function g, provided that the size of the wave√ packet ∼ 2 is smaller than the oscillation scale of the exponential ∼ 2/ε, i.e. provided ε>21/2 . Then the squared modulus of fplat (207) reads as


 2

1

iε CIS d ∗

, Fplat d


exp − ,
) (209) w(j

2d

2 (2 ) where superscript CIS is abbreviation for coherent initial state, i.e. localized wave packet. This equation is the expression for the plateau for arbitrary residual perturbation. The integral over angles cannot be done in general but if ε < 2 then CIS we obtain only the linear response expression for the plateau is needed. Expanding Fplat CIS = 1 − Fplat

ε2 2

 , 2 CIS

CIS =



|wm (j∗ )|2 .

(210)

m =0

The integral in Eq. (209) can be done analytically if the perturbation w(j,
) (203) has a single nonzero Fourier mode, say w±m0 . In that case the integral gives ε

CIS Fplat = J02 2 |wm0 (j∗ )| , (211) 2 were J0 is the zero order Bessel function. The plateau for a single mode residual perturbation is therefore given by a Bessel function. As a simple illustration of the above theory we again take the kicked top with a one-step propagator given in Eq. (B.9). Using classical w(j, ) (B.11) and a single-mode formula (211) we get    1 − j ∗2 CIS 2 Fplat = J0 εS . (212) 2 sin(j ∗ /2) Comparison of theory with numerics is shown in Fig. 29. We can see that up to time t1 (208) quantum fidelity follows the classical one (circles in Fig. 29). After t1 the correspondence breaks as the classical fidelity decays as a power law, while quantum fidelity freezes. For details about classical fidelity see Section 6. The value of the plateau agrees with the linear response formula (210) for weak perturbations and with the full expression (212) for strong perturbation. Vertical chain lines show theoretical values of t2 (calculated later), which is the time when the plateau ends. Also, at certain times quantum fidelity exhibits resonances or strong revivals. These “spikes” occurring at regular intervals can be seen also in a long time decay of fidelity in Fig. 30. These are called the echo resonances and are particular to one-dimensional systems and localized initial packets, so that the variation of the frequency derivative  = d(j )/dj over the wave packet is small. Due to the discreteness of quantum energies, at certain times a constructive interference occurs, resulting in a revival of fidelity. These echo resonances must not be confused with the revivals of fidelity due to the beating phenomenon (194) which is classical, whereas echo resonances are a quantum phenomenon. The time of occurrence of a resonance tres is given by tres =

2

, 2

(213)

or at fractional multiples of basic tres . Echo resonances have been described and explained in detail in Ref. [16] and they can also be observed in the numerical results shown in Refs. [167,171]. For a more mathematically oriented derivation

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T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

1 0.95

F

0.9 0.85 0.8 0.75 (a)

S = 200, εS = 0.32

0.7 1

1/6

1/4 1/3 1/2

1

2/5

F

0.1 1/5 0.01 S = 1600, εS = 3.2

(b) 0.1

1

10

100

1000

t/S1/2 Fig. 29. Fidelity freeze for coherent initial states in a regular kicked top (B.9) [from [16]]. Symbols connected with dashed lines denote the corresponding classical fidelity while the solid line represents a quantum simulation. After the initial decay, when the quantum fidelity agrees with the classical one (until t1 ), quantum fidelity freezes at the plateau (horizontal chain line, Eq. (211)). This plateau lasts long (note the log scale) until at t2 (217), denoted by vertical chain line, decay starts again. In (b) we also indicate fractional k/p resonances with k/p marked on the figure.

1 S = 1600, εS= 0.064

F

0.1

0.01

0.001

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

ε2 S1/2 t Fig. 30. Long time ballistic decay after the freeze plateau for the regular kicked top (B.9) with coherent initial state and residual perturbation [from [16]]. Chain curves indicate theoretical Gaussian decay (216) with analytically computed coefficients. The decay is the same as for non-residual perturbations but with a “renormalized” perturbation strength ε 2 /2 and perturbation r¯ instead of v. ¯

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97

of the condition under which they occur see also Ref. [128]. The same resonant times are also found for revivals of the wave-packet after just forward evolution studied in Refs. [172,173], see also the review [174]. For the quantum kicked rotor in a regime of quantum resonance similar phenomenon as freezing in regular systems can be observed due to the formal similarity of propagator [175]. It is important to realize that freeze is pure quantum phenomenon; classical fidelity does not exhibit freeze (see Fig. 29) even though the perturbation is exactly the same as in the quantum case! The operator (t) determines the plateau. Long time decay on the other hand will be dictated by the operator R (t) (33). For long times, when this decay will take place, we can define a time-averaged operator R (32),  R (t) 1 t  ¯ R = lim dt R(t  ), (214) = lim t →∞ t →∞ t 0 t ¯ This approximation is justified, because the fidelity decay will happen on a long time scale and approximate (t) ≈ Rt. ∼ 1/ε2 , whereas the average of R converges in a much shorter classical averaging time tave . For R the semiclassical limit r can be calculated using the Poisson brackets instead of commutators, r = −{w, dw/dt}. When we average r over time, only the zeroth Fourier mode survives resulting in  (J)/t ¯r (J), r¯ (j) = − m · jj {|wm (j)|2 m · (j)}. (215) m =0

Provided ε/2 is sufficiently small (i.e. the plateau is high) the echo operator can be for long times approximated by ¯ Mε exp(−iε 2 Rt/22). Long time decay of fidelity can then be calculated by evaluating the ASI in analogy to the procedure developed for non-residual perturbations (188), replacing the perturbation ε with ε 2 /2 and v¯ with r¯ . Using formula F (t) = exp(−(t/r )2 ) (192), we obtain long-time fidelity decay for coherent states and residual perturbations,  82 1 F (t) exp{−(t/rr )2 }, rr = 2 . (216)  ε r¯ · −1 r¯ The semiclassical value of r¯ is given in Eq. (215) and its derivative is r¯ = jj r¯ . The decay time scales as rr ∼ 21/2 ε −2

and is thus smaller than the upper limit ta ∼ 20 ε −2 of the validity of the ASI approximation. Remember that the above formula is valid only if the plateau is close to F = 1, such that the term (t) can be neglected. For such small ε the crossover time t2 from the plateau to the long time decay can be estimated by comparing the linear response plateau √ (210) with√the long time decay (216), resulting in t2 ≈ rr ε CIS /2 ∼ 2−1/2 ε −1 . For stronger perturbations, namely up to ε ∼ 2 time t2 can be estimated as rr . We therefore have * ε + 1/2 t2 = min 1, CIS rr = min{const 21/2 ε −2 , const 2−1/2 ε −1 }. (217) 2

The theoretical prediction for the departure point t2 (217) of fidelity from the plateau is plotted as a vertical chain line in Fig. 29. In Fig. 30 we show results of long time numerical simulation for the kicked top (B.9). The agreement with theoretical Gaussian decay (216) is good. 5.3.2. Freeze in a harmonic oscillator The time t1 (208) when the plateau starts can diverge for systems with vanishing frequency derivatives . Such is the case for instance in harmonic oscillator where  = 0. One can show that in this case the plateau is not constant but oscillates and is in the linear response regime by factor 1/2 higher than in systems with  = 0. The full expression for the plateau for the harmonic oscillator and coherent initial state is [170] 
2 ε CIS Fplat = exp − har , (218) 2 where har is some time-dependent coefficient. This expression must be compared with the Bessel function (211) for systems with non-zero . Because Bessel functions decay asymptotically in an algebraic way, the plateau for large perturbations is smaller in harmonic oscillators than in general systems (note that for small perturbations it is larger). As opposed to general situation, for harmonic oscillator classical fidelity agrees with the quantum one despite the freeze. Because of inequality (78) there is also freeze in reduced fidelity and purity. The plateau in reduced fidelity scales the

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T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

same as for fidelity (218) whereas is does not depend on 2 for purity. The long time decay of fidelity or of reduced fidelity and purity after the plateau ends does not show any peculiarities for the harmonic oscillator. For further details as well as for a numerical example in the Jaynes–Cummings system see Ref. [170]. 5.3.3. Rotational orbits So far we have always talked about the operator V¯ being either zero or not, resulting in a qualitatively different decay of quantum fidelity. Similar behavior as for V¯ = 0 might occur for particular localized initial states provided the expectation value of V¯ and its powers are zero, V¯ n  ≡ 0, even if on the operator level we have V¯ = 0. Such situation might occur in regular systems having rotational orbits in classical phase space. For these we may have perturbations which have v ¯ class = 0 and correspondingly quantum freeze occurs. Note that for classical fidelity only the functional dependence will change from a Gaussian to a power-law as one puts the initial coherent state in the rotational part of phase space, in both cases depending on εt [63]. Quantum fidelity on the other hand decays as a Gaussian on a time scale r ∼ 1/ε in the case of V  = 0, while it displays a freeze and later decays on a much longer time scale rr ∼ 1/ε 2 for V  = 0. Freeze of quantum fidelity and correspondingly short correspondence between quantum and classical fidelity for an initial state put in the rotational part of phase space has been observed in Ref. [167]. Interesting behavior might emerge at borders between regions with V¯ n  = 0 and V¯ n  = 0, for instance close to separatrices that separate rotational parts of a phase space from the rest. The decay of quantum fidelity at such a border has been studied in Ref. [171]. While far away in the rotational part of phase space the freeze has been observed, even though on the operator level one has V¯ = 0, interesting and non-trivial behavior has been seen in the transition region. There fidelity decays as a power law, depending on the product εt, i.e. on a time scale  ∼ 1/ε. Such power-law decay is actually not a complete surprise, as the average fidelity (Section 5.2.3) also decays as a power-law due to regions of phase space with diverging time-scale r of Gaussian decay (192). Still, exact theoretical understanding of the fidelity decay close to separatrices is lacking. 5.3.4. Random initial states Similarly as for non-residual perturbations (Section 5.2.2) we again assume a finite Hilbert space of sufficient size, such that the difference between fidelity for one specific random state and an average over all states can be neglected. We 1 shall replace quantum averages with classical averages, g(J,
) V dd jdd
g(j,
). First we calculate the plateau. Calculating the plateau for coherent initial states, we argued that after time t1 (208) the fidelity is time-independent and we have a plateau. For random initial states the same happens, only the time t1 is different. One can use the same argumentation as for coherent states, taking into account that the effective size
j of random states is
j ∼ O(1), resulting in t1 ∼

1 ∼ ε0 20 . |m|

(219)

Compared to coherent states, t1 is now independent of 2 and is therefore smaller. After averaging time t1 we can use the general time-averaged expression for the value of the plateau (207), arriving at

  2




(2 )d iε dd
RIS d

fplat

d j

exp − w(j,
)

, (220) d V 2 (2 ) where RIS stands for random initial state. Interestingly, the plateau for a random initial state is the average plateau for a coherent state squared, where averaging is done over the position of the initial coherent state. If we denote the plateau CIS (j∗ ) (209), then the plateau for a random initial state F RIS is for a coherent initial state centered at j∗ by Fplat plat

2

(2 )d 



RIS CIS

Fplat dd jFplat

(j) .

V

(221)

One word of caution is necessary though concerning the above formula. The w(j,
) (204) needed in the calculation of CIS (j∗ ) (209) diverges at points in phase space where m · (j) = 0. For a coherent initial state this was no real problem Fplat as divergence would occur only if we placed the initial packet at such a point. For random initial state though, there is an average over the entire action space in the plateau formula (221) and if there is a single diverging point somewhere in

T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

99

1 S=1600, εS=0.32

0.8

F

0.6

0.4

0.2

0 1

10

100 t

1000

10000

Fig. 31. Fidelity plateau for a random initial state for the regular kicked top (B.9) [from [16]]. The chain line shows the theoretical value of the plateau as computed from Eq. (222). Due to near resonant terms with small m · (j) the plateau is lower than in Fig. 29, even though the perturbation is of the same strength.

the phase space it will cause divergence. The solution to this is fairly simple. If we compare the semiclassical formula for wm (204) with its quantum version for matrix elements Wj k (30), we see that there are no divergences in Wj k as the matrix element Vj k is for residual perturbations zero precisely for those j, k which would give divergence due to RIS over j is actually an approximation for a vanishing Ej − Ek . Remembering that the integral in the formula for Fplat sum over 2n the remedy is to retain the original sum over the eigenvalues of the action operator excluding all diverging terms. The formula for the plateau in the case of such divergences is therefore

2



m· (2n) =0



1 RIS

CIS Fplat

Fplat (2n)

. (222) N



n In the summation over n we exclude all terms in which, for any constituent Fourier mode m, we would have m·(2n)=0. To summarize, if there are no points with m ·  = 0 in phase space on can use the integral formula (221), otherwise a summation formula (222) must be used. To illustrate the summation formula for the plateau, we use the same kicked top as for coherent states (B.9). As one can see in Fig. 31 the agreement with numerics is very good. Compared to Fig. 29(a) for coherent state (where the same perturbation strength is used) the plateau is here much lower due to near-diverging terms. For a numerical example of a system with no singularities, where the integral formula (221) can be used, see Ref. [16]. For long times, after the plateau ends at t2 ∼ 1/ε, the fidelity again starts to decrease due to increasing (t). The calculation using ASI is analogous to the case of non-residual perturbations, Eq. (197), one just has to replace v¯ with r¯ and ε with ε2 /2. We shall only give the result,
 tran d 2 F (t)  , tran = const × 2 . (223) t ε For comparison with numerical simulation see Ref. [16]. Finally, we make a short comment on the behavior of average fidelity (averaged over positions of coherent states) for residual perturbations. For non-residual perturbations we have seen in Section 5.2.3 that the asymptotic decay is determined by the divergences of the decay time r . For residual perturbation the situation is very similar. During the plateau, until time t2 , averaging of the plateaus for coherent states gives the plateau for random states. Thus in the average fidelity one still has a plateau of order 1 − F ∼ (ε/2)2 (in the linear response). After the plateau, we have to average Gaussian decay (216), where in general divergences of rr will cause a power-law asymptotic decay. The power will depend on the nature of the divergence and therefore on the perturbation The only difference to general

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T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

perturbations is that the decay depends on the parameter ε 2 t and not on εt as for non-residual perturbations. Note though that some researchers [176] have predicted a universal t −3d/2 decay of average fidelity for residual perturbations, that is with the power depending only on the dimension d and not on perturbation specifics. The reason for the discrepancy between these results is presently unclear. 5.4. Composite systems and entanglement In case of composite systems one might not be interested in the influence of perturbations on the whole system but just on some “central” subsystem of interest. Stability of a subsystem is quantified by reduced fidelity FR (t) (72) and by echo purity FP (t) (74). We shall mostly discuss coherent initial states, i.e. a direct product of localized packets for central subsystem and environment, for which semiclassical treatment is the simplest. At the end we shall also comment on other possible choices of initial states. The calculation of the reduced fidelity is quite analogous to the one for fidelity (192). We again employ a classical timeaveraged perturbation v(j), ¯ where the action vector j has dc components jc for the central system and de components e j for the environment, j = (jc , je ), if dc and de are the number of DOF of the central subsystem and environment, respectively. The shape of the initial packet is given by squeezing matrices c and e for the central subsystem and the environment, respectively. The derivatives of the classical v¯ with respect to actions can be written as v¯  = (¯vc , v¯ e ),

v¯ c =

j¯v(j∗ ) , jjc

v¯ e =

j¯v(j∗ ) . jje

(224)

Performing Gaussian integrals in action space gives reduced fidelity [50] 
2 ε FR (t) = exp − 2 C¯ R t 2 , 2

1 ¯ c ). C¯ R = 2(¯vc −1 c v 2

(225)

/ . ¯ c + v¯ e −1 ¯ e A single parameter C¯ R determines the reduced fidelity decay. This must be compared with C¯ = 21 2 v¯ c −1 c v e v (192) which governs fidelity decay. Compared to fidelity, reduced fidelity depends understandably only on perturbation derivatives with respect to actions of the central system. But note that the average coupling v¯ depends also on the dynamics of the environment and so C¯ R itself does depend on the environment. The derivation of echo purity (or in special case of uncoupled unperturbed dynamics of purity) is a little more involved. We have to calculate averages of the form
  εt FP (t) 2−2d dj d˜j exp −i  d (j)d (˜j), 2 c e c e e c e ˜ ˜  = v(j ¯ , j ) − v( ¯ j , j ) + v( ¯ j , ˜j ) − v(j ¯ c , ˜j ), (226) with d (j) being the initial density (191) of the wave packet. When expanding v(j) ¯ around the position j∗ of the initial packet, the constant term, the linear terms as well as the diagonal quadratic terms cancel exactly, regardless of the position of the initial packet. The lowest non-vanishing contribution comes from the non-diagonal quadratic term −i

εt c ˜c e  (j∗ )(je − ˜j )], [(j − j ) · v¯ce 2

(227)

 is a d × d matrix of mixed second derivatives of v¯ evaluated at the position of the initial packet, where v¯ce c e  )kl = (v¯ce

j2 v¯ . j(jc )k j(je )l

(228)

After performing integrations we get the final result (see Ref. [177] for details) FP (t) = 

1 det(1 + (εt) u) 2

,

 −1  u = −1 c v¯ ce e v¯ ec ,

(229)

T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

101

100

F,FR,I

I

10-1

F FR

10-2 10

100

1000

10000

t Fig. 32. Decay of fidelity F (t), reduced fidelity FR (t) and purity I (t) for integrable dynamics and localized initial packets [from [170]]. The theoretical Gaussian decay for the fidelity (192) and the reduced fidelity (225) overlaps with the numerics (symbols). Purity decays asymptotically as a power law (229) and also nicely agrees with the numerics (full curve), until a finite size saturation level is reached. Note that purity decays on a much longer time scale than fidelity and reduced fidelity due to independence of its decay time on the small parameter 2.

 and its transpose v¯  . Note that the matrix u is a classical quantity (independent where u is a dc × dc matrix involving v¯ce ec of 2) that depends only on the observable v¯ and on the position of the initial packet. For small εt we can expand the determinant and we get initial quadratic decay in the linear response regime (special case of Eq. (84)),

FP (t) = 1 − 21 (εt)2 tr u + · · · .

(230)

The most prominent feature of the expression (229) for the echo purity (or purity) decay for initial product wave packets is its 2 independence. In the linear response calculation this 2-independence has been theoretically predicted in Ref. [162] as well as numerically confirmed [50]. We see that the scaling of the decay time P of FP (t) is P ∼ 1/ε. This means that echo purity will decay on a very long time scale and so localized wave packets are universal pointer states [178], i.e. states robust against decoherence [91]. For large times we use the fact that det(1 + zu) is a polynomial in z of order r = rank(u) and find asymptotic power law decay FP (t)  const(εt)−r . Note that the rank of u is bounded by the smallest of the subspace dimensions, i.e. 1 r  min{dc , de }, since the definition (74) is symmetric with respect to interchanging the roles of the subspaces “c” and “e”. Let us give two simple examples: (i) For dc = 1 and for any de we shall always have asymptotic power law decay with r = 1. If a single DOF of the subsystem “c” is coupled √ with all DOF of the subsystem “e”, e.g. v¯ = j c ⊗ (j e1 + j e2 + · · ·), then |v¯  |2 ∝ de and we have FP (t)  1/(εt de ); (ii) Let us consider a multidimensional system where the matrix u is of rank one so it can be written as a direct product of two vectors,u = x ⊗ y. The determinant occurring in FP (t) is det(1 + (εt)2 u) = 1 + (εt)2 x · y. Such is the case for instance if we have a coupling of the same strength between all pairs of DOF. The dot product is then x · y ∝ dc de and √ we find FP (t)  1/(εt dc de ). The power of the algebraic decay is independent of both dc and de . The bottom line is, that the power law of the asymptotic decay depends on the perturbation. In Refs. [124,179] the author predicted ∼ t −d decay of purity for intermediate times and universal ∼ t −2d decay for large times. While the intermediate decay is in accord with our theory, its long time decay is not. For numerical examples where asymptotic decay of purity exhibits different powers see Ref. [177]. To demonstrate the theory of reduced fidelity and echo purity decay we again take two coupled kicked tops, using a regular propagator in Eq. (B.18). We take S = 100, coupling ε = 0.01 and coherent initial state. As the unperturbed dynamics is uncoupled FP (t) is actually equal to purity I (t). The results of the numerical simulation are plotted in Fig. 32 together with the theoretical predictions (225) for reduced fidelity and for purity (229). No fitting is involved. In the above paragraph we have discussed coherent initial states. What about other choices? We shall consider two possibilities: (i) superposition of two coherent states—so called cat states; and (ii) random initial states. In both cases

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we assume that the unperturbed evolution is uncoupled, so echo purity equals purity. Let us first describe the decay of purity I (t) for cat states, |(0) =

√1 (|c1 (0) + |c2 (0)) ⊗ |e (0), 2

(231)

where the initial positions of the wave packets c1 (0), c2 (0) and e (0) are j∗c1 , j∗c2 and j∗e , respectively. Assuming that the initial packets of the central subsystem are orthogonal, the initial reduced density matrix is c (0) = 21 (|c1 (0)c1 (0)| + |c2 (0)c2 (0)| + |c2 (0)c1 (0)| + |c1 (0)c2 (0)|),

(232)

and represents coherent superposition of two macroscopically separated packets—superposition of “dead and alive cat” [180]—see also a popular account in Ref. [91]. What we would like to illustrate is decoherence. Decoherence is a process where off diagonal matrix elements of the initial reduced density matrix disappear due to the coupling with the environment, so that the resulting reduced density matrix mix c (dec ) is a statistical mixture of two diagonal contributions only, 1 mix c (dec ) = 2 (|c1 (dec )c1 (dec )| + |c2 (dec )c2 (dec )|),

(233)

with dec being decoherence time. Here we already used the fact that purity for individual localized packets decays on a long 2 independent time scale (229) and because we expect dec to be much shorter, we can use propagated packets as pointer states. During the process of decoherence purity will decay from I (0) = 1 for the initial reduced density matrix c (0) to I (dec ) = 1/2 for decohered matrix mix c (dec ). Using similar techniques as for the calculation of echo purity for individual coherent states (229), one can also calculate purity for cat states, the result being [181] I (t) = 41 (I1 (t) + I2 (t) + 2Fe (t)),

(234)

where I1,2 (t) are purities (229) for individual constituent coherent states, that is for c1,c2 (0) ⊗ e (0), while Fe (t) is a cross-correlation quantity similar to reduced fidelity. Explicitly, it is given by  Fe (t) = exp(−t 2 /2dec ), dec = 2/Ce /ε,  ∗c1 Ce = 21 (v¯e (j∗c1 ) − v¯e (j∗c2 ))−1 ) − v¯e (j∗c2 )). (235) e (v¯ e (j Decoherence for regular systems can now be understood from the very different time scales on which I1,2 and Fe decay. Fe (t) decays fast as a Gaussian on an 2-dependent time scale while I1,2 (t) decay slowly on an 2-independent time scale. At the point when Fe (t) becomes negligibly small, purity will be simply I (t) ≈ (I1 (t) + I2 (t))/4 and therefore approximately equal to 1/2. The decay of Fe (t) therefore signifies decoherence with the decoherence time scale dec being simply equal to the decay time of Fe (t) (235). On the other hand, the decay of I1,2 (t) is a signal of relaxation of individual packets to equilibrium. In the simple case when v¯ has coupling terms between all pairs of DOF  one can see [181] that Ce scales as Ce ∼ dc de (j∗c1 − j∗c2 )2 and therefore the decoherence time scales as dec ∼ 2/(dc de )/(ε|j∗c1 − j∗c2 |). It therefore goes to zero in any of the following four limits: the semiclassical limit 2 → 0, the thermodynamic limit of large environment and/or large central subsystem, dc,e → ∞, the limit of macroscopic superpositions, i.e. of large packet separation, |j∗c1 − j∗c2 | → ∞ and finally the limit of strong coupling. Note that these are exactly the limits one needs to explain the lack of cat states in classical macroscopic world of large objects! One should note that irreversible decoherence is possible only in the thermodynamic limit. For a finite integrable system there will be partial revivals of purity at large times. This revivals get less prominent and happen at larger times with increasing dimensionality d or decreasing 2. Also, due to high sensitivity of systems to perturbation, reversing the process of decoherence will be in practice close to impossible. Therefore, for all practical purposes one can observe decoherence also in a sufficiently large but finite regular system. For the theory presented here (234) we needed that the average perturbation V¯ converged, i.e. time is larger than the classical averaging time tave . But for sufficiently separated packets decoherence time dec will eventually get very small [182–184], even smaller than tave . We can circumvent this difficulty by noting that for very short times one can instead of the time-averaged V¯ use just an instantaneous time independent V (0). Then our theory can again be used [181]. The theory developed here therefore applies to truly macroscopic superpositions of localized wave packets (for which dec is very small) regardless whether the dynamics are regular or chaotic.

T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

103

1 0.8

I(t)

0.6 0.4 0.2 0 1

10

100

1000

t Fig. 33. Decay of purity I (t) (234) in a regular system for an initial cat state (231) [from [181]]. The dotted curve shows the theoretical prediction (234) with no fit parameters, and the full curve shows numerics. The initial decay of purity until dec ∼ 30 is given by a fast Gaussian decay of Fe (t) (235). At the end of decoherence purity is I (dec ) = 1/2, then the relaxation of individual packets begins, visible as a slow power-law decay of purities I1,2 (t) (229).

Let us now demonstrate the decay of purity for cat states (234) by a numerical experiment. We take a system of two coupled anharmonic oscillators with 2 = 1/500 and coupling ε = 0.01, for details see Ref. [181]. The results are shown in Fig. 33, where one can nicely see two different regimes of purity decay. In the first one fast Gaussian decoherence happens until purity reaches I = 1/2 at dec ∼ 30, and in the second one a slow algebraic relaxation takes place. In the context of quantum information perhaps more interesting than coherent states are random initial states. For regular dynamics echo purity will initially decay quadratically, but on 2 dependent scale that is typically much shorter than for coherent initial states. Echo purity for random states in a spin chain (a system having no classical limit) has been studied in Ref. [49]. In the linear response regime a quadratic decay has been observed and interestingly the full functional form of the decay approached a Gaussian in the thermodynamical limit. Quadratic decay of concurrence, a quantity describing entanglement for mixed states, has been observed for an integrable spin chain [110]. In Ref. [185] the initial entanglement production rate (time derivative of purity) as well as its asymptotic saturation value have been studied for random state averaging and for coherent state averaging. In the integrable regime the initial entanglement production rate is much larger for random state averages than for coherent state averages due to a different 2 dependence of the corresponding correlation functions. Decoherence in a system of two 21 spins coupled to a bath composed of a number of spin 21 particles has been studied in Ref. [186]. It has been numerically found that decoherence is faster when the internal bath dynamics is chaotic than when it is regular. Interestingly, in a certain regime of parameters decoherence decreases by increasing coupling strength to the bath. An interesting question is to what extent the decay of e.g. purity can be described classically. The classical analog of decoherence has been considered in Refs. [187–190]. Using a perturbative approach the influence of the type of the perturbation and of the dynamics on the quantum-classical correspondence were explored, see also Ref. [191] for a study of quantum-classical correspondence of entanglement in coupled harmonic oscillators. We shall not discuss the decay of reduced fidelity or of echo purity in the case of freeze. Based on the general inequality between the three quantities (78), we can immediately state that reduced fidelity and echo purity will also exhibit freeze. For an example of purity freeze in a harmonic oscillator see Ref. [170] and also Section 5.3.2. 5.5. Mixed phase space For classically mixed situation, with coexisting regular and chaotic regions in phase space, much less is known than for purely regular (Section 5) or chaotic (Section 3) dynamics. If a localized packet is started completely within a region of phase space having regular/chaotic structure one can use the appropriate theory for quantum fidelity. One should be cautious though as there can be different time-scales involved that are not present in purely regular or chaotic situations. The behavior of fidelity in the border region between regular and chaotic components might be particularly

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tricky. Two effects can be important at such a border: (i) classical dynamics in border region is “sticky”, i.e. diffusive, causing a slow power-law decay of various classical quantities [192–194]. Here the value of 2 might be very important for the impact this classical phenomenon has on quantum fidelity decay. On the other hand, there are also pure quantum effects that will influence fidelity decay, (ii) the transition between V¯  = 0 for regular region (in typical situation) and V¯  → 0 in chaotic region occurs at the border. We have seen that for regular dynamics the fidelity typically decays as a power-law at points with vanishing average perturbation (actually vanishing v¯  ), for instance around separatrices. Similarly, average fidelity (200) also has a power-law decay. Therefore, both effects suggest a power-law decay of quantum fidelity at the border between chaotic and regular regions of phase space. The fidelity decay in such region has been numerically studied in Ref. [195]. They were able to fit a two-parameter power-law formula to the numerically obtained fidelity decay, although the results were less than convincing as the power-law covered less than one order of magnitude. To numerically evaluate quantum fidelity for systems with mixed classical phase space on can use the dephasing representation. In Refs. [57,58] it has been shown that the dephasing representation works well in such situations. Other numerical results are reported in Ref. [148]. Another point worth mentioning is the actual transition from regular to chaotic behavior as some parameter of the system is varied. Due to finite 2 the transition will generally not happen at the same parameter value as for classical dynamics. In particular, in Ref. [105] the parameter region of weak chaoticity in a saw-tooth map has been studied and deviations from the Fermi golden rule and Lyapunov regime of quantum fidelity decay have been observed. The authors studied fidelity decay around the transition point between the Fermi golden rule and Lyapunov regime, occurring at ε ∼ εr ∝ 2. By increasing ε/2 the deviation from the Fermi golden rule starts first, after which the decay rate of quantum fidelity approaches the classical Lyapunov exponent in an oscillatory way. The reason for this transitional behavior are strong fluctuations in the fidelity amplitude, causing deviations between |f |2 and F . 6. Classical echo-dynamics In this section we shall define the corresponding classical quantity, characterizing classical Loschmidt echoes, namely the classical fidelity. One can use a strategy which is completely analogous to the quantum case to derive a theoretical description of the decay of the classical fidelity. The results found here may be of interest for themselves, or can serve as a reference for the quantum results. For example, within the time scale of the order of Ehrenfest time (∼ log 2 for chaotic systems), classical fidelity and quantum fidelity should strictly agree. Classical fidelity has been considered in several papers. In [46] a definition and a linear response treatment has been given. Benenti and Casati [59] considered it in the context of quantum classical correspondence in a diffusive system with dynamical localization. Eckhardt [60] treated its generalization to stochastic flows, whereas Benenti et al. [63,61] considered the asymptotic decay of the classical fidelity, both in the case of regular and chaotic dynamics. In Refs. [62,196] a theory has been developed for the short-time regime of classical fidelity decay and its relation to the Lyapunov spectrum. The classical fidelity Fcl (t) can be defined as  dxε (x, t)0 (x, t), (236) Fcl (t) = 

where the integral is extended over the phase space, and 0 (x, t) = Uˆ 0t (x, 0),

ε (x, t) = Uˆ εt (x, 0)

(237)

give the classical Liouvillean evolution for time t of the initial phase space density (x, 0). In order to have Fcl (t) = 1, we have to require that the phase space density is square normalized dx2 (x, 0) = 1. We note that classical dynamics of L2 phase space densities is unitary due to phase volume conservation, i.e. the Liouville theorem. Therefore, we can also build a classical echo picture and form a unitary classical echo propagator that composes perturbed forward evolution with the unperturbed backward evolution which is also unitary and is given by Uˆ E (t) = Uˆ 0† (t) Uˆ ε (t).

(238)

Writing the phase space density after an echo as E (x, t) = UE (t)0 (x, 0)

(239)

T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

we rewrite the classical Loschmidt echo as  Fcl (t) = dxE (x, t)0 (x, 0).

105

(240)



The above definitions (236) and (240) can be shown to correspond to the classical limit of quantum fidelity, if written in terms of Wigner functions [46,47]. In the ideal case of a perfect echo (ε = 0), fidelity does not decay, Fcl (t) = 1. However, due to chaotic dynamics, when ε = 0 the classical echo decay sets in after a time scale 1  tε ∼ ln , (241)

ε required to amplify the perturbation up to the size  of the initial distribution. Thus, for t?tε the recovery of the initial distribution via the imperfect time-reversal procedure fails, and the fidelity decay is determined by the decay of correlations for a system which evolves forward in time according to the Hamiltonians H0 (up to time t) and Hε (from time t to time 2t). This is conceptually similar to the “practical” irreversibility of chaotic dynamics: due to the exponential instability, any amount of numerical error in computer simulations rapidly erases the memory about the initial distribution. In the present case, the coarse-graining which leads to irreversibility is not due to round-off errors but to a perturbation in the Hamiltonian. Another somehow related concept named quantum-classical fidelity has recently been introduced and studied [197], which measures L2 distance between the Wigner function and the corresponding classical phase space density as a function of time. It is a very suitable tool for precise definitions of various Ehrenfest time scales and a detailed study of quantum-classical correspondence. Quantum-classical fidelity can also be interpreted as a “classical” fidelity if the Plack constant 2 is treated as a perturbation. 6.1. Short time decay of classical fidelity The analysis presented below follows Refs. [46,62]. The propagation of classical densities in phase space is governed by the unitary Liouville evolution Uˆ ε (t) d ˆ Uε (t) = Lˆ Hε (x,t) Uˆ ε (t), dt

(242)

where Lˆ A(x,t) = (∇A(x, t)) · J∇, A is any observable, and Hε (x, t) = H0 (x, t) + εV (x, t),

(243)

is a generally time-dependent family of classical Hamiltonians with perturbation parameter ε. The matrix J is the usual symplectic unit. Similarly, dUˆ ε† (t)/dt = −Uˆ ε† (t)Lˆ Hε (x,t) . Using Eqs. (242), (243), and (238) and writing Uˆ ε (t) = Uˆ 0 (t)Uˆ E (t) we get d ˆ UE (t) = {Uˆ 0† (t)Lˆ εV (x,t) Uˆ 0 (t)}Uˆ E (t). dt

(244)

Classical dynamics have the nice property that the evolution is governed by characteristics that are simply the classical phase space trajectories, so the action of the evolution operator on any phase space density is given as Uˆ 0 (t) =  ◦ −1 t , where −1 denotes the backward (unperturbed) phase space flow from time t to time 0. Similarly, the backward evolution t is given by Uˆ 0† (t) =  ◦ t , where t represents the forward phase space flow from time 0 to time t. Here and in the following we assume the dynamics to start at time 0. We note that echo-dynamics (238) can be treated as Liouvillean dynamics in the interaction picture, since {Uˆ 0† (t)Lˆ A(x,t) Uˆ 0 (t)}(x) = Uˆ † (t)(∇x A(x, t)) · J∇x (−1 t (x)) 0

= (∇t (x) A(t (x), t)) · J∇t (x) (−1 t (t (x))) ˆ = (∇x A(t (x), t)) · J∇x (x) = {LA( (x),t) }(x). t

(245)

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T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

In the last line we used the invariance of the Poisson bracket under the flow. This extends Eq. (244) to the form (242) d ˆ UE (t) = Lˆ HE (x,t) Uˆ E (t), dt

(246)

where the echo Hamiltonian is given by HE (x, t) = εV (t (x), t).

(247)

The function HE is nothing but the perturbation part εV of the original Hamiltonian, which, however, is evaluated at the point that is obtained by forward propagation with the unperturbed original Hamiltonian. It is important to stress that this is not a perturbative result but an exact expression. Also, even if the original Hamiltonian system was time independent, the echo dynamics obtains an explicitly time dependent form. Trajectories of the echo-flow are given by Hamilton equations x˙ = J∇HE (x, t).

(248)

At this point we limit our discussion to time independent Hamiltonians and perturbations. The slightly more general case of periodically driven systems reducible to symplectic maps shall be discussed later. Inserting (247) into Eq. (248) yields x˙ = εJ∇x V (t (x)) = εJMtT (x)(∇V )(t (x)).

(249)

Here we have introduced the stability matrix Mt (x), [Mt (x)]i,j = jj [t (x)]i . 6.1.1. Linear response regime for Lipschitz continuous initial density Let us first discuss the classical echo dynamics in the linear response regime, i.e. in the case where the total echo displacement  t  t
x(t) = x˙ dt  = ε dt  JMtT (x)(∇V )(t  (x)) (250) 0

0

is small compared to all other classical phase space scales. Writing classical fidelity in the symmetric representation we immediately arrive at the simple linear response result  Fcl (t) = dx(x +
x(t)/2)(x −
x(t)/2) 
2   t 1 dx ∇(x) · dt  JMtT (x)(∇V )(t  (x)) . (251) = 1 − ε 2 dt  4  0 Now it’s easy to specialize the above formula to the cases of chaotic and regular dynamics. In the case of chaotic dynamics, the stability matrix Mt (x) will after a short time t expand any initial vector with exponential exp( max t) where max is the maximal Lyapunov exponent of the flow. Since the time integral of an exponential is still an exponential, the expression in the big bracket on the RHS of Eq. (251) can be written as 2C[] exp( max t) where C[] is a constant depending only on the initial density. Hence Fclch (t) = 1 − ε 2 C[]2 exp(2 max |t|) + O(ε 4 ).

(252)

For sufficiently strong perturbations, ε > 2, and typical initial packets quantum fidelity decays in a similar way [73]. On the other hand, for regular dynamics, the stability matrix is of parabolic type (all eigenvalues 1 but having defective Jordan canonical form) so it always stretches an arbitrary initial vector linearly in time and Fcl (t) = 1 − ε 2 C  []2 t 2 + O(ε 4 ), reg

(253)

where C  [] is another constant depending on the initial density  only. However, please note that in deriving the expression (251) we have assumed that the initial density (x) was smooth and differentiable. In fact, the above result still holds for general Lipschitz continuous densities [196].

T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

107

6.1.2. Linear response regime for discontinuous initial density It is interesting though, that the result is quite different for discontinuous initial densities, like for example characteristic functions on sets in phase space which may be of considerable interest in classical statistical mechanics. In such a general case it is useful to rewrite the classical fidelity as  1 Fcl (t) = 1 − [(x) − (x +
x(t))]2 . (254) 2  This representation is a trivial consequence of the square normalizability of phase-space density and Liouville theorem for the echo dynamics x +
x(t). Now it is easy to see that for Lipschitz continuous densities the previous results (252) and (253) follow, while for discontinuous densities the main contribution to the integral, for small ε, i.e. for small |
x|, comes from the set of phase space points x where (x) is discontinuous, where the contribution is proportional to discontinuity gap times |
x|, namely  dxD (x)|
x(t)|, (255) Fcl (t) = 1 − 

where D (x) is a certain distribution taking account for the discontinuity gaps of the density (x). So, the main distinction from the continuous case is that the fidelity drop is proportional to the length of the echo displacement |
x| instead of its square. Since the latter asymptotically expands as e max |t| or |t| for chaotic and regular dynamics, respectively, we obtain universal linear response formulae for fidelity decay of discontinuous densities Fclch,dis (t) = 1 − |ε|C  [] exp( max t) + O(ε 2 ), reg,dis

Fcl

(t) = 1 − |ε|C  []|t| + O(ε 2 ),

where C  [] and C  [] are constants depending only on the details of the initial density . 6.1.3. General (multi-)Lyapunov decay for chaotic few body systems From now on we assume that the flow t has very strong ergodic properties, e.g. it is uniformly hyperbolic and Anosov, and consider to what extend can classical echo dynamics be explored in order to understand classical fidelity decay beyond the linear response approximation. To understand the dynamics (249) we need to explore the properties of Mt . We start by writing the matrix MtT (x)Mt (x) =



e2 j t dj2 (x, t)vj (x, t) ⊗ vj (x, t)

(256)

j

expressed in terms of orthonormal eigenvectors vj (x, t) and eigenvalues dj2 (x, t) exp(2 j t). After the ergodic time te necessary for the echo trajectory to explore the available region of phase space, Oseledec theorem [198] guarantees that the eigenvectors of this matrix converge to Lyapunov eigenvectors being independent of time, while dj (x, t) grow slower than exponentially, so the leading exponential growth defines the Lyapunov exponents j . Similarly, the matrix Mt (x)MtT (x) =



e2 j t cj2 (xt , t)uj (xt , t) ⊗ uj (xt , t),

(257)

j

where xt = t (x), has the same eigenvalues [cj2 (xt , t) ≡ dj2 (x, t)], and its eigenvectors depend on the final point xt only, as the matrix in question can be related to the backward evolution. The vectors {uj (xt )}, {vj (x)}, constitute left, right, part, respectively, of the singular value decomposition of Mt (x), so we write for t?te Mt (x) =

N  j =1

exp( j t)ej (t (x)) ⊗ fj (x)

(258)

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assuming that the limits ej (x) = limt → ∞ cj (x, t)uj (x, t), fj (x) = limt → ∞ dj (x, t)vj (x, t) exist. Rewriting Eq. (249) by means of Eq. (258) we obtain x˙ = ε

N 

exp( j t)Wj (t (x))hj (x),

(259)

j =1

where hj (x) = Jfj (x), and introducing new observables Wj (x) = ej (x) · ∇V (x).

(260)

Note that for general hyperbolic systems the signs of the vector fields ej (x), fj (x) may actually not be unique, so they rigorously exist only as director fields, i.e. rank one tensor fields. But since from a physics point of view this represents only a technical difficulty [196], we shall ignore the problem in the following discussions. For small perturbations the echo trajectories remain close to the initial point x(0) for times large in comparison to the internal dynamics of the system (te , Lyapunov times, decay of correlations, etc), and in this regime the echo evolution can be linearly decomposed along different independent directions hj (x(0)) x(t) = x(0) +

N 

yj (t)hj (x(0)).

(261)

j =1

For longer times, the point x(t) moves away from the initial point, but the dynamics is still governed by the local unstable vectors at the evolved point. Therefore, the decay of fidelity is governed by the spreading of the densities along the conjugated unstable manifolds defined by the conjugate vector fields hj (x) = J fj (x). Inserting (261) into (259) we obtain for each direction hj y˙j = ε exp( j t)Wj (t (x)).

(262)

For stable directions with j < 0, after a certain time, the variable yj becomes a constant of the order ε. For unstable directions with j > 0, we introduce a new variable zj as yj = ε exp( j t)zj and rewrite the above equation as z˙ j + j zj = Wj (t (x)).

(263)

The right hand side of this equation is simply the evolution of the observable Wj starting from a point in phase space x=x(0). Due to assumed ergodicity of the flow t , Wj (t (x)) has well defined and stationary statistical properties such as averages and correlation functions. The linear damped Eq. (263) is formally equivalent to a Langevin equation (with deterministic noise) and hence its solution zj (t) has a well defined time- and ε-independent probability distribution Pj (zj ). Its moments can be expressed in terms of moments and correlation functions of the deterministic noise Wj , in particular Wj = 0. The analysis generalizes to the case of explicitly time-dependent Wj . Going back to the original coordinate yj we obtain its distribution as Kj (yj ) = Pj (zj ) dzj /dyj , or Kj (yj ) = Pj (exp(− j t)yj /ε) exp(− j t)/ε. This probability distribution tells us how, on average, points within some initial (small) phase space set of characteristic diameter  spread along a locally well defined unstable Lyapunov direction j and therefore represents an averaged kernel of the evolution of such densities along this direction. Starting from the initial localized density 0 , of small width  such that (261) does not change appreciably along  the decomposition 0 0 , echo dynamics yields for the densities t (y) = dN y 0 (y ) j Kj (yj − yj ). For the stable directions j we set Kj (yj ) =
(yj ), as the shift of yj (of order ε) can be neglected as compared to unstable directions. This also implies that the assumption ε> is necessary in order to get any echo at all after not too short times. Classical fidelity (240)  can now be written as Fcl (t) = dN y0 (y)t (y). As long as the width j of 0 along the unstable direction j is much larger than the width of the kernel Kj , there is no appreciable contribution to fidelity decay in that direction. At time tj = (1/ j ) log(j /(εj )),

(264)

where j is a typical width of the distribution Pj , the width of the kernel is of the order of the width of the distribution along the chosen direction. After that time, the overlap between the two distributions along the chosen direction starts

T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

109

100 10-1 λ1

10-2

2λ1 (loxodromic)

F

10-3 10-4 10-5 10-6

λ1+λ2

10-7 10-8 0

2

4

6

8

10

12

14

16

t Fig. 34. Decay of classical fidelity for two examples of 4D cat maps perturbed as explained in text [taken from [62]]. Triangles refer to doubly-hyperbolic case where initial set was a 4-cube [0.1, 0.11]4 , and ε = 2 × 10−4 , whereas circles refer to loxodromic case where initial set was [0.1, 0.15]4 , and ε = 3 × 10−3 . In both cases initial density was sampled by 109 points. Chain lines give exponential decays with theoretical rates,

1 = 1.65, 1 + 2 = 2.40 (doubly-hyperbolic), and 2 1 = 1.06 (loxodromic).

to decay with the same rate as the value of the kernel in the neighborhood of yj = 0, which is ∝ exp(− j t). The total overlap decays as 1

Fcl (t) ≈

exp[− j (t − tj )],

(265)

j ;tj
where only those unstable directions contribute to the decay for which tj > t. As the time tj is shorter the higher the corresponding Lyapunov exponent j , fidelity will initially decay with the largest Lyapunov exponent 1 . In chaotic systems with more than two degrees of freedom we, however, expect to observe an increase of decay rate after the time t2 , i.e. Eq. (265) provides good description for classical fidelity as long as Fcl (t) does not approach the saturation value F∞ ∼ N where the asymptotic decay of classical fidelity is then given by the leading eigenvalue of Perron–Frobenius operator [61]. The result (265) does not only explain a simple Lyapunov decay of Loschmidt echoes as observed in many numerical experiments on two dimensional systems, but also predicts a cascade of decays with increasing rates given by the sums of first few Lyapunov exponent for few body chaotic systems. For example, in the first non-trivial case of a four dimensional dynamics with two positive Lyapunov exponents one can have two distinct behaviors. In the simple doubly hyperbolic case of well separated individual Lyapunov exponents 1 > 2 the decay is expected to go through a cascade of increasing rates, first 1 and then 1 + 2 , whereas in the loxodromic case 1 = 2 the rate is 2 1 . We illustrate this numerically for fourdimensional cat maps, i.e. 4-volume preserving automorphisms on a 4-torus, x = Cx (mod 1), x ∈ [0, 1)4 . The choices ⎡

2 ⎢ −2 C d -h = ⎣ −1 2

−2 3 2 −2

⎤ −1 0 1 0⎥ ⎦, 2 1 0 1

⎡

0 ⎢ 0 Clox = ⎣ 1 −1

1 1 −1 −1

0 1 1 −2

⎤ 0 0⎥ ⎦ 1 0

are two examples representing the doubly-hyperbolic and loxodromic case. The matrix Cd−h has the unstable eigenvalues ≈ 5.22, 2.11, while the largest eigenvalues of Clox are ≈ 1.70 exp(±i1.12). The perturbation for both cases was  done by applying an additional map at each time step x¯1 = x1 + ε sin(2 x3 ) (mod 1), x¯2,3,4 = x2,3,4 . In Fig. 34 we show the two types of decay which agree with theoretical predictions.

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100

100 10-1

10-1

g(t)

10-2

10-2 10-3 0

2

4

6

8

10

10-3

10-4 0

20

40

60

80

100

t √ Fig. 35. Decay of the fidelity g(t) for the sawtooth map with the parameters K0 = ( 5 + 1)/2 and ε = 10−3 for different values of L = 1, 3, 5, 7, 10, 20, ∞ from the fastest to the slowest decaying curve, respectively [taken from [61]]. The initial phase space density is chosen as the characteristic function on the support √ given by the (q, p) ∈ [0, 2 ) × [− /100, /100]. Note that between the Lyapunov decay and the exponential asymptotic decay there is a ∝ 1/ t decay, as expected from diffusive behavior. Inset: magnification of the same plot for short times, with the corresponding Lyapunov decay indicated as a thick dashed line.

6.2. Asymptotic long time decay for chaotic dynamics However, our theory above is valid only up to a time-scale in which the echo-dynamics spreads all over the available phase space. After that time, one should use a different approach to understand asymptotic (long time) decay of classical fidelity. In this regime, there does not yet exist any rigorous approach to fidelity decay, hence we describe a heuristic approach of the paper [61], also demonstrating different regimes of classical fidelity decay in chaotic classical maps. We illustrate the general phenomena in a standard model of classical chaos, namely the sawtooth map which is defined by p = p + F0 ( ),

= + p (mod 2 ),

(266)

where (p, ) are conjugated action-angle variables, F0 =K0 ( − ), and the over-bars denote the variables after one map iteration. We consider this map on the torus 0  < 2 , − L p < L, where L is an integer. For K0 > 0 the motion is completely chaotic and diffusive, with Lyapunov exponent given by =ln{(2+K0 +[(2+K0 )2 −4]1/2 )/2}. For K0 > 1 one can estimate the diffusion coefficient D by means of the random phase approximation, obtaining D ≈ ( 2 /3)K02 . In order to compute the classical fidelity, we choose to perturb the kicking strength K =K0 +ε, with ε>K0 . In practice, we follow the evolution of a large number of trajectories, which are uniformly distributed inside a given phase space region of area A0 at time t = 0. The fidelity f (t) is given by the percentage of trajectories that return back to that region after t iterations of the map (266) forward, followed by the backward evolution, now with the perturbed strength K, in the same time interval t. In order to study the approach to equilibrium for fidelity, we consider the quantity g(t) = (f (t) − f (∞))/(f (0) − f (∞)).

(267)

Thus, g(t) drops from 1 to 0 as t goes from 0 to ∞. We note that f (0) = 1 while, for a chaotic system, f (∞) is given by the ratio A0 /Ac , with Ac the area of the chaotic component to which the initial distribution belongs. √ The behavior of g(t) is shown in Fig. 35, for K0 = ( 5 + 1)/2 and different L values. One can see that only the short time decay is determined by the Lyapunov exponent √ , f (t) = exp(− t). In this figure we demonstrate that the Lyapunov decay is followed by a power law decay ∝ 1/ Dt, a manifestation of deterministic diffusion taking place on the cylindrical phase space (for large L) [59], up to the diffusion time tD ∼ L2 /D, and then the asymptotic relaxation to equilibrium takes place exponentially, with a decay rate  (shown in Fig. 36) ruled not by the Lyapunov exponent but by the largest Ruelle–Pollicott resonance.

T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

111

γ

1

0.1

1

10 L

√ Fig. 36. Asymptotic exponential decay rates of fidelity for the sawtooth map (K0 = ( 5 + 1)/2, ε = 10−3 ) as a function of L [taken from [61]]. The rates are extracted by fitting the tails of the fidelity decay in the Fig. 35 (triangles) and from the discretized Perron–Frobenius operator (circles). The line denotes the ∝ 1/L2 behavior of the decay rates, as predicted by the Fokker–Planck equation.

These resonances have been determined for the sawtooth map by diagonalizing a discretized (coarse-grained) classical propagator [61] and determining the spectral gap between the modulus of the largest eigenvalue and the unit circle. In Fig. 36 we illustrate the good agreement between the asymptotic decay rate of fidelity (extracted from the data of Fig. 35) and the decay rate  as predicted by the gap in the discretized Perron–Frobenius spectrum. More generally, one can conclude that the long time asymptotics of the (shifted) classical fidelity g(t) behaves in the same way as a typical temporal correlation function of the (forward) dynamics at time 2t. This is true even for systems where the decay of correlations is given by power laws and the spectral gap of the Perron–Frobenius operator vanishes, e.g. for the stadium billiard [61].

6.3. Asymptotic decay for regular dynamics As for the decay of classical fidelity in the regime of regular classical dynamics, the theory has been developed in Ref. [63]. We shall not outline the details here but just repeat the main result. The evolution of initial phase space density (x, 0) under regular dynamics is quite trivial, as it evolves quasiperiodically on the set of neighboring tori as can be expressed explicitly in the action-angle coordinates [63]. We note that a perturbation of an integrable system, being of KAM (Kolmogorov–Arnold–Moser) type, i.e. sufficiently smooth in the phase space variables, can cause two main effects on the KAM tori of an unperturbed system: (i) either the shape of the tori changed, or (ii) the frequencies of the motion changed. Depending on whether one of the two effects is dominant, we have two different types of decay: (i) algebraic decay f (t) ∝ t −d , where d is the number of degrees of freedom if mainly the shape of the tori is changed, (ii) ballistic decay, which is basically determined by the shape of the initial density (x, 0), i.e. Gaussian if the latter is a Gaussian, if mainly the frequencies of the tori are changed. In Fig. 37 we show an example of classical fidelity decay in an integrable rectangular billiard, with the action-angle Hamiltonian (d = 2) H0 =

1 2 2 2 I + I2 2 1 2

(268)

which is perturbed by a generic perturbation of the form V = cos( ) cos(1 ) cos(2 ) + sin( )I1 I2 . Depending on the value of the parameter , the perturbation mainly affects either the shape of the tori or their frequencies.

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100

f(t)

10-1

10-2

10-3

10-4 102

103 t

Fig. 37. Fidelity decay for the rectangular billiard for various values of the parameter = 0√ (full line), 0.232 (dashed), 0.3 (dot-dashed) and /2 (dotted) [taken from [63]]. The parameters of the system have been chosen as follows: 1 = ( 5 + 1)/2, 2 = 1. In all cases the initial phase space density is a hyper-rectangle centered around I1 = 1, I2 = 1, 1 = 1 and 2 = 1 with all sides of length I1 = I1 = 1 = 2 = 0.02. The perturbation parameter is ε = 3 × 10−4 and the number of trajectories N = 105 . The ∝ 1/t 2 decay is shown as a thin dotted line.

6.4. Classical fidelity in many-body systems In systems with increasing, macroscopic number of DOF the above theoretical considerations can still be applied though somehow qualitatively different behavior is obtained. The details can be found in Ref. [196]. The central result of that analysis is that the fidelity of a many-body system (in the larger d limit), which is perturbed by a global perturbation (i.e. the one which perturbs all degrees of freedom in an approximately uniform way), and for which Lyapunov exponents around the maximal one max are smoothly distributed with a well defined density in the thermodynamic limit d → ∞, decays with a doubly exponential law Fcl (t) = exp(−A dε exp( max t)).

(269)

A is some system specific constant which does not depend on d, ε, t. As we have discussed already within the classical linear response approximation, it turns out that it is of crucial importance whether the initial phase space density is (Lipschitz) continuous or not, namely we have index = 1 for discontinuous distribution, and = 2 for Lipschitz continuous one. 6.5. Universal decay in dynamically mixing systems which lack exponential sensitivity to initial conditions Recently, another possibility of relaxation and mixing behavior in classical dynamical systems has been investigated [199,200], which does not require exponential sensitivity on initial conditions. Even though it may appear quite exotic at the first sight, such a situation can arise quite often, for example in polygon billiards of two and higher dimensions, hard point gases in one dimension, etc. The decay of classical fidelity for such situations has been addressed in Ref. [201]. It has been found that under quite general dynamical conditions of mixing and marginally stable (parabolic) dynamics (being a consequence of the lack of exponential sensitivity to initial conditions), the fidelity decay exhibits a universal form (for small perturbations ε) namely Fcl (t) = (B|ε|2/5 t),

(270)

where B is some constant depending on details of temporal correlation decay of the perturbation and (x) is a universal function with the following asymptotic properties: 1 − (x) ∝ |x|5/2 , for |x| → 0, and (x) = exp(−|x|), for |x| → ∞.

ln(1/)

T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

p (a) (b)

s

113

c E

mix

(c)

(d)

ln (1/h) Fig. 38. Schematic view of different fidelity decay regimes for mixing dynamics [from [46]]. For instance, by increasing the perturbation fidelity goes through a Gaussian perturbative regime (a), then exponential Fermi golden rule (c) and finally for ε > εmix it decays almost instantly. The limits 2 → 0 and ε → 0 do not commute! For details see text.

7. Time scales and the transition from regular to chaotic behavior It is instructive to summarize all different time scales involved in the decay of fidelity for regular and chaotic dynamics. As for generic perturbations one does not have freeze, we shall limit our discussion to the situation with a non-zero time-averaged perturbation. 7.1. Chaotic dynamics The exposition here closely follows the one in Ref. [46]. For chaotic dynamics there are six relevant time scales (even seven for coherent initial states): the (short) Zeno time, the classical mixing time tmix on which correlation function decays; the quantum decay time of the fidelity m (90); the Heisenberg time tH (95) after which the system starts to “feel” finiteness of Hilbert space and effectively begins to behave as an integrable system; the decay time p (98) of perturbative Gaussian decay present after tH ; the time t∞ when the fidelity reaches the finite size plateau; for coherent initial states we have in addition the Ehrenfest time tE up to which we have quantum-classical correspondence. Depending on the interrelation of these time scales, i.e. depending on the perturbation strength ε, Planck’s constant 2 and the dimensionality d, we shall also have different decays of fidelity. All different regimes can be reached by e.g. fixing 2 and increasing ε. Let us follow different decay regimes as we increase ε (shown in Fig. 38): (a) For ε < εp we will have tH < m . This means that at the Heisenberg time, the fidelity due to exponential decay (90) will still be close to 1, F (tH ) ≈ 1, and we will see mainly a Gaussian decay due to finite Hilbert space (98). The critical εp below which we will see this regime has already been calculated and is (99) εp = √

(2 )d/2 2 = 2d/2+1 √ . cl N cl V

For ε < εp the fidelity will have Gaussian decay with the decay time p (98)  V 21−d/2 p = . ε 4cl (2 )d

(271)

(272)

(b) For εp < ε < εs we will have a crossover from the initial exponential decay (90) to the asymptotic Gaussian decay (98) at time tH illustrated in Figs. 13 and 14. This regime occurs if m < tH < t∞ . With increasing perturbation, t∞ will decrease and the upper border εs is determined by the condition t∞ = tH . Denoting a finite size plateau by F (t) ∼ 1/N , with  lying between 1 and 2, depending on the initial state (see Appendix A), we have the condition exp(−(tH /p )2 ) = 1/N which gives εs = √

 2   ln N = εp  ln N. cl N

(273)

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Further increasing the perturbation, we reach perhaps the most interesting regime, in which quantum fidelity can decay slower the more chaotic the systems is. In this regime the exponential decay persists until the plateau is reached. (c) For εs < ε < εmix (εE ) we will have an exponential decay until t∞ . The upper border εmix is determined by the condition m = tmix which is a point where the argument leading to the factorization of n-point correlation function breaks down. For random initial states we get  N 2 εmix = √ = εp . (274) 2tmix 2cl tmix √ Note that the relative size of this window εmix /εs = N/2tmix ln N increases both in the semiclassical 2 → 0 and in the thermodynamic d → ∞ limit. For coherent initial states the quantum correlation function relaxes on a slightly longer time scale than tmix , namely on the Ehrenfest time tE ∼ − ln 2/ . Until tE quantum packet follows the classical trajectory and afterwards interferences start to build leading to the breakdown of quantum-classical correspondence. Equating m = tE gives the upper border for coherent states   2

tmix εE = √ = εmix . (275) − ln 2 − ln 2 2cl (d) For ε > εmix the perturbation is so strong that the quantum fidelity decays before tmix , i.e. perturbed and unperturbed dynamics are essentially unrelated and fidelity decays almost instantly. For coherent initial states the upper border of regime (c) is at εE which is smaller than the lower border εmix of regime (d), which opens up the possibility of another regime between (c) and (d), namely for εE < ε < εmix the fidelity will decay within the Ehrenfest time which grows logarithmically with the number of contributing states N (i.e. with log(1/2)). In this regime the decay of quantum fidelity is the same as the decay of classical fidelity  and can be explained in terms of classical Lyapunov exponents [62]. The relative width of this regime ε /ε = ln(1/2)/ tmix again grows mix E √ logarithmically with 1/2. In regime (d), that is for 2 < ε < 2, quantum dynamics actually becomes “hypersensitive” to perturbations [73]. For typical initial packets the decay is double exponential, F ∝ exp(−const × e2 t ). Due to large fluctuations between different initial conditions for times smaller than the Ehrenfest time the average fidelity depends on the way we average. For instance, averaging fidelity one obtains exponential Lyapunov-like decay, F (t) ∝ e− 1 t , whereas by averaging the logarithm of the fidelity one gets a double exponential decay, log F (t) ∝ e2 −2 t , see Ref. [73] for details. Similar information as from Fig. 38 can be gained from Fig. 6 in Ref. [116]. For composite systems we have seen in Section 3.3 that reduced fidelity and echo purity decay semiclassically on the same time scale as fidelity, the difference between time scales being ∼ 1/Nc + 1/Ne if Nc,e are dimensions of two subspaces. Therefore, time scales involved in the decay of either reduced fidelity or echo purity (purity) are exactly the same as for fidelity. Similar regimes as for the fidelity decay were also obtained in Refs. [202,26,203] when studying energy spreading in parametrically driven systems, see also Ref. [204] for a connections between fidelity and local density of states. 7.2. Regular dynamics For regular dynamics the situation is much simpler. We have only three relevant time scales: (i) the classical averaging time tave in which the average perturbation operator converges to V¯ (176), (ii) the quantum fidelity decay time r and, (iii) the time t∞ when the fidelity reaches a fluctuating plateau due to the finite dimension of Hilbert space. For times decay is smaller than tave decay is system and state specific and cannot be discussed in general. After tave the fidelity √ first quadratic in time as dictated by the linear response formula (179). The decay time r scales as ∼ 2/ε (192) for coherent initial states and as ∼ 2/ε (197) for random initial states. Beyond linear response the functional dependence of the decay is typically a Gaussian for coherent initial states and power law for random initial states with the before mentioned decay time r (by the decay time in the case of power law dependence we mean the time when the fidelity

T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

115

reaches a fixed value, e.g., 21 ). For weak perturbation this decay persists until t∞ when the finite size effects take in and fidelity approaches a time-averaged saturation value (42). However, for stronger perturbations and very small 2, such that the time-scale ta (189) of applicability of action space integral approximation may be shorter than t∞ , one may observe another regime of asymptotic fidelity decay after ta [171], which is typically again a power law (however, of different origin than for random initial states). Detailed semiclassical theory of this regime can be found in Ref.[205]. The time t∞ again depends on the initial state as well as on the Hilbert space dimension N. For random initial states the power law decay gets faster with increasing dimensionality d of the system, and is conjectured to approach a Gaussian decay in the thermodynamic limit [42]. In contrast to chaotic systems, for regular systems the decay time scale of reduced fidelity and of echo purity for composite systems differs √ from that of fidelity. For coherent initial states the reduced fidelity decays as a Gaussian on a time scale  ∼ 2/ε (225), i.e. with the same scaling as fidelity (192) but with a different prefactor. Echo purity (or purity) for coherent initial states on the other hand initially decays quadratically but then asymptotically goes into a power law decay, FP ∼ 1/(εt)r (229), where the power r depends on the perturbation and is bounded by 1 r  min(dc , de ) if dc,e are the numbers of degrees of freedom of two subsystems. The important point though is that the whole decay does not depend on 2. For small 2 the decay of echo purity is therefore much slower than that of fidelity. If the initial state of the central subsystem is a superposition of two packets, the so-called cat state, the decay time of√ purity is much smaller and scales the same as for reduced fidelity or fidelity for a single coherent state, i.e. as dec ∼ 2/ε (234). For random initial states echo purity and reduced fidelity decay on a time scale  ∼ 2/ε. 7.3. Comparison, chaotic vs. regular Let us compare decay time scales of chaotic and regular systems. One might expect that quantum fidelity will decay faster for chaotic systems than for regular, at least such is the case for classical fidelity (Section 6). As we shall see, this is not necessarily the case. Quantum fidelity decay can be faster for regular systems! The fidelity decay time scales as ∼ 1/ε for regular systems, while it is ∼ 1/ε 2 for mixing dynamics. This opens up an interesting possibility: it is possible that the fidelity decays faster for regular system than for chaotic one. Demanding r < m we find that for sufficiently small ε one will indeed have faster fidelity decay in regular systems. This will happen for  εr = 2 C¯ 1/2 random init. state, /2cl ∝ 2 ε< (276) 3/2 3/2 −1   2 v¯ ·  v¯ /8cl ∝ 2 coherent init. state. εc = 2 We explicitly wrote the result for random initial states εr and coherent initial states εc as the two have different scaling with 2. We can see that for random initial states εr scales in the same way as εmix and so one has faster decay of fidelity in regular systems provided ε < εr ∼ εmix . For a coherent initial state this can be satisfied above the perturbative border ε > εp only in more than one dimension d > 1. In one dimensional systems εp and εc have the same scaling with 2 and whether we can observe faster decay of fidelity in regular systems than in chaotic ones depends on the values of cl and v¯  . We stress that our result does not contradict any of the existing findings on quantum-classical correspondence. For example, a growth of quantum dynamical entropies [206,207] persists only up to logarithmically short Ehrenfest time tE , which is also the upper bound for the Lyapunov decay of quantum fidelity [7] and within which one would always find F reg (t) > F mix (t) (for coherent states) above the perturbative border ε > εp , whereas the theory discussed here reveals new nontrivial quantum phenomena with a semiclassical prediction (but not correspondence!) much beyond that time. If we let 2 → 0 first, and then ε → 0, i.e. we keep ε?εr,c (2), then we recover the result supported by classical intuition, namely that the regular (non-ergodic) dynamics is more stable than the chaotic (ergodic and mixing) dynamics. On the other hand, if we let ε → 0 first, and only after that 2 → 0, i.e. satisfying Eq. (276), we find the somewhat counterintuitive result saying that chaotic (mixing) dynamics is more stable than the regular one. How can we understand this? Finite 2 causes that there is a lower limit on the size of structures, see for instance figures of Wigner functions in Section 7.5. Therefore, for sufficiently small perturbations quantum fidelity cannot exhibit Lyapunov decay because the latter implies occurrence of smaller and smaller structures as time progresses. So what comes into play for quantum fidelity are correlations between perturbations applied at different times. If correlations are small, like in a chaotic system, perturbations will “add up” in a slow random way, causing a slow decay of quantum fidelity.

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10000

εc

τ

1000

εE

εmix

(c)

100

(d)

(a) reg

10

ula

r

chao tic

1 0.0001

0.001

0.01 ε

0.1

1

Fig. 39. Numerically calculated decay time of the quantum and classical fidelity for the double kicked top (B.20) in regular and chaotic regimes. The two solid lines show numerically calculated decay times of quantum fidelity for chaotic and regular regimes. Symbols are decay times of classical fidelity, diamonds for the regular and squares for the chaotic regime. Vertical lines show the position of perturbation borders (277). The shading and the letters (a), (c) and (d) correspond to the regimes described in Fig. 38. The Zeno regime corresponds to very short times  < 1 (i.e. strong perturbations ε > 0.4).

Let us now demonstrate the above regimes by a numerical example (Fig. 39). As already noted, for a one dimensional system (d = 1), the ‘surprising’ behavior of the regular decay time being smaller than the mixing one, r < m , is for coherent initial states possible only around the border εp (unless cl is very small) where the exponential decay in the mixing regime goes over to a Gaussian decay due to finite N. However, for more than one degree of freedom, such behavior is generally possible well above the finite size perturbative border εp . To have general situation we shall therefore use double kicked top (d = 2). The propagator is given in Appendix B.1, Eq. (B.20). Depending on the parameters the corresponding classical system is√chaotic √or regular. As initial state we take a product of spin coherent states (B.22) |ϑ,  ⊗ |ϑ,  with (ϑ, ) = ( / 3, / 2) and S = 100. We numerically calculated the dependence of the decay time  (when fidelity reached 1/e ≈ 0.37) on the perturbation strength ε for chaotic and regular cases. We then compared these numerical data with theoretical predictions. Using classically calculated  = 0.058 one gets chaotic decay time m (90) m = ε8.6 2 S 2 . To determine the decay time of quantum fidelity for regular situation, Eq. (192), √ −1   √ , which gives one needs the coefficient v¯  v¯ . We determined it by fitting the dependence of r , such that r = 4.5 ε S √ √ v¯  −1 v¯  = 0.31. Note that the coefficient v¯  −1 v¯  has been obtained by numerical fitting only for convenience. In principle it could be obtained from classical dynamics, but we would again have to resort to numerical calculations as √ the regular system is not completely integrable but is rather in a mixed KAM-like regime. The values of  and v¯  −1 v¯  can then be used to calculate various perturbation borders as discussed before. For our numerical values we get (271), (276), (275), and (274), εp = 0.0005,

εc = 0.0019,

εE = 0.013,

εmix = 0.029.

(277)

In addition to the decay time of quantum fidelity we also numerically computed the decay time of classical fidelity. All these data are shown in Fig. 39. We can see that in the regular regime the quantum and the classical fidelity agree in the whole range of ε. In the chaotic regime things are a bit more complicated. By decreasing perturbation from ε = 1 we are at first in regime of very strong perturbation where the fidelity decay happens faster than any dynamical scale and it does not depend on whether we look at chaotic or regular system or quantum or classical fidelity. There the fidelity decays within the Zeno times scale, see Section 2.1.1. For smaller ε the regular and chaotic decays start to differ. In chaotic situation the quantum and classical fidelity still agree. This correspondence breaks down around εmix where the quantum fidelity starts to follow the theoretical m , while the classical fidelity decay is clas = log(0.25/ε)/ , with 0.25 being a fitting parameter (depending on the width of the initial packet) and = 0.89 is the Lyapunov exponent, read from Fig. 40. For an explanation of this classical decay see Section 6. Incidentally, in our chaotic system the classical mixing time is very short, tmix ∼ 1, and we see that the correspondence breaks down already slightly before εE . The quantum fidelity decay time m is valid until a perturbative border εp is reached, when finite Hilbert space dimension

ε=0.0005

1000

ε=0.001

2.5

ε=0.002

τ

100

117

quantum

T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

ε=0.01

2

ε=0.04

1.5 classical

λ

10 ε=0.1

1 0

1

2

3

4

5

κ regular

1 0.5 0 0

chaotic

2

4

6

8 10 12 14 16 18 20 κ

Fig. 40. Numerically calculated dependence of quantum (solid lines) and classical (symbols) fidelity decay times on the parameter  for the double kicked top (B.20). Different curves are for different perturbation strengths ε. By increasing  the classical dynamics goes from regular to chaotic, see also the right panel showing the dependence of the largest Lyapunov exponent on . By decreasing ε, on the other hand, we go from the regime of quantum-classical correspondence for ε > εE , towards a genuinely quantum regime in which for chaotic dynamics we can increase the decay time by increasing chaoticity—shaded region for the three smallest ε.

effects become important and the decay times become equal to p . Note that for ε < εc we indeed have faster fidelity decay for chaotic than for regular dynamics. 7.4. Increasing chaoticity In Fig. 39 we saw there is a range of perturbations for which quantum fidelity decay is faster for regular than for chaotic situation. Another interesting aspect of our correlation function formalism can also be seen for chaotic dynamics alone. Because the decay rate of the fidelity in a chaotic situation is proportional to the integral of the correlation function , a stronger chaoticity will typically result in a faster decay of the correlation function C(t) and therefore in smaller , resulting in slower fidelity decay. This means that increasing chaoticity (of the classical system) will increase quantum fidelity, i.e. stabilize quantum dynamics. Of course, for this to be observable we have to be out of the regime of quantum-classical correspondence. This phenomenon is illustrated in Fig. 40, where we show similar decay times as in Fig. 39, i.e. the same system and initial condition, but this time depending on the parameter  of the double kicked top (B.20). The parameter  controls the chaoticity of the classical dynamics, i.e. at  = 1 we are in the quasi-regular regime and for larger  we get into the chaotic regime. This can also be seen from the dependence of the Lyapunov exponent on  in the right panel of the figure. Data is shown for six different perturbation strengths ε. We can see that in the regular regime ( < 2) the classical fidelity agrees with the quantum one regardless of ε. In the chaotic regime though, the agreement is present only for the two largest ε shown, where we have ε > εmix (277). For ε < εc and for chaotic dynamics (three smallest ε) we get into the non-intuitive regime (shaded region in Fig. 40) where the quantum fidelity will increase if we increase chaoticity. Note that this growth of the decay time stops at around  ∼ 4 because the classical mixing time tmix gets so small that the transport coefficient is given by its time independent first term  = C(0)/2 alone and so  cannot be decreased any further. From the figure one can also see that the decay time in the transition region, where the corresponding classical system has a mixed phase space, does not change monotonically with . Such a behavior is system specific; for some additional results for the kicked top, see Ref. [148]. At the end let us make a brief comparison of decay times of purity for coherent initial states. In chaotic situation purity decays in the same way as fidelity,  ∼ 22 /ε 2 , so there is nothing new. On the other hand for regular dynamics purity decays on an 2 independent time scale,  ∼ 1/ε, which is therefore for small 2 much larger than for chaotic dynamics. Still, because of different dependence on perturbation strength ε (i.e. coupling) one can have situation where purity decay in regular system is faster than in chaotic one. Comparing time scales of decays (229,111) within the range

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of linear response we see that in order to have such a situation we must have √ ε < εcI = 22 tr u/(4cl ) ∝ 22 ,

(278)

where u is 2-independent matrix of second derivatives used in (229). 7.5. Echo measures in terms of Wigner functions In Section 3.4 we have derived a semiclassical expression for fidelity in terms of the Wigner function of the initial state (124). That was just an approximation. On the other hand, one can also express fidelity in terms of two Wigner functions exactly. Wigner functions, or more generally Weyl symbols for quantum operators, have a nice property that the trace of a product of two operators is equal to the overlap integral of two corresponding Weyl symbols, tr AB = WA WB d, where WA,B are the corresponding Weyl symbols and the integral is over the whole phase space. Here we will need a special case of this equality for density matrices,  tr(A B ) = WA WB d, (279) where A,B are now two density matrices and WA,B the corresponding Wigner functions. For more information about Wigner functions for systems described by a canonical variables [q, p] = i2 see, e.g. book [208], whereas for the definition of Wigner function for spin systems (e.g. our kicked top models) see Ref. [209]. Remembering that the fidelity can be written as a trace of the product of the initial density matrix (0) and density matrix after an echo, M (t)=Mε (t)(0)Mε† (t), or equivalently, as a trace of two forward propagated density matrices ε (t)=Uε (t)(0)Uε† (t) and 0 (t) = U0 (t)(0)U0† (t), we have   F (t) = tr (0)M (t) = W(0) WM (t) d = W0 (t) Wε (t) d. (280) Similarly, one can write the reduced fidelity (72) and echo purity (74) in terms of Wigner functions of the reduced density matrices c = tr e [],  FR (t) = Wc (0) WM dc , (281) c (t) and echo purity  FP (t) = [WM ]2 dc . c (t)

(282)

All these expressions are exact. The classical quantity analogous to Wigner function is just the classical density in phase space, therefore these expressions are handy for comparison with classical quantities. But note that the Wigner function is not necessarily positive whereas the classical density is (the positivity of Wigner function is a necessary but not sufficient condition for the classicality of a quantum state). In the next subsection we are going to illustrate the decay of fidelity and echo purity in terms of Wigner functions. Specifically, we shall compare chaotic and regular dynamics to see how is different decay of fidelity reflected in the corresponding Wigner functions. 7.5.1. Illustration with Wigner functions Again we will take our standard kicked top (B.1) with parameters  = /2 and  = 30 for the chaotic case√ and  = √0.1 for the regular one. The spin size is S = 100 and we choose a coherent initial state at (ϑ∗ , ∗ ) = (1/ 3, 1/ 2). The perturbation strength is ε = 1.5 × 10−2 . In Fig. 41 we show the fidelity decay for both cases (regular case is the same as in Fig. 26) and an illustration of states in terms of Wigner functions. For the chaotic (triangles in the fidelity plot and pictures above) and the regular case (pluses in the fidelity plot and pictures below) we show two series of Wigner functions at times t = 0, 60, 120: the Wigner function after the unperturbed forward evolution (row labeled “forward”) and the Wigner function after the echo (row labeled “echo”). In the first frame we also show the structure

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Fig. 41. Fidelity decay for chaotic (top curve and pictures) and regular (bottom curve and pictures) kicked top (B.1). Initial conditions and the perturbation are the same in both cases (see text for details). Wigner functions after forward and echo evolution are shown.

of the classical phase space being either regular with KAM tori or chaotic. In the inset the data for the fidelity decay is shown on a longer time scale and the vertical line shows the theoretical position of the beating time tb (194). In terms of Wigner functions the fidelity can be visualized as the overlap between the echo Wigner function and the initial Wigner function. The initial Wigner function shows two maxima (white regions of high value) because we have projected the initial coherent state to the invariant OE subspace, resulting in a certain symmetry of the resulting Wigner function. For chaotic dynamics the forward Wigner function develops negative values around the Ehrenfest time after which the quantum-classical correspondence is lost. For regular dynamics this correspondence persist much longer, namely until the Ehrenfest time of regular dynamics ∼ 2−1/2 after which the initial wave packet of size ∼ 21/2 spreads over the phase space. For a detailed study of Wigner functions in chaotic systems see Refs. [210,211] and references therein. The echo Wigner function for regular dynamics moves ballistically from the initial position, causing the Gaussian decay of fidelity. We also see that for regular dynamics the echo Wigner function does not necessarily have negative values even if they occur in the forward Wigner function. In our case quantum fidelity agrees with the classical one for regular dynamics. In a chaotic case on the other hand, the echo image stays at the initial position and “diffusively” decays in

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Fig. 42. Echo purity decay in the Jaynes–Cummings model for regular and chaotic dynamics [from [162]]. In the inset, the same data is shown for smaller times. Full curves are numerics while dashed ones are theoretical predictions. For small ε = 0.005 and large 2 = 1/4 used here, echo purity decay can be faster in regular than in chaotic system. Top set of pictures shows the square of the Wigner function of the reduced density matrix M c (t) for chaotic dynamics and the bottom one for regular dynamics. Its integral gives echo purity.

amplitude, causing the fidelity to decay slower than in the regular case. Classical fidelity follows quantum fidelity in the chaotic regime only up to Ehrenfest time. The previous figure demonstrated the possibility of faster fidelity decay in regular than in chaotic systems, provided inequality (276) is fulfilled. Let us now show that the same can happen also for echo purity. From inequality (278) we see, that either 2 has to be large (i.e. strong quantum regime) or the perturbation has to be weak. As a numerical model we use the Jaynes–Cummings model [212,213], describing a system of harmonic oscillator coupled to a spin. As opposed to kicked top systems, the Jaynes–Cummings model has a time independent Hamiltonian, and time is a continuous variable. Regular (integrable) dynamics is obtained if only the co-rotating coupling term is present, while predominantly chaotic dynamics takes place in the presence of both co- and counter-rotating terms, see Ref. [51] for details about numerical parameters. In Fig. 42 we show the decay of echo purity for relatively small perturbation ε = 0.005 in spin energy, i.e. perturbation is the so-called detuning, and large 2 = 1/4, so that the condition ε < εcI (278) is satisfied. In addition, we show a set of pictures for times t = 50, 100, 150 and 200 representing square of the Wigner function of the reduced density matrix M c (t) after an echo. The integral of this quantity directly gives echo purity (282). The central subsystem is chosen to be the spin DOF, so the phase space is the surface of a sphere. The top set of pictures is for chaotic and the bottom one for regular dynamics. From the Wigner function plots one can see that the blob of maximal value of the Wigner function changes its position with time for regular dynamics while it stays at the same place for chaotic dynamics. More importantly, the height of this peak decays faster for regular dynamics than for chaotic one, reflected in a faster decay of echo purity. We should not forget that this regime of faster decay for regular dynamics is reached only for sufficiently large 2 (or small ε), see Eq. (278), and that in general, echo purity in regular systems will decay very slowly due to its 2 independence (229). 7.5.2. Wigner function approach We have seen that the Wigner function often gives an alternate and sometimes very physical view of echo evolution, particularly if we start with a coherent state as an initial state. Note that for a random state the Wigner function is

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essentially a mess and does not present, to our knowledge, any useful insight. In this context we wish to discuss two papers that reach similar results. In Ref. [115] the authors develop an analogy to the formalism used in the treatment of decoherence, despite of the fact, that they do not consider coupling to some environment. For this purpose the authors consider an ensemble of perturbations. Yet they proceed quite differently than in the case of the RMT treatment offered in Section 4, in that they do not consider unitary evolution and average over the resulting fidelity amplitudes or fidelities. The Wigner function is the density operator in the phase space representation of quantum mechanics. In this picture we obtain a clear distinction between the evolution of the diminishing coherent part of the Wigner function and the one of the emerging rapidly oscillating part. The fidelity is then given as the trace of the product of the average of the perturbed density matrices (t) with the 0 (t) evolving according to the unperturbed evolution  Fz (t) = tr ((t)0 (t)) =

dp dq W (q, p, t)W (q, p, t),

(283)

which in phase space representation can be written as an integral over the Wigner function W (q, p, t) and the average Wigner function W (q, p, t), averaged over an ensemble of perturbations. The use of an average density matrix approaches us to concepts usually used in decoherence and induces the authors of this paper to introduce an ensemble of perturbations solely in terms of the time dependence where they assume white noise. This allows them to propose a master equation which can be solved. It is important to note, that the fidelity decay for each element of the ensemble is dominated by the decay of correlations in time through the white noise. As an initial condition a coherent state is used, and they conclude that with perturbation strength above some threshold, they recover the exponential decay by limiting the integration to the area where the Wigner function varies slowly. The rate of this decay is perturbation independent and is determined by the (maximal) Lyapunov exponent of the underlying classical mechanics. An integration over the remaining area, where we have the well known fast oscillations developing rapidly, will produce a second term, that yields the perturbation dependent Fermi golden rule decay. The Gaussian perturbative regime is never reached in this analysis, because strong perturbations are assumed from the outset. While the approximations in this paper are uncontrolled and several assumptions are very special, it provides very important qualitative insight: first the somewhat surprising fact that the Lyapunov decay is independent of the perturbation strength becomes intuitively clear, and second a close relation to the dynamics of decoherence is established. A similar picture was used in Ref. [116] to describe the transition between the two regimes in more detail. Regarding fast oscillations of Wigner functions for sufficiently extended states, it has been argued in Refs. [214,47] that such rapid oscillations enhance sensitivity of quantum systems to perturbations. When an initial state  has been prepared by a “preparation” evolution starting from a localized wave packed 0 ,  = exp(−iH0 t)0 , an enhanced sensitivity has been observed when increasing the preparation time t [47]. This has been explained as being due to the small structure in Wigner function of the initial state . The observed dependence though is just due to the dependence of fidelity decay on the initial state and can be explained by classical Lyapunov exponents [215]. Therefore, it cannot depend on quantum intereference effects caused by the rapid oscillations in Wigner function, see also discussion in Ref. [216].

8. Application to quantum information Quantum information theory is a relatively recent endeavor, for a review see Refs. [217–219,13]. Its beginnings go back to the 1980s and in recent years theoretical concepts have been demonstrated in experiments. While quantum cryptography, a method of provably secure communication, is already commercially available, quantum computation is still limited to small laboratory experiments. Errors and error correction are among the main concerns in quantum information. In the present section we are going to discuss, how the techniques and results developed for fidelity decay can be used to design quantum operations that are less prone to errors. We shall divide the whole exposition into two parts. In the first part we shall show how a straightforward application of fidelity theory can help us understand the dependence of errors on various parameters. In the second part we are going to describe how one can actually decrease errors.

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8.1. Fidelity studies We present a partial overview of results obtained by numerically studying fidelity in quantum computation for various kinds of errors. Note that due to the extensiveness of the literature on the subject the list is by no means exhaustive. One of the first studies of the sensitivity to errors of quantum algorithms appeared shortly after Shor’s discovery of the factoring algorithm. In Ref. [220] the influence of Markovian errors (e.g. due to the coupling with the environment) on the workings of Shor’s algorithm has been studied. At random instances of time a random qubit was chosen to decay with a certain probability to its zero state. In subsequent work [221] the stability of Shor’s algorithm running on an ion trap quantum computer is analyzed in the presence of random phase drift errors due to pulse length inaccuracies. Fidelity is found to depend exponentially on the perturbation strength ε 2 and time t as one expects for uncorrelated errors, see Section 2.1.2 or Eq. (290). In a series of papers [222,223] the influence of unwanted intrinsic qubit–qubit couplings has been studied. The dependence of fidelity on the computation time has been found to be Gaussian. This agrees with a Gaussian fidelity decay for perturbations with non-decaying correlation function. Note that such “regular” (i.e. Gaussian) decay of fidelity is not unexpected as the individual one or two-qubit gates, due to their simplicity, can decrease correlation function only in a very limited way, thus causing a slow decay of correlations and consequently a fast Gaussian-like decay of fidelity. A number of numerical studies has been done by the groups of Shepelyansky and Casati. In Refs. [224,225] the question was addressed, how inter-qubit couplings change the eigenstate structure of a quantum computer, see also Refs. [226–228]. Above some threshold coupling chaos sets in and the authors advocated that quantum computation fails. This result looks like the opposite to our conclusions, where more chaos can mean more stability. One should bear in mind though, that there is no straightforward connection between eigenvector statistics and the actual dynamical fidelity which is a natural quantity measuring the stability. In [225] as well as in [229] survival probability of register states (i.e., fidelity of an unperturbed eigenstate) has been studied. The decay rate has been found to be given by the local density of states. That the eigenstate mixing is not the most relevant quantity is confirmed also by data from nuclear experiments [230]. In Ref. [231] the stability of a quantum algorithm simulating saw-tooth map has been numerically considered for static and random errors, pointing out that static imperfection are more dangerous. Similar study has been performed in Ref. [232] for the quantum kicked rotator. For random errors the fidelity has been found to decay exponentially with ε2 and the number of gates. Such exponential decay can be shown to be general for uncorrelated (in time) errors [43], independent of the quantum algorithm, see also Eq. (290). For static errors a faster Gaussian decay has been observed, which can again be understood by the linear response formalism. In Ref. [233] the influence of random and static imperfection has been studied, confirming that static imperfections can be more dangerous [43,231]. The influence of errors on the working of quantum computer has been studied also in Ref. [136]. Static imperfections can be decreased by doing rotations between the gates, i.e., introducing “randomizing” gates during the evolution [43]. Similar idea is used in the so-called Pauli random error correction (PAREC) method [234,235]. The group of Berman studied in detail the stability of the Ising quantum computer, see Refs. [236–239] and references therein. The Ising quantum computer consists of a series of spin 21 particles placed in a magnetic field with a strong gradient. The magnetic gradient allows selective addressing of individual qubits having different resonant frequencies. The nearest neighbor Ising coupling on the other hand enables the execution of two-qubit gates. All gates are performed by applying rectangular pulses of a circularly polarized electro-magnetic field lying in the plane perpendicular to the applied magnetic field. Different gates can be applied by choosing appropriate frequencies, phases and lengths of pulses. Perturbation theory has been used to explain the dependence of fidelity on errors in various physical parameters of the Ising quantum computer. In Ref. [164] the stability of a quantum algorithm simulating the quantum saw-tooth map is studied with respect to errors in the parameters of the map (classical errors) and with respect to uncorrelated random unitary errors of gates (quantum errors). As expected from the theory [43], the fidelity decay for uncorrelated errors is found to be exponential and independent of the dynamics of the classical map. Similar results have been obtained also for the decay of concurrence in the presence of uncorrelated unitary errors [240]. In Ref. [241] noisy unitary errors were also considered theoretically. A quadratic bound (in the number of applied errors) on the error is obtained, the result being a trivial consequence of the linear response expression for fidelity, Eq. (13). Unitary noise in Grover’s algorithm has been studied in Ref. [242]. Summarizing, the plethora of numerical investigations has shown that the dependence of errors on various parameters like e.g. perturbation strength ε, the number of gates, etc., can be readily understood within the linear response formalism

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of fidelity decay. Furthermore, as one is predominantly interested in large values of fidelity, linear response is all that is needed. 8.2. Decreasing the errors Once we have identified the errors and their dependence on various parameters, we can try to decrease or even better to entirely eliminate such errors. We shall more closely discuss two possibilities: (i) We can change the algorithm, i.e. apply a different set of quantum gates, so that the resulting sequence of gates is more resistant against a given kind of errors; (ii) If one is able to perform certain unitary operations with sufficiently small errors, one can do the actual quantum calculation in a transformed frame, called the logical frame, in which the influence of errors is suppressed. Such procedure is known as dynamical decoupling [14]. There are other approaches to reduce errors like quantum error correction, decoherence-free subspaces and procedures using the quantum Zeno effect. In the limit of ideal and infinitely short measurements quantum Zeno procedures have certain similarities [243] to the so-called “bang–bang” [244] limit of dynamical decoupling. One should be careful though as dynamical decoupling consists of unitary manipulations of dynamics while quantum Zeno effect involves quantum measurements. Fidelity theory could help to understand errors present in these methods in the non-ideal limit, for some studies see Refs. [15,245]. Another possibility to combat errors is by quantum error correction [246–249]. We will not discuss it in detail as we have to first be able to implement individual gates sufficiently well for the error correction to work. Also, quantum error correction can efficiently cure only errors that are correlated over just a few qubits. Its success relies on the fact that errors are local, so that information can be “hidden” in nonlocal states, whose change then serves to diagnose the error and correct them. The efficiency therefore sharply decreases for correlated multi-qubit errors. For study of error correcting codes and decoherence-free subspaces in the presence of errors see Refs. [250,251]. Using decoherence-free subspaces [252–255] one exploits certain symmetries of the coupling (like e.g. permutation symmetry) enabling a part of the Hilbert space to be free of decoherence for a given coupling. In the next section we shall show an illustrative, though not very realistic example of the use of fidelity theory to improve the stability of the Quantum Fourier transformation against a certain kind of perturbations. 8.2.1. Modifying the algorithm We shall first briefly present the language of quantum computation which is slightly different than the so far used framework of continuous-time Hamiltonian dynamics. We shall focus on a standard quantum computation where the propagator is decomposed into quantum gates. There are other approaches to quantum computation, for instance, utilizing the ground state to perform computation, like in adiabatic quantum computation [256,257] or using holonomies in degenerate subspaces, as in holonomic quantum computation [258]. For applications of the linear response approach to the stability of holonomic quantum computation see [259–261]. A quantum computer is composed of n elementary two-level quantum systems—called qubits. The union of all n qubits is called a quantum register |r. The size of the Hilbert space N and therefore the number of different states of a register grows with the number of qubits as N = 2n . A quantum algorithm is a unitary transformation U acting on register states. In the standard framework of quantum computation a quantum algorithm U which acts on N dimensional space is decomposed into simpler units called quantum gates, Ut , where t is a discrete time counting gates. Typically, quantum gates act on either a single qubit or on two-qubits. Ideal quantum computation with no errors then consists of a series of ideal gates: U (T ) = UT · · · U2 U1 .

(284)

A quantum algorithm is called efficient if the number of needed elementary gates T grows at most polynomially in n = log2 N, and only in this case it can generally be expected to outperform the best classical (digital) algorithm. Decomposition of a given algorithm U into individual gates Ut is of course not unique. There are many decompositions with different Ts, resulting in the same algorithm U = U (T ). At present only few efficient quantum algorithms are known. The above ideal situation without errors cannot be achieved in practice. Generally, at each time step there will be errors caused by perturbations (i.e. unwanted evolution). These can result from the coupling of qubits with external

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degrees of freedom or from the non-perfect evolution of qubits within the real algorithm in contrast to the ideal one. We shall model both perturbations on a given gate Ut by a unitary perturbation 12 of the form Utε = exp(−iεV (t))Ut .

(285)

We set 2 = 1 and use the superscript ε to denote a perturbed gate. We allow for different perturbations V (t) at different gates. The perturbed algorithm is a product of perturbed gates, U ε (T ) = UTε · · · U1ε . Please note that Uj (subscript) is a single gate, that is a transformation acting between time t = j and t = j + 1, while we use U (t) for a propagator from the beginning to time t, i.e. U (t) = Ut · · · U1 . Fidelity will again serve as a measure of stability so we have Mε (T ) = U † (T )U ε (T ).

F (T ) = |Mε (T )|2 ,

(286)

Note that here the discrete total time T is not a free parameter but is fixed by the number of gates, although we could in principle also look at the fidelity as it changes during the course of execution of the algorithm. We have seen in Section 7 that we can generally have higher fidelity in chaotic systems than in regular ones. For chaotic evolution fidelity decays linearly in time, while it decays quadratically for regular dynamics. Therefore, one might expect that an algorithm will be the more stable (have higher fidelity) against a given perturbation the more “chaotic” it is. A nontrivial important question then is, for which decomposition of the algorithm U shall we have the highest fidelity? In the present section we give a concrete example of such optimization. The presentation closely follows Ref. [43]. As quantum computer will work adequately only if fidelity is extremely high, the linear response expansion of F (T ) is all we need. Writing the echo operator in the interaction picture we have, ˜

˜

˜

Mε = e−iεV (T ) · · · e−iεV (2) e−iεV (1) ,

(287)

where V˜ (t) = U † (t)V (t)U (t) is the perturbation of the tth gate V (t) propagated with the unperturbed gates U (t) = Ut · · · U1 . Be aware that V˜ (t) is time dependent for two reasons; due to the use of the interaction picture (propagation with U (t)) and due to the explicit time dependence of the perturbation itself, i.e. the fact that different perturbations occur at different gates. To assess the typical performance of a quantum algorithm independent of a particular initial 1 state of the register, we average over Hilbert space of initial states as • = N tr (•). Expanding fidelity we obtain to lowest order F (T ) = 1 − ε 2

T 

C(t, t  ),

C(t, t  ) =

t,t  =1

1 tr[V˜ (t)V˜ (t  )]. N

(288)

Two situations can now be distinguished depending on the time dependence of the perturbation V (t) at different gates. The extreme case of time-dependent V (t) is uncorrelated noise. This can result from coupling to a many body chaotic system, for which the matrix elements of V (t) may be assumed to be Gaussian random variables, uncorrelated in time, Vj k (t)Vlm (t  )noise =

1
j m
kl
tt  . N

(289)

Hence one finds C(t, t  )noise =
tt  , where we have averaged over noise. In fact the average of the product in Mε (287) equals the product of the average and yields the noise-averaged fidelity to all orders F (T )noise = exp(−ε 2 T ),

(290)

which is independent of the quantum algorithm U (T ). This result is completely general provided that the correlation time of the perturbation V (t) is smaller than the duration of a single gate. On the other hand, for a static perturbation, V (t) ≡ V , one may expect slower correlation decay, depending on the “regularity” of the evolution operator U (T ), and hence faster decay of fidelity. Note that in a physical situation, where 12 We treat only unitary errors here. Quantum computation is expected to require probability conservation. To treat non-unitary effects we would have to include “environment” into our picture.

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the perturbation is expected to be a combination V (t) = Vstatic + Vnoise (t), the fidelity drop due to a static component is expected to dominate long-time quantum computation T → ∞ (i.e. large numbers of qubits n) as compared to the noise component, as soon as the quantum algorithm exhibits long time correlations of the operator Vstatic . The fact that static imperfections can be more dangerous than time-dependent ones (e.g. noise) has been also observed in numerical experiments done in Ref. [231]. Therefore, in the following we shall concentrate on static perturbations as they are more dangerous. We concentrate on the quantum Fourier transformation algorithm (QFT) [262,263] and shall consider its stability against static random perturbations. The perturbation V (t) ≡ V will be a random hermitian matrix from a Gaussian unitary ensemble (GUE) [144]. The GUE matrices have been extensively used to model quantum statistical properties of classically chaotic Hamiltonians without time-reversal invariance [19,86,20]. Note that a GUE matrix acts on the whole Hilbert space and therefore represents a perturbation that affects correlations between all n qubits. Quantum error correction methods for instance do not work for this type of errors. Second moments of a GUE matrix V are normalized as Vj k Vlm GUE =
j m
kl /N,

(291)

where the averaging is done over a GUE ensemble. Using this in the general expression for the correlation function (288) we arrive at the correlation function for a static GUE perturbation,

2

1

tr U (t, t  )

, (292) C(t, t  )GUE =

N where U (t, t  ) is the unperturbed propagator from gate t  +1 to t, U (t, t  )=Ut · · · Ut  +1 , with the convention U (t, t) ≡ 1. The correlation sum must then be evaluated for the QFT algorithm. For the usual decomposition of the QFT algorithm 2 into gates we have T ∼ n2 /2 gates, namely n one-qubit Hadamard gates and two qubit “B” gates, nT /2 diagonal k−j Bj k = diag{1, 1, 1, exp(i j k )}, with j k = /2 . The correlation sum  = t,t  =1 C(t, t  ) scales as QFT ∝ n3 . Cubic growth can be understood from the special block structure of the QFT. There are n blocks, each containing one Hadamard gate and j “B”-gates, j = 0, . . . , n − 1. These “B”-gates, which are almost identities, are responsible for the correlation function to decay very slowly, resulting in  ∝ n3 , see Ref. [43] for details. The idea how to modify the QFT algorithm is to get rid of bad “B”-gates. This can in fact be done [43], obtaining the so-called improved QFT (IQFT) algorithm. The IQFT has twice as many gates as the QFT, but retains the same scaling TIQFT ∼ n2 . Although the perturbation is applied twice as often during the evolution of IQFT the correlation sum is smaller, because the correlation function is greatly decreased. More importantly, the correlation sum grows only quadratically with n, IQFT ∝ n2 . The dependence of fidelity F (T ) on the number of qubits n can be seen in Fig. 43. Both the QFT and the IQFT are represented, and we see the considerable advantage of the modification for large n. A few comments are in place at this point. The high fidelity of the IQFT algorithm depends on the perturbation being a GUE matrix. The optimization is therefore somewhat perturbation specific. We should point out that the optimization of the QFT algorithm becomes more difficult if the perturbation results from a two-body (two-qubit) GUE ensemble [86,264]. This is connected with the fact that quantum gates are two-body operators and can perform only a very limited set of rotations on full Hilbert space and consequently have a limited capability of reducing the correlation function in a single step. For such errors fidelity will typically decay with the square of the number of errors T, i.e. the number of gates, like ∼ exp(−ε 2 T 2 ), that is the same as for regular systems. This means that the very fact that the algorithm is efficient, having a polynomial number of gates, makes it very hard to reduce the correlation function and therefore causes fast fidelity decay. In this section we presented a result for optimizing the QFT if the perturbation acts after each individual gate. In experimental implementations each gate is usually composed of many simple pulses. Provided the perturbation is GUE, the IQFT will be more stable even if the perturbation acts after each pulse (i.e. many times during a single gate). This is for instance the case in Ref. [239] where the same gain as here has been numerically verified for a QFT algorithm running on the Ising quantum computer. 8.2.2. Dynamical decoupling Our approach to the analysis of fidelity consisted in exploring how F (t) decays for different perturbations and/or initial states for a given unperturbed dynamics H0 . The question can be turned around though. One can ask, what can we infer about the fidelity F (t) if we have a given perturbation V, but are free to choose the unperturbed evolution H0 .

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1 0.95 0.9 0.85 F(T)

0.8 0.75 0.7 0.65 0.6 0.55 0.5 3

4

5

6

7

8

9

10

n Fig. 43. Dependence of the fidelity F (T ) on the number of qubits n for the QFT (pluses) and the IQFT algorithms (crosses), for fixed ε = 0.04 [from [43]]. The full curve is exp(−ε 2 {0.47n3 − 0.76n2 + 2.90n}) and the dashed one exp(−ε 2 {1.22n2 + 1.78n}). An “improved” IQFT algorithm, being more random, has a higher fidelity, which furthermore decreases slower with n than for the original QFT algorithm. All is for a static random matrix perturbation.

This question is of immediate interest if we want to suppress the unwanted evolution caused by V (t). By appropriately choosing H0 (t) we want to suppress unwanted effects of V (t) as much as possible. Of course, just setting H0 (t)=−V (t) will do the trick. But the rule of the game is, that we are not able to generate an arbitrary H0 , but just some subset. One can, for instance, imagine that V (t) is an unwanted coupling with the environment. In such case V (t) includes also environmental degrees of freedom which we are not able to control. So one can allow H0 to act only on the the central system and not on the environment. Such a dynamical suppression of unwanted evolution is known as a dynamical decoupling [14,15,244,265,266]. Let us assume that we are able to perform operations U0 (t) from a group G without errors and infinitely fast.13 In the case of a discrete group G time t is a discrete index counting the elements of a group. U0 can be thought of as generated by some Hamiltonian H0 (t). Usually cyclic conditions are assumed, U0 (t + Tc ) = U0 (t), where Tc is the period. Due to the presence of the unwanted term V (t), the evolution is actually given by Uε (t) applying both H0 (t) and V (t). Because U0 (t) can be executed exactly, we can define a new computational frame called the logical frame (interaction frame in fidelity language), in which there is no evolution due to U0 . Mathematically, the propagator in the logical frame is Ulog (t) = U0† (t)Uε (t).

(293)

We can see, that Ulog (t) is nothing but the echo operator Mε (t) (6), so we have
  t i  ˜  ˆ Ulog (t) = T exp − ε dt V (t ) , 2 0

(294)

where Tˆ denotes a time-ordering and V˜ (t) = U0† V (t)U0 (t). The goal of dynamical decoupling is to make evolution in the logical frame Ulog as close to identity 1 as possible or more generally, to make it a tensor product of evolution of the central system and the environment, Ulog (t) → Usys ⊗ Ubath , i.e. to decouple our system from the environment. While in the laboratory frame there are contribution due to H0 and V (t), in the logical frame there is only a contribution due to logical V˜ (t). By appropriate choice of U0 (t) such that V˜ (t) “averages” out, suppression of evolution due to V (t)

13 The condition of infinitely fast, so-called “bang–bang” [244] execution, can be relaxed [265].

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can be achieved in the logical frame. Two approaches are possible, depending on how we choose the correcting dynamics U0 (t): – The first one is called a deterministic dynamical decoupling [14], because U0 (t) is chosen to deterministically traverse all the elements of the group G. Applying the BCH formula (Section 2.1.4) on Ulog we get  
i 1 (295) ε V¯ t + ε 2 (t) + · · · Ulog (t) = exp − 2 2 with the known expressions for V¯ (176) and (t) (25). If we are able to choose H0 (t) such that V¯ ≡ 0 dynamical decoupling is achieved and is said to be of the 1st order. Dynamical decoupling therefore exactly corresponds to the freeze of quantum fidelity, either in integrable systems (Section 5.2) or in chaotic ones (Section 3.2). The fact that going into a different computational frame can dramatically change the influence of errors has been also numerically verified in the Ising quantum computer [239]. – Even if one might not be able to make V¯ = 0, one might be able to reduce it. A standard measure of success of such a suppression is fidelity, i.e. the expectation value of Ulog (t). Learning from the results of fidelity decay, we can expect that the suppression will be the larger, the more “chaotic” the evolution U0 (t) is [42,43,46]. In the maximally random case the U0 (t) at different times are totally uncorrelated. Dynamical decoupling for such uncorrelated U0 (t) has been recently proposed [15] and named random dynamical decoupling. Lowest order of the error probability has been estimated [15] and is equal to the linear response approximation of fidelity for chaotic dynamics (87), see also Section 2.1.2 on fidelity decay for uncorrelated perturbations. Summarizing, the idea of dynamical decoupling is to suppress unwanted evolution V (t) by applying unitary corrections U0 (t) and doing quantum operations in the logical frame defined by U0 (t). This can be achieved either by making V¯ = 0 or by making U0 (t) uncorrelated in time, so the error correlations are decreased. The idea of applying some additional pulses to correct for unwanted evolution is extensively used in nuclear magnetic resonance (NMR). In the quantum computer context it has been for instances used by the group of Berman to suppress near-resonant errors in the Ising quantum computer, see Ref. [267] and references therein. 9. A report on experiments The first experiments of echo dynamics go back to Hahn [8] in NMR and in the introduction we have given a historical rundown of NMR techniques. In this section we shall limit our discussion to a few experiments or potential experiments, that relate directly to the theory presented in this paper. In this context recent experiments with microwaves and elastic waves are remarkable, in that they yield a very good agreement with random matrix theory, but we shall actually start with some recent NMR experiments by the Cordoba group, which at this point cannot be explained and provide some intriguing riddles. Finally we shall discuss extensively potential experiments in atomic optics expanding on an idea by Gardiner et al. [33,34,48]. 9.1. Echo experiments with nuclear magnetic resonance The first real interacting many-body echo experiment was done in solids by Rhim et al. [9]. Time reversal, i.e. changing the sign of the interaction, is achieved for a dipolar interaction whose angular dependence can change sign for a certain “magic” angle, that causes the method to be called magic echo. Still, the magic echo showed strong irreversibility. Much later a sequence of pulses has been devised enabling a local detection of polarization [10,268] (i.e. magnetic moment). They used a molecular crystal, ferrocene Fe(C5 H5 )2 , in which the naturally abundant isotope 13 C is used as an “injection” point and a probe, while a ring of protons 1 H constitutes a many-body spin system interacting by dipole forces. The experiment proceeds in several steps: first the 13 C is magnetized, then this magnetization is transferred to the neighboring 1 H. We thus have a single polarized spin, while others are in “equilibrium”. The system of spins then evolves freely, i.e. spin diffusion takes place, until at time t the dipolar interaction is reversed and by this also spin diffusion. After time 2t the echo is formed and we transfer the magnetization back to our probe 13 C enabling the detection of the polarization echo (66). Note that in the polarization echo experiments the total polarization is

L=200mm

T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

L=200mm

128

B = 475mm

B = 438mm

Fig. 44. Geometry of the billiards (figure taken from [69]). In the billiard in the right figure bouncing ball orbits have been avoided by inserting additional elements. j

j +1

conserved as the dipole interaction has only “flip-flop” terms of the form S+ S− , which conserve the total spin. To detect the spin diffusion one therefore needs a local probe. With the increase of the reversal time t the polarization echo—the fidelity—decreases approximately exponentially. The nature of this experiment has been elaborated further by Pastawski et al. [38]. The group of Levstein and Pastawski then performed a series of NMR experiments where they studied in more detail the dependence of the polarization echo on various parameters [11,39,40]. They were able to control the size of the residual part of the Hamiltonian, which was not reversed in the experiment and is assumed to be responsible for the attenuation of the polarization echo. In presence of strong perturbations, such as magnetic or quadrupolar impurities, the echo decays according to a perturbation dependent exponential law as prescribed by the Fermi golden rule. However, in pure systems, where residual interactions are made small, the decay enters a Gaussian regime where the decay rate is perturbation independent. Notice that the experiments are constrained to move from a naturally big perturbation and, while one can reduce it, the full cancellation of perturbation cannot be achieved. A similar situation was observed in liquid crystals where a molecule involves a few tens of spins [269]. This is consistent with the original interpretation [38,39] that the many body spectrum is so dense, that almost any small perturbation can place the Echo dynamics in a regime where it is perturbation independent. These surprising results triggered a number of numerical studies and theoretical investigations. In Ref. [270] the quantum Baker map has been experimentally implemented on a 3-qubit system within the NMR context. They also studied the influence of perturbations on the dynamics and could by measuring the resulting density matrices calculate the fidelity. For some other NMR measurements of fidelity see also [271]. 9.2. Measuring fidelity of classical waves The dynamics of classical electromagnetic waves in a thin resonator is equivalent to a single-particle two dimensional Schrödinger equation, i.e. single particle quantum mechanics. This is exploited in microwave resonator experiments, where properties of two dimensional quantum billiards can be studied. The Marburg group considered a cavity consisting of a rectangle with inserts that assure a chaotic ray behavior, Fig. 44. Both situations with and without parabolic manifolds leading to so-called bouncing ball states were considered. One wall was movable in small steps, this change causing the perturbation occurring in an echo experiment. It is important to note that the shift of the wall changes the mean level density and thus the Heisenberg time. This trivial perturbation, which would cause very rapid correlation decay is eliminated in this case by measuring all times in proper dimensionless units, i.e. in terms of the Heisenberg time as proposed in Section 4. Two antennae allow to measure both reflection and transmission amplitudes. The experiment was carried out in the frequency domain. Afterwards the Fourier transform is used to obtain correlation functions in the time domain. Recall (Section 2.6.1) that these functions are used to define scattering fidelity which, for weak coupling and chaotic dynamics, agrees with standard fidelity. In Fig. 45 the correlation functions are shown for two different systems. Observe that the cross-correlation agrees with the random matrix prediction in time up to six times the Heisenberg time and in magnitude of the cross-correlation function over five orders of magnitude. In the case with bouncing ball states the agreement with RMT is understandably much poorer. In Fig. 46, the scattering fidelity itself is reproduced; as we are in the range of isolated resonances, i.e. in a weak coupling situation this should be standard fidelity, and indeed the agreement with fidelity as obtained from RMT is excellent. The perturbation strength in this case was determined independently from the level dynamics and, understandably, for low energies it does not agree with its semi-classical limit. It is more surprising that the shape of fidelity as obtained for RMT in Eq. (130) holds for some stretch beyond the weak coupling limit. This can be seen in Fig. 47, where a wide range of data for chaotic systems has been averaged over, and the validity of scaling with 2 C(t) is demonstrated. This in turn implies scaling

10 −2

10 −2

10 −3

10 −3

10 −4

10 −4

* ,S '] C [S11 11

* ,S '] C [S11 11

T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

10 −5 λ =0.047 5 .. 6 GHz n =5

10 −6 10 −7

129

10 −5 λ=0.047 5 .. 6 GHz n=5

10 −6 10 −7

10 −8

10 −8 0

2

4 t

6

8

0

2

4 t

6

8

ˆ ∗ , S  ] for the billiard with bouncing balls (left), and for the Fig. 45. (Figure taken from [69]) Logarithmic plot of the correlation function C[S 11 11 one without bouncing-balls (right). The experimental results for the auto correlation are shown in black, while the correlation of perturbed and unperturbed system are shown in gray. The smooth solid curve corresponds to the theoretical autocorrelation function, and the dashed curve to the product of autocorrelation function and fidelity amplitude.

1.0

f11(t)

f11(t)

1.0

λ =0.047 5 .. 6 GHz n =5

λ=0.045 5 .. 6 GHz n =5

0.1

0.1 0

2

4 t

6

8

0

2

4 t

6

8

Fig. 46. Logarithmic plot of the corresponding fidelity amplitudes (figure taken from [69]). The smooth curve shows the linear-response result. For the billiard without bouncing balls, the perturbation parameter was obtained from the variance of the level velocities; in the other case it was fitted to the experimental curve.

with perturbation strength 2 as well as with the time dependence and the dependence on Heisenberg time as it appears in the correlation integralC(t). Indeed if we scale the time dependence with the function obtained in the linear response approximation of RMT and average over all results that are available for different systems and frequency ranges we find that the exponentiation of the linear response result yields an excellent approximation in the range of the present experiment. Lobkis and Weaver [68] have performed an interesting experiment on cross-correlations of the long time signal, usually called “coda”, of elastic signals at different temperatures for a number of aluminum blocks, Fig. 48. Both the excitation and the readout were transmitted with the same piezo-element, that connected the specimen with the pulse generator and analyzer isolating the latter from the former by a reed relay. The sample was kept in vacuum and with strict control of the uniformity of temperature. The purpose of the measurement was to investigate the normalized cross-correlation of these signals for the same block at different temperatures T1 and T2 ,  dST1 ()ST2 ((1 + ε)) X(ε) =  . (296)  dST21 () dST22 ((1 + ε))

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1.0

0.1 0

1

2

3

4

4π2λ2C(t) Fig. 47. (Color online) (Figure taken from [69]) Average of the experimental fidelity amplitude on a rescaled axis x = 4 2 2 C(t). The solid line corresponds to g(x) = exp(−x) with  = 2exp / 2 = 0.36.

Fig. 48. Different aluminum block specimens (figure taken from [68]).

The correlation integral is calculated around some time t, which the authors call the “age” of the sample. Temperature steps of 4◦ were chosen. The biggest effect termed dilation is simply due to changes in volume and in the speeds of propagation of the waves. This effect should show up in the correlation functions as a maximum of size 1 at a time difference determined by factors well known for elastic media. In fact we find a maximum at the predicted time, but it is smaller than 1 (see Fig. 49). These maxima as a function of the age t of the sample, are called distortions D(t). The correlation function (296) suggests that the distortion can be related to the logarithm of scattering fidelity and indeed it is easy to show [83] that D(t) = − ln fs (t) holds. The timeshift in the correlation maximum exactly compensates for the trivial dilation effect, and in this sense is equivalent to measuring time in units of Heisenberg time, as was done in the microwave experiment. As the piezo-element provides weak coupling and the environment of the samples is evacuated, we are in a weak coupling limit, which implies that distortion is also equal to fidelity. In [83] this equivalence is shown in detail. Furthermore for a block for which we expect “chaotic” ray splitting good agreement with RMT results is demonstrated. In this case perturbation strength as obtained from a fit of RMT is in fair agreement with an estimate resulting from elastomechanics assuming chaotic ray splitting dynamics.

T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

131

0.8 0.6

X(ε)

0.4 0.2 -0.0 -0.2 -0.4 0.0000

0.0004

0.0008 dilation ε

0.0012

Fig. 49. The typical form a correlation function computed according to Eq. (296) (figure taken from [68]).

1.6

D(t)

1.2

0.8

0.4

0 0

20

40

60 t / [msec]

80

100

Fig. 50. (Color online) The distortion as a function of time, for the cuboid (figure taken from [83]). The thick solid lines correspond to measurements in the frequency ranges from 100 to 800 kHz, in steps of 100 kHz (from bottom to top). The thin solid lines show the best fits with − ln[f (t)] where f (t) is given in Eq. (130) with V = 1. The fit values for 0 can be found in [83].

Actually RMT also fits very well the results for a regular aluminum brick (upper right sample in Fig. 48); see Fig. 50. While this seems at first sight surprising it is in line with findings in Ref. [272], which suggest that apart from a number of symmetries, the system displays wave chaos. 9.3. Fidelity decay and decay of coherences Under special circumstances the time evolution of two coupled systems can show an interesting relation between decay of coherences in one subsystem and fidelity decay in the other one. Basically two types of coupling are described in [48] for which this occurs. On one hand, we can have what we might call a dephasing situation, with a Hamiltonian of the form H = H c + H e + O c Ve ,

[Oc , Hc ] = 0.

(297)

Here as usual operators with indices c and e operate only on either of two subsystems termed somewhat arbitrarily central system and environment, and Oc is some operator commuting with the Hamiltonian with eigenvalues √ o when applied to some eigenfunction  with Hc  =
  . Then it is easy to prove, that for an initial state (1/ 2) (j + k ) ⊗ 0

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T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

the elements of the density matrix of the central system, i.e. the coherences relate to fidelity decay in the environment as cj,k (t) = e−i(
j −
k ) 0 |e−it (oj Ve +He ) eit (ok Ve +He ) |0  = e−i(
j −
k ) f (t).

(298)

Up to the phase factor the right-hand side is clearly the fidelity amplitude resulting for the function 0 which can be chosen arbitrarily in the environment with an unperturbed Hamiltonian oj Ve + He and a perturbed Hamiltonian ok Ve + He . If we can measure the coherences, we can measure fidelity, which does not depend on the phases. Moreover, if we know the phases, we actually can measure the fidelity amplitude. A particularly simple situation occurs if the internal evolution of the central system for some reason is irrelevant. This can occur, if the two states defining the density matrix element we consider, are degenerate, or if the evolution of the internal system is so slow, that coherence decays on a much shorter time scale. It has been argued [182–184] that this can occur if |oj − ok | is just large enough. On the other hand Gardiner, Cirac and Zoller [33,34] proposed long ago an experiment to test the sensitivity to changes of dynamics of chaotic systems. Actually the idea they proposed can be generalized and contains much of what was exposed above. Experiments along the lines given above can be carried out in atomic optics. The non-evolution of the central system as assumed in the original papers by the above authors typically holds, though we saw, that it is not essential. We shall describe below a proposition for such an experiment for a kicked rotor in more detail following Ref. [148]. Yet there is another interesting option presented in [48]. As usual quadratic Hamiltonians present additional opportunities. A harmonic oscillator is coupled linearly to a bath of oscillators as H = H c + H e + H int = 2a † a +



2 b † b +



2g (ab† + a † b ).

(299)

Here a, a † are the annihilation and creation operators for the central oscillator and b , b † the corresponding operators for the bath oscillators.  and  are the corresponding frequencies and g the coupling constants. Now a coherent state basis will produce an adequate factorized basis, which will remain factorized for very long times, and allow a similar development. The distance between complex “positions” z of the coherent states in the central system will determine the strength of the perturbation. We find that the decay of the coherences in the density matrix determines the fidelity decay of an initially not excited environment of oscillators. As we are dealing with an eigenstate of the unperturbed oscillator we expect Fermi golden rule behavior, i.e. exponential decay with the square of the strength of the perturbation, and indeed recover the famous relation |c12 (t)|2 = exp(−t|z1 (0) − z2 (0)|2 )|c12 (0)|2 , [273,91,125]. Historically, it is quite interesting to notice, that the Paris experiment [274] that shows decoherence explicitly can be reinterpreted as a measurement of fidelity decay of the environment at least for large angles. This follows directly from the fact, that the relation between the decoherences and fidelity decay is equally true for well separated Gaussian wave packets. 9.3.1. The Gardiner–Cirac–Zoller approach and the kicked rotor In [33,34] the authors proposed an experiment to study quantum dynamics of the center of mass motion of a trapped atom. Indeed they use the internal DOF of the trapped atom, both to disturb the dynamics and to measure the effect of this perturbation. While they do not use the word explicitly, measurement of the fidelity amplitude using the properties of particular couplings of two-level quantum systems is proposed. As mentioned above their method is a special case of the general theory outlined in [48] and reported above. It is though important to note that the basic idea is given in this reference. A more detailed proposition along these lines to measure fidelity decay for the kicked rotor e.g. is given in [148], which we shall follow in this subsection. The kicked rotor can be modeled experimentally by an atom in a standing light wave [275–277]. For such an experiment typically an atom of mass M is chosen, which has two electronic levels with spacing 0 . The states shall be denoted by |1 and |e for the lower and the excited one. They are driven by two counter–propagating laser fields. The dipole µ of this transition is coupled to the electromagnetic field E(x, t) = E
cos(kL x + L )e−it + c.c. of the lasers with wave number kL , frequency , complex amplitude E, polarization
and phase L . In the rotating wave

T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

133

Fig. 51. Level scheme for the measurement of the Loschmidt-echo (taken from [148]). A classical laser drives the transitions |e ↔ |1 and |e ↔ |2 of a three-level atom with excited electronic state |e and two hyperfine ground states |1 and |2 which have a level spacing of hf . The laser is detuned by  from the transition |e ↔ |1 and by  + hf from the transition |e ↔ |2.

approximation the Hamiltonian describing this interaction reads as ( ) pˆ 2 2 −it ˆ Hint = + 20 |ee| + cos(kL x + L )e |e1| + H.c. , 2M 2

(300)

with the Rabi-frequency  = µ
E/2. A large detuning  = 0 −  allows to adiabatically eliminate the excited state |e leading to the Hamiltonian pˆ 2 Hˆ 1 = + 21 [sin(2kL x) + 1]
T (t) 2M

(301)

for the state |1 in a frame rotating with the laser frequency. The interaction strength is given by 1 = 2 /(8) and we choose the phase L appropriately. Moreover, the periodic kicks theoretically described by the train of
-functions in Eq. (301) can be approximately realized by rapidly switching on and off the laser fields with period T. Therefore, the Hamiltonian equation (301) corresponds to the Hamiltonian of the kicked rotor. In order to measure fidelity a similar setup can be applied using atom interferometry [278,277]. For this reason we take again an atom with excited electronic state |e but two hyperfine ground states |1 and |2 separated by the hyperfine-splitting hf , as illustrated in Fig. 51. As above for large detuning  we can eliminate the excited level and find for the lower hyperfine doublet the Hamiltonian Hˆ g = Hˆ 1 |11| + Hˆ 2 |22| with pˆ 2 Hˆ 2 = + {22 [sin(2kL x) + 1] + 2hf }
T (t). 2M

(302)

Here we analogously defined a second interaction strength 2 =

2 = 1 − d, 8( + hf )

(303)

where d = 1 hf /( + hf ). We mention that the physical quantity d is then the perturbation strength, which has to be rescaled if results are to be compared e.g. with a random matrix model. The state for the motion of the atom, prepared in a internal superposition of states |1 and |2, propagates in two different potentials described by the Hamiltonians Hˆ 1 and Hˆ 2 , Eqs. (301) and (302), respectively. Indeed, this is exactly the situation we need, in order to realize a measurement of the fidelity amplitude. f˜N = 0 | Uˆ 2−N Uˆ 1N |0 ,

(304)

where Uˆ i describes the time evolution with the Hamiltonian Hˆ i . To achieve this the atomic state must be initially in a superposition of the internal states |1 and |2, | =

√1 (|1 + |2)|0  2

(305)

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of the composite system of internal and external degrees of freedom. Here |0  represents the initial state of the center-of-mass motion. The time evolution leads to |(t = NT ) =

√1 [|1|N  + |2|N ]. 2

(306)

 Fidelity can now be extracted by determining the probability WN ( ) ≡ dx |j ( )|x| N |2 to find the atom in the internal state |j ( ) ≡ √1 [|1 + e−i |2]. Here we used the position states |x in order to trace over the external degrees 2 of freedom. We find WN ( ) = 1 + Re[e−i N | N ]

(307)

from which we can finally calculate the real and imaginary part of the fidelity amplitude Re f˜N = WN (0) − 1,

Imf˜N = WN ( /2) − 1.

(308)

Due to the constant potential terms in the Hamiltonians Hˆ 1 and Hˆ 2 , Eqs. (301) and (302), the measured fidelity amplitude f˜N picks up a constant phase factor exp[−i(hf − d)N ]. Thus the desired fidelity amplitude as defined in Eq. (4) is related to the measured fidelity amplitude via fN = ei(hf −d)N f˜N .

(309)

For a specific implementation of the above scheme, the authors in [276] propose to use the D2 level of Sodium. More technical details are given in [148]. It is important to note, that we have specifically selected the example of the kicked rotor, because it has recently received attention [276,175] and because of the paradigmatic character it acquired in chaos theory due to many important contributions of Chirikov and others. Yet the original proposal referred to the movement in some trap [33,34] and due to the flexibility that lasers permit, many different Hamiltonians can be studied in this way. Also the perturbation strength can easily be varied. Similar experiments have been performed [279–281], using certain variants of the idea of Gardiner, Cirac and Zoller, leading to related quantities. 10. Summary Echo-dynamics and studies about fidelity decay have received considerable additional attention as we were writing this paper, and we apologize, if we missed some recent developments, though we believe that the basic ideas are covered and we are more likely to miss some applications. Echo-dynamics is basically defined by the forward evolution with some unitary dynamics, followed by a backward evolution with a perturbed one. We have focused on Hamiltonian time evolution and based our treatment on the fact that the two-point time correlation function in the interaction picture essentially describes the process, at least for the functions or observables that are usually considered. The most common object of research is fidelity, i.e. the autocorrelation function of some initial state under echo-dynamics. This quantity is particularly important, because it is equivalent to the cross correlation function of some initial state evolved with an ideal and a perturbed Hamiltonian, which often serves as a benchmark for the reliability of quantum information processes. The behavior of other quantities, such as expectation values of operators, purity or S-matrix elements, were also considered. The basis of the entire analysis resides in the observation, that a system is the more stable under perturbations, the shorter the temporal correlations of the perturbation operator in the interaction picture. This leads directly to the relative harmlessness of noisy perturbations, but also to the slightly counterintuitive result that typically a time independent or time periodic perturbation will affect stability more strongly in integrable systems then in chaotic ones. It also leads to a way of stabilizing quantum information processes by introducing chaotic or random elements, or outright engineering rapid decay of the correlations of unavoidable errors by adequate gates. Most of the standard results are obtained from linear response approximation or in other words from the fist nonvanishing term in a Born expansion. The remarkable fact arises, that in most instances the simple exponentiation of this term gives a result of wide validity, which can be proven in some instances, and in others is confirmed by comparison with numerics. Our treatment considers integrable and chaotic situations, as well as random matrix models for the latter. The concept of integrability and chaos carries over to situations with no classical analogue by considering the behavior of correlation functions but is supported by spectral statistics.

T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

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When surveying experiments, we find that measurements of the fidelity amplitude are possible in quantum optics due to a proposal by Gardiner et al., which is somewhat dated, but has never been strictly implemented, though related more complicated quantities have been measured. On the other hand, fidelity of S-matrix elements has been tested directly or via Fourier transform of measurements in the energy domain in elastic and microwave experiments, respectively. For chaotic systems scattering fidelity is a good approximation for the standard fidelity, particularly if ensemble averages are taken as was the case in both experiments. Then a good agreement with the results of random matrix theory was observed. The experiments performed show clearly, that effects as observed in the energy domain have no clear signature, thus justifying the general interest in the time domain studies. The experiments are at this point limited to weak and intermediate perturbation strength. The intriguing Lyapunov regime, which shows decay independent of perturbation strength for some time range and sufficiently strong perturbation has not yet been reached, and neither has the revival at Heisenberg time, predicted by exact solutions of the random matrix models. Both experiments will be quite challenging, as they imply measurements of very low fidelities. While we have concentrated on the correlation function approach, we mention other approaches, that were used to obtain results in different regimes. Vanicek’s semi classical approach seems to be the most flexible, as he is also able to obtain essentially all the regimes. Yet we have treated systems such as kicked spin chains that have no known classical analogue and nevertheless can be treated with the correlation function method. Fidelity decay of mixed systems has been treated on the margin only, as only limited studies exist, and the matter seems in first approximation to result in a separate treatment of integrable and chaotic parts. For future developments, the evolution of expectation values of relevant operators seems to be a promising field, but also scattering fidelity has at this point only been used where it is equivalent to fidelity in a chaotic system. Perturbations of scattering channels are an interesting alternative. Also the use of quantum freeze of fidelity in information processes presents interesting perspectives. Acknowledgments We acknowledge the hospitality of Dr. Brigitte Ibing and of the Centro Internacional de Ciencias, Cuernavaca, which hosted us during a significant part of the effort to write this article. We are grateful to L. Kaplan, H.M. Pastawski, H.-J. Stöckmann and J. Vaníˇcek who sent us many useful comments after reading the manuscript. We also wish to thank G. Benenti, R. Blatt, G. Casati, B. Georgeot, R. Jalabert, H. Häffner, H. Kohler, T. Kottos, C. Lewenkopf, F. Leyvraz, C. Pineda, M. Raizen, W. Strunz and G. Veble. Financial support from the Slovenian Research Agency under program P1-0044 and project J1-7437, CONACyT projects # 41000-F and 44020-F as well as UNAM-DGAPA project IN-100803 are acknowledged. T.G. acknowledges financial support from the EU Human Potential Programme no. HPRN-CT-2000-00156 at the initial stage of the work. M.Ž. acknowledges a fellowship of the Alexander von Humboldt Foundation and the hospitality of the Department of Quantum Physics of the University of Ulm. Appendix A. Time-averaged fidelity In this appendix we discuss and illustrate the time-averaged fidelity for different initial states (eigenstates of H0 , random pure states, or completely mixed states) and in the crossover regime between weak and strong perturbation [46,282]. The point of crossover
rm from weak (43) to strong (44) perturbation regime is system dependent and cannot be discussed in general apart from expecting it to scale with 2 similarly as a mean level spacing
rm ∼ 2d . We will discuss the value of F¯ for three different initial states: (i) First, let us consider the simplest case when the initial state is an eigenstate of U0 say,  = |E1 E1 | with matrix elements lm =
l,1
m,1 . For weak perturbations this gives (43) F¯weak = 1, therefore the fidelity does not decay at all. This result can be generalized to the case when  is a superposition or even a mixture of a number of eigenstates, say K of them, all with approximately the same weight, so that one has diagonal density matrix elements of order ll ∼ 1/K, resulting in F¯weak ∼ 1/K. On the other hand, for strong perturbations ε?εrm we get F¯strong = (4 − )/N for an initial eigenstate. Here = 1 for systems with anti-unitary symmetry where the eigenstates of H0 can be chosen real and = 2 for the case where anti-unitary symmetries are absent.

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1

S•F

0.1 S = 100 S = 200 0.01

S = 400 S = 800

1e-05

1e-04

0.001

0.01 ε

0.1

1

10

Fig. A1. Dependence of time-averaged fidelity (multiplied by the Hilbert space size N = S) F¯ on ε is shown for a chaotic kicked top system and Hilbert space average  = 1/N , i.e. our case (iii). The transition from weak to strong perturbation regime is seen (A.3). Horizontal full lines are the theoretical predictions F¯strong (A.3), while the theoretical result for the weak regime corresponds to 1.

Summarizing, for an initial eigenstate we have time-averaged values of fidelity F¯weak = 1,

F¯strong = (4 − )/N.

(A.1)

With this simple result we can easily explain the numerical result of Peres [31] where almost no decay of fidelity was found for a coherent initial state sitting in the center of an elliptic island, thus being a superposition of a very small number of eigenstates (it is almost an eigenstate). The behavior in a generic case may be drastically different as described in the present work.  (ii) Second, consider the case of a random pure initial state | = m cm |Em , giving ml = cm cl∗ . The coefficients cm are independent random complex Gaussian variables with variance 1/N , resulting in averages |lm |2  = 1/N 2 for m = l and 2ll  = 2/N 2 (average is over Gaussian distribution of cm ). Using this in the expressions for average fidelity (43) and (44) we get F¯weak = 2/N,

F¯strong = 1/N.

(A.2)

The result for weak perturbations agrees with case (i) where we had F¯weak ∼ 1/K if there were K participating eigenvectors. (iii) Taking a non-pure initial density matrix  = N −1 1, we have F¯weak = 1/N,

F¯strong = (4 − )/N 2 .

(A.3)

As expected, the fluctuating plateau is the smallest for maximally mixed states and strong perturbation. Observe that the average fidelity F¯ (42) is of fourth order in matrix elements of O, the same as the inverse participation ratio (IPR) of the perturbed eigenstates. Actually, in the case of initial eigenstate, our case (i), the average fidelity (42) can  be rewritten as F¯ = m |O1m |4 , which is exactly the IPR. We recall that Omn = Em |Elε  is the overlap matrix between perturbed and unperturbed eigenstates. The inverse of the IPR, i.e. the participation ratio, is a number between 1 and N which can be thought of as giving the approximate number of unperturbed eigenstates represented in the expansion of a given perturbed eigenstate. For an average over the mixed states, case (iii), we have instead F¯ = l,m |Olm |4 /N 2 , i.e. the average IPR divided by N . The time-averaged fidelity is thus directly related to the localization properties of eigenstates of Uε in terms of eigenstates of U0 . Therefore, fidelity will decay only until it reaches the value of finite size fluctuations and will fluctuate around F¯ thereafter. The time t∞ when this happens, F (t∞ ) = F¯ , depends on the decay of fidelity and is discussed in Section 7.

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137

1

ε =4 S•F

0.1 S = 100 S = 200 0.01

S = 400 S = 800

1e-05

1e-04

0.001

0.01 ε

0.1

1

10

Fig. A2. The same as Fig. A.1 but for a regular kicked top.

To illustrate the above theory we have calculated the average fidelity (42) for a kicked top with a propagator (B.1). As an initial state we used  = 1/N, i.e. the case (iii), where the dimension of the Hilbert space is determined by the spin magnitude: N = S (OE subspace). We calculated the dependence of S F¯ on ε for two cases: a chaotic one for kicked top parameters  = 30,  = /2 shown in Fig. A.1 and a regular one for  = 0.1,  = /2 shown in Fig. A.2. In both cases one can see a transition from the weak perturbation regime F¯weak = 1/N to the strong regime F¯strong = 3/N 2 for large ε. In the chaotic case the critical εrm can be seen to scale as εrm ∼ 2 = 1/S. In the regular situation, the strong perturbation regime is reached only for a strong perturbation ε ∼ 4, where the propagator Uε itself becomes chaotic. (The transition from the regular to chaotic regime in the kicked top happens at around  = 3, see e.g. [31].) Still, if one defines εrm as the points where the deviation from the weak regime starts (point of departure from 1 in Fig. A.2) one has scaling εrm ∼ 1/S also in the regular regime. Appendix B. Models for numerics Here we shall give a brief overview of models we used to perform numerical experiments to demonstrate various theoretical results. Except for several occasions, we use kicked systems, for which the Hamiltonian is time periodic, consisting of free time evolution, interrupted periodically by instantaneous “kicks”. For such systems, time is a discrete variable, given by the number of kicks. These systems can frequently be simulated much more efficiently, than timeindependent systems, if chaoticity is required. Most extensively, we use the kicked top (Section B.1) and the kicked Ising chain (Section B.2). In some occasions, we use the sawtooth map [283] and the famous kicked rotor [284]. In addition to kicked models we also use the Jaynes–Cummings system, well known in quantum optics [285], and a system of coupled anharmonic oscillators. These are the only specific numerical models with a time independent Hamiltonian we are discussing in this review. Jaynes–Cummings model is used only once (Fig. 42), and for further details, we refer the interested reader to Ref. [162], from which Fig. 42 has been taken. Anharmonic oscillators are used in Section 5.4 with more details available in the original Ref. [181]. B.1. The kicked top The kicked top has been introduced by Haake et al. [286] and has served as a numerical model in numerous studies ever since [19,206,207,287–289]. The kicked top might also be experimentally realizable [290]. We shall use different versions of the kicked top leading to different propagators depending on the phenomenon we want to illustrate. All of them are composed as a product of unitary propagators, each depending on standard spin operators, Sx,y,z . The latter fulfill the following commutator relations: [Sk , Sl ] = iεklm Sm (εklm is the Levi–Civita symbol). The half-integer

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(integer) spin S determines the size of the Hilbert space, since N = 2S + 1, and therefore the value of the effective Planck constant 2 = 1/S. The semiclassical limit corresponds to the limit of large spins S → ∞. Mostly we use the standard kicked top whose one step propagator (Floquet operator) is given by   Sz2 Uε = exp(−iSy ) exp −i( + ε) . (B.1) 2S Here ε determines the perturbation strength, while the parameters  and  are two parameters, which allow to change the dynamical properties of the system. The propagator for t steps is simply a power (Uε )t . The Hamiltonian which generates Uε is given by
 ∞ Sy  1 Sz 2 Hε = ( + ε) +
(t − k), (B.2) 2 S S k=−∞

where the perturbation is
 1 Sz 2 , V = 2 S

(B.3)

which has the classical limit V → v = z2 /2 (we use lower case letters for corresponding classical observables). In the limit S → ∞, the area preserving map corresponding to Uε can be written as a map on the unit sphere. Its explicit form can be obtained from the Heisenberg equations for the spin operators. The angle  in Eq. (B.1) is usually set to /2, whereas we shall either use  = /2 or /6. For these two cases, the decay of the classical correlation function displays two different behaviors, monotonic decay for  = /6 and oscillatory decay for  = /2, see Fig. 1. In addition, the symmetries are different. For  = /2 the propagator U0 commutes with the operator exp(−i Sy ), which describes a

rotation around the y-axis. The Hilbert space can be decomposed into three invariant subspaces. Following Peres’s book [31] we denote them with HEE , HOO and HOE , with the basis states √ EE : |0, {|2m + | − 2m}/ 2, √ NEE = S/2 + 1, (B.4) 2, NOO = S/2, OO : {|2m − 1 − | − (2m − 1)}/ √ √ OE : {|2m − | − 2m}/ 2, {|2m − 1 + | − (2m − 1)}/ 2, NOE = S, where we assume S to be even. Here, m runs from 1 to S/2 and |m are standard eigenstates of Sz . For  = /2 the subspaces HEE and HOO coalesce and we have just two invariant subspaces. Unless stated otherwise, we always work in the subspace HOE . Regardless of the angle , the kicked top propagator (B.1) has an anti-unitary symmetry. The parameter  determines the degree of chaoticity of the corresponding classical map. The classical map is mostly regular for small values of  (by regular we mean close to integrable), at  ∼ 3 (see e.g. Ref. [31]) most of the tori disappear while for larger  > 3 the system becomes numerically fully chaotic. We use  = /2 and  = 0.1 to simulate regular dynamics. The small value of  allows to use the integrable dynamics for  = 0, to compute v¯ in Eq. (176) in an approximate way: v¯ ≈ (1 − z2 )/4. To simulate chaotic dynamics, we use  = /2 or /6 and  = 30. For  = /6 the integral of the classical correlation function is cl = 0.0515 while it is much smaller, cl = 0.00385, for  = /2 due to the oscillating nature of the correlation function. In Section 5.2.3 we describe fidelity decay averaged over the position of initial coherent states for regular dynamics and a perturbation with non-zero time average. For that purpose we use a different perturbation (as compared to the propagator in Eq. (B.1)):   Sz2 Uε = exp(−i( + ε)Sy ) exp −i , (B.5) 2S with  = 0.1 and  = /2 as before. A third version of the kicked top is used to demonstrate the fidelity freeze. For the chaotic case discussed in Section 3.2 we use the following propagator:     Sz2 Sx2 − Sz2 Uε = exp −i exp −i Sy exp −iε , (B.6) 2 2S 2S

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139

with  = 30. For that perturbation, there is an associated operator W such that V is given as a time-derivative [see Eq. (29)] of
 1 Sz 2 . (B.7) W= 2 S Note that in contrast to the time-independent case, considered in Section 2, for kicked systems one has [17] V =W1 −W0 , where Wt = U0−t W U t0 . For the semiclassical calculation of the plateau height the classical limit w = z2 /2 is needed. The long time decay beyond t2 , the duration of the freeze, is governed by the operator R (32). In the present case, it is given by R=−

Sx Sy Sz + Sz Sy Sx . 2S 3

(B.8)

The classical correlation integral corresponding to this operator is R = 5.1 × 10−3 . In Section 5.3, fidelity freeze in regular systems is discussed. There, we use the propagator   Sz2 Uε = exp −i exp(−iεSx ), 2S

(B.9)

with  = 1.1. For the calculation of the plateau, one needs to represent the unperturbed system in action angle variables. In a spherical coordinate system,these variables are given by   z = j. (B.10) x = 1 − j 2 cos , y = 1 − j 2 sin ,  The classical limit of the perturbation, expressed in terms of the action j and the angle is v = 1 − j 2 cos , while the classical limit w of the associated operator W reads as  1 sin( − /2) w(j, ) = 1 − j2 , (B.11) 2 sin(/2) where  = j is the frequency of the unperturbed system. In Section 4.3.4, discussing quantum freeze in RMT, we need a propagator having an anti-unitary or a unitary symmetry, i.e., belonging to COE or CUE class. So far all kicked top models described had an orthogonal symmetry. Specifically, to demonstrate freeze for a perturbation that breaks time-reversal symmetry we used   Sz2 sym 1/2 1/2 Uε = P exp(−i Sy /2.4)P exp(−iεSx ), P = exp −i (B.12) − iSz , 2S with standard  = 30. For ε = 0 the propagator belongs to COE symmetry class while only unitary symmetry remains for nonzero perturbation ε. Symmetrized version of the propagator is chosen (half of twist P before and half after the sym rotation) in order to have zero time average of the perturbation Sx . In the eigenbasis of unperturbed U0 the matrix of Sx is complex antisymmetric. The classical correlation integral of the perturbation is cl = 0.16. To see the radical difference between the freeze and non-freeze fidelity decay we also show numerics for an unsymmetrized propagator, nosym

Uε

= P exp(−i Sy /2.4) exp(−iεSx ),

(B.13) nosym

nosym

for which the matrix for Sx does not have zeroes on the diagonal in the eigenbasis of U0 . Evolution with Uε sym will therefore not result in a quantum freeze, whereas for a symmetrized version Uε we are going to have freeze. To show the agreement between RMT for the unitary case and numerics we used the following one step propagator: 


Sx2 Uε = P exp −i Sy exp −i10 − i(1 + ε)Sx , (B.14) 2.4 2S with the same P as for the orthogonal case (B.12). In this case the integral of the correlation function is slightly larger, cl = 0.17. Note that in order to see freeze for this case we have to set the diagonal elements of the perturbation to zero by hand.

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B.1.1. Two coupled kicked tops On several occasions we will use a double kicked top. Its propagator is composed of the coupling term and two single kicked top propagators. To illustrate the dependence of fidelity freeze in chaotic systems (described in Section 3.2) on the number of degrees of freedom we use two coupled kicked tops with the one step propagator Uε = exp(−iSzc Sze ) exp(−i Syc /2) exp(−i Sye /2) exp(−iεV S),

(B.15)

with a strong coupling  = 20 to ensure chaotic dynamics. The perturbation V is chosen similar to that of a single kicked top (B.6) as V =

(Sxc )2 − (Szc )2 + (Sxe )2 − (Sze )2 . 2S 2

(B.16)

This perturbation can be generated from W=



 1 Szc 2 1 Sze 2 + . 2 S 2 S

(B.17)

With the superscripts c and e, we denote the Hilbert spaces (“central” system or “environment”), on which the respective operator acts. For numerical simulations we desymmetrized the system. For that purpose, we project the initial state to an invariant subspace of dimension N = S(S + 1) spanned by {HOE ⊗ Hr }sym , where Hr = H\HOE and {·}sym is a subspace symmetric with respect to the exchange of the two tops. The integral of the classical correlation function corresponding to the operator R is R = 9.2 × 10−3 . To study the reduced fidelity and echo purity for regular systems (Section 5.4), we use Uε = exp(−ic Syc ) exp(−ie Sye ) exp(−iε(Szc )2 (Sze )2 /S 3 ),

(B.18)

√ where it is important to choose incommensurate parameters, c = /2.1 and e = / 7 to avoid resonant behavior. Note that here the unperturbed dynamics is uncoupled. Therefore echo purity and purity coincide. For this system, the classical actions are given by the coordinates yc,e , so the classical time-averaged perturbation is 1 v¯ = (1 − jc2 )(1 − je2 ). 4

(B.19)

In Sections 7 and 3.3 we use Uε = Uc Ue exp(−i( + ε)Szc Sze /S),

(B.20)

with standard single kicked top propagators
2  1 Szc,e . −ic,e 2 S

 Uc,e = exp(−ic,e Syc,e ) exp

(B.21)

In Section 7 where we compare timescales of fidelity decay in regular and chaotic system we use c,e = 0, c,e = /2 and the coupling  = 5 for chaotic and  = 1 for regular dynamics. For chaotic dynamics the integral of the correlation function is cl = 0.058. On the other hand, in Section 3.3 demonstrating purity decay for chaotic dynamics we use c,e = 30 and c,e = /2.1 and uncoupled unperturbed system,  = 0, giving the integral of the correlation function cl = 0.056. In all single kicked top simulations, the initial state |(0) used for the computation of fidelity decay, is either a random state with the expansion coefficients cm =m|(0) being independent Gaussian complex numbers or a coherent state. For double kicked top the initial coherent states are products of coherent states for each top. A coherent state

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141

centered at the position r∗ = (sin ϑ∗ cos ∗ , sin ϑ∗ sin ∗ , cos ϑ∗ ) is given by |ϑ∗ , ∗  = =


1/2 S  ∗ 2S cosS+m (ϑ∗ /2) sinS−m (ϑ∗ /2)e−im |m S+m

m=−S ∗ e−i S

(1 + ||2 )S

∗

 = ei tan(ϑ∗ /2),

exp(S− )|S,

(B.22)

with S± = Sx ± iSy . In the semiclassical limit of large spin S the expansion coefficients of the above state tend to ∗ cm = m|ϑ∗ , ∗   exp(−S(m/S − z∗ )2 /2(1 − z∗2 ))e−im , therefore the squeezing parameter is =

1 . sin2 ϑ∗

(B.23)

Coherent states have a well defined classical limit and this enables us to compare quantum fidelity for coherent initial states with the corresponding classical fidelity. The initial classical phase space density corresponding to a coherent state is [288]  2S (B.24) clas (ϑ, ) = exp{−S[(ϑ − ϑ∗ )2 + ( − ∗ )2 sin2 ϑ]}.

 The above density is square-normalized as 2clas d = 1. B.2. The kicked Ising-chain As an example of a generic interacting quantum many body model, which may be of some interest also for illustrations relevant for quantum information, we consider the kicked Ising (KI) model [42,49] with the Hamiltonian HKI (t) =

L−1 

j j +1

{Jz z z

j

j

+
p (t)(hx x + hz z )},

(B.25)

j =0

 j where
p (t) = m
(t − mp) is a periodic train of delta functions with period p. The operators x,y,z are the standard j j Pauli spin matrices satisfying the canonical commutation relations [ , k ] = 2i
j k
   . The model represents a chain of L interacting spins 1/2 with periodic boundary conditions subject to a tilted magnetic field. Integrating over one period p we arrive at the Floquet operator: ⎞ ⎛ ⎞ ⎛   j j +1 j j z z ⎠ exp ⎝−i (hx x + hz z )⎠ , (B.26) U = exp ⎝−iJz j

j

where we chose units such that p = 2 = 1. The KI model depends on three independent parameters (Jz , hx , hz ). It is integrable for longitudinal (hx = 0) and transverse (hz = 0) magnetic fields. The KI model is likely to be non-integrable everywhere else. Numerical investigations of the general case of a tilted magnetic field have revealed finite regions of parameter space where the system has an ergodic and mixing behavior, or non-ergodic quasi-integrable behavior, when approaching the thermodynamic limit. For the numerical illustrations of this paper we consider only three typical cases—points on a line in 3d parameter space with fixed parameters J = 1, hx = 1.4 and a varying parameter hz exhibiting three different types of dynamics [42]: hz = 0 (integrable), hz = 0.4 (intermediate, i.e. non-integrable and non-ergodic), and hz = 1.4 (ergodic and mixing or simply “chaotic”). The non-trivial integrability of a transverse kicking field, which somehow inherits the solvable dynamics of its wellknown autonomous version [291,292], is quite remarkable since it was shown [293] that the Heisenberg dynamics can 0j  <j y be calculated explicitly for observables which are bilinear in the Fermi operators cj = (j − izj ) j  xj  with time correlations decaying to the non-ergodic stationary values as ∼ t −3/2 [293].

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Appendix C. Scattering fidelity in different approximations

C.1. Limit of weak coupling, diagonal S-matrix elements For non-diagonal S-matrix elements, the weak coupling limit has been treated in Section 2.6.1, with the result, s (t) tends to the standard fidelity Eq. (63). That was the important case, since it makes sure that the scattering fidelity fab of a closed chaotic system. For a = b, however, we find ⎧ ⎫ ⎨ ⎬     2 2 i(Ej −El )t Sˆaa (t)∗ Sˆaa (t) ∼ 4 2 wa wb (t) e−W t |vj a |4 Oj2l e2 i(Ej −El )t + |vj a |2 |vka |2 Okl e ⎩ ⎭ j,l j =k,l ⎧ ⎛ ⎞⎫ wa wb 1  ⎨ 2 i(Ej −E  )t ⎝ 2  2 ⎠⎬ l = 4 2 3Oj l + e Okl (t) e−W t ⎩ ⎭ N N j l k =j ⎧ ⎫ ⎨ 1  2 i(Ej −E  )t  2 ⎬ wa wb l = 4 2 (C.1) e Okl , (t) e−W t 3f (t) + ⎩ ⎭ N N k =j

jl

whereas

⎧ ⎨

⎫ ⎬  1 |Sˆaa (t)|2  ∼ 4 2 wa wb (t) e−W t 3 + e2 i(Ej −El )t . ⎩ ⎭ N j

(C.2)

l =j

While the first term separates again as desired into an auto-correlation part and the fidelity amplitude, the second term, 2 =
and treat the perturbed spectrum in general, does not. Only in the perturbative regime, where me may assume Okl kl to first order in , we again find the factorization into fidelity amplitude and autocorrelation function. C.2. Rescaled Breit–Wigner approximation and the perturbative regime For this purpose we refer to the rescaled Breit–Wigner approximation [294], which amounts to a treatment of the anti-Hermitian part in Heff in first order perturbation theory. The main point is that due to the rescaling this gives useful results even at relatively strong coupling to the continuum. Here, we also treat the perturbation W in first order perturbation theory. In the eigenbasis of H0 we have      −i /2)t ∗  2 ∗ 2

i(E ∗ 2 −2

i(E +i  /2)t j j l l Sˆab (t) Sˆab (t) = 4 (t) Vka , Vj a e Vj b Vkb Okl e (C.3) jk

l

where O is again the orthogonal matrix that transforms the eigenbasis of H0 into the eigenbasis of H0 . The numbers  in the respective bases. As we treat j and j denote the diagonal element of the anti-Hermitian part of Heff and Heff  2 

W in first order perturbation theory we have Okl =
kl , j = j , and El = El + Wll . Therefore    ∗ Sˆab (t)∗ Sˆab (t) = 4 2 (t) Vj∗a Vj b Vkb Vka e− (j +k )t e−2 i(Ek −Ej )t =



jk

e

−2 i Wkk t



k

∗ Vj∗a Vj b Vkb Vka e− (j +k )t e−2 i(Ek −Ej )t .

(C.4)

j

Under the assumption that W is a random perturbation, uncorrelated with the dynamics of the unperturbed system, we may average over exp[2 i Wkk t] independently, which yields the fidelity amplitude f (t). What remains is just the autocorrelation function of the S-matrix element Sab and we find  Sˆab (t)∗ Sˆab (t) = f (t)|Sˆab (t)|2 .

(C.5)

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143

Note that it has not been necessary to actually perform the rescaling (of the Breit–Wigner approximation). It just serves as an argument for the enlarged region of validity of the approximation presented. C.3. Rescaled Breit–Wigner approximation and linear response Here, we use the rescaled Breit–Wigner approximation to obtain the decaying dynamics for the unperturbed system. We then compute the scattering fidelity in linear response approximation.  Sˆab (t)∗ Sˆab (t) = 4 2 wa wb (t)vb |U0† |va va |U1 |vb ,   |j e−2 i(Ej −ij /2)t j |. U1 = e−2 iHeff t , U0 = e−2 iHeff t =

(C.6)

j

Only U1 depends on the perturbation W (assumed random), so we may write U1 in linear response approximation and average the result over W:   t  t   2 2  ˜  ˜ ˜ dW () − 4 d d W ()W ( ) , (C.7) U1 = U0 1 − 2 i 0

where W˜ (t) = U0 (t)

0

0

−1

W U0 (t). Note that U0 is not unitary, so that U0 (t)−1 = U0 (t)† . Averaging over W yields: # "  t    2 2  −2 i(Ej −Ek ) (j −k ) , (C.8) U1 = U0 [1 − 4 C(t)], C(t) = d d diag 1 + e e 0

0

k

where we assumed that W is taken from the Gaussian orthogonal ensemble, such that Wij Wkl  =
ik
j l +
il
kj . Apart  from the additional factor e (j −k )  we have here precisely the linear response expression of the fidelity amplitude as obtained in [52]. This factor, which depends on the widths of the resonances, tends to smooth out possible level correlations. However, the two dominant features, the quadratic decay for small (perturbative regime) and the linear decay for larger (Fermi golden rule regime) remain present. In [52], we found that the level correlations have at most a 15% effect on the fidelity decay curve. Note that both, U0 and U1 are diagonal in the eigenbasis of H0 . Therefore, we obtain for the cross correlation function above:   ∗ −2 i(Ek −ik /2)t (t) = 4 2 wa wb (t) vj∗b e2 i(Ej +ij /2)t vj a vka e [1 − 4 2 2 Ck (t)]vkb . (C.9) Sˆab (t)∗ Sˆab jk

In the case a = b this simplifies to  (t) = 4 2 wa wb (t) Sˆab (t)∗ Sˆab



|vj b |2 |vj a |2 e−2 j t [1 − 4 2 2 Cj (t)].

(C.10)

j

If we neglect the dependence of C(t) on the resonance widths, we may average independently over the “autocorrelation part” in each term of he sum. In this way, we obtain ⎡ ⎤   (t) = |Sˆab (t)|2  ⎣1 − 4 2 2 N −1 Cj (t)⎦ = |Sˆab (t)|2 f (t), (C.11) Sˆab (t)∗ Sˆab j

i.e. a product of the autocorrelation function and the fidelity amplitude. Note that again it was not necessary to actually perform the rescaling (of the Breit–Wigner approximation). For the case a = b we obtain an additional term in Eq. (C.10):   (t) = 4 2 wa2 (t) |vj a |4 e−2 j t [1 − 4 2 2 Cj (t)] Sˆaa (t)∗ Sˆaa j

+ 4 wa wb (t) 2

 j =k

|vj a |2 |vka |2 e−2 i(Ej −Ek )t e− (j +k )t [1 − 4 2 2 Ck (t)].

(C.12)

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As long as level correlations are not important in the fidelity amplitude f (t), we may average over the autocorrelation part and the fidelity amplitude part independently, and obtain the desired result:  Sˆaa (t)∗ Sˆaa (t) = |Sˆaa (t)|2 f (t).

(C.13)

It will be interesting to investigate situations where the separate averages are still possible, but effects of the width fluctuations on the fidelity amplitude, Eq. (C.8) become noticeable. Appendix D. Echo purity in RMT: technicalities In Eq. (167), 1, I˜, and J˜ denote operators (matrices) on the Hilbert space H. Note that in general, I˜ is hermitian while J˜ is not. Keeping only terms of order up to (2 ε)2 , we find M (t) ⊗ M (t) = [(1 − 2 iε I˜ − (2 ε)2 J˜)0 (1 + 2 iε I˜ − (2 ε)2 J˜† )] ⊗ [. . .] + O(ε 3 ) = 0 ⊗ 0 − 2 iε[I˜0 ⊗ 0 − 0 I˜ ⊗ 0 + 0 ⊗ I˜0 − 0 ⊗ 0 I˜] + (2 ε)2 [J˜0 ⊗ 0 + 0 J˜† ⊗ 0 + 0 ⊗ J˜0 + 0 ⊗ 0 J˜† ] + (2 ε)2 [I˜0 I˜ ⊗ 0 − I˜0 ⊗ I˜0 + I˜0 ⊗ 0 I˜ + 0 I˜ ⊗ I˜0 − 0 I˜ ⊗ 0 I˜ + 0 ⊗ I˜0 I˜] + O(ε3 ), where 0 is completely general. If the initial state is a product state: 0 = c ⊗ e then it can be shown that the linear terms vanish. If, in addition, the initial state is pure (in the full Hilbert space) than the above expression can be reduced to Eq. (84) in Section 2.6.6. In what follows the linear terms are ignored, because they become zero when averaged over the perturbation V. The tensor form p[ . ] obeys a number of symmetry relations, which may be used to reduce the number of terms in the above expression. By writing this form, given in Eq. (165), explicitly in the product basis of the full Hilbert space H, we find p[A ⊗ B] = p[A† ⊗ B † ]∗ = p[B ⊗ A] = p[B † ⊗ A† ]∗ .

(D.1)

Therefore, we have FP (t) ≈ p[0 ⊗ 0 ] − 2(2 ε)2 (AJ − AI ) AJ = 2 Re p[J˜0 ⊗ 0 ] AI = p[I˜0 I˜ ⊗ 0 ] − Re p[I˜0 ⊗ I˜0 ] + p[I˜0 ⊗ 0 I˜].

(D.2)

If the initial state is pure and separable, this expression reduces to the one in Eq. (84) in Section 2.6.6 term by term. We start by averaging over V. In the case of AJ , this is particularly simple. Let us use Greek letters , , . . . as indices which run over the basis states in the full Hilbert space H (these basis states need not be product states). In view of Eq. (169) and applying the rule V V
 =


+


 we obtain ⎛ ⎞  t     J =
 C (t), C (t) = (D.3) d d ⎝1 + e−2 i(E −E )(− ) ⎠ . 0

0



We finally average J˜ over O ∈ O(N ) the orthogonal group and obtain ⎛ ⎞ ⎛ ⎞      1 1 J˜ = | O C (t)O | = | ⎝ C (t)⎠ | = ⎝ C (t)⎠ 1, N N   



(D.4)



such that AJ = 2C(t)p[0 ⊗ 0 ],

C(t) = Re

1  1  C (t) = C (t), N N 



(D.5)

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145

as this sum is always real. This is precisely the same correlation integral, which appeared in the linear response study of the fidelity amplitude, Section 4.1. As in Eq. (128), we may replace the sum of phases with the spectral form factor. This leads to the expression in squared brackets of Eq. (129) for C(t). As I (0) = p[0 ⊗ 0 ] factors out from AJ , we find FP (t) = I (0)[1 − 4(2 ε)2 C(t)] + 2(2 ε)2 AI .

(D.6)

If we could neglect the term AI , the echo-purity decay of an initially pure state would be given by the decay of the fourth power of the fidelity amplitude. For the second part of the echo purity, we have to average the matrix I˜0 I˜, as well as the higher rank tensors: ˜ I 0 ⊗ I˜0 and I˜0 ⊗ 0 I˜. The average over V yield  t  t  + − ± I I
= C


+ C


 , C = d d e2 i(E −E )(± ) , (D.7) 0

I˜ I˜
=



0

O I O . . . O I O
.

(D.8)



Generally, these terms appear within sums which run over all indices , , ,
, and for that case we find    + − I˜ I˜
X 
= O O (C O O
+ C O O
)X 
 


 


=A



X +

 =

=



(BX   + CX   ) + D

 =

 

X

  (AX  + BX   + CX   ) + (D − A − B − C) X ,

where X is an arbitrary tensor, and  + − A= (C + C )O O O O ,

B=



D=

 + − 2 2 (C + C )O O , 

(D.9)





C=

 

 

+ 2 2 − (C O O + C O O O O ),

+ − 2 2 (C O O O O + C O O ).

(D.10)

Distinguishing between the cases  =  and  =  allows to evaluate the group integrals. In this way, we obtain ( ( ) ) 1 1 1 1 2 2 A= 2t − (C+ + C− ) , B = 2t + ((N + 1)C+ − C− ) , N +2 N −1 N +2 N −1 ) ( 1 1 1 D= (D.11) [6t 2 + C+ + C− ], C = 2t 2 + ((N + 1)C− − C+ ) . N +2 N +2 N −1 The new coefficients C± are related to double integrals over the spectral form factor as follows:   t  t 1  ± 0 : + case, C± = C ∼ −B± (t) + B± (t) = d d b2 (E,  ±  ). t : − case, N 0 0

(D.12)

 =

Due to the symmetric sums, the coefficients C± are strictly real quantities—and so are A, B, C and D. As the two point form factor may depend on the energy range, we include here a global average over the full length of the spectrum. Due to D = A + B + C we have   (AX  + BX   + CX   ). (D.13) I˜ I˜
X 
=  




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T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

Let us start with I˜0 I˜. With the help of Eq. (D.13), we find   |I˜ 0  I˜

| = (A|0  | + B |0   | + C|0 |) I˜0 I˜ = 

 


= A0 + BT0 Next, we consider I˜0 ⊗ I˜0 =

+ C1.

   
ε

=

 ε



(D.14)

|I˜ 0 ε ε| ⊗ |I˜
0
 |

(A|0ε ε| ⊗ | 0  | + B|0 ε ε| ⊗ |0  | + C|0 ε ε| ⊗ | 0 |)

= A0 ⊗ 0 +



(B| |0 ⊗ | |0 + C| |0 ⊗ | |0 ).

(D.15)



Finally we find I˜0 ⊗ 0 I˜ =

   
ε

=

 ε



|I˜ 0 ε ε| ⊗ |0 I˜

|

(A|0ε ε| ⊗ |0  | + B|0 ε ε| ⊗ |0  | + C|0 ε ε| ⊗ |0 |)

= A0 ⊗ 0 +



(B| |0 ⊗ 0 | | + C| |0 ⊗ 0 | |).

(D.16)



Collecting all terms, we obtain AI = p[X], X = A0 ⊗ 0 + BT0 ⊗ 0 + C1 ⊗ 0 − Re[| |0 ⊗ (B| |0 + C| |0 )] + | |0 ⊗ (B0 | | + C0 | |),

(D.17)

where summation over  and is assumed. There are seven different tensors, appearing in the expression for X. We compute the purity functional for each of them: p[0 ⊗ 0 ] = I (0),

(D.18)

p[T0 ⊗ 0 ] = I  (0),

(D.19)

p[1 ⊗ 0 ] = ne = dim He , ⎤ ⎡   | |0 ⊗ | |0 ⎦ = (0 )2 = tr(T0 0 ), p⎣

(D.20)



⎡



p⎣

(D.21)



⎤ | |0 ⊗ | |0 ⎦ = tr e [(tr c 0 )2 ] = pdual [0 ⊗ 0 ] = Idual (0),

(D.22)



⎤ ⎡   | |0 ⊗ 0 | |⎦ = ij |0 |klkj |0 |il = I2 (0), p⎣ 

⎡



p⎣



(D.23)

ij ,kl

⎤ | |0 ⊗ 0 | |⎦ = nc tr 20 ,

mc = dim Hc ,

(D.24)

T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

147

where we have used nothing but the fact that 0 is a density matrix, i.e. 0 = †0 and tr 0 = 1. Latin indices are used to indicate basis states of the factor spaces Hc and He . With the help of the symmetry relations of the purity functional, Eq. (D.1), we can show that all quantities except for (D.21) are real. We find that AI = AI (0) + B[I  (0) − Re tr(T0 0 ) + I2 (0)] + C[nc + ne tr 20 + Idual (0)].

(D.25)

This expression must be invariant under arbitrary unitary transformations in the factor spaces Hc and He . Note that p[0 ⊗ 0 ] = pdual [0 ⊗ 0 ]. If 0 is a pure state, i.e. if 0 = ||, some of the above quantities simplify, and we obtain AI = (A − C)I (0) + C(nc + ne ) + B[I  (0) + I2 (0) − Re tr(T0 0 )].

(D.26)

Further simplifications rely not so much on the separability of the initial state, but rather on the possibility to choose a product basis, in which the initial state 0 has real elements. If this is possible, I  (0) = I2 (0) = I (0) and tr(T0 0 ) = 1, such that AI = (A + 2B − C)I (0) + C(nc + ne ) − B.

(D.27)

If, finally, the initial state is also separable, we have I (0) = 1. Then using A + B = D − C, we find: AI = D + (nc + ne − 2)C.

(D.28)

In the limit of large N, D and some parts of C may be neglected. We then find AI =

nc + n e − 2 2 (2t + t − B− (t)), N +2

 B− (t) = 2

 

t

d 0

d b2 (E, ).

(D.29)

0

Due to the latter relation, we obtain for the purity echo decay:
 
 t   n c + ne − 2 2 2  FP (t) = 1 − 4(2 ε) 1 − d d b2 (E, ) . t + t/2 − N +2 0 0

(D.30)

This decay is of the same form as the decay of the fourth power of the absolute value of the fidelity amplitude but faster by a factor of [1 − (nc + ne − 2)/(N + 2)]. D.1. The decoupled case For that case, we work in the eigenbasis of H0 . It might be strictly separable, but another interesting situation would be a weak coupling, treatable by first order perturbation theory and thus breaking the separability of the Hamiltonian without affecting its separable eigenbasis. For echo purity in the linear response approximation, Eq. (D.2) remains valid, but when calculating AJ and AI we have to replace J˜ and I˜ with their bare counterparts J and I. We find that Eq. (D.6) remains valid (if we may perform an independent spectral average), and that only AI needs to be modified. Due to the (almost) separable Hamiltonian H0 it is not realistic to assume that H0 corresponds to chaotic dynamics. The best we can then do with RMT, is to assume that H0 has an uncorrelated Poisson spectrum. In our approach this amounts to setting the spectral two-point form factor to zero. For AJ , this yields: AJ = 2I (0)(t 2 + t/2).

(D.31)

In order to compute AI we use Eq. (D.7), which gives the average of the product of two matrix elements of I and find   + − I I
X 
= (C X  + C X  ). (D.32)  




148

T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

This gives I 0 I =

 

+ − (C |0   | + C |0 |),

I 0 ⊗ I 0 =

 

I 0 ⊗ 0 I =

 

+ − (C | |0 ⊗ | |0 + C | |0 ⊗ | |0 ), + − (C | |0 ⊗ 0 | | + C | |0 ⊗ 0 | |).

(D.33)

+ − Neglecting level correlations in the spectrum of H0 , we may set C =
 t 2 and C =
 t 2 + (1 −
 )t/N . Note + − however, in the case of separable H0 , the matrices C and C are also separable. That might need special treatment. Disregarding this possibility, we find


 t  t I 0 I = 2t 2 − ||0 || + 1, N N 
 t I 0 ⊗ I 0 = 2t 2 − ||0 ⊗ ||0 + N 
 t  2 I 0 ⊗ 0 I = 2t − ||0 ⊗ 0 || + N 

t  | |0 ⊗ | |0 , N 

t  | |0 ⊗ 0 | |. N

(D.34)



Collecting all terms, and applying the purity functional, we get


t AI = 2t − N 2

⎤ ⎡     t 2 2 2 ⎣ (tr e 0 )ii (0 ) + |ij |0 |kj | ⎦ + [ne − Idual (0) + nc tr 20 ]. N  i

(D.35)

ij k

If 0 is a pure initial state this simplifies to


t AI = 2t − N 2

 [Ipr(tr e 0 ) − Ipr 0 + Ipr(tr c 0 )] +

t [nc + ne − I (0)], N

(D.36)

where Ipr  is the sum of the diagonal elements squared of . As such it is a certain generalization of the inverse participation ratio from pure to mixed states. For the purity, we obtain FP (t) = 1 − 2(2 ε)2 [AJ − AI ]  ( ) nc + ne − I (0) − Ipr 2 0 t 2 2 = 1 − 4(2 ε) t [I (0) − Ipr 2 0 ] + 1− + O(ε 4 ), 2 N

(D.37)

where Ipr 2  = Ipr(tr e ) + Ipr(tr e ) − Ipr . If the initial state is a basis state of the product basis, this gives
 nc + ne − 2 t FP (t) = 1 − 4(2 ε) 1− + O(ε 4 ). 2 N 2

(D.38)

If the initial state is a product state of two random states in Hc and He , we find  ( ) t nc + ne − 1 − Ipr 2 0 FP (t) = 1 − 4(2 ε)2 t 2 [1 − Ipr 2 0 ] + 1− + O(ε4 ), 2 N where Ipr 2 0 = 3/(nc + 2) + 3/(ne + 2) − 9/((nc + 2)(ne + 2)).

(D.39)

T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

149

Appendix E. Higher order terms in the Born series We will calculate averages of products of real symmetric matrices with zero diagonal and imaginary antisymmetric matrices with Gaussian independently distributed off-diagonal elements. Both ensembles have been used in the RMT formulation of the fidelity freeze in Section 4.3. Non-zero matrix elements of the perturbation V are independent random numbers with Gaussian distribution, zero mean, and second moments equal to  1
il
j k +
ik
j l − 2
ij
ik
il ; symmetric, Vij Vkl  = (E.1) N
il
j k −
ik
j l ; antisymmetric, for the two cases considered. Equivalently, for i = k the last equation can be stated as  1 1 1; symmetric, Vij Vj i  = , Vij Vij  = N N −1; antisymmetric.

(E.2)

The bracket • denotes an average over a Gaussian distribution of matrix elements. Using the Born expansion of the echo operator (11) the fidelity amplitude is expressed as a sum of integrals of m-time correlation functions, i.e., terms of the form 1 tr V˜ (t1 ) · · · V˜ (tm ), N

(E.3)

where we use a trace for the initial state average. What we would like to do is to get an estimate for the difference between both ensembles for the above average (E.3). Using the eigenbasis of the unperturbed evolution U0 (t) and averaging over the ensemble of V’s, the above correlation function is 1 Vj k Vkl · · · Vrj  exp{i(Ej − Ek )t1 /2 + · · ·}, N

(E.4)

where a summation over indices j, k, . . . , r is implied and we did not write in full the exponential factor involving eigenvalues of U0 , as it is unimportant in the present context. As far as the second order correlation function is concerned, we have already seen that it is the same for real symmetric and for complex antisymmetric ensembles. We shall show that the same holds also for the 4th order term. Forgetting about the exponential phase term, we would like to calculate Vik Vkl Vlp Vpi .

(E.5)

Note that no summation is implied here. We use Wick contraction, giving three contributions, (E.6) denoted terms (i), (ii) and (iii), respectively. By renaming indices as i → k, k → l, l → p and p → i, we see that terms (iii) and (i) are equal. We are mainly interested whether there is a difference between the symmetric and the antisymmetric case. From Eq. (E.2) we see, that the difference could come only from the terms of form Vij Vij , differing in sign between the two ensembles. Let us look if there are any such terms present in (i) or (ii) (E.6). In term (i) the presence of such contraction would immediately mean that we have also diagonal element, e.g. Vii , which though are zero by construction for our perturbations. Similarly, in term (ii) we see that the nonzero term is Vij Vij Vj i Vj i . For the antisymmetric case the minus sign occurs twice. Thus the average is the same for both ensembles. Altogether there are no terms involving one minus sign, i.e. of the form Vij Vij Vkl Vlk , and therefore the 4th order average is the same for both ensembles. For the 4th order term (E.5) one can actually relatively quickly explicitly calculate the average by using (E.1). The result is (
il −
il
ip −
il
ik +
ik
il
ip )/N 2 ; term (i), (
il
kp −
ik
il
ip )/N 2 ; term (ii), regardless of the ensemble.

(E.7)

150

T. Gorin et al. / Physics Reports 435 (2006) 33 – 156

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Physics Reports 435 (2006) 157 – 182 www.elsevier.com/locate/physrep

Parity doubling among the baryons R.L. Jaffe∗ , D. Pirjol, A. Scardicchio Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Accepted 20 September 2006 editor: J.A. Bagger

Abstract We study the evidence for and possible origins of parity doubling among the baryons. First we explore the experimental evidence, finding a significant signal for parity doubling in the non-strange baryons, but little evidence among strange baryons. Next we discuss potential explanations for this phenomenon. Possibilities include suppression of the violation of the flavor singlet axial symmetry (U (1)A ) of QCD, which is broken by the triangle anomaly and by quark masses. A conventional Wigner–Weyl realization of the SU (2)L ×SU (2)R chiral symmetry would also result in parity doubling. However this requires the suppression of families of chirally invariant operators by some other dynamical mechanism. In this scenario the parity doubled states should decouple from pions. We discuss other explanations including connections to chiral invariant short distance physics motivated by large Nc arguments as suggested by Shifman and others, and intrinsic deformation of relatively rigid highly excited hadrons, leading to parity doubling on the leading Regge trajectory. Finally we review the spectroscopic consequences of chiral symmetry using a formalism introduced by Weinberg, and use it to describe two baryons of opposite parity. © 2006 Elsevier B.V. All rights reserved. PACS: 11.30.Rd; 11.40.Ex; 12.39.Fe

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. What is the experimental evidence? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Possible origins of parity doubling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Effective restoration of SU (2)L×SU (2)R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Dynamical suppression of U (1)A symmetry violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Parity doubling and chiral symmetry restoration at short distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Parity doubling and intrinsic deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Implications from mended chiral symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Conclusion and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

∗ Corresponding author.

E-mail address: [email protected] (R.L. Jaffe). 0370-1573/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physrep.2006.09.004

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1. Introduction The possibility that excited hadrons occur in nearly degenerate pairs of opposite parity—a phenomenon known as parity doubling—has been considered from time to time since the 1960s. Early arguments were based on Regge theory [1]. There have been speculations that dynamical symmetries, not appearing in the underlying QCD Lagrangian, might be responsible for this phenomenon [2,3]. Most recently it has been suggested that parity doubling can be explained by “restoration” of the underlying SU (2)L × SU (2)R chiral symmetry of QCD at high energies [4–8] and/or in specific sectors of the spectrum [9,10]. In this paper we carry out a detailed analysis, both phenomenological and theoretical, of parity doubling among the baryons made of u, d, and s quarks. We do not consider mesons here because there is less data and more controversy. Given a concordance on meson resonances, our methods could easily be applied. We address three questions: First, what is the experimental evidence for parity doubling? Second, could a symmetry of the underlying QCD Lagrangian be responsible for parity doubling? And third, even if the origins of parity doubling lie elsewhere, SU (2)L × SU (2)R is of fundamental importance for QCD, and is widely discussed in connection with spectroscopy. What, if any, are its implications for the classification of hadrons? On the first question, we find significant evidence for parity doubling among non-strange baryons (nucleons and
s), but only weak evidence among hyperons (
s and s) with strangeness (S) minus one. There is not enough data to carry out an analysis for s (S = −2) or − s (S = −3). Better information on the existence and classification of baryon resonances would help significantly here. Turning to the possible origins of parity doubling, the most elegant explanations proposed make use of the fundamental symmetries of the QCD Lagrangian: chiral SU (2)L × SU (2)R and the singlet axial symmetry U (1)A . Any underlying symmetry behind parity doubling must involve charges that transform as pseudoscalars, since the multiplets involve states of opposite parity. The natural candidates are the charges of the isospin axial currents, Qa5 (a = 1, 2, 3), and the flavor singlet axial charge, Q5 . U (1)A connects states of opposite parity, but the same flavor and spin. SU (2)L ×SU (2)R charges change both parity and isospin, so multiplets in general contain states of opposite parity and different isospin. Chiral symmetry restoration has been proposed as a mechanism for parity doubling for high excitations in the baryons spectrum [4–8]. This phenomenon was given the name “effective chiral restoration”, which we also use in the following for ease of comparison with the literature. It is equivalent to realizing chiral symmetry linearly on the excited states. As we pointed out in a recent paper [11], SU (2)L × SU (2)R chiral symmetry, when applied to the effective Lagrangian describing the interactions of excited baryons and matter, does not, by itself, lead to parity doubling. One can write down what appear to be Wigner–Weyl representations, where hadrons of equal mass and opposite parity transform into one another under the action of the SU (2) axial charges. However SU (2)L × SU (2)R invariant operators that couple the would-be parity doublets to pions can be added to the effective Lagrangian describing these particles. These operators have the effect of removing the degeneracy of the previously degenerate states and spoiling any relationship between their axial charges. Non-linear field redefinitions make the situation clear (and do not change the physics): The only irreducible representations are multiplets of definite isospin and parity that transform according to the non-linear realization [12–15]. To obtain degenerate pairs of opposite parity (and hence to achieve parity doubling) it is necessary to suppress the families of SU (2)L × SU (2)R invariant pion couplings that split them. The existence of these operators and the need to suppress them dynamically has not been systematically examined in work on effective chiral symmetry restoration [4–6]. A somewhat different approach is taken in Refs. [9,10], where hadrons are assigned to specific representations of SU (2)L × SU (2)R in the context of a linear sigma model. Chiral symmetry is subsequently broken when the sigma field takes a vacuum expectation value. By assigning hadrons to minimal (though not irreducible) SU (2)L × SU (2)R representations, and allowing only low dimension operators in the linear sigma model Lagrangian, a phenomenological framework for describing baryon resonances is constructed. In some examples, Refs. [9,10] take parity doubling as an input when they choose representations and interactions. In fact the example we discuss at length in Section 3 was first introduced in Ref. [16] as a “mirror” model of opposite parity baryons. Because Refs. [9,10] do not attempt either to marshall evidence for parity doubling in the data or explain it in terms of some more fundamental dynamics, we do not discuss them further here. If “effective chiral symmetry restoration” is responsible for parity doubling, then states, to the extent they are parity doubled, must decouple from pions. This prediction can be tested experimentally. Furthermore, only the simplest SU (2)L × SU (2)R multiplets involve only parity doubled pairs of states. Generic representations include states of different isospin as well as parity, which should be characteristic of this mechanism for parity doubling.

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In contrast, the U (1)A symmetry of QCD is broken explicitly by a term in the quantum action induced by instantons [17] and, of course, by quark masses. Thus, its generator, Q5 , the associated singlet axial charge, does not commute with the QCD Hamiltonian [17]. It is also not a symmetry of the QCD vacuum, since, as Coleman showed long ago [18], a charge that does not commute with the Hamiltonian cannot annihilate the vacuum. Thus Q5 |0  = 0. Note, however, that this does not mean that U (1)A is “spontaneously broken”. In a theory like QCD with a non-perturbative qq ¯ condensate in addition to instantons and quark masses, the question of whether a chiral symmetry is explicitly or spontaneously broken is a matter of degree. At one extreme, for zero up and down quark mass, SU (2)L × SU (2)R is clearly spontaneously broken. As an example of the other extreme, consider QED: Since me  = 0, the axial U (1)A associated with the current e ¯  5 e is not a symmetry of QED. Its charge does not commute with the Hamiltonian and does not annihilate the vacuum. In fact one can compute the ee ¯ condensate in perturbation theory. Specifically, in the MS scheme this is ee = m3 /(42 )f (), where f () = 1 +
+ /(2)(5 + 5
+ 3
2 ) + O(2 ), with
= log(2 /m2 ) and  is the renormalization point [19]. Despite the fact that ee  = 0, we say this symmetry is explicitly broken. In QCD with three colors, the importance of explicit U (1)A symmetry violation can be gauged by the magnitude of the  mass, which comes entirely from explicit symmetry violation and is much greater than the scale of spontaneous symmetry breaking, f . Therefore, as discussed by ’t Hooft [17], U (1)A behaves like an explicitly broken symmetry. This is one place where it is not useful to take the large Nc limit in QCD. There the situation becomes reversed because the source of explicit U (1)A symmetry violation disappears and the  becomes a ninth pseudoscalar Goldstone boson (when mu,d,s = 0). In that limit the U (1)A symmetry should be regarded as spontaneously broken. However, we live in a world with Nc = 3 and explicit breaking of U (1)A dominates. Like any explicitly broken symmetry, if the matrix elements that violate U (1)A symmetry are small in a sector of the spectrum, some of its consequences will follow. In the case at hand, parity doubling will be seen in a sector of the spectrum if the matrix elements of the divergence of the flavor singlet axial current are dynamically suppressed in that sector [20]. Other consequences of the symmetry are not obtained: for example the parity doubled states do not necessarily form representations of U (1)A . For the sake of clarity, we refer to this phenomenon as “dynamical suppression of U (1)A symmetry violation” rather than “U (1)A symmetry restoration”.1 To summarize, either chiral symmetry or U (1)A could be responsible for parity doubling. In both cases extra dynamical mechanisms must be at work. For SU (2)L × SU (2)R , chirally invariant operators must be suppressed. If so, hadrons form chiral multiplets, not necessarily limited to parity doublets, and they decouple from pions. U (1)A is more conventional: parity doubling can result if operators that violate the symmetry are suppressed. We also briefly explore some other proposed dynamical origins for parity doubling. We consider the possibility that parity doubling is not directly associated with a symmetry of the underlying Lagrangian, but instead is a consequence of an intrinsic deformation of excited hadrons. If baryons can be accurately described as intrinsically deformed systems, if the intrinsic state spontaneously violates parity, and if the intrinsic state is rigid, i.e. the matrix element between the state and its parity image is small, then when collectively quantized, the spectrum would exhibit parity doubling. This is, in fact, an old idea [1,2,22], developed in early geometrical models of baryons. The arguments in support of this mechanism seem to apply best to states of high angular momentum, and there is evidence for parity doubling at high J among non-strange baryons. However we also find strong evidence for parity doubling among baryons with J = 1/2 and 3/2, which is hard to motivate from this point of view. We also discuss the arguments given by Shifman [8] that the chiral invariance of perturbative QCD, valid at short distances, together with certain dynamical assumptions motivated by the Nc → ∞ (Nc is the number of colors in QCD) limit, lead to parity doubling among mesons, and perhaps by extension, baryons. Finally, although SU (2)L × SU (2)R is in general not apparent in the spectrum, it is not devoid of implications for the spectrum of baryons. Weinberg has shown that hadrons can be classified into multiplets under an SU (2)L × SU (2)R symmetry where the axial generators are associated with pion–hadron vertices, provided hadron–hadron scattering amplitudes obey certain constraints at asymptotic energies [23,24]. Weinberg’s “mended” chiral symmetry does not naturally lead to parity doubling, but it can accommodate it for special choices of the hadrons’ assignments into SU (2)L × SU (2)R representations. We explore the possible spectroscopic consequences of chiral symmetry in the context of Weinberg’s work on “mended symmetries”, and comment on the relation of this approach to the contracted spin-flavor symmetry emerging in the large Nc limit [25,26]. 1 In an earlier version of this paper we argued that suppression of U (1) violation would be less likely among strange baryons. We no longer A believe this to be required. We thank Manohar for discussions on this point.

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The remainder of the paper is organized as follows. In the next section we examine the experimental situation. Despite all the theoretical work, we know of no systematic attempt to quantify the evidence for parity doubling. Typically, papers on the subject include graphs or tables where the reader is invited to see some evidence for a correlation between states of opposite parity (for a recent example, see Ref. [7]). It is difficult, however, to factor in the (often large) widths of states and the fact that some states are much better established than others. Also it is not clear which state to pair with which. Some previous attempts to quantify the evidence for (or against) parity doubling can be found in Refs. [27,28]. We introduce a measure of the correlation between states of opposite parity, (I, J, S), that has some statistical basis. Specifically, we test the hypothesis that two functions, positive and negative parity spectral densities for each isospin (I), strangeness (S), and spin (J), are identical within a tolerance , which measures the extent of symmetry breaking. We construct the spectral functions for each I and J from the baryons accepted by the Particle Data Group [29], including widths, and weighting resonances to reflect their reliability. We have no a priori standard with which to compare Nature’s value of (I, J, S). Instead we construct a set of “alternative realities” by scrambling the parities of the known states. As an ensemble, this set is free from correlations, so the comparison between (I, J, S) computed over this ensemble with the value obtained from Nature’s assignment of parities gives a measure of the significance of the correlation. The method and our results are discussed in detail in Section 2. In Section 3 we look at the fundamental symmetries of QCD to see which, if any, could be responsible for the phenomenon of parity doubling. We review the realizations of chiral symmetry and briefly summarize the arguments of our recent paper [11] on the way that massless Goldstone bosons transforming non-linearly undermine the predictions of linearly realized chiral symmetry for baryon masses and couplings. We go on to state the conditions on the chiral effective Lagrangian that must be satisfied in order to obtain parity doubling, irrespective of the underlying dynamical mechanism responsible for it. We describe the phenomenon of “dynamical suppression of U (1)A breaking”, and show that it can lead to parity doubling in a sector of the Hilbert space of QCD. We also summarize and review arguments for parity doubling from chiral symmetry restoration at short distances, and due to intrinsic hadron deformation. For completeness we summarize the spectroscopic predictions of Weinberg’s approach in Section 4, where we introduce a new example that reproduces naturally some of the predictions of SU (2)L × SU (2)R restoration, without fine tuning of the coefficients in the chiral Lagrangian. Finally in Section 5 we summarize our results and discuss areas for further work. 2. What is the experimental evidence? In this section we evaluate the experimental evidence for parity doubling among the baryons. We have chosen to study the baryons as listed by the Particle Data Group (PDG) [29]. There is still much uncertainty concerning the spectrum of light (u, d, s) baryons and mesons, and other authors have chosen to study different data [4,5]. We chose the PDG listings because they provide a critical summary of many experiments over many decades, and also because they treat baryon states uniformly, assigning masses, widths, and “reliabilities” using their famous “star” system. Data on the meson spectrum accepted by the PDG is less complete, more controversial and lacks the reliability assignments. This point requires some discussion. The PDG has assigned reliabilities to 44 non-strange baryon resonances [30]. For each angular momentum there are two isospin channels, I = 1/2 (nucleon) and I = 3/2 (). The PDG non-strange meson listings are harder to interpret. The “Meson Summary Table” lists (coincidently) 44 well-established non-strange mesons. However those states are distributed over four channels, I = 0 and 1 with C = ±1. Many more states appear in the full meson listings: 70 states can be found in the non-strange meson listings and another 96(!) can be found under the heading “Other light unflavored mesons”. The latter list includes some that have been seen in only one experiment and some that other groups have claimed to exclude. Recently, Bugg [31] has made an independent analysis of the meson spectrum including an attempt at assigning reliabilities, with results quite different from the PDG analysis. It would be very interesting to repeat our statistical analysis on a well-defined set of meson resonances. We find that there is reasonably strong evidence for parity doubling among the non-strange baryons (I = 1/2 nucleons and I = 3/2 s), and weaker, inconclusive evidence among the S = −1 baryons (I = 0 s and I = 1
s). There is not enough data on cascades (S = −2) or s (S = −3) to perform our analysis. The results are summarized in Figs. 1–5, which should be consulted after an explanation of our method. It will be clear that the study of parity doubling is hampered by our incomplete knowledge of baryon resonances, especially those with strangeness. Certainly a new experimental attack on hadron spectroscopy could greatly improve matters.

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225

σ ( MeV)

200 175 150 125 100 75 50 0.5

0.75

1

1.25

1.5

1.75

Ω Fig. 1. The value of  for Nucleons (I = 1/2, S = 0) with J = 1/2 for varying . The value of  with the Nature’s parity assignments is the heavy black line.

7/2

5/2

5/2

J

J

7/2

3/2

3/2

1/2

1/2 0

0.4

0.8

1.2

Ω

(a)

0

0.1

0.2

0.3

0.4

0.5

Ω

(b)

Fig. 2. The value of  at = 125 MeV for nucleons (left) and ’s (right) of different spins J. The value of  with Nature’s parity assignments is marked with an open square.

7/2

5/2

5/2

J

J

7/2

3/2

3/2

1/2

1/2 0.2

(a)

0.4 Ω

0.6

0.8

0.2 (b)

0.6

1

1.4

Ω

Fig. 3. The value of  at = 125 MeV for
’s (left) and ’s (right) of different spins J. The value of  with Nature’s parity assignments is marked with an open square.

Despite many papers on the subject over many years, we know of no attempt to make a (semi-) quantitative evaluation of the evidence for parity doubling. Typically authors simply show the spectrum of states with the same flavor and spin quantum numbers and opposite parity, and rely on the eye of the reader to recognize a propensity for states of opposite

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60 150

50 40

100

30 20

50

10 0

0 1.75

2.00

2.25

2.50

2.75

0.90

0.98

1.06

Ω

1.14

1.23

1.31

Ω

Fig. 4. Histograms of frequency of the value of  in all the possible realizations of parity assignments in the nucleons (left) and ’s (right). All spins less than or equal 27 are included. The value of  with the parities as observed in Nature is indicated with a star. The area on the left of the bar represents the ‘confidence level’ (C.L.) attached to the given . For nucleons we have 95% C.L. and for the ’s 86% C.L.

140 120

40

100

30

80 60

20

40

10

20

0

0 0.91

1.20

1.48 Ω

1.77

1.55

1.91

2.27

2.63

Ω

Fig. 5. Histograms of frequency of the value of  in all the possible realizations of parity assignments in the
(left) and  (right). All spins J = 1/2, . . . , 7/2 are included. The value of  with the parities as observed in Nature is indicated with a star. The area on the left of the bar represents the ‘confidence level’ (C.L.) attached to the given . For
we have 60% C.L. and for the  65% C.L.

parity to cluster nearby in mass. Such an “eyeball” approach has several shortcomings: • baryon resonances have large widths that are hard to display graphically, • different resonances have different statistical significance. Overlap of two well established resonances should be given more weight than overlap of two doubtful states, • there is no reason to expect parity doubling to be an exact symmetry, so spectra must be compared with an allowance for symmetry breaking, and • because of symmetry breaking and resonance widths, in channels with many states it is often not clear which states are to be compared. Recently Glozman and Klempt [27,28] have made more quantitative analyses, which however still suffer from some of the shortcomings listed above. In our analysis we will address all of these difficulties. We know of no way to obtain a rigorous measure of the significance of parity doubling in the physical spectrum, which perhaps explains the absence of work in this direction. However we do believe it is possible to improve on the eyeball. To do this one needs two ingredients: (1) A measure of how much correlation there is between negative and positive parity states; and (2) A control set of “alternative worlds” on which the correlation can be measured and compared with our own. A realistic approach to generating control spectra has to respect several features of the baryon resonance spectrum: (a) states are sparse at low mass, where there is no evidence for parity doubling; (b) the data in each channel show roughly equal numbers of positive and negative parity states; (c) reliabilities decrease and widths increase in general with resonance mass. Reference spectra that ignore these features would not give fair comparisons with the real world. We do not know an algorithm that would generate a priori satisfactory comparison spectra free of these problems.

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Instead we have chosen to use the physical spectrum itself to generate comparisons. Specifically, in each channel of definite J, I, and S, we keep the masses, widths, and reliabilities fixed, and reshuffle the parities, keeping the number of positive and negative parity states fixed. This method generates comparison spectra which respect the characteristics listed in the previous paragraph, but which, as an ensemble, is free from any additional parity correlation beyond the desirable feature that the number of states of opposite parities is roughly equal in each channel. Comparing the real world with these alternative realities tests whether the correlation between states of opposite parity in the real world beats a random ensemble with otherwise similar spectroscopic features. We consider “channels” of specific total angular momentum, J, isospin, I, and strangeness, S. In each channel we define positive and negative parity spectral functions, ± J I S (m), which we take to be a sum over normalized Breit–Wigner resonances with masses and widths taken from the PDG.2 Specifically, we include only the states listed in the “Note on Nucleon and  Resonances” [30]. We weigh the resonances by a confidence factor, Wj , proportional to the number of stars designated by the PDG, because we believe more weight should be given to correlations between well established states, + I J S (m, {C}) =


Wj j

− I J S (m, {C}) =

j

2 (m − mj )2 + 2j /4


Wj j

Cj ,

j (1 − Cj ). 2 (m − mj )2 + 2j /4

(1)

The sum ranges over all states with a definite J, I , and S, and Wj = 1.0, 0.75, 0.50, 0.25 for 4∗ , 3∗ , 2∗ , and 1∗ resonances, respectively. We have also performed the analysis with an exponential weighting, Wj = 1.0, 0.50, 0.25, 0.125, respectively. The effect on our results is negligible except for the
s (I = 1, S = −1), where the shape of the distribution shifts significantly enough to warrant comment (see discussion below). The factor Cj assigns a parity to the j th state: Cj = +1 for positive parity, and Cj = 0 for negative parity. Of course, Nature prescribes a parity for each state as a set of Cj ’s. Notice that the ± I J S are normalized,   ∞
Cj for + , ± ± (2) dm I J S (m, {C}) ≡ NI J S ({C}) = Wj (1 − C ) for − , j −∞ j

so the norm measures the number of states weighted by our confidence that they are real. We do not expect parity doubling to be an exact symmetry of QCD. Instead we expect it to be approximately valid, parameterized by some symmetry breaking scale , which is expected to be of order (say) 50–200 MeV. Our aim, then, is to test the hypothesis that the masses of positive and negative parity states are compatible within fluctuations with a variance . A simple indicator we can use to test the above hypothesis is    |m1 − m2 | + I J S ({C}) = dm1 dm2 I J S (m1 , {C}) erfc − (3) I J S (m2 , {C}),

∞ 2 where erfc(z) = √2 z dt e−t is the complementary error function. erfc(0) = 1 and erfc(z) → 0 when z → ∞. Eq. (3) then can be interpreted as follows. For a given m1 and m2 , erfc(|m1 − m2 |/ ) is the probability that a gaussian random number  with average m1 and variance gives a | − m1 |
|m2 − m1 |, i.e. the probability that m2 is closer to m1 than a randomly generated number (with variance ). We can interpret as the order of magnitude of the breaking of the symmetry (the argument is symmetric in m2 ↔ m1 ). The possible values of m1 and m2 ± are then distributed according to ± I J S (m1,2 , {C}), normalized not to unity, but to NI J S ({C}). I J S ({C}) measures the correlation of the spectral functions with a “forgiveness factor” .  is going to be larger in channels with more states, greater reliability, and, of course, better correlation between states of opposite parity. Since erfc(x) 1 the maximum possible value of I J S occurs if the typical separations |mi − mj | and widths i are much smaller than 2 We do not include the quoted uncertainties in masses and widths since they are generally smaller than the widths of the states, and including them would complicate our analysis without adding much value.

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the chosen . In that case we can set erfc(|m1 − m2 |/ ) ≈ 1 for all m1 and m2 and the two integrals decouple giving and ∗I J S ≡ Max[I J S ({C})] = NI+J S ({C})NI−J S ({C}). Some other features of I J S are worth noting: There is no penalty for the absence of parity doubling low in the spectrum. A state like the nucleon with no parity partner simply does not contribute. Any overlapping states of opposite parity (and the same IJS) contribute to I J S . No choice of multiplet assignments is needed. The contribution of broad states is insensitive to shifts and/or correlations on scales small compared to their widths. We do not know how to judge the a priori significance of the experimental value of I J S because the underlying distributions ± I J S are not normalized to unity, but instead weighted by the number and reliability in that channel—features that we regard to be important in the analysis. So we are led to check the value of  realized in Nature against an ensemble of control spectra generated by scrambling the parities among the otherwise fixed states for each J, I, and S. The crucial feature of these control spectra is that the parities are reassigned randomly. So the comparison between Nature and the control set frames the question whether the parities of the states with a given I, J, S are more correlated than a random assignment of parities. Fig. 1 shows the results in a typical channel, nucleons with J =1/2. 1/2,1/2,0 is plotted for Nature’s parity assignments (in red) and the control sample (in gray), for a range of . The ordering of instances turns out not to depend significantly on , so we lose little by choosing a fixed value of for the rest of the analysis. We have chosen = 125 MeV, though we cannot attribute any physical significance to this choice. Fig. 2 presents the data for nucleons and s (for J 7/2) and Fig. 3 shows the data for
s and s. Data are most abundant for nucleons and s, and for these two channels Nature shows typically more correlation than the alternative realities. To give a quantitative answer, in terms of confidence level, we construct a histogram of the frequency of a given value of . Because the I J S is normalized in a way that reflects the significance of each channel, it makes sense to combine data for sets of channels, thereby obtaining a larger control ensemble. This has been done in Fig. 4 for nucleons and for s. From these figures it is clear that although a small number of parity reassignments would do better, Nature’s correlation between states of opposite parity is quite striking. In numbers the ‘confidence level’ (the area on the left of the starred bin in Fig. 4) for the value of  in the Nature is 95% for the nucleons and 86% for the s. The evidence for parity doubling among s and
s, channel by channel, is shown in Fig. 3, and the statistic is summed over all S = −1 channels in Fig. 5. For the strange baryons we find much smaller values of the confidence level for : 58% for the
s and 72% for the s. We conclude that it is not possible to make a definitive statement in this sector. There may be some correlation, but more data are needed. Fig. 6 shows histogram using the “exponential” weighting factor for the
s. Here, the change in weighting makes a difference: the signal for parity doubling is stronger when well established resonances are more strongly weighted.

200

150

100

50

0 0.47

0.65

0.83

1.02

Ω Fig. 6. Frequency histogram of the value of  for the
’s using the “exponential” weighting of the significance of states. Compare with Fig. 5, left box.

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3. Possible origins of parity doubling In this section we examine some possible explanations for the phenomenon of parity doubling among light quark baryons. In the classical limit, the Lagrangian of QCD with nf light quark flavors has the symmetry SU (nf )L × SU (nf )R × U (1)A × U (1)V .

(4)

This consists of the product of chiral symmetry, SU (nf )L × SU (nf )R , with the singlet axial symmetry, U (1)A , and the baryon number invariance U (1)V . Our understanding of QCD is profoundly shaped by these symmetries, so in the following, we review their possible role in connection with the phenomenon of parity doubling. Since we are studying light quark spectroscopy but wish to keep track of strange quark mass effects, we take nf = 2. We start by reviewing the implications of chiral symmetry for the masses and couplings of hadrons. This part also introduces the notation to be used in the remainder of the paper. We also briefly review the arguments we presented in Ref. [11] that parity doubling does not follow from the underlying SU (2)L × SU (2)R symmetry of QCD. That symmetry can only be restored in the spectrum if classes of SU (2)L × SU (2)R invariant operators are suppressed for some dynamical reason. Next, we consider the possibility that parity doubling is a consequence of dynamical suppression of flavor singlet axial symmetry violation. As is usually the case with explicitly broken symmetries, the symmetry can be approximately realized in the spectrum if the matrix elements of operators that violate the symmetry are suppressed. We then discuss two other possibilities: first Shifman’s attempt to relate parity doubling to chiral symmetry restoration at short distances in QCD, and second the possibility that parity doubling is associated with an intrinsic deformed baryon state, a mechanism familiar from nuclear and molecular physics. 4. Effective restoration of SU (2)L × SU (2)R The Lagrangian of QCD with nf massless quarks has an exact chiral SU (nf )L ×SU (nf )R symmetry, with generators given by the vector T a and axial X a charges. The generators satisfy the commutation relations [T a , T b ] = if abc T c ,

[T a , Xb ] = if abc X c ,

[Xa , Xb ] = if abc T c ,

(5)

where a, b, c = 1, . . . , n2f − 1. For the case of nf = 2 of most interest here, fabc = εabc . The vector and axial isospin currents are exactly conserved, j ja = j j5a  = 0. The generators have different transformation properties under parity: the vector generators, T a , commute with parity, but the axial charges anticommute {X a , P } = 0. This suggests that parity is embedded nontrivially in representations of the chiral group, in a way which could reproduce the observed parity doubling in the hadron spectrum. It is instructive at the outset to review the consequences of current conservation for the spectrum. Conservation of the vector current requires hadrons to lie in degenerate multiplets of definite isospin. The consequences of axial current conservation are embodied in generalized Goldberger–Treiman relations. Consider two hadrons of identical flavor and spin but opposite parity—without loss of generality we can consider baryons of spin and isospin 1/2—and expand the matrix elements j5a  in terms of form factors, ¯ s)( gA (q 2 ) + q gP (q 2 ) + i  q  gM (q 2 ))t a u(p  , s  ), B+ |j5a  |B−  = u(p,

(6)

where q  = p  − p   . Conservation of the axial current demands that q  B+ |j5a  |B−  = 0, which gives
mg A (q 2 ) + q 2 gP (q 2 ) = 0,

(7)

gP (q 2 ) has a pion pole at q 2 = 0 with residue f gB+ B− , where gB+ B− is the s-wave B+ B−  coupling constant. The prefactor of q 2 cancels the pole, leaving a smooth limit as q 2 → 0. For states at rest ( q = 0), where q 0 =
m = m+ − m− and small
m,
mg A + f gB+ B− + O(
m2 ) = 0, where gA = gA (0).

(8)

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Suppose for some reason that B+ and B− were degenerate. Then
m = 0 and Eq. (8) requires gB+ B− = 0—the pion must decouple at zero momentum. The argument goes through for states of different isospin with
I = 1, as well as for mesons and strange and heavy hadrons. Thus any states of the same spin, isospin, etc., but opposite parity that have the same mass must decouple from pions at zero momentum. This is a consequence of axial current conservation. So chiral symmetry does not guarantee the states to be degenerate, but if they are, it forces them to decouple from pions at zero momentum, i.e. in the s-wave. The thesis of Refs. [9,10,4–6] is that hadrons show parity doubling because at masses of order 1.5–3 GeV (where the data show evidence of parity doubling) chiral symmetry is approximately restored in the hadron spectrum. By this they mean that the symmetry is realized (approximately) in Wigner–Weyl mode with the appearance of (approximately) degenerate multiplets that transform into one another linearly under chiral rotations. The irreducible representations of SU (2)L × SU (2)R are labelled by the Casimirs of the left/right-handed group a = T a ∓ X a . The represenfactors (IL , IR ) which are related to the usual isospin and its axial partner by IL,R tation (IL , IR ) contains (2IL + 1)(2IR + 1) states, with total isospin taking all possible values compatible with |IL − IR | I IL + IR . Parity exchanges the left and right-handed isospins IL ↔ IR , such that these irreducible representations are in general not parity invariant, unless IL = IR . We are interested in the baryon states such as N,  with isospins 1/2 and 3/2. These states can be contained only in chiral multiplets, (1/2, 0), (1/2, 1), (3/2, 0), (3/2, 1), . . . , which are not parity invariant. It is easy to form parity eigenstates by taking sums and differences of mirror chiral representations H+ ∼ (IL , IR )(IR , IL ),

H− ∼ (IL , IR )(IR , IL ).

(9)

In this example, H± represents parity-even(odd) hadrons. Therefore a Wigner–Weyl representation of SU (2)L ×SU (2)R generically includes degenerate multiplets of both positive and negative parity and a range of isospins. In addition to degeneracy, the linear realization of the SU (2)L × SU (2)R predicts relations between axial and vector charges typical of a non-Abelian symmetry. In Ref. [11] we showed that decoupling of pions implied by Eq. (7) in the parity doubling limit is not a curiosity. Instead hadrons can only form Wigner–Weyl representations of SU (2)L × SU (2)R if they completely decouple from pions. Indeed this is just the mechanism proposed in Refs. [4–6], where it is argued that states high in the spectrum are generically insensitive to the effects of spontaneous symmetry breaking. Whether or not one accepts these arguments, this dynamical picture makes many predictions that can be tested. First and foremost is the prediction that the parity doubled states should decouple from pions, not only in the s-wave at zero momentum as required by the Goldberger–Treiman relation, Eq. (7), but in higher partial waves as well. Also, the arguments of Refs. [4–6] in the parity doubled domain of the spectrum apply whether or not the particular states are degenerate. All the pion transition matrix elements should be small to the extent that chiral symmetry is “restored”. Second, larger multiplets including states of different isospin should be found in the spectrum. Glozman and Klempt [27,28] have made some initial attempts to classify states into such multiplets, with varying amounts of success. Third, strange and heavy hadrons should show the same phenomenon—the emergence of explicit SU (2)L × SU (2)R symmetry high in the spectrum—since the heavy quarks are merely spectators to this dynamical mechanism. Finally, the phenomenon should be generic. If some well established states high in the spectrum cannot be classified into representations of SU (2)L × SU (2)R , it is evidence that other dynamics is at work. For the sake of completeness, we review the argument of Ref. [11] that pions must decouple from states that exhibit the full structure of SU (2)L × SU (2)R symmetry. Parity invariance of QCD requires that physical states are parity eigenstates. As explained above, this can be realized only if the physical states transform according to sums and differences of chiral representations as in Eq. (9). Thus the combined consequence of chiral symmetry realized in WW mode and parity invariance of QCD, is the appearance of parity doubling in the hadron spectrum. However, we know that in Nature chiral symmetry is realized in Nambu–Goldstone mode with the appearance of Goldstone bosons. Actually two distinct possible NG realizations have been discussed in the literature, distinguished by the invariance of the vacuum. In the usual realization SU (2)L × SU (2)R breaks to isospin. Long ago, Dashen pointed out that SU (nf )L × SU (nf )R could break to SU (nf )V × ZnAf , where ZnAf is the center of axial SU (nf ) [32]. We consider the usual scenario first, and return to Dashen’s alternative at the end of this subsection.

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Consider for definiteness two spin-1/2 baryons of positive (B+ ) and negative (B− ) parity, and unspecified isospin I.3 Assume that they transform linearly under chiral symmetry j

i [T a , B+ ] = −tija B+ , j

i [Xa , B+ ] = tija B− ,

j

i [T a , B− ] = −tija B− , j

i [Xa , B− ] = tija B+ ,

(10) (11)

with i, j isospin indices. This corresponds to the SU (2)L × SU (2)R representations B+ ∼ (I, 0) + (0, I ) and B− ∼ (I, 0) − (0, I ). The most general chirally and parity invariant effective Lagrangian containing only B± , reads L = B¯ + ij/B+ + B¯ − ij/B− − m0 (B¯ + B+ + B¯ − B− ) + · · · ,

(12)

where the ellipses denote terms of higher order in the chiral expansion. The two hadrons are degenerate, in agreement with the arguments of [6]. In addition, the matrix elements of the axial current are also fixed by the symmetry, via the Noether theorem. So far we have neglected the possible coupling of B+ and B− to pions. The pion field transforms nontrivially under the chiral group. This has the consequence that additional operators, constructed from the baryon fields B± and a , can be added to the effective Lagrangian, Eq. (12). For example, using Weinberg’s choice [13,14] for the pion non-linear transformation under axial rotations, [Xa , b ] = −if ab () = −i(ab 21 (1 −  2 ) + a b ),

(13)

(where we expressed the pion field in units of f = 131 MeV) another dimension three, chirally invariant operator can be added to Eq. (12),   1 − 2 4ia t a ¯ ¯ L = m1 B+ B+ − B − B+ − (B+ ↔ B− ) . 1 + 2 1 + 2 Rediagonalizing the terms quadratic in B± we see that the new term has the effect of splitting the degeneracy: the new mass eigenvalues are m0 ± m1 . m1 is the coupling constant set to zero by the generalized Goldberger–Treiman relation if the hadrons B+ and B− are to be degenerate. The effects of pion couplings are most conveniently displayed by redefining the baryon fields, B+ − 2ia t a B− B˜ + = , √ 1 + 2

B− − 2ia t a B+ B˜ − = . √ 1 + 2

(14)

The new fields have also positive (B˜ + ) and negative (B˜ − ) parity, but transform nonlinearly under chiral rotations. Specifically, [Xa , B˜ ±i ] = v0 (2 )εabc c tijb B˜ ±j ,

(15)

with v0 (2 ) a function which depends on the particular definition of the pion field. We use here and in the following the definition adopted in Ref. [14], which corresponds to v0 (x) = 1. Most importantly, the new states B˜ ± do not transform into one another under chiral transformations. Instead of taking B+ ↔ B− as in Eqs. (10), the axial chiral rotation of the fields defined in Eq. (14) transforms each of B˜ i into itself, plus a number of pions. This agrees with the intuitive idea that an axial rotation does not transform hadrons within a chiral multiplet, but just rotates their isospins and creates pions from vacuum. It is easy to construct the most general chirally invariant effective Lagrangian describing the dynamics of the baryons B˜ ± following the usual rules of chiral effective field theory [12–14]. When rewritten in terms of the original fields, B± , we discover several other possible chirally invariant terms, allowed when pions transform nonlinearly, but omitted from Eq. (12). They take the simplest form when expressed in terms of the B˜ ± fields, and are given by L = B¯˜ + (ij/ − ε abc a D /b t c )B˜ + − B¯˜ + (D /a )t a B˜ − + (B˜ + ↔ B˜ − ) − (m0 − m1 )B¯˜ + B˜ + − (m0 + m1 )B¯˜ − B˜ − + L2 , 3 The example which follows is equivalent to the “mirror model” presented in Ref. [16] in the context of the linear sigma model.

(16)

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where D a = 2j a /(1 + 2 ) is the covariant derivative of the pion field. The additional terms L2 contain three operators which couple the pions derivatively to the baryons L2 = c2 [B¯˜ + (D/a )5 t a B˜ + + B¯˜ − (D/a )5 t a B˜ − ] + c3 [B¯˜ + (D/a )t a B˜ − + B¯ − (D/a )t a B˜ + ] + c4 [B¯˜ + (D/a )5 t a B˜ + − B¯˜ − (D/a )5 t a B˜ − ].

(17)

Note that the baryons B˜ ± are not degenerate, but are split by 2m1 . Similar results are obtained for the matrix elements of the axial current, which is obtained from the Noether theorem Aa = (c2 + c4 )B¯˜ +  5 t a B˜ + + (c2 − c4 )B¯˜ −  5 t a B˜ − + (1 − c )(B¯˜  t a B˜ + h.c.) + (pion terms). + 

3

(18)

−

No predictions are obtained from chiral symmetry for these couplings, which can take any values. The scenario of “effective chiral symmetry restoration” discussed by the authors of Ref. [6] requires that the coefficients of the chirally invariant operators m1 , c2−4 in this effective Lagrangian be suppressed. The alternative explanations for parity doubling described in the following subsections, would, in the language of SU (2)L × SU (2)R symmetry restoration, amount to a dynamical mechanism for the suppression of m1 and c2−4 . For further discussion of parity doubling and SU (2)L × SU (2)R symmetry restoration, see Ref. [11]. Note that these results are specific to the assignment of B± to the chiral representations (0, 21 ), ( 21 , 0), and a different chiral assignment will give different predictions for the pion couplings. Finally, for completeness, we describe the alternative scenario suggested in 1969 by Dashen [32], and revisited recently by Kogan et al. [33], wherein the vacuum is left invariant not only by the vector generators {T a }, but also by a discrete symmetry ZnAf (the center symmetry of SU (nf )A ). This scenario has no implications for parity doubling unless nf
3, so we restrict our consideration to the minimal case nf = 3. Although at least three light flavors are required, this scenario would have nontrivial implications for the nonstrange baryons, which are our main focus here. The center of axial SU (3) contains the operator Z = exp((2i/3)X8 ) along with Z 2 and the identity. Since Z does not commute with parity, the complete discrete symmetry of the vacuum is the dihedral group D3 =(1, Z, Z 2 , P , P Z, P Z 2 ). According to Coleman’s theorem [18], the same symmetry should also be apparent in the hadron spectrum. Since D3 has both one- and two-dimensional irreducible representations, the Dashen scenario of chiral symmetry realization could give rise to parity doubling in the hadron spectrum.4 More precisely, there are one-dimensional representations of either parity, and two-dimensional representations, containing parity doublets. ⎯

+

⎯

+ Fig. 7. Typical baryon mass spectrum in the Dashen scenario for chiral symmetry breaking. Shown are two singlets of positive and negative parity, and a parity doublet. Pion transitions h1 → h2  between singlets and doublets are forbidden by the symmetry (dashed arrows), while transitions among singlets and among doublets are allowed (solid arrow). 4 This does not occur in the two flavor case because Z × parity is Abelian and has only one dimensional irreducible representations. 2

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Since mesons are singlets under the discrete D3 group, this symmetry gives a selection rule for the pion strong decays between hadrons: the h∗ → h decay is allowed only between singlets, or only between doublets, but not between singlets and doublets as illustrated in Fig. 7. No clear evidence for this selection rule is observed among the baryons—in fact Dashen points out several problems with it that still pertain today—although symmetry breaking effects through mass terms could obscure it. We refer to the original work [32] for details. In the Dashen scenario the qq ¯ condensate vanishes in the massless quark limit, and f ∼ mq as well. Phenomenological analysis of the pattern of chiral symmetry breaking [33,34] does not rule out this behavior. However Kogan et al. have shown that it is in conflict with rigorous QCD inequalities [35]. So we do not consider the Dashen scenario further as a possible origin for parity doubling. 5. Dynamical suppression of U (1)A symmetry violation In this section we adopt the standard picture of U (1)A symmetry violation in QCD as summarized, for example, by ’t Hooft [17]. The U (1)A symmetry is broken in the quantum Hamiltonian by quark masses and by the triangle anomaly. Since we are interested in baryons made of u, d, and s quarks, we consider QCD with three flavors. On account of the   explicit symmetry breaking, the flavor singlet axial current, j5 , has a divergence, and its charge, Q5 (t) = d3 xj 05 (t, x ), does not commute with the Hamiltonian, 
d 3g 2   2mq iq ¯ 5q − i Tr F˜  F . (19) [H, Q5 ] = i Q5 = i d3 xj j5 , j j5 = dt 162 q=u,d,s



Because there are no massless particles coupled to j5 , the singlet axial charge is a well defined operator. This contrasts with the isospin axial charges, Qa5 , a = 1, 2, 3, which become ill-defined in the chiral limit. Some of the consequences of U (1)A invariance, including parity doubling, can be obtained if the matrix elements of [H, Q5 ] are suppressed in a subspace of the space of states where the spectrum is sparse. The operator Q5 can create  mesons from the vacuum, so it clearly contains frequency components of order 1 GeV. We believe this obstructs restoration of U (1)A symmetry in a strong, operator, sense. However a weak form, applied only to matrix elements within the subspace, seems possible and could lead to parity doubling. Suppose first, for simplicity, that there were a subspace, h, of the Hilbert space of QCD, in which the matrix elements  of j j5 were exactly zero. One possibility is that Q5 also has no non-zero matrix elements in this subspace. In that case the symmetry has no implications for the spectrum in h. Alternatively, suppose Q5 has a non-zero matrix element between two mass eigenstates |B+  and |B−  in h. B+ and B− must have the same flavor and angular momentum quantum numbers but opposite parity, denoted by the ± sign. According to Eq. (19) they must also have the same mass  if j j5 can be neglected as we have assumed, B+ |[Q5 , H ]|B−  = (m(B− ) − m(B+ ))B+ |Q5 |B−  = 0.

(20)

If B+ |Q5 |B−   = 0, as we have assumed, then m(B+ ) = m(B− ). So the spectrum of states with a given J, I, and S, consists either of isolated states which have no non-zero matrix elements of Q5 within h, or of degenerate families of both positive and negative parity connected by Q5 . Excluding accidental degeneracies, the result is parity doubling. We do not believe that stronger conditions can be imposed: a. [H, Q5 ] = 0 and Q5 |0 = 0. This condition is not consistent with the QCD lagrangian. Phenomenologically, it leads to parity doubling throughout the baryon spectrum. Since parity doubling is at best an approximate symmetry in a subspace, and clearly does not hold for the lightest states, this condition is not tenable. b. [H, Q5 ]|B = 0 if B ∈ h. It is easy to show that this condition leads to parity doubling of the state |B, however we believe it is too strong. In particular, it implies that the state Q5 |B must be an eigenstate of H with the same mass as |B. We expect that the operator Q5 generically creates  mesons, so the amplitude B   |Q5 |B, where |B   might be the original state |B or some other baryon, should not be small. The weak condition we have adopted does not preclude the possibility that the axial charge acting on one member of a parity doublet has a significant amplitude to produce an  .

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Table 1 Comparison of the physical manifestations of the scenario discussed here of “suppression of U (1)A symmetry breaking (SB)”, with those of “U (1)A restoration”, which would be obtained by attempting to realize U (1)A in Wigner–Weyl mode Prediction

U (1)A restoration

Suppression of U (1)A SB

Q5 |0 = 0 Parity doubling Mass states are Q5 eigenstates

Yes Yes Yes

No Yes No



If the matrix elements of j j5 are small, but not zero, in h, then the degeneracies are no longer exact. It is interesting to consider this case from the point of view of the U (1)A Goldberger–Treiman relation and to contrast it with the SU (2)L × SU (2)R case.5 Again taking B± to be spin-1/2 baryons, the matrix element of the flavor singlet axial current and its divergence can be parameterized in terms of invariant form factors, ¯ + )( gA (q 2 ) + q gP (q 2 ) + i  q  gM (q 2 ))u(p− ), B+ |j5  (0)|B−  = u(p ¯ + )dA (q 2 )u(p− ). B+ |j j5  |B−  = u(p

(21)

Taking the states to be at rest, q 2 = (q 0 )2 =
m2 . Then contracting q  with the first of Eqs. (21) we obtain
m(gA (
m2 ) +
mg P (
m2 )) = dA (
m2 ).

(22)

Note that the quantity in parentheses is the matrix element of the axial charge: B+ |Q5 |B−  ∝ gA (
m2 ) +
mg P (
m2 ). The flavor singlet case differs from the flavor non-singlet case in that there is no zero mass pole in the induced pseudoscalar form factor, gP . Indeed, the nearest significant singularity in the pseudoscalar channel is  the  at nearly 1 GeV. Therefore, if the matrix elements of j j5 are small, the induced pseudoscalar contribution is negligible (second order in
m2 ), leaving,
mg A (0) ≈ dA (0).

(23)

Eq. (23) allows us to formulate the problem of “suppression of U (1)A symmetry violation”: Are there states for which dA (0) is small and gA (0) is not? While we cannot answer the question decisively, it is well posed. At least in principle, this question can be answered using lattice QCD computations of the form factor dA (q 2 ). In the chiral limit, this coincides with the form factor of the gluonic term in the anomaly. We emphasize that the scenario of “suppression of U (1)A symmetry violation” is very different from that of U (1)A restoration, or U (1)A realized in Wigner–Weyl mode. For convenience we summarize in Table 1 the similarities and differences between these two scenarios. In particular, we do not require the invariance of the vacuum under U (1)A transformations. Quark models suggest that the matrix elements of the singlet axial current between states with similar quark configurations but opposite parity are of order unity. For example the singlet axial current in the bag model changes a quark with angular momentum j and positive parity into a quark with angular momentum j and negative parity with amplitude of order unity. For such states
m ∼ dA (0), and suppression of the matrix elements of the divergence of the singlet axial current would result in approximate parity doubling. Other states, for which gA (0) is smaller, need not be nearly degenerate, even though dA (0) may be small. The divergence of the flavor singlet axial current gets contributions from the anomaly and from quark masses. Is it possible that the matrix elements of F˜ F and ms s¯ 5 s are unusually small at moderate excitations in the baryon spectrum? Perhaps instanton fluctuations are suppressed at high excitation in the baryon spectrum as they are at short distances.

5 The U (1) Goldberger–Treiman relation was considered for nucleons in [36]. A

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171

We are not aware of any work along this direction.6 We do not know how to estimate the matrix elements of F F˜ between excited hadrons, and leave it for further study. The generalization of suppression of U (1)A symmetry violation to strange hadrons is not straight forward. In the limit ms = 0 SU (3)V becomes exact and therefore suppression of F F˜  would lead to parity doubling among strange as well as non-strange baryons in octets and decuplets. When the strange quark mass is increased, the flavor singlet current has an increasing divergence proportional to ms s¯ 5 s. However there is a linear combination of flavor singlet  ¯  5 d, whose divergence comes only and non-singlet currents that includes only u and d quarks, J¯5 ≡ u ¯  5 u + d from the axial anomaly, 2g 2  Tr F˜  F . j J¯5 = −i 162 The discussion of parity doubling among non-strange hadrons can be repeated without modification using J¯ instead of  the flavor singlet current, j5 . However N ± , ± |F F˜ |N ∓ , ∓  ≈ 0 does not imply that F F˜ ≈ 0 for their strange SU (3) partners. For example, one might model the effects of the anomaly by the ’t Hooft determinant, det q¯R qL + det q¯L qR , which is a six-quark operator of the form u¯ d¯ s¯ u ds [41]. This operator has tree level matrix elements in ’s and
’s, but only contracted loop contributions in non-strange baryons. Without a better model of hadron matrix elements of F F˜ , it is not possible to conclude that parity doubling among non-strange baryons implies doubling among the analogous strange states. Finally, it is worth emphasizing that this sort of symmetry restoration occurs frequently in quantum systems. Since the phenomenon involves explicit rather than spontaneous symmetry violation, we can find examples in ordinary quantum mechanics. Consider, for example, an atom in an external electric quadrupole field, HSB ∝ e(ri rj − 13 ij )Qij .Although rotational invariance is broken in general, it is unbroken among j = 1/2 states where  1 ,m |[HSB , J ]|1 ,m  = 0 even though [HSB , J ]| 1 ,m   = 0 in general.

2

2

2

6. Parity doubling and chiral symmetry restoration at short distances The idea that SU (2)L × SU (2)R chiral symmetry is restored at asymptotically large Euclidean momenta (short distances) in current correlators in QCD dates back to Weinberg’s 1967 paper on spectral function sum rules. These sum rules relate moments of the spectral functions of vector and axial currents. However, turning them into relations among individual meson states is a different problem, and requires additional theoretical input. Recently, such relations have been discussed by several authors [8,47], who argued that parity doubling might be a consequence of the chiral invariance of current correlators at short distances in QCD and other dynamical assumptions. Usually these arguments are framed in terms of “chiral symmetry restoration”, so it is interesting to ask, in the spirit of Ref. [11], what additional dynamics are responsible for suppressing the chirally invariant operators necessary to obtain a Wigner–Weyl realization of chiral symmetry in the spectrum. Refs. [8,47] discuss mesons at high excitation and large Nc . The arguments do not go over to the baryon sector unchanged [48] since baryons have different large Nc systematics than mesons, and baryon correlators have different short distance behavior than meson current correlators. Furthermore our phenomenological analysis pertains to hadrons in the 1.0–2.5 GeV region, and includes many resonances that are the lightest states in a particular isospin and angular momentum channel, where such an analysis certainly would not apply. Nevertheless it remains interesting to examine the assumptions required to obtain parity doubling in this simpler limiting case. Ref. [47] discusses relations among vector and axial mesons, while Ref. [8] presents similar relations for pseudoscalar and scalar mesons. For simplicity, we phrase our discussion in terms of the latter case. Shifman compares the correlators of two scalar currents, (Q2 ), with the correlator of two pseudoscalar currents, ˜ 2) ˜ 2 ), in the Euclidean domain, Q2 = −q 2 > 0. Shifman assumes that the difference,
(Q2 ) ≡ (Q2 ) − (Q (Q

6 In Ref. [37] Shifman’s analysis [8] is extended to flavor singlet correlators, where instanton contributions determine the convergence of sum rules analogous to Weinberg’s [40]. Our analysis suggests that the proper quantity to analyze would be matrix elements of F˜ F between baryon states of identical J, I, and S, but opposite parity, possibly using instanton methods along the lines of Ref. [38].

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obeys an unsubtracted dispersion relation, 
( 2 ) 2 2 2 ˜
(Q ) ≡ (Q ) − (Q ) = − d 2 2 , Q + 2

(24)

where the spectral functions ( 2 ) and (

˜ 2 ) (with
≡ − ) ˜ receive contributions from physical intermediate states of mass created by the currents from the vacuum. Well established arguments based on the operator product expansion and QCD sum rules [49] require that
(Q2 ) vanishes like 1/Q4 as Q2 → ∞, 2


(Q2 )

 , 2 Q →∞ Q4 ∼

(25)

˜ individually go like where  is the chiral symmetry violating condensate. Each of the correlators,  and , 2 2 2 4 Q log Q as Q → ∞, so the fact that their difference vanishes as 1/Q is evidence of the rapid restoration of chiral symmetry at short distances in QCD. Combined with the unsubtracted dispersion relation, the condition that there is no term ∼ 1/Q2 in
(Q2 ) requires a superconvergence relation,  d 2
( 2 ) = 0. (26) Eq. (26) is the only input from chiral symmetry in the argument of Ref. [8]. Convergence of the integration requires that the spectral functions, each of which grows like 2 must quickly become equal at large . Naively one would think that this constraint applies at asymptotically large masses where broad, overlapping resonances build up continua in and . ˜ Strong additional assumptions are needed to make inferences about parity doubling. The principal assumption is that QCD may be described by a limit in which the number of colors, Nc , is taken to infinity before considering high radial excitations (labeled by a radial quantum number n). To explore the implications of this unusual order of limits, we analyze the problem at large but fixed Nc and extract the leading order predictions at the end. First, let us recap the argument of Ref. [8], taking Nc → ∞ first. Then, the spectral function ( ) ˜ receives contributions only from zero width scalar (pseudoscalar) meson resonances, with masses Mn (M˜ n ) and coupling to the current fn (f˜n ). Continuum contributions are suppressed by powers of Nc . For reference, the squared masses, squared widths, and couplings of the mesons are assumed to have the following dependence on n and Nc : Mn2 , M˜ n2 ∼ 2 n, ˜ 2n ∼ 2 n/Nc2 ,

2n , fn , f˜n ∼ Nc 4 n,

(27)

where  is the dynamical mass scale of QCD. The n dependence is motivated by string analogies, the spectra of linear potentials and the ’t Hooft model [8,51]. The factors of Nc are in accord with standard large-Nc counting arguments. Ignoring the widths or equivalently, exchanging the limit Nc → ∞ with the sum over n, the spectral functions are sums of poles on the real axis, so

( 2 ) = (fn ( 2 − Mn2 ) − f˜n ( 2 − M˜ n2 )), (28) n

and the superconvergence relation reads,
Nc 2 (n Mn2 − ˜ n M˜ n2 ) = 0,

(29)

n

where we have defined fn = Nc 2 n , etc. This sum must converge at large n. Taking n = ˜ n = , independent of n (an assumption later relaxed in Ref. [8]) the weakest behavior large n behavior consistent with the convergence of the sum in Eq. (29) is 1 Mn2 − M˜ n2 ∼ alt , n→∞ n

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173

√ where “alt” denotes an alternating sign. Since Mn ∼ M˜ n ∼ n at large n, Shifman finds
Mn ≡ Mn − M˜ n ∼ alt n−3/2 , indicating that the spectrum achieves parity doubling rather rapidly at large n.7 For any fixed Nc the widths of resonances grow at the same rate as their masses (see Eq. (27)), so the spectral functions approximate a continuum as n → ∞ at fixed Nc . Let us keep track of these 1/Nc corrections. The effect of a finite width for the nth meson on the spectral functions was analyzed in Ref. [52]. When the widths are finite the meson poles in the correlation functions move into the lower half complex q 2 -plane, and the correlation functions develop cuts with branch points at the thresholds for the decay of the nth meson. We refer to Ref. [52] for details. It is straightforward to find the modification of Eq. (29)  
 1 1 Nc 2 n (Mn2 − n Mn ) − ˜ n M˜ n2 − ˜ n M˜ n = 0. (30)   n Repeating the same arguments that followed Eq. (29), we find
Mn −

1 1
n ∼ alt 3/2 , n→∞  n

√ ˜ n . Referring back to Eqs. (27) we see that , ˜ ∼ n/Nc to leading order in n and Nc . However where
n ≡ n − we have no independent argument to fix the large n behavior of the difference
n . It could, in principle go like a constant at large n, or like some power of n larger than n−3/2 . If we parameterize our ignorance by an exponent , presumably smaller than 1/2, we can summarize the 1/Nc corrections to the analysis of Ref. [8],
Mn ∼ alt n→∞

1 n3/2

+C

n , Nc

where both  and the constant C are unknown. Thus the effect of 1/Nc (remember Nc = 3 in our world) corrections could completely destroy parity doubling. While further analysis or more assumptions might indicate that these 1/Nc effects can be neglected, this example serves well to illustrate the fact that parity doubling does not follow from the chiral invariance of QCD alone, nor from suppression of chiral symmetry violating effects. In the example of Ref. [8] the assumptions include those necessary to give the asymptotic behavior summarized in Eqs. (27) and to suppress the effects of hadron widths at large but finite Nc . 7. Parity doubling and intrinsic deformation Parity doubling in the baryon spectrum might be a consequence of a phenomenon well known in nuclear and molecular physics. If a system can be described by the collective quantization of an intrinsic state and (a) the intrinsic state (spontaneously) violates parity, and (b) the deformation is relatively rigid, then the low lying excitations of the system will display parity doubling. The phenomenon is well known, for example, in the spectrum of ammonia. Clear examples occur in nuclear physics as well. The existence of parity doubled states in nuclei like 225 Ra, with intrinsic octupole (“pear shaped”) deformation, plays an important role in proposed sensitive searches for T-violating electric dipole moments [42–44]. Linear superpositions of the intrinsic state, | and its parity image, Pˆ | form parity eigenstates, |±  = √1 (| ± Pˆ |). If the deformation is rigid then the Hamiltonian matrix element between 2 | and Pˆ | is small and the states of opposite parity, but otherwise identical structure, are nearly degenerate. This degeneracy survives quantization of the collective coordinates that restores rotational symmetry resulting in a tower of parity doubled excitations. One can imagine dynamics that might lead to a similar phenomenon in hadron spectroscopy [2], though a convincing discussion is beyond our present understanding of hadrons in QCD. If baryons on the leading Regge trajectory were 7 This derivation has been criticized in Ref. [50], where it is pointed out that the N → ∞ correlator,
(Q2 ), does not satisfy an unsubtracted c dispersion relation, or equivalently that certain interchanges of limits and sums/integrals, required by the derivation, are not allowed by the asymptotic behavior of
( 2 ) when Nc is infinite. By keeping finite Nc corrections (see the following paragraph), which replace the -functions in
with finite width resonances (in the manner proposed in Ref. [52]), we maintain the unsubtracted superconvergence relation and avoid any problems with interchanges of limits. By this approach we include important physics, namely the fact that the resonant contributions to the spectral functions merge into a continuum at large n for any finite Nc , and, happily, agree with the conclusions of Ref. [50] that asymptotic chiral symmetry and conventional large Nc arguments alone do not require parity doubling.

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Table 2 Parity doubled I = 1/2 and 3/2 baryons with angular momentum 5/2 or greater. Data from the PDG [29] B(J P )

PDG reliability

Mass (MeV)

N (5/2− ) N (5/2+ )

∗ ∗ ∗∗ ∗ ∗ ∗∗

1675 1680

N (7/2+ ) N (7/2− )

∗∗ ∗ ∗ ∗∗

1990 2190

N (9/2+ ) N (9/2− )

∗ ∗ ∗∗ ∗ ∗ ∗∗

2220 2250

(5/2+ ) (5/2− )

∗ ∗ ∗∗ ∗∗∗

1905 1930

(7/2+ ) (7/2− )

∗ ∗ ∗∗ ∗

1950 2200

(9/2+ ) (9/2− )

∗∗ ∗∗

2300 2400

well described as a dumbbell-like structure with a quark on one end and a diquark on the other [45], then the intrinsic state would violate reflection symmetry. If tunneling of the odd quark from one end to the other were suppressed enough, then parity doubling would result. If correct, this argument would predict parity doubling for states of high angular momentum. There is some evidence for this among nucleon (I = 1/2) and  (I = 3/2) states, although the
s are not very well known (see Table 2). Hyperons (s and
s) are too poorly known to provide much information. If this mechanism is responsible for parity doubling, one expects the evidence for parity doubling should improve with increasing angular momentum, and should be found in the spectrum of s and
s as well. These predictions could be tested in an experimental re-examination of baryon spectroscopy. As a first attempt in this direction, we have repeated the analysis of Section 2 on the non-strange baryons with the states in Table 2 removed. The significance of the signal for parity doubling remains virtually unchanged when the high angular momentum states are removed. Thus the phenomenon cannot be entirely explained by collective coordinate quantization of a parity-violating, enlongated intrinsic state. 8. Implications from mended chiral symmetry We explore in this section some implications of chiral symmetry using a comparatively less well-known approach. While not directly relevant to the topic of parity doubling, this approach has potentially interesting implications for the properties of baryon resonances of opposite parity. As discussed in Section 2, chiral symmetry realized in Nambu–Goldstone mode does not impose any constraints on the masses and couplings of hadrons, beyond those required by isospin. However, in a series of interesting papers, Weinberg showed that the full chiral symmetry group may remain manifest in a very special sense [23,24]. The symmetry is realized on the matrix of the couplings of the Goldstone bosons, rather than on the mass eigenstates, as typical for a symmetry realized in Wigner–Weyl mode. Following Ref. [24], we will refer to chiral symmetry realized in this fashion as “mended chiral symmetry”. When combined with the large energy asymptotics of Goldstone boson scattering amplitudes following from Regge theory, the approach of Refs. [23,24] gives also nontrivial implications for the mass spectrum. In this section we give a brief review of the approach of Refs. [23,24], and use it to construct a chirally invariant model for a system of two nucleons of opposite parity. In this model the two nucleons of opposite parity are not necessarily degenerate, but their pion couplings are exactly the same as predicted from a linear realization of chiral symmetry. This approach provides such a natural realization of such predictions of chiral symmetry “restoration” without actual parity doubling. The mended chiral symmetry approach is equally applicable to mesons and baryons. In the original work Ref. [23] it was used to describe the quartet of J P = 0± , 1± mesons ( , , , a1 ), and the (N, , N1 ) system of the nucleon, delta and Roper baryons, respectively. However, in the sector of mesons with zero helicity, this approach is considerably more

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predictive than in the baryon sector with nonzero helicity. This is due to the existence of an endomorphism following from the parity transformation of the pion couplings. We will restrict our discussion in this section to the predictions which are common to both baryons and mesons. We start by introducing some notation. For simplicity, we consider nf = 2 with isospin generators T a . Consider the amplitude Ma for pion coupling between two states h () → h ( )a with h , h baryon states with helicities ,  , respectively. This amplitude is related to the matrix element of the axial vector current between the two baryon states by PCAC as A(h (p, ) → h (p  ,  )a ) ≡ Ma =

1 (p − p  ) h (p  ,  )|Aa  |h (p, ). f

(31)

The form of the pion coupling simplifies in the collinear limit, where the baryon momenta are parallel to a common direction p p  eˆ3 , which can be chosen for convenience to define the +z direction. In such a frame, the pion coupling given by the expression Eq. (31) is related to the matrix element of a light-cone component of the axial current, and is helicity conserving Ma =

1 | q |h (p  ,  )|Aa0 − Aa3 |h (p, ) ≡ [Xa ()]  . f

(32)

This defines the transition operator Xa () for pion couplings to hadrons. For simplicity we will omit the  dependence of the operators Xa (), unless explicitly required. The pion couplings X a and the isospin operators T a satisfy the commutation relations [T a , T b ] = iabc T c ,

[T a , Xb ()] = iabc X c ().

(33)

In addition to this, another commutation relation can be obtained by considering the set of all Adler–Weisberger sum rules [53] on baryon targets. They can be derived from an unsubtracted dispersion relation for the h a → h b scattering amplitude. Saturating the dispersion relation with one-body intermediate states, the sum rule can be expressed in operator form as [X a (), Xb ()] = iabc T c .

(34)

The commutation relations Eqs. (33), (34) form the Lie algebra of the SU (2)L ×SU (2)R symmetry, which is furthermore explicitly realized on the hadronic states. This realization may be different for each distinct helicity . The resonance saturation approximation of the Adler–Weisberger sum rule, required to derive the commutation relation Eq. (34), can be justified, for example, by working at leading order in the 1/Nc expansion. The absence of the
I = 2 Regge trajectories with intersect (0) > 0 for the pion-nucleon scattering amplitude h  → h  gives another commutation relation. This connects the pion coupling operator X a with the mass spectrum [Xa (), [X b (), m2 ]] ∝ ab .

(35)

This commutation relation implies that the baryon mass operator has very simple transformation properties under SU (2)L × SU (2)R . It consists of two terms ˆ 20 () + m ˆ 24 () m ˆ 2 () = m

(36)

ˆ 24 transforming as the with m ˆ 20 transforming as a singlet (0L , 0R ) under the chiral group SU (2)L × SU (2)R , and m 1 1 2 I = 0 component of a chiral four-vector ( 2 L , 2 R ). In the absence of the m ˆ 4 term, the chiral symmetry would have been realized explicitly in the mass spectrum, and baryons would be grouped into irreducible representations of SU (2)L × SU (2)R . The m ˆ 24 term prevents such a simple picture, and implies that in general, mass eigenstates are mixtures of different irreducible representations of SU (2)L × SU (2)R and do not exhibit the pattern of degeneracies associated with a Wigner–Weyl representation of SU (2)L × SU (2)R . Note that m ˆ 24 is not connected with explicit chiral symmetry breaking, which has been set to zero from the outset. Thus Weinberg’s mended symmetry scenario provides a good example of the point emphasized in Ref. [11], that “restoration” of SU (2)L × SU (2)R in the spectrum requires suppression of chirally invariant operators. To summarize, Weinberg’s dynamical arguments, based on assumed Regge asymptotics, suffice to prove that hadrons with given helicity  fall into reducible representations of the chiral group. These representations in general may be

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different for different helicities. The reducible nature of the representations and the absence of degeneracies distinguish Weinberg’s approach from the “chiral symmetry restoration” picture discussed in Section 3. The crucial feature is the presence of the chiral nonsinglet m ˆ 24 piece in the Hamiltonian, which connects different irreps, according to selection rules imposed by its quantum numbers. The representation content of a given hadron state is not fixed by these general arguments. The mended symmetry approach, in effect, replaces the arbitrariness of the coupling constants and masses in the effective Lagrangian with the equally unknown representation content of a given state. This framework is predictive only when supplemented with a prescription for the chiral irreps content. Once the representation content is specified, many physical parameters (masses and couplings) are predicted in terms of just a few input parameters. This lends such an approach considerable predictive power. For applications of this formalism to heavy hadron physics and exotics see Ref. [56–59]. We comment here on the relation of this approach to the large Nc picture of the baryons following from the work of Dashen, Jenkins and Manohar [25,26]. In the large Nc limit, a new symmetry emerges in the baryon sector of QCD. This is the contracted spin-flavor symmetry SU (2nf )c , with nf the number of light quark flavors. The generators of the symmetry are the spin J i , isospin I a , and the operator Gia 0 defined in terms of p-wave pion couplings to the baryons in the large Nc limit (Gia → Gia ) 0 Ma =

1 i q Nc [Gia ] . f

(37)

The connection to the operators introduced in Eq. (32) is Xa () ∼ [G3a ] . The operators J i , T a , Gia 0 satisfy the commutation relations defining the algebra of the contracted SU (2nf ) symmetry [J i , J j ] = iεij k J k , j ka [Gia 0 , J ] = iεij k G0 ,

[T a , T b ] = iεabc T c , b ic [Gia 0 , T ] = iεabc G0 ,

(38) jb

[Gia 0 , G0 ] = 0.

(39)

Furthermore, the mass operator in the large Nc limit m20 commutes with all the generators of the contracted symmetry. The contracted SU (2nf ) symmetry relates states with different spins and isospins, analogous to the Wigner symmetry in nuclear physics. For example, in the symmetry limit the nucleon N and the  belong to the same irreducible representation of the SU (4)c , along with infinitely many other states with spin and isospin J = I = 1/2, 3/2, 5/2, . . . . The pion couplings among these states are fixed by the contracted symmetry, such that they are precisely the same as the predictions of the constituent quark model [25,26]. Both the mended chiral symmetry and the large Nc contracted symmetry approach impose constraints on the form of the pion couplings to the baryons. In general these constraints are different, although they can agree in special situations. The reason for the difference is the commutation relation Eq. (35). In the large Nc limit such a commutator would vanish, since the mass operator commutes with the pion couplings X a (). This leads to a nontrivial chiral structure of the mass operator in the mended symmetry approach, which does not follow from the large Nc contracted symmetry. In certain particular cases this difference becomes immaterial, and the predictions of the two approaches are identical. This happens when the hadrons in a sector of given helicity  are assigned to one single irrep (IL , IR ) of SU (2)L × SU (2)R , which renders the double commutator Eq. (35) zero, in agreement with the large Nc prediction. Such a situation in illustrated in Scenario I below. Another important difference between the two approaches lies in their range of applicability: while the mended chiral symmetry method is restricted to the couplings of the Goldstone bosons to baryons, the contracted large Nc symmetry is equally applicable to the couplings of any light mesons to baryons. We proceed to an explicit discussion of the constraints on the mass spectrum and pion couplings from the mended chiral symmetry approach. The hadronic states with given helicity  can be taken to transform as linear combinations of irreducible representations (jL , jR ) of SU (2)L × SU (2)R |h() ∼


jL ,jR

cjL jR ()|jL , jR ,

(40)

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where we have suppressed all labels on the states except for their SU (2)R × SU (2)L representation content. In general, the mass operator m ˆ 24 will mix different representations (barring accidental degeneracies), such that the states |h() will be nondegenerate, unless the sum over representations contains only one term. Parity invariance relates the representation contents of hadronic states with opposite helicity.8 Acting with the parity operator Pˆ on a hadron state h (p, ) gives a state of opposite helicity and momentum Pˆ |h (p, ) = h |h (−p, −), y with h the intrinsic parity of the state h. It is more convenient to consider the normality operator  = Pˆ R , the product of parity and a 180◦ rotation, which restores the momenta to their original direction. The action of this operator on the ) = h (−)jh |h (p, −). hadron states is |h (p, The generators of the symmetry transform under  as Xa () = −X a (−),

T a  = T a .

(41)

The mass operator commutes with the normality operator. Using this relation, it is easy to see that the states |h(−) furnish also a reducible representation of the chiral algebra, containing the representations (jR , jL ) obtained from those in Eq. (40) by exchanging jL ↔ jR
|h(−) ∼ cjR jL ()|jL , jR . (42) jR ,jL

To illustrate the application of this approach, we consider two particular cases. The first scenario describes a N,  pair with the same mass, which reproduces the N N , N
,

 couplings of the quark model. The second scenario describes a pair of spin 1/2 nucleons of opposite parity. Scenario I. A model for a nonstrange nucleon and delta baryons can be constructed by placing the hadrons into minimal chiral multiplets. This model was first proposed in Ref. [23] and discarded because it has the unphysical feature that the nucleon and delta are degenerate. It was proposed again in Ref. [55] as a possible justification for the quark model with SU (4) spin-flavor symmetry. This scenario comes closest to the large Nc picture of the ground state baryons, where the nucleon- mass splitting is of order 1/Nc . We place each helicity sector into irreducible representations of SU (2)L × SU (2)R  = + 21 : (1L , 21 R ) ⊃ N, ,

(43)

 = + 23 : ( 23 L , 0R ) ⊃ .

(44)

The transformation properties of the negative helicity states are obtained from these via the parity relation Eq. (42) and are  = − 21 : ( 21 L , 1R ) ⊃ N, ,

(45)

 = − 23 : (0L , 23 R ) ⊃ .

(46)

We show in Fig. 8 the weight diagrams of the chiral representations corresponding to  = ± 21 , and the associated hadron states. The mass matrix (see Eq. (36)) for the helicity =+1/2 is proportional to the unit matrix in the basis (N+1/2 , +1/2 ), since the m ˆ 24 term does not have diagonal matrix elements on a irreducible representation (IL , IR ). Thus the nucleon and delta are degenerate in this model. Collinear pion emission connects only states with the same helicity, and the corresponding axial matrix elements can be obtained by expressing the axial current in terms of the generators of the chiral group X a = 21 (IRa − ILa ). Defining the reduced matrix elements for p-wave pion emission as 1 (47) X(J  , J )J  |J 1; 0I  I3 |I 1; I3 a, +1 √ √ one finds X(1/2, 1/2) = −5/ 2, X(1/2, 3/2) = −4, X(3/2, 3/2) = −5 2, in agreement with the predictions of the quark model with SU (4) spin-flavor Wigner symmetry. In particular, the result for X(1/2, 1/2) is equivalent to J  I  ; I3 |X a |J I ; I3  =

2J 

8 Of course Lorentz invariance constrains the representation content so that states of the same hadron with different || are degenerate.

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IR

3

IR ++

0

∆

n ,∆

∆

++ 3

IL

∆−

p , ∆+ λ =+

1 2

p, ∆+

∆−

3

IL

0

n,∆

λ =− 1 2

Fig. 8. Weight diagrams for the chiral representations (1, 21 ) and ( 21 , 1) for states with helicity  = + 21 (left) and  = − 21 (right), respectively. The states on the same diagonal have the same third component of the isospin I 3 = 21 (IL3 + IR3 ), and correspond to the nucleons and  states shown.

gA /gV = 5/3. The agreement with the experimental result gA /gV = 1.2695 ± 0.0029 can be improved by refining the model and adding a second chiral representation ( 21 L , 0R ) in the  = + 21 sector. The mixing angle of the two representations is chosen appropriately such as to reproduce the data on gA /gV [23]. We do not consider this further, and refer the reader to Weinberg’s paper for details [23]. Scenario II. Consider a different model, containing the two chiral representations (0, 21 ) and ( 21 , 0). In the helicity +1/2 sector, there are two spin-1/2 nucleons NA , NB which are linear combinations of the two chiral representations, with a mixing angle   = + 21 : NA = cos (0L , 21 R ) + sin ( 21 L , 0R ),

(48)

NB = − sin (0L , 21 R ) + cos ( 21 L , 0R ).

(49)

We take these states to be mass eigenstates, with eigenvalues mA , mB . They are related to the chiral singlet and chiral quartet terms as follows. Consider the squared mass matrix in the basis of the irreducible representations ( 21 , 0) and (0, 21 )  m2=+1/2 =

2L 

 2R

 ,

(50)

ˆ 20 , and  is the off-diagonal matrix element of the where 2L,R are the diagonal matrix elements of the chiral singlet m chiral quartet term in the mass-squared operator of Eq. (36). Their relation to the hadronic parameters is 2L = m2A sin2  + m2B cos2 ,

2R = m2A cos2  + m2B sin2 ,

 = 21 (m2A − m2B ) sin 2.

(51)

The helicity −1/2 sector can be constructed from these states by applying the normality transformation, as explained above. Choosing the two nucleons to have opposite parities NA = +1, NB = −1, the representation content of the helicity − 21 states is =−

1 : NA = sin (0L , 21 R ) + cos ( 21 L , 0R ), 2

NB = cos (0L , 21 R ) − sin ( 21 L , 0R ).

(52) (53)

Next, consider the couplings of these states to pions. The pion coupling operator X a = 1/2(IRa − ILa ) is related to the generators of the symmetry, and thus couples only states in the same chiral representations. The simplest way to derive the pion couplings is to consider the transitions pi↑ → nj ↑ + , which are mediated by the operator X − = X1 − iX2 .

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For example, the diagonal pion transition NA → NA  is computed as A(pA↑ → nA↑ + ) ∼ (cos (1, 0)L , ( 21 , − 21 ) R | + sin ( 21 , − 21 ) L , (1, 0) R |)

× X− (cos |(1, 0) L , ( 21 , + 21 ) R  + sin |( 21 , + 21 ) L , (1, 0) R )

= cos2 ( 21 , − 21 ) R |IR− |( 21 , + 21 ) R  − sin2 ( 21 , − 21 ) L |IL− |( 21 , + 21 ) L  = cos 2,

(54)

− where in the last line we show only the chiral isospin (and its third component) changed by the IL,R operator. According to the effective Lagrangian of Eq. (16), this amplitude is proportional to the coupling c2 + c4 . In a similar way one can obtain all the other couplings in this effective Lagrangian, with the results

c2 + c4 ∼ cos 2,

(55)

c2 − c4 ∼ − cos 2,

(56)

1 − c3 ∼ − sin 2.

(57)

This illustrates how a special chiral representation content in the framework of the mended chiral symmetry can lead to specific relations among pion couplings in the effective Lagrangian. In particular, choosing  = 3/4 reproduces the predictions of the chiral doubling model with linearly transforming baryons (c2−4 = 0), but without mass degeneracy. The mass parameters of this model are 2L = 2R = (m2A + m2B )/2 and  = −(m2A − m2B )/2, with nondegenerate baryons m2A − m2B  = 0. Of course, any such explanation leaves open the dynamical origin of this (or any other) particular value of the mixing angle, as well as the representation content. More complicated representation contents are also possible, e.g. including also (IL , IR ) = (1, 1/2), (1/2, 1) as in Scenario 1. Such questions could be addressed for example using lattice QCD methods, as discussed recently in Ref. [60]. 9. Conclusion and discussion Although parity doubling was observed in the baryon spectrum more than 35 years ago, a complete understanding of its dynamical origin is still lacking. In this paper we both evaluate the empirical evidence for parity doubling and consider some possible theoretical explanations. We propose a new mechanism, based on the dynamical suppression of the U (1)A violation, and point out a prediction that distinguishes it from SU (2)L × SU (2)R restoration.9 Most of the evidence previously offered in favor of parity doubling has been obtained by visually grouping states into multiplets. Given the number of broad, closely spaced states in each channel, this procedure is both ambiguous and imprecise. It does not take into account the resonance widths, nor the fact that different resonances are established with varying degrees of certainty. We propose in this paper a quantitative measure of parity doubling, which is free of these problems. First, we introduce an overlap integral I J S ({C}) of the spectral densities in channels with the same spin J, isospin I and strangeness S, but opposite parity. This is computed on an ensemble of control samples (each of them denoted by {C}), obtained from the real world by arbitrarily permuting the parities of the hadronic states. Finally, the value of the overlap integral I J S in the real world is compared against the statistical distribution on the ensemble, which assigns a probability measure (p-value) to the statement of parity doubling. The results of our study gives strong evidence for parity doubling in the nonstrange baryon sector (both N and ), with spins from J = 1/2 to 7/2. For larger spins the number of states decreases, and we do not expect the control samples to be representative statistically. Still, there is good evidence for parity doubling among high spin, non-strange baryons as can be seen from Table 2. On the other hand, the correlations between positive and negative parity strange baryons are not significant. However, some of the states with strangeness expected in the quark model have not yet been seen, which could partly be responsible for the weaker correlations observed in these sectors. A new experimental attack on baryon spectroscopy could shed significant light on the systematics and origins of parity doubling. In principle our statistical test could be used to search for more complex correlations, such as those expected in models of “chiral symmetry restoration”. In such schemes, additional degeneracies involving hadrons of different isospins 9 If parity doubling arises from suppression of U (1) violation, strange hadrons should not show significant parity doubling. A

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(but the same spin) are expected on top of parity doubling. The pattern of the mass spectrum in this case depends on the representation, (jL , jR ), of SU (2)L × SU (2)R assumed to be realized. For example, in the (1/2, 1)(1, 1/2) − scheme of Ref. [54] one expects the baryons to be grouped into “quartets” of the form (NJ+ , NJ− , + J , J ) with the same spin J. We have not used our statistical method to look for more complicated correlations. This would require higher dimensional integrals analogous to Eq. (3). In any case no general correlations of this form are observed by eye. However, it is possible that more complicated chiral representations may not appear uniformly in the spectrum, but be localized in some mass region. This would render our method inapplicable, due to the sparseness of the corresponding control sample. We have examined a number of possible explanations of parity doubling. First we have tried to clarify the relationship between the usual SU (2)L × SU (2)R chiral symmetry of QCD and parity doubling. The terminology used in the literature discussion has been confusing. In an important sense (for massless u and d quarks) SU (2)L × SU (2)R is not broken at all, but instead represented in the Nambu–Goldstone form by massless pions and isospin multiplets of hadrons that transform non-linearly under chiral transformations. There is nothing to “restore”. Instead, we frame the question as follows: “What is needed in order that baryons fall into Wigner–Weyl representations of SU (2) × SU (2)with the usual degeneracies and relations among couplings”? The answer, presented in Ref. [11] and reviewed here, is that a set of SU (2)L × SU (2)R invariant operator matrix elements must be suppressed, basically amounting to decoupling the pion from the parity doublet states. If this decoupling were complete, it would, in effect, leave two independent SU (2)L × SU (2)R symmetries: one under which the pion and non-parity-doubled states transform non-linearly, and the other under which the parity doubled states transform linearly. A toy model of this type has recently been proposed by Cohen and Glozman [61]. The question is what dynamics are responsible for suppressing these operator matrix elements and giving rise to this additional symmetry? If the dynamics behind effective chiral symmetry restoration applies generically to hadrons high in the spectrum, as advocated, for example, in Ref. [27], then parity doubling should be universal in that domain. It would be puzzling if, for example, some (e.g.) highly excited mesons are parity doubled, but some are not. A search for such exceptions would be a particularly effective way to test this explanation of parity doubling. We emphasize that parity doubling alone is not conclusive evidence for effective chiral symmetry restoration. As discussed in Section 3A, chiral symmetry realized linearly on a set of baryons requires that pions decouple from the parity doubled states. In addition, one would have to check the validity of certain relations for axial couplings between the members of the putative chiral multiplet implied by the symmetry algebra implemented in the Wigner–Weyl mode. Next, we considered the possibility that parity doubling is a consequence of dynamical suppression of U (1)A symmetry breaking in a subsector h of the Hilbert space of QCD. Note that this is not equivalent to U (1)A restoration, as realized for example in large Nc QCD. For example, the effects of the anomaly are still present, and give the ,  the masses observed in nature. Rather, we require only that the matrix elements of the divergence of the axial singlet  current are very small on a subset of baryon states B1 |j j5 |B2  ≈ 0, with B1 , B2 ∈ h. 10 This condition is satisfied if the matrix elements of the gluon anomaly term taken between non-strange baryons of opposite parity are very small.11 The hypothesis that the matrix elements of F˜ F are suppressed among excited baryons may be amenable to study using lattice QCD methods. Although it predicts parity doublets, the mechanism of dynamical suppression of U (1)A symmetry breaking does not make predictions for the pion couplings to the doublet states (except that the pions must decouple from parity doubled states at zero momentum, as required by the Goldberger–Treiman relation). Such relations would be obtained, for example, if we required the doublet to be left invariant under U (1)A and parity transformations. In addition to suppression of U (1)A symmetry violation, we review two other possible origins for parity doubling. First we examine arguments recently put forward that parity doubling in the spectrum can be related to chiral symmetry restoration at short distances, a phenomenon well grounded in perturbative QCD. The only input from short distance chiral symmetry is the superconvergence relation, Eq. (26). The rest of the argument requires input from phenomenological models (e.g. linear excitation spectra) and large Nc implemented in an unusual way—examining the large n (excitation) limit, but taking Nc → ∞ first. O(1/Nc ) corrections seem capable of invalidating the conclusions. In any event this exercise provides an example of the extent of additional dynamics required to obtain effective chiral 10 In the absence of strange quark contributions. 11 Strange quark contributions would seem to defeat this explanation among strange hadrons, leading to the intriguing prediction that parity doubling, if due to this mechanism, will not be significant among hyperons and other strange hadrons.

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symmetry restoration. Finally, we look at the possibility that parity doubling may be a consequence of an intrinsic deformed state that spontaneously violates parity. A mechanism familiar from nuclear and molecular physics, this leads one to expect parity doubling at high spin for both strange and non-strange baryons, but gives little insight into why spin 1/2 and 3/2 nucleons and s should show statistically significant evidence for the symmetry. An alternative way of obtaining relations among masses and axial couplings which adds dynamical input in a minimal way is offered by the mended chiral symmetry approach of Weinberg [23,24]. In this approach the set of all pion couplings among baryon states X a (in sectors of given helicity), together with the isospin operators T a , are shown to satisfy the SU (2)L × SU (2)R algebra. However, input from Regge theory shows that the mass operator is not a singlet under this symmetry, but contains in addition also a chiral 4-vector term. This implies that the hadrons transform in reducible representations of the SU (2)L × SU (2)R symmetry. Models of hadrons can be constructed by postulating a representation content, and the corresponding mixing angles. Even minimal models containing one or two such representations can reproduce the properties of the lowest lying baryons remarkably well [23,24]. As examples of this approach we presented two such models. First, to introduce the ideas we describe a model containing a (N1/2 , 3/2 ) degenerate pair of the same parity, which reproduces the quark model predictions for pion couplings. This model was previously considered in Refs. [55,57]. The second model is new and contains two nucleons of opposite parity and spins J = 1/2. Their masses and couplings are determined by just three mass parameters and one mixing angle among chiral representations. We show that, for a particular value of the mixing angle, the pion couplings to the parity doublet take precisely the same values as those expected if chiral symmetry were realized linearly on the two states, but without mass degeneracy in the spectrum. This illustrates a natural realization of such relations which does not require chiral symmetry restoration. Acknowledgments We thank W. Bardeen, S. Beane, D. Bugg, T. Cohen, M. Golterman, L. Glozman, A. Manohar, M. Shifman and A. Vainshtein for conversations and correspondence relating to this work. This work is supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under cooperative research agreement DE-FC02-94ER40818. References [1] V. Barger, D. Cline, Phys. Lett. B 26 (1967) 85; V. Barger, D. Cline, Phys. Rev. Lett. 20 (1968) 298; See also Y. Iwasaki, Prog. Theor. Phys. 44 (1970) 777; L.Y. Glozman, arXiv:hep-ph/0105225. [2] See, for example, F. Iachello, Phys. Rev. Lett. 62 (1989) 2440; D. Robson, Phys. Rev. Lett. 63 (1989) 1890; F. Iachello, Phys. Rev. Lett. 63 (1989) 1891. [3] M. Kirchbach, Mod. Phys. Lett. A 12 (1997) 3177 [arXiv:nucl-th/9712072]. [4] L.Y. Glozman, Phys. Lett. B 475 (2000) 329 [arXiv:hep-ph/9908207]; L.Y. Glozman, Phys. Lett. B 539 (2002) 257 [arXiv:hep-ph/0205072]. [5] T.D. Cohen, L.Y. Glozman, Phys. Rev. D 65 (2002) 016006 [arXiv:hep-ph/0102206]. [6] T.D. Cohen, L.Y. Glozman, Int. J. Mod. Phys. A 17 (2002) 1327 [arXiv:hep-ph/0201242]; L.Y. Glozman, Acta Phys. Polon. B 35 (2004) 2985 [arXiv:hep-ph/0410194]. [7] A.E. Inopin, arXiv:hep-ph/0404048. [8] M. Shifman, arXiv:hep-ph/0507246. [9] D. Jido, T. Hatsuda, T. Kunihiro, Phys. Rev. Lett. 84 (2000) 3252 [arXiv:hep-ph/9910375]. [10] For a summary with further references, see D. Jido, M. Oka and A. Hosaka, Prog. Theor. Phys. 106 (2001) 873 [arXiv:hep-ph/0110005]. [11] R.L. Jaffe, D. Pirjol, A. Scardicchio, Phys. Rev. Lett. 96 (2006) 080403 [arXiv:hep-ph/0511081(v2)]. [12] S. Weinberg, Physica 96A (1979) 327; J. Gasser, H. Leutwyler, Nucl. Phys. B 250 (1985) 465. [13] S. Weinberg, Phys. Rev. 166 (1968) 1568. [14] S. Weinberg, The Quantum Theory of Fields, vol. 2, Modern Applications. [15] S.R. Coleman, J. Wess, B. Zumino, Phys. Rev. 177 (1969) 2239; C.G. Callan, S.R. Coleman, J. Wess, B. Zumino, Phys. Rev. 177 (1969) 2247. [16] C. DeTar, T. Kunihiro, Phys. Rev. D 39 (1989) 2805. [17] For a review, see G. ’t Hooft, Phys. Rep. 142 (1986) 357.

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Contents of Volume 435 P. Busch, C. Shilladay Complementarity and uncertainty in Mach–Zehnder interferometry and beyond T. Gorin, T. Prosen, T.H. Seligman, M. Zˇnidaricˇ Dynamics of Loschmidt echoes and fidelity decay

1

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R.L. Jaffe, D. Pirjol A. Scardicchio, Parity doubling among the baryons

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Contents of volume

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