Fundamental Aspects Fundamental Aspects Fundamental Aspects
QP-PQ: Quantum Probability and White Noise Analysis Managing Editor: W. Freudenberg Advisory Board Members: L. Accardi, T. Hida, R. Hudson and K. R. Parthasarathy
QP-PQ: Quantum Probability and White Noise Analysis VOl. 17:
Fundamental Aspects of Quantum Physics eds. L. Accardi and S. Tasaki
Vol. 16:
Non-Commutativity, Infinite-Dimensionality, and Probability at the Crossroads eds. N. Obata, T. Matsui and A. Hora
Vol. 15:
Quantum Probability and Infinite-Dimensional Analysis ed. W. Freudenberg
Vol. 14:
Quantum Interacting Particle Systems eds. L. Accardi and F. Fagnola
Vol. 13: Foundations of Probability and Physics ed. A. Khrennikov
QP-PQ VOl. 10:
Quantum Probability Communications eds. R. L. Hudson and J. M. Lindsay
VOl. 9:
Quantum Probability and Related Topics ed. L. Accardi
Vol. 8:
Quantum Probability and Related Topics ed. L. Accardi
VOl. 7:
Quantum Probability and Related Topics ed. L. Accardi
Vol. 6:
Quantum Probability and Related Topics ed. L. Accardi
QP-PQ Quantum Probability and White Noise Analysis Volume XVII
Proceedings of the Japan-Italy Joint Workshop on Quantum Open Systems, Quantum Chaos and Quantum Measurement
Fundamental Aspects Fundamental Aspects Fundamental Aspects Fundamental Aspects Waseda University, Tokyo, Japan
27 - 29 September 2001
Edited by
Luigi Accardi University of Roma II, Italy
Shuichi Tasaki Waseda University, Japan
World Scientific New Jersey London Singapore Hong Kong
Published by World Scientific Publishing Co. Pte. Ltd.
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British Library Cataloguing-in-PublicationData A catalogue record for this book is available from the British Library.
FUNDAMENTAL ASPECTS OF QUANTUM PHYSICS QP-PQ: Quantum Probability and White Noise Analysis Vol. 17
-
Copyright 0 2003 by World Scientific Publishing Co. Re. Ltd.
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V
Preface In spite of the long history since its foundation, several fundamental aspects of quantum mechanics are not yet fully understood. On the other hand, the recent development of modern technology, especially nanotechnology, allows to realize several experiments in quantum measurement theory which, for several years were only thought experiments and to design quantum systems whose classical counterparts are chaotic. In other words, several fundamental conceptual questions have now become experimental problems. Moreover several advances in our theoretical and mathematical techniques allow a much more precise, microscopic, description of the parameters involved hence a greater control with respect to purely phenomenological models. This volume collects the invited lectures of the “Japan-Italy Joint Workshop on Quantum Open Systems, Quantum Chaos and Quantum Measurement”, held in Waseda University, from 27 to 29 September, 2001 within the “Japan-Italy Joint Forum: Quantum Probability, Physics and Information Theory”, which was a part of the scientific activities of “Italia in Giappone 2001”. A common feature of the systems considered in this range of phenomena is that they are typically open systems, i.e. involving the interaction of a system with relatively few degrees of freedom with a system with many degrees of freedom, such as a field or a measurement apparatus or more generally an ”environment”. Therefore, the general theory of open systems provides a natural unifying framework for the three main topics of the Workshop. We thank the Italian embassy to Tokyo, in particular the italian ambassador Gabriele Menegatti and Dr. Angelo Volpi, for financial support to the workshop. We are grateful to Prof. Ichiro Ohba for additional support provided by the COE of Waseda University: Establishment of Molecular Nano-Engineering by Utilizing Nanostructure Arrays and its Development into Micro-Systems (Research Leader: Iwao Ohdomari) promoted by the Ministry of Education, Culture, Sports, Science and Technology of Japan, and to Prof. Hiromichi Nakazato for partial support from Waseda University Grant for Special Research Projects from Waseda University. Also the workshop was partly supported by Grant-in-Aid for Scientific Research (C) from Japan Society for the Promotion of Science. 11 November 2002 Luigi Accardi (Universita di Roma Tor Vergata) Shuichi Tasaki (Waseda University)
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vii
Contents
I. Quantum Open Systems Onsager Relation with the “Slow” Degrees of the Field in the White Noise Equation Based on Stochastic Limit L. Accardi, K. Imafuku and Y. G. Lu Quantum Stochastic Differential Equations in View of Non-Equilibrium Thermo Field Dynamics T. Arimitsu
1
24
Non-Equilibrium Local States in Relativistic Quantum Field Theory I. Ojima
48
Stability of Quantum States of Finite Macroscopic Systems A. Shimizu, T. Miyadera and A. Ukena
68
Delocalization and Dissipative Property in 1D Disordered System with Oscillatory Perturbation H. Yamada Fluctuation Theorem, Nonequilibrium Steady States and MacLennan-Zubarev Ensembles of a Class of Large Quantum Systems S. Tasaki and T. Matsui
80
100
11. Quantum Chaos Weak Chaos: Classical and Quantum Features R. Artuso
120
Quantum Transport in Quantum Billiards: From Kelvin through Arnold K. Nalcamura
136
Dynamical and Energetic Barrier Tunneling in the Presence of Chaos in Complex Phase Space A. Shudo, T. Onishi, K. S. Ikeda and K. Takahashi
157
On Quantum-Classical Correspondence and Chaos Degree for Baker’s Map K. Inoue, M.Ohya and I. V. Volovich
177
111. Quantum Measurements and Related Topics Welcher-Weg Puzzle with a Decaying Atom S. Talcagi
188
viii
Quantum Zen0 Effect, Adiabaticity and Dynamical Superselection Rules P. Facchi
197
Unstable Systems and Quantum Zen0 Phenomena in Quantum Field Theory P. Facchi and S. Pascazio
222
The Tunneling Time Problem and the Consistent History Approach to Quantum Mechanics N. Yamada
247
To Decay Or Not To Decay? Temporal Behavior of a Quantum System - Analysis Based on a Solvable Model H. Nakazato
267
Quantum Decomposition and Quantum Central Limit Theorem A . Hora and N. Obata
284
An Introduction to the EPR-Chameleon Experiment L. Accardi, K. Imafuku and M. Regoli
306
Description of the Damped Oscillator by a Singular Fkiedrichs Operator W. von Waldenfels
322
1
ONSAGER RELATION WITH THE “SLOW” DEGREES OF THE FIELD IN THE WHITE NOISE EQUATION BASED ON STOCHASTIC LIMIT L. ACCARDI, K. IMAFUKU, Y.G. LU Centro Vito Volterra, Universita’ di Roma Tor Vergata, 00133 Rome, Italy Stimulated by the many works on the non-equilibrium physics, we apply a novel technique of stochastic limit which enables us to deal degrees of the freedom of the field, to simple examples and show that the relevant definition of the current with the degrees of the fields, which holds Onsager’s reciprocal relation, is possible.
1
Introduction
Microscopic understanding of the non-equilibrium phenomena, for example heat conductivity, microscopic entropy and so on, is one of the fundamental problems of modern physics Recently several papers have been devoted to this problem and many models have been studied either analytically or n ~ m e r i c a l l y l - ~In ~ .these models, one considers various systems with the reservoirs which drive the system to a non-equilibrium stationary state. The effects of reservoirs on the dynamics of the system are modeled by random forces, boundary conditions or dynamical systems. For example, Spohn and Lebowitz showed that a harmonic chain placed between two reservoirs with different temperatures has a nonequilibrium stationary state with nonvanishing currents, whose properties are consistent with the macroscopic theory of non-equilibrium physics. In these papers, the so-called open system approach, which is well established and fruitful for this problem, has been often used. The fact that not only system but also environment system have important informations to understand this subject has been also pointed out by several authors. The stochastic limit approach allows us to extract notable dynamics not only of the system but also of the environment. Therefore it is natural to expect that a satisfactory description of the non-equilibrium currents could be obtained. In order to realize this programme, one should be able to derive the quantum Langevin equation from a fundamental microscopic Hamiltonian, not only for the system, but also for the field obserbables. To this goal recently the authors of the present paper explored a new approach to deal with degrees of freedom of the environment 17, and showed that the stochastic limit works not only for operators in the system space but also for some operators of the field space describing the “slow degrees of the freedoms”. They survive in the
’.
2
stochastic limit and give rise to nontrivial commutations relations with the white noise operators. In this paper we apply this new technique to simple examples and show that it gives a microscopic insight into non-equilibrium physics, for example Onsager’s relations. The rest of this paper is arranged as follows: In Sec. 2 we give a general scheme of the stochastic limit. In Sec. 3, we apply the stochastic limit to the simple model which describes the system interacting one field. The physical meaning of our approach is made clear in these sections. In Sec. 4, we apply our scheme to a typical situation of non-equilibrium physics and we show that the relevant definition of the non-equilibrium current is possible with the field degrees of freedom. In Sec.5, the relation of the current to Onsager’s relation is explained. 2
Stochastic Limit
In the present section, we briefly describe the general scheme of the stochastic limit technique for the Hamiltonian of an form = Ho
+ XV
(1)
where X is real parameter, HOis the free Hamiltonian and V is the interaction is to take Hamiltonian. The general idea of the stochastic limit approach the time rescaling 18119
in the solution u,(X)= eitHo e --itH(’)
(3)
of the Schrodinger equation in interaction picture associated to the Hamiltonian H(’), i.e.
The rescaling (2) gives the rescaled equation
0 (which is equivalent to X 0 and t 4 co under the and the limit X condition that X2t tends to a constant) captures the dominating contributions ---f
--f
3
to the dynamics, which, under appropriate assumptions on the model shown to converge to the solution of d -ut
dt
=
1 -ihtUt, ht = lim - V ( t / X 2 ) ,
-
A-0
x
l9
is
U ( 0 )= 1
We can also obtain the limit of the Heisenberg evolution
where Ut is the solution of (6) and X is an observable belonging to a certain class (slow observables, cf. 17). The main result of this theory is that the time rescaling induces a rescaling
of the quantum field, defining the Hamiltonian (l),which in the present paper will be assumed to be a scalar boson field: ([uk,ai,]= 6 ( k - k ’ ) ) (the meaning of w ( k ) and R will be described in the next section) and, in the limit A -+ 0, the rescaled field becomes a quantum white noise (or master field) bn(t,k) satisfying the commutation relations
[bn(t,k ) , bA,(t’, k’)] = Gn,n,27rS(t - t’)S(k - k’)6(w(k) - 0 ) .
(9)
Moreover, if the initial state of the field is a mean zero gauge invariant Gaussian state p f ( 0 ) with correlations: (UiUk,)
= N ( k ) d ( k - k’)
(10)
then the state of the limit white noise will be of the same type with correlations
(bA(t,Ic)bn,(t’,k’)) = Sn,n,27r6(t - t’)6(k - k’)d(w(k) - 0 ) N ( k )
(bn(t,k)bA,(t’,k‘)) = 6n,n127r6(t - t’)6(k
-
(11)
k’)b(w(k) - 0 ) ( N ( k+ ) 1). (12)
It is now well understood that this scheme plays an important role in the analysis of the limit (7) when X is a system operator. In the following section, we show the application of the stochastic limit to the slow degrees of freedom of the field and we explain the physical meaning of the dynamics extracted with the stochastic limit.
4
3
System+l-field
In the present paper we consider the following simple Hamiltonian system describing a single boson harmonic oscillator interacting with a scalar boson field:
H
= Ho
+ xv
(13)
where
Ho = Out,
+ /dkw(k)aLuk, [a,at]= 1,
V
=
[Uk,UL,]
g*(k)atak + g(k)a u:) = b(k - k’).
(14) (15)
a and ut are system operators and the field, described by ak and aL, may be interpreted as environment. With this interpretation the Hamiltonian describes the time evolution of a system which is affected by the environment. These effects have been widely studied l8>l9.On the contrary, our interest in this paper is the consideration of the dynamics of some operators of the field. It is important at this point to underline the difference between our approach and the existing literature on the so-called output field 19. In these papers the observable X = ( X t ) is an adapted process of the master field, typically
for which the right hand side of (7) makes sense, but the left hand side does not. Our programme instead is to study the limit (7) for appropriate observables X of the original q-field. In this paper, we assume that the initial state of the total system (the system and field) is decoupled, i.e.
where psysis arbitrary and p f ( 0 ) is given by (9), (lo), (11).
3.1
Vacuum initial state
First, let us consider the case that the initial state of the field is the vacuum (i.e. N ( k ) = O in (10)) or p f ( 0 ) = lO)(Ol,
aklO)
= 0 for all k
(17)
5
With the stochastic limit we obtain the followingwhite noise Hamiltonian equation for time evolution operator Ut
with
-a)
2 3(w(k)
The causal commutator rule of the stochastic limit implies the following commutation relations;
where (P.P. denoting the principal part integral):
Now, let us consider the time evolution of the number operator
Since N ( k ) = 0, we can extend the relation (10) to the case k = k' by continuity. With this convention we have that ( N ) = 0 in the vacuum state. Notice that this is the number operator of the original, not that of the master field. At first sight this might appear somewhat weird because, in the stochastic limit, while the system space remains unaltered the original field converges to the white noise (9), (lo), (11). Therefore the right hand side of (7) is perfectly well defined if X has the form Xs@l ( X s a system observable) but if X = 1s@N,a priori it makes no sense to apply the Ut-evolution, which acts on the master space, to the number operator (23) which acts on the original space of the ak-field. However one can find a strict mathematical argument in 17. In this paper, we apply the result of the argument of the paper to the simple cases and discuss the physical meaning of the time evolution of (23). According to the number operator converges, in the sense of correlations,
''
6
still denoted N , living on the master space and the operator (23) satisfying the following commutation relations with bt
[bt,NI = bt, We shall investigate the outer Lungewin
", l8
b,tl = btt evolution of (23), i.e.
N ( t ) = U-tNUt,.
(24)
(25)
Using (21), (24) and (25) we obtain d -N(t) dt
=i
(abt,
+ atb-t)
= ibL,aN(t)
-
+y-ataN(t) +fat,
U-tNU!,
iN(t)atb-t
-
iU-tNU!t ( a b t ,
+ iatb-t
+
- ibt,u
+ y?N(t)ata - 2Rey-atN(t)a
+ y-at,
(26)
Taking partial trace over the field,
(.)
= trfield (Pf(0)
'
),
(27)
(26) becomes d ( N ( t ) )= - y - a f u ( N ( t ) ) - y'(N(t))ata dt +2Rey-at(N(t))a y*atu - y-ata.
-
+
(28)
p f ( 0 ) is the initial state of the field and ( N ( t ) )is operator of the system.
We look for a solution of (28) of the form
( N ( t ) )= a(t)atu + c ( t )
(29)
for some real functions a(t) and c(t). Notice that these functions should satisfy the initial condition c(0) = ( N ( 0 ) )= 0 (since the initial state is vacuum)
a(0)= 0,
(30)
By substitution of (29) into (as), we obtain
d -a@) dt d
-c(t) dt
= -2Rey_a(t)
+ 2Rey-
(31)
=0
and we can get
a(t)= 1 - exp (-2tRey-),
c ( t ) = ( N ( 0 ) )= 0
(33)
7
or
( ~ ( t =) )(1 - e-2tRey-) utu
(34)
This solution describes the phenomenon that all the energy of the initial state of the system is exponentially released to the degrees of the freedom described by the original field operator a k . In this sense, we can understand that the represent the “slow degrees” (systematic motion) of the field.
3.2 Finite temperature case In finite temperature case (with inverse temperature introduce two additional independent Fock fields
[ti,‘$1
= 0,
ak = J
Ek@c
m
E
P-l),
it is convenient to
= ‘$k@i = 0, k f
m‘$L,
+m
‘
$
k
(35) (36)
(37)
and to express the field u t , in the representation given by the state (lo), (11) in terms of the (&,&’)-doublet (cf. 19, chap. 2): [ E k , t L , ] = b(k
- k’),
[‘$k,‘$l~] = 6 ( k - k’)
(38)
In the following we shall restrict our considerations to equilibrium states of the field, i.e. we assume that the initial state p j ( 0 ) of the field is charxterized by (lo), (11) with:
Notice that, in this case, N ( k )# 0 for any Ic. Therefore the expectation value ( U t a k ) is not well defined and, in order to speak of “mean number of photons’’ in this situation, some renormalization is needed. This remark will play an important role in the following discussion. With the same argument in previous section, we can obtain
d --Vt dt where
=
-i ( u
(ci
+ 4)+ ut (ct + d i ) ) lJt
In this representation the commutation relations (21) become:
where
These (43) and (44) correspond to (21). On the other hand the generalization of (24), i.e. of the commutation relations between the field and the number operator, become i.e. of more complex:
with new operators
9
where
Notice that
YM- - YM+ = T-i
T N - - Y N + = Y+.
The outer Langevin equation for the time evolution of N is written as
10 = i y - a N(t)
+ U-,(CM +
-
+
= i y - a N ( t ) - iyYN(t)a - iy&-a
[&Nuit, ct-,] =
= ([CW,
-iy+N(t)a
U-,NUitI)
t
+ iy-atN(t) + i 7 M - d
-iy:N(t)at
[U-tNU!,, dt,] =
iylN(t)a iyM+a
= ([d-,,
U - , N q
+ iyTaN(t)
-
iy;v-a
- iy&+aj
t
+ iyN+a
and by substituting (62)-(65) into (61), we obtain
+dt,atN(t)
(61) = i (c!,uN(t)
-
N(t)ad-t - N(t)atc-t
+ iy,-at - iyl;+ut) iy&-a + iyM+a)
+ia ( + i y p J N ( t ) - i y + N ( t ) d
+iat ( i y - a N ( t ) - i y Y N ( t ) a -i (-iy"(t)d + i y - d N ( t ) iyM-at
+
-i (-iy+N(t)a + i y ; a N ( t ) =i
-
iy;v-a
-
iy&+a t ) a
+ iyrv+a) a t
( c t , u N ( t ) + d L , a t N ( t ) - N(t)ad-t - N(t)a+c-
4
+
-y-ataN(t) - y r N ( t ) a t a 2Rey-atNa
-y+aatN(t) +2 (ReyM-
-
y;N(t)aat
ReyM+) uta
+ 2Rey+aN(t)at -
2 (ReyN-
-
ReyN+) .at
(66)
Taking partial trace over the degrees of the field (see (27)), (66) becomes d - ( N ( t ) ) = - y - a t u ( N ( t ) ) - y Y ( N ( t ) ) u t u 2Rey_at(N(t))a dt -y+aat ( N ( t ) )- y; ( N ( t ) ) a a t 2Rey+a(N(t))at
+ +
=
+2 (ReyM- - ReyM+) at, - 2 (ReyN- - ReyN+) .at -y-ata(N(t)) - y T ( N ( t ) ) a t a 2Rey-at(N(t))a -y+aat ( N ( t ) )-
+2Rey-ata
-
+ y;(N(t))aat + 2Rey+a(N(t))at
2Rey+aat
(67)
(68)
From (67) to (68), we used the relation (60). Formally, (68) can be solved with a same way as (28). Substituting
+
( N ( t ) )= cr(t)atu c(t)
11
we obtain d = -2 (Rey- - Rey+) a ( t ) 2 (Reydt d -c(t) = ZRey+a(t) - 2Reydt
+
-a(t)
-
Re?+)
(69)
(70)
and
a ( t )= 1 - exp (-2t (Reyc(t> =
( N ( o ) )- Rey-
-
-
Rey+)) ,
(71)
Rey+ (1 - exp (-2t (Rey-
-
Rey+))
).
(72)
with initial condition a(0)= 0,
c(0) = ( N ( 0 ) ) .
With notion of Rey-
-
Re?+ = 2a/dk~g(k)~26(w(k) - 0) =: y
(73)
and
(71) and (72) become
a ( t )= 1 - exp (-2ty),
(75)
or
( N ( t ) )= (1 - exp (-2ty)) (a+,,-
1 m) + (“0))
(77)
Of course this solution has only a formal meaning since in the initial condition ( N ( 0 ) )is infinite in the finite temperature case. In this sense, the solution should be written in the form 1 A(N(t)) := (“t)) - (“0)) = (1 - exp (-2ty)) ( a t , (78)
m)
The physical meaning of this solution is also as clear as the one for (34). Also in this case the energy of the system is exponentially released to the “slow degrees” of freedom of the field. However in this case not all the energy of the system is transferred to the field because the system is left not in the vacuum state by in a thermal state (with temperature p-’).
12
For the following discussion in the next section, let us comment on (67). The terms +2 (ReyM-
-
ReyM+) ata - 2 (ReyN- - ReyN+) .at
(79)
in (67) are due to the fact that N does not commute with ct and d t , whereas the oher terms
+
-y-utu(N(t)) - y T ( N ( t ) ) a t a 2Rey-at(N(t))a
-y+aat(N(t)) - y ; ( N ( t ) ) a a t
+ 2Rey+a(N(t))at
are due to the non-commutability of Ut with q and dt. Therefore, when we consider the equation for some observable X which commutes with ct and dt (for example an operator in the system space), the equation of motion becomes
d dt
d dt = -y-utaX(t) - y?X(t)uta
-X(t) = - (U-txUq -y+aat X (t)
-
+ 2Rey_atX(t)a y$ X (t)aa + 2Rey+a X ( t )at
(80)
which does not contain the terms (79). 4
System+2-fields
In this section, we consider a system interacting two fields with the following Hamiltonian;
H=Ho+XV
(81)
+
(82)
where
Ho = flat,
/dkw(k)&an,k, n=1,2
and
v=
n=1,2
/ d k (&(k)atUn,k f g n ( k ) a a L , k )
(83)
13
We suppose that the two fields are both Gibbsian at different temperatures, i.e., = & , n ' & ( k ) d ( k - k'). (an,kanf,k') t
(85)
with
Our interests in this model is that the behavior of the slow degrees of freedom of the field operator in the stochastic limit. Let us consider the number operator of each field
Nn =
J
dkaA,kan,k
(87)
and its time evolution
and we obtain the limit noise operators
Notice that
With the same argument used in the 1-field case in the previous section, we obtain
+
-dN l ( t ) = i ( ~ ? ) ~ a N ~ ( d?;'atN1(t) t) - Nl(t)ad?> dt n=1,2
-
Nl(t)a
14
+
(-y?)utuN2(t)
-
+
yT(n’N2(t)ata 2Rey?)atN2(t)a
n=1,2
-yp).at
N2 ( t )- yf-( n )N2 (t)aat + 2ReyY)aNz(t ) a t )
+2 (ReycL
-
Re?$),) at,
-
2 (Rey,_ (2) - Reygi) uut
(95)
Using (60) for each field, (94) and (95) can be written respectively d (-yF)ata(N1(t)) -dt( N l ( t ) ) = n=1,2
-
yT(n)(N1(t))ata 2Rey(n)at(Nl(t))a
d (-y?’utu(N2(t)) -(N2(t)) = dt n=1,2
-
yT‘n’(N2(t))atu
+
+ 2Rey(n’at(N2(t))a
+ 2Reyp’a(N2(t))at)
- y Y ) u u t (N2(t)) - yf-(”’(N2(t))aat
+2Rey?)ata - 2Reyy)aat The formal solution of (96) and (97) can be obtained
(Nn(t))= a n ( t ) a L + cn(t) where
(97) (98)
15
Notice that these are only formal solutions for the same reasons as in the case of (77), that is, (Nl(0)) and ( N z ( 0 ) )in cl(t) and cz(t) are infinite. This solution would be interpreted, in the renormalized sense, as
+
A(Nn(t)) := (N,(t)) - (Nn(0))= an(t)ata cb(t)
(103)
with
It is important to notice that, unlike the 1 field case, the time dependence of the each number operator remains even after the time goes to infinity in the following sense. First of all the system is driven by the field to the stationary state psys(co) := trfield (U-mp3 8 pfu!,)
where the parameter
p'
= $e-"P'a'a
(106)
is given by:
The time dependence of the number operators cancel each other at t i.e.,
--f
co,
and with this time dependence we can describe a non-vanishing current crossing the stationary state of the system. The relation between this current and non-equilibrium thermodynamics is considered in the following section.
16
It is also important to notice that the stationary state of the system is nothing but the equilibrium with the temperature p’-l. This temperature de>> 1 it can be approximated pends on 0 and in the high temperature case bY
(especially, in the symmetric case y(’) = y(’) it becomes p’-l (0;’ + ,8c1)/2.)Moreover, when we take into account only on degree of freedom of the system, it is impossible to distinguish between the non-equilibrium stationary state of the system interacting with the two fields and the equilibrium state interacting with a field of temperature This means that we need some additional interpretation to the master equation for the reduced density operator to define any current, as was done in Our definition of the current is based on the dynamics of not the system but the fields and this would be more relevant way to describe the non-equilibrium physics from microscopic point of view. N
’.
5
Stationary current
In this section, we introduce the notion of current and discuss its relation to thermodynamics. Using (94) and (95), let us introduce a current operator j 1 + 2 ( t )
dt where
This current can be considered as describing the flpw of quanta from the field 1 to 2. In addition we introduce the energy flow J,E,,(t) from 1 to 2 ji-2
where
( t ):=
d (
~
( 2t )- ~1 ( t ) )
(112)
17
5.1 Average of the current Using the solution (98), the partial average of this current, in the initial state of the master field, is obtained d (J1+2(t)) =
((N2(t))- (Nl(t)))
= 2 (y(2) - y(1)) exp
(
-
2(y(l) + y(2))t)utu
By the replacement of (86) with
we can take into account the chemical potentials p n (n = 1 , 2 ) of each field. The current (115) becomes
With the same computation we can estimate
Here, according to the ordinary thermodynamic argument we define the heat current (J;+,(W)) with (117) and (118) as ( JL 2( 4)
= (5;+2(W))
+ PO(Jl+2(W))?
Po =
P I + P2 2
(119)
These currents (Jfh2(m))( k = ~ , q are ) possitive when p1 = p2 and ,& < /32 (the temperature of field 1 is higher than 2) and it is consistent to the physical intuition. Let us focus on these stationary currents.
18
Now let us consider the linear transport regime which makes clear the relation with linear non-equilibrium themodynamics. Let
PL + PR
Po = ___ , SP=PR-PL
and
PO=
PL
+ PR
~
6 ~ P=L - P R
(120)
or
thus we obtain
J ~ = + m~ p
+ L ~ST-
= L16p J?4
TO
+ La-6T TO
with
This result is similar to the one which was derived by Sivan and ImryaO and Tasaki" for a different situation and from a different point of view: Onsager's reciprocal relation between the particle current (110) and the heat current (118)
holds.
19
6
Summary
In the present paper, we have studied the dynamical evolution in the stochastic limit of the field operator of a Bose oscillator interacting with one or two Bose fields and investigated its relation to non-equilibrium physics and we have obtained the Langevin and master equations for the associated currents. As in l 7 it was found that some operators of the field space, describing the “slow degrees of freedom” of the field survive even after the time rescaling and the stochastic limit extracts the slow dynamics of the field. In particular, the master equation for the number operator of the field was studied. In the case of a system two fields, nonvanishing currents in the stationary state of the system were obtained. These currents were defined directly in terms of the fields degrees of freedom. They are consistent with non-equilibrium thermodynamics in the linear regime and for them Onsager’s reciprocal relation holds. A similar situation (open system) and the role of the scaling limit were considered in the previous papers, especially in However we would like to emphasize that we defined the currents in terms of the slow degrees of the field which were extracted by application of the stochastic limit to the field, whereas their the definition of the current was given in terms of the reduced time evolution of the system. This should give a new insight to nonequilibrium physics from more microscopic point of view. Our approach can be easily appried to more general systems: this will be shown in a forthcoming paper.
+
3910.
Acknowledgements The authors acknowledge kind and helpful informations from Prof. Tasaki. Kentaro Imafuku is grateful to Centro Vito Volterra and Luigi Accardi for kind hospitality. Kentaro Imafuku is supported by an overseas research fellowship of Japan Science and Technology Corporation.
Appendix Here we sketch the computation of the case where, instead of a single harmonic oscillator, one consides a more complicated system, for example a chain of oscillators. Let us consider the following system Hamiltonian H, and interaction
20
V between the system and fields;
is such that the system Hamiltonian H , has
It is assume that U(q0,...qN-l) discrete spectrum, i.e.
The interaction Hamiltonian can be rewritten as
where
Q
E,,Zj t
n=L,R i<j R E F
=
C
PR,IG)(EjlPR,-R,
C P R ?
=1
r
%€F,
F := {R = R j - Ri : hRj, hRi E Spec H,} FR := {har E Spec H,} = {hRr E Spec H , : 3hRrr E Spec H,, Or - Rr,
= R}
In an interaction picture, (135) becomes
V ( t )= h
x
n=L,R 0
(aij,n(R,
t ) + aJi,n(-Or t ) ) + h.c
21
where
In the generic case, i.e. where 1. The spectrum Space H , is non degenerate 2. For any R IFnl = 1, i.e. there exists a unique pair of energy levels R,, R j E SpecH, such that R = R j - Ri, the interaction Hamiltonian (140) can be simplified as
V ( t )= fL
c
n=L,R%F
where an(Q)t ) =
EL ( a n ( R , t )+ a i ( - R , t ) ) +h.c
s
(142)
-i(w(k)-Cl)t
&(k)ak,ne
(143)
With time rescaling t + t / X z and X + 0, the white noise operator associated with (143) is obtained as 1 x an(^, t / x z ) bn,R,t (144) +
with the correlations = 2r6nnlbnJ6(t- t’) (bn,n,tb,,,nr,t,) t
x /’dk
Ign(k)lZ(Nn(k)+ 1)6(w(k)
-
0) (145)
= 2r6nn~hnf6( t t’) (b,,n,tbnl,nt,t<) t
x s d k
1gn(k)12Nn(k)6(w(k) - 0).
(146)
With the exactly same argument as in the harmonic oscillator case, we can compute the time evolution of the number operator of the field and we obtain
A ( N n ( t ) )=
c
ClEF
where
(An,n(t)@En
+ C&,n(t))
(147)
22
The non-vanishing current relevant to non-equilibrium thermodynamics can be seen in this case too.
References 1. F. Bonetto, J.L. Lebowitz and L. Rey-Bellet, math-ph/0002052 and reference therein. 2. A.J. O’Connor and J.L. Lebowitz, J. Math. Phys. 15,692 (1974). 3. J.L. Lebowitz and H. Spohn, J. Stat. Phys. 19,633 (1978). 4. G. Casati, J. Ford, F. Vivaldi, and W.M. Visscher, Phys. Rev. Lett. 52, 1861, (1984). 5. S. Goldstein, C. Kipnis and N. Ianiro, J. Stat. Phys. 41,915 (1985). 6. U. Zurcher and P. Talkner, Phys. Rev. A42, 3278 (1990). 7. W.R. Frensley, Rev. Mod. Phys. 62, 745 (1990). 8. G. Klein and I. Prigogine, Physica 19, 74, 89, 1053 (1953); R. Brout and I. Prigogine, Physica 22, 621, (1956), Prigogine “Nonequilibrium statistical mechanics” New York, Wiley( 1962). 9. H. Spohn and J.L. Lebowitz, Commun. math. Phys. 54,97 (1977). 10. H. Spohn and J.L. Lebowitz, Adv. Chem. Phys. 38. 109 (1979). 11. J. Bafaluy and J.M. Rubi, Physica A 153, 129 (1998); ibid 153, 147 (1998) 12. J.-P. Eckmann, C.-A. Pillet and L. Rey-Bellet, Commun. math. Phys. 201,657, (1999); J. Stat. Phys. 95,305 (1999). 13. S. Tasaki and P. Gaspard, J. Stat. Phys. 81, 935 (1995); Theor. Chem. Acc. 102,385 (1999). S. Tasaki, Proceedings of “the 3rd Tohwa University Conference on Statistical Physics”. 14. P. Gaspard, Phys. Rev. E53,4379 (1996). 15. T. Antal, Z. Racz and L. Sasvari, Phys. Rev. Lett. 78,167 (1997);
23
16. 17. 18.
19. 20.
T. Antal, Z. Racz, A. Rakos and G.M. Schutz, Phys. Rev. E57,5184 (1998); ibid 59,4912 (1999). S. Tasaki, Chaos, Solitons and Fractals, 12, 2657 (2001). L. Accardi, K. Imafuku, and Y . G. Lu, Volterra Preprint N . 5 0 ~(2002) L. Accardi, A. Frigerio, and Y. G. Lu, Commun. Math. Phys. 131, 537 (1990); L. Accardi, J. Gough, and Y. G. Lu, Rep. Math. Phys. 36, 155 (1995); L. Accardi, S. V. Kozyrev, and I. V. Volovich, Phys. Lett. A 260, 31 (1999); L.Accardi, S.V.Kozyrev, Quantum interacting particle systems, Lecture Note of Levico school, September 2000, Volterra Preprint N.431 L. Accardi, Y. G. Lu, and I. V. Volovich, Quantum Theory and Its Stochastic Limit Springer-Verlag (in press). U. Sivan and Y . Imry, Phys. Rev. E33,551 (1986).
24
QUANTUM STOCHASTIC DIFFERENTIAL EQUATIONS IN VIEW OF NON-EQUILIBRIUM THERMO FIELD DYNAMICS TOSHIHICO ARIMITSU Institute of Physics, University of Tsukuba, Ibaraki 305-8571, Japan E-mail:
[email protected] Most of the mathematical approaches for quantum Langevin equation are based on the non-commutativity of the random force operators. Non-commutative random force operators are introduced in order to guarantee that the equal-time commutation relation for the stochastic annihilation and creation operators preserves in time. If it is true, it means that the origin of dissipation is of quantum mechanical. However, physically, it is hard to believe it. By making use of the unified canonical operator formalism for the system of the quantum stochastic differential equations within Non-Equilibrium Thermo Field Dynamics, it is shown that it is not true in general.
1
Introduction
The studies of the Langevin equation for quantum systems were started in connection with the development of and are still continuing in order to develop a satisfactory f o r m ~ l a t i o n ~(see - ~ comments in lo). Most of the mathematical approaches for quantum Langevin equation are based on the non-commutativity of the random force operators. For dissipative systems, for example, we have equations for the operators ( a ( t ) )and ( a t ( t ) )averaged with respect to random force operators of the forms
d - ( a ( t ) ) = - i u ( a ( t ) ) - /G(a(t)), dt d - ( d ( t ) ) = i u ( a T ( t ) )- K ( d ( t ) ) , (2) dt with the initial condition (a(0))= a and ( u t ( 0 ) )= a t , where a and at satisfy the canonical commutation relation [a, a t ] = 1.
(3)
The equal-time commutation relation for these operators decays in time:
[ ( a ( t ) ) ,( a + ( t ) )= ] e-2nt.
(4)
Random force operators d f ( t ) and d f t ( t ) are introduced in order to rescue this situation. If the random force operators in the Langevin equations
d a ( t ) = -iua(t)dt - Ka(t)dt
+ &df(t),
(5)
25
the equal-time commutation relation for the stochastic operators a ( t ) and a t ( t ) preserves in time: d ( [ a ( t ) ,a t ( t ) ] )= 0 , meaning with (3) that
[ a ( t ) , a t ( t ) ] = 1.
(8)
The above argument is of zero temperature related only t o the zero-point fluctuation. However, it has been extended t o include the situations for finite temperature. Then, we have a crucial question. Should we interpret that the origin of thermal dissipation is quantum mechanical? In this paper, we will investigate this question with the help of the system of the stochastic differential equations within Non-Equilibrium Thermo Field Dynamics (NETFD)11-38. NETFD is a canonical operator formalism of quantum systems in farfrom-equilibrium state which provides us with a unified formulation for dissipative systems by the method similar t o the usual quantum field theory that accommodates the concept of the dual structure in the interpretation of nature, i.e. in terms of the operator algebra and the representation space. The representation space of NETFD (named thermal space) is composed of the direct product of two Hilbert spaces, the one for non-tdde fields and the other for tilde fields. It can be said that NETFD is a framework which gives a foundation of Green’s function formalisms, such as Schwinger’s closed-time path method, Keldysh-method, and so in terms of dissipative quantum field operators within the representation space constructed on an unstable vacuum. In the extension to take account of the quantum stochastic p r o c e s s e ~ ~ ~ - ~ ~ , NETFD again allowed us t o construct a unified canonical theory of quantum stochastic operators. The stochastic Liouville equations both of the Ito and of the Stratonovich types were introduced in the Schrodinger representation. Whereas, the Langevin equations both of the Ito and of the Stratonovich types were constructed as the Heisenberg equation of motion with the help of the time-evolution generator of corresponding stochastic Liouville equations. The Ito formula was generalized for quantum systems. NETFD has been applied to various systems, e.g. the dynamical rearrangement of thermal vacuum in s u p e r c o n d ~ c t o rspin ~ ~ , relaxation3’, various transient phenomena in quantum ~ p t i c s ~ l non-linear -~~, damped harmonic o~cillator~ the ~ , tracks in the cloud chamber (a non-demolition continuous mea~urement)~’, microscopic derivation of the quantum Kramers equation38.
26 In the next section, the framework of NETFD will be briefly explained. In section 3, two systems of the stochastic differential equations will be introduced, one with non-unitary time-evolution generator and the other with unitary time-evolution generator. Both systems are constructed to be consistent with the same quantum master equation. The key is the existence of the fluctuation-dissipation theorem between the multiple of martingale operators and the imaginary part of hat-Hamiltonian. In section 4,two systems will be applied to the model of damped harmonic oscillator interacting with irrelevant random force system by the linear dissipative coupling. It will be shown that both systems are consistently applicable. The existence of the non-commutative random force operators is essential for the system with unitary time-evolution generator. In section 5, two systems will be applied to the model of damped harmonic oscillator interacting with irrelevant system by the position-position interaction. It will be shown that the system with unitary time-evolution generator cannot produce the framework which is consistent with the master equation, since there appear only commutative random force operators in martingale. Section 6 will be devoted to summary and discussion. 2
Framework of NETFD
The dynamics of physical systems is described, within NETFD, by the Schrodinger equation for the thermal ket-vacuum lO(t)):
a
-lO(t)) at
= -iiqO(t)).
(9)
The time-evolution generator H is an tildian operator satisfying
The tilde conjugation
N
is defined by
where cl and c2 are c-numbers. The tilde and non-tilde operators at an equal time are mutually commutative:
[A, B] = 0.
(15)
27
The thermal bra-vacuum (11 is the eigen-vector of the hat-Hamiltonian H with zero eigen-value:
(llI? = 0.
(16)
This guarantees the conservation of the inner product between the bra and ket vacuums in time: (llO(t)) = 1. Let us assume that the thermal vacuums satisfy (11" = (11 and l O ( t ~ ) = )~ 10(to))at a certain time t = to. Then, (10) guarantees that they are satisfied for all the time: (11" = (11, lO(t))"
= lO(t)).
(17)
The tilde operator and the non-tilde operator are related by the thermal state condition for the bra vacuum:
(11.1 = (lIAt,
(18)
which reduces the numbers of the degrees of freedom to the original ones. The numbers of the degrees of freedom were doubled by the introduction of tilde operators. The observable operator A should be an Hermitian operator consisting only of non-tilde operators.
3 3.1
Two Systems of Stochastic Differential Equations
Quantum Master Equation
Let us consider the system of quantum stochastic differential equations which is constructed to be consistent with the quantum master equation (the quantum Fokker-Planck equation)
-iHIO(t)),
(19)
with the hat-Hamiltonian
where I?s = H s - I?s with H s being the Hamiltonian of a relevant system. It is easily seen that fis satisfies (113s = 0. It is assumed that the imaginary part fi of the hat-Hamiltonian can be divided into two parts, i.e., the relaxational part ~ Iand R the diffusive part ID:
28
and each of them satisfies
( I [ & = 0.
(ll& = 0 ,
(22)
Introducing the time-evolution operator c ( t ) by
d -
-V(t) dt
=
-ikV(t),
(23)
with the initial condition c(0) = 1, we can define the Heisenberg operator
A(t) = V - ' ( t ) A c ( t ) ,
(24)
which satisfies the Heisenberg equation d
-A(t) dt
= i [ H ( t ) ,A ( t ) ] ,
(25)
for dissipative systems. The equation of motion for the averaged quantity (lIA(t)lO)is derived by means of the Heisenberg equation (25) by taking its vacuum expectation:
d
-(1IA(t)lO) dt
= i(1I[kW7 A(t)llO).
(26)
The same equation can be also derived with the help of the master equation (19) as d
- (1IAlO(t))= -i ( 1I AH lO(t)).
(27) dt We would like to emphasize here that the existence of the Heisenberg equation of motion (25) for coarse grained operators is one of the notable features of NETFD. This enabled us to construct a canonical formalism of the dissipative quantum field theory, where the coarse grained operators a ( t )etc. in the Heisenberg representation preserve the equal-time canonical commutation relation
[a(t),a+(t)] = 1,
[a@),a+(t)l= 1.
(28)
3.2 Non- Unitary Time-Evolution The system of stochastic differential equations with non-unitary timeevolution is constructed by the following general procedures. The stochastic Liouville equation
dlOf (t)))= -ifif,tdtlOf (t))) 7
(29)
29
of the Ito type is specified with the stochastic hat-Hamiltonian
fif,tdt = H d t +dMt = Hsdt
+ifidt +dikt,
(30)
+
where fi = f i ~ fiD is thesame that appeared in the master equation (19). The martingale operator d M t annihilates the bra-vacuum (11 of the relevant system:
(lid&& = 0 ,
(31)
which means that the stochastic Liouville equation (29) preserves its probability just within the relevant system. This feature is the same as the one within the system of stochastic differential equations for classical systems. The martingale operator satisfies the fluctuation-dissipation theorem of the second kind:
dMtdMt = - 2 D ~ d t ,
(32)
which should be interpreted as a weak relatioma Applying the bra-vacuum (I of the quantum Brownian motion (see Appendix B) to (29), we have an equation for lO(t)) = (l0f ( t ) ) ) ,
(33)
which is nothing but the quantum master equation (19). Introducing the stochastic time-evolution operator
dPf(t)= -ifif,tdtVf(t),
(34)
we can define the stochastic Heisenberg operator
A ( t )= P;'(t)AVf(t),
(35)
which satisfies the stochastic Heisenberg equation (the Langevin equation)
d A ( t ) = i [ f i f ( t ) d t , A(t)]- dh;l(t) [dh;r(t), A ( t ) ] ,
(36)
of the It0 type. Here, we introduced the martingale operator in the Heisenberg representation by
d&(t)
= d (VF1(t)MtVf(t)) .
(37)
dM(t) = d'M(t),
(38)
Note that
aIt is similar to the classical cases where the fluctuation-dissipation theorem of the second kind is specified within the stochastic convergence.
30 with
d ' k ( t ) = Pr'(t)dMtVf(t). (39) Making use of the relation of the Ito-Stratonovich stochastic calculus (see Appendix A), we can derive from (29) the stochastic Liouville equation dlOf (t)))= -&,tdt
lOf(t))),
(40) of the Stratonovich type, where the symbol o indicates the Stratonovich stochastic multiplication. The stochastic hat-Hamiltonian Hf,tdt = Hsdt
O
+ iIjRdt + d M t ,
(41)
contains only the relaxational part f i ~ . With this hat-Hamiltonian, we can write down the stochastic Heisenberg equation
d A ( t ) = i [ H f ( t ) d t A(t)],
(42)
of the Stratonovich type. Note that it does not have the term producing diffusive time-evolution, which is the same characteristics that appeared in the system of classical stochastic differential equations. This is a minifestation of the fluctuation-dissipation theorem (32).
3.3
Unitary Time-Evolution
The system of stochastic differential equations with unitary time-evolution is constructed by the following general procedures. The stochastic Liouville equation
d l O f ( t ) ) )= -ifi:tdto IOf(t))), (43) of the Stratonovich type is specified with the stochastic hat-Hamiltonian Hxtdt = Hsdt
+d k ; ,
(44)
with the Hermitian martingale operator
(dMy) =d k f
(45)
Note that (43) does not satisfy generally the conservation of probability just within the relevant system, i.e.,
(lldh;l,U
# 0,
(46)
but it does within whole the system, the relevant and irrelevant systems, i.e.,
( ( l l d k : = 0.
(47)
31
Here, ((11 = (11(1 with (I being the bra-vacuum of the quantum Brownian motion (see Appendix B). The martingale operator satisfies the fluctuationdissipation theorem
dM;dI@
= -2fidt,
(48)
of the second kind. Introducing the unitary stochastic time-evolution operator
fif ( t )by
dUf((t)= -iHxtdt o O f ( t ) ,
(49)
with the initial condition U f ( 0 )= 1, we can define the stochastic Heisenberg operator
A(t) = U f l ( t ) A O f ( t ) ,
(50)
which satisfies the stochastic Heisenberg equation (the Langevin equation)
d A ( t ) = i [ H y ( t ) d t7 A(t)],
(51)
of the Stratonovich type. Note that the time-evolution generator U f ( t ) is a unitary operator:
q ( t )= U F l ( t ) .
(52) By making use of the relation between the Ito and Stratonovich stochastic calculus, we can derive from (43) the stochastic Liouville equation
dlOf(t)))= -i"r&dtlOf(t)h
(53)
of the Ito type with the stochastic hat-Hamiltonian
??Kt = H d t + d M y .
(54) Applying (I to (53), we see easily that it reduces to the quantum master equation (19). Within the Ito calculus, the time-evolution operator U f ( t )satisfies
d U f ( t ) = -i"r:,dtUf(t),
(55)
with the initial condition f i f ( 0 ) = 1. The stochastic Heisenberg operator A(t) defined by (50) satisfies the stochastic Heisenberg equation
d A ( t ) = i["ry(t)dt, A(t)]- d G u ( t ) [dAk'(t), A(t)],
(56)
of the Ito type. Here, we introduced the martingale operator in the Heisenberg representation by
d G u ( t ) = d (!?;l(t)M:Uj(t))
.
(57)
32
Note that
37
dh;r’(t) = d’h;r’(t), with
d ’ h U ( t )= OF1( t ) d M y O f ( t ). 4
Application to Quantum Damped Harmonic Oscillator
4.1
Q u a n t u m Master Equation
The hat-Hamiltonian of the semi-free field is bi-linear in (a, 6,at, lit), and is invariant under the phase transformation a --+ aeie:
H
= glut,
+ ggZitz + 93aG + g4atat + go,
(60)
where g’s are time-dependent c-number complex functions. The operators a , Gt , etc. satisfy the canonical commutation relation? [ak,
a t ] = 6k,k‘,
[Gk,
&t]
= 6k,k’.
(61)
The tilde and non-tilde operators are mutually commutative. Throughout this paper, we confine ourselves to a one-mode case, for simplicity, and do not label explicitly the operators a, 6t, etc. with a subscript k for specifying a momentum and/or other degrees of freedom unless it is necessary. The generalization to a dissipative quantum field is straightforward. The tildian nature (ik)”= i H makes (60) tildian:
H
+ ili,
= W ( U t U - &)
(62)
with
h = C&tU + 6t6) + cga6 + C3UtGt + c4,
(63)
where w = 8 e g1 = -8e gg, c1 = S m 91 = S m 92, cg = S m g3, c3 = S m 94 and c4 = S m go. With the help of (18) for A = a:
(64)
(116 = ( l J U t ,
the property (112= 0 gives us the relations 2c1
+ cg + c3 = 0,
c3
+ c4 = 0.
(65)
bThroughoutthis paper, we confine ourselves to the case of boson fields, for simplicity. The extension to the case of fermion fields are rather straightforward.
33
Then, (63) reduces t o
L'
= cl(ata
+ Gt6)
-
+ C Z ) atiit + ( 2 ~ +1 ~ 2 ) .
~ 2 -~( 26~ 1
(66)
Let us write down here the Heisenberg equations for
att(t)= eiHtate-iHt.
a(t)= eifitae--iHt
(67)
They are given by
d - a ( t ) = -iwa(t)
dt d -afl(t) dt
+
ClU(t)
= i w a t y t ) - c&t)
- (2Cl
+ c2) i i f ( t ) ,
- c&(t).
(68) (69)
Since the semi-free hat-Hamiltonian f? is not necessarily Hermite, we introduced the symbol lt in order t o distinguish it from the Hermite conjugation t. However in the following, we will use t instead of tt, for simplicity, unless it is confusing. By making use of the Heisenberg equations (68) and (69), we obtain the equation of motion for a vector ( l l a t ( t ) u ( t )in the form d -dt( l l a t ( t ) a ( t ) = -2K(llat(t)a(t)
where we introduced
K
+iC<(lI,
(70)
and C< respectively by
In deriving ( 7 0 ) ,we used the thermal state condition (18) in order t o eliminate tilde operators. Applying the thermal ket vacuum 10) t o ( 7 0 ) , we obtain the equation of motion for the one-particle distribution function
n(t)= ( l l a t ( t ) a ( t ) l o )= (11utulo(t)),
(73)
as d dt
-n(t)
=
-2nn(t)
+ iC<.
(74)
The equation (74) is the Boltzmann equation of the system. The function C< is given when the interaction hat-Hamiltonian is specified. The initial ket-vacuum 10) = lO(t = 0 ) ) is specified by
34
with a real quantity f . Here, we are neglecting the initial correlation 42. The initial condition of the one-particle distribution function n = n(t = 0 ) can be derived by treating (IlaLiIO)as follows. In the first place, (1lUiqO) = ( 1 l U f U t J O ) = f ((11. a 10) + (110)) = f (n+ 1) 7
(76)
where we used the tilde conjugate of (75)for the first equality, and the canonical commutation relation (61) for the second. On the other hand, (1laClO) = (1ILialO)=
(llU+UlO)
= 72.
(77)
Here, for the first equality, we used (15), i.e., the commutativity between the tilde and non-tilde operators, and, for the second equality, (64). Equating (76) and (77),we see that
n=- f 1 - f' If it is assumed that there is only one stationary state, we can refer the stationary state as a thermal equilibrium state. We will assign the thermal equilibrium state to be specified by the Planck distribution function with temperature T :
Then, we have from (74)
In this case, the Boltzmann equation (74) reduces to d -n(t) = -2K (n(t)-it). dt Solving (71) and (72) with respect to c1 and cg, and substituting (80) for C<, we finally arrive at the most general form of the semi-free hatHamiltonian I?, (62),corresponding to the stationary process 13:
H=Hs+ii?,
(82)
at&)- 2 (1 -k 6)a6 - 2'6at6'] - 2Kn.
(84)
where
fi
1- K
[(I + 2'6) (at, f
This hat-Hamiltonian is the same expression as the one derived by means of the principle of correspondence 11,12 when NETFD was constructed first by
35 referring to the projection operator formalism of the damping theory The hat-Hamiltonian (82) with (83) and (84) describes time-evolution of the system of a damped harmonic oscillator. It may be worthwhile to write down here an attractive expression which leads us to a new concept, named spontaneous creation of dissipation The expression is given by 43344145.
14115150,51,52.
where
is the order parameter for dissipative time-evolution of the unstable vacuum. The annihilation and creation operators
They satisfy the canonical commutation relation [Ti,,, Y,&,t] = 6k,k‘bfiu,
(91)
and annihilate the bra- and ket-vacuums at time t: yk,t)O(t)) = 0,
(117:
(92)
= 0.
We can obtain the results (85) only by algebraic manipulations. This technical convenience of the operator algebra in NETFD, which is very much similar to that of the usual quantum field theory, enables us to treat open systems in far-from-equilibrium state simpler and more transparent The vacuum (85) is the functional of the one-particle distribution function n k ( t ) . The dependence of the thermal vacuum on n k ( t ) is given by 22 30,31932733134135.
36
We see that the vacuum lO(t)) represents the state where exists the macroscopic object described by the one-particle distribution function n k ( t ) . The master equation (19) can be rewritten as 22
This shows that the reference vacuum, in this case, is migrating in the superrepresentation space spanned by the one-particle distribution function {nk ( t ) } with the velocity { d n k ( t ) / d t } as a conserved quantity.
4.2 Non- Unitary Time-Evolution Confining ourselves to the case where the interaction hat-Hamiltonian between the relevant system and the irrelevant system of Brownian motion is bi-linear in ( a , a t , and their tilde conjugates) and ( d B t , d B i , and their tilde conjugates), and is invariant under the phase transformation a + aeio, and dBt + dBt ez', the martingale operator satisfying (31) is given by
where we introduced the random force operator
dWt = 6 ( p d B t
+udBi) ,
and the annihilation and creation operators
+ uc+, y? = a+ a, (97) of the relevant system with p + u = 1, which satisfy the commutation relation yu = pa
-
[Yu,
r?]= 1,
(98)
and annihilate the relevant bra-vacuum: (1lyT = 0,
(ll??= 0.
(99)
Note that the normal ordering : . . . : in (95) is defined with respect t o the annihilation and the creation operators. Making use of the annihilation and the creation operators, we can rewrite and h~consisting of introduced in (84) as
h~
BR = - K (7'7~-k ???u) , f i =~ 2 K (a + U)r???.
h
(100) (101)
37 The Langevin equations for a ( t ) and a t ( t ) are given by
+
+ +
d a ( t ) = -iwa(t)dt - ~ [ ( -p ~ ) a ( t ) 2vGt(t)]dt d W t , d a t ( t ) = i w a t ( t ) d t - K[apa(t)- ( p - v ) a t ( t ) ] d t d W t ,
(102) (103)
where we used the facts
d W ( t )= dWt,
d W ( t ) = dWt.
(104)
If we put v = l / 2 , then p = l/2, (102) and (103) reduce, respectively, t o
&(t) = -iwa(t)dt
-kit(t)dt
dat(t) = iwat(t)dt - k i ( t ) d t
+ dWt,
+ dWt.
(105) (106)
Although d W t and d W t are commutative, we have the conservation of the equal-time canonical commutation relation
d ( [ a ( t ) ,a w l ) = 0.
(107)
Applying the bra-vacuum (( 1 I t o (102) and (103), we have, for any value of v ,
d((lla(t)= -iw((lla(t)dt
+ /c((llat(t)dt + & ( ( l l d B f .
-
d ( ( l ( a t ( t )= i w ( ( l l a t ( t ) d t-
r;((lla(t)dt &((lIdBt,
(108) (109)
Note that these Langevin equations for the vector ((lla(t)and ( ( l l a t ( t ) have, respectively, the same structure as (5) and (6).
4.3
Unitary Tame-Evolution
The unitary martingale operator satisfying (45) is given by :d
My :
= &i
-
: (ntdBt - d B f u
i (Y9dWt
+T?dWt)
-
+ t.c .> : i (dWt7,
+d W f 7 ,
with t.c indicating tilde conjugate. Note that there is no cross term between tilde and non-tilde operators. Here, we introduced new random force operators
&(dB2 - d B t ) , which annihilates the bra-vacuum (1 of the irrelevant dW2
=
(ldW$
= 0,
(ldW$
= 0,
(111) system:
(112)
38 and satisfies the commutation relation
The expression (110) is consistent with the microscopic Hamiltonian of the linear dissipative coupling. The martingale operator (110) satisfies the fluctuation-dissipation theorem (48) with (84). Therefore, we conclude that there exists the system of stochastic differential equations with unitary time-evolution operator consistent with the quantum master equation (19) with the hat-Hamiltonian (82). The Langevin equations for a ( t ) and at@) are given by
+ 6 dB(t), = i w d ( t ) d t + 6 dB+(t),
d a ( t ) = -iwa(t)dt d&)
(114) (115)
where the operators dB(t) and dBt(t) in the Heisenberg representation are defined by
d G dB(t) = Oyl(t)o 6dBt Of@)= 6dBt 6 dBt(t) = Oy1(t) 6 dBf O f ( t )= 6 dBf 0
0
0
-
~ a ( t ) d t , (116) Kat(t)dt. (117)
In deriving (116) and (117),we used the properties 1 [Of@) ? 6dBt] = [Of@), 6 dBt]+ ,[dOf(t), 6 dBt] = KUUf(t)dt,
(118)
and
which comes from the characteristics of the Ito multiplication:
(lOf(t)dBtl) = ( I O f ( W t l ) = 0.
(121)
Substituting (116) and (117), and applying ((11, (114) and (115) reduce to (108) and (109) having the same structures as (5) and (6), respectively.
39 5
Application to Quantum Kramers Equation
Quantum Master Equation
5.1
Let us find out the general structure of hat-Hamiltonian which is bilinear in (x,p , 2 , F ) . x and p satisfies the canonical commutation relation
[x, p ] = i.
(122)
Accordingly, 2 and @ satisfies
[2, @] = -i. (123) = ik, and (1Ik= 0 give us the general expression The conditions, (ik)”
H=Hs+iri,
(124)
where
Hs
= H s - iis,
1 2m
H s = -p2
mw2 + -22 2
’
ri = ri, + r i D ,
(125) (126)
with 1 (x - 2 ) ( p +@) , 2 1 2 fiD = - - m w ( l + 2%)(x - 2 ) . 2 Here, we neglected the diffusion in 2-space. The Schrijdinger equation d -lO(t)) = -iHIO(t)), at gives the quantum Kramers equation 46. The Heisenberg equation for the dissipative system is given by d 1 1 -x(t) = i [ H ( t ) ,x ( t ) ]= -p(t) --K { x ( t )- .(t)} , dt m 2 d 1 - p ( t ) = -mw2x(t) - --K {p(t) ~ ( t ) } dt 2 +irCw(l+an) { x ( t )- 2 ( t ) } . I7R = -i--K
+ +
(127)
(128)
(129)
(130) Applying the bra-vacuum (11 of the relevant system, we have the equations for the vectors: d 1 -(llx(t) dt = ;(1Ip(t),
d -dt( 1 J p ( t ) = - w 2 ( 1 l x ( t ) - -K(llp(t).
(131)
40
5.2 N o n - Unitary Time-Evolution The stochastic Liouville equation within the Ito calculus becomes
dlOf ( t ) ) = ) -i"rif,tdtlOf (t))),
(132)
with the stochastic hat-Hamiltonian
"rif,tdt = H d t
+dMt.
(133)
Here, the martingale operator d M t satisfying ( 3 1 ) is defined by
(134) with dKmW
dXt = 2 (dBt
+dBf) ,
(135)
where d B t , d B f and their tilde conjugates are the operators representing quantum Brownian motion (see Appendix B). The generalized fluctuationdissipation theorem is given by
dh;r,dh;r, = - 2 f i ~ d t .
(136)
Taking a random average, the stochastic Liouville equation (132) reduces to the quantum master equation (128) with lO(t)) = ( l O f ( t ) ) ) . The stochastic Heisenberg equation (the Langevin equation) for this hatHamiltonian is given by
d z ( t ) = i [ " r i f ( t ) d t ,~ ( t ) ]d A k ( t ) [dJG(t), ~ ( t ) ] 1 1 = -p(t)dt - K { z ( t )- 2 ( t ) }d t , m 2
+
dp(t) = -mw2z(t)dt
(137)
1
-
--IF.{p(t) + p ( t ) } d t - ( d X t + d X t ) , 2
(138)
where we used the properties
dX(t) =dXt,
d X ( t ) = dXt.
(139)
Applying the bra vacuum ((11 to (137) and (138), we have the Langevin equations for vectors
1 d ( ( l l 4 t ) = ,((1lP(t)dt,
(140)
d ( ( l I p ( t )= - m w 2 ( ( 1 J z ( t ) d t - n ( ( l l p ( t ) d t - 2((11dXt.
(141)
41
The averaged equation of motion is given by applying (141) in the forms
/)lo) to (140) and
where ((. . .)) = (lI(I . . . I) 10). The vacuums (1 and I) are introduced in Appendix B. These averaged equations can be also derived from (129) and (130) by taking the average ((. . .)).
5.3 Unitary Time-Evolution The martingale operator representing position-position interaction may be given by
dMF
= XdXt
-
2dXt.
(144)
We did not include the crossing terms between tilde and non-tilde operators to be consistent with the microscopic interaction Hamiltonian. The fluctuation-dissipation theorem for this martingale operator is given bY
dMydMF
=
-2Pdt,
(145)
with
fiu
= -Kmw
2
-(1 +an) (x - 2 ) . 8 Then, the Ito stochastic hat-Hamiltonian becomes
fixtdt
= Hudt
+d M y ,
(146)
(147)
is the hat-Hamiltonian for the master equation. The master equation is different from (128). The stochastic Heisenberg equations (the Langevin equations) for z ( t ) and p ( t ) become
1 d x ( t ) = -p(t)dt, m dp(t) = -mw2x(t)dt
-
dXt,
42
where we used the fact
d X ( t ) = dXt. (151) Applying ((11to (149) and (150), we have the Langevin equations for the vectors ((11x(t)and ( ( l l p ( t )in the forms
6
Summary and Discussion
Within the system of non-unitary time-evolution generator (non-unitary system), the time-evolution generator V f ( t )is constituted by the commutative random force operators dWt and dWt. Therefore, the random force operators dW(t) and d X ( t ) in the Heisenberg representation is, respectively, equal to dWt and dXt in the Schrodinger representation, i.e.,
dW(t) = dWt,
d X ( t )= dXt.
(154)
In the application of the system of unitary time-evolution generator (unitary system) to the damped harmonic oscillator where the martingale operator is constituted by non-commutative random force operators manifesting the linear dissipative coupling between the relevant and irrelevant subsystems, the random force operators in the Heisenberg representation are related to those in the Schrdinger representation by
d W ( t ) = dWt
-
Kyv(t)dt,
dWP(t)= d W t - r;ye(t)dt. The second terms show up because of the non-commutativity. The appearance of these terms is essential in order to make the unitary system consistent with corresponding master equation. On the contrary, in the application of the unitary system to the quantum Kramers equation where the martingale operator is constituted only by commutative random force operators manifesting the position-position coupling between the relevant and irrelevant subsystems, the random force operators in the Heisenberg representation is equal to those in the Schrodinger representation, i.e.,
d X ( t )= dXt.
(157)
43 Therefore, the unitary system cannot be consistent with corresponding master equation. The above applications tell us that the origin of dissipation cannot be quantum mechanical. In spite of this unsatisfactory nature of the unitary system, it is attractive since hat-Hamiltonian for microscopic system is Hermitian and there is no mixing terms between tilde and non-tilde operators. The hat-Hamiltonian should have the structure
for microscopic systems. In fact, we succeeded to extract the correct stochastic hat-Hamiltonian for the stochastic Kramers equation by an appropriate coarse graining of operators (the stochastic mapping) in time and corresponding renormalization of physical quantities 38. The simple limit 49 does not give us the correct Kramers equation. This something touchy situation should be investigated based on the unified system of stochastic differential equations shown in this paper. It will be reported in the future publications.
Acknowledgments The authors would like to thank Dr. N. Arimitsu, Dr. T . Saito, Dr. A. Tanaka, Mr. T. Imagire, Dr. T. Indei and Mr. Y. Endo for their collaboration with fruitful discussions.
Appendix A
Ito and Stratonovich Multiplications
The definitions of the Ito 47 and the Stratonovich 48 multiplications are given, respectively, by
X ( H ) ( t .)d Y ( H ) ( t )= X ( H ) ( t [Y(H)(t ) + d t ) - Y ( " ) ( t ) ],
+
d X ( H ) ( t .) Y ( H ) ( = t ) [ X ( H ) ( t d t ) - X ( H ) ( t )Y ] (H)(t), and
x [Y(H)(t
+dt)
-
Y(H)(t)],
(159) (160)
44
for arbitrary stochastic operators X ( H ) ( t and ) Y ( H ) ( in t ) the Heisenberg r e p resentation. From (159), (160) and (161), (162), we have the formulae which connect the Ito and the Stratonovich products in the differential form 1 X ( H ) ( t o) d Y ( H ) ( t )= X ( H ) ( t ) d Y ( H ) ( t )s d X ( H ) ( t .)d Y ( H ) ( t ) , (163)
+ 1 d X ( H ) ( t o) Y ( H ) ( t=) d X ( H ) ( t .) Y ( H ) ( + t ) - d X ( H ) ( t ). d Y ( H ) ( t )(164) . 2
The connection formulae for the stochastic operators in the Schrdinger representation are given, in the same form as (163) and (164), by 1 X(’)(t) o dY(’)(t) = X(‘)(t)dY(’)(t) sdX(’)(t) . d Y ( S ) ( t ) , (165)
+ 1 dX(‘)(t) o Y(‘)(t) = dX(’)(t) . Y(‘)(t)+ - d X ( S ) ( t ). dY(’)(t), 2
(166)
where the operators X(’)(t) and dX(’)(t) in the Schrodinger representation are introduced respectively through X ( H ) ( t )= Vi’(t)X(’)(t)Vj(t) and
d X ( H )( t )= v;1 (t)dX(’) ( t )v j ( t ). B
Quantum Brownian Motion
Let us introduce the annihilation and creation operators bt, bL and their tilde conjugates satisfying the canonical commutation relation:
The vacuums
(I
[bt,
bi,] = 6 ( t - t’),
I)
are defined by
and
&I)
btl) = 0,
[it,
= 0,
bill = 6(t - t’).
(167)
(Ibi = ( l i t .
(168)
dt’ bt,,
(169)
dt‘ bi,,
( 170)
The argument t represents time. Introducing the operators
1 1
t-dt
Bt
=
B!
=
dBt1 =
t-dt
1 t
dBj, =
and their tilde conjugates for t 2 0, we see that they satisfy B(0) = 0, Bt(0)= 0,
[B,, B!]= min(s,t),
(171)
45 and their tilde conjugates, and that they annihilate the vacuum thermal state condition for (I: dBtI) = 0,
d&l) = 0 ,
( I d B j = (Id&.
I)
with the
(172)
These operators represent the quantum Brownian motion. Let us introduce a set of new operators by the relation
( 173)
d C f = B'""dB,", with the Bogoliubov transformation defined by B'"u =
-") '
(1 +" -1 1
(174)
where f i is the Planck distribution function. We introduced the thermal doublet: dBf=' = d B t , -dBJ,
dJf=l
dBf=2 = d B J , dBF=2 = -dBt,
(175) (176)
and the similar doublet notations for d C f and d C f . The new operators annihilate the new vacuum (I, and have the thermal state condition for I):
I).
dCtI) = 0 , dCtl) = 0 , (IdCf = (IdCt. (177) We will use the representation space constructed on the vacuums (I and Then, we have, for example,
(178)
(IdBtI) = ( I d B f I ) = 0, (IdBJdBtI)= " d t , They can be written
+
( I d B t d B J I ) = (1 f i ) d t .
dBjdBt = dBtdBt = "dt, dBtdBi = dBfdBf
=
(1 + " ) d t ,
(179) (180) (181)
as weak relations. References
1. I. R. Senitzky, Phys. Rev. 119, 670 (1960). 2. M. Lax, Phys. Rev. 145,110 (1966). 3. H. Haken, Optics. Handbuch der Physikvol. XXV/2c (1970), [Laser Theo r y (Springer, Berlin, 1984)], and the references therein. 4. R. F. Streater, J. Phys. A: Math. Gen. 15,1477 (1982).
46 5. H. Hasegawa, J. R. Klauder and M. Lakshmanan, J. Phys. A: Math. Gen. 18,L123 (1985). 6. L. Accardi, Rev. Math. Phys. 2, 127 (1990). 7. R. L. Hudson and K. R. Parthasarathy, Commun. Math. Phys. 93,301 (1984). 8. R. L. Hudson and J. M. Lindsay, Ann. Inst. H. Poincark 43,133 (1985). 9. K. R. Parthasarathy, Rev. Math. Phys. 1,89 (1989). 10. R. Kubo, J. Phys. SOC.Japan 26 Suppl., 1 (1969). 11. T. Arimitsu and H. Umezawa, Prog. Theor. Phys. 74,429 (1985). 12. T. Arimitsu and H. Umezawa, Prog. Theor. Phys. 77,32 (1987). 13. T. Arimitsu and H. Umezawa, Prog. Theor. Phys. 77,53 (1987). 14. T. Arimitsu, M. Guida and H. Umezawa, Europhys. Lett. 3,277 (1987). 15. T. Arimitsu, M. Guida and H. Umezawa, Physica A148, 1 (1988). 16. T. Arimitsu and H. Umezawa, J. Phys. SOC.Japan 55, 1475 (1986). 17. T. Arimitsu, H. Umezawa and Y. Yamanaka, J. Math. Phys. 28,2741 (1987). 18. T. Arimitsu, J. Pradko and H. Umezawa, Physica A135,487 (1986). 19. T. Arimitsu, Physica A148,427 (1988). 20. T. Arimitsu, Physica A158 (1989) 317. 21. T. Arimitsu, J. Phys. A: Math. Gen. 24,L1415 (1991). 22. T. Arimitsu, Physics Essays 9,591 (1996). 23. T. Arimitsu, Thermal Field Theories, eds. H. Ezawa, T. Arimitsu and Y. Hashimoto (North-Holland, 1991) 207. 24. T. Arimitsu, Phys. Lett. A153,163 (1991). 25. T. Saito and T. Arimitsu, Modern Phys. Lett. B 6,1319 (1992). 26. T. Arimitsu, Condensed Matter Physics (Lviv, Ukraine) 4 (1994) 26, and the references therein. 27. T. Saito and T. Arimitsu, J. Phys. A: Math. Gen. 30,7573 (1997). 28. T. Imagire, T. Saito, K. Nemoto and T. Arimitsu, Physica A 256,129 (1997). 29. T.Arimitsu, Y. Sudo and H. Umezawa, Physica 146A,433 (1987). 30. T. Tominaga, M. Ban, T. Arimitsu, J. Pradko and H. Umezawa. Physica 149A,26 (1988). 31. M. Ban and T. Arimitsu, Physica 146A,89 (1987). 32. T. Tominaga, T. Arimitsu, J. Pradko and H. Umezawa, Physica A150, 97 (1988). 33. T. Iwasaki, T. Arimitsu and F.H. Willeboordse, Thermal Field Theories, ed. H. Ezawa, T. Arimitsu and Y. Hashimoto (North-Holland 1991) 459. 34. T. Arimitsu, F.H. Willeboordse and T. Iwasaki, Physica A 182,214 (1992).
47 35. T. Arimitsu and N. Arimitsu, Phys. Rev. E 5 0 , 121 (1994). 36. T. Saito and T. Arimitsu, Mod. Phys. Lett. B 7 , 623 (1993). 37. T. Arimitsu, Stochastic Processes and their Applications, Ed. A. Vijayakumar and M. Sreenivasan (Narosa Publishing House, Madras, 1999) 279. 38. T. Satio and T. Arimitsu, Stochastic Processes and their Applications, Ed. A. Vijayakumar and M. Sreenivasan (Narosa Publishing House, Madras, 1999) 323. 39. J. Schwinger, J. Math. Phys. 2, 407 (1961). 40. L. V. Keldysh, Sov. Phys. JETP 20, 1018 (1965). 41. K. Chou, Z. Su, B. Hao and L. Yu, Phys. Rep. 118, 1 (1985). 42. S. Fujita, Introduction to Non-Equilibrium Quantum Statistical Mechanics (Robert E. Krieger Pub. Comp., Malabar, Florida 1983). 43. F. Haake, Springer P a c t s in Modern Physics, vol. 66 (Springer-Verlag, 1973) 98. 44. F. Shibata and T. Arimitsu, J. Phys. SOC.Japan 49, 891 (1980). 45. T. Arimitsu, J. Phys. SOC.Japan 51,1720 (1982). 46. T. Arimitsu, J. Phys. SOC.Japan 51,1054 (1982). 47. K. Ito, Proc. Imp. Acad. Tokyo 20, 519 (1944). 48. R. Stratonovich, J. SIAM Control 4, 362 (1966). 49. L. Accardi, J. Gough and Y. G, Lu, Rep. Math. Phys. 36, 155 (1995). 50. T. Arimitsu and H. Umezawa, in Advances on Phase Transitions and Disordered Phenomena, eds. G. Busiello, L. De Cesare, F. Mancini and M. Marinaro (World Scientific, Singapore 1987) 483. 51. H. Umezawa and T. Arimitsu, in Foundation of Quantum Mechanics I n the Light of New Technology, eds. M. Namiki, Y. Ohnuki, Y. Murayama and S. Nomura (Physical Society of Japan, Tokyo 1987) 79. 52. T. Arimitsu, H. Umezawa, Y. Yamanaka and P. Papastamatiou, Physica A148,27 (1988). 53. T. Arimitsu, Mathematical Sciences [S.iiri Kagaku], June (1990) 22-29, in Japanese.
48
NON-EQUILIBRIUM LOCAL STATES IN RELATIVISTIC QUANTUM FIELD THEORY * IZUMI OJIMA Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan E-mail: ojimaQkurims.lcyoto-11.ac.jp A novel method is established in relativistic QFT for distinguishing nonequilibrium states with a local thermodynamic interpretation, by comparing them with KMS states by means of local thermal observables. With the help of such observables, the states can be ordered into classes of increasing local thermal stability. It is also possible to identify states exhibiting certain specific thermal properties of interest, such as a local temperature or entropy density. The method is illustrated in a simple model describing the spatietemporal evolution of a “big heat bang”.
1 Introduction In contrast to global equilibrium states specified by the KMS condition in a general and clear-cut way2i3, there has so far been no general mathematical characterization of local thermal states to describe such nonequilibrium situations as those only locally in equilibrium, e.g., a situation with an x-dependent temperature distribution. Once one leaves the realm of global equilibrium, one necessarily encounters a variety of different thermal situations which make a simple-minded unification impossible. Thus, what is most crucial in discussing local thermal situations is to find pertinent concepts and their mathematical formulations, so as to discriminate different non-equilibrium states and to describe their relevant local thermal properties. We emphasize that the states of interest should be identified in connection with microscopic theory which is taken here as relativistic quantum field theory (QFT) suited for describing local properties of physical states. In this setting, we propose a new mathematical framework for treating local thermal states, on the basis of a suitable set of Lilocal thermal observables” defined at each point x, which enables us to compare at x any given state with the family of global equilibrium states regarded as “thermal reference states”. To understand the meaning of this comparison, such parallelism might be helpful that the latter is compared with the notion of a model space appearing universally in mathematics, e.g., R” in the theory of manifolds, where local charts describe situations in local neighbourhoods by mapping them onto the model *TALK BASED UPON A JOINT PAPER BY D.BUCHHOLZ, 1.0. AND H. ROOS’
49
space, which is similar for “local thermal observables”. Another picture will also be possible in relation with quantum statistical inference to determine unknown states in relation with some family of known states. To explain the essential points of the formalism, we start in Sec. 2 with a brief survey of a few necessary ingredients of relativistic QFT and identify the set of thermal reference states with the relativistic KMS states. The characterization of local thermal observables is given in Sec. 3. In terms of these two ingredients, analysis and interpretation of local thermal properties of non-equilibrium states are given in Sec. 4. Here we identify a hierarchy of locally thermal states ordered in increasing degree of accuracy approaching to equilibrium. A model illustrating our general scheme will be discussed in Sec. 5. This model involves spontaneous breaking of t i m e reversal s y m m e t r y due to positivity of states in nonequilibrium. 2
Thermal reference states
Among the basic ingredients of relativistic QFT4,we need the following ones: i) Local observables describing the basic physical variables (e.g., stress energy tensor, conserved currents, etc.), denoted generically by &x). Here a quantum field &x) is a singular object to be treated as an “operator-valued distribution”, which becomes an operator after being smeared with a test function f E D(R4) or S(R4):
4(f
:= / d & )
fb).
(1)
Later we directly treat &x) at a point x by taking it as a quadratic f o r m on a suitable domain of state vectors (or more directly, a s a form on the space A* of linear functionals on the algebra A defined below). The polynomials A in smeared fields of such a form as
constitute a *-algebra A of local observables (with scalar multiples of 1 understood to be included) equipped with the involution * defined by
fio
where &(x) = (complex conjugate).a For a finitely extended spacetime region O ( c R4) the subalgebra of A generated by such A’s as above with aA purely algebraic definition of A can be made precise mathematically in terms of a Borchers algebra5 defined by the tensor algebra of smearing functions (divided by certain ideals related to the local commutativity, etc.).
50
supp(fi)c 0 is called a local subalgebra d ( U ) of d localized in 0. ii) The general notion of states definedAmathematicallyas normalized positive linear functionals w on A: w(clA1 c2A2) = c l w ( A 1 ) ~ 2 w ( A 2 ) , c1, c2 E @, w ( A * A ) 2 0 , and w(1) = 1. The conventional description of states in terms of state vectors is recovered in each of the GNS representation spaces (7rw,fjw,C2w)4 determined by w s.t. ,(A) = (0, I 7rw(A)C2,), fj, = rw(d)O,. iii) Relativistic covariance described by the automorphic action of the Poincar6 group P i = R4 M Lk on A: the group P i is a semi-direct product of the group R4 of spacetime translations a, x Hx a and of the (proper orthochronous) Lorentz group L! (as the connected component of the group) of Lorentz transformations A, x = (xp) Ax = (AEx”), preserving the Minkowski metric x . y := xOyo- ? . y’ G gpvxpyv invariant: Ax . A y = x . y . The action a of P i on A: P i 3 X = ( a , A ) H ax E Aut(d) is defined by
+
+
-
+
where f x ( x ) := D ( A ) f ( A - l ( x- a ) ) with a matrix representation D of L l determined by tensor characters of For a pure translation (1,a ) , a E R4,we denote a ( l , a=: ) a, for short. We have Lyx(d(0))= d(X(0)) = d ( A 0 a)). iv) Local commutativity as a mathematical expression of Einstein causality valid for observables: [d(01),A(Oz)] = 0 if 0 1 and 0 2 are spacelike separated in the sense that ( x - y)’ < 0 for V x E 0 1 , V y E 0 2 . v) Global equilibrium states identified with relativistic KMS states: A relativistic KMS state wp (with inverse temperature p > 0 in a Lorentz frame specified by a timelike unit vector e E V+ := {x E R4;x 2 = x . x = ( x ’ ) ~- (5)’> O,xo > 0)) is characterized by the relativistic KMS condition6: for each pair A,A’E d there is a function h = hA,A, analytic in R4 i (V+n (Pe - V+))=: Dp, and continuous on with boundary conditions s.t. for Va E R4,
4.
+
+
h ( a ) = wp(A’a,(A)), h(a
+ i p e ) = wp(a,(A)A’).
(5)
Any KMS state wp describes thermal equilibrium in the rest frame of the system. We use a notation combining an inverse temperature l / k ~ T =@ and a rest frame e = ( e p ) = ( P p / @ ) into one 4-vector p = (pp) = p e E V+. The set of KMS states is known to be a simplex3: any state in the convex set Cp of all KMS states wp at ,L3 E V+ is decomposed uniquely into a convex combination of extremal points(=pure thermodynamic phases) in Cp. Thus states in Cp can be distinguished by “classical” (=central) observables (e.g., chemical potential, mean densities distinguishing different phases).
51
Lorentz behaviour of KMS states: From the definition of KMS condition we obtain Cp 3 wp H wp o ax1 E CAP for X = (A,a) E P l . While the temperature is unchanged: 1 / f l = the state wp is changed, wp # wp o ax’, by the change of the reference frame e = under the boost (: spontaneous breakdown of Lorentz invariance in thermal equilibrium7). Just for simplicity we adopt here a working hypothesis of the absence of phase transitions, i.e. Cp for ‘doconsists of a unique KMS state wp. To avoid mathematical subtleties, we also assume the weak continuity of KMS states wp in p, that is, all functions p H w p ( A ) , V A E A, are continuous, as is expected quite generally except phase transition points. From this uniqueness assumption a simple transformation formula follows, wp 0 ail = W A ~ which , implies that the KMS states considered here are isotropic in the rest frame and are invariant under space-time translations.
l/dm,
Family of thermal reference states as “model space”: On the basis of these ingredients, we define the family of thermal reference states as follows. While the inverse temperature p is sharply fixed for any KMS state wp, we need to admit, in the discussion of non-equilibrium, such possibilities that p is not precisely known or statistically fluctuating even at a point x , which can be described by mixtures of KMS states. Denoting such mixtures with /3 fluctuating in a compact set B c V+ by CB := { w g =
J
d p ( P ) wp; dp: probability measure s.t. supp(dp) c B } , ( 6 )
we define the thermal reference states for our thermal interpretation of nonequilibrium local systems by
c :=
u
CB.
(7)
B : cpt CV+
This set C should play the role of a “model space”, in terms of which unknown local nonequilibrium thermal states are examined and described. To implement this idea, we need to map a given unknown state to the known space C, which is accomplished by “coordinatization maps”. Thus, our next task is to find such maps.
3 Local thermal observables = “coordinatization maps” Our basic strategy is as follows: we measure thermal observables {(a,} in an unknown state w of our interest and compare the measured values w ( @ %= ) @.i(w)regarded as “@~-coordinates”of w , with the corresponding ones
52 @i(wp) =: (ai(p) (wp E Cp) or ( a i ( w ~ (:= ) Jdp(p)(ai(@),B c V+, W B E C ) in the known states wp E Cp or WB E C. Then, such equalities as @ i ( w ) = @i(wp) (or (ai(w) = @ i ( w g ) ) will allow the unknown w to be identified with wp (or W ~ ) C E as long as the properties described by { Q i } are concerned: w = wp or W B (mod {ail). So, our aim in this section is to achieve this goal so as to exhibit the locally thermal properties of w , by selecting out suitable thermal observables {(ai} from the formalism of relativistic QFT. Note that smeared fields &f) E A are not well suited for describing local properties of physical states because of the fuzzy localization of a test function f . We also need to exclude many elements in A of no relevance to thermodynamic interpretations. Here we focus on a treatment of point-like fields &x) regarded as (quadratic) forms on states (or, on pairs of state vectors) at the cost of losing product structure, the latter of which can effectively be compensated by normal products in operator-product expansion (OPE).lo This induces a transition from the algebra A to a linear space Q, consisting of normal products of fields 4(x) at a spacetime point x E R4. Then, local thermal obseruables at x are identified in Q, with its suitable subspace 7,. Thermal interpretation of elements in 7,as thermal observables U Cg. is settled by pairing them with thermal reference states E C = B: cpt CV+
3.1 Point fields as idealized local obseruables
For a precise formulation, we need some energy bounds' of the form, zt$(f) 5 v(f) (1+ H ) k , valid in the vacuum sector H of many models. (Here u is some Schwartz norm of test functions, H a (positive) Hamiltonian, and k > 0 determined by Physical idea behind this relation is that observed values of &f) can become large only with large energy.) In a more refined formg: for Vl > 0 , there should exist a sufficiently large m > 0 s.t.
6.
(8)
with 11.11: operator norm in X,A: Laplacian on R4.From this it follows that there exist a sequence of test functions 6i tending to the Dirac measure 6, at x, hi + h,, and a sufficiently large m > 0 such that ,
a+cc
lim (1
i+m
+ HI-"
(1
+ H)-"
=: (1
+ HI-"
&XI (1
+
.
(9)
Then a field &x) at a spacetime point x becomes well-defined as a form on such states w in the vacuum sector that w ( ( 1 < 00. We denote
+
53
Q,,, the linear spaces of such point fields $(z) satisfying the condition 11(1+ H)-"$(z)(l + H)-"ll < 00, whose hermitian elements can be regarded as idealized observables at z meaningful for states w s.t. w ( ( 1 H ) 2 m ) < CO. For m 5 m', Q,,, c Q,,,,and Q,,, is invariant under the subgroup of P r stabilizing the point z and is in general finite dimensional. 111 Q,,,,the meaningless product of fields at a point are substituted by normal products defined by OPE method (whose mathematical justification has recently been given in a satisfactory way by Bostelmann"). For instance, the ill-defined square $ ( z ) ~is replaced by the subspaces N ( $ 2 ) q ,ofx Q,,, which are generated by such 6j(z), j = 1;.. , J ( q ) , called normal products as appearing in OPE of $(z <)&z -
+
+
This estimate holds for any $ E Q,,, for spacelike C(E R4)+ 0 with any q > 0, by choosing suitable finite number of fields 6j(z)E Q,,,, sufficiently large n E No,and suitable set of analytic functions C H cj(C), j = 1 , .. . , J ( q ) . In the similar manner, the spaces ~ V ( $ p ) ~ , , ( cQ,,,)of normal products of higher powers p can be defined. The convergence in (10) can be made sufficiently rapid so that the spaces N ( J 2q), x (with large q ) of normal products contain also "balanced" derivatives
8, [$(z + ()$(z - <)] with 8, any monomial of partial derivatives w.r.t. the relative coordinates C:
L
J=1
J
(11) which holds for V r > 0 and with 3q > 0 and 3n E No sufficiently large. As mentioned at the beginning, the validity of the above inequalities are restricted so far in the v a c u u m sector, whereas we need in the present context the quantities in thermal situations. This gap between vacuum and thermal sectors can be filled up by means of the universal localizing map12 which allows us to define local and representation-independent Hamiltonian H o ( 2 0) satisfying & ( A ):= $at(A)It=o = [ i H , A ]= [ i H o , A ] for VA E d(O)(nDom(6)) in a local region 0 c R4.By replacing H with this H o , all the above results can be made applicabLe to our present thermal situations: e.g., expectation values of point fields @(z)E Q,,,, z E 0, become well-defined in a thermal
54 state w s.t. w ( ( 1 +Ho)’,) < 00 (as a constraint on a state w compatible with a locally thermal interpretation of w requiring locally finite energy). So our states w can be extended from A to Q,,,, n > 0, for x in some local region 0. In particular for thermal reference states W B E C, the equality WB(&Z)) = W B ( f j ( f ) ) valid for vf s.t. s d x f(x) = 1 shows that expectation values of intensive observables in equilibrium can be determined in any small space-time cells. 3.2
Local thermal observables & their hierarchy
In A and Q,,, there are many elements irrelevant to thermal interpretation of states at z. For instance, 8, &x) which is sensitive to space-time variations of states to be examined vanishes in all thermal reference states because of their translation invariance. Thus, it is necessary to select out from Q,,, suitable subspaces consisting of local thermal observables. This task is done as follows: 0
0
0
0
For small n, we have no freedom because of Q,,, = C1. Increasing n, we have, at certain no basic field of the underlying theory.
> 0, Q,,,,
= C & ( x ) with
&(x) a
Next, a meaningful substitute for &(x)’ are the normal product spaces JV(&)~,,, q > 0. Similarly, spaces N ( c & ~ , , ,q > 0 play the role of higher powers $ ~ ( x ) p~ ,E N . Elements of all these spaces are to be regarded as thermal observables, constituting a proper subspace 7,in the space of all point fields:
I, :=
c
N(d{) q,x
1
P>q
where N ( @ )q,x := C 1 and N ( & )q,x := C &(x), q
> 0.
As a result, we arrive at the set 7,of all local thermal observables whose hierarchical structure is self-evident by construction. In any thermal reference state W B E C, 2-point- functions are determined by expectation values of observables @j(x) E ~ V ( q 5 g ) ~ , , , > q 0 in W B E C, and similarly the spaces N ( @ )q , x , q > O fix ppoint functions for any p E N. Since ppoint functions for large p are shown to govern those for smaller p (whose proof is based on the relativistic KMS condition, Edge-of-the-Wedge Theorem and OPE), we see that this hierarchy has an operationally intrinsic meaning in such a form as “the larger p , the finer resolution of thermal properties is provided
55
by N ( @ ) q , x ,q > 0”. In particular, I, of all thermal observables at II: s e p arates states in C and is big enough to determine all properties of thermal states. Therefore, macroscopic properties of thermal states can be analyzed with some subset of thermal observables in N ( @ )q,x c ‘& for small p and q.
3.3 Macroscopic interpretation of local thermal observables Now local thermal observables in 7,at hand, we clarify how they provide information about macroscopic thermal properties of states in C.
Thermal functions as macroscopic observables: Under the uniqueness assumption on the KMS states, all intensive thermal parameters associated with states in C can be represented by functions /3 ++ F(/?),which we call thermal functions (in parallel with the physical quantities appearing in thermodynamics as thermodynamic functions, such as internal energy, entropy, etc., whose structural features are exhibited through their temperature dependence; when the uniqueness assumption is not valid, we simply need to add some order-parameters sufficient for discriminating thermodynamic pure phases, such as chemical potentials). We describe the correspondence between quantum local thermal observables and classical macroscopic obseruables by means of a map G associating such a thermal function to each quantum observable by
G : A or ?j 3 A
-
G ( A )E C(V+) with G ( A ) ( P ):= w p ( A ) .
(13)
Note that the thermal function G(&(z)) corresponding to any ~ ( I I : )E I, is an x-independent Lorentz tensor in3!, (determined by tensorial character of 6) as a result of wp 0 a;,,,, = W A ~ .The thermal interpretation of each local
-
@(p)= thermal observable &(x) is given by this thermal function P G(&(z))(p)= wp(&(z)) (which amounts to recording the mean values of a local thermal observable &(x) in all equilibrium states wp). Since G defined on A is easily seen to be linear, positive, G(A*A)2 0, and normalized, G ( l ) = 1, its dual map G* (from the set Ml(V+) of probability to the set Ed of states of d)is a classical-quantum measures dp@) on E (c-q) channel, p H G*(p), mapping a probability p to a quantum state G * ( p ) belonging to the space of thermal reference states C by
v+
by means of which thermal interpretation of a quantum state G * ( p ) is given
56 by the probability dp(P). Applying this to &(z) E I, we obtain
which fixes some (generalized) moment of d p ( P ) , unknown a priori in the actual situation. Since the image of 7,under G is dense in C(V+),all the moments of measure dp(@ are determined, from which dp(P) itself can be reconstructed. Thus we now have the following crucial observation:
Quantum local thermal observables 6 ( x ) E I , provide the same information o n the thermal properties of states in C as the corresponding classical macroscopic observables @ = G(&(z)) do (such as energy density, entropy density, etc.). For any W B E C , there is a probability measure dp(P) s.t. W B = G * ( p ) ,determined uniquely by 7,(which separates thermal states). Moreover, any continuous function F on compact subsets B c V+ can be approximated by thermal functions Q with arbitrary precision even i f F itself is not an image of G. This allows entropy density s ( p ) to be treated as a thermal function in spite of the absence of such G(z) E 7,that w p ( i ( z ) ) = s ( p ) . In the dual form of G as the c-q channel G*, the problem of finding its eflective inverse images involves interesting implications just for the characterization of non-equilibrium local states (see Sec.4): when restricted t o a subspace S, of T,, even a state w $ C may have its inverse image p~ under G*, which provides the locally thermal interpretation of w (i.e., conditional existence of a q-c channel for locally thermal states inverse to the c-q channel G*). 4
Thermal properties of non-equilibrium states
With thermal reference states and local thermal observables specified, we have now at our disposal the necessary tools for the analysis of locally thermal properties of non-equilibrium states. 4.1
Characterization of locally thermal states
Thermal properties of a state is determined by comparing it with thermal reference states in C by means of local thermal observables: given a subspace S, c I, at x E EX4, a state w is said to be S,-compatible with a thermal interpretation at z (S,-themal, for short) if 3wg = G * ( p g ) E C s.t. w ( i ( z ) ) = w B ( & ( z ) ) for ~ & ( zE)
s,,
(16)
57
to be denoted by w = G * ( ~ B(mod ) S,). Namely, w looks like a thermal state w.r.t. all observables in S,. Then, the mean values of thermal functions @ := G ( & ( x ) )E G ( S x )in w at x are defined consistently by
WB
w ( @ ) ( x ):= W(&(.))
= ps(@),
&(z) E
s,,
(17)
Thus, the relation w 3 G * ( ~ B(mod ) S,) can be “solved” conditionally in favour of p~ for any Sx-thermal state w , in the form of “(G*)-’”(w) := w ( @ ) ( x )= p~ (mod G(S,)), which provides the locally thermal interpretation for w a t x . (Viewed operationally, p~ may enter in the statistical inference as the probability of estimation whose Fischer information is specific heat . I 3 ) As a calibration procedure, this is a physically natural characterization of states with a locally thermal interpretation, whereas for practical use it is not convenient because of the necessity to refer each time to all members of the family C . Therefore, it is desirable to replace it with an equivalent characterization relying entirely on thermal observables 7,.By introducing on 7,a family of seminorms Q for all compact subsets B c V+ by
(meaningful due to continuity of a), our desire is materialized by Criterion 1 For a subspace S, of T, containing 1, a state w on A is S,thermal iff there is a compact set B c V+ s.t.
Iw(&(x))l I T B ( 6 ( 2 ) ) , E sx. (19) Remark 1 This can be compared with the step in manifold theory where the treatment of smooth functions on a manifold is given algebraically and intrinsically without recourse to the model space R”. A simple physical interpretation of this criterion is that the mean values of local thermal observables should not exceed their maximal possible values in thermal states. This constraint can be checked more easily in applications. Mathematical justification of the criterion relies on the standard measuretheoretic arguments14 as follows. If w is S,-thermal, there is a probability measure dp supported in B s.t.
for &(x) E S,, from which the inequality Iw(&(x))l I T B ( & ( z ) ) follows easily. Conversely, assume the inequality for w and some B . Since @ = 0 implies ~ ( & ( x= ) )supBGBI@(P)I = 0 and hence, w ( & ( x ) ) = 0, the linear functional w ( @ ) ( x ) := w ( & ( x ) ) on the space of thermal functions @ corresponding to
58 &(z) E S, is well-defined. By lw(@)(z)l I T B ( ~ ( Z )= ) S U P ~ E B I@(P)I and Hahn-Banach Theorem, this functional can be extended from a's to all the continuous functions F on B in such a way as
lw(F)(z)l
5
PEB
IF(P)I
= llFllB.
(21)
Then, w ( .)(z) defines a positive, normalized ( w ( l ) ( z )= 1) functional on a commutative unital C*-algebra of functions F with norm 11 . l l ~ so , that it can be represented by Gelfand theorem in the form w ( F ) ( z )= /,dp,(P) F ( P ) with a probability measure dp, supported by B. Then W B := G*(p,) E C is seen to coincide with w for S,: w = G*(p,) (mod S,). Thus w is S,-thermal. Thermal states W B describing locally thermal properties of w are not uniquely fixed by w = W B (mod S,) if the size of S, is too small. However, when w admits a thermal interpretation in the sense of w = W B (mod 7,) w.r.t. all thermal observables in 7z,then the corresponding state W B E C (and hence the corresponding measure dp,) is unique. 4.2
Existence of locally thermal non-equilibrium states
We show here the existence of locally thermal non-equilibrium states by verifying the following statement: For any finite-dimensional subspace S, of local thermal observables and any compact set B c V+, there exists an S,-thermal state w on A satisfying w = W B (mod S,) with some thermal state W B E C B , but is n o t I,-thermal. Such an w describes a non-equilibrium situation at x whose thermal interpretation is provided by the subset of thermal functions corresponding to S,. To find such a state, we consider the case where TB is a n o r m on S,, namely, T B ( & ( z ) ) = o for &(z) E S, implies i ( z ) = 0. Since all linear functionals on a finite-dimensional space are continuous and all norms are equivalent, we have for an arbitrary state wo on A some constant c s.t. IwO(@(z))l I c T B ( & ( z ) ) , &(z) E
sx.
(22)
Thus wo can be lifted to the space of thermal functions through W O ( @ ) ( X ) := & ( x ) E S,. By Hahn-Banach Theorem, this functional can be extended to the space of all continuous functions F on B s.t.
WO(&(X)),
Iwo(F)(.)l
I c SUP IF(P)I. PEB
(23)
This extension wo(.)(z) can be assumed to be hermitian by replacing, if necessary, w o ( F ) ( z ) with 2-'(wo(F)(z) wo(F*)(x)). For any such linear,
+
59
hermitian and continuous functional wo(.)(z) on the C*-algebra of continuous functions on B , there exists a signed measure do, supported by B s.t. w o ( F ) ( z ) = Sda,(P) F(P). Decomposing da, into its positive and negative parts do,*, da, = da,+ - dax-, and setting w ( A ) := (1 +a,-(B))-l
+
(WO(A)
J
da,-(p)wp(A))
, A E A,
(24)
i.e. w is S,-thermal. Since the choice of wo here is completely arbitrary, it is not '&,-thermal in general. The above expression for w shows that w can be interpreted as a perturbation of a thermal background state in CB. If TB is a seminorm there is a non-trivial subspace S,, 0 c S, annihilated by TB. We can reduce this case also to the above one in generic situations, but, we omit the details of it here (see '). Thus we conclude that there exist non-equilibrium states admitting a thermal interpretation f o r any finite-dimensional S, of thermal observables.
4.3 Degree of thermal stability In Sec. 3 we found a hierarchical structure in the space T, of thermal observables. Using it we can classify our locally thermal states into classes of increasing thermal stability as follows: 0
0
0
For trivial subspace S, = C1,all states are S,-thermal since only their normalization is tested. Adding the basic field &(z) to S, already results in a non-trivial constraint on the choice of states w , since the condition lu(&(z))I5 T B ( & ( x ) ) &(x) , E S, is not automatically satisfied because of unboundedness of the form &(z). Next, we add to S, the normal products belonging to N ( C $ ; ) ~ with ,, increasing q which provide us with information about correlations of &(z) at neighboring points. Using these normal products in the relation w = WB (mod S,), we can select states having the same correlations as W B ,E C. Along this line, we analyze a given state by means of elements in N ( & )q,x for higher powers p .
,
60
In this way, the hierarchy among Sxls is transcribed in the series of selected states tending to a genuine equilibrium. Thus, for an &-thermal state, the size of S, can be taken as a measure of its degree of thermal stability.
4.4 Determination of specific thermal properties Our framework allows us also to judge whether a thermal function ip has locally a definite value in a state w or is statistically fluctuating: to this end we need to examine the S,-thermality of w with sufficiently large S, such that the m e a n value of ip together with its fluctuations can be determined. For instance, if S, contains observables i p l ( x ) and @2(x) corresponding to ip and @, respectively, then the observable
corresponds to the thermal function (ip - K 1)2. This is non-negative in all thermal reference states E C and vanishes only in those states in which @ has a sharp value K . Thus if w is &-thermal and w(6@,(z)) = 0 for some K , we can conclude that @ has the sharp value K at x in w . So, choosing suitable spaces S,, one can select those states w having locally a sharp temperature vector p, i.e., w = wp (mods,). The minimal spaces S, admitting such an analysis are finite dimensional in generic cases (see Sec. 5 . ) . If w = wp (mods,), all thermal functions @ corresponding to 6(z) E S, have locally definite values in w. However, by enlarging S,, w may cease to have a thermal interpretation. If so, w shares only certain gross thermal properties with a KMS state wo with its non-equilibrium nature revealed by a more refined analysis. In this way, the method of analyzing locally thermal properties of states via suitable subsets S, of thermal observables amounts to some procedure of coarse graining. So, this suggests the possibility to identify a variety of states having locally certain definite thermal properties. Such an approach seems natural in treating nonequilibrium systems.
4 . 5 Space-time evolution of thermal properties Extending our formalism from a point x to a (finitely extended) subregion U c R4,we can now incorporate local states with thermal interpretation in U . For simplicity, we keep the set of thermal functions fixed in the respective regions, by identifying the spaces S,, x E 0 with the aid of translations:
61
With this convention understood, we say that a state w is So-thermal in 0 , if there exists WB(,) E C for each x E 0, s.t. w = WB(,) (mod S,). The resulting functions 0 3 x H w ( @ ) ( z )= w ( + ( z ) ) describe the spacetime behaviour of mean values of thermal functions @. Hence they provide the link between microscopic dynamics and the evolution of macroscopic thermal properties, i.e. thermo-dynamics of states. Thus, the above selection criterion for non-equilibrium states can be regarded as a localized and hierarchized form of the 0-th law of thermodynamics, by which the conceptual problems are solved of identifying non-equilibrium states admitting locally a thermal interpretation and of describing their specific thermodynamic properties.
4.6 Field equations and quasiparticles: a low energy theorem In states w close to thermal equilibrium, microscopic dynamics is seen to lead to linear evolution equations for thermal functions, as a generalized low energy theorem. Starting from the field equation O,&(x) = go(%)for basic field do(x),we are led to O,w(q50)(x) = 0 under the following assumptions: i) $o(z) is a linear combination of normal products in h/(@) po and qo (hence is a thermal observable).
qo,,
for some
ii) A state w is So-thermal with S, (x E 0) being any space of thermal observables s.t. S, 3 N(@') (hence &(z) and &(z) belong t o S,). Now i) implies that $(x) has vanishing thermal function EO= 0 because of w p ( g o ( z ) ) = wp(OZq5o(z)) = O,wp(&(s)) = 0 due to the translation invariance of KMS states. From ii) we have OZw(q5o)(z) = Ci,w(&(z)) = w(O,&(x)) = w ( $ o ( z ) ) = w(=o)(x) = 0. Similar behaviours can be found for many other thermal observables in suitable states: e.g., there exist thermal observables +(x) in N(&)q,, s.t. 0,6(x) = g ( x ) is also a thermal observable. Repeating the same arguments as above, we have O,w(@)(x)= 0 for such an If w ( g ( x ) )is different from 0 but small So-thermal w that S, 3 JV(&')~,,. for an So-thermal state w , the above equations are valid in an approximate sense. This result is consistent with the familiar quasi-particle interpretation of perturbations of equilibrium states. 5
A model example
The preceding abstract notions and results are illustrated here in the simple example of a free massless scalar field. After a brief outline of the model
62
we will determine the structure and physical significance of its local thermal observables. We then exhibit interesting examples of non-equilibrium states, describing a “big heat bang”, for which a definite temperature, thermal energy and entropy density can be defined at every space-time point in the future cone of some given initial point. The basic thermal observable in this model is the free massless scalar field $0 (z) characterized by the field equation and commutation relation G j o ( z ) = 0,
[iO(.l),
io(z2)l =
-
22)
. 1,
(28)
-~ &(p0)6Cp2). It generates a polynomial with i D ( z ) := ( 2 ~ ) J”dpe-i(”1-”2)P *-algebra A of local observables. A has the actions of the Poincar6 group P i , dilation group R+, and 2 2 , given respectively by a a , ~ ( & ( z=) )&,(Ax a), 6,(&(z)) = s&(sz), and k ( i o ( z ) ) = - - i o ( ~ ) . We restrict attention to Zz-invariant states w, i.e., w o y = w, so their n-point functions vanish for odd n. Simple examples of this type are quasifree states determined by 2-point functions through the familiar Wick formula:
+
w(do(z1)60(52)
._ .-
4 O ( ~ C n ) )
w(do(zil)6o(zi2))...w(io<~i,-,)ia(xi,)) n even,
(29)
pairings
(o
n odd.
Given any distribution A2(z, y) s.t. CI,A2(2, y) = O,A2(z, y) = 0, A2(z, y) -
A,(y,z) = i D ( z - y) and SJ”dzdyAz(z,y)f(z)f(y) 2 0 (Vtest functions f),-we can construct a state w on A as the linear extension of (29) with w(q5o(z)&(y)) := A,(z,y). This gives a large supply of simple states from which we will also draw the non-equilibrium states considered below. 5.1
Thermal reference states
As a well-known fact, the algebra A has a unique Zz-invariant (relativistic) KMS state wp for each temperature vector ,L? E V+. This state is quasifree and is determined by its 2-point function given by
As outlined in Sec. 2, the KMS states wg, p E V+, fix the convex set C of thermal reference states which enters into our analysis of the thermal properties of (Zz-invariant) non-equilibrium states.
63
5.2 Local thermal observables Now we determine the local thermal observables in our model. We will consider primarily the spaces N ( & )q , x , q > 0, which are generated by the Wick square of J0,its balanced derivatives and the unit operator 1. Introducing the multi= index notation p = ( p l , p z , . . . p m ) and setting ...a,,m, these balanced derivatives are defined by
v:J:: (z) := lim c-
0
[bo(z
+ ~ ) ~ o ( xC) - w, -
(bo(z
+ ~ ) J o ( -z c))11 ,
(31) where w, is the vacuum state (as recovered from (30) in the limit of large timeC)&(x - <) like p). For odd m the balanced derivatives vanish, since &(x is even in C as a consequence of locality. The thermal functions corresponding to these observables are given by
+
p H W p ( a q : ( z ) ) =c,d;(p2)-1,
(32)
+
where c, = 0 for odd m and c, = (-1)”/2(4~)”(rn 2)!F1B,+z (&: Bernoulli numbers) for even m. Among these functions for smaller m we find: (a) The thermal function associated with : : (x) is ,L? +-+ = ( 1 2 / ~ , ~ ) - ~this T ~ qualifies ; the temperature not as a parameter extrinsic to a physical system but as a physical quantity to be measured. (b) The balanced derivatives i””(x) := -( 1/4) P”:4 : : (x)give rise to
&
p
EP”(@)
:= (T2/90) (4P”P” - P2g””)
(33)
(@)-3,
which coincide with the expectation value in wp of symmetric traceless stress energy tensor f%‘”(z)given by
8’””(2)= Y ( z ) + (l/l2) ( a p ;
:&:(z).
- g P V 0,)
(34)
6:
Being a coboundary of a vector-valued 2-form (1/12) (8,. gP”-djcP gp”) : :(z), the second tensor on RHS gives no contribution to the total energy of any such states as deviating only locally from equilibrium, whereas, in non-equilibrium, it describes the transport of energy driven by sources localized at the boundary of the system. We call i””(x) the thermal energy tensor. It is conserved and symmetric, but its trace i,”(z)= (1/4) 0,: $8 : (z) vanishes not identically but in all states sufficiently close to equilibrium (according to the discussion in Sec. 4 on field equations). The thermal energy tensor and the Wick square can be used to distinguish KMS states in C corresponding to a given temperature vector: a thermal
64 function given with a fixed x E V+ by /3
H
(1/3) (4(/3x)’ - /3’x2)(/3’))-3
-
2(/3’)-’
+ (.c’)-l
(35)
associated with the thermal observable
6,i(x) := (30/.rr2)x,xut,ipu(~) - 24
:& : (x)+ ( x 2 ) - ’ .1,
(36)
is positive on V+ except for the point /3 = x (because of the “inverse” Schwarz inequality (Px)’ 2 P2x2valid in V+ of the Minkowski space). Hence W B ( ~ , ~ ( Z ) ) = 0 for WB E C implies the equality WB = w,: a KMS state with inverse temperature vector x . Thus in this model a finite number of thermal observables determine whether a state has locally a definite temperature and a well-defined rest system. (c) Because of (32) and Up (/3’)-’ = 0 valid on V+, all thermal functions corresponding to higher balanced derivatives are annihilated by Up. So the subspace generated by them does not separate the states in C. Yet it is large enough to approximate the entropy current s p , /3 s”(,B) := (2.rr2/45),P(/3’)-’, on any compact B C V+. Thus any S,-thermal state with Sxgenerated by these derivatives has a well-defined entropy density at x.
-
5.3 Examples of non-equilibrium states In Sec. 4 we proved abstractly the existence of non-equilibrium states admitting a thermal interpretation on finite-dimensional subspaces of local thermal observables. Here we exhibit non-equilibrium states which are substantially closer to thermal equilibrium: they admit a thermal interpretation on the infinite-dimensional spaces S, of thermal observables generated by N ( 6 )4,2, p = O , . . . , 3 , q > 0, for all x in a future lightcone in spacetime, at each point of which they have a definite temperature, thermal energy and entropy density. In spite of their simplicity the examples carry many physically interesting points. They describe the spatio-temporal evolution of systems which have infinite temperature at some space-time point (corresponding to a “big heat bang”). In view of the secondary role of masses of particles and their interaction near the singular points, our results may provide some hints to the dynamical effects of such singularities in more realistic theories although we are dealing with a massless free field. The non-equilibrium state considered here is quasi-free with a 2-point function defined for 5 1 , x2 E V+ in the form with y > 0 fixed:
65 which is consistent with the field equation and commutation relation in (28) and also is invariant under Lorentz transformations and dil?tions but not under translations. Since the positivity condition Wbhb (+0(~)40(f)) 2 o also holds if suppf c V+,it defines a (possibly singular) state of A. By direct computation we have the expectation values of the balanced derivatives in this state for even m and x E V+
where wP(,) is the KMS state corresponding to the temperature vector p ( x ) := 2yx. Thus Wbhb is &+-thermal, S, being the subspaces of thermal observables p = o,... , 3 , q > 0; in fact, Wbllb = up(,) (mod generated by JV(&)~,,, N ( & q,,) for odd p and q > 0 is trivial, 0 = 0, due to the &-invariance, but it no longer holds for p,= 4, q > 0, as checked by direct computation. So the higher correlations of $0 in wbhb are of a non-thermal nature. But the state approaches equilibrium for large time-like translations a E V+ in the sense of for Vlocal observables A E A,
lim wbllbo cra(A)= wm(A),
(39)
i.e. wbhhlooks asymptotically like the vacuum w,. Let us turn now to the locally thermal interpretation of this state. As 6,i(x) E S, (see (36)), and Wbhb (b,i(z)) = O for M = p ( x ) , we conclude wbhb= up(,) (mod S,) for b’x E V+ (which is also clear from (38)). Its local temperature T ( x ) = ( p ( x ) z ) - ’ / z = (4y2x2)-1/2decreases monotonically in time-like directions and all Lorentz observers moving along a world line R+ e register the same temperature at a given time after the big heat bang at 0. The thermal energy density and pressure in Wbhb can be read off from wbhb(cpu)(x)
= wbhb
(;””(.))
=
(r2/90) (4pp(x)pu(z)- g p y p ( x ) 2 )( p ( x ) 2 ) - 3
= (7r2/144Oy4) (4xpx’
-
gp’z2) ( x 2 ) - ’ ,
(40)
describing a flow of massless particles in V+ which is isotropic for the above Lorentz observers. They also find the relation between the thermal energy density and temperature in agreement with Stefan-Boltzmann’s law for massless scalar particles and the familiar relation valid between the thermal energy density and pressure due to b&b(EL)(x) = 0. From i ~ ( x=) (1/4) U, : : (x) follows the equation U, w b h b ( : : (x)) = 0, so the density : : (x) propagates like a massless particle, in accordance with the general results in Sec. 4. As a matter of fact, relation (38) implies
&
that 0,wbhh(W :
4::(x))
&
=0
for all balanced derivatives.
66
It is an interesting fact that the full energy density of the state wbhb is larger than its thermal energy density. Making use of relation (34) one obtains Wbhb(e’”(z))
= ((n2/1440y4)f (1/288y2)) (4zpzv- gp”z2)(z2))-3 . (41)
The coboundary term in Eq.(34) leads to an additional contribution due to the transport of energy from the hot boundary of the lightcone into its interior. Note that for y = 1 this term is of the same order of magnitude as the thermal energy, but it is not visible in the locally thermal properties of the state. In other words, the energy density of the state inferred from its temperature based upon Stefan-Boltzmann’s law can give only an underestimated value. This feature of an apparently “missing energy” may be of significance in cosmological models. As wbhh is N(&),,,-thermal for all q > 0, one can consistently attribute to it the entropy current wbhb(sp)(z)= (2n2/45) / P ( z ) ( p ( ~ ) ~=) - ~ (n2/180z3)z ” ( ~ ’ ) ) - for ~ 2 E V+. For the lack of interaction causing dissipa= 0, tive effects, it is no surprise that this current is conserved, dpwbhb(sfi)(z) without entropy production. While the entropy current decreases monotonically in time-like directions, the total entropy within spacelike sections of V+ of the form 121 < vlzol for fixed v < 1 stays constant for z o > 0. We conclude our discussion of the state wbhb with the remark that one can generate from it other states which are still &+-thermal. Because of a,(Sv+) c SV, valid for a E V+ and the invariance of states in C under translations, the state Wbhb 0 a, has this property, too. Moreover, since c is stable under convex combinations, the states J d v ( a )wbhbo a, with arbitrary probability measure u with compact support in V+ are also &+-thermal. If v differs from the Dirac measure, these states have, however, locally no longer a definite temperature vector and the corresponding expectation values of the thermal functions exhibit a more complex space-time behavior. While our attention here has been restricted to the simplest class of theories, our arguments can be extended to more complex situations with little more efforts. So we believe that our approach provides a natural setting for the analysis of the thermodynamic properties of non-equilibrium states in concrete models as well as in the general framework of QFT, to shed a new light on the complex features of non-equilibrium systems. A particularly interesting issue is the phenomenon of thermalization related to the problem of the arrow of time. Owing to the PCT symmetry of local QFT under which thermal observables transform covariantly, there is no such arrow encoded in its algebraic structures. As seen above, however, the states can break this symmetry in the sense that their thermal interpretation is compatible in some lightcone either future or past, but not in all of Minkowski space (unless they
67
are in global equilibrium): in fact wbhb does not have a thermal interpretation extending beyond V+, and similarly the state wbhbo 19, obtained from wbhb by the action of the anti-automorphism 19 implementing the P C T symmetry, is thermal only in the past light cone -V+. A proof establishing such a one-sidedness of thermalization in generic cases would be an important step towards the understanding of the microscopic origin of the arrow of time.
Acknowledgments
I would like to thank Prof. S.Tasaki and the organizers for having invited me to this fruitful workshop and Profs. L.Accardi, T.Hida and M.Ohya for their encouragement and interest in this work. References 1. D. Buchholz, I. Ojima and H. ROOS,Thermodynamic Properties of NonEquilibrium States in Quantum Field Theory (hep-ph/0105051), to appear in Ann. Phys. (N.Y.). 2. R. Haag, N.M. Hugenholtz and M. Winnink, Comm. Math. Phys. 5 , 215( 1967) 3. 0. Bratteli and D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics, Vol.2, Springer, 1996 4. R. Haag, Local Quantum Physics, Springer, 1996 5 . H.-J. Borchers, Nuovo Cim. 24, 214(1962) 6. J. Bros and D. Buchholz, Nucl. Phys. B429,291(1994) 7. I. Ojima, Lett. Math. Phys. 11, 73(1986) 8. W. Driessler and J. Frohlich, Ann. Inst. H. Poincar6 27,221(1997) 9. K . Fredenhagen and J. Hertel, Comm. Math. Phys. 80, 555(1981) 10. K. Wilson and W. Zimmermann, Comm. Math. Phys. 24, 87(1972) 11. H. Bostelmann, Lokale Algebren und Operatorprodukte am Punkt, PhD Thesis, Universitat Gottingen, 2000 12. D. Buchholz, S. Doplicher and R. Longo, Ann. Phys. (N.Y.) 170, 1(1986) 13. K. Matsumoto, A Geometical Approach to Quantum Estimation Theory, PhD Thesis, University of Tokyo, 1997 14. P.R. Halmos, Measure Theoq, Springer, 1974
68
STABILITY OF QUANTUM STATES OF FINITE MACROSCOPIC SYSTEMS AKIRA SHIMIZU, TAKAYUKI MIYADERA* AND AKIHISA UKENA Department of Basic Science, University of Tokyo 3-8-1 Komaba, Tokyo 153-8902, Japan E-mail: shmzQASone.c.u-tokyo.ac.jp We study the stabilities of quantum states of macroscopic systems, against noises, against perturbations from environments, and against local measurements. We show that the stabilities are closely related to the cluster property, which describes the strength of spatial correlations of fluctuations of local observables, and to fluctuations of additive operators. The present theory has many applications, among which we discuss the mechanism of phase transitions in finite systems and quantum computers with a huge number of qubits.
1 Introduction The stability of quantum states of macroscopic systems, which are subject to noises or perturbations from environments, have been studied in many fields of physics, including studies of ‘macroscopicquantum coherence’ and quantum measurement.2 However, most previous works assumed that the principal systems were describable by a small number of collective coordinates. Although such models might be applicable to systems which have a non-negligible energy gap to excite ‘internal coordinates’ of the collective coordinates, there are many systems that do not have such an energy gap. As a result of the use of such simple models, the results depended strongly on the choices of the coordinates and the f o r m of the interaction Hint between the principal system and a noise or an environment. For example, a robust state for some Hint can become a fragile state for another Hint. However, macroscopic physics and experiences indicate that a more universal result should be drawn for the stability of quantum states of macroscopic systems. Moreover, the stability against measurements were not studied well. Although one might conjecture that effects of measurements would be equivalent to effects of noises or environments, the conjecture is wrong as we will show in section 9 of this paper. In this paper, we study these stabilities using a general model with a macroscopic number of degrees of freedom N . In addition to the fact that N is huge, we make full use of the locality - ‘additive’ observables must be the sum of local observables over a macroscopic region (Eq. (7) below), the interaction hintmust be local (Eqs. (12) and (14)), and measurement must be local. By noticing these points, which were absent or ambiguous in most
69
previous works, we derive general and universal r e s ~ l t s We . ~ also propose a new criterion of stability; the stability against local measurements. We present a general and universal result also for this ~ t a b i l i t y . ~ The present theory has many applications because it is general and universal. We here mention applications to phase transitions in finite systems, and quantum computers with a huge number of qubits. 2
Macroscopic quantum systems
We consider “macroscopic quantum systems.” We first describe what this means. As usual, we are only interested in phenomena in some energy range A E , and describe the system by an effective theory which correctly describes the system only in A E . For a given A E , let M be the number of many-body quantum states in that energy range. Then,
N
N
1nM
(1)
is the degrees of freedom of the effective theory. Here, the symbol ‘N’ means that corrections of o ( N ) , such as ln(AE) = O(lnN), are neglected. For weakly interacting systems, for example, N becomes the number of singlebody quantum states which constitutes the M many-body states. Note that N sometimes becomes a small number even for a system of many degrees of freedom when, e.g., a non-negligible energy gap exists in A E . Such systems include some SQUID systems at low temperatures, and a heavy atom at a meV or lower energy range. W e here exclude such systems, because they are essentially systems of small degrees of freedom. Namely, we consider systems whose N is a macroscopic number. Otherwise, the difference between O ( N ) and O ( N 2 ) , which plays the central roles in macroscopic physics and in the following discussions, would be irrelevant. Since A E sets a minimum length scale e, the system extends spatially over a finite volume
V
N
Ned
in d dimension. Since V is proportional to the macroscopic number N, we say V is also macroscopic, disregarding the magnitude of e. In short, we consider macroscopic quantum systems for which N and V are macroscopic for a given energy range A E . Note that V is essentially equal to N because of Eq. (2). We therefore use V and N (and the words “volume” and “degrees of freedom”) interchangeably in the following discussions.
70 3
Cluster property
As we shall show later, correlations between distant points are important in the study of stability. As a measure of the correlations, we first consider the cluster property. Although we are considering finite systems, we first review the case of infinite system^.^ A quantum state of an in infinite system is said to have the cluster property if
( ~ ~ L L ( z ) ~ &+( Y o )as)
1 5
- y~ + 00
(3)
for any local operators &(x) and & ( y )at z and y , respectively, where
Here, by a local operator at x we mean a finite-order polynomial of field o p erators and their finite-order derivatives at position z.a The cluster p r o p erty should not be confused with the lack of long-range order: A state with a long-range order can have the cluster property. In fact, symmetrybreaking vacua of infinite systems have both the long-range order and the cluster p r ~ p e r t y . ~ ,Many ~ , ~ theorems , ~ , ~ ~ ~have been proved regarding the cluster property. 4,5 Among them, the following is most important to the present theory : K n o w n theorem: In infinite systems, a n y pure state has the cluster property. In oder to study finite systems, we generalize the concept of the cluster property to the case of finite systems. For a small positive number E , let Q ( E ) be the minimum size of the region over which correlations of any local operators become smaller than E . Namely, o ( ~=) supz ~ O ( C , Z ) )where , I ~ ( c , x ) ( denotes the size of the region O ( E , ~ Here, ). O ( C , ~is) defined by its complement O(C,z)', which is the region of y in which
for any local operators &(x) and b ( y ) . Then, we say that a quantum state of a finite macroscopic system has the cluster property if O(C) << V for sufficiently large V . More strictly, we consider a sequence of states for various values of V such that they are essentially equivalent to each other except for the values aTo express and utilize the locality of the theory manifestly, we use a field theory throughout this paper.
71
of V.’ We say that the states (for large V) of the sequence have the cluster property if a(€)for any E > 0 becomes independent of V for sufficiently large V. It is clear from the known theorem above that a sequence of pure states of finite V that do not have the cluster property approaches a mixed state of an infinite system as V + 00. 4
Anomalously-fluctuating states
As a second measure of correlations between distant points, we consider fluctuations of additive operators. Here, by an additive operator we mean an operator of the following form:
A=
c qx),
(7)
XEV
where &(x) denotes a local operator at x. When we regard the system as a composite system of subsystems 1 and 2, then A = A(1) A(’), hence the name “additive.” In thermodynamics, it is assumed that
+
(&A2)I o(V2) for all additive quantities.c In particular, if a state (classical or quantal) satisfies
(&A2)I O(V)
(9)
for all additive quantities, we call it a normally-fluctuating state (NFS). In quantum theory of jinite macroscopic systems, however, there exist pure states for which s o m e of additive operators have anomalously-large fluctuations; ( S A 2 ) = O(V2).
(10)
We call such a pure state an anomalously-fluctuating state (AFS). It is easy to show that a n AFS does n o t have the cluster property. Hence, according to the known theorem of section 3, an AFS cannot be a pure state in infinite systems. Considering also that in thermodynamics any state in a pure phase is an NFS, we see that an AFS can exist only jn finite macroscopic *For example, the ground-state wavefunctions of many particles for the same particle density, for various values of V. Cotherwise, it would be meaningless to talk about the average value of A, which is O(V). dIt is possible to change an AFS into an NFS by enlarging the system by adding an extra system of volume Vextra V2, in which the quantum state is an NFS, because this leads to ( 6 A 2 ) = O(V2) O(Vextra) = O(Kxtra) N O(Vextra V). We here exclude such an artificial and uninteresting case.
+
-
+
72
quantum systems. Since AFSs are such unusual states, they might be expected to be unstable in some sense. Our purpose is to study this conjecture, by formulating the stability definitely, and thereby present general theorems about the stability. 5
Fragility of quantum states of macroscopic systems
We say a quantum state is fragile if its decoherence rate
r
r satisfies
KV~+~,
(11)
where K is a function of microscopic parameters, and 6 is a positive constant. To understand the meaning of the fragility, consider first the non-fragile case where S = 0. In this case, the decoherence rate per unit volume is independent of V . This is a normal situation in the sense that the total decoherence rate I? is basically the sum of local decoherence rates, which are determined only by microscopic parameters (i.e., independent of V ) . On the other hand, the case 6 > 0 is an anomalous situation in which the decoherence rate per unit volume behaves as K V 6 . Note that this can be very large even when K is small, because V is huge. This means that a fragile quantum state of a macroscopic system decoheres due to a noise or environment at an anomalously great rate, even when the coupling constant between the system and the noise or environment is small.
-
6
Fragility in weak noises
The most important assumption of the present theory is the locality. For the interaction with a noise, the locality requires that the interaction Hamiltonian should be the sum of local interactions;
Here, f (z, t ) is a random noise field with vanishing average f(z,t ) = 0 , and &(z) is a local operator at z. (See section 3 for the meaning of the local operator.) We assume that the statistics of f (z, t ) is translationally invariant both spatially and temporally, i.e., f (z,t)f(z’,t’) is a function of z - z’ and
t - t’. We also assume that the time correlation of the noise is short. The total Hamiltonian is
73 Here, H denotes the Hamiltonian of the principal system, which can be a general Hamiltonian including, e.g., many-body interactions. Using this general local model, we can show the following for the fragility that is defined in section 5: Theorem 1: Let IQ) be a pure state, whose time evolution b y H is slow,of a macroscopic system. If IQ) is an AFS, then it is fragile in the presence of some weak noise. If IQ) is an NFS, then it is not fragile in any weak noise. It follows from this theorem that an AFS decoheres (hence collapses) at an anomalously great rate if external noises contain such a noise component, whereas a NFS does not decohere at such an anomalously great rate in any weak noise.
7 Fragility under weak perturbations from environments The physical realities of noises are perturbations from environments. We can show a similar theorem for the effects of perturbations from environments. Again, the most important assumption is the locality of the interaction between the principal system and an environment. Namely, the interaction Hamiltonian should be the sum of local interactions; XEV
Here, f(z) and &(z) are local operators at z of an environment and the principal system, respectively. Similarly to the case of noise, we assume that (in the interaction picture) (f(x,t ) ) E = 0, and that (f( 2 ,t )f (d, t ’ ) )is~ a function of x --5’ and t -t’, where (. . .)E denotes the expectation value for the state of the environment E. We also assume that the correlation time of (f(2, t )f (d, t’))E is short. The total Hamiltonian is Htotal =
H + Hint + H E ,
(15)
where H and k~denote the Hamiltonians of the principal system and the environment, respectively. Here, k can be a general Hamiltonian including, e.g., many-body interactions. Using this general local model, we can show the following for the fragility that is defined in section 5 : Theorem 2: Let IQ) be a pure state, whose time evolution by H is slow,of a macroscopic system. If IQ) is an AFS, then it is fragile under some weak perturbation from some environment. If IKP) is an NFS, then it is not fragile under any weak perturbations f r o m environments. It follows from this theorem that an AFS decoheres (hence collapses) at an anomalously great rate if perturbations from environments contain such a
74 perturbation term, whereas a NFS does not decohere at such an anomalously great rate under any weak perturbations from environments. 8
Do relevant perturbations always exist?
By theorems 1 and 2, we have shown that NFSs are not fragile in any noises or environments, which interact weakly with the principal system via any local interactions. This should be contrasted with the results of most previous works, according to which a state could be either fragile or robust depending on the form of the interaction. We have obtained the general and universal conclusion because we have made full use of the locality as well as the huge degrees of freedom. Regarding AFSs, on the other hand, theorems 1 and 2 show only that they are fragile in some noise or environment, which interact weakly with the principal system via local interactions. In other words, for any AFS it is always possible to construct a noise (or an environment) and a weak local interaction that make the AFS fragile. These theorems do not guarantee the existence of such a relevant noise (or an environment) and a relevant interaction in real physical systems. We discuss this point in this section. As described in section 2, we are only interested in phenomena in some energy range AE, and describe the system by an effective theory which correctly describes the system only in AE. The effective theory can be constructed from an elementary dynamics by an appropriate renormalization process. In this process, in general, many interaction terms would be generated in the effective interaction Hence, it seems quite rare that a relevant noise or an environment and a relevant interaction are completely absent. Even when the coupling constant to the relevant noise (or environment) is, say, ten times smaller than those to other noises (or environments), the relevant noise (or environment) would dominate the decoherence process of the AFS because the decoherence rate grows anomalously fast with increasing V, except when the noise (or the perturbation from the environment) is negligibly weak such that its intensity is, e.g., O(l/V). Namely, an AFS should be fragile apart from such an exceptional case. However, for general systems, we cannot exclude the exceptional case where the relevant noise is negligibly weak. Therefore, we cannot draw a definite conclusion on whether AFSs are always fragile in real physical systems. This motivates us to explore another stability, which will be described in the expect, in accordance with experiences, that by an appropriate renormalization process Htot can be made local in the relevant space-time scale.
75 next section. 9
Stability against local measurements
We can prove a stronger statement by considering the stability against measurement. Suppose that one performs an ideal measurement of a local observable &(z) at t = t , for a state j3 (pure or mixed) of a macroscopic system, and obtains a value a. Subsequently, one measures another local observable b ( y ) at a later time t b , and obtains a value b. Let P ( b ; a ) be the probability distribution of b, i.e., the probability that b is obtained at t b under the condition that a was obtained at t,. On the other hand, one can measure b ( y ) at t = t b without performing the measurement of &(z) at t,. Let P ( b ) be the probability distribution of b in this case. We say b is stable against local measurements if for any E > 0
(P(b;a ) - P ( b ) J5
E
for sufficiently large
13: - yI,
(16)
for a n y local operators ;(z) and b ( y ) and their eigenvalues a and b such that
P(a)2 E. This stability is stronger than the stability against noises and perturbations from environments. In fact, the latter stability is related to I C, P ( b ; a ) P ( b ) ( . There are many examples of states for which 1 C , P ( b ; a ) - P ( b ) J 5 E is satisfied whereas I P ( b ; a ) - P(b)I 5 E is not. For the simple case t b - t , + 0 , we can show the fo1lowing:f Theorem 3: Let 6 be a pure o r mixed state of a macroscopic system. If j3 i s stable against local measurements, t h e n it has the cluster property, and vice versa. It follows from this theorem that any AFS is unstable against local measurements. 10 Mechanism of symmetry breaking in finite systems
AFSs generally appear in, e.g., finite systems which will exhibit symmetry breaking if V goes to infinity. In such systems, we can find states (of finite systems) which approach a symmetry-breaking vacuum as V -+ 00. We call such a state a pure-phase vacuum. It has a finite expectation value (M) = O ( V )of an additive order parameter M , and has relatively small fluctuations (&A2)5 O ( V ) for a n y additive operator A (including 2) Hence, the 5969718,9.
fResults for more general cases will be described elsewhere.
76 pure-phase vacua are NFSs. In a mean-field approximation, the pure-phase vacua have the lowest energy. However, it is always possible to construct a pure state(s) that does not break the symmetry, = 0, and has an Although such states equal or lower energy than the pure-phase vacua cannot be pure in infinite systems, they can be pure in finite systems When [I?,M] # 0, in particular, the exact lowest-energy state is generally such a symmetric ground To lower the energy of a pure-phase vacuum, a SB field is necessary. However, an appropriate SB field would not always exist in laboratories. The symmetric ground state is composed primarily of a superposition of pure-phase vacua with different values of and, consequently, it has an anomalously large fluctuation of M; ( S M 2 ) = 0(V2).697>8>9 Therefore, if one obtains the exact lowest-energy state (e.g., by numerical diagonalization) of a finite system, which will exhibit symmetry breaking if V goes to infinity, the state is often an AFS. The present results suggest a new origin of symmetry breaking in finite systems.” Although symmetry breaking is usually described as a property of infinite systems, it is observed in finite systems as well. The results of sections 6 and 7 suggest that although a pure-phase vacuum (which is an NFS) has a higher energy than the symmetric ground state (an AFS), the former would be realized because the latter is fragile in some noises or environments. This mechanism may be called “environment-induced symmetry breaking,” a special case of which was discussed for interacting m a n y - b o s o n ~ The .~ result of section 9 suggests more strongly that only a pure-phase vacuum should be realized, because an AFS is changed into another state when one measures only a tiny part of the system, and such drastic changes continue by repeating measurements, until the state becomes an NFS. This mechanism may be called “measurement-induced symmetry breaking.” We consider that these scenarios explain the symmetry breaking (i.e., realization of an pure-phase vacuum) in finite systems, much more naturally and generally than the idea of the symmetry breaking field: It seems quite artificial to assume that an appropriate static symmetry breaking field would always present in real physical systems,g although it is true that symmetry breaking fields are a convenient mat hematical tool.
(M)
639.
4,679710.
(M),
gFor example, it is quite unlikely that a SB field for antiferromagnets could exist in laboratories.
77 11
Stability of quantum computers with many qubits
Quantum computers are useful only when the number of qubits N is huge. Hence, useful quantum computers are macroscopic quantum systems. Various states appear in the course of a quantum computation. Some state may be an NFS, for which (6A2) = O ( V ) for a n y additive operator A. This means that correlations between distant qubits are weak. Properties of such states may be possible to emulate by a classical system with local interactions. We therefore conjecture that other states - AFSs should appear in some stages of the computation for a quantum computer to be much faster than classical computers.13 In fact, two of the authors confirmed this conjecture in Shor's algorithm for f a ~ t 0 r i n g . l ~ The present results suggest that the decoherence rate of quantum computers can be estimated by fluctuations of additive operators, which depend strongly on the number of qubits N and the natures of the states of the qubits.15 Since AFSs are used in some stages of the fast quantum computation, the state of qubits can become fragile in some noise or environment, for quantum computers with many qubits. Note that the dominant perturbation for the case of huge N can be different from that for small N , because the decoherence rate of an AFS grows anomalously fast with increasing N . Therefore, the quantum computer should be designed in such a way that it utlizes AFSs for which the intensities of the relevant noises are 0 ( 1 / N ) or smaller. Since the error corrections are not almighty, we think that one must consider both such optimization and the error corrections to realize a quantum computer with a large number of qubits. ~
12
Discussions
The present results show that the stabilities of quantum states of finite macroscopic systems are closely related to the cluster property, which describes the strength of spatial correlations of fluctuations of local observables, and to fluctuations of additive operators. Note that the stabilities are defined as dynamical properties of an open system, whereas the cluster property and fluctuations of'additive operators are defined as static properties of a closed system. Hence, it is non-trivial - may be surprising - that they are closely related to each other. We stress that the approximate stability against all local interactions (between the principal system and environments) would be more important than the exact stability against a particular interaction, which was frequently discussed in previous works. As discussed in section 8, many types of interactions
78
would coexist in real physical systems, and the exact stability against one of them could not exclude fragility to another. In this paper, we did not mention temperature. It is clear that similar conclusions can be drawn for thermal equilibrium states (KubeMartinSchwinger state^^>^), because thermal equilibrium states can be represented as vector states by introducing an auxiliary field.12 We also point out that the present results may be important to study the foundations of non-equilibrium statistical physics. For example, in the linear response theory of Kubo,l' he assumed the unitary time evolution of a closed system. However, actually, the system is continuously measured over a time period longer than l / w when one measures, say, the AC conductivity at frequency w . Therefore, it is necessary for the validity of the linear response theory that the non-equilibrium state under consideration is stable against measurements. Theorem 3 suggests that such states must have the cluster property. This observation may become a foundation not only of nonequilibrium statistical physics but of non-equilibrium field theory, which is not established yet.
Acknowledgments The authors thank Prof. I. Ojima for discussions and suggestions.
References
[*I
Present address: Department of Information Sciences, Tokyo University of Science, Chiba 278-8510, Japan. 1. A. J. Leggett, Suppl. Prog. Theor. Phys. (Kyoto) 69, 80 (1980). 2. W. H. Zurek, S. Habib and J. P. Pax, Phys. Rev. Lett. 70,1187 (1993), and references cited therein. 3 . More details will be presented in A. Shimizu and T. Miyadera, Phys. Rev. Lett., in press. 4. R. Haag, Local Quantum Physics (Springer, Berlin, 1992). 5 . D. Ruelle, Statistical Mechanics: Rigorous Results (Benjamin, Reading, 1969). 6. P. Horsh and W. von der Linden, Z. Phys. B72,181 (1988). 7. S. Miyashita, Qunatum simulations of condensed matter phenomena (eds. D. Dolland and J. E. Gabernatis, World Scientific, Singapore, 1990) p. 228. 8. T. Koma and H. Tasaki, J. Stat. Phys. 76,745(1994) 9. A. Shimizu and T. Miyadera, Phys. Rev. E 64, 056121 (2001).
79
10. A. Shimizu and T. Miyadera, J. Phys. SOC.Jpn. 71 (2002) 56. 11. A. Shimizu and T. Miyadera, Phys. Rev. Lett. 85, 688 (2000) 12. H. Umezawa, H. Matsumoto and M. Tachiki, Thermo Field Dynamics and Condensed States (North Holland, Amsterdam, 1982). 13. A, Shimizu, talk presented at The 4th Symposium on Quantum Eflects and Related Physical Phenomena (December 20-21, 2000, Tokyo, Japan). 14. A. Ukena and A. Shimizu, in preparation. 15. This point was first suggested by G. M. Palma, K.-A. Suominen and A. K. Ekert, Proc. Roy. SOC.Lond. A (1996) 452, 567. 16. R. Kubo, J. Phys. SOC.Jpn. 12,570 (1957).
80
DELOCALIZATION AND DISSIPATIVE PROPERTY IN 1D DISORDERED SYSTEM WITH OSCILLATORY PERTURBATION HIROAKI YAMADA Department of Material Science and Technology, Faculty of Engineering, Nzigata University, Ikarashi 2-Nocho 8050, Niigata 950-2181, Japan E-mail:
[email protected]. ac.jp Energy relaxation dynamics in a simple quantum model of electron-phonon system is numerically investigated. We show delocalization in one-dimensional disordered electronic system with coherent harmonic perturbations. The appearance of the delocalization implies that the system has potential for irreversibility and dissipation. Next, we investigate dissipative property of the dynamically delocalized state and show that an irreversible quasistationary energy flow indeed appears in the form of a "heat" flow when we couple the perturbed system with an autonomous mode.
1
Introduction
There have been some attempts examing quantum dissipation '. The most popular approaches are heat-bath approach (with infinite number of phonon modes) 2,3, random matrix bath approach 4 , linear response theory and Landauer formula 7,8,9, and so on. In the orthodox approaches, stochastization mechanisms and heat reservoir consisting of infinite number of degrees of freedom are explicitly or implicitly assumed in advance 10!11*2. A more interesting scenario of the electronic stochastization is the possibility that the stochastization mechanism is generated in the system without a n y help of the time-dependent stochastic source. In the present paper, it is demonstrated that, contrary to traditional theories, infinite number of phonon modes are not necessary and just few phonon modes are sufficient for the delocalization and irreversible energy transfer from the scattered electron to the phonon modes, if the scattering potential is spatially irregular 12,13,14. In the concrete, a possibly simplest situation is modeled by a one-dimensional disordered system (1DDS) coupled with finite number of harmonic time-dependent perturbation and/or harmonic oscillators. Delocalization phenomena and energy relaxation dynamics of the system are numerically investigated. Some of the further details of the numerical . we give a short review of results have been reported in references l 3 > l 5 Here the papers and some new results. Note that classically chaotic system with quasi-periodic perturbation have also been used in order to investigate the 576,
81
localization-delocalization transition by several authors 16,17,18. The outline is given as follows. In the next section we explain the two kinds of the models we used. In section 3, we give some numerical results about quantum diffusion of initially localized wave packet in lDDS with oscillatory perturbation. In section 4, we show numerical results about occurrence of dissipative phenomena in a closed quantum system, which is an autonomous system, consisting of lDDS and oscillators. In section 5 , we investigate a quantum state of the autonomous mode when the energy flow exists, and give a simple phenomenological interpretation for the thermalization phenomena. The last section is devoted to summary and discussion. Derivations of some equations are added in appendices.
2
2.1
Model Autonomous model
We use 1-DDS coupled with a few harmonic oscillators in order to investigate the energy transfer between the electron and the phonon modes. The total Hamiltonian Htot is consisting of tightly-binding electronic part H e l , harmonic oscillators Hph,.v with incommensurate frequencies { w i } and interaction part Hint between them with coupling strength { b j } as follows:
N
M
n=l j = 1
V ( n ) is the onsite energy of electron at the site n, which varies at random in the range [-W, W ]from site to site. If the number of phonon modes goes to infinity with an analytical frequency spectrum, then the phonon system becomes a heat reservoir implicitly or explicitly supposed in orthodox theories, but in our treatment the number of phonon modes is finite.
82 2.2 Nonautonomous models If the harmonic oscillators are highly excited, the model Htot becomes equivalent to a simple time-dependent Hamiltonian perturbed by a classical driving force. Indeed, oscillatory external perturbation V ( n ,t ) = V(n)[l cos(flit)] can be mathematically identified with highly excited quantum harmonic oscillators Accordingly if the approximation is used for some autonomous phonon modes, we can replace them by external perturbation. Then the time-dependent Shradinger equation that we generally simulate becomes
+
3xf=l
1g91712.
N
M
where Qtot(n, { q j } , t ) represents the wave function of the whole system in a site basis. One of the advantage of this model is that although the number of the autonomous modes A4 is limited due to computer power, we can freely control the number of the frequency components L of the harmonic perturbation. In the simulation we set A4 =O or 1 and/or L =0,1,...,5. For convenience, we refer Hamiltonian in eq.(5) as Hzh in the following sections.
3
Dynamical Delocalization in Nonautonomous System
In our previous paper l 3 we showed that the lDDS exhibits a remarkable delocalization behavior when it is perturbed by classical oscillating forces with several frequency components. The model system is just the model (5) without the harmonic oscillators, i.e., M=O. Such a delocalization phenomenon is a key to understand the occurrence of irreversibility and dissipation. 9.1
Dynamical Delocalization
When oscillatory harmonic perturbations are applied to lDDS, an initially spreads unlimitedly, and localized wave packet of electron ( @ ( t= 0) = S,,o) we called such a quantum state dynamically delocalized state. It is very interesting that such a non-localized state can be easily realized only by applying
83
2
3
I
I
I
l
4
5
6
7 8 9 '
l
1
.
1000
*
"
I
2
3
4
t Figure 1. Logarithmic plots of time-dependent MSD of some cases ( L =0,1,2,5), where t =0.5, W = 0.9 and A = 1. 20 different configurations are used for the ensemble average. The insert is real scale one.
a weak coherent perturbation. The delocalization property can be quantitatively characterized by the mean square displacement (MSD) of the wave >, where fi 3 N nln >< nl is the position packet: mz(t) =< q(t)Ifi'l@(t) operator and Q ( t )is the time-dependent wave packet. We performed longer time simulation for larger system than the previous ones 1 3 . The results are shown in Fig.1 for monochromatically ( L = 1) and polychromatically ( L 2 2) perturbed cases. It is found that the wave packet, which is localized without the interaction with the oscillatory perturbation 20,21, spreads beyond the original localization length in the unperturbed lDDS ( L =0) as time elapses. The diffusive behavior is observed within the time scale accessible by numerical computations, and the diffusion process is not in general the normal diffusion but a subdiffusion, which is characterized by
cn=l
84
a power law increase:
mz(t) t'",(0 < a 5 1). N
(6)
The subdiffusive behavior approaches the normal diffusion ( a = 1) promptly as the number L of the frequency and/or the perturbation strength E increase. However, we note that in the monochromatic case ( L = 1)the diffusive behavior is suppressed at a certain level which is longer than the original localization length in the unperturbed case ( L = 0 ) . In the next subsection, we consider further details of the monochromatically perturbed cases ( L = 1). The appearance of the diffusive behavior of the wave packet in the real space implies that through the dynamic interaction with the coherent perturbation a quantum-mechanical pure state is transformed into a complex pure state which may be called "stochastic" state. The Anderson localized state is thus unstable against weak dynamical perturbations, and a delocalization manifests itself in the form of an unlimited diffusion. Further it has been also revealed that the dynamical delocalization properties obey remarkable spatiotemporal scaling laws. It was discussed extensively in our previous paper and we do not repeat it in the present paper 1 3 . 3.2 Monochromatically perturbed case (L=l)
We consider the localization in the monochromatically perturbed 1DDS. First, we numerically obtain quasieigenstates of the time-periodic system. The quasieigenstates are defined by eigenstates {\ary>} of one-periodic timeevolution operator U ( T = as,
%)
where T+ denotes the time ordering product. The property of quasieigenstates are directly related to the localization properties of the wave packet @ ( nt, = k T ) =< n l U ( T ) k l @ ( t= 0 ) > in site representation. In Fig.2, we show some typical quasieigenstates Iucl(n)l= I < n1aa > I for the I-DDS with monochromatic perturbation. The quasieigenstates are exponentially localized, and it is consistent with the appearance of the suppression of the diffusive behavior at certain time in Fig.1. In appendix A, a method to calculate numerically the quasieigenstates is given. Next we consider the localization phenomena by transforming the timedependent Schrodinger equation to sationary Schrodinger equation. Inserting
85
16'
16'
1o
-~
10.4
200
400
600
I
1 ' 1 '
800 1000
I
l
;
0
200 400 600 800 1000
l
l
l
l
l
l
4
II I
10-l 1o-2 1o
-~
1o
-~
I
0
200
400
600
800 1000
l
l
I
400 600 800 1000
0
200
\
., I . l l l l l l 200 400 600 800 1000
0.1
10-l
0.01
10.' 1o
-~
1o
-~
0.001
0
200
400
n
600 800 1000
0
n
Figure 2. Some quasieigenstates Iua(n)I = 1 < n(Q, > 1 for one-periodic time evolution operator U ( T = 2n/Ql) of monochromatically perturbed lDDS in Fig.1.
ck
Q(n,t)= exp(-iqt) C n , k exp(-ikRt) to equation (5), the amplitude obey following equation.
Cn,k
86
where s2 = R1. (See appendix B.) This equation is equivalent to twodimensional tightly binding system with a static electric field and off-diagonal randomness in k-direction. Roughly speaking, we found that at least in monochromatically perturbed IDDS, complicated phenomena due to two kinds of the localization, Anderson localization and Stark-ladder localization, are mixed 22,23,24. It is difficult to analytically get the exact evidence for the localization because our system is a nonseparable system. 3.3 Energy transfer
The non-autonomous model HF,=,(t) in eq.(5) without the autonomous mode can be transformed into an autonomous version Hiut composed of 1DDS and linear oscillators as follows.
where 53 = -ih&. The relation between eigenstates of the autonomous version HFut and quasieigenstaes of the nonautonomous model HF,=,(t) is given in appendices A and C. It is worth noting that the highly excited harmonic oscillators in the autonomous model are equivalent to the harmonic perturbation in the nonautonomous model. Based upon the autonomous picture, we can discuss the exchange of energy between the electron and the ”phonon modes” of the linear oscillators. The backaction of the delocalization of electron to the phonon modes will result in an excitation and/or deexcitation in the phonon modes. We discuss on the exchange of energy between the electronic system and the perturbing system. Since we can introduce an autonomous version of the non-autonomous model, we can explicitly compute the energy which flows from the electronic system to the perturbing mode, which is represented by the expectation value < QaUt(t)l C,”=,s 2 j J j 1 9 a u t ( t ) >= E J , where QaUt(t) is the time-dependent wave function of the autonomous system HEut(t). By using the formula (22) in an appendix C, it immediately follows that
87 2
4 ’ P C
W
O
a
-1
z c
-2XlO3
0
I 1000
,
I
2000
I
I
3000
4000
5000
t
Figure 3. Time-dependence of ansemble-averaged phonon energy < E,(t) > in polychromatically perturbed cmes ( L = 2,5). The parameters are same values to Fig.1.
where Q ( t )is the time-dependent wavefunction of the nonautonomous system HyM=o(t). The derivation is given in appendix C. Figure 3 shows the ensemble averaged energy < E J ( ~>) transferred to the phonon modes during the time evolution process depicted in Fig.1. It is evident that the phonon energy fluctuates around a certain level, and do not show any signature of net energy transfer between the phonons and the electron. This is because the backaction of the electron makes the phonons excite and deexcite symmetrically around the initial Fock state. In other words, the phonons show a diffusive motion around the initial state along the ladder of Fock basis. Another important reason is that the initial electronic state (Q(t= 0) = & o ) is a mixture of almost all the localized basis with negative and positive energy eigenvalues, and the expectation value of energy is close t o zero. In other words, the electron has no excessive energy. Indeed, if we choose the initial electronic state to the highest-energy localized state (or the lowestenergy localized state), the symmetry of the diffusion of phonons around the initial phononic state is broken and the phonon absorbs (or emits) energy as is depicted in Fig.4, although the net energy flow is not very intensive because the excited diffusion flow and the deexcited diffusion flow almost cancel out each other. This fact implies that, if a phonon is prepared initially not in a highly excited state but in the ground states of the harmonic oscillators,
88
0
I
I
I
I
200
400
600
800
t
I 1wO
I
I
i
IZW
1400
1Mx)
Figure 4. Time-dependence of phonon energy E J ( t )of the nonautonomous modes in polychromatically perturbed caw ( L = 3). The initial states of the electron are set in eigenstates at high energy ( n , = ~ 3) and low energy (n,i = 123), where net is number of energy level from the top of the energy level. The other parameters are same values to Fig.1.
the deexcitation is forbidden, and the phonon mode will be excited diffusely toward the higher Fock states, which means the onset of a one-way energy transport from the electronic system t o the phonon system. This is just the main subject of the subsequent sections.
4
Energy Relaxation of Delocalized States
As has been seen in the last section, the additional monochromatic perturbation enhances the localization length of lDDS, and polychromatic perturbation makes the localization length diverge. In this section we observe the energy relaxation of the lDDS perturbed by coherently oscillatory force. In the concrete, we couple the dynamically perturbed lDDS with a harmonic oscillator in order to investigate energy flow between electron and the autonomous modes. We prepare the electron initially in a sufficiently high excited eigen&ate and set the autonomous phonon in the ground state, and compute the time-dependent phononic energy Eph( t ) ,electronic energy Eel ( t ) ,and MSD of the electron.
89
-....c=o 1
0.2 0.4
......
- 8.1 - 8.2
__
.....
-
-
8.1
Figure 5. Time-dependence of (a) an electronic energy, (b) a phononic energy and (c) MSD of electron in monochromatically perturbed cases (A4= 1, L = l), where W = 0.9, h. = 1/8, b = 1.0, w = 0.8, n,=fi and the varius perturbation strength E = 0.1,0.2,0.4.
4.1
Monochromatically perturbed case (M = 1,L = 1)
As shown in Fig.5, if the coupling strength is large enough ( b = 1.0) the diffusion and a one-way energy transfer continues until it reaches a fully relaxed state even at smaller values of perturbation strength E . The behavior is quite different from the non-perturbed case ( L = 0) where the energy flow saturates at certain level before the packet reach the fully relaxed state with the equal weight at the each site 13. It seems that at an early stage of the
90
time-evolution the phonon energy exhibits a nice linear increase, while the electronic energy decreases monotonously until it almost vanishes. Flow rate of energy increases as the increment in the perturbation strength E . The MSD of the electronic state also approaches the maximum length allowed by the finite system size. The final electronic state with almost zero energy can be regarded as an equilibrium state which contains all the localization basis, whose energies are distributed symmetrically around zero, with even statistical weight. We can judge that the system becomes completely dissipative in such a coupling strength regime. Furthermore, we show some cases with different electronic initial states in Fig.6. We numbered the localized states of the isolated lDDS from the top of the energy level and denoted the number by n,~. The eigenstates n , ~ =4, 14 and 21 are used as the initial excited states. The other parameters are set the same as the case of the Fig.5. The behavior of the energy relaxation is almost similar to the other case in Fig.5. As a result, it seems that a stationary energy transport continues slowly for long time before the spread of the wave packet satures when the coupling strength is large enough to cause sufficient mixing in the system.
4.2 Polychromatically perturbed case (M = 1,L 1 2) When the number of the frequency components of the perturbation is larger than or equal to two ( L 1 2), the lDDS exhibits typical symptom of dynamical delocalization. In this subsection we examine dissipative property for the polychromatically perturbed 1DDS. Typical examples of time-dependent energy transfer between a polychromatically perturbed lDDS ( L = 2,4,5) and the autonomous mode are depicted in Fig.7. In all cases the MSD grows up to the maximum scale and a complete delocalization is achieved, and the electronic energy shows a very nice relaxation behavior even in the cases with small coupling strength. In the early stage of time-evolution, the electron loses its energy linearly in time. In such a quasi-stationary regime the emission rate of energy per unit time can be well defined 15. Monotonic increase of phonon energy continues until the wave packet spreads over the system size and the electronic energy approaches to zero level, which indicate a complete delocalization. We can confirm the energy fluctuation in the nonautonomous modes during the energy transfer from electron to the autonomous mode. As shown in Fig.8, energy of each of the nonautonomous modes Ej,(t) fluctuates around certain level due to exchange of the energy between each of the modes and electron. In conclusion, all the above features indicate that a complete dissipation
91 I
I
I
I
(a)
2.5-
M=I,L=I
2.0-
b=l.O.~=O.4
- ".31=4 ....... ....
14
-
21
I
I
I
I
0
1W
200
300
400
500
20
15
J" 05 0.o 0
0
I
I
I
I
1W
200
300
400
500
400
500
1M)
Figure 6. Time-dependence of (a) an electronic energy, (b) a phononic energy and (c) MSD of electron in monochromatically perturbed cases (A4= 1, L = l), where W = 0.9, R = 1/8, b = 1.0, w = 0.8,E = 0.4 and the frequency Q,=fi, for three different initial excited eigenstate of electron (n,l =4, 14 and 30).
is realized in case of L 2 2.
92 I
2.1
-
I
I
I
I
I
I
(a 1
1.5-
z
w-
I..-
1.5-
2.I 1.s
-
QI.I
0.5 0.I
Figure 7. Time-dependence of (a) an electronic energy, (b) a phononic energy and (c) MSD of electron in monochromatically and polychromatically perturbed cases ( M = 1, L = 1,2,4,5), where W = 0.9, h = 1/8, b = 0.4, w = 0.8 and E = 0.4. The frequency components of the perturbation {ni} are chosen within a range [0.5,1.5] randomly.
5
Quantum State of Autonomous Mode
In this section, we pay attention to the quantum state of the autonomous phonon mode during the (quasistationary) energy flow. We show the phonon distribution in polychromatically perturbed case ( L = 5 ) as a typical example. Figure 9 shows the semi-log plots of the probability distribution P(Enp,)= I < n,hIQtot(t) > l2 of the autonomous phonon mode as a func-
93 8.6
I
1
I
I
I
- =, 11 12
8.4
I
......... 13
1.1
8.8
8.2
m i 8.8
-8.2
-9.4
-8.6
Figure 8. Time-dependence of energy E J of the total nonautonomous modes and each of the modes (#1,#2,#3) in polychromatically perturbed cases ( L = 3), where W = 0.9, 7% = 1/8,b = 0.8,w = 0.8 and E = 0.4. The frequency components of the perturbation {ni}are chosen within a range [0.5,1.5] randomly. The insert shows the Eel,EphrEJ in the time evolution.
tion of the energy En,, . The Boltzmann-type distribution appears only when quasistationary energy transfer from electron to phonons is observed. As a result, the phonon mode reaches promptly a ” thermalized state” characterized by a well defined time-dependent temperature T ( t ) . A simple phenomenological interpretation is possible for the appearance of the Boltzmann-type distribution as we use a harmonic oscillator. We express the total Hamiltonian in Fock space.
94
16’
1o-2
1o
-~
1o
-~
0
1
3
2
4
P ‘h
Figure 9. Phonon distribution P ( E n p h )= I < n,hlQ(t) > l2 V.S. Enph at several time (t = 100,200,300,400,500) in the polychromatically perturbed case (M = 1, L = 5). The parameters are b = 0.4 and E = 0.4.
M
.
where the b denotes the coupling strength. The b i , b, are creation and annihilation operators of an electron at n-th site in real space, which satisfy the usual anticommutation relations for Fermions, [b,, bL]+ = bn,m. The a;,uj are creation and annihilation operators for the j t h energy eigenstate of the autonomous modes. The Heisenberg equation for the creation operator of the
95 autonomous mode becomes the following linear equation.
Here we assume some statistical property of the second term of RHS, R(t) = ib&& Ut(t)bkV(n,t)b,U(t). Neglecting the weak dependence of the correlation function < Rt(t2)R(tl)>= G(t1,ta - t l ) ( t 2 > t l ) on tl and rapid decay for time region, t 2 - tl > t,, the expectation value of the autonomous mode increases in proportion to time. If a; is an integration over the stochastic source R ( t )with the very short characteristic time t,, the amplitude a$ is a sum over statistically independent uantities and hence should obey a Gaussian stochastic process. Regarding a j as c-number, the distribution function of
9
a; should be the Gaussian distribution P(aj,a;) 0: exp{-const x la,I2}, which is equivalent to the Boltzmann-type distribution. Moreover, if the Heisenberg equation for density operator of the electron,
can be effectively transformed into a diffusion-type equation, the MSD En2< nlp(n> increases linearly in time.
<
n2 >=
6
Summary and Discussion
Diffusive and dissipative property of lDDS perturbed by a time-dependent harmonic driving force is numerically investigated. We have shown that the lDDS is sensitive to a coupling with other degrees of freedom and that the 1DDS driven by periodically time-varying perturbation exhibits a subdiffusive behavior. We investigated dissipative properties of the dynamically delocalized states by coupling the system with another simple system prepared in the ground state. An irreversible flow of energy from electron to phonons is induced spontaneously even though the number of phonon modes is only two. AN electron scattered by an irregular potential emits its energy to the phonon modes, and moreover the phonons are excited to a thermal state characterized by a well defined temperature. In this report, we did not show the results in stochastically perturbed cases, but the results are almost similar to that of polychromatically perturbed cases with more than four colors. Spatial irregularity also plays a crucial role as an origin of quantum irreversibility when it is combined with a dynamical interference arising from some
96
other degrees of freedom. Such a mechanism may provide a simple dynamical modeling t o understand the origin of resistivity in solid state materials 25,26. Note that there are, recently, some interesting reports concerning roles of the chaotic system as a "heat bath" in quantum system with small degrees of freedom 27,28,29,30
Acknowledgments This study is based on the collaboration with Professor K.S.Ikeda. Author would like to thank him for valuable suggestions and stimulating discussion. Author appreciates Professor S.Tasaki for inviting him to Japan-Italy Joint Waseda Workshop on: Fundamental Problems in Quantum Physics. Author also thanks the participants of the workshop for useful comments and discussions.
Appendix A
Relation between quasieigenstates and eigenstates
In this appendix we show a relation between eigenstates of the autonomous version HEzl and Floquetstates of the nonautonomous system HFLl,M=O.Let us consider eigenvalue equation of time-evolution operator U ( t )for one-period
T
= STIR,
U(T)da(n)= exp(-iva)da(n),
(16)
where qa and &(n) are a t h quasieigenenergy and quasieigenstate in site n-representation, respectively. To get the quasieigenstates numerically, we diagonalize matrix < llU(T)(m> which is created by one-periodic time evolution for each of the unit vectors, where the Im > denotes unit vector with nonzero element only in the rn-th site as Im >= ( l O , O , . . . , l , .. . , O >)t. We define a function da,e(Q,n)by using the operator, the quasieigenstate and quasieigen energy, &,e(Q,n)= exp(-ivae/R
+ iWJ(e/fl)d,(n),
(17)
where l represent an arbitrary integer. It can be found that this function becomes the eigenstate of an autonomous version H:zl of the monochromatically perturbed Hamiltonian H p & M ( t ) , by inserting the function into the eigenvalue equation, H ~ ~ l ~ a , e ( Q = ,E,,g$,,e(Q,n). n) As a result we see that e means a quantum number which characterizes the eigenstate of the linear
97
oscillator, and the eigenenergies of the autonomous version are given by,
E,,e B
= 77,
+2 d .
(18)
Floquet States
We derive a stationary ShrBdinger equation by inserting Qtot(n,t) of eq.(l9) in the time-dependent SchrBdinger equation ( 5 ) for H r M Z 0 . L
The amplitude
C n , k l , k 2...,kL , obeys
the following equation.
j=1
+
( C n + l , k ~ , k ~ , . . . , k ~Cn-l,kl,kz,...,kL)
+ Cn,k~-l,kz,..,,k~ C n , k i , k z + l , . ~ ~ ,+ k ~ Cn,kl,kz-l ,...,kr. (Cn,kl+l,kz,...,k~
+ ... + cn9 k l kz~ ,...,k L 4-1 + c n ,ki ,kz ,...,kL 1 1 -
(20)
It can he regarded as a (L+1)- dimensional tight-binding system with disorder under external field.
C
Derivation of eq.(12)
In this appendix we derive the expression of the phonon energy in the nonautonamous system. We show the expression only in monochromatically perturbed case. An extension to polychromatically perturbed cases is easy. First we consider the following autonomous Hamiltonian: aut H L = 1 = Hel+
HPZi(4)+ RJ.
(21) The term R J ( = H J = -ihR&) represents a linear oscillator, where J and R are action variable and frequency of the motion, respectively. 4 is aqn angle variable which is conjugate to J . The time evolution operator is given as follows: Uaut = -
exp(-i-)
HaUtt h
98
ds{~,L
=
+
4)}).
H O ~ ~ ( R S +
(22)
where T is the time-ordering operator. We set an initial state,
IPtot(t = 0)
>= Ji> @Iq50 >,
(23)
where li > and 140 > are initial state of electron a n d eigenstate of the phase operator, respectively. Here we consider a derivative of < Qtot(t))HJ)Ptot(t) > by time,
-iJOt A
> 8140 >
x exp(-)U(t)li
In t h e last equality we used following relations.
iJRt eXP(+-
af(4)exp(-) - i J R t ad
A
=
af(d+Rt)
a4 -- _1 af(4+ fit) R
at
.
(25)
As a result, we can get integration of above equation (24), i.e., EJ(t), by calculating of t h e time-evolution of the initial s t at e Ji> of the electron. References 1. K. Ikeda, Ann. Phys. 227, 1(1993), and refereces therein. 2. A. 0. Caldera and A. J. Leggett, Ann. Phys. (N.Y.) 149, 374(1983); E. Shimshoni and Y . Gefen, Ann. Phys. (N.Y) 210, 16(1991); Y . C. Chen and J. L. Lebowitz, Phys. Rev. Lett. 69, 3559(1992). 3. D. Cohen, Phys. Rev. E55, 1422(1997). 4. A. Bulgac, G.D.Dang and D.Kusnezov, Phys. Rev. E58, 196(1998). 5. R. Kubo, Can. J. Phys. 34, 1274(1956); D. A. Greenwood, Proc. Phys. SOC. 71, 585(1958).
99
6. M. Toda, R. Kubo, N. Saito, Statistical Physics I, (Springer-Verlag, 1991); R. Kubo, M. Toda and N. Hashistume, Statistical Physics II, (Springer-Verlag, 1991). 7. R. Landauer, IBM J. Res. Develop. 1,2338(1957). 8. R. Landauer, Philos. Mag. 21, 863(1970); M. Buttiker, Y. Imry and R. Landauer, Phys. Lett. A96, 356(1983); Y.Gefen and D.J.Thouless, ibid 59, 1752(1987); R. Landauer, Phys. Rev. B33,6497(1986); M. Cahy, M.Mclennan and S.Datta, ibid B37, 10125(1988). 9. S. Tasaki, Chaos, Soliton and Feactals, 12,2658(2001). 10. H.Haken and G.Strob1, Z. Phys. 262,135(1973). 11. A. M. Jayannavar, Phys. Rev. E48,837(1993). 12. H. Yamada and K. S. Ikeda, Phys. Lett. A222, 76(1996). 13. H. Yamada and K. S. Ikeda, Phys. Lett. A248, 179(1998); Phys. Rev. E59, 5214(1999). 14. H. Yamada: Physica E9,389(2001). 15. H. Yamada and K.S.Ikeda: Phys. Rev. E65,046211(2002). 16. G. Casati, I. Guarneriand D. L. Shepelyansky, Phys. Rev. Lett. 62,345(1989). 17. G. Casati, I. Guarneri, M. Leschanz, D. L. Shepelyansky and C. Sinha, Phys. Lett. A154, 19(1991). 18. F. Borgonovi and D.L.Shepelyansky, Nonlinearity 8, 877(1995); Phys. Rev. E51,1026(1995); Physica D 107,24(1997). 19. J. S. Howland ; Math. Ann. 207,315(1974). 20. K. Ishii, Prog.Theor. Phys. Suppl. 53,77(1973). 21. E. Abraham, P.W.Anderson, D.C.Licciadello and T.V.Ramakrishnan, Phys. Rev. Lett. 42,673(1979). 22. M.Holthaus, G.H.Ristow and D.W.Hone, Phys. Rev. Lett. 75,3914(1995). 23. M.Holthaus and D.W.Hone, Phil. Mag. 74,105(1996) and references therein. 24. A.L.Burin, Y.Kagan and I,Y.Polishchuk, Phys. Rev. Lett. 86,5616(2001). 25. N.F. Mott and E. A. Davies, Electronic Processes in Non-Crystalline Materials (Clarendon Oxford, 1979). 26. See, for example, P. Sheng, Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena (Academic press, 1995). 27. M. Toda, S. Adachi and K. Ikeda, Prog. Theor. Phys. Suppl. 98,323(1989). 28. H. Kubotani, T . Okamura and M. Sakagami, Physica A214 560(1995); M. Sakagami, H. Kubotani, T. Okamura, Prog. Theor. Phys. 95, 703(1996). 29. A. R. Kolovsky, Europhys. Lett. 27, 79(1994); Phys. Rev. Lett. 76, 340(1996). 30. D. Cohen, Ann. Phys. 283,175(2000).
100
FLUCTUATION THEOREM, NONEQUILIBRIUM STEADY STATES AND MACLENNAN- ZUBAREV ENSEMBLES OF A CLASS OF LARGE QUANTUM SYSTEMS SHUICHl TASAKI Advanced Institute for Complex Systems and Department of Applied Physics, School of Science and Engineerings, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, JAPAN E-mail:
[email protected] TAKU MATSUI Graduate School of Mathematics, Kyushu University, 1-10-6 Hakozaki, Fukuoka 812-8581, JAPAN E-mail:
[email protected] For an infinitely extended system consisting of a finite subsystem and several reservoirs, the time evolution of states is studied. Initially, the reservoirs are prepared to be in equilibrium with different temperatures and chemical potentials. If the time evolution is L1-asymptotic abelian, (i) steady states exist, (ii) they and their relative entropy production are independent of the way of division into a subsystem and reservoirs, and (iii) they are stable against local perturbations. The explicit expression of the relative entropy production and a KMS characterization of the steady states are given. And a rigorous definition of MacLennan-Zubarev ensembles is proposed. A noncommutative analog to the fluctuation theorem is derived provided that the evolution and an initial state are time reversal symmetric.
1
Introduction
The understanding of irreversible phenomena including nonequilibrium steady states is a longstanding problem of statistical mechanics. Various theories have been developed so far'. One of promising approaches deals with infinitely extended dynamical system^',^>^. Not only equilibrium properties, but also nonequilibrium properties has been rigorously investigated. The latter include analytical studies of nonequilibrium steady states, e.g., of harmonic crystal^^>^, a one-dimensional gas7, unharmonic chains8, an isotropic XYchaing, a one-dimensional quantum conductor" and an interacting fermionspin system". Entropy production has been rigorously studied as well (see [ll-171 , and the references therein). Based on the idea of Ichiyanagi18, Ojima, Hasegawa and Ichiyanagi" derived a formula relating the relative entropy to the ther-
101
modynamic entropy production for an infinitely extended driven system:
where wt is the state at time t , w is the initial equilibrium state and S (wlwt) is the C* generalization of the relative entropy19-22i2a . Ojima13 generalized this formula to include initial states w where reservoirs are in different equilibria. Convergence of the entropy production to the steady-state value was investigated as well. Recently, JakSi&Pi1letl5 and Ruelle14 rediscovered and extended his results. Also J a k X and Pillet obtained a condition for strict positivity of the entropy production'' (see also [16]). On the other hand, recent progress in dynamical systems approach to classical nonequilibrium statistical mechanics reveals a new symmetry of entropy production fluctuations, known as the fluctuation theorem. It was found numerically by Evans, Cohen and Morrisz3 and shown rigorously for thermostated systems by Gallavotti and C ~ h e n Roughly ~ ~ . speaking, this theorem asserts that the probability of observing the entropy production to be a(>O) during a time interval t is exp(at) times larger than the probability of observing it to be -a. asymptotically in the limit of large t. It was then extended to transient states25,to stochastically driven systemsz6~z7~28 and to open conservative ~ y s t e m s ~ The ~ y ~related ~. topics have been extensively investigated (see e.g., references in [29,30] ). However, its quantum generalization has not been well studied. In this article, the time evolution of states is investigated for a C' algebraic system consisting of several (infinitely extended) heat reservoirs and a finite subsystem with L1 asymptotic abelian property, which means that the time evolution *-automorphism Tt satisfies
for enoughly many dynamical variables A and B . Note that this is one of mixing conditions. Along the line of thoughts by Spohn and Lebowitz5, we follow the evolution of states starting from initial states where heat reservoirs are in equilibrium with different temperatures and chemical potentials. Then, nonequilibrium steady states are derived as t -+ f m limits in the weak sense. Weak convergence of the states is guaranteed by the L1 asymptotic abelian property. When a few conditions are satisfied in addition, the steady states are shown to be KMS (Kubo-Martin-Schwinger) states with respect to certain aThroughout this article, we follow Araki's definition of relative entropy S ( w l w t ) l g . It is slightly different from the one, SBR ( w t l w ) , e.g., used in 2 : : SBR (wtlw)= - S ( w l w t ) .
102
*- automorphism. More interestingly, the steady states correspond to the ones proposed by MacLennan3' and Z ~ b a r e v ~ ~ . In addition, a quantum analog to the fluctuation theorem is derived for the relative entropy production. As will be explained later, the relative entropy is the average of the logarithm of the so-called relative modular operator, which acts on states but cannot be reduced to left nor right multiplications (such operators are sometimes called super operator^^^). Hence, this superoperator may be regarded as a relative entropy operator although it is not a standard dynamical variable. We then study fluctuations of the logarithm of the relative modular operator and show that their distribution has a symmetry claimed by the fluctuation theorem. Note that Prigogine and his coworkers34 have been continuously investigating a realization of entropy as a superoperator, i.e., an operator acting on density matrices but not represented as standard dynamical variables. Also Ojima, Hasegawa and Ichiyanagi12 studied a free energy operator represented by the relative modular operator. As previously mentioned, we mainly consider quantum dynamical systems with L1 asymptotic abelian property. This condition is vaild for free Bose and Fermi Gasses in dimensions greater than or equal to three2, certain coupled quantum oscillator^^^. It is known that the condition is not valid for the one dimensional XY model, and for some interacting f e r m i o n ~ ~Thus ~ . in view of mathematical rigour, our analysis is restricted to special class of quantum systems. However recent results of spin fermion models due to JakSiC and Pillet in [ll]suggest that what we describe here is physically generic. Moreoever, by sticking to the condition of L1 asymptotic abelian property, we may exhibit an overview of nonequilibrium steady states in a concise manner. The rest of this paper is arranged as follows. Sec. 2 is devoted to the description of a C* algebra corresponding to the system. We specify precisely the decomposition of the system into several heat reservoirs and a finite subsystem. Corresponding to each decomposition, initial states are prepared as KMS states, where heat reservoirs are in equilibrium with different temperatures and chemical potentials. Then, the L1-asymptotic abelian property and other assumptions on dynamics are explained. In Sec. 3, the convergence of states at time t to steady states as t 4 fco is shown. The steady states do not depend on the choice of initial states of the finite subsystem nor on the way of division. The steady states at t = f m are time reversal of each other. And the steady states are ergodic in a sense that they are stable against local perturbations in both directions of time. We remark that the steady states are related to the initial states via Moller morphisms. In Sec. 4, the definition of the relative entropy in C" algebra and the implications of the previous ~ o r k s ' ~are 1~ summarized. ~ And steady-state entropy production is shown to
103 be independent of the way of division of the system into heat reservoirs and a finite subsystem. Then, the fluctuation theorem for the logarithm of the relative modular operator is derived. In Sec. 5 , the existence of a system division with invertible Moller morphisms is assumed and we show that the steady states may be characterized as MacLennan-Zubarev nonequilibrium ensembles in a sense that they are KMS states with respect to a *-automorphism, whose generator is represented by a linear combination of Zubarev's local integrals of motion33 in a certain sense. Sec. 6 is devoted to the summary. Here we give only the results and the proofs will be given elsewhere3g. 2
Large Quantum Systems
2.1 Field algebra The system S in question is described by a field algebra Namely, F is a C' algebra where the following *-automorphisms are defined: (i) a strongly continuous one-parameter group of (t E R), which describes time-evolution.
*- automorphisms Tt
(ii) a strongly continuous L-parameter group of *- automorphisms a d ((p' E RL)satisfying a ~ = a~ d I + d 2a, which ~ represent ~ the gauge transformation.
(iii) an involutive *-automorphism 0, which is represented as 0 = with some $0 E RL.
The groups Tt, a+ and 0 are interrelated as 0Tf
=Tf0,
@a+= a@,
TfCXyd
= Urp'Tt
for all t E R and (p' E RL. A subalgebra A c F which is invariant under the action of a$ ((p' E RL) is called the observable algebra, which describes observable physical quantities. The *-automorphism 0 defines the even and odd subalgebras, respectively, F+ and F-:
F+ = { A E F';O(A)= &A} . When the system involves fermions, even and odd subalgebras correspond to dynamical variables which are sums of products of, respectively, even and odd number of fermion creation and/or annihilation operators. Let ZAE RL be the unit vector whose Xth element is 1, then, because of (ii), the *-automorphisms aSzAdefines a strongly continuous group and its
104
generator will be denoted as gx (A = I, . . . L )
where D(gx) is the domain of g x and the limit is in norm. And we assume (iv) D(6) c D(gx) (A = I , 2,. . . L ) , where 6 is the generator of the time evolution *-automorphism rt and D(6) is its domain dense in F. In addition to the gauge symmetry, the system is assumed to possess time reversal symmetry: (v) There exists an involutive antilinear *-automorphism LrtL=r-t
2.2
L
such that
.
(4)
Decomposition of the system and initial states
We consider the situation where the system S can be decomposed into N independent infinitely extended subsystems Sj ( j = 1, . . . N), which play a role of heat reservoirs, and a finite-degree-of-freedom subsystem So interacting with all the others. More precisely, the algebra .F is represented as a tensor product of N infinite dimensional subalgebras Fj ( j = I, . . . N) of S j , and a finite dimensional subalgebra Fo of So:
such that the following conditions are satisfied:
(Sl) There exists a gauge-invariant time evolution group ry (t E R) which is a perturbation t o rt by a selfadjoint element -V E An D(6) and which is a product of strongly continuous groups ?t(j) ( j = 1, . . . N) independently acting on subalgebras Fj ( j = 1, . . . N )
.,“ =
. ..
?i‘”
.
(6)
Namely, ?t(j) leaves the other subalgebras Fk invariant and it commutes with the other groups T i k ) :
?P’(A) = A
(vAEFk, k # j )
(7)
- ( J ) - ( k ) = -(k)-(A Tt 7 s 7 s rt
(t,SERk#j)
(8)
105
(S2) The gauge *-automorphism a$ is a product of strongly continuous groups &$) ( j = 0, I , . . . N ) independently acting on subalgebras Fj ( j = 0,1,. , - N ) : ffg=
&(O) --(I). . . - ( N ) $
a$
a$
(9)
7
and they satisfy
&d( j ) ( A= )A
&(j)&(t) = &(’c)&($ dl ’pz
(02
The groups
‘pl
(‘A E Fk
($6, @z
1
/C
#j)
E RL
#j)
(10)
(11)
and 6:) are interrelated as
for all j , k = 1, . . . ,N , t E R and (p’ E RL. And an assumption is made for the domains of the generators 8j and jf),respectively, of the strongly continuous ( t ,s E R): groups and
) ~ ( j ffor) all ) j = 0,1,.. . N , ( ~ 3 )~ ( 6c) ~ ( 8 , ) ~ ( 6c
x = I , . . . L.
Then, as the condition (Sl) implies that the domain of the generator T: is equal to D(6): D(SV) = D ( S ) ,one has
S(A) = Sv(A) + i[V,A]
Sv of
(for A E D ( 6 ) )
( 12)
(for A E D ( 6 ) )
(13)
N
bV(A)= X 8 j ( A ) j=1
Individual time evolutions and gauge transformations are assumed to be time reversal symmetric:
(S4)
Lfy) = $2
’
,&(A = 3
‘p
Note that one may assume L ( V )= V without loss of generality. Indeed, any V can be decomposed into an even and odd elements with respect to the time reversal operation L:
v = v,+ v, (14) where V, = ${V + L ( V ) }and V, = ${V ~(v)}. On the other hand, when -
the conditions (v) and (S4) are satisfied, one has
[Vo,A]= 0 ,
(VA E F)
(15)
106
and the odd part V, does not contribute to ry. As in the previous works5>9-13>15, we are interested in the evolution of initial states where N infinitely extended heat reservoirs are in equilibrium with different temperatures and different chemical potentials and the finite subsystem is in an arbitrary state, which is described by a nonsingular density matrix. As discussed in [13,15],such states are specified as a KMS state:
(S5) Let a,“ (x E R) be a strongly continuous group defined by N
a;(A) =
IJ ? - p J x a p j / z J x ( e i ~ ~ X A e -)i ~ ~ (XA E F ) (j)
-(j)
7
(16)
j=1
where
/3j
and ,iij = ( p y ) ,. . . p(jL )) are, respectively, the inverse temper-
ature and a set of chemical potentials of the j t h heat reservoir. The operator DS ( E Fo n d)is selfadjoint and exp(Ds) represents an initial state of the finite-degree-of-freedom subsystem So. Then an initial state w is a KMS state with temperature -1 with respect to a,”. Namely, w is a state such that, for any pair A, B E F,there exists a function F A , B ( ~ ) of x analytic in the stripe {z 6 C ;0 > Imx > -1) and satisfies the KMS boundary condition:
FA,B(z)= w(Aa,W(B))
FA,B(Z- i)
Because of (S3), the domain of the generator and 8, is given by
= w(a,W(B)A)
(x E R) (17)
iU of a$ satisfies D(&) 3 D(S)
Note that a decomposition without the finite subsystem is possible as well. We note that the boundaries among subsystems can be changed in an arbitrary way and, in some cases, it is necessary to compare two situations corresponding to different divisions. For this purpose, we introduce a notion of locally modified states. Consider a decomposition different from ( 5 ) :
and a KMS state w‘ of temperature -1 with respect to
107
where the temperatures pj and chemical potentials i;j are the same as those of a,“. Then the state w’ is said to be a locally modified state of w if the generators 8, and lw, of, respectively, a$ and a,”’are related as
8,,(A) - i W ( A= ) i[W,A] (“A E D ( j w ) )
(21)
where W E A is selfadjoint and D ( i Wis ) the domain of i,. Note that, if there exist several KMS states, locally modified states w and w’ may be globally different. Note that the state w”, which corresponds to the same division (5), but to a different initial state exp(Dg) of the finite subsystem, is a locally modified state of w because the generators of the defining groups of w and w“ differ by a bounded derivation:
i w I l ( A-) b ( A )= i[Dg - D s , A ]
(“A E D(8,))
(22)
2.3 Assumptions on initial states and dynamics The state wt at time t starting with the initial state w is given by
wt=wort,
(23) and its weak limits for t + f o o are expected to be nonequlibrium steady states. Of course, the limits do not exist in general. As one of sufficient conditions for the existence of the limits, we assume that the evolution is L1(GL)-asymptotically abelian:
( A l ) L1 (6L)-asymptotically abelian property: There exists a norm dense *-subalgebra GL such that
L L
+m
dtIl[A,~t(B)]11 < -too
( A E G‘L, B E GL nF+)
(24)
( A , B E G L nF-)
(25)
+m
dtII[A,rt(B)l+lI < +oo
where [., .]+ is the anticommutator and
F* are even/odd
subalgebras
Note that there may exist more than two KMS states at low temperatures, for example, if the quantum system undergoes the phase trasition with symmerty breaking, the KMS states should not be unique. However, because a heat reservoir in thermodynamics is fully characterized by its temperature and chemical potentials, we assume that reservoir states are uniquely determined by the KMS condition:
108
(A2) Uniqueness of initial states: There is a division of the system: 3 = 3 0 8 3 1 8 . . . @ FN into N heat reservoirs and a finite subsystem such that, for each set of temperatures { D j } , chemical potentials {&}, and an initial subsystem state e D S ,there exists a unique KMS state w of aW , with temperature -1. And the perbelongs to GL. turbation V in the time evolution *-automorphism
ry
Assumption (A2) implies the invariance of the state w under the perturbed time evolution Indeed, as seen from (Sl), (S2) and (S5), aW , and commute. Hence, the state wory is again a KMS state of aW , with temperature -1 and, by assumption (A2), it is equal to w : w o r: = w . The L1(GL)-abelian property (Al) and V E GL implies the existence of Moller morphisms -yk defined by (cf. Prop. 5.4.10 of Ref. [2] )
ry.
ry
lim ry-'rt(A) = y+(A) .
t-fw
To prove certain properties, the invertibility of Moller morphisms is necessary and, in stead of (A2), we assume (A3) Uniqueness of initial states and invertibility of Mmller morphisms: There is a division of the system: 3 = . . 8 3into ~ N heat reservoirs such that, for each set of temperatures {&} and chemical potentials {&}, there exists a unique KMS state w of a: with temperature -1. And the belongs to BL. perturbation V in the time evolution *-automorphism In addition, the Mmller morphisms -yk defined in (26) are invertible.
ry
ry
admits a finite-dimensional invariant s u b If the perturbed time evolution algebra, Mmller morphisms are not invertible. Hence, the decomposition of the system in (A3) should not contain the finite-dimensional subalgebra 3,.
3
Steady states
3.1 Properties of steady states
Theorem I: Existence of steady states When the evolution rt satisfies (Al) the L1(Gt)-asymptotic abelian property, the weak limits
~ exist for each initial state w explained in (S5). The states w + are rt-invariant 17.
109
In view of thermodynamics, steady states are expected to depend only on the global boundary conditions such as the temperatures and chemical potentials of the reservoirs. Indeed. we have
Theorem 2: Independence of steady states o n division and D s When rt is ( A l ) L’(GL)-asymptotically abelian and (A2) the KMS state for uW , is unique, for any locally modified state w‘ of w , one has lirn w’
t-fm
o rt(A)=
lim w 0 q ( A ) = w+,(A)
t++m
(‘A E F)
This implies that the steady states w f m are determined only by the temperatures and chemical poteitials of the heat reservoirs, but does not depend on the way of division into subsystems nor on the initial state of the finite-degree-of-freedom subsystem.
For Spin Fermion models, the same result is obtained by JakSiC, and Pillet in [111. As an immediate consequence of Theorem 2, one has Proposition 3: T i m e reversal property of the steady states Under the assumption of Theorem 2, the two steady states are time reversal with each other:
*
w c c
(29)
= L WTcc
where the time reversal operation
L*
on a state w is defined by
L*w(A) = w ( L ( A * ).)
(30)
Under stronger assumptions, steady states have certain ergodicity.
Theorem 4: Stability of steady states against local disturbance When rt is ( A l ) L1(GL)-asymptoticallyabelian, (A3) the KMS state for uW , is unique and the Moller morphisms y+ are invertible, the steady states w+, are stable against local pertubation in the sense lim w+m (B*rt(A)B)= w+,(A) w+,(B*B)
t+*m
. (‘A,B
E
F)
(31)
The same is true for the state w - ~ This . corresponds to the ’return to equilibrium’ property of equilibrium states’ and implies certain ergodicity of the steady states.
110 4
4.1
Relative entropy, its production and fluctuation theorem Relative entropy of states over c“ algebra
For a finite dimensional C* algebra, the relative entropy S(pzlp1) of two states represented by density matrices p1 and p2 is given by S(P2lPl) =Tr{Pl(logPl-logPz)}
I
(32)
where Tr stands for the trace. A generalization to states over a C* algebra is carried out with the aid of GNS (Gelfand-Naimark-Segal) representation and Tomita-Takesaki theory of von Neumann algebras. We summarize the outline following [ 121. For a given C* algebra A, there exist a Bilbert space K ,a vector R E K and a *-morphism r : A + B(K)from A to a set B(K) of all bounded linear operators on K ,such that (i) w(A) = ( R , r ( A )0) (‘A E A) and (ii) the set {n(A)RlAE A} is dense in K (cyclicity of the state R). The triple (K,0,n) is called the GNS representation. A set of all B E B(K)which commute with every element of n(A) is denoted as n(d)’ (commutant of r ( A ) ) . r(A)’ is again an algebra. Let M be a double commutant of r ( A ) :M = r(A)”,then M” = M . An algebra like M is called a von Neumann algebra. Given a von Neumann algebra M c B(K),a vector R E K is called separating if AR = 0 for A E M implies A = 0. If a vector R is separating and cyclic with respect to M , there exist antilinear operators S and F satisfying
SAR = A*R (‘A E M ) ,
FA’R
= A‘*R
(‘A’ E M’) .
(33)
The closure S of S admits a polar decomposition:
S = JAW (34) where A = S*S is positive and self-adjoint, and J is an antilinear involution. Moreover, they satisfy J M J = M‘ and AitMAPat= M . This is the outline of Tomita-Takesaki theory. The set
P
= {AJAJRIA E M } c K ,
(35) is called the natural positive cone, where the bar stands for the closure. For two vectors !P,R E P which are both cyclic and separating, one defines an operator Sq,n by
Sq,nAR = A*Q . ( A E M )
(36)
Arakilg defined the relative entropy of Q and C2 by
S(Rl@)
= (Q,ln&,n@)
,
(37)
111
where A Q , =~ S;?,Sq,n is called the relative modular operator with Sq,n the closure of Sq,n. For any faithful states w1 and w2 on a C* algebra, when both of them are represented by separating and cyclic vectors, rk and fl respectively, belonging to the same natural positive cone in a GNS representation, their relative entropy S(w2lwl) is defined by S(W2lW1)
= S(fllQ)
(38)
'
In the next subsection, we investigate the temporal change of the relative entropy S(wlwt) between the initial and present states.
4.2 Relative entropy and its change Explicit expression of the relative entropy production was obtained by Ojima et a1.12,13and JakSiC and Pillet15i11.
Theorem 5: Relative entropy [Ojima et al.12113and JakSic and Pillet15i11] The relative entropy S(wlwt) between the initial and present states is given by N
S(wlwt) =
CPj [w,(J;)ds j=1
where w, z w reservoir:
o
- Wt(DS)
0
+ W(J9.s)
(39)
rs and Jj" corresponds to the heat flow to the j t h L
Moreover if (Al) the time evolution q is asymptotically abelian, (i) the relative entropy production Ep(wt) = $S(wlwt) at time t converges to the steady state values in the limit o f t + f m : N t-fcc lim &(wt)
= c,oJw*m(J;)
= ~P(W+m)
,
(41)
j=1
(ii) they do not depend on the initial states of finite dimensional 5 0. Note that the subsystem, (iii) Ep(w+,) 2 0 and Ep(w-,) positivity of Ep(w+,) is consistent with thermodynamics.
NB 5.1 For finite-degree-of-freedom systems, the generators jj and ijf)are as commutators: given by local Hamiltonians Hj and number operators
112
& ( A ) = i[Hj, A ] and @ ( A )
= i [ N3 ( ” ,A ] , where
Hj and
each other. And the total Hamiltonian H is H = C,”=,Hj because of - [ H j , V] = [ H , H j ]and
+
commute with V. Therefore,
V ]= [ H ,N,”)],
which, indeed, represents nonsystematic energy flow to the j t h reservoir.
NB 5.2 Since D s corresponds to the logarithm of density matrix describing the initial state of the finite system, it is interesting to rewrite (39) as
which may read as follows: The entropy change of the finite subsystem -wt(Ds) is the sum of entropy flow from the resevoirs and the entropy production $ S ( w ( w t ) = E p ( w t ) . However, as discussed elsewhere38, such an interpretation is not correct in general, but E p (w t ) can be identified with thermodynamic entropy production only for very large It]. Under stronger assumptions, one can show the independence of the limits on the way of division. Theorem 6: Division independence of Ep(w*,) Let w’ be a locally modified state of w by W . Then, if q is (Al) L1(GL)-asymptotically abelian, (A2) the KMS state for g,” is unique and Ds, Dk, W E D(S), one has lim Ep(w’
t+*m
or Ep(w+,)
o
q )= t-fm lim E p ( w 0 q ) ,
(43)
is independent of the way of division of the system.
4.3 Fluctuation theorem In view of (37) and (38),the logarithm of the relative modular operator between the present and initial states divided-by the duration t can be regarded as the mean entropy production operator &: 1 IIt = - Inan,,,
t
(44)
113
where Rt and R are vector representions of wtand w,respectively, in a GNS representation. Since Ant,n is positive, fit is selfadjoint and admits a spectral decomposition:
s_,
+W
fit =
where Pt(X) is a spectral family of Pt ( [a, b])= Pt(b)- Pt(u - 0):
XdPt(X)
fit.
(45)
Then the expectation value of
( O t , p t ( [ a , b l ) o t )= P r ( [ a , b l ; w t )
(46)
may be regarded as the probability of finding the values of the mean relative entropy production within an interval [u,b] at the state wt. As seen in the proof of Theorem 7, the probability is uniquely determined by the initial state and the time evolution automorphism. As a result of the time reversal , w t ) enjoys a simple symmetry symmtery, the probability distribution Pr ( [ a b]; property analogous to the Gallavotti-Cohen fluctuation t h e ~ r e m ’ ~ - ~ ~ .
Theorem 7: Fluctuation theorem Let Pr ( [ a b]; , wt) (t > 0 ) be the probability of finding the values of the mean relative entropy production within the interval [a,b] as defined above. Then, if the initial state w is time reversal symmetric, the probability satisfies an inequality
for 0 5 a 5 b.
NB 7.1 If the probality measure Pr ( ( a , b ] ; w t is ) absolutely continuous with respect to a reference measure V R with a density function p(X;wt):
P r ( ( a ,bl; W t ) = Theorem 7 implies
Lb
P ( X ;d t ) d V R ( X ) ,
p ( a ; w t ) = ,at
.
P(-a; W t ) This is a noncommutative extenion of the transient fluctuation theorem of E~ans-Searles’~ and of the detailed fluctuation theorem of Jarzynski”. Also if X = a is a discrete point, one has
114
For a particular value a = Ep(w+,), Theorem 7 implies that the probability of finding the mean relative entropy production at the steady-state average Ep(w+,) is exponentially larger than the probability of finding it at the o p posite value -Ep(w+,).
4.4 Outline of the proof
of Theorem 7
Because w is 7:-invariant
, one has w 0 7t(A)= w o ( X A q * )
(48)
where a unitary element Yt is defined as a norm convergent series:
In order to give a simple explanation, yt is assumed to be a:-analytic. Let 0 and Rt be the vector representations of w and w t , respectively, and let An,,n be the relative modular operator, then the characteristic function for the mean entropy production operator fit is given by
@ ( E ) = (Rt,exp(ifitE)ot)
= (Rt,A$/,$t)
=w
(Y"&&K*)).
On the other hand, if w is time reversal symmetric, the time reversal symmetry of a$ and the KMS boundary condition give
@(-E)
(q*) Y")
= w (@?
= w (Y"u&i
(V))= @(E + it) .
(50)
In terms of the cumulative distribution function F ( X ) = Pr ((-co, A]; w t ) , this reads as
1,
+cc
-
1,
+cc
eiEXdF(-X)=
ei<xe-xtdF(A) ,
which gives b+O
l-0
-a+O
d F ( A )= 1 b - O
eCXtdF(X).
The desired result immediately follows from this.
115
5
Characterization of steady states
5.1 KMS Characterization of states Under the setting (Sl)-(S5), the evolving state wt is characterized as a KMS state.
Theorem 8: K M S characterization of evolving states The state wt at time t is a KMS state at temperature -1 with respect to the strongly continuous *-automorphism -1
w
ugt 3 7 t U r Y t
where Tt = ry-
1
rt, and
,
its generator is given by
@ ( A ) = 8 , ( A ) + i L d s [rS( t ( V ) ), A ] for all A E
(51)
(52)
~(62)) =~(6,).
When the Moller morphisms Y+ exist and are invertible, the steady states w*m are characterized as KMS states:
Theorem 9: KMS characterization of steady states When (Al) the time evolution *-automorphism rt is L1(B~)-asymp totically abelian and (A3) the Moller morphisims T* are invertible, the steady states wfoo are KMS states at temperature -1 with respect to the strong continuous *-automorphism uw* 32
Furthermore, if &,(V) E
-1
w
-Y* UZY*.
~
(53)
GL, its generator 8; satisfies
) GL. for all A E D ( j Wn
NB 9.1
For finite systems, the KMS state w with respect to the automorphism u," corresponds to the density matrix
*-
116
where Z is the normalization constant, 03,Hjrp y ) and N j X )are, respectively, the local temperature, local energy, local chemical potential and local number operator of the j t h reservoir. As discussed in NB 5.1,
where J,” is a heat flow to the j t h reservoir. Therefore, because of Theorem 8, the density matrix pwL correponding to wt is given by
f i v
r
L
X=l
J
-t
Note that one has pw, = ~ - ~ ( and p ~this ) is consistent with the Liouville-von Neumann equation for the density matrices.
NB 9.2 For infinite systems, the case when the right-hand side of (54) generates og* is most interesting. Then, if the integral
would converge, w&- would be a perturbed KMS state of the initial state w by a self-adjoint operator &. Moreover, NB 9.1 suggests that the corresponding density matrices would be N
1
z
L
.
(56)
Note that such statistical ensembles for steady states were introduced by MacLennan3’ and Z ~ b a r e v ~ ~ . Howe_er, if the steady state carries nonvanishing entropy production, the integral V* does not converge since the w-average of its integrand does not vanish at infinities:
Thus, the original proposal (56) by MacLennan and Zubarev cannot be justified. Rather, the KMS states with respect to oz* generated by (54) should be regarded as a precise definition of the MacLennan-Zubarev ensembles. Note that the above observation is consistent with the result by JakSii: and Pillet”, which asserts that the relative entropy production between the steady state and the initial state vanishes if the former is a normal state of the latter.
117
6
Summary
For an infinitely extended system consisting of a finite subsystem and several reservoirs, we have studied the time evolution of states. Initially, the reservoirs are prepared to be in equilibrium with different temperatures and chemical potentials. If the time evolution is L1-asymptotic abelian and a few more conditions are satisfied, (i) steady states exist, (ii) they and their relative entropy production are independent of the way of division into subsystem and reservoirs, and (iii) they are stable against local perturbations. The relative entropy production is calculated explicitly and we have given a KMS characterization of the steady states, which provides a rigorous meaning to the MacLennan-Zubarev steady state ensembles. In addition, a noncommutative analog to the fluctuation theorem is derived provided that the evolution and an initial state are time reversal symmetric. Before closing, we remark that the unidirectional evolution is consistent with the time reversal symmetry of the system. To see this, let us consider the following thought experiment: At t = 0, the system is prepared to be in an i*-invariant state w . Until time t = tl(> 0), the system evolves according to the *-automorphism rt. At t = tl, time reversal operation i* is applied. After t = t l , the system evolves according to rt again. The state wt at time t is given by
where we have used L*W = w . Because of Theorem 1, the initial state w evolves towards the steady state w+, and, for large t l , the state just before the time reversal operation w t l - = w o rtl is close to w+,. On the other hand, as -tl < 0, the state w o r-tl just after the time reversal operation is close to the other steady state w-,. Afterwards, the state wt = w 0 r t t - 2 t l deviates from w-, and Thus, reaches w at time t = 2tl. Then, the state wt again approaches w+,. the time reversal operation discontinuously changes a state w t l - (- w+,) to but does not invert the evolution. In this way, the a state wtl+(- W-,), unidirectional state evolution is consistent with the time reversal symmetry. A similar view was given by Prigogine et al.34 for the behavior of entropy under time reversal experiments, where dynamics was considered to increase entropy and the time reversal operation was thought to induce a discontinuous entropy decrease.
118
Acknowledgments T h e authors thanks Professors L. Accardi, H. Araki, T. Arimistu, H. van Beijeren, J.R. Dorfman, P. Gaspard, T. Hida, I. Ojima, A. Shimizu, K. Saito (Meijo Univ.), K . Saito (Tokyo Univ.), S. Sasa, M. Toda for fruitful discussions and valuable comments. This work is partially supported by Grant-in-Aid for Scientific Research (C) from the Japan Society of t h e Promotion of Science.
References 1. For example, see, M. Toda, R. Kubo and N. Saito, Statistical Physics Z (Springer, New York, 1992); R. Kubo, M. Toda and N. Hashitsume, Statistical Physics ZZ (Springer, New York, 1991). 2. 0. Bratteli and D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics vol.1 (Springer, New York, 1987); v01.2, (Springer, New York, 1997). 3. I.P. Cornfeld, S.V. Fomin and Ya.G. Sinai, Ergodic Theory, (Springer, New York, 1982); L.A. Bunimovich et al., Dynamical Systems, Ergodic Theory and Applications, Encyclopedia of Mathematical Sciences 100, (Springer, Berlin, 2000). 4. D. Ruelle, Statistical Mechanics: Rigorous Results, (Benjamin, Reading, 1969); Ya. G. Sinai, The Theory of Phase Transitions: Rigorous Results, (Pergamon, Oxford, 1982). 5. H. Spohn and J.L. Lebowitz, Commun. Math. Phys. 54, 97 (1977) and references therein. 6. J. Bafaluy and J.M. Rubi, Physica A153, 129 (1988); ibid. 153, 147 (1988). 7. J. Farmer, S. Goldstein and E.R. Speer, J. Stat. Phys. 34, 263 (1984). 8. J.-P. Eckmann, C.-A. Pillet and L. Rey-Bellet, Commun. Math. Phys. 201, 657 (1999); J . Stat. Phys. 95, 305 (1999); L. Rey-Bellet and L.E. Thomas, Commun. Math. Phys. 215, 1 (2000) and references therein. 9. T.G. Ho and H. Araki, Proc. Steklov Math. Institute 228, (2000) 191. 10. S. Tasaki, Chaos, Solitons and Fractals 1 2 2657 (2001); in Statistical Physics, eds. M. Tokuyama and H. E. Stanley, 356 (AIP Press, New York, 2000); Quantum Information ZZZ, eds. T. Hida and K. Saito, 157 (World Scientific, Singapore,2001). 11. V. JakSiC, C.-A. Pillet, Commun. Math. Phys. 226, 131 (2002). 12. I. Ojima, H. Hasegawa and M. Ichiyanagi, J. Stat. Phys. 50, 633 (1988). 13. I. Ojima, J . Stat. Phys. 56, 203 (1989); in Quantum Aspects of Optical Communications, (LNP 378,Springer,1991). 14. D. Ruelle, Comm. Math. Phys. 224 , 3 (2001). 15. V. JakSiC and C.-A. Pillet, Commun. Math. Phys. 217, 285 (2001). 16. V. JakSiC and C.-A. Pillet, J. Stat. Phys. 108, 269 (2002). 17. D. Ruelle, J . Stat. Phys. 98,57 (2000). 18. M. Ichiyanagi, J. Phys. Soc. Japan 5 5 , 2093 (1986).
119
19. H. Araki, Publ. RIMS, Kyoto Univ. 11, 809 (1976); 13,173 (1979). 20. A. Uhlmann, Commun. Math. Phys. 54,21 (1977). 21. M. Ohya and D. Petz, Quantum Information and Its Use, (Springer, Berlin, 1993). 22. R.S. Ingarden, A. Kossakowski and M. Ohya, Information Dynamics and Open Systems, (Dordrecht, Kluwer, 1997). 23. D.J. Evans, E.G.D. Cohen and G.P. Morriss, Phys. Rev. lett. 71,2401 (1993). 24. G. Gallavotti and E.G.D. Cohen, Phys. Rev. Lett. 74, 2694 (1995); J. Stat. Phys. 80, 931 (1995). 25. D.J. Evans and D.J. Searles, Phys. Rev. E 50,1645 (1994); D.J. Searles and D.J. Evans, J . Chem. Phys. 112,9727 (2000); ibid. 113,3503 (2000). 26. J. Kurchan, J.Phys. A 31,3719 (1998): 0.Mazonka and C. Jarzynski, archived in cond-mat/9912121 (1999). 27. J.L. Lebowitz and H. Spohn, J . Statist. Phys. 95,333 (1999). 28. C. Maes, J. Stat. Phys. 95,367 (1999). 29. C. Jarzynski, J . Stat. Phys. 98,77 (2000) 30. L. Rey-Bellet and L.E. Thomas, Ann. H. Poincare' 3,483 (2002): C.Maes and K. NetoEni, J . Stat. Phys. 110,269 (2003). 31. L. Rondoni, T. T61 and J. Vollmer, Phys. Rev. E 61,R4679 (2000). 32. J.A. MacLennan, Jr., Adv. Chem. Phys. 5 , 261 (1963). 33. D.N. Zubarev, Nonequilibrium Statistical Thermodynamics, (Consultants, New York, 1974). 34. I. Prigogine, From being to becoming, (Freeman, New York, 1980) and references therein. 35. F. Fidaleo and C. Liverani, J. Stat. Phys. 97,957 (1999). 36. Z. Ammari,Non-existence of the Mdler Morphism f o r the spin fermion dynamical system, archived in mp-arc/02-195; Scattering theory for the spin fermion model, archived in mp-arc/02-196. 37. H. Araki and A. Kishimoto, Commun. Math. Phys. 52, 211 (1977); H. Araki, R. Haag, D. Kastler and M. Takesaki, Commun. Math. Phys. 53,97 (1977). 38. S. Tasaki, in Dynamics of Dissipation, eds. P. Garbaczewski, R. Olkiewicz, 395 (LNP 597, Springer, Berlin, 2002). 39. S. Tasaki and T. Matsui, in preparation.
120
WEAK CHAOS: CLASSICAL AND QUANTUM FEATURES ROBERTO ARTUSO Center for Nonlinear and Complex Systems Dipartimento d i Scienze Chimiche, Fisiche e Matematiche, Via Valleggio 11, 22100 Como, Italy and I. N. F.M., Unit6 di Como, I.N. F. N., Sezione d i Milano E-mail:
[email protected] We consider the standard mapping in the regime of weak classical chaos, where transport exhibits anomalous features, such as nontrivial scaling properties of the moments’ distribution. We then study the corresponding quantized map, focusing on asymptotic properties, that from many respects fit into the standard picture of quantum dynamical localization.
1 1.1
Introduction W h y weak chaos is of interest
After more than twenty years of intense theoretical efforts and remarkable experimental activity, the discipline of chaotic dynamics has reached a noticeable scientific maturity. Complexity quantifiers like Kolmogorov-Sinai entropy or physical phenomena like the period-doubling route to chaos are by now textbook issues, and part of the conceptual vocabulary of modern physics. While, during the early stage, dynamical systems theory focused on a classical setting, we also have now a fairly extensive set of theoretical as well as experimental results involving quantum mechanics, from universality properties of spectra to the mechanism by which interference effects destroy classical transport through the so called dynamical localization1. What still stands as a major problem, despite a number of partial results, is a proper treatment of systems which display intermediate properties between integrability (complete order) and full chaoticity: such systems are often called mixed systems, as their classical phase space shows the coexistence of regular regions (elliptic islands) and hyperbolic, chaotic portions. Typical trajectories originating in the chaotic sea come close to stable structures and are sticked to them for quite long periods: the anomalies of mixed systems are due to the influence of these trapped segments on the overall dynamics. Quite recently the physical importance of such systems has been remarked in a number of different contexts: this strongly motivates us to deepen our theoretical understanding of the basic mechanisms of dynamics and to explore further physical situations
121
in which such effects play a fundamental role. For instance weak chaos has been invoked in the context of quantum fractal fluctuations: when scattering problems are dominated by a statistically large number of metastable states, they display extremely erratic response curves (like cross section us some relevant parameter), that may display a fractal character over a wide range of scales. What is probed by such response curves is the layout of resonance poles in the complex energy plane, so the erratic behaviour of fluctuations is closely linked to distribution properties of the poles: in particular lack of smoothness has to be related to resonance poles clustering in the vicinity of the real energy axis, which should dynamically manifest as a slow algebraic decay of survival probability inside the interaction region. Such a behaviour is -on the classical level- typical of a mixed system, and this lead to the prediction of fractal fluctuations for transmissions of electrons through mesoscopic cavities2: such a conjecture is in accord with some experimental findings3, and accurate numerical simulations4.
1.2
P l a n of the paper
One of the most remarkable breakthroughs in the discipline of quantum chaos has been the observation that chaotic systems, displaying deterministic diffusion in the classical framework, show, after a typical time scale, a quantum suppression of diffusion, in a way that bears remarkable analogies to localization theory for disordered systems (see various contributions in ’). The relevant quantum parameter, namely the localization length E , depends both on typical quantum scales, through the effective Planck’s constant, and on classical indices, as it is proportional to the classical diffusion constant D. On the other side it is known that mixed systems often display anomalous diffusion, where the typical spreading of trajectories does not grow linearly with time5>6. This opens the question whether major departures from the standard picture are also observed in a quantum-mechanical framework. The paper is organized as follows: we first describe the classical setting, where we consider the standard map for particular parameter values that induce anomalous dynamical behavior. We provide numerical evidences that such a behavior is encoded in a non-trivial spectrum of “spreading” exponents: in particular the momentum variance grows faster than linearly (accelerated diffusion). Next we take into account properties of the corresponding quantum mapping: as a diffusion coefficient is not defined classically, we investigate the asymptotic regime, to check for departures from the usual properties of quantum dynamical localization. We provide numerical evidences that qualitatively the asymptotic dynamics remain the same, and moreover some of the
122
relevant scaling features remain unaltered. 2
2.1
The standard map in the anomalous regime Transport in the standard mapping
Area preserving maps represent the simplest setting where chaotic dynamics appear in hamiltonian systems7. They correspond to time-dependent periodic hamiltonians with one spatial degree of freedom: a prototype example is the so-called standard map, whose Hamiltonian is n
where the periodic delta function is defined as
c +cc
ST(t)
=
S(t-T.m)
m=-cc
We will take T = 1, so that the only relevant classical parameter is the kicking strength K . The discrete mapping relates position and momentum after one period of the perturbation
We take the position q to be an angle, so that the phase space is the cylinder T1x R: the transport properties we will consider will be along the momentum direction. In the integrable limit ( K = 0) the momentum is a constant of motion and it just labels the invariant circles on which motion takes place; for small K we are in the KAM regime, where deformed invariant circles still prevent unbounded transport: this can take place just after K = K,,, where the last global invariant circle is destroyed8. For very large values of K the motion looks completely chaotic and motion along pdirection provides a striking example of deterministic diffusion: ((Pn -PO)’)
N
2 ~ n
(4)
where (. . .) denotes an average over a set of initial conditions, and where the diffusion constant D can be expressed in terms of the nonlinear parameter K (the leading behavior is D K2/4, with modulations that can be expressed in terms of Bessel functions). Strong deviations from such a behavior are expected around particular nonlinear parameter K , for which stable accelerator
-
123 modes exist. Period Q accelerator modes are orbits for which Pn+Q -Pn = 2re K sin(qn+i) = 2re
xy=l
(5)
Such orbits sustain ballistic propagation: the influence on global dynamics is particularly strong when bifurcations lead to a self similar chain of elliptic islands around an accelerator mode6, where typical orbits, originating from the chaotic sea, may remain trapped for arbitrarily long times. This mechanism may greatly enhance transport properties, as we show in Fig. (l),where the asymptotic behavior is of the form
with p > 1. This means that diffusion is anomalous: the diffusion constant diverges. The nonlinear parameter has been chosen in the region where sticking effects have been shown to be most
2.2 A closer look at anomalous transport and correlations When we consider dynamical behavior for such anomalous nonlinearity parameters, we find that not only we get accelerated diffusion with a nontrivial transport exponent p, but that, differently from the standard gaussian case, a whole set of independent scaling exponents arise. To characterize the general behavior, we introduce the scaling function p(q) (which is sometimes called the intermittency function12) through the asymptotics of growth laws:
In Fig. (2) we show the scaling function p(q) corresponding to the parameter value K = 6.905, which displays a rich structure (in particular it goes closer and closer to ballistic behavior for high q values): the same function is reported for a typical normally diffusive case, where the numerical results are extremely close to the gaussian case P(q) = 1/2 Vq. The anomalies induced by long sticking times do not only influence transport properties, but obviously appear when considering other types of dynamical properties, like correlation functions. One expects a departure from the exponential decay law, characterizing genuine chaotic systems, in the form of power-law long time tails, that can eventually account for transport properties through Kubo-like formulas. We remark that direct numerical investigations of correlations are generically hard to control in the long time regime, due to aThough anomalous diffusion has been observed in a variety of dynamical systems, stringent theoretical arguments have been formulated only for a few exampled1.
124
strong statistical errors. Both phase and time average imply a l/a error ( M being the number of sample points), which is extremely hard t o beat by just brute force computing. An efficient way t o explore the asymptotic behavior of correlation functions is via the analysis of return time statistic^'^. In the present case the procedure works as follows: we partition into two pieces I'O and rl the phase space of the system (without any reference t o an for underlying symbolic dynamics*), which we may take on the two-torus T2, instance by the q = 7r line. We then run a very long trajectory { q m r p m } and
c
' '
F
I 1 1 1 1 1 1 1 1 1
1x10''
"""I'
1x10'
' '
"""I'
I 1 1 1 1 1 1 1 1 1
lXlO2
' ' """I'
I
' '
I I 1 1 1 1 1 1 1
I
I
1x10'
Figure 1. Momentum spreading for the standard map ( K = 6.905). The average is taken over lo7 initial conditions: the estimated growth exponent is p = 1.534.
bWewarn the reader that symbolic dynamics for the standard map is exceedingly complex14.
125
P
Figure 2. Spectrum of moments’ exponents (full line K = 6.905, dashed line K = 11). The exponents are estimated from averages over 2 . lo5 initial conditions, iterated up to n = lo4.
associate to it the string € 0 , €1 . . . E M where Ek = 0 if ( q k , p k ) E ro and Ek = 1 if ( q k , p k ) E rl. From this string we extract a sequence of residence times t k , by simply recording the length of all substrings where the same symbol is repeated. For instance from the string
11110001111111000011000000001...
(8)
we get tl = 4, t2 = 3, t 3 = 7 , t 4 = 4, t 5 = 2, t 6 = 8,. . .. From the set of t j values we build up a probability distribution of residence times p(t), which
126 we suppose to yield a finite average
m=l
It is easy to realize that p ( m ) / ( t )is now the probability that a point chosen at random in our sequence €1 ...€k . . . is the startang (or ending) point of a residence sequence of length m, so that the probability that a point chosen at random in the symbol sequence belongs to a residence sequence of length m is m . p ( m ) / ( t ) .We then introduce a measure of correlations through a function C(n),which measures the probability that two points in the symbol sequence, n steps apart and chosen at random belong to the same residence substring. We then relate C ( n ) to p(m): as a matter of fact
(10)
This equation shows how the asympotics of C(m) and p(m) are closely related: so if p(m) decays exponentially we expect exponential decay for correlation functions as well, while if p(m) decays according to a power law p(m) m--(’+P) we expect a power law decay, of the form m--’p,for correlation functions. We remark that such an approach has been theoretically refined in 15, and discussed in detail in a number of definite examples16. In Fig. (3) we show the numerical results for the standard map ( K = 6.905): the behavior of p ( t ) leads to expect a power law decay of correlation functions as t-0.46.In view of Kubo formula (see for example 17) N
k=O
where Cffis the force autocorrelation function] a divergence of the diffusion constant is predicted
which means an anomalous spreading ( ( P n - POI2)
n1.54
in agreement with the direct numerical experiment] see Fig. (1).
127 I
1x10'
E
1x106
:
I
I I I l l 1 I
lX1o5E-
p(t> I lX1o4 E-
1x10~E
: F
-
1x102
I
I 1 1 1 1 l 1 1 1
I
I
I 1 1 1 1 1 1 1
I
I 1 1 l 1 1 1 1 1
I
1
1
Figure 3. p ( t ) for I B standard map (K = 6.905). The statistics is uilt from 10'~iterations of a single initial condition.
3 9.1
Quantum behavior of the standard map in the anomalous regime General setting and early numerical observations
Quantum dynamics of the system described by Eq. (l),is conveniently studied via the (unitary) quantum evolution operator over one perturbation period. In the present case it can be written in the following form (see, for instance")
ii = exp ( - 5 6i2 )
exp ( - i ~ c o s ( q ) )
(14)
128
h a -. We notice that quantum behavior is ruled by two parameters: 2 aq the classical nonlinearity K , and the effective Planck’s constant h. Peculiar features (quantum resonance) arise when h is commensurate t o 27r (see 19120), as U is then characterized by periodicity properties that lead t o a Bloch band spectrum, and ballistic transport. We now address the generic case, far from quantum resonance, and take wherep =
h=
2?T
m + PGM
where PGM = (&- 1)/2 is the golden mean. The standard picture that arises in such a case, when K is in the regime of strong classical chaos, is that the variance (*t
$t)
(16)
(where gt = Ut$0 and $0 is usually taken as a wave packet centered around the zero momentum state), follows the classical diffusion up t o a break time t*: after that time the spreading is suppressed and the variance fluctuates around a constant valuelg. This phenomenon (called quantum dynamical localization) is closely linked t o spectral properties of the evolution operator U :
(w are called quasienergies, and 4w quasienergy eigenfunctions): the suppression of classical diffusion is due t o pure point character of the spectrum of U , an observation which is strongly supported by the fact that the system bears remarkable analogies with Anderson localization in condensed matter physics2lC.An important role in the quantum analysis is thus played by the localization length [, which gives the rate of exponential decay of the a s y m p totic wave function in momentum space: equivalently it quantifies the number of momentum states that have a nonvanishing overlap with the wave function. There is a standard heuristic argument23 that allows t o predict the scaling of both E and t* with the physical parameters: if we start from a single momentum state and initially quantum spreading follows classical diffusion, the time needed for an wave packet, initially localized, t o spread over a number E of momentum states will be
E2h2
N
D . t*
~
‘We remark that, even if strong arguments theoretically support the picture of quantum dynamical localization, a rigorous theory has not yet been established. For recent developments, using supersymmetric techniques see 22.
129
the break time must be the time scale over which quantum effects become relevant (the point nature of the spectrum is revealed), thus, on the basis of the uncertainty principle we have the estimate
-Ti. t *
5
and thus t*
N
E , so that
N
fi
from Eq. (18) we get the scaling
It is remarkable how a genuine quantum quantity like the localization length is strictly related to a truly classical index, the diffusion constant: we notice that related scalings appear also in the standard theory of localization in random media24. From our perspective the scaling relation of Eq. (20) opens interesting questions, as we are dealing with classically anomalous behavior, where D diverges. As a matter of fact a pioneering numerical study reported lack of dynamical localization for K = 6.905 up to rather large timesz5. This calls for two major questions: 0
0
is the asymptotic dynamics still compatible with quantum dynamical localization? does localization length still retain scaling properties as in the fully chaotic case?
3.2 Asymptotic dynamics for the classically anomalous case We now report a series of numerical experiments26 about the asymptotic nature of quantum dynamics for the classically anomalous case K = 6.905. The first numerical observation (see Fig. (4)), is that in all cases we examined a strong evidence about dynamical localization has been found (even though for small Ti values this may take quite a long time). When considering then the final distribution over momentum states we also observe (see Fig. (5)) exponentially localized distribution, like in the standard cases, with a completely chaotic classical limit. Thus we have evidence (at least in the regime where the quantum action scale h. is bigger than the size of self similar accelerator islands) that the long time behavior is qualitatively consistent with quantum dynamical localization. As the classical parameter in Eq. (20) is not defined, the question is whether the scaling of E with Ti deviates from the standard picture or not. To this end we considered three different measures of localization length: the first
130
1x106
iX1o3
I ,1 1x10~
I ( , , ,
lXlO0
1 1 1
J
n Figure 4. ( p i ) for K = 6.905 and ii= 0.27 (upper curve) and h. = 2.7 (lower curve).
comes from the form of the final wave function: expanding on momentum states +m
+t(q) =
C
m=-m
for long times we have (cfr. Fig. (5))
I
nt(m)eirng
(21)
131
I
-10000
,
,
,
,
I
-5000
,
,
,
,
I
0
,
,
,
,
I
5000
,
,
,
,
10000
n Figure 5. Final probability distribution over momentum states for K = 6.905 and f i = 0.599.
Another way to give a measure of the localization length is via the inverse participation ratiod
Finally we considered the entropy localization lengthls
t s = exP(Sm) dFor a discussion on inverse participation ratio in quantum chaos, see 27.
132
0
is.,. 5x10'
1
0 lX1O0
I
I
,
I
I
I
lxlo-'
I
, , , , I
lXlO0
Figure 6. Scaling of localization length with ti: circles refer to entropy estimates, stars to inverse participation ratio and diamonds to exponential decay rate of asymptotic distribution over momentum states.
where +W
For all measures of localization length we get results that are consistent with the scaling
E N 7=L-2 (26) like in the fully chaotic case: see Fig. (6). This is somehow surprising: though the scaling relation Eq. (20) cannot hold, as the classical diffusion coefficient
133
diverges in the present case, nevertheless the scaling with the quantum parameter li is preservede. We remark that the break time t* is not the only relevant time scale to be considered when dealing with quantization of chaotic systems: in particular a logarithmic time scale28
where I0 is some characteristic action and A is the Lyapunov exponent, signals the breakup of semiclassical approximation: there is some evidence2Qthat a transition from logarithmic to power-law dependence for this time scale takes place due to classical anomalous behavior (as anticipated in 30). 4
Conclusions
We have considered classical and quantum features of the standard map at particular nonlinearity K values, for which a large chaotic sea coexists with self-similar accelerator islands. The structure of the phase space has striking classical consequences: diffusion is anomalous, moments’ growth is not described by a single exponent, and correlation functions exhibit long time tails, decaying according to a power law, which is consistent, by Kubo relation, with the accelerated diffusion. When going to the quantum mechanical case we observed that, though the standard theory of dynamical localization cannot be applied, as the classical diffusion constant diverges, still the asymp totic regime is strongly reminiscent of the purely chaotic one, and moreover the localization length is still inversely proportional to li2, like in the usual case.
Acknowledgments We acknowledge support from MURST, through the PRIN 2000 research project Chaos and localization in classical and quantum mechanics. The present work originated from very pleasant collaborations with Michele Rusconi and Daniel Alonso. ‘A related observation31 is that the probability to remain trapped near ballistic modes is strongly suppressed by tunnelling in the quantum framework, turning from power-lay to exponential decay, even though recently possible deviation from such a picture have been suggested32
134
References 1. G.Casati and B.V.Chirikov (eds), Quantum Chaos (Cambridge Univer-
sity Press, Cambridge, 1995). 2. R. Ketzmerick, Phys.Rev. B 54, 10841 (1996). 3. H. Hegger, B. Huckenstein, K. Hecker, M. Janssen, A. Freimuth, G. Reckziegel and R. Tuzinski, Phys.Rev.Lett. 77,3885 (1996); AS. Sachrajda, R. Ketzmerick, C. Gould, Y. Feng, P.J. Kelly, A. Delage and Z. Wasilewski, Phys.Rev.Lett. 80, 1948 (1998). 4. G. Casati, I. Guarneri and G. Maspero, Phys.Rev.Lett. 84, 63 (2000). 5. J.P. Bouchaud and A. Georges, Phys.Rep. 195, 127 (1990). 6. S. Benkadda, S. Kassibrakis, R.B. White and G.M. Zaslavsky, Phys.Rev. E55, 4909 (1997); P. Castiglione, A. Mazzino, P. Muratore-Ginanneschi and A. Vulpiani, Physica D134, 75 (1999). 7. A.J. Lichtenberg and M.A. Lieberman, Regular and Stochastic Motion (Springer-Verlag, New York, 1983); J.D. Meiss, Rev.Mod.Phys. 64, 795 (1992). 8. J.M. Greene, J.Math.Phys. 20, 1183 (1979). 9. J.R. Cary, J.D. Meiss and A. Bhattacharjee, Phys.Rev. A23,2744 (1981); A.B. Rechester, M.N. Rosenbluth and R.B. White, Phys.Rev. A23, 2644 (1981). 10. G.M. Zaslavsky, M. Edelman and B.A. Niyazov, Chaos 7,159 (1997). 11. T. Geisel, J. Nierwetberg and A. Zacherl, Phys.Rev.Lett. 54, 616 (1985); R. Artuso, G. Casati and R. Lombardi, Phys.Rev.Lett. 71, 62 (1993); P. Cvitanovib, R. Artuso, R. Mainieri, G. Tanner and G. Vattay, Classical and Quantum Chaos, www .nbi .dk/ChaosBook/,Niels Bohr Institute (Copenhagen 2001). 12. G. Mantica, J.Phys. IV France 8, (1998). 13. B.V. Chirikov and D.L. Shepelyansky, Physica D13, 395 (1984); C.F.F. Karney, Physica D8, 360 (1983); R. Artuso, Physica D131, 68 (1999). 14. F. Christiansen and A. Politi, Nonlinearity 9, 1623 (1996); P.-M. Binder and J.M. Pedroza, Phys.Rev. E62, R5883 (2000). 15. V. Baladi, J.-P. Eckmann and D. Ruelle, Nonlinearity 2, 119 (1989). 16. R. Artuso, G. Casati and I. Guarneri, J.Stat.Phys 83, 145 (1996); P. Dahlqvist and R. Artuso, Phys.Lett. A219, 212 (1996). 17. P.M. Bleher, J.Stat.Phys. 66, 315 (1992). 18. F.M. Izrailev, Phys.Rep. 196, 299 (1990); F.M. Izrailev and D.L. S h e p elyansky, Sov.Phys.Dok1. 24, 996 (1979). 19. G. Casati, B.V. Chirikov, J. Ford and F.M. Izrailev, in Springer Lecture Notes in Physics, 93, 334 (1979).
135
20. S.-J. Chang and K.-J. Shi, Phys.Rev. A34, 7 (1986). 21. S. Fishman, D.R. Grempel and R.E. Prange, Phys.Rev.Lett. 49, 508 (1982). 22. A. Altland and M.R. Zirnbauer, Phys.Rev.Lett. 76, 3420 (1996); see also the Comment by G. Casati, F.M. Izrailev and V.V. Sokolov, Phys.Reu.Lett. 80, 640 (1998) and the authors’ reply, Phys.Rev.Lett. 80, 641 (1998). 23. B.V. Chirikov, F.M. Izrailev and D.L. Shepelyansky, Sov.Sci.Rew. Sect. A2, 209 (1981). 24. D.J. Thouless, Phys.Rep. 13, 93 (1974). 25. B. Sundaram and G.M. Zaslavsky, Phys.Rev. E59, 7231 (1999). 26. R. Artuso and M. Rusconi, Phys.Rev. E64, 015204(R) (2001). 27. T. Dittrich and U. Smilansky, Nonlinearity 4, 59 (1991). 28. G.P. Berman and G.M. Zaslavsky, Physica A91, 450 (1978). 29. A. Iomin and G.M. Zaslavsky, Phys.Rev. E63, 047203 (2001). 30. Y.-C. Lai, E. Ott and C. Grebogi, Phys.Lett. A173, 148 (1993). 31. J.D. Hanson, E. Ott andT.M. Antonsen, Jr., Phys.Rev. A29, 819 (1984); T. Geisel, G. Radons and J. Rubner, Phys.Rev.Lett. 57, 2883 (1986); R. Artuso and M. Rusconi, unpublished. 32. M. Stefancich, P. Allegrini, L. Bonci, P. Grigolini and B.J. West, Phys.Rev. E57, 6625 (1998); S. Fishman, private communication.
136
QUANTUM TRANSPORT IN QUANTUM BILLIARDS: FROM KELVIN THROUGH ARNOLD KATSUHIRO NAKAMURA Department of Applied Physics, Osaka City University, Sumiyoshi-b, Osaka 558-8585, Japan E-mail: nakamuraoa-phys.eng.osaka-cu.ac.jp Dynamics of billiard balls and its role in physics have received a wide attention since the monumental lecture by Lord Kelvin at the turn of the 19-th century. Billiards can nowadays be created as quantum dots or antidots in the microscopic world, and one can envisage the quantum-mechanical manifestation of chaos of billiard balls. We show two interesting themes bridging between nonlinear dynamics and quantum transport in these mesoscopic billiards: For antidot lattices, the experimentally-observed anomalous fluctuations in the magneto-resistivity are attributed to orbit bifurcations; for 3-dimensional quantum dots, the Arnold diffusion is pointed out to have a possibility to yield the enhanced magnet-resistance beyond the weak localization correction.
1
Birth of physics of billiards
Dynamics of billiard balls and its role in physics have come to receive a very wide attention since the monumental lecture by Lord Kelvin at the turn of the 19-th century. We shall begin with a brief review of his lecture. On April 27(Fri.), 1900, at Royal Institution of Great Britain, he delivered a lecture entitled as ”The 19-th century clouds over the dynamical theory of heat and light” . The first cloud was a question on the existence of ether propagating the light. He denied a possibility of the earth to move through the ether. The second one was a question on the validity of Maxwell-Boltzmann (MB) distribution leading to the equi-partition of energy and he ultimately doubted the ergodicity hypothesis behind MB distribution. Five years later after Kelvin’s lecture, the first cloud was swept away by Einstein’s ”special theory of relativity”. By the way, how did the second cloud disappear? The ergodicity hypothesis means an assumption that a long time average of a given physical quantity should accord with its phase-space average. Choosing as an example the ideal gas consisting of atoms with no internal degree of freedom, Lord Kelvin addressed a discrepancy of the ratio of two kinds of its specific heats(at constant pressure and at constant volume) between the theoretical issue predicted by the equi-partition of energy and the experimentally-observed value. Noting further this discrepancy to be enhanced for molecules with rotational degrees of freedom as well as translational
137
Figure 1. Kelvin’s billiards: (a) triangle; (b) flower.
ones, he insisted on a breakdown of the ergodicity ansatz. To demonstrate more explicitly the breakdown of ergodicity hypothesis, Kelvin investigated a point-particle motion bouncing with the hard wall of a triangular billiard (see Fig.1 (a)). Measuring each line segment between successive bouncings and each reflection angle at the wall repeatedly, he showed the breakdown of equi-partition of energy, i.e., inequivalence between longtime averages of transverse and perpendicular components of kinetic energy. Next, he chose a flower-like billiard in Fig.l(b), carried out a similar pursuit, and again showed the long-time averages of radial and angular parts of kinetic energy not to satisfy the equi-partition of energy. This investigation implies a birth of physics of billiards. Physics of billiards was thus launched on April 27, 1900. Hence, in order to sweep the 19-th cloud over ergodicity hypothesis, it had become indispensable to envisage complex features of nonlinear dynamics of a particle in billiards. In particular, an accumulation of studies on billiards (by Birkhoff, Krylov, Sinai and others) during the 20-th century since Kelvin’s lecture were devoted to those on nonintegrable and chmtic billiards with the shape like in Fig.l(b). In fact, concave and convex billiards as prototypes of conservative chaotic systems have received a growing theoretical and experimental interest in the fields of nonlinear dynamics and statistical mechanics [l].Dynamics of a billiard ball is chmtic, i.e., extremely sensitive to initial conditions: a very slight variation in initial coordinates or momenta yields a thoroughly different orbit. The sensitivity to initial conditions causes a cluster of initial points with similar initial conditions to exhibit mixing in phase space as time elapses, and thereby to show an ergodic property. In this way, chmtic
138 billiards like in Fig.l(b) have resolved the second cloud of Kelvin, and provide an essential playground by which to consolidate the foundation of statistical mechanics. Since new turning years around 1990, the physics of billiards has developed in every direction of science and technology. Billiards are nowadays fabricated as q u a n t u m dots or antidots in ballistic microstructures where the ~ and larger than system size is much less than the mean free path l ( 20pm) the Fermi wavelength (A 50nm)[2]. One can envisage quantum-mechanical manifestations of chaos of billiard balls (: electrons)[3]. Many puzzling experiments on resistance fluctuations in these quantum billiards are raising a fancy of exploring the effect of billiard-ball dynamics on ballistic quantum transport. In the following, we shall demonstrate two interesting examples. N
2
Orbit bifurcations in triangular antidot lattices
The antidot lattice is the two-dimensional electron gas(2DEG) at GaAs/Al,Gal-,As heterostructure with imposed periodically- arranged potential peaks. The motion of electrons can be considered as ballistic and semiclassical. In recent experiments, rich phenomena of magnetoresistivity revealed complicated classical trajectories of the electron moving between antidots [4].In a later experiment, triangular antidot lattices were fabricated into a ballistic regime where the sample size was less than the length of mean free path, and some novel phenomena are found [5]. In this experiment, the mean free path d 20pm, size of sample L 15pm and the period a 0.2pm, so the electronic motion between antidots can be considered as ballistic. Moreover, as the radius of etched hole d a/2, the magnetoresistivity is indicated to reflect properties of the chaotic orbits in the presence of low but intermediate magnetic field B less than 0.5 Tesla. The result of the experiment [5] is displayed in Fig. 2(a), where the magnetoresistivity at different temperatures shows a monotonic decrease with respect to B and is accompanied by several distinguished peaks, which looks to have a periodicity of almost h/2e divided by the unit-cell area of the lattice. We shall systematically investigate the puzzling issue of the above experiment from the viewpoint of orbit bifurcations in nonlinear dynamics. A schematic illustration of trangular antidots is given in Fig.3. The Hamilte nian describing the electron dynamics reads fi = &(P :A(?))' U(i), where A(i), U ( i ) and m* are the vector potential, the potential of triangular antidots and the effective mass of electron, respectively. For each of antidots, we use the soft-wall potential U ( T ) = EF(T/TO - 1 - 5)' for T 5 ro(1 s) and U ( T )= 0 for T > r o ( 1 s) where T is the distance of electron from the N
N
N
N
+
+
+
+
139
i
@)
- _- - - _ _ _ 0
0.05
0.1 0.15 B (Tala)
0.2
0.00
0.05
0.10 0.15 B flesh)
0.20
Figure 2. (a) Magneto-resistivity p,,(B) of triangular antidot lattice at different temperatures. Solid, dotted, dashed, dot-dashed, and 2dots-dashed lines correspond to T = 0.07,0.31,0.6,1.0and 1.4K, respectively; (b) Computed results for p z I ( B ) at T = 0.07K. Solid line is the experimental result. Dotted line is the theoretical result from Eq. (7) excluding the effect of bifurcation of P.0.s. A series of circle symbols is theoretical result including the effect of bifurcation of P.0.s.
+
center of each antidot (at nuel maea with n,m integers and el = ( 1 , O ) and e2 = (1/2, &/2)), EF is Fermi energy, and TO and s are other potential parameters[6]. 2.1
Semiclassical conductivity
The semiclassical theory can describe Kubo formula for the ballistic quantum transport[7] . In the very strong field, electrons will move only along the Landau orbit and the magnetoresistivity obviously shows the Shubnikov-de Haas oscillation. On the contrary we shall here concentrate ourselves on the low but intermediate field regime from zero t o 0.3 Tesla. Commensurate cyclotron orbits [4]in this field regime, which have relatively large radii comparable to the sample size, can not contribute t o the quantum transport because of the lack of coherence and the finite temperature effect. So we can conclude that
140
Figure 3. Left panel: Schematic illustration of the triangular antidot lattice and typical periodic orbits. While P.0.s 1,2,5-7 are completely confined by surrounding antidots, the confinement of P.0.s 3 and 4 is incomplete and polygons depicted by these orbits include corners with obtuse angle outside the confining region; Right panel: Example of tangent bifurcations induced by varying the B field.
the conductivity fluctuations in [5]can be explained neither by the Shubnikovde Haas oscillation nor the commensurate cyclotron orbit theory. Noting the period of lattice is only 0.2pm [5],we have to consider the change of periodic orbits (P.0.s) which are shorter than the sample size. In the semiclassical formalism, czz for chaotic systems is given as a sum of the smooth part co and the oscillation part b e described by P.0.s [7]:
141
where N ( 0 ) is the mean density of states at EF,and < . . . >pr implies 0 , Mpo are the the phase space average. gs(= 2 ) , A , L p o , 7 , V ~ , S p o , ~ pand spin factor, area of the system, the length of P.O., scattering time, Fermi velocity, action of P.O., Maslov index and stability matrix, respectively. R(LPo/LT) = sinh[L,,/LT) with thermal cutoff length LT = V ~ h / ( x l c ~ T ) is a thermal dampimg factor. The velocity-correlation function reads O I L T
with rpothe period of P.O.. It should be pointed out that Eq.(3) is only valid for completely chaotic systems in which all P.0.s are isolated. 2.2
Quantum correction without orbit bifurcations
The triangular antidot lattice under consideration is completely chaotic in the B field less than 0.5 Tesla, i.e., in the low but intermediate B field. In other words, the measure of chaotic orbit is unity (:loo%) and does not change with the magnetic field. Then the smooth part 'a is determined by chaotic scattering orbits, which we shall calculate first. In Eq.(2), it is enough to do the integration from t = 0 to the time of the first collision, because after the collision the electron will lose its memory and give no contribution to the integration. So the integration in Eq.(2) will be replaced by V; dt < >pr e-tfi e - t / r - iV;/($ $), where eVt/' is the
Jr 9
+
-
probability for the electron to show a free flight up to the first collision time >pr over the direction gives a factor d/VF and the averaging of < VF 1/2. Although in Eq.(2) N ( 0 ) l / h2, other factors there are very small, so we will see the magnitude of a:, is comparable to that of ha,,. Using the experimental values [5] , we obtain a:, = 1.94 x 10-4R-'. Subtracting a:, from the experimental value u,,(B = 0) x l/p,,(B = 0) obtained from Fig.2, we can evaluate ha,,(B = 0) = -1.61 x 10-4(negative value!) at 0.07K. The B field does not enter into the mean density of states. Moreover, the system remains fully chaotic, keeping the portion of scattering orbits unchanged with respect to B field up to B = 0.5 Tesla. Therefore a:, should be independent of B field and we should proceed to investigate the influence of B field on ha,, by using the semiclassical P.O. theory. To simplify our analysis, we first investigate B dependence of the averaged magnitude (variance) of ho,, in the diagonal approximation, and then the precise structure of fluctuations. In the experimental temperature region T >> h 2 / ( 2 m * L 2 k ~ ) 0.01K with L the characteristic length of the system,
t
N
N
142
the thermal fluctuations smear out fine structures on the energy scale of mean level spacing, and the asymptotic form R(Lp,) x 2(Lp,/LT) exp(-L,,/LT) can be employed. Under these circumstances, the variance < 602, > in , 2 Eq.(5) can be divided into two parts < ha:, >= 6a,,2+ < &a,, > : the r 2 B-dependent part < 6ux, > reads
x cos(2eBAp,/fi), -
(5) J
2
while the B-independent part is 6uXx2 =< 6uxx(B = 0) >= Sa:,(B = 0)/2; A,, is the area enclosed by P.O., taking a positive (negative) value for a clockwise (counter-clockwise) P.O.; Noting the secondary integration over t in Eq.(4) to yield almost the same value for different P.O.s, the averaged value C ,; is used to replace C$i2. As long as there is no degeneracy among P.O.s, Eq.(3) is valid and we can use the Hannay and Ozorio de Almeida (H-OdA) sum rule for P.0.s (81, i.e., Cpo l/ldet(kpo - 1)1... = J,”dL/L... . In Eq.(5), cos2(igp0Pdr - qP07r/2) = 1/2 cos(i IpoPdr - qp07r)/2 , but using HOdA sum rule, cos(g PO Pdr - qp0r)vanishes since the terms of even and odd Maslov numbers will cancel each other, and only the average part can survive. Then, substituting into Eq.(5) the probability distribution of the and, area A enclosed by P.O. with length L, PL(A) = &exp(-&) integrating it, we have
I
with Gamma function the conductivity as
+
r, arriving at the amplitude of the oscillation part of
Soxx(B) = Sa,,(B
= 0).\/1/2+
<
> /6uXx2(B = 0).
(7)
and the negative value 6u,,(B = 0) derived already, Using in Eq.(7) we evaluate resistivity p,,(B) x l/(u:, 6a,,(B))a t T = 0.07K, yielding the result plotted in dotted line in Fig.2(b), where the monotonic decrease of resistivity predicted from Eqs.(6) and (7) fits the experimental result well except at several distinct peaks. Equations (6) and (7) also show the higher
+
143
the temperature, the more slowly p,, decreases, the same feature as found in Fig.S(a). But it should be noticed that there are several anomalous peaks on the experimental curve, lying at B = 0.05,0.105 and 0.155Tesla.
2.3 Orbit bifurcations and anomalous resistivity fluctuations As mentioned before, Eq.(3) is valid only when all P.0.s are isolated. In the low but intermediate field regime where there is no degenerate family of P.O.s, the B field will bend P.0.s and some P.0.s will overlap each other at some special values of B, where the bifurcation of orbit occurs. At these points TrMpo = 2, namely, Idet(&fpo - 1)l = 0 and the amplitude factor in Eq.(3) becomes divergent. At bifurcations, the uniform approximation should be employed [9]. From Eq.(3), one finds that only the bifurcation of several shortest orbits is needed since that of long orbits is suppressed by finite temperature. To judge whether the bifurcation occurs, we should compute 4 x 4 stability matrix Mpo for P.0.s because TrMpo = TraPo 2=4 at bifurcations. The definition for Mpo reads Mpo = M ( t = Tpo)which characterizes a growth of the variation 6; ( t ) = Mb-y(t = 0); the equation of
+
motion for M is
% = J ~ J Y F D ( twhere )M,J
=
I:(
,!)
with I the 2 x 2
unit matrix and y denotes a pair of conjugate coordinates, (r, p - :A). The initial A4 is a unit matrix since the unit displacement is conceived at t = 0. From Fig.3, one find that P.0.2 will be bent by magnetic field and become to overlap with P.O.l, and the similar phenomenon occurs between P.0.s 3 and 4. When the symmetry of P.0.5 becomes higher, it will overlap with P.0.6. Furthermore, P.0.6 will change into P.0.7 at a certain special value of B. Taking the potential parameters TO = 0.093pm and s = 0.35, we compute TrM for these P.0.s and display the results in Fig.4. From Fig.4(a), we find that P.0.s 1 and 2 merge each other and simultaneously disappear at B = 0.0574T[Tesla], and in Fig4.(b), so do P.0.s 3 and 4 at B = 0.1087T. Therefore they show tangent bifurcations. Moreover, P.0.s 5 and 6 in Fig.4(c) merge each other and only P.0.6 survives at B = 0.144T; subsequently P.0.6 meets P.0.7 at B = 0.152T and only P.0.7 survives. So they show two successive pitchfork bifurcations. We should emphasize that each of P.0.s shown in Fig.3 corresponds to a pair of bent P.0.s under the B field. One is its inflation toward outside, and the other is its contraction toward inside. Their direc:ions of electronic motion are opposite. Here only the inflation orbits play an irr.portant role in the bifurcation of orbits and the contraction orbits, which are very stable, are irrelevant. Computing all necessary variables and using the uniform approximation [9], we can derive the new oscillation part ba,, at
144
g
-150
-300 ? O
-450
I
0.052
0.050
0.054 B (Tcsla)
0.056
0.058
(b)
Po4
- 1500
1
m3 I
*! ++
0.104
..
0.106 B f?ksla)
0.108
0.15 B (Tala)
0.16
Po5
40
0.14
Figure 4. Values of T r M : (a) P.0.s 1 (triangle) and 2 (square); (b) P.0.s 3 (square) and 4 (triangle); (c) P.0.s 5 (cross), 6 (square) and 7 ( downward triangle). Horizontal line represents TrM = 4. Magnitude of TrM is very much affected by the geometric nature of P.O., and the anomalously large scale of ordinate in Fig.l(b) is due to the incomplete confinement of P.0.s 3 and 4.
145
these bifurcations as: 6aXx & R ( L p o / L T ) C ~cos(Spo/tL ~ - qp07r/2) at the tangent bifurcations and &axx &R(Lpo/LT)Cig cos(Sp,/tL - qP07r/2) at the pitchfork bifurcations, which are much larger than contributions of O(1/6) from each of isolated P.0.s in Eq.(3). Incorporating the interference terms leading to the above anomalous corrections, the calculated results for the resistivity p x x = 1/(g2x +6axx(B)) are plotted in a series of circle symbols in Fig.Z(b), where the minima of 6oxx, which are responsible to the bifurcation points, correspond to the three peaks in p x x observed in the experiment [5]. Obviously, the lengths of P.0.s 1 and 2 at the bifurcation point are the shortest. Since the stronger magnetic field leads to the larger inflation of P.O., the orbits at the bifurcation in Fig.4(c) are the longest and the ones in Fig.4(b) are intermediate. Bifurcation of other longer P.0.s are suppressed due to the finite temperature effect. One can find that the first peak at B = 0.05T is the highest, the one at 0.1T is the second highest, and the peak at 0.15T is the lowest. One can also predict that the higher temperature suppresses the height of these peaks. Both the locations and heights of all these peaks are in very good agreement with the experimental issue in Fig.2(a). It should be noted: (i) a deformation of the potential for each of antidots does not affect positions of bifurcation because these positions are determined by the symmetry of P.0.s in Fig.3; (ii) the anomalously large oscillations in resistivity analyzed here may be observed only in the ballistic regime. The observed small oscillation in resistivity in diffusive square antidot lattices was attributed to the change of the level density of systems with respect to B field [4], which corresponds to the change of phase factors in nondiagonal terms beyond Eq. (5). N
N
3
Arnold diffusion and enhanced magneto-resistance in 3-d quantum dots
Let us move to the second subject. Since the striking experiments on the magneto-conductance of nanoscale stadium and circle billiards with a pair of conducting leads [2], the semiclassical conductance formula was derived for two-dimensional (2-d) chaotic billiards by combining the semiclassical Green function with the Landauer formula [lo], and some salient features of experiments well agreed with the semiclassical theory. Subsequently, the semiclassical conductance formula was extended for integrable 2-d billiards [ll]. Recently, the dependence of the conductance on the lead orientation was found experimentally in a ballistic square billiard [12].
146
At present, another important object of research on quantum transport is the three-dimensional (3-d) billiard [13]. The ballistic 3-d quantum dots will be fabricated, for instance, by exploiting drying etching processess with focused-ion beams applied to Al,Gal-, As/GaAs/AI,Gal-,As double heterostructures, which will be less than both the elastic mean free path and phase coherence length. In the remaining half of this review, we shall extend the semiclassical conductance formula for 2-d billiards to 3-d billiards and investigate the quantum transport in open 3-d billiards from a viewpoint of the Arnold diffusion, a key concept in high-dimensional nonlinear dynamical systems. While Arnold diffusion is a classical phenomenon, an anomalous phenomenon like this in classical mechanics should have a quantum counterpart, which can only be analytically revealed by using the semiclassical theory. 3.1 Semiclassical conductance for open 3-dimensional billiards
The conductance for open 3-d billiards connected to a pair of rectangular parallel-piped lead wires can be expressed by the transmission coefficient T as G=2ez= ~ 2e2 Cn,m=l Itnm12, where t,, is the S-matrix element between h the incoming mode m = ( m z my) , and the outgoing mode n = (n5,ny),and the double summation is taken over all propagating modes. By extending the formula for 2-d open billiards [14], the transmission amplitude connecting those modes at the Fermi energy EF reads as
(8)
where u, and un are the longitudinal velo2ities of electrons for the incoming and outgoing modes, respectively. x’,y ,z‘ and x ,y ,z are the local coordinates for the transverse (x, y) and longitudinal ( z ) directions (inward the billiard for z’ and outward fo’; z y ) of the incoming and outgoing leads, ,, respectively, and G(x ,y ,z ;x ,y ,z ; E p ) is the Green function for an electron propagating from the entrance to the exit. +m(x’,y)) and $ ~ ~ ( x ” , y ) ‘ ) are the transverse components of the wave functions at the leads. To simplify the problem, the cross sectiqn pf the leads is ass,umed to be square with side length 1. Then gm(x ,y ) = sin(rnz7rx / I ) sin(m,ny / 1 ) and &(;,’ y”) = sin(nz7rx”/I) sin(nZ7ry”/l). In the ballistic regime, the Green function from entrance (i)to exit (1”)can be replaced by the Gutzwiller semiclassical path-integral expression I,
It
3
,I
I
3
,I
,I
147
i
,I
I
, r , E F )- i ~ ~ , / 2 ) , (9)
x exp(-S,(r TL
where S, and us are the action and Maslov index for a trajectory s, and 'UF is the Fermi velocity. Here, to avoid direct transmission without any bouncing at the billiard wall we may place a stopper inside the billiard. In the integration in Eq.(8) we introduce the entrance angle 0' between the initial direction of the orbit and the z' axi;, an$ the angle b$ween the projection of the initial direction onto the x - y plane and the x axis. Angles 0" and cp" are defined for the exit in the same way. Then, we can
6
rewrite 1 det($))
,
+
for each trajectory s as I det(
a(P" .P2)
)
)I
= P%
where
A , = Is($$$ - $$)I. For large mode numbers, the integration over the transverse coordinates in Eq.(8) can be performed in the stationaryphase approximation, and the transmission amplitude becomes
with
B,
=
ad' a@" I(sin2cpI , (-?T--r~os2@ a y ax I,
,,
"
ay" ax"
acp" ad'
-
I
I
ay
ax
-ay" ax,,sin2
$41
,
' - -acp acp sin2@')
x [sinzp (,,ca@' os28 a@' ay ax
+sin28 (7,cos2cp
ae" acp"
,,
+sin28 (---r;-cos2(p ay ax
acp' a@'
I!
acp dP sin2 0") - --
ay' ax'
I
- 77sin2cp')ll a y ax
,
(11)
where mx = fm,,fi, = &my, fi, = fnz,fi, = *nV, and 3, = S(r ,, r s, E F ) h m x x ' , / l + h.rrm,yb/l- h n , x p / l - tirrfi,y~/l, and V is the Maslov index which includes an extra phase coming from the possible sign change in each of the four Fresnel integrals. The above result means that only those isokted tfajectories connec4ing a pair of transverse p!anes $,discrete angles sin @ cos cp = m,.rr/kpl, sin 8 sin cp = m Y ? ? / k F 1 , sin @ cos cp = n z T / k F 1 , and sin 0'' sin 9'' = f i , T / k F l dominate the conductance. Using Eq.(lO), we shall first analyze a completely chaotic 3-d billiard with a pair of conducting leads. In this case, the electron injected from the incoming lead will bounce at the billiard boundary in an ergodic way before reaching the I!
!
+
148
exit, leading to amplitudes of the same order for both transmission and reflection coefficients. In the large mode number case, the summations over modes are replaced by integrations and the transmission coefficient T = Itnrn[' is rewritten as
En,,
T
256ki
-p-
1,
d&
sinO'd(sin0') -7T
J 1 sinO"d(sinO") -1
1:
d&' C c o s O ' cosO" S,U
Noting the dimension of ( $&)l/' to be l', and using the diagonal approximation, the above integration can be evaluated as T c( ( k ~ l ) ' .The conductance for 3-d chaotic billiards is thus given by G 0: g(kF2)' without any dependence on the lead orientations. 3.2
Completely or partially broken-ergodic 3-d billiards
However, the situation will be dramatically changed for the case of completely or partially broken-ergodic 3-d billiards. As examples, we choose S0(2)-symmetric billiards: A completely integrable (: broken-ergodic) 3-d billiard is available by rotating, e.g., the 2-d ellipse billiard (on the Y - 2 plane) around the 2 axis (see Fig.5). Note: X , Y ,2 are the global coordinates; A partially broken-ergodic 3-d billiard is also available in a similar way by rotating, e.g., the 2-d stadium. All these SO(2)-symmetric 3-d billiards have the angular momentum LZ as a constant of motion. In an open system version of each of these billiards, one may place in its inside a smaller stopper of the similar shape, to avoid direct transmission of electrons. The resultant 3-d shell billiards still retain the SO(2) symmetry. Firstly, consider the vertical case (i) when the incoming and outgoing leads are connected with the billiard vertically at points A and C , respectively (see Fig.5). Electrons to reach the exit C should have the vanishing angular mcmentum, Lz = 0. Therefore, only the electrons with the initial velocity vector at the entrance A lying in the Y - 2 plane can reach C because of the conservation of L z ; Other incoming electrons falling into the trajectories out of this plane cannot reach C and should return to A . In deriving the transmission coefficient for this lead orientation, the integration prior to Eq.(11) is reduced to the line integration over 0' and 0" only, yielding the conductance G 0: $ k ~ 1 (linear in k F ! ) . (The reflection coefficient, which is obtained by applying the same procedure as in the chaotic billiard, is proportional to ( k d ) ' . ) We shall
149
2 A
- +) YY
rs
riem
..
Entrance Q Figure 5. Schematic illustration of the 3-d elliptic shell billiard. Dual 2-d ellipses sharing the same foci in Y - 2 plane are rotated around the Z axis to generate a 3-d elliptic shell billiard. The longer radii of outer and inner ellipses are taken as 2 8 and 1.858. The shorter radius of the outer ellipse is 0.8%. The side length 1 of the square lead is taken as R/20.
then consider the parallel case (ii) when the incoming and outgoing leads are parallelly placed at points Q and C , respectively (see Fig.5). Each electron incoming from the entrance Q has the vanishing angular momentum, Lz = 0, and thereby can reach C or return to Q with probability of the same order. Therefore, for this lead orientation, G oc g ( k p 1 ) ’ . Generally, in completely or partially broken-ergodic 3-d billiards, we have G 0: g ( k ~ 1 ) Ywith 0 < y 5 2, and the exponent y is determined by how the billiard symmetry is broken by the lead orientation. It should be noted: open 2-d billiards always share G c( c k p 1 [lo], independently of both integrability and lead orientations.
3.3 Effects of symmetry-breaking weak magnetic field The magneto-resistance is also a very important problem in 3-d billiards. It is well known that in ballistic 2-d billiards the decrease of resistance with increasing magnetic field (: negative magneto-resistance) occurs due to the
150 weak localization based on the quantum interference between a pair of timereversal symmetric orbits[lO]. What kind of additional novel phenomena will be expected in SO(2)-symmetric 3-d billiards with the vertical lead ( A ,C) orientation when a symmetry-breaking weak magnetic field B will be applied along the X axis? In contrast t o phase-space structures in 2-d systems where each of chaotic zones generated by weak B field is mutually separated by the KolmogorovArnold-Moser (KAM) tori, those in 3-d systems consist of chaotic zones mutually connected via narrow chaotic channels called the Arnold web (AW) [15]. In this case, electrons incoming in an arbitrary direction, which permeate (chaotic) zone by zone in an ergodic way through the AW, can show a global diffusion, i.e., Arnold diffusion (AD). Therefore, a B field increases the transmission channels and the number of stationary points in ,the integration in Eq.(lO), and the resultant integration in Eq.(12) over cp and as well as 0’ and 0” indicates G 0: $ ( k ~ l ) ’ . In general we shall expect a more general result, G 0: ( k ~ 1 ) Owith 1 < ,Ll 5 2, and the exponent p depends on the width of AW. A crucial point is that a B field has increased the exponent in the power-law behavior of the semiclassical conductance in case of the vertical lead orientation. This anomalous phenomenon is just the negative magneto-resistance beyond the weak localization. We here keep to employ the terminology of the negative magneto - resistance wherever a B field reduces the zero-field resistance. On the other hand, for the parallel lead ( Q , C ) orientation where each ~ electronic trajectory lies almost in the vertical plane, G 0: $ ( k ~ l is) retained, irrespective of the absence or presence of B field; We can expect only the B field-induced increment of the proportionality constant, which implies a conventional negative magneto-resistance caused by suppression of the weak localization. Originally, the negative magneto-resistance was evidenced in the numerical analysis of the conductance of 2-d quantum dots [lo]. The accompanying theory, based on the resistance, manipulated backscattering orbits and attributed this phenomenon to the weak localization [lo]. Later works [16] based on the conductance, which introduced the small-angle-induced diffraction effect together with an idea of interference between a pair of partially time-reversal symmetric orbits [16], swept away a cloud (i.e., breakdown of unitarity) hanging over the above theory.
/
151
3.4
Numerical results for open 3-d elliptic billiard
To numerically verify the above prediction, we shall choose a S0(2)-symmetric 3-d elliptic shell billiard with the vertical lead orientation. This ellipsoid is a typical example of completely integrable S0(2)-symmetric 3-d billards. The system in the presence of a symmetry-breaking field B parallel to X axis will be examined. To understand the classical dynamics of electrons, we analyze as a PoincarC surface of section the X - 2 plane with longitude S and velocity component wz/uo as Birkhoff coordinates. (uo = is the velocity unit corresponding to the lowest mode.) For B = 0, the trajectory from the entrance A with an arbitrary initial velocity vector lying out of 2 - Y plane proves to be confined to a torus (see Fig.G(a)), failing to reach the exit C , i.e., S = ~ / 2 .Let r/ (E8 / W c ) be the ratio between the cyclotron radius !Rc and the characteristic length 8 of the cross-sectional ellipse (see the caption of Fig.5). Then B field can be expressed as B = r/Bo with Bo = mewo/e8 (Bo corresponds to 8).In case of a weak field ( B = 0.02Bo), Poincar6 sections for the trajectory with the same initial velocity vector as in Fig.G(a) are given in Figs.G(b) and (c) for increasing numbers of bouncings at the outer billiard wall, and similarly those for the trajectory with the initial velocity vector lying in the X - Y plane are given in Figs.G(d)-(f). One can find: as the B field is switched on, the orbit trajectory first leaves the initial torus, enters an outer layer, and, repeating a similar process, diffuses over more and more distant layers, exhibiting a phenomenon of the Arnold diffusion (AD). Even though a symmetry-breaking field is weak enough to keep the orbit almost straight, AD makes it possible for the trajectories with initial angular momenta Lz # 0 to reach the exit C , i.e., S = 7r/2 (see Figs.G(c) and (f)) and eventually to contribute to conductance. To proceed to the numerical calculation of semiclassical transmission amplitudes in Eq.(lO) for the 3-d elliptic billiard with the vertical lead orientation, let us define the 4 x 4 monodromy matrix M as
(i’i) PI
M
( ”A).
At the entrance we introduce a pair of mu‘P, tually orthogonal coordinates and r f lying in the transverse plane =
perpendicular to the initial direction of each orbit.
Similarly r i
and
I,
are chosen at the exit. Using elements ( M i j ) of M matrix, we rewrite the necessary factors for the numerical computation. For exam-
ri
ae” pie, 2 sin$‘Mz2)
a(e”,y”)
=
a(r1’
,T?’)
,, ,,
a(ri’,rt’)
9r: ) a(r;” , r : ’ / ) a(z” ?Y”)
- M42(COS(p”
cose”M11
-
=
L [ M ~ ~ cose”Mlz ( ~ ~ ~- ~ ” PF
sin(p”Mz1)]/(M11M~z- MIZMZI) and
152
Figure 6. Poincard sections ( X - Z plane) for the trajectories emanating from point A . Longitude S (-7r/2 5 S 5 ~ / 2 )and velocity component vz/vo (vo = &) are chosen as Birkhoff coordinates. S = 0 and S = s / 2 imply entrance A and exit C, respectively. (a) B = 0 and initial velocity v x / v 0 = 0.821,vy/v0 = 0.549,vz/vo = 0.157; (b),(c) B = 0.02Bo and the same initial velocity as (a): ( b ) P = 500; ( c ) P = 17000. (d)-(f) B = 0.02Bo and initial velocity v x / v o = 0.843,vy/v0 = 0.538,vz/v0 = 0: ( d ) P = 500; (e)P = 4100; (f)P= 16000. P is the number of bouncings at the outer billiard intersecting X - Z plane. For the meaning of 50, see the text.
153
8
10
12
14
16
IS
20
k,Jl K
Figure 7. Numerical results for semiclassical conductance versus Fermi wavenumber for A, C leads orientation. Circle and square symbols are for B = 0.0ZBo and B = 0, respectively. Solid quadratic and dashed linear lines are the corresponding fitted curves for their means.
I
first and last rotation angles in M should be determined by using r l
, r:
and r: , r i chosen as above, since, after folding the orbit, the initial and final coordinates should become identical. Combining all these factors with Eq.(lo), the semiclassical conductance is calculated explicitly. The result, which includes both the classical contribution and the quantum correction in an inseparable way, is plotted in Figz7. Although the data show small fluctuations, the fitted curves for their means are described by G 0: $ ( k F l ) 2 and G 0: $ k F l in cases of B = 0.02Bo and B = 0, respectively, confirming the analytic issue suggested in the previous section. The increase of the exponent in the k~ dependence of G upon switching on a weak field, which cannot be seen in 2-d cases, provides a numerical evidence that the negative magnetoresistance (: a difference between two lines in Fig.7) in 3-d systems comes from AD as well as the weak-localization correction. We should note the following:(i)In an extremely weak field case when the width of Arnold web (AW) is thin enough, we shall see G 0: ( k ~ l ) Pwith 1 < /3 < 2. Our numerical calculation in the previous section suggests that,
154
to have ,f3 = 2 in a very large k F regime, %/%, should be no less than (ii) role of a stopper is also essential. Reflection at an inner convex wall should increase the instability of orbits, which is favourable for a genesis of AD; (iii) to clearly observe AD, the symmetry-breaking field should destroy all constants of motion except for the energy. From this viewpoint, it is more advantageous to choose a S0(2)-symmetric 3-d chaotic shell billiard generated from a fullychaotic 2-d billiard with a hollow by its rotation around the 2 axis. The billiards of this kind, which are partially broken-ergodic, are experimentally more accessible than a (completely broken-ergodic) elliptic shell billiard, and all conclusions for the latter should hold for the former as well.
4
Summary and discussions
Within a semiclassical framework, we have first analyzed two parts of conductivity of fully-chaotic triangular antidots in the low but intermediate magnetic field. Taking into account both the smooth classical part evaluated by mean density of states and the oscillation part evaluated by periodic orbits, we find that resistivity of the system yields a monotonic decrease with respect to magnetic field. But when including the effect of orbit bifurcation due to the overlapping between a couple of periodic orbits, several distinguished peaks of resistivity appear. The theoretical results accord with the interesting issue of the recent experiment of NEC group. We have also investigated the semiclassical conductance for threedimensional (3-d) ballistic open billiards. For partially or completely brokenergodic 3-d billiards such as SO(2) symmetric billiards, the dependence of the conductance on the Fermi wavenumber is dramatically changed by the lead orientation. Application of a symmetry-breaking weak magnetic field brings about mixed phase-space structures of 3-d billiards which ensures a novel Arnold diffusion that cannot be seen in 2-d billiards. In contrast to the 2-d case, the anomalous increment of the conductance should inevitably include a contribution arising from Arnold diffusion as well as a weak localization correction. More details of Sections 2 and 3 are given in Refs. [17]. Thus, while classical billiards launched by Kelvin are means by which to verify the foundation of statistical mechanics, quantum billiards (quantum dots and antidots) fabricated by nanotechnology provide stages where to c a p ture via quantum transport quanta1 signatures of orbital bifurcations, Arnold diffusions, and other interesting phenomena in nonlinear dynamics.
155 Acknowledgments This review has emerged from joint works with my former student, Dr. Jun Ma with whom I had enjoyed a very fruitful period. References 1. N. S. Krylov, Works on the Foundation of Statistical Physics (Princeton University Press, Princeton , 1979); Ya. G. Sinai, Russ. Math. Surv. 25, 137 (1970); P. Gaspard and G. Nicolis, Phys. Rev. Lett. 65, 1693 (1990). 2. C. M. Marcus, A. J . Rimberg, R. M. Westervelt, P. F. Hopkins and A. C. Gossard, Phys. Rev. Lett. 69, 506 (1992); A. M. Chang, H. U. Baranger, L. N. Pfeiffer and K. W. West, Phys. Rev. Lett. 73, 2111 (1994). 3. Special issue on Chaos and Quantum Transport (ed. by K. Nakamura): Cham, Solitons and Fractals 8, No.7 and 8 (1997); K. Nakamura, Quantum versus Chaos: Questions Emerging from Mesoscopic Cosmos (Kluwer Academic Publisher, Dordrecht, 1997). 4. D. Weiss et al., Phys. Rev. Lett. 66, 2790 (1991); 70, 4118 (1993). R. Fleischmann, T. Geisel and R. Ketzmerick, Phys. Rev. Lett. 68, 1367 (1992). 5. F. Nihey, S. Hwang and K. Nakamura, Phys. Rev. B 51, 4649 (1995). 6. G. Kirczenov et al., Phys. Rev.B 56, 7503 (1997). 7. G. Hackenbroich and F. von Oppen, Z. Phys. B 97, 157 (1995); Europhys. Lett. 29, 151 (1995); K. Richter, Europhys. Lett. 29, 7 (1995). 8. J . H. Hannay and A. M. Ozorio de Almeida, J. Phys.A 17, 3429 (1984). 9. H. Schomerus and M. Sieber, J. Phys. A 30, 4537 (1997). 10. R. A. Jalabert, H. U. Baranger and A. D. Stone, Phys. Rev. Lett. 65, 2442 (1990). H. U. Baranger, R. A. Jalabert and A. D. Stone, Phys. Rev. Lett. 70, 3876 (1993); Chaos. 3,665 (1993). 11. W. A. Lin and R. V. Jensen, Phys. Rev. B 53, 3638 (1996); W. A. Lin, Chaos, Solitons & Fractals 8, 995 (1997). 12. J. P. Bird, R. Akis, D. K . Ferry, D. Vasileska, J. Cooper, Y. Aoyagi and T. Sugano, Phys. Rev. Lett. 82, 4691 (1999). 13. H. Primack and U. Smilansky, Phys. Rev. Lett. 74, 4831 (1995); Phys. Rep. 327, 1 (2000). 14. D. S. Fisher and P. A. Lee, Phys. Rev. B 23, 6851 (1981); A. D. Stone and A. Szafer, IBM J. Res. Dev. 32, 384 (1988). 15. B. V. Chirikov, Phys. Rep. 52, 263 (1979); A. J. Lichtenberg and M. A. Lieberman, Regular and Stochastic Motion (Springer, New York, 1983);
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G. M. Zaslavsky, R. Z. Sagdeev, D. A. Usikov and A. A. Chernikov, Weak Chaos and Quasi-Regular Patterns (Cambridge University Press, Cambridge, 1991). 16. I. L. Aleiner and A. I. Larkin, Chaos, Solitons & Fractals 8, 1179 (1997); Y. Takane and K. Nakamura, J . Phys. SOC.Jpn. 66, 2977 (1997)). 17. J. Ma and K. Nakamura, Phys. Rev. B 62,13552 (2000); condmat/0108276 (2001); K. Nakamura and J. Ma, J. Phys. SOC. Jpn. 72, No. 1 (in press).
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DYNAMICAL AND ENERGETIC BARRIER TUNNELING IN THE PRESENCE OF CHAOS IN COMPLEX PHASE SPACE A. SHUDO AND T. ONISHI Department of Physics, Tokyo Metropolitan University Minami-Ohsawa, Hachioji, Tokyo 192-0397, Japan E-mail: [email protected], [email protected]
K.S. IKEDA Faculty of Science and Engineering, Ritsumeikan University Noji-cho 1916, Kusatsu 525-0055, Japan E-mail: [email protected]
K. TAKAHASHI The Physics Laboratories, Kyushu institute of Technology Kawazu 680-4, Iizuka 820-8502, Japan E-mail: [email protected] Recent developments of complex semiclassical description of quantum tunneling in non-integrable systems are discussed. In chaotic systems, the hierarchical generation structure is generally formed in the complex phase space and chaos in the complex domain controls the nature of tunneling in multi- dimensions. The organization principle determining the tunneling paths which are relevant to the complex semiclassical description of tunneling is encoded in the self-similar structure of complex phase space. In view of structure of the complex phase space, it is shown that no qualitative difference exists between dynamical and energetic barrier tunneling.
1
Introduction
Phase space of multi-dimensional Hamiltonian systems is generally composed of infinitely many invariant components. Total energy, which is a trivial conserved quantity of Hamiltonian systems, specifies a family of invariant sets in the whole phase spece. In completely integrable systems, even after an energy being fixed, n-dimensional torus forms the invariant subset of (2n - 1)-dimensional phase space specified by the energy, while lower dimensional invariant manifolds can appear in generic non-integrable systems. Chaotic trajectories have the largest dimension as an invariant set, while the periodic orbits have the lowest. Coexistence of qualitatively different ergodic components, which are usually intermingled in a self-similar way in the phase space, characterizes a generic situation which is so complicated that our understanding is far from accomplished. The orbits in classical mechanics are
158
always confined on the corresponding invariant set by definition, in particular, except in case of ideal chaotic systems, there are orbits with positive measure that move only on the limited subspace whose dimension is less than that of the full phase space. On the other hand, the wavepacket of quantum mechanics is not forced to stay on a certain limited classical manifold, but spreads over or share different invariant subsets simultaneously. The spreading is a consequence of the wave effect which is the most marked difference between classical and quantum mechanics. There is not any obstacle in principle preventing the transition between arbitrary two points in the phase space and the quantum wavepacket can penetrate into any kinds of barriers. Such a classically forbidden process does not have no classical counterparts. The penetration into the energy barrier is especially called tunneling, which is understood as the most typical quantum effect and play important roles in many physical and chemical phenomena. However, a s mentioned here, the barrier preventing the classical transition is not limited to the energetic one in the system with multi-dimensions. It was pointed out' that tunneling between the quasi-doublets whose supports are symmetrically formed tori in the phase space should exist in an analogous way as the doublets of a symmetric double well potential. When the localized states associated with the tori appear as the congruent pairs, approximately degenerate pairs are formed in a similar way as the localized states on one side or the other side of an energy barrier. This sort of tunneling carrying a completely multidimensional origin of the system is termed dynamical tunneling'. The simplest case realizing dynamical tunneling is the completely integrable system in which the energy splitting between symmetrically formed congruent tori can explicitly be evaluated. The argument analogous to the one-dimensional systems is possible. A series of works by Wilkinson have been the first systematic attempt to examine what happens if the integrable system, in which invariant tori fill the entire phase space, is perturbed weakly to a nearly integrable regime2. The analysis employing Herring's formula together with continuation of KAM tori into a classically forbidden region derives a non-trivial but smooth h dependence of the energy splittings caused by tunneling between KAM tori. These studies show that even purely wave effect like tunneling strongly reflects the underlying classical dynamics though it has no classical counterpart. We also learn that multi-dimensionality can make the problem considerably rich and complicated. In particular, as the strength of perturbation is increased, chaos begins to dominate and the phase space become a mixture of qualitatively different invariant structures. Since one can regard all the tran-
159
sitions due to the wave effect as quantum tunneling, there can appear different kinds of tunneling according to the variety of underlying classical structures. If one considers the situation where chaotic sea lies between two islands, which are assumed to be symmetrically formed for simplicity, it would be reasonable to consider that approximately three states, the two are the states supporting stable islands, and the other is chaotic one, are coupled to each other in the splitting problem between two island states. This modeling has been used to explain an irregular oscillation of tunneling splitting observed by changing an external parameter of the system. The tunneling process via a route as torus tf chaos tf torus has been called chaos-assisted tunnelzn$i4. If one admits this simple modeling in which the coupling among three relevant states are represented as a 3 x 3 matrix, one can develop a statistical argument of the tunneling splitting associated with the random matrix description4. This situation can actually be prepared in a non-concentric annular billiard problem first introduced in order to study an aspect of classical chaos5. Not only numerical evidence for the scenario mentioned above?, but also a real experimental measurement of microwave spectra in a superconducting cavity have been performed?. Also, an experimental setting has been proposed for cold atoms trapped in temporaly modulated periodic potential generated by two laser beams8. Chaos-assisted tunneling is concerned with the tunneling transition between torus states that are separated by the chaotic sea. The scenario described there would intuitively be feasible, but we should note that mechanism involved in it is entirely non- trivial. As stressed above, the problem of tunneling in multidimensional or especially in nonintegrable systems is concerned with, by its nature, the mixed phase space, otherwise any dynamical barrier does not exist. The fact that an infinitely many and qualitatively different invariant structures are embedded in a classical phase space inevitably gives rise to the question of how such inhomogeneous invariant structures are quantizable in a self-consistent way. The coupling between torus and chaotic states, which is assumed in the chaos-assisted tunneling scenario, must exist and should be taken into account if one is going to quantize a mixed phase space, but it is by no means clear how it is done. 2
Semiclassical approach to tunneling in nonintegrable systems
As was argued in the study of integrable or nearly-integrable systems ’,’, the most hopeful approach to attack the problem addressed here, is to resort to the semiclassical analysis, which has particularly developed in recent quantum chaos contexts. The semiclassical study of chaotic systems is mainly pursued
160 in the energy domain, where the trace formula due to Gutzwiller'' play a central role, and its validity has been confirmed beyond the naturally expected limitation of the WKB approximation. However the original Gutzwiller's formula is derived for the hyperbolic system, and it cannot cover the situation where dynamical tunneling between tori is in question. In addition, the connection between the underlying classical dynamics and purely quantum effect can be made only if the WKB approach, which is our unique tool to interpret the wave effect in term of the classical trajectory concept, can be applied even to the case where purely quantum effects are dominant. The instanton method is most well known technique to fulfill this requirement. An instanton is semiclassical orbit describing quantum tunneling by means of the imaginary time''. It surely represents classically forbidden process on an constant energy surface, but it is restricted to the case in which the degree of freedom of tunneling path is one dimension, though the coupling to the heat bath or some environment with infinite degrees of freedom is allowed in a formulation extended appropriately. A crucial point that is missing in the semiclassical treatment taken so far is that it cannot provide us with any prescription how to describe the correlation or connection between qualitatively different invariant components. As for EBK(Einstein-Brillouin-Keller) quantization scheme, it is assumed that each torus is quantized independently to give an individual energy eigenstate. Gutzwiller's trace formula can treat non-separable systems, but it can also quantize the case where a single ergodic component exists in the phase space. (More precisely, one should say that a set of periodic orbits are the objects of quantization in the trace formula, but each periodic orbit itself forms an ergodic component that is different from the chaotic component with a positive measure. The closure of a set of periodic orbits gives a single connected set.) The statistical description of energy eigenvalues in mixed systems is also related to this problem. Derivation of intermediate level spacing distribution12 is based on the idea that the eigenstates supporting the torus and chaotic regions are superposed independently. No interaction between these different ergodic components are assumed. It is clear, on the other hand, that the issue we want to consider is not concerned with quantization of such independent ergodic components, but we should make clear what controls the coupling between these different invariant components in the classical phase space. The quantum wave effect we are interested in here comes genuinely from the coupling between them. We can therefore say that the problem of tunneling in chaotic systems and quantization of a mixed phase space are strongly related to each other and a root of difficulty can be the same.
161
The scattering matrix approach has been applied to the non-concentric annular billiard model to give a classical picture13. The authors gave an argument that the tunneling splitting understood as a manifestation of chaosassisted tunneling is determined not by a direct path from a torus to chaos, but mainly by the route from the torus regime to chaotic one via beach states which correspond to the sticky domain close to the KAM torus and orbits are trapped for a very long time. Such a path gives the shortest path, which means the minimum imaginary action being expected. They developed their argument based on the one-step quantum propagator in which only short time classical paths are used as classical inputs, so one might call it a semiquantum approach. The coupling between torus and chaotic region can only be implemented quantum mechanically in their approach. In order to take into account the connection between different ergodic components purely classically, or alternatively, to describe dynamical tunneling within a completely classical language, full use of complex orbits would be inevitable. As already mentioned, instanton is an idea to realize it in one dimensional potential system, and it provides us with a simple picture of tunneling by means of a classical trajectory. The original instanton technique can only be applied to the limited case, but even in the system with multi-dimensions its usefulness remains and is still be helpful, especially in the energetic barrier tunneling problem. In fact, Creagh and Whelan have investigated semiclassically the energetic barrier tunneling in two dimensions basically along the line of trace formula, and found that monotonic and average behavior of the tunneling splitting caused by quasi-doublet states, each of which is supported by the chaotic seas, is well described by the complex orbit passing through the potential well via the pure imaginary time, which gives the minimum imaginary action14. This orbit is an analog of instanton in the standard context. The instanton orbit thus survives even in the scenario in multi- dimensions and still effective a t least for a part of the aspects of tunneling. This is, however, not the end of the story. They found that an oscillatory component found in the splitting sequence, the origin of which is expected to come from the existence of chaos on both potential wells, cannot be explained as long as only an instanton orbit is taken into account, since its extension to the real regime gives only a simple behavior, not picks up any chaotic nature in its itinerary. To reconcile this somewhat contradictory aspect, they have focused on the homoclinic orbits in chaotic seas15. More precisely, they showed that oscillatory component observed in the tunneling splitting distribution can successfully be reproduced by including the trajectories which pass through the potential barrier very close to the instanton path and are connected with
162 the orbits homoclinic to the real orbit extended from the original instanton. The existence of compound orbits which bear both characters of instanton and chaotic orbits just represent the coupling between torus and chaotic states. This simultaneously shows that not only a single instanton orbit but a bunch of instanton-like orbits run under the potential barrier. Mechanism of how the instanton-like orbit and the orbits outside the potential barrier are connected has been studied in detail for a one-dimensional barrier tunneling model subject to periodically oscillatory perturbation16. A semiclassical expression for the wave matrix is represented as a sum over the complex trajectories’’, for which not only dynamical variables but time are also fully complexified. This is in contrast to a well-known instanton orbit which is an extension of time to the pure imaginary domain. As already indicated in the study of classical Smatrix theoryI8, the complex trajectories are necessarily accompanied by the singularities on the complex t-plane. However, the role of those singularities was not fully recognized in generic multidimensional tunneling problems. A close study has revealed that the position of singularity on the complex t-plane moves as a function of an initial phase of the perturbation. Rather, it moves to infinity at a certain initial phase, and such a divergent movement of singularities brings about complex-domain heteroclinic entanglement between the stable manifold of a barrier-top unstable periodic orbit and incoming trajectories. As a result of interference between those multiple tunneling paths, the fringed pattern is created in tunneling tail of the scattering wavefunction16. Homoclinic or heteroclinic entanglement implies creation of chaos. Hence, this event induced by divergent movement of singularities can be interpreted as a sign of coupling between complex tunneling paths and chaotic orbits. The model system taken in their analysis does not show genuine homoclinic or heteroclinic chaos on the real plane, as the scattering map model introduced below, but the mechanism found there concerning how the qualitatively different orbits are connected can be considered as genericI6. It might be still controversial whether the tunneling in 1.5-dimensional degree-of-freedom system can be regarded as genuine energetic barrier one since we should interpret this tunneling process as dynamical one if the original 1.5-dimensional system is mapped to an equivalent 2-dimensional system by using the extended phase space. The result presented in the following sections is intended to clarify how tunneling in chaotic systems is influenced by the underlying classical phase space structure in simpler situations. We here employ two-dimensional map models, each of which is given as a discretized reduction from the time evolution of a Hamiltonian system under &spiked periodic perturbation.
163 We here apply the semiclassical analysis in the time domain. As compared with the energy domain approach, the (complex) classical orbits appearing in the time domain semiclassical analysis are not the objects compatible to invariant quantities of the dynamical system such as periodic orbits, since they depend on the representation we take, or on the initial and final conditions. Complementally, however, calculation of the time domain propagator explicitly gives the quantum wavefunction which contains much information on how the wavefunction in the tunneling regime reflects complex classical manifolds under consideration. We can obtain the correspondence between phase space structure and the shape of the wavefunction more directly. For the map system with a fixed time step, in contrast to the continuous flow systems, it is relatively easier to enumerate whole candidate complex paths which can contribute to the semiclassical propagator. As compared with the case of real semiclassics, the amount of task for path search is more tedious and sometimes difficult, because the dimension of initial space to be scanned becomes simply twice as much as that in the real one. But, this is an unavoidable step, since we do not know a priori a definite selection rule or some general organization principle controlling which complex orbits are relevant and which are negligible, or more generally we do not have enough knowledge of the role of complex orbits especially in chaotic Hamiltonian systems. In order to establish scenario describing tunneling in the presence of chaos, making a connection to some canonical basis, i.e., the theory of complex dynamical systems, which would not be so familiar with physicists as the real one, is strongly desired. One more advantageous point in using the map model is that it is relatively easier, and in a certain simple case almost po~sible'~,to handle the Stokes phenomenon than in the Hamiltonian flow case. In the semiclassical treatment on the real domain, all the classical orbits satisfying the stationary phase condition contribute to the semiclassical superposition. In contrast, not all of the complex orbits given by the saddle point condition, though it is formally equivalent to the stationary phase condition, necessarily contribute. This contributing and non-contributing problem has its origin in the Stokes phenomenon, which is a discontinuous change of asymptotic solutions in the saddle point analysis, or more originally, differential equations with some large parameter. It is a general phenomenon, and one cannot avoid it if the asymptotics in the complex plane is to be treated. It looks a technical problem, but as mentioned below a correct treatment of the Stokes phenomenon is crucial to establish the selection of complex orbits which finally remain to contribute to the semiclassical sum. The map model has, however, a clear limitation in the treatment of tun-
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neling problem. The complex trajectories for the map models are obtained by extending all the dynamical variables, but the time variable still remains discrete. On the other hand, the instanton orbit is the one following the complex time. So we have to explore the possibility of whether or not there are some alternatives in the map models to describe tunneling between energetically splitted domains for which the instanton is the most natural object to bridge them. In the following, we shall present how the time-domain complex semiclassical analysis for map models reveals mechanism of tunneling in the presence of chaos in a quite transparent manner 20,22. We here intend to show; (1) dynamical tunneling and energetic barrier tunneling in the presence of chaos take place under the same principle from the viewpoint of the structure of the complex phase space, (2) a self-similar structure of manifolds, which forms a hierarchical, or multi-generation structure, gives a common frame in complex phase space both in dynamical and energetic tunneling, (3) selection of semiclassically relevant orbits can be done according to the rule encoded in the multi-generation structure.
3 Map model for dynamical and energetic tunneling The standard map has originally been introduced to study generic features of Hamiltonian systems with mixed phase space. It is a one-dimensional pendulum perturbed by a series of delta spikes:
The most well studied map is given by making a standard choice as
where K controls the degree of nonlinearity. Typical mixed phase space is realized in some intermediate range of the kicking strength. In order to extract the tunneling amplitude from KAM torus to chaotic sea as purely as possible, we may replace the kinetic term by some appropriate form in order that the initial wavepacket is completely included in the KAM banded region”. In either case, KAM torus and its secondary generated islands form dynamical barriers in the phase space. Another model which is designed to study energetic barrier tunneling is
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given by putting,
Hob) =
P2 2
1
V ( q )= KexP(-yq2),
(3)
where K and y are some positive parameters. The repulsive potential is localized around the origin q = 0 and the strength of force decays exponentially around the origin. This would be a simplest possible scattering map which show several characteristics typically observed in chaotic tunneling problems”. Classical mechanically, it should be noted that this map creates no chaos in the real phase space, in spite of periodically kicked perturbation like the standard map. It has only a single periodic orbit at ( q , p ) = (O,O), which is a fixed point of the map. Although this fixed point is unstable, there is neither homoclinic nor heteroclinic entanglement in the real phase space. Any manifold initially placed on the real plane is stretched but not folded perfectly so that it leaves away to infinity along the unstable manifold of the fixed point ( q , p ) = (0,O). The situation is analogous to the billiard model with two disk scatterers, in which there exists only one unstable periodic orbit bouncing between two disks and no chaotic scattering occurs. 4
Initial value representation of semiclassically contributing orbits
In the time domain approach, to specify on which situation one focuses, the initial and final states of the wavepacket propagation should carefully be chosen. In particular, as emphasized, tunneling transition beyond the dynamical barriers can occur between qualitatively different classical invariant components, and so the story might drastically be changed depending on from which state we start to develop wavepacket propagation, especially for the first model which is intended to examine tunneling over dynamical barriers. However, surprising enough, and it is one of our primary claims that the candidates of dominating complex tunneling paths will be specified, not depending on the initial and final states of the wavepacket. For the reason explained below, it is a generic property of the system with chaos in the complex plane, but its justification requires some deep knowledge on the theory of multidimensional complex dynamical systems. So for the moment, we will follow the standard approach. The wavepacket for the standard map (2) is initially put on an eigenstate of the momentum state p = P O , i.e.,
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At an intermediate perturbation strength, for some part of the initial conditions, the corresponding classical orbits are trapped by the tori, and the others are bathed in the chaotic sea. A typical wavefunction in the momentum representation is displayed in Fig. l(a) where quantum wavefunction at a certain fixed time step is plotted on the logarithmic scale. While the corresponding classical Lagrangian manifold spread over some finite region within a finite time step, quantum wavefunction has non-zero amplitude everywhere in the phase space. An interesting observation is that the tail of wavefunction does not decay monotonically even in the region where the classical trajectories cannot reach. The structure thus appearing in the completely tunneling region is a typical characteristic in tunneling in the presence of chaos20*22.Similarly, in case of the modified standard map where the initial wavepacket is put completely inside KAM torus band20, more complicated structures such as the plateau accompanied with irregular interference pattern, cliff-like structure, and crossover of several slopes emerge outside the region where classical trajectories are not accessible and thus monotonical decay of the tail is naturally expected. The structures observed in tunneling tails are not only the property owned by the standard map. Figure l ( b ) displays the wavefunction on the coordinate space for the scattering map model (3). The initial state is chosen as the Gaussian wavepacket given as
where u is the squeezing parameter, and qa,pa are configuration and momentum of the center of mass, respectively. The wavepacket is initially placed on the negative q region far from the origin and the initial kinetic energy is set far less than the potential maximum to suppress the direct propagation process beyond the potential barrier. At the instance of n = 10 shown in Fig. l ( b ) , the wavepacket is already reflected by the potential, so the peak -120 represents the center of the reflected wave. On the other around q hand, several structures appearing in the tail of the peak are caused by the tunneling penetration. In spite of the tunneling regime, there again appear plateaus with some interference patterns and staircase structures. The origin of the observed non-monotonic tunneling tail in fully quantum mechanical calculation can be well explained as a superposition of the semiclassical waves associated with complex orbits. The semiclassical approximation we carry out here is just the saddle point evaluation of the multi-fold integral which represents the discretized quantum propagator20. The resultN
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P
9
Figure 1. Quantum wavefunction for (a) the standard map at TI = 5, with K = 1.5, = 10, with K = 500, R = 1, and R = 2n x 7/512, and (b) the scattering map at y = 0.005,u= 10, . For the standard map, the initial wavepacket is put on po = 0, and for the scattering map, the parameter specifying the initial Gaussian packet are given as pa = -123,pa = 23.
ing form after the approximation is called the Van-Vleck propagator, the expression in case of the standard map being given as, (6) po=a Pn=P
where the summation is taken for all (PO,40) which satisfy the boundary conditions for the initial and final momenta as po = a and p , = ,B, where a and ,B are real. Here,
is the action functional along a classical trajectory with given PO and p,, and
represents the amplitude factor associated with its stability. If we wish t o obtain the formula in case of the scattering map model, the action functional is to be switched by
168
where Q = ( q - i p a 2 ) / ( & ) , P = (p-iqo-2)/(fio-1) are a pair of redefined canonical variables2'. For the semiclassical approximation in the real domain, all the real orbits satisfy a given boundary condition. In contrast, in the complex semiclassical treatment, as mentioned above, even the orbits satisfying the boundary condition do not necessarily contribute to the final semiclassical summation as a result of the Stokes phenomenon in the complex domain. It is crucial to deal especially with the situation where multiple complex solutions satisfy the saddle point condition. Although recent development of the exact WKB method give a possibility even to treat the Stokes phenomenon in higher dimensions21, general methods, including the situation in which infinity many saddle point solutions exist, have not yet been known. So we here follow a phenomenological prescription whose validity has been tested in ref. 23. The candidate complex orbits are numerically searched by solving a shooting problem. For the standard map, the momentum p is an observable, so in addition to the initial momentum the final momentum should be real, i.e., Imp, = 0. Also, for the scattering map model, the final coordinate is set to be real, i.e., Imq, = 0. The saddle point equations to be solved is therefore expressed as,
for the standard and scattering map, respectively. Here AQo = QO - Qa. In either case, for a given ct or qa,pa and a given /3 the equations are solved by scanning the complex initial plane (J,q). We remark that the number of solution is infinite even in the finite time step, since the equations (11) and (12) contain transcendental function, whereas a finite number of root searching is sufficient in the real semiclassics within a finite time step. If the transcendental potential is replaced by the polynomial one, the equation determining the shooting problem is an algebraic equation, so the number of solution at a time step n is always given as d" where d is the degree of polynomial.
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Changing the final statep, or q,, we obtain the initial value representation of the contributing solutions for the standard and scattering map, respectively:
M t ) { ( 6 , ~I PO ) = Q , Im P, = 0 } MF’Pa) = { ( 6 , d I pa = P ( q a , p a ) , Imqn
(13) =0
)
(14)
Since for a fixed ,B there are infinite number points which are the solutions of (11)or (12), M , set thus defined is composed of infinitely many strings, each of which is obtained by changing ,B from -m to +m. Indeed as demonstrated in Figs. 2(a) and 2(b), a bunch of strings appear in the M , set. These are the initial value representation of the semiclassical propagator approximating quantum calculations displayed in Figs. l(a) and l ( b ) , respectively. We should remember that the standard map creates chaos and tunneling occurs dynamically, while the scattering map generates no chaos on the real plane and energetic barrier prevents the diffusion of classical orbits. Nevertheless, in either case, we notice that self-similar aspect of these initial value representations is common irrespective of the kind of tunneling. 5
Structure of initial Value representation of tunneling paths
As mentioned above, it is not a result of chaos in the complex space that an infinitely many complex paths appear. In case of the system with polynomial potential like HCnon map, the number of solutions is always finite in a finite time stepz4. two states polynomial But, in any case, an important point is that not a few of candidate tunneling paths are running not only beyond the dynamical barriers, but also the potential barrier. The existence of multiple tunneling paths is a direct reflection of chaos in the complex space . It is needless to say, however, that not all of these complex paths contribute to the semiclassical sum in equal weights, but their amplitude are distributed in a wide range. Therefore, we could not claim that multiple tunneling paths actually contribute in chaotic tunneling problem until1 when their weights of contributions are found to be comparable. Each path has its amplitude which is mainly controlled by the imaginary part of its action S,. But it is not necessarily true that only a single complex orbit with the smallest imaginary action survives as a contribution and dominates the others, because difference of ImS, should now be measured in the unit of h and the paths whose imaginary part of the action is within the width of h are not negligible. Therefore, the question becomes; does the number of such substantially contributing complex paths increase with time? The answer is yes, and moreover such relevant complex paths are clearly distinguishable from the others on the initial value planez0-2z.
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Figure 2. The initial value representation M n for (a) the standard map at n = 40 with K = 1.5, and (b) the scattering map at n = 10, with K = 500, A = 1,and y = 0.005, u = 10. The figure (a) is a magnification of the whole (&q)-plane, and the time step is different from the one shown in Fig. l(a). In (a) the strings running in the vertical direction and in (b) in the horizontal direction look as if they cross with each other,but actually they avoid with very narrow gaps, that is, the strings form a serial chain-like structure connected via narrow gaps. In (b), several kinds of dotted or dashed lines represent the semiclassical waves which dominate the others in a certain range of q when constructing the final superposition shown in Fig. 4(b).
In caSe of the standard map, among quite densely aggregated strings, we can identify a sequence of strings which form a chain like object as a whole and run in the vertical direction. Also in the scattering map, a similar chain-shaped sequence, a piece of which is a string, can be recognized in the horizontal direction. Each string forming a chain structure has relatively small Im S, and thus give large contribution to the semiclassical sum. The readers can find a detailed description in refs. 20 and 22 of how the contributions from the chain-shaped strings give rise to interfere and create characteristic structures such as plateau and cliff patterns. In ref. 22, we use more refined terminology to redefine the chain structure here specified only in a phenomenological manner. If we follow a correct interpretation of the self-similar structure, which is unambiguously identified using the stable
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001
-
ow1 om1.
0
2
4
6
8
,
0
1
2
1
4
0
2
time 4 step 6
time step
Figure 3. The distance from the real plane as a function of time is displayed for several orbits whose initial conditions are put on several chain-shaped structures. The case of the standard map is shown in (a) and the scattering map for (b). In (a), the bold and dashed lines are the orbits for K = 1.5. The initial small fluctuation seen in the bold line is due to the presence of rotational domains in complex space space. The dotted line which shows a monotonical decay corresponds to the case for K = 4.0,in which the influence of rotational domains is small.
manifold of a fixed point located at the origin, we should regard that the strings contained in the chain-shaped structure here explained do not belong to a single chain but to several different chains22. But, here we use the chain as the one which is most prominent and distinguishable on the initial value plane. The reason why these special set of strings provide small value of Im S, is that the orbits launching there approach the real plane straightforwardly. A direct approach gains only smaller imaginary actions as compared to the paths going round about the complex space. Furthermore, it can be confirmed that the number of the direct paths increases exponentially as a function of time while the minimum difference of Im S, between the contributing orbits decreases e x p ~ n e n t i a l l y ~Therefore, ~ , ~ ~ . substantially relevant complex paths increase with the time step. This holds both in case of the dynamical and energetic t ~ n n e l i n g ~ ~ . ~ ~ . The other paths not contained in the most distinguishable chain do not monotonically approach the real plane, but itinerate in the pure imaginary pace^',^^. In Figs. 3(a) and 3(b), we show such orbits not going to the real plane directly, together with the orbits with direct approach to the real plane. However, it may always be true that monotonical decrease of the imaginary part of each canonical variable makes ImS, smaller, but this is not the necessary condition. It is possible that after some itinerary in complex phase
172
space, ImS, might happen to take a small value due to cancellation among some terms in the action functional. We cannot exclude such a possibility, but it appears to be hopeless, as it stands, to pick up such paths which will be embedded deeply in highly aggregated strings. Self-similarity commonly observed in the standard and scattering map models provides a clue to solve this problem. The strings are so densely bunched that there seems to be no rule to generate them, but this is not the case but rather there is a definite mechanism to make this M , set self-similar. We do not enter into the details here, but we should say that a key object to understand the reason why the self-similar structure appears on the initial value plane is the (forward) Julia set of the complex dynamical systems24. The hierarchical or multi-generation structure appears as its natural consequence. In particular, because of simplicity of chaotic dynamics in the complex phase space, the scattering map model allows us to analyze closely the multigeneration structure by means of a symbolic dynamics 2z*25. It is shown through this analysis that the chain-formed sequence is not unique, but similar chain structures are located in a self-similar way. Furthermore, itinerary of orbits in the complex phase space reflects self-similarity of the chain-shaped structures and one can read the history of the individual orbits by decoding information in the self-similar s t r u c t ~ r e ~ ~ ~ ~ ~ . We can give, on the basis of the theory of complex dynamical systems, a quite simple reason for the existence of orbits which itinerate in purely complex plane, nevertheless give the small ImS, due t o cancellation along their i t i n e r a r ~ ~ Furthermore, ~j~~. in case of the scattering map, using the symbolic dynamics, we can provide a systematic way to pick up such orbits from densely aggregated stringsz5. In spite of the existence of those orbits which should be included in the semiclassical sum, as shown in Figs. 4(a) and 4(b), the semiclassical wavefunctions, which show quite good agreement with purely quantum mechanical calculations presented in Fig. 1, are obtained only by including the orbits belonging to the primary chain in which the orbits monotonically approach the real plane. If one includes the itinerating orbits, the resulting semiclassical wavefunctions are completely destroyed. This implies that the itinerating orbits must not be included in the final semiclassical superposition. Only possibility to explain why they are to be excluded is that they should be removed as a result of the Stokes phenomenon. We can give an argument based on a certain phenomenological criterionz3, but a final solution has not yet been obtainedz5.
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P Figure 4. Semiclassical wavefunction for (a) the standard map, and (b) the scattering map. The same parameters and the same initial conditions are chosen as purely quantum calculations shown in Fig. l ( a ) and (b), respectively.
6
Concluding remarks
As for the structure of initial manifolds in the complex plane, any significant difference between dynamical and energetic barrier tunneling could not be found. The manifolds are arranged in a self-similar way in the initial value representation. The reason why we can claim that it is a common property is that this hierarchical structure appears as a result of fractality of the Julia set in the complex phase space. Chaotic nature appearing in wavefunctions in tunneling regime is a reflection of chaotic behavior of complex orbits on the Julia set24. The Julia set exists commonly as long as the complex phase space exhibits chaos. Even in case where there is no chaos on the real domain, it is possible that the Julia set spread in the complex plane, as the scattering map model does so. In this sense, it does not matter whether tunneling occurs dynamically or energetically, rather the existence of the Julia set is a common language to describe the mechanism of chaotic tunneling, and the qualitative difference of Julia sets, if any, could become more important to determine the nature of tunneling. The best we can predict concerning which complex paths are most dominant and finally determine the wavefunction is to indicate a group of complex paths which have almost similar amplitude and can potentially become the most dominant ones. Conversely speaking, we cannot specify a particular tunneling orbit which dominates the others. The role of chain-shaped structure identified on the initial value representation should be understood in this way. It can be said, therefore, that a bunch of candidates compete each other and
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this just characterizes the situation in chaotic tunneling. The reason why there are exponentially large number of complex paths whose classical action have nearly equal imaginary part can also be explained by the fact that the semiclassically contributing orbits are dense on the Julia setz4. The discussion given in ref. 15 may also be related to this fact. In ref. 15, it has been shown that not a single tunneling path is sufficient, but homoclinic orbits combined with nearly instanton paths which connect chaotic seas, which are separated by an energetic potential well, are essential to explain the features of tunneling splitting distribution. The orbits on the (forward) Julia set can have both properties, the instanton and chaotic (homoclinic) trajectories. In fact, it is particularly important to point out that the claim that the Julia set is dense in the semiclassically contributing candidates holds even in case of the system with mixed phase specez4. Furthermore, if the rotational domains in complex plane, which contain not only the KAM tori on the real plane but their extension to the complex space, or possibly Siege1 disks and Hermann rings, have null volume in full complex plane, the rotational domains themselves are contained in the Julia set, which implies that KAM tori are elements of the Julia set and so they form the unstable sets, meaning that an arbitrary nearby complex orbit does not stay in its vicinity. Due to the transitivity of the Julia set, which has rigorously been proved for the complex HCnon mapz6, and conjectured for the standard or semi-standard mapsz7, there is a point on the Julia set which can come arbitrarily close to any other points on the Julia set. Therefore, the complex orbit on the Julia set which is not trapped by a rotational component can escape from the region where KAM tori dominate, even if it is initially placed on arbitrarily close to a certain KAM torus on the real plane. In other words, in the system with mixed phase space, the orbits on the Julia set could have both rotational and chaotic characters. Such coexistence is a remarkable property in the complex dynamical systems, and it might be quite helpful to make it possible to carry out the semiclassical quantization. This nature in the complex phase space also justifies our assertion that selection of the initial and final states does not matter when considering tunneling between classically inaccessible domainsz4.
Acknowledgments The authors acknowledge Y. Ishii for crucial suggestions on the complex dynamical systems in higher dimensions. The present work was supported by Grand-in-Aid for Scientific Research No. 13640410, from the Ministry of Education, Culture, Sports, Science and Technology of Japan.
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ON QUANTUM-CLASSICAL CORRESPONDENCE AND CHAOS DEGREE FOR BAKER’S MAP KEI INOUE, MASANORI OHYA Department of Information Sciences Science University of Tokyo Noda City, Chiba 278-8510 Japan IGOR V. VOLOVICH Steklov Mathematical Institute Russian Academy of Sciences Gubkin St. 8, 117966 Moscow, Russia Quantum baker’s map is a model of chmtic system. We study quantum dynamics for the quantum baker’s map. We use the Schack and Caves symbolic description of the quantum baker’s map. We find an exact expression for the expectation value of the time dependent position operator. A relation between quantum and classical
trajectories is investigated. Breakdown of the quantum-classical correspondence at the logarithmic time scale is rigorously established ’.
1
Introduction
The quantum-classical correspondence for dynamical systems has been studied for many years, see for example and reference therein. A significant progress in understanding of this correspondence has been achieved in a mathematical approach when one considers the Planck constant FL as a small vari0 quantum theory is able parameter. It is well known that in the limit ti reduced to the classical one 4,5. However in physics the Planck constant is a fixed constant although it is very small. Therefore it is important to study the relation between classical and quantum evolutions when the Planck constant is fixed. There is a conjecture that a characteristic time scale t h appears in the quantum evolution of chaotic dynamical systems. For time less then t h there is a correspondence between quantum and classical expectation values, while for times greater that t h the predictions of the classical and quantum dynamics no longer coincide. An important problem is to estimate the dependence t h on the Planck constant tz. Probably a universal formula expressing t h in terms of tz does not exist and every model should be studied case by case. It is expected that certain quantum and classical expectation values diverge on a time scale inversely proportional to some power of ti . Other authors suggest that for chaotic systems a breakdown may be anticipated on a much smaller logarithmic time 213
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61738
178
scale (see 2,9 for a discussion). Numerous works are devoted to the analytical and numerical study of classical and quantum chaotic s y s t e m ~ ’ ~ - ~ ~ . Most results concerning various time scales are obtained numerically. In this paper we will present some exact results on a quantum chaos model. We compute explicitly an expectation value for the quantum baker’s map and prove rigorously the appearance of the logarithmic time scale. The quantum baker’s map is a model invented to study the chaotic behavior 16. The model has been studied in 17-25. In this paper quantum dynamics of the position operator for the quantum baker’s map is considered. We use a simple symbolic description of the quantum baker’s map proposed by Schack and Caves 2 3 . We find an exact expression for the expectation value of the time dependent position operator. In this sense the quantum baker’s map is an exactly solvable model though stochastic one. A relation between quantum and classical trajectories is investigated. For some matrix elements the breakdown of the quantum-classical correspondence at the logarithmic time scale is established. Here we would like to note that in fact the notion of the time scale is not a uniquely defined notion. Actually we will obtain the formula
where qm and qm are quantum and classical positions respectively at time m. This formula will be interpreted as the derivation of the logarithmic time scale (see discussion in Sect.5). The main result of the paper is presented in Theorem 1 in Sect. 4. In another paper 34, semiclassical properties and chaos degree for the quantum baker’s map are considered. 2
Classical Baker’s Tkansformation
The classical baker’s transformation maps the unit square 0 5 q , p 5 1 onto itself according to
This corresponds to compressing the unit square in the p direction and stretching it in the q direction, while preserving the area, then cutting it vertically and stacking the right part on top of the left part. The classical baker’s map has a simple description in terms of its symbolic dynamics 12. Each point ( q , p ) is represented by a symbolic string with a dot
where Ek E ( 0 , I}, and W
00
The action of the baker’s map on a symbolic string E is given by the shift map (Bernoulli shift ) U defined by U c = E‘, where 6 ; = Em+l. This means that, at each time step, the dot is shifted one place to the right while entire string remains fixed. After m steps the q coordinate becomes
k=l This relation defines the classical trajectory with the initial data
k=l 3
Quantum Baker’s Map
Quantum baker’s maps are defined on the D-dimensional Hilbert space of the quantized unit square. To quantize the unite square one defines the Weyl unitary displacement operators U and V in D - dimensional Hilbert space, which produce displacements in the momentum and position directions, respectively, and the following commutation relation is obeyed
UV = EVU, where E = exp ( 2 7 ~ i l D We ) . choose D = 2 N , so that our Hilbert space will be the N qubit space C g N . The constant TL = 1 / D = 2 - N can be regarded as the Planck constant. The space C2 has a basis
180
then
5 = 0,1, ...,2N - 1 and denote 1s) = lElE2 . 'h" = 151) @ "$2) @ . '
' '
@ [EN)
We will use for this basis also notations (177) = (771772. . . 7 7 ~ ,) q k = 0 , l ) and { l j ) = Ij 132 . . . jN ) j k = 0 3 1 ) . The operators U and V can be written as '
,A
fi = ,Z?rig
v = e2?rip A
7
where the position and momentum operators t and 5 are operators in CBN which are defined as follows. The position operator is 2N-I
and
+
j 112 qj = 2N , j = o , 1 , . . . 7 2 N - l
The momentum operator is defined as
5 = FNI~FI; where FN is the quantum Fourier transform acting to the basis vectors as
.
D-1
here D = 2 N . A quantum baker's map is the unitary operator T in CBN with the following matrix elements
where I[) = I [ ~ J ~ . . . E N ) , 17) = I V ~ Q . . . ~ N ) and 6(z) is the Kronecker symbol, S(0) = 1; S(z)= 0, z # 0. This transformation belongs to a family of quantizations of baker's map introduced by Schack and Caves 23 and studied in 24,25
181
4
Expectation Value
We consider the following mean value of the position operator m = 0,1, ... with respect to a vector It) :
riN' = ([I TmeT-m It),
4
for time (5)
where It) = I&& . . . [ N ) . First we show that there is an explicit formula for the expectation value r i N ) . In this sense the quantum baker's map is an explicitly solvable model. Then we compare the dynamics of the mean value r?) of position operator 4 with that of the classical value qm , Eq. ( 2 ) . We will establish a logarithmic time scale for the breakdown of the quantum-classical correspondence for the quantum baker's map. jFrom Eq. (4)one gets for m = 0,1, ...,N - 1
and for m = N
Using this formula we will prove the following Theorem 1. One has the following expression for the expectation valule (5) of the position operator N-m
r i N ) = (
1 C -+2N-m+l tm+k
2k
(8)
k=l
for 0 5 m
< N . For m = N
we have
.p = -21 Proof. By a direct calculation, we obtain
rc,"'(
(9)
182
Using the Kronecker symbols one gets
We can write it as
+A(. c 3 N - m + 1 1 " ' 3 N,
1)
183 Finally we obtain (8) for 0 5 m
\ k=l
In the case m = N we have
c = c+ 7(?)"IZ
f 1/2 rjvN) = 2 N - 1 j2N j =O 2N-1
j
j=O
1
1/2
=
p
1
2N-1
c
(j+1/2)
=
-. 2
j =O
The theorem is proved. 5
Time Scales
We consider here the quantum-classical correspondence for the quantum baker's map. First let us mention that 2N = l / h and the limit ti -+0 corresponds to the limit N -+ 00. Therefore from Theorem 1 and Eq. ( 2 ) one has the mathematical correspondence between quantum and classical trajectories as ti + 0:
Now let us fix the Planck constant fl = 2 - N and investigate on which time scale the quantum and classical expectation values start to differ from each other. From Theorem 1 and Eq. ( 2 ) we obtain the following Proposition 1. Let rLN) be the mean value of position operator at the time m and qm is the classical trajectory Eq. ( 2 ) . Then we have
for any 0 5 m 5 N . Let us estimate the difference between the quantum and classical trajectories.
184
Proposition 2. Let qm and Then we have
for any string [ = Proof. Note that
T?)
be the same as in the Proposition 1.
... and any time 0 I m 5 N
I2N-m+l 1 ( 1 + 51+
(;)
z+...)
=1p .
Using the above inequality, one gets from Eq. (10)
This means that we have
for any 0 5 m 5 N . Proposition 2 shows an exact correspondence between quantum and classical expectation value for baker’s map. We can write the relation (11) in the form
):.1
1
- qml
=hy-1 I2N-mfl
(12)
since the Planck constant h = 2 -N. In particular for m = 0 we have
,.I
(N)
h -Qol 5 %
(13)
for any 5 = 5152 .... Now let us estimate at what time m = tt, there appears an essential difference between classical trajectory and quantum expectation value. From Eq. (12) we can expect that the time m = tt, corresponds to the maximum of the function 2m/2N-1 for 0 5 m 5 N , i.e. th
=N
1 log, h
1
(14)
185
For time 0 5 m < t h the difference between classical and quantum trajectories in (12) is bounded by 1/4 since 1 1 ft2m-l = 2N-m+l
4
One can see that the bound is saturated. Indeed let us take a string arbitrary & , ...,[N but with EN+^ = 0,&+2 = 0, .... Then one has
rLN)- qm
=
< with
m = 0,1, ...,N
Therefore we have established the logarithmic dependence of the time scale on the Planck constant h.
6
Conclusions
In this paper we have computed the expectation values for the position o p erator in the quantum baker’s map. Breakdown of the quantum-classical correspondence at the logarithmic time scale is rigorously established. For better understanding of the quantum-classical correspondence and the decoherence process it is important to perform similar computations for more general matrix elements which include also the momentum operators and ccherent vectors. Only the simplest quantization of the baker’s map was considered in the paper. It would be interesting to extend the computations to the whole family of quantizations of quantum baker’s map proposed in 23. Some of these questions will be investigated in another paper 34.
Acknowledgements We are grateful to R. Schack, Ya.G. Sinai and W. Zurek for interest to the work and helpful remarks. The main part of this work was done during the visit of I.V. to the Science University of Tokyo. He is grateful to JSPS for the Fellowship award. His work was also supported in part by RFFI 99-0100866 and by INTAS 99-00545. K.1 would like to thank organizer very much for giving him a chance to talk in Japan-Italy joint forum Waseda workshop fundamental problems in quantum mechanics.
References 1. K.Inoue, M.Ohya and I.V.Volovich, On quantum-classical correspondence for baker’s m a p , quant-ph/0108107, 2001.
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2. Wojciech H. Zurek, Decoherence, einselection, and the quantum origins of the classical, quant-ph/0105127. 3. D. Giulini, E. Joos, C. Kiefer, J. Kupsch, I.-0. Stamatescu, and H.D. Zeh, Decoherence and the Appearance of a Classical World in Quantum Theory, Springer, Berlin,1996. 4. V.P. Maslov, Perturbation theory and asymptotic methods, MGU, Moscow, 1965. 5. K. Hepp, The classical limit f o r quantum mechanical correlation functions, Commun.Math.Phys., 35, 265-277, 1974. 6. G.P. Berman and G.M. Zaslavsky, Physica, 91A, 450 (1976). 7. M.V. Berry and N.L. Balazs, J. Phys. A12, 625 (1979). 8. W.H. Zurek and J.P. Paz, Phys. Rev. Lett. 72, 2508 (1994). 9. Zbyszek P. Karkuszewski, Jakub Zakrzewski, Wojciech H. Zurek, Breakdown of correspondence in chaotic systems: Ehrenfest versus localization times, nlin.CD/0012048. 10. D.V. Anosov and V.I. Arnold (eds.), Dynamical Systems, VINITI, Moscow, 1996. 11. Ya.G. Sinai, Introduction to Ergodic Theory, Fasis, Moscow, 1996. 12. V.M. Alekseev and M.N. Yakobson, Symbolic dynamics and hyperbolic dynamic systems, Phys. Reports, 75, 287-325, 1981. 13. G.M. Zaslavskii, Stochasticity of Dynamical Systems, Nauka, Moscow, 1984. 14. M.V. Berry, Some quantum-to classical asymptotics, Les Houches Summer School “chaos and quantum physics”, Edits. Giannoni, M.J. Voros, A. and Justi, Zinn, North-Holland, Amsterdam, 1991. 15. G. Casati and B.V. Chirikov (eds.),Quantum Chaos: between Order and Disorder, Cambridge Univ. Press, Cambridge, 1995. 16. N.L. Balazs and A. Voros, The quantized baker’s transformation, Ann. Phys., 190, 1-31, 1989. 17. A.M. Ozorio de Almeida and M. Saraceno, Periodic orbit theory f o r the quantized baker’s map, Ann. Phys., 210, 1-15, 1991. 18. F.M. Dittes, E. Doron and U. Smilansky, Long-time behavior of the semiclassical baker’s map, Phys. Rev. E, 49, R963- R966,1994. 19. M. Saraceno and A. Voros, Towards a semiclassical theory of the quantum baker’s map, Physica D, 79, 206-268, 1994. 20. L. Kaplan and E.J. Heller, Overcoming the wall in the semiclassical baker’s map, Phys. Rev. Lett., 76,1453-1456, 1996. 21. M.G.E. da Luz and A.M.Ozorio de Almeida, Path integral f o r the quantum baker’s map, Nonlinearity, 8 , 43-64, 1995. 22. R. Schack and C.M. Caves, Hypersensitivity to perturbations in the quan-
187
tum baker’s m a p , Phys. Rev. Lett., 71, 525-528, 1993. 23. R. Schack and C.M. Caves, Shifts o n a jinite qubit string: A class of q u a n t u m baker’s maps, Applicable Algebra in Engineering, Communication and Computing, AAECC 10, 305-310, 2000. 24. A.N. Soklakov and R. Schack, Classical limit in t e r m s of symbolic dynamics f o r the quantum baker’s m a p , quant-ph/9908040. 25. Andrei N. Soklakov, Ruediger Schack, Decoherence and linear enLropy increase in the quantum baker’s m a p , quant-ph/0107071. 26. M. Ohya, Complexities and their applications t o characterization of chaos, International Journal of Theoretical Physics, 37, No.1, 495-505, 1998. 27. R.S. Ingarden, A. Kossakowski and M. Ohya, Information Dynamics and Open Systems, Kluwer Academic Publishers, 1997. 28. K . Inoue, M. Ohya and K.Sato, Application of chaos degree t o some dynamical systems, Chaos, Soliton and Fractals, 11, 1377-1385, 2000. 29. Salman Habib, Kosuke Shizume, Wojciech Hubert Zurek, Decoherence, Chaos, and the Correspondence Principle, Phys. Rev. Lett. 80,4361 (1998). 30. I.Y. Arefeva, P.B. Medvedev, O.A. Rytchkov and I.V. Volovich, Chaos in M(aLrix) T h e o q , Chaos, Solitons & Fractals, 10, No.2-3,213-223, 1999. 31. A. Mariano, P. Facchi, S. Pascazio, Decoherence andfluctuations in quantum interference experiments, quant-ph/0106027. 32. J. Emerson, L.E. Ballentine, Quantum-Classical Correspondence f o r the Equilibrium Distributions of Two Interacting Spins, quant-ph/0103050. 33. Diana Monteoliva, Juan Pablo Paz, Decoherence in a classically chaotic quantum system: entropy production and quantum-classical correspondence, quant-ph/0106090. 34. K. Inoue, M. Ohya and I.V. Volovich, Semiclassical properties and chaos degree f o r the quantum baker’s map, J. Math. Phys., 43-2, 734-755, 2002.
188
WELCHER-WEG PUZZLE WITH A DECAYING ATOM SHIN TAKAGI Fuji Tokoha University, Fuji 4 1 7-0801, Japan E-mail: takagi@fuji-tokoha-?1.ac.jp The interference experiment with Cso performed successfully by the Wien group is remarkable in two respects. First, the molecule c60 is far bigger than previously used particles. Second, most of the molecules were prepared in excited states and thought to have decayed during the flight from the slit to the detector. The second feature raises the possibility that the “Welcher-Weg observation” could have been performed by the emitted photon. An elementary indeterminacy-relation argument says that the observation is effectively performed (or not performed) if A0 < d (or A0 > d ) , where A0 and d are the wavelength of the emitted photon and the inter-slit separation, respectively. This article is to discuss how this elementary expectation may be quantified and qualified, for the case of a decaying 2-state atom (instead of c60 for the sake of simplicity), on the basis of the Schrodinger equation.
1
Welcher-Weg Puzzle
As is well-known, the Welcher-Weg puzzle lies at the heart of quantum mystery. It is vividly described, for instance, by Tomonaga’s article entitled “The Judgement on the Case of a Photon” This long-standing Gedanken experiment has since been realized in many laboratories. An impressive demonstration has been made, for instance, by Tonomura and coworkers2. Apparently, in a Young-type experiment, a photon (or an electron) behaves as if it were a wave when it passes through a double slit, and as if it were a particle when it arrives at a screen. During the last two decades or so in the 20th century, Young-type interference experiments (or their variants) have been carried out successfully not only with photon, electron and neutron but also with atoms such as He, Ne, Na and so on as well as a molecule 12. In the face of these remarkable progress, it was natural to wonder how far the Young-type experiment could be extended to a bigger object. I remember a dialogue between Prof.von Weizskker and Prof.Glauber at a meeting on quantum mechanics in the early go’s, when von Weizskker said “However far the technology should advance, one would not be able to see an interference pattern with tennis balls,” and Glauber remarked “It might be possible with soccer balls, though.”
’.
189
2
Young-type experiment with
c60
In spite of the prediction by Prof.Glauber and the subsequent knowledge of a plan3 of a soccer-ball experiment, it came to me quite as a surprise to hear in 1999 that a c60 interference pattern had been observed by the Wien physicists4. The set-up of the experiment was roughly as follows a:
= incident speed of c60 N 220m/s, d = inter - slit distance 2: 100nm, w
= distance between double slit and screen 'v 1.3m, X = de Broglie wavelength of C6, = 27rh/M~,~w N 2.5pm, 7 l - 0 ~ = time of flight from double slit to screen = D/w N 5.7ms. D
After passing the double slit, c60 is described by the superposition of a wave *+(R;t ) that came through the upper slit and another wave * - ( R t ) that came through the lower slit, where R is the center-of-mass position of 0: 1 -{Q+(R;t)
+ *-(R;t)}.
Jz
(1)
-
Suppose that c60 passes the double slit at t = 0 and arrives at the screen at around t TTOF. At the latter time, *+(R; t ) may be approximately given by spherical wavepackets centered at the upper/lower-slit position f d / 2 , respectively :
where & ( R ; t ) are spherical-shell like wavepackets peaked at R N w t with width much larger than A. Thus, the probability P ( t , x ) for Cso to be found at a position x on the screen is given as 1
P ( t , x )= - { * + ( x ; t ) +
IJz 1
=-
2
*-(x;t)}
l2
{ p + ( X ; t ) ( 2 + (*-(x;t)(2} +I(t,x),
(3)
where
I(tlx) = R! { q
( X ; t)*-(x;
t ) }.
(4)
aAlthough the experiment used a grating rather than a double slit, we deal with the latter for the sake of simplicity.
190
Hence, for x around the center of the screen, n+
d
2
Thus, the peak-tepeak separation A of the interference pattern is given as
D Acu -A. d The experimental result was in good accord with this elementary expectation. 3
Sloppy view on decoherence
What is surprising is that c 6 0 survived various possible sources of decoherence. In general, one would expect some disturbance by environment. A naive traditional treatment of disturbance would argue that the wavefunction (1) is modified to 1 -{9+(R; t)eie+ 9-(R; t)eie-}, (7)
Jz
+
where O+ are uncontrollable random phases due to the disturbance (or by some sort of “Welcher-Weg observation ” ). The interference term in (3) would then be modified to
I ( t ,x) =
{e-’(’+-’-)@*
+(x ;t,*-(x;t)} >
(8)
which would vanish when averaged over the “random phases”:
I(X)= R { e-’(e+-e-)9T(x;t)9-(x;t)} 0. N
(9)
Unfortunately, it is by now well recognized that this treatment is unacceptable; one should not regard environment as something outside the physical system but include it within the quantum mechanical treatment. The experiment was carried out in a high vacuum. Hence the probability of collision with gas particles is negligible. On the other hand, being prepared in an oven of 900 K or so, each of c 6 0 is expected to be in an excited state (with respect to internal vibration modes, etc) when it encounters the double slit, and is likely to decay to the ground (or a lower excited) state by emitting photons. These internal modes and electromagnetic field act as environment for the center-of-mass of 0 . Let us therefore focus our attention on the effect of a decay process. For simplicity, we consider only a single excited state le), and assume that c 6 0 is in Ie) and the electromagnetic field is in the vacuum state Ivac) at t = 0
191
right after c 6 0 has passed the double slit. Then the initial state of the entire system is given by
1
-{Q+(R;O)
Jz
+Q-(RO)}
le,vac),
(10)
where
(11)
le,vac) e 1e)Ivac). One might think that by the time t to 1
-{@+(Rt)
Jz
where
N
TTOF the
above state would have evolved
+ @-(R;t)}IglY),
(12)
\k*(R; t) are the same as in ( I ) , and ILY)
= MIY)
(13)
with 17) being a one-photon state. If this were the case, the probability for c60 to be found at x would again be given by Eqs.(3) and (4), that is, the decay process would not affect the coherence at all. On the other hand, one might think of the following possibility:
1 -{*+(R;t)
Jz
k,Y)
+ *-(R;t)
le,v4>.
(14)
If this were the case, the probability in question would be given by (3) with the interference term now replaced by
I(t,x) =
{(g,Yle,vac)Q;(x;t)Q-(x;t)}
,
(15)
which vanishes since (g, Yle, v a ~ )= 0. Hence, the coherence would be completely destroyed by emission of a photon. However, Eq.(14) is fundamentally wrong; the decay process should proceed in essentially the same way regardless of the center-of-mass position. But, as explained below, Eq.(12) is not quite correct either. 4
Recoil accompanying decay
When c60 decays by emitting a photon, it should recoil in order to conserve momentum. This opens up a possibility of determining the position of c60 by “observing ” the emitted photon. This circumstance might be summarized as (quotation from Ref.[5] as adapted to the present situation) :
192
“The result of such a determination (of position) must be either the whole c60 or nothing at all. Thus the c60 must change suddenly from being partly in one path and partly in the other to being entirely in one of the paths. This sudden change is due to the disturbance in the translational state of the c60 which the observation necessarily makes. . . . . . . (The) possibility (of interference) disappears when the c 6 0 is forced entirely into one of the paths by an observation. The other path then no longer enters into the description of the c 6 0 , so that it counts as being entirely in one of the paths in the ordinary way for any experiment that may subsequently be performed on it.” Let us clarify the meaning of this statement, in which the phrase “observation” is ambiguous, with a careful quantum mechanical treatment of the decay process. Let (k) be the state with a plane-wave photon of wavenumber k. Also, let ko
= (E, - Eg)/tzc,
be the lifewhere Ee(Eg)is the energy of the excited(ground)state, and time of the excited state. First, consider c 6 0 that passes through the upper slit at t = 0. Let the position of the upper slit be R+(= d/2). Under the condition
koctB 1 A
rlc<< min{ko,d-’},
the state at time t is given approximately by
e-rt/29+(R;t ) (e,vac; t )
+
Dk(t) *Y)(R;t ) ,-ik.R+ Ig, k; t ) , k
and SY’(R;t ) is the same as Q + ( Rt ) in Eq.(2) with b+(R;t ) replaced by b?’(R; t ) ,the latter being a somewhat deformed version of the former. However, this deformation due to recoil is significant only towards the tail of the wavepacket. Thus, for the purpose of discussing the interference pattern, it
193
is legitimate to suppose that
d+(k) ( R ; t )N ++(R;t).Hence, Eq.(17) may
be
simplified as
{
Q + ( R ; ~ e) - r t l 2 le,vac; t )
+
(21)
e-iEgt/hlg)IT+(t))},
where
Ir+(t))= C
Dk(t) e-ik'R+--ikctIk).
(22)
k
We have a similar expression for c60 that passes through the lower slit. Hence, for Csothat passes the double slit, the state at time t is neither (12) nor (14) but 1 e--rt/2-
&
+
{ Q+(R;t) Q-(R;t)} le,vac;t)
+z 1
{Q+(R;t ) I-Y+(t)) + Q-(R;t ) IT-@))
1 Id- i E g t / h
(23)
lk),
(24)
7
where I%t(t))
Dk(t)
e
Tik.d/Z-ikct
k
k
Note that 17) in Eq.(12) has been replaced by Ir*(t))for the upper and the lower wave, respectively. IT+( t ) )represent photon wavepackets, which are superpositions of plane-wave photons multiplied by the phase factors eFik'd/2 that originate from the recoil effect. In the terminology of classical electromagnetic waves, these phase factors may be understood as follows: Adopt the convention that the phase of the emitted plane wave is 0 at the position of 0.Then, the planewave is given by ei(x-R)'k, which is equal to eTik'd/2eik.x for R = f d / 2 , respectively. In summary, Welcher-Weg information is inscribed into the emitted photon in the form of the phase factor eFik'd/2. We therefore find that
2 ~ ( t , x= ) e-rt I{Q+(x; t)
+ ~ - ( xt))I2 ;
+ ll{Q+(R;t)Ir+(t))+ @-(R;t)Ir-(t)))Il2 9
(26)
which may be cast into the form (3) with
I ( t , x )= IR (C(t)Q;(x; t)Q-(x; t ) },
(27)
194
where the magnitude of C(t), to be called the coherency, represents a measure of coherence being maintained:
where n is a unit vector, and we have made use of the property
lDk(t)I2 6(C - CO),
(31)
which is justifiable under the condition (16). We thus arrive at the following conclusion: If r q O F << 1, then ~ ( I T T O F1)1 1. Obviously, if the lifetime is long enough, no photon is emitted and the coherence is maintained in tact. 0
If
r V o F
>> 1, then
that is, if the wavelength Ao(- 27r/Co) of the emitted photon is much longer than d , the coherence is maintained in tact, while if A0 is much shorter than d , the coherence is destroyed. This result may also be interpreted as follows : If no photon has been emitted, the electromagnetic field has not made any observation and the interference pattern is not affected. On the other hand, if a short-wavelength photon is emitted, the electromagnetic field has made a Welcher-Weg observation and the interference pattern is washed out, while if a long-wavelength photon is emitted, the electromagnetic field has failed to made a Welcher-Weg observation and the interference pattern survives. In short, one may use the phrase “observation”, but “observation” in this context has nothing to do with (the existence or not of) a conscious observer; “observation ” is automatically performed or non-performed by the ambient electromagnetic field. In the experiment by the Wien group, one expects 7pm < A0 < 19pm for a photon emitted during the time of flight. Therefore, kod/7r < 0.029 and ICI > 0.9986. Hence, the decoherence effect if any is on the order of lop3, which is too small to impair the interference pattern.
195 5
N a t u r e of decoherence a n d recovery of coherence
Note that the result (32) is what one expects from an indeterminacy-relation argument. However, the detailed nature of decoherence is more subtle than this; for h-TOF >> 1, the coherence is totally destroyed if kodl7r is an integer and only partially destroyed if kodl7r is a little greater than 1. This oscillatory behavior, which could not have been foreseen by an indeterminacy-relation argument, is a consequence of the fact that the coherency is governed by a photon wavepacket that is a collection of plane-wave photons superposed with phase factors efik.d/2 as superposition coefficients. In real life, there are many candidates for the initial excited state of c60 and hence for ko. When averaged over various ko’s, the above-mentioned oscillatory behavior may be smeared out. For instance, if ko’s were distributed in accordance with the Boltzmann factor ePhckolT without degeneracy, then (30) would be replaced by
where k~ = T1h.c. In this case, the result (32) would be valid provided that ko is replaced by k ~ Although . both (30) and (33) are too idealized to apply to c 6 0 , some such dependence of the coherency on d will be demonstrated by future experiments3. Finally, suppose that the situation FTTOF >> 1 A A0 << d has been set up. Will the coherence be lost forever? No, it will not be the case. Recall that the state (23) is of the form
The coefficient of Ig, k; t ) determines the joint probability P (t ,x,k) for c 6 0 to be found at x with its internal state in the ground state and a photon of wavenumber k to be detected;
I
P ( t ,x,k) 0: e-ik.d/2Q + ( R ; t) + eik’d/2!P-(R;t)
I. 2
(35)
Thus, by observing a plane-wave photon k, one does see the same interference pattern as (3) with its center shifted by (k.d/kd)D. This is an example of the “quantum eraser” 6 ; the Welcher-Weg information possessed by the photon wavepacket is “erased” by detection of a plane-wave photon. No mystery is involved here. It is just a matter of selection of data. Of course, if one does not measure a plane-wave photon, one does not have a necessary data to do the selection and misses an interference pattern.
196
Acknowledgments I would express my sincere thanks to Prof.S.Tasaki and other members (both Italian and Japanese) of the organizing committee of the Japan-Italy Joint Forum Waseda Workshop for their invitation to give a talk summarized in this article, which is an enlarged version of a popular article7 published previously. It is a pleasure to note that a paper with an essentially same content has been written also by a triplet of Italian colleagues', who also give an interesting discussion on the blackbody aspect of (260.
References 1. S.Tomonaga, in Quantum-Mechanical View of the World (Kobundo, Tokyo, 1965), in Japanese. 2. A.Tonomura et al, Am.J.Phys 57, 117 (1989). 3. M. Arndt et al, in Epistemological and Experimental Perspectives on Quantum Physics( eds. D. Greenberger, W.L.Reiter and A. Zeilinger) (Kluwer Academic, Dordrecht, 1999), p.221-224. 4. M. Arndt et al, Nature 401,680 (1999). 5. P.A.M.Dirac, The Principles of Quantum Mechanics, 4th ed., (Oxford at the Clarendon Press, 1958), p.8-9. 6. M.O. Scully et al, Phys. Rev. 25,2208 (1982); Nature 351,111 (1991). 7. Shin Takagi, Parity 15 No.4, 22 (2000), in Japanese. 8. P.Facchi, A.Mariano and S.Pascazio, Recent Res. Develop. Physics 3,1 (2002).
197
QUANTUM ZEN0 EFFECT, ADIABATICITY AND DYNAMICAL SUPERSELECTION RULES PAOLO FACCHI Dipartimento da Fasica, Universita di Bari I-70126 Bari, Italy and Istituto Nazionale di Fisica Nucleare, Sezione di Bari, 1-70126 Bari, Italy The evolution of a quantum system undergoing very frequent measurements takes place in a proper subspace of the total Hilhert space (quantum Zen0 effect). When the measuring apparatus is included in the quantum description, the Zen0 effect becomes a pure consequence of the dynamics. We show that for continuous measurement processes the quantum Zen0 evolution derives from an adiabatic theorem. The system is forced to evolve in a set of orthogonal subspaces of the total Hilbert space and a dynamical superselection rule arises. The dynamical properties of this evolution are investigated and several examples are considered.
1
Introduction
The quantum Zen0 effect1i2 is a direct consequence of general features of the Schrodinger equation that yield quadratic behavior of the survival probability at short time^.^,^ It consists in the hindrance of the evolution of a quantum system when very frequent measurements are performed, in order to ascertain whether it is still in its initial state. Following an interesting idea by both the experimental and theoretical investigations of the last decade have dealt with oscillating (mainly, two-level) systems.6 However, a few years ago, the presence of a short-time quadratic region was experimentally confirmed for an unstable quantum mechanical system7 and the existence of the Zeno effect (as well as its inverse8i9) has been recently proved.” Interestingly, the quantum Zeno effect (QZE) does not necessarily freeze everything. On the contrary, for frequent projections onto a multi-dimensional subspace, the system can evolve away from its initial state, although it remains in the subspace defined by the “measurement.” This continuing time evolution within the projected subspace has been recently investigated and called quantum Zeno dynamics.” It involves interesting, yet unsettled, physical and m a t h e r n a t i ~ a l ~issues. ~.~~,~ The above-mentioned investigations deal with “pulsed” measurements, according to von Neumann’s projection postulate. l 4 However, from a physical point of view, a “measurement” is nothing but an interaction with an external system (another quantum object, or a field, or simply a different degree of freedom of the very system investigated), playing the role of apparatus. In
198
this sense von Neumann’s postulate can be considered as a useful shorthand notation, summarizing the effect of the quantum measurement. By including the apparatus in the quantum description, several author^,^%^^-*^ during the last two decades, have demonstrated the QZE without making use of projection operators (and non-unitary dynamics). In particular, the QZE has been reformulated in terms of “continuou~” measurement^,^-'^ obtaining the same physical effects (as well as a quantitative comparison with the “pulsed” situation18) in terms of a continuous (eventually strong) coupling to an external agent. The studies of the last few years pave the way to interesting possible applications of the QZE. Indeed, we have a physical and mathematical framework that enable us to analyze the modification of the evolution of a quantum system and possibly to tailor the interaction in order to slow the evolution down (or eventually accelerate it). The potential importance of such a scheme cannot be underestimated. It is therefore important to understand in more details which features of the coupling between the “observed” system and the “measuring” apparatus are needed to obtain a QZE. In other words, one wants to know when an external quantum system can be considered a good apparatus and why. The purpose of the present article is to clarify these issues and cast the quantum Zen0 evolution in terms of an adiabatic theorem. We will show that the evolution of a quantum system under the action of a continuous measurement process can be profoundly modified: the system is forced to evolve in a set of orthogonal subspaces of the total Hilbert space and an effective superselection ~~ rule arises in the strong coupling limit. These quantum Zeno s u b ~ p a c e sare just the eigenspaces (belonging to different eigenvalues) of the Hamiltonian describing the interaction between the system and the apparatus: they are subspaces that the measurement process is able to distinguish. The general ideas will be applied to some relevant examples. Some interesting issues and possible applications will be discussed in details. 2
Pulsed measurements
Let Q be a quantum system, whose states belong to the Hilbert space ‘H and whose evolution is described by the unitary operator U ( t )= exp(-iHt), where H is a time-independent lower-bounded Hamiltonian. Let P be a projection operator and RanP = ‘ l i p its range. We assume that the initial density matrix po of system Q belongs to ‘Hp: Po = PPOP,
Tr[poP]= 1.
199
Under the action of the Hamiltonian H (i.e., if no measurements are performed in order to get information about the quantum state), the state at time t reads A t ) = U(t)poU+(t)
(2)
and the survival probability, namely the probability that the system is still in H p at time t, is P ( t ) = TI- [U(t)PoUt(t)P].
(3)
No distinction is made between one- and many-dimensional projections. The above evolution is ‘%undisturbed,”in the sense that the quantum systems evolves only under the action of its Hamiltonian for a time t , without undergoing any measurement process. Assume, on the other hand, that we do perform a selective measurement at time T , in order to check whether Q has survived inside H p . By this, we mean that we select the survived component and stop the other one. The state of Q changes (up to a normalization constant) into Po
+
P(.)
and the survival probability in
= PU(-r)POU’(~P
H p
p ( T ) = Tr [U(T)poUt(,)PI =
(4)
is
v(T)
[V(T)poVt( T ) ] ,
PU(7)p. (5)
We stress that the measurement occurs instantaneously (this is the essence of von Neumann’s projection postulate) .14 The QZE is the following. We prepare Q in the initial state PO at time 0 and perform a series of P-observations at time intervals T = t / N . The state of Q at time t reads (up to a normalization constant) p y t ) = V,(t)poVi(t),
and the survival probability in
H p
VN(t)E [P U ( t / N) P ]N
(6)
is given by
dN)(t= ) Tr [VN(t)POVL(t)]
’
(7)
Equations (6)-(7) are the formal statement of the QZE, according to which very frequent observations modify the dynamics of the quantum system: under general conditions, if N is sufficiently large, all transitions outside H p are inhibited. Notice that the dynamics (6)-(7) is not reversible. We emphasize that close scrutiny of the features of the survival probability has clarified that if N is not too large the system can display an inverse Zen0 e f f e ~ t , *by ~ ~which , ~ ~ decay is accelerated. Both effects have recently been seen in the same experimental setup.1° We will not elaborate on this here.
200
2.1
Misra and Sudarshan’s theorem
In order to consider the N + 03 limit (“continuous observation”), one needs some mathematical requirements: assume that the limit operator
V(t)
lim V N ( ~ ) “ 0 3
exists (in the strong sense) for t
> 0. The final state of Q is then
and the probability to find the system in
7-tp
is
By assuming the strong continuity of V ( t )at t = 0 lim V ( t )= P,
t+o+
one can prove that under general conditions the operators
V ( t ) exist for all real t and form a semigroup.
(12)
Moreover, by time-reversal invariance
v + ( t )= V ( - t ) , so that V t ( t ) V ( t = ) P. This implies, by (l),that
If the particle is “continuously” observed, in order to check whether it has survived inside H p , it will never make a transition to 7-t; (QZE). Two important remarks are now in order: first, it is not clear whether the dynamics in the N -+ 03 limit is time reversible. Although one ends up, in general, with a semigroup, there are concrete elements of reversibility in the above equations.” Second, the theorem just summarized does not state that the system remains in its initial state, after the series of very frequent measurements. Rather, the system evolves in the subspace H p , instead of evolving “naturally” in the total Hilbert space 7-t.
201
2.2
Complete measurements
Let us first consider the particular case of complete selective measurements. HFlphas dimension 1 and the initial state is a pure (normalized) state la): p a = la)(al,
PO = Ia)(aI.
(15)
The time evolution operator (6) after N measurements in a time interval t reads vN(t) = [PaU(t/N)PaIN = IPa(aIu(t/N)Ia)IN= PaA(t/NIN,
(16)
where A(t) is the (undisturbed) survival amplitude in state la) at time t A(t) = (a(e-iHtla).
(17)
Therefore, in this case the problem of the existence of the limit operator V(t) is reduced to the existence of the limit function limN A(t/N)N. Let T = t / N be the time interval between two successive measurements. We can write N
A
(k)
] exp = exp [ N l o g A ( ~ )= =
where 1
y(7) = -7 1ogIA(T)l2,
1
W(T)
= -- argA(T), T
(19)
and the “observed” survival probability has a purely exponential decay with an effective rate ~ ( 7 ) : pLN)(t)= Id(T)12N = exp[-y(~)t].
(20)
From (16) and (18) one sees that V(t) in (8) exists (in the strong sense), for t > 0, if and only if y(O+) and w(O+) exist and are finite [if y(O+) = +m, the existence of the limit w(O+) is irrelevant], and it reads
Notice that, for one-dimensional projections, when V ( t )exists [and y(O+) < 00 so that V(t) # 01, the strong continuity in the origin V(O+) = Pa [see Eq. (ll)]follows from the very existence of (21) and need not be assumed as an independent hypothesis.
202
A sufficient condition for the existence of V ( t ) is that the initial state belongs to the domain of the Hamiltonian, la) E D ( H ) . Indeed, in such a case, the first and second moment of the Hamiltonian exist,
(alHla) = Ea < 00,
(alH21a)= l l H l ~ ) 1 1<~00,
(22)
and the survival amplitude has the following asymptotic behavior at short times:a r2 r + 0, 2 i.e. the survival probability exhibits a short-time quadratic behavior,
d(r) 1 - iEar - (alH21a)-, N
where
rz
=
1
J ( a I H 2 l 4 - (alH142
is called the Zeno time. Therefore, by plugging (23) into (19), ~ ( 7N )
Ea,
r
+ 0,
and one gets
V ( t )= Pa exp (-iE,t),
p ( t ) = PO,
P,(t) = 1.
More and more frequent measurements hinder the evolution and eventually freeze it. This is the quantum Zeno paradox: “A watched pot never boils”. Notice that V ( t ) in (27) form a strongly continuous one-parameter unitary group within ‘Hp. Therefore, starting from the dynamics (16), which is irreversible and probability-nonconserving, one ends up with a fully reversible evolution. Reversibility is recovered in the limit. For one-dimensional projections, interesting behaviors can be obtained (in the very limit N + 00) only by relaxing the hypothesis la) E D ( H ) . In such a case the survival probability, at variance with (24), can be no longer quadratic at short times and one can obtain different results depending on its short-time behavior.21 Assume, for example, that
aMore precisely, without further information on the third moment, A ( r ) = 1 - iE,r ( C Z ~ H ~ ~ ~ o) (Tr 2~ )/whence ,Z $7) = r/ri o(r).
+
+
-
203 for some a > 0. The effective decay rate (19) becomes
When a > 1, y(O+) = 0 and the limit (27) is recovered again. On the other hand, for 0 < cy < 1, y(O+) = +co, and the limit operator V(t) in Eq. (21) vanishes. In such a case irreversibility is still present in the limit and probability is immediately lost. Notice that in this case the limiting operator is not continuous at t = 0, V(O+) = 0 # V ( 0 ) = Pa,whence Misra and Sudarshan's theorem does not apply. [Note: from (16), V ( 0 )= limN v ~ ( 0 = ) pa .I An interesting case arises at the threshold value cy = 1, when y(O+) = l / r c # 0. Irreversibility is still present in the limit, but it manifests itself in a gentler way, through the limit operator for t 2 0 :
V ( t )= Pa exp(-t/2rc),
(30)
probability is not conserved and the Zen0 paradox does not arise. Notice that in Eq. (30) we had to assume w(O+) < 00, in order to assure the existence of
V(t). 2.3 Incomplete measurements In the case of incomplete measurements some outcomes are lumped together, for example, because the experimental equipment has insufficient resolution.22 Therefore, the projection operator P , which selects a particular lump, is manydimensional. Let us first consider a finite dimensional ' l i p = RanP,
TrP=s,
with
s
< 00.
(31)
The resulting time evolution operator is just a generalization of (16) to a finite dimensional matrix and has the explicit form [see (27)]
As shown in Sec. 2.2, if
' l i p C D ( H ) , then V(t) in (32) is unitary within and is generated by a resulting self-adjoint Hamiltonian PHP.23 Again, reversibility is recovered in the limit. For infinite dimensional projection, s = 00,one can always formally write the limiting evolution in the'form (32) but has to define the meaning of P H P . In such a case the time evolution operator V ( t ) may be not unitary and interesting phenomena can arise, related to the self-adjointness of the limiting 'lip
204
Hamiltonian PHP. This an interesting probleml1*l2that will not be discussed here. In general, for incomplete selective measurements, the system Q does not remain in its initial state, after the series of very frequent measurements. Rather, the system is confined (and evolves) in the subspace F l p , instead of evolving “naturally” in the total Hilbert space X. 2.4
Nonselective measurements
We will now consider the case of nonselective measurements and extend Misra and Sudarshan’s theorem’ in order to accomodate multiple projectors and to build a bridge for our subsequent discussion. For nonselective measurements the measuring apparatus functions, but no selection of outcomes is and all “beams” go through the whole Zen0 dynamics. Let
n
be a (countable) collection of projection operators and RanP, = l i p , the relative subspaces. This induces a partition on the total Hilbert space
‘H = @FlP*.
(34)
n
Consider the associated nonselective measurement described by the superoperator Pp=
c
PnpPn.
(35)
n
The free evolution reads 6 t p 0 = U(t)poUt(t),
U(t) = exp(-iHt)
(36)
and the Zen0 evolution after N measurements in a time t is governed by the superoperator
which yields
where
205 We follow Misra and Sudarshan2 and assume, as in Sec. 2.1, the existence of the strong limits (t > 0)
Vn . Then V n ( t )exist for all real t and form a semigroup,2 and
VA(t)V,(t) = P,.
(41)
Moreover, it is easy to show that for n' # n. Therefore the final state is with
C VL ( t) V , (t ) =
CPn= 1. (43)
n
n
The components V,(t)poVA(t) make up a block diagonal matrix: the initial density matrix is reduced to a mixture and any interference between different subspaces l i p n is destroyed (complete decoherence) . In conclusion,
P,(t) = Tr "Pnl
=T r [POP,] = Pn(O),
vn.
(44)
In words, probability is conserved in each subspace and no probability "leakage" between any two subspaces is possible. The total Hilbert space splits into invariant subspaces and the different components of the wave function (or density matrix) evolve independently within each sector. One can think of the total Hilbert space as the shell of a tortoise, each invariant subspace being one of the scales. Motion among different scales is impossible. (See Fig. 1 in the following.) If TrP, = s, < 00, then the limiting evolution operator V,(t) (40) within the subspace lip,, has the form (32),
V,(t)
= P,exp(-iP,HP,t),
(45)
is unitary in lip,, and the resulting Hamiltonian P,HP, is self-adjoint, provided that l i p , c D ( H ) . The original limit result (14) is reobtained when p , = 1 for some n, in (44): the initial state is then in one of the invariant subspaces and the survival probability in that subspace remains unity. However, even if the limits are the same, notice that the setup described here is conceptually different from that of Sec. 2.1. Indeed, the dynamics (39) allows transitions among different subspaces lip,, + lip,,,, while the dynamics (6) completely forbids them. Therefore, for finite N , (39) takes into account the possibility that
206 one subspace lip,, gets repopulatedZ5s4after the system has made transitions to other subspaces, while in (6) the system must be found in ‘Flp, at every measurement.
3
Dynamical quantum Zen0 effect
Our whole discussion has dealt so far with “pulsed” measurements, according to von Neumann’s projection p ~ s t u l a t e . ’However, ~ from a physical point of view, a “measurement” is nothing but an interaction with an external system (another quantum object, or a field, or simply another degree of freedom of the very system investigated), playing the role of apparatus. We emphasize that in such a case the QZE is a consequence of the dynamical features (i.e. the form factors) of the coupling between the system investigated and the external system, and no use is made of projection operators (and non-unitary dynamics). The idea of “continuous” measurement in a QZE context has been proposed several times during the last two decade^,^,'^,^^ although the first quantitative comparison with the “pulsed” situation is rather recent.18 We consider therefore a purely dynamical evolution, by including the detector in the quantum description. In one can consider the Hamiltonian
where H is the Hamiltonian of the system under observation (and can include is the inthe free Hamiltonian of the apparatus, H = Hsys Hdet) and H,,, teraction Hamiltonian between the system and the apparatus, K representing the strength of the measurement or, equivalently, the inverse response time of the apparatus (see examples in Sec. 4):
+
3.1 A theorem We now state a t h e ~ r e m , ~which ~ > ’ ~is the exact analog of Misra and Sudarshank theorem for a dynamical evolution of the type (46). Consider the time evolution operator
U K ( ~=) exp(-iHKt).
(47)
We will prove that in the “infinitely strong measurement” (“infinitely quick detector”) limit K + co the evolution operator
U ( t )= lim U K ( ~ ) , K-CU
207
becomes diagonal with respect to Hme,:
[U(t),Pn] = 0 , where Hrneaspn = VnPn, (49) Pn being the orthogonal projection onto ‘Flp,, , the eigenspace of Hmembelonging to the eigenvalue qn. Note that in Eq. (49) one has to consider distinct eigenvalues, i.e., qn # qm for n # m, whence the ‘Flp,’s are in general multidimensional. Moreover, the limiting evolution operator has the explicit form
where
is the diagonal part of the system Hamiltonian H with respect to the inter. action Hamiltonian H,, It is worth noticing that the limiting evolution (48) is understood in the sense of the intertwining relations (49), that is
while, strictly speaking, each single addend has no limit, due to a fast oscillating phase. In other words, one should read Eq. (50) as
U K ( t )= exp[-i(Hdiag + K H ~ ~ , + ) ~o (]l ) ,
for K
-+
co.
(53)
3.2 Dynamical superselection rules
Before proving the theorem of Sec. 3.1 let us briefly consider its physical implications. In the K -+ co limit, due to (49), the time evolution operator becomes diagonal with respect to H,, namely
[ U ( t ) Hrneasl , = 0, (54) a superselection rule arises and the total Hilbert space is split into subspaces ‘Flp,, which are invariant under the evolution. These subspaces are simply defined by the Pnk, i.e., they are eigenspaces belonging to distinct eigenvalues qn: in other words, subspaces that the apparatus is able to distinguish. On the other hand, due to (51), the dynamics within each Zen0 subspace %pn is governed by the diagonal part PnHPn of the system Hamiltonian H . The evolution reads p ( t )= U(.t)poUt(t)= e-i(Hd’ag+KHmB,. )tpOei(Hd’ag+KHm,,,)t (55)
208
and the probability to find the system in Ftp, pn(t) = =
[p(t)pn]= ~r [ ~ ( t ) p O ~ ~ ( t )~r ~ n[ ]~ ( t ) p o ~ n ~ ~ ( t ) ] [POP,]= P, ( 0 ) (56) 1
is constant. As a consequence, if the initial state of the system belongs to a specific sector, it will be forced to remain there forever (QZE): +o E ~ t --$ ~ +(t) , E
np,.
(57)
More generally, if the initial state is an incoherent superposition of the form po = Ppo, with P defined in (35), then each component will evolve separately, according to
p ( t ) = U ( t ) p o U t ( t )= =
C
c
nPo pn ei(Hdiag+KHme,.)t
e - i ( H d i a g+KH,,,.)tp
n
,-iP,HP,tp
nPo p n eiP,HPnt - CVn(t)poVi(t),
n
(58)
n
with V n ( t )= Pn exp(-iPnHPnt), which is exactly the same result (43)-(45) found for the case of nonselective pulsed measurements. This bridges the gap with the description of Sec. 2.4 and clarifies the role of the detection apparatus. In Fig. 1 we endeavored to give a pictorial representation of the decomposition of the Hilbert space as K is increased. 3.3 Proof of the theorem
We will now use perturbation theory and provez6 that the limiting evolution operator has the form (50). From that, property (49) follows. In the next section we will give a more direct proof of (49), which relies on the adiabatic theorem. Rewrite the time evolution operator in the form
U K ( ~=) exp(-iHKt)
= exp(-iHAr) = UA(T)
(59)
where
A = 1/K,
T = Kt = t / A ,
HA= AHK = Hmem+ AH,
(60)
and apply perturbation theory to the Hamiltonian HAfor small A. To this end, choose the unperturbed degenerate projections Pn,
H,,P,
= qnPna,
Pn
=
c a
pna,
(61)
209
K
Figure 1. The Hilbert space of the system: a superselection rule appears as the coupling K to the apparatus is increased.
whose degeneration a is resolved at some orger in the coupling constant A. This means that by denoting with qna and Pn, the eigenvalues and the orthogonal projections of the total Hamiltonian HA
-
-
HAP,, = VnaPna, they reduce to the unperturbed ones when the perturbation vanishes
-
pna
’3Pn,,
-
qna
A-0 +
qn.
(62)
(63)
Therefore, by applying standard perturbation theory,27 we get the eigenprojections
-
Pn, = Pn,
+ AP;;) + O(A2)
where
The perturbative expansion of the eigenvalues reads
210
where
Write now the spectral - decomposition of the evolution operator (59) in terms of the projections Pna
and plug in the perturbation expansions (64), to obtain uA(.)
=
C e-iqnaTpna n,a
Let us define a new operator
as
where Eqs. (66)-(68) were used. By plugging Eq. (71) into Eq. (70) and making use of the property
CP,H-=-~-HP Qn
Qn
n
an
n
n,
an
we finally obtain
Now, by recalling the definition (60), we can write the time evolution operator U K ( t ) as the sum of two terms 1
U K ( t )= K d , K ( t ) + $Lm,K(t),
(74)
21 1
where
is a diagonal, adiabatic evolution and
is the off-diagonal, nonadiabatic correction. In the K adiabatic term survives and one obtains
U K ( t ) = Uad,K(t)
+o
-+
( ~ - 1 )= e-i(KHmeas+Cn P n H P n ) t
00
limit only the
+ o ( ~ - 1 ),
(77)
which is formula (50) [and implies also (49)]. The proof is complete. As a byproduct we get the corrections to the exact limit, valid for large, but finite, values of K .
3.4 Zen0 evolution from an adiabatic theorem We now give an alternative proof [and a generalization to time-dependent Hamiltonians H ( t ) ]of Eq. (49). The adiabatic theorem deals with the time evolution operator U ( t ) when the Hamiltonian H ( t ) depends slowly on time. The traditional formulationz7 replaces the physical time t by the scaled time s = t / T and considers the solution of the scaled Schrodinger equation
d i-uT(S) ds
= TH(s)UT(s)
(78)
in the T + 00 limit. Given a family P ( s ) of smooth spectral projections of H ( s )
H ( s ) P ( s )= E ( s ) P ( s ) ,
(79)
the adiabatic time evolution U A ( S ) = limT,,UT(s) pr~perty’~,~~
has the intertwining
UA ( s ) P ( o )= P(s)uA(s)
that is, U A ( S )maps X p ( 0 ) onto ‘ X P ( ~ ) . Theorem (49) and its generalization,
valid for generic time dependent Hamiltonians,
212
are easily proven by recasting them in the form of an adiabatic theorem.lg In the H interaction picture, given by d
i-us(t) dt
=~
~ s ( t ) Hkea(t) , = uit(t)HmeasUs(t),
(83)
the Schrodinger equation reads d
i-UL(t)
= KHLea,(t)U L ( t ) . (84) dt The Zen0 evolution pertains to the K + 00 limit. And in such a limit Eq. (84) has exactly the same form of the adiabatic evolution (78): the large coupling K limit corresponds to the large time T limit and the physical time t to the scaled time s = t / T . Therefore, let us consider a spectral projection of Hikeas ( t )I
P ? m= u i ( t ) P n ( t ) w t ) ,
(85)
such that I
Hmeas(t)Pn(t)= ~ n ( t ) P n ( t ) .
Hme,(t)~A(t)= r/n(t)PA(t),
(86)
The limiting operator
has the intertwining property (80)
UI(t)P;(o) = P;(t)UI(t), i.e. maps
X p ( 0 ) onto
(88)
'H~lpt,(~):
$; E
XPt,(O)
+
$'(t) E
XPL(t).
(89)
In the Schrodinger picture the limiting operator
U ( t )= K-03 lim ~ s ( t ) ~ &=( t~)s ( t ) ~ ' ( t )
(90)
satisfies the intertwining property (81) [see (85)]
U (t)Pn( 0 ) = us (t)U' (t)Pn(0)= us (t)U' ( t p ; (0) =
U,(t)PA(t)UI(t) = Pn(t)Us(t)U'(t) = Pn(t)U(t),
(91)
and maps Xp,(o) onto X p , ( t ) : $0
E XP,(O)
+
$(t) E %,(t).
(92)
213
is constant: if the initial state of the system belongs to a given sector, it will
be forced to remain there forever (QZE). For a time independent Hamiltonian H,,,(t) = H,,,,, the projections are constant, P,(t) = P,, hence Eq. (81) reduces to (49) and the above property holds a f o r t i o r i and reduces to (56). 3.5
Generalizations
The formulation of a Zen0 dynamics in terms of an adiabatic theorem is powerful. Indeed one can use all the machinery of adiabatic theorems in order to get results in this context. An interesting extension would be to consider time-dependent measurements
Hme, = Hmeas (t) ,
(94)
whose spectral projections P, = P,(t) have a nontrivial time evolution. In this case, instead of confining the quantum state to a fixed sector, one can transport it along a given path (subspace) rFlpn(t),according to Eqs. (92)-(93). One then obtains a dynamical generalization of the process pioneered by Von Neumann in terms of projection operator^.^^^^^ 4
Applications
As a first example, consider the Hamiltonian
H&v=H+KHmem=
(95)
describing a two-level system, with Hamiltonian
+
H = R((11)(21 l2)(ll) = R
(a t a) 1 0 0
,
coupled to a third one, that plays the role of measuring apparatus:
214 This example was considered by Peres.16 One expects the third level to perform better as a measuring apparatus when the coupling K becomes larger. Indeed, if initially the system is in state Il),the survival probability reads4 1
+
p o ( t ) = - [ K 2 R2 cos(Klt)] K,4
K1 = d
2 K-w
+ 1,
m.
(98)
In spite of its simplicity, this model clarifies the physical meaning of a “continuous” measurement performed by an “external apparatus” (which can even be another degree of freedom of the system investigated). Also, it captures many interesting features of a Zen0 dynamics. Indeed, as K is increased, the Hilbert space is split into three invariant subspaces ‘H = @ ‘lipn,the three eigenspaces of Hmeas:
‘H~,= ((12) + 13))/Jz),
‘ H =~{II)}, ~
8p-, = ((12) - 13))/Jz},
(99)
corresponding to projections 1
0
0
0 0 0o ) ,
.=q0 0
0
0
0
”).
0
P - , = ; ( o 0-1
0 11 1 ) ,
0
1
(100)
with eigenvalues 170 = 0 and q+l = fl. Therefore the diagonal part of the system Hamiltonian H vanishes, Hd‘%= C P,HP, = 0, the Zen0 evolution is governed by Hdiag
+KH,,,=
(1: 1) 0
0
K
(101)
and any transition between 11) and 12) is inhibited: a watched pot never boils. Second example: consider
O H41ev = 00,
R
0
+ KT, + KIT; =
(102)
where states 11) and 12) make Rabi oscillations,
/o
1 0
\o
0
o\
0 01
21 5
while state 13) “observes” them,
/o
0 0
o\
and state 14) “observes” whether level 13) is populated, 0 0 0 0
K’7:
+ 13)(41) = K‘
= K’(14)(31
(105)
If K >> R and K’, then the total Hilbert space is split into the three eigenspaces of 71 [compare with (99)]:
zpo= { I I )14)), ,
zp_, = {(12)-13))/fih
’H~,= {(12)+13))/fi),
(106)
the Zen0 evolution is governed by
and the Rabi oscillations between states 11) and 12) are hindered. On the other hand, if K’ >> K >> R, the total Hilbert space is instead divided into the three eigenspaces of 7; [notice the differences with (106)l:
f i ~=; {II), 12)),
XP:,
EP; = {(13)+14))/Jz),
= {(13)-14))/fi)7
(108)
the Zen0 Hamiltonian reads
O R R O
diagt H4lev - ‘01
+K’7; = ( 0
0
0 0 0
0
0
K’
i t )
(109)
and the 0 oscillations are fully restored (in spite of K >> R).30 A watched cook can freely watch a boiling pot. Third example (decoherence-free s ~ b s p a c e sin~ quantum ~ computation). The H a m i l t ~ n i a n ~ ~ 2
Hmeas= Z g x ( b 12)ii(ll - bt ll)ii(21) - Z d t b i=l
(110)
216
describes a system of two (i = 1 , 2 ) three-level atoms in a cavity. The atoms are in a A configuration with split ground states l0)i and ll)% and excited state I 2 ) i , while the cavity has a single resonator mode b in resonance with the atomic transition 1-2. Spontaneous emission inside the cavity is neglected, but a photon leaks out through the nonideal mirrors with a rate n. The excitation number N ,
N=
c
12)ii(21+b+b,
i=1,2
commutes with the Hamiltonian,
Therefore we can solve the eigenvalue equation inside each eigenspace of N . A comment is now in order. Strictly speaking, the Hamiltonian (110) is non-Hermitian and we can not apply directly the theorem of Sec. 3.1. (Notice that the proof of the theorem heavily hinges upon Hermiticity of Hamiltonians and unitarity of evolutions.) However, we can enlarge our Hilbert space Z, by including the photon modes outside the cavity a, and their coupling with the cavity mode b. The enlarged dynamics is generated by the Hermitiun Hamiltonian
+
J
dwwaLa,+
6J
d~ [aLb+ u,bt] .
It is easy to show that the evolution engendered by H,,,, when projected back to Z, is given by the effective non-Hermitian Hamiltonian (110), provided the field outside the cavity is initially in the vacuum state. Notice that any complex eigenvalue of H,,, engenders a dissipation of Z into the enlarged Hilbert space embedding it. On the other hand, any real eigenvalue of H,,, which preserves probability within Z. Hence it generates a unitary dynamics is also an eigenvalue of H,,, and its eigenvectors are the eigenvectors of the restriction Hmeasl~. Therefore, as a general rule, the theorem of Sec. 3.1 can be applied also to non-Hermitian measurement Hamiltonians Z,,,, , provided one restricts one’s attention only to their real eigenvalues. The eigenspace So corresponding to N = 0 is spanned by four vectors
217
where 10jlj2)denotes a state with no photons in the cavity and the atoms in state ljl)llj2)2. The restriction of Hmeasto SO is the null operator Hrneas Is0 = 0 ,
(115)
hence SOis a subspace of the eigenspace 'Ftp, of Hmea belonging to the eigenvalue ~0 = 0
SO c ZP,,
Hmeaspo = 0.
(116)
The eigenspace S1 corresponding to N = 1 is spanned by eight vectors
s1 = {1020), l o w , IlOO), IlW, IW,1021), l o w , 1111)},
(117) to Sl is represented by the 8-dimensional matrix
and the restriction of H,,,
' 0 0 0 0 0 0 -ig 0 0 -ig 0 0 0 0 0 0
,
0 ig 0 0 0 0 0 ig 0 0 i n 0 0 0 0 0 -ir; 0 0 0 0 0 -in 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -ig -ig
0 0 0 0 0 ig ig -iK
The eigenvector (1021) - 1012))/& has eigenvalue 770 = 0 and all the others have eigenvalues with negative imaginary parts. Moreover, all restrictions Hme,ls, with n > 1 have eigenvalues with negative imaginary parts. Indeed they are spanned by states containing at least one photon, which dissipates through nonideal mirrors, according to -inbtb in (110). The only exception is state 10,2,2) of S2,but also in this case it easy to prove that all eigenstates of HmeasIs2 dissipate. Therefore the eigenspace 'Ftp, of Hmeasbelonging to the eigenvalue 70= 0 is 5-dimensional and is spanned by 'FlP, = {IOOO), IOOl), lolo), l o w , (1021) - l O W / J Z } , (119) If the coupling g and the cavity loss K are sufficiently strong, any other weak Hamiltonian H added to (110) reduces to PoHPo and changes the state of the system only within the decoherence-free subspace (119). Fourth example. Let
Hdecay= H
+ KH,,,
=
(e T;'
-1
-i2/$7
);
.
(120)
0
This describes the spontaneous emission 11) + 12) of a system into a (structured) continuum, while level 12) is resonantly coupled to a third level 13).4
21 8 This case is also relevant for quantum computation, if one is interested in protecting a given subspace (level 11)) from d e c o h e r e n ~ e ~by~ vinhibiting ~~ spontaneous emission.33 Here y represents the decay rate to the continuum and TZ is the Zen0 time (convexity of the initial quadratic region). Notice that, in a certain sense, this situation is complementary to that in (110); here the measurement Hamiltonian Hmeasis Hermitian, while the system Hamiltonian H is not. Again, we have to enlarge our Hilbert space, apply the theorem to the dilation and project back the Zen0 evolution. As a result one can simply apply the theorem to the original Hamiltonian, for, in has a complete set of orthogonal projections that univocally this case, H,,, defines a partition of ‘FI into quantum Zen0 subspaces. We shall elaborate further on this interesting aspect in a future work. As the Rabi frequency K is increased one is able to hinder spontaneous emission from level 11) (to be protected) to level 12). However, in order to get an effective “protection7’ of level Il), one needs K > I / T z . More to this, when the presence of the inverse Zen0 effect is taken into account, this . these requirement becomes even more stringentg and yields K > 1 / ~ i y Both conditions can be very demanding for a real system subject to d i s ~ i p a t i o n . For instance, typical values for spontaneous decay in vacuum are y N 7; N 10-2gs2 and 1 / ~ g y N 1020s-1.34
5
Conclusions
If very frequent measurements are performed on a quantum system, in order to ascertain whether it is still in its initial state, transitions to other states are hindered and the QZE takes place. This formulation of the QZE hinges upon the notion of pulsed measurements, according to von Neumann’s projection postulate. However, as we have seen by means of several examples, a “measurement” is nothing but an interaction with an external system (another quantum object, or a field, or simply another degree of freedom of the very system investigated), playing the role of apparatus. This enables one to reformulate the QZE in terms of a (strong) coupling to an external agent and to cast the quantum Zen0 evolution in terms of an adiabatic theorem. There are many interesting examples, varying from quantum computation to decoherence-free subspaces to “protection” from decoherence. Additional work is in progress, also in view of possible practical applications.
Acknowledgments
I am grateful to S. Pascazio for helpful discussions.
21 9
References
1. A. Beskow and J. Nilsson, Arkiv fiir Fysik 34,561 (1967); L. A. Khalfin, JETP Letters 8, 65 (1968). 2. B. Misra and E. C. G. Sudarshan, J . Math. Phys. 18,756 (1977). 3. H. Nakazato, M. Namiki, and S. Pascazio, Int. J. Mod. Phys. B 10,247 (1996); D. Home and M. A. B. Whitaker, Ann. Phys. 258,237 (1997). 4. P. Facchi and S. Pascazio, Progress in Optics, edited by E. Wolf (Elsevier, Amsterdam, 2001), Vol. 42, Ch. 3, p. 147. 5. R. J. Cook, Phys. Scr. T 21,49 (1988). 6. W. H. Itano, D. J. Heinzen, J. J. Bollinger, and D. J. Wineland, Phys. Rev. A 41,2295 (1990); T.Petrosky, S. Tasaki, and I. Prigogine, Phys. Lett. A 151, 109 (1990); Physica A 170,306 (1991); A. Peres and A. Ron, Phys. Rev. A 42,5720 (1990); S.Pascazio, M. Namiki, G. Badurek, and H. Rauch, Phys. Lett. A 179,155 (1993); T. P. Altenmuller and A. Schenzle, Phys. Rev. A 49,2016 (1994); J. I. Cirac, A. Schenzle, and P. Zoller, Europhys. Lett. 27, 123 (1994); S. Pascazio and M. Namiki, Phys. Rev. A 50,4582 (1994); P. Kwiat, H. Weinfurter, T. Herzog, A. Zeilinger, and M. Kasevich, Phys. Rev. Lett. 74,4763 (1995); A. Beige and G. Hegerfeldt, Phys. Rev. A 53,53 (1996); A. Luis and J. Periiia, Phys. Rev. Lett. 76,4340 (1996). 7. S. R. Wilkinson, C. F. Bharucha, M. C. Fischer, K. W. Madison, P. R. Morrow, Q. Niu, B. Sundaram, and M. G. Raizen, Nature 387,575 ( 1997). 8. A. M. Lane, Phys. Lett. A 99,359 (1983); W. C. Schieve, L. P. Horwitz, and J. Levitan, Phys. Lett. A 136,264 (1989); A. G. Kofman and G. Kurizki, Nature 405,546 (2000). 9. P. Facchi, H. Nakazato, and S. Pascazio, Phys. Rev. Lett. 86, 2699 (2001). 10. M.C. Fischer, B. Gutikrrez-Medina, and M.G. Raizen, Phys. Rev. Lett. 87,040402 (2001). 11. P. Facchi, V. Gorini, G. Marmo, S. Pascazio, and E. C. G. Sudarshan, Phys. Lett. A 275,12 (2000); P. Facchi, S. Pascazio, A. Scardicchio, and L. S. Schulman, Phys. Rev. A 65,012108 (2002). 12. C. N. Friedman, Indiana Univ. Math. J. 21,1001 (1972). 13. K. Gustafson, Irreversibility questions in chemistry, quantum-counting, and time-delay, in Energy storage and redistribution in molecules, edited by J. Hinze (Plenum, 1983), and refs. [10,12] therein. See also K. Gustafson and B. Misra, Lett. Math. Phys. 1,275 (1976). 14. J. von Neumann, Mathematical Foundation of Quantum Mechanics
220 (Princeton University Press, Princeton, 1955). 15. P. Facchi and S. Pascazio, Quantum Zeno effects with “pulsed”and ‘%ontinuous” measurements”, in Time’s arrows, quantum measurements and superluminal behavior, edited by D. Mugnai, A. Ranfagni, and L. S. Schulman (CNR, Rome, 2001) p. 139; Fortschr. Phys. 49,941 (2001). 16. A. Peres, Am. J . Phys. 48,931 (1980). 17. K. Kraus, Found. Phys. 11,547 (1981); A. Sudbery, Ann. Phys. 157, 512 (1984); A. Venugopalan and R. Ghosh, Phys. Lett. A 204,11 (1995); M. P. Plenio, P. L. Knight, and R. C. Thompson, Opt. Comm. 123,278 (1996); M. V. Berry and S. Klein, J. Mod. Opt. 43, 165 (1996). E. Mihokova, S. Pascazio, and L. S. Schulman, Phys. Rev. A 56,25 (1997); A. D. Panov, Phys. Lett. A 260,441 (1999); J. RehbEek, J. Pefina, P. Facchi, S. Pascazio, and L. MiSta, Phys. Rev. A 62,013804 (2000); P. Facchi and S. Pascazio, Phys. Rev. A 62,023804 (2000); B.Militello, A. Messina, and A. Napoli, Phys. Lett. A 286,369 (2001). 18. L. S. Schulman, Phys. Rev. A 57,1509 (1998). 19. P. Facchi and S. Pascazio, Phys. Rev. Lett. 89,080401 (2002). 20. P. Facchi and S. Pascazio, Quantum Zeno phenomena in quantum field theory, these proceedings (2002). 21. See, for example, J. G. Muga, G. W. Wei, and R. F. Snider, Europhys. Lett. 35,247 (1996); H.Nakazato, To Decay Or Not To Decay? Temporal behavior of a quantum system -Analysis based on a solvable model-, these proceedings (2002). 22. A. Peres, Quantum Theory: Concepts and Methods (Kluwer Academic Publishers, Dordrecht, 1998). 23. For an interesting example, see K. Machida, H. Nakazato, S. Pascazio, H. Rauch, and S. Yu, Phys. Rev. A 60,3448 (1999). 24. J. Schwinger, Proc. Nat. Acad. Sc. 45, 1552 (1959), reproduced in J. Schwinger, Quantum kinematics and dynamics (Perseus Publishing, New York, 1991). 25. H. Nakazato, M. Namiki, S. Pascazio, and H. Rauch, Phys. Lett. A 217, 203 (1996). 26. P. Facchi, Quantum Time Evolution: Free and Controlled Dynamics, PhD Thesis (October 2000) Bari. 27. A. Messiah, Quantum mechanics (Interscience, New York, 1961). 28. M. Born and V. Fock, Z. Phys. 51,165 (1928); T.Kato, J . Phys. SOC. Jap. 5,435 (1950). 29. Y. Aharonov and M. Vardi, Phys. Rev. D 21,2235 (1980). 30. B. Militello, A. Messina, and A. Napoli, Fortschr. Phys. 49,1041 (2001). 31. G. M. Palma, K. A. Suominen, A. K. Ekert, Proc. R. SOC. Lond. A
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222
UNSTABLE SYSTEMS AND QUANTUM ZEN0 PHENOMENA IN QUANTUM FIELD THEORY P. FACCHI AND S . PASCAZIO Dipartimento di Fisica, Universitd di Bari and Istituto Nazionale di Fisica Nucleare, Sezione d i Bari 1-70126 Bari, Italy E-mail: paolo.facchiOba.infn.it E-mail: [email protected] We analyze the Zen0 phenomenon in quantum field theory. The decay of an unstable system can be modified by changing the time interval between successive measurements (or by varying the coupling to an external system that plays the role of measuring apparatus). We speak of quantum Zen0 effect if the decay is slowed and of inverse quantum Zen0 (or Heraclitus) effect if it is accelerated. The analysis of the transition between these two regimes requires close scrutiny of the features of the interaction Hamiltonian. We look in detail at quantum field theoretical models of the Lee type.
1
Introduction
The seminal formulation of the quantum Zen0 effect by Misra and Sudarshan’ deals with unstable systems, i.e. systems that decay following an approximately exponential law.2 Such a formulation was implicit also in previous where the features of the “nondecay” amplitude and probability were investigated. The attention was diverted to oscillating systems, characterized by a finite Poincar6 time, when Cook published a remarkable paper,4 proposing to test the quantum Zen0 effect (QZE) on a two-level system undergoing Rabi oscillations. Although oscillating systems are somewhat less interesting in this context, they are also much simpler to analyze and indeed motivated interesting experiments5 and lively discussions,6 giving rise to new ideas.7 However, interesting new phenomena occur when one considers unstain particular, other regimes ble systems, whose PoincarC time is become possible, in which measurement accelerates the dynamical evolution, giving rise to an inverse quantum Zeno effect (IZE).11312113y14 The study of Zen0 effects for bona fide unstable systems requires the use of quantum field theoretical techniques and in particular the Weisskopf-Wigner appr~ximation’~ and the Fermi “golden” rule:16 for an unstable system the form factors of the interaction play a fundamental role and determine the occurrence of a Zen0 or an inverse Zen0 regime, depending of the physical parameters describing the system. The occurrence of new regimes is relevant
223 from an experimental perspective, in view of the beautiful experiments recently performed by Raizen’s group on the short-time non-exponential decay (leakage through a confining p ~ t e n t i a l ) ’and ~ on the Zen0 effects for such (nonoscillating) systems.18 We analyze here the transition from Zen0 to inverse Zen0 in a quantum field theoretical context, by looking in particular at the Lee model. This is a good prototype for other field-theoretical examples and is more general than one might think.’g’10 The usual approach to QZE and IZE makes use of “pulsed” observations of the quantum state. However, one obtains essentially the same effects by performing a “continuous” observation of the quantum state, e.g. by means of an intense field, that plays the role of external, “measuring” apparatus. This is not a new idea,20,14but has been put on a firmer basis only recently.” The “continuous” formulation of the QZE has been discussed in detail e l ~ e w h e r e ~ and ’ ? ~ will ~ be briefly reconsidered here, by focusing in particular on quantum field theory and its interesting peculiarities, leading to new effects. 2
Notation and preliminary notions: pulsed measurements
Let H be the total Hamiltonian of a quantum system and la) its initial state a t t = 0. The survival probability in state la) is
P ( t )= Id(t))’ = and a short-time expansion yields a quadratic behavior P(t)
N
1 - ?/Ti,
Ti2
(1)
= ( a ( H 2 ( a-) ( a ( H ( a ) 2 ,
(2)
where 7-2 is the Zen0 time. Observe that if the Hamiltonian is divided into a free and an interaction parts
H
+ Hint,
= HO
with
Hola) = w,la),
(alHintla) = 0,
(3)
the Zen0 time reads
(aIHktla) (4) and depends only on the (off-diagonal) interaction Hamiltonian. Let us start from “pulsed” measurements, as in the seminal approach.’ The notion of “continuous” measurement will be discussed later. Perform N (instantaneous) measurements at time intervals 7 = t / N , in order to check whether the system is still in state la). The survival probability after the measurements reads 7;’
p ( N ) ( t=) ~
(
7= P )
=
( ~t / N ) N exp ( - ~ ’ / T ; N ) N
N-03 -+
1.
(5)
224
Figure 1. Evolution with frequent “pulsed” measurements: quantum Zen0 effect. The dashed (full) line is the survival probability without (with) measurements. The gray line is the interpolating exponential (6).
If N = M the evolution is completely hindered. For very large (but finite) N the evolution is slowed down: indeed, the survival probability after N pulsed measurements (t = N T ) is interpolated by an exponential law’’
~ ( ~ )=(~t )(
7= exp(NlogP(7)) ) ~ = exp(-yeff(T)t),
with an eflective decay rate 1 2 %R(T)= -logP(.r) = --log 7 7 For
T +
-
2
IA(T)I= --Re[logA(~)] 20. 7
(6)
(7)
0 one gets P ( T ) exp(-T’/TZ), whence
T/TZ
reff(T) (T 0) (8) is a linear function of T . Increasingly frequent measurements tend to hinder the evolution. The physical meaning of the mathematical expression “T + 0” is a subtle that will be touched upon also in the present article. The Zen0 evolution is represented in Figure 1. N
3
+
From quantum Zen0 to inverse quantum Zen0 (“Heraclitus”)
Let us concentrate our attention on truly unstable systems, with a “natural” decay rate y, given by the Fermi “golden” rule.16 We ask: is it possible to find a finite time T * such that yeff(7*) = Y? (9) If such a time exists, then by performing measurements at time intervals T * the system decays according to its “natural” lifetime, as if no measurements were done. The quantity T* is related to Schulman’s “jump”
225
Figure 2. 2 < 1. (a) Determination of (“Heraclitus”)effect.
T*.
(b) Quantum Zen0 vs inverse quantum Zen0
By Eqs. (9) and (7) one gets
i.e., T* is the intersection between the curves P ( t ) and e - y t . In the situation outlined in Figure 2(a) such a time T* exists: the full line is the survival probability and the dashed line the exponential e-Yt [the dotted line is the asymptotic exponential 2e-Yt, that will be defined in Eq. (31)]. The physical meaning of T* can be understood by looking at Figure 2(b), where the dashed line represents a typical behavior of the survival probability P ( t )when no measurement is performed: the short-time Zen0 region is followed by an approximately exponential decay with a natural decay rate y. When measurements are performed at time intervals T , we get the effective decay rate r e f f ( ~The ) . full lines represent the survival probabilities and the dotted lines their exponential interpolations, according to (6). If T = 7 1 < T * one obtains QZE. Vice versa, if T = 7 2 > T * , one obtains IZE. If T = T* one recovers the natural lifetime, according to (9): in this sense, amusingly, T* can be viewed as a transition time from Zeno (who argued that a sped arrow, if observed, does not move) to Heraclitus (who replied that everything flows). Heraclitus opposed Zen0 and Parmenides’ (Zeno’s master) philosophical vie^.^^,^^ Sometimes (interestingly) T* does not exist: Eq. (9) may have no finite solutions. In such a case only QZE is possible and no IZE is attainable. This is in contrast with some recent claims,26according to which the inverse Zen0 regime is ubiquitous, as opposed to the quantum Zen0 one. We can get a qualitative idea of the meaning of these statements by looking a t the survival probability of an unstable system for sufficiently long times2
226
t Figure 3. 2 > 1. The full line is the survival probability, the dashed line the renormalized ~ If. P ( t ) and exponential e-rt and the dotted line the asymptotic exponential 2 ~ 7 (a) e-rt do not intersect, a finite solution T* does not exist. (b) If P ( t ) and e--7* intersect, a finite solution T * exists. (In this case there are always at least two intersections.)
where 2, the intersection of the asymptotic exponential with the t = 0 axis, is the square modulus of the residue of the pole of the propagator (wave function renormalization)2,2a,12 and will be defined in the next section [Eq. (31)]. A sufficient condition for the existence of a solution T * of Eq. (9) is that 2 < 1. This is easily proved by graphical inspection. The case 2 < 1 is shown in Figure 2(a): P ( t ) with the property (11) and e - Y t must intersect. The other case, 2 > 1, is shown in Figure 3: a solution may or may not exist, depending on the features of the model investigated. There are also situations (e.g., oscillatory systems, whose PoincarC time is finite) where y and 2 cannot be defined.22 The transition from Zen0 to inverse Zen0 is therefore a complex, model-dependent problem, that requires careful investigation. We shall come back to this issue in the following sections, where (11) will be derived for a particular field theoretical model.
4
The Lee Hamiltonian
Some of the most interesting Zen0 phenomena, including the transition from a Zen0 to a Heraclitus regime, arise in a quantum field theoretical f r a m e w ~ r k We .~~ will ~ ~now ~ ~study the time evolution of a quantum system in greater detail, by making use of a quantum field theoretical techniques, and discuss the primary role played by the form factors of the interaction. The Consider a generic Hamiltonian H and an initial normalized state.).1 total Hilbert space can always be decomposed into a direct sum '7-l = 'Ha$'7-ld, with 'Ha= PaN and '7-ld = pd'7-l, where Pa = /.)(a/ and p d = 1 - pa. Let us
227
accordingly split the total Hamiltonian into a free and an interaction part
H = HO+Hint,
(12)
where
Ho = P,HP,
+ PdHPd,
Hin+,= P,HPd -I- PdHP,.
(13)
This decomposition can always be performed,lg even in relativistic quantum field theory,1° the only “problem” being that the decomposition itself depends on the initial state la). Let {In)}be the eigenbasis of HO in ‘Fld
(ala) = 1, (aln) = 0 , (nln’)= bnnJ, Hola) = wala), Holn) = wnln).
(14) (15)
The interaction Hamiltonian Hint is completely off-diagonal and has nonvanishing matrix elements only between X, and %d, namely
Equations (14)-(16) completely determine the free and interaction Hamiltonians in terms of the chosen basis. Indeed we get
~o=wala)(al +):wnIn)(nI,
Hint = C ( g n I a ) ( n I +gLIn)(aI). (17) n
n
This is the Lee HamiltonianZ7 and was originally introduced as a solvable quantum field model for the study of the renormalization problem. The interaction of the normalizable state \ a ) with the states In) (the formal sum in the above equation usually represents an integral over a continuum of states) is responsible for its decay and depends on the f o r m factor gn. The Fourier-Laplace transform of the survival amplitude d(t)in (1) is the expectation value of the resolvent
the Bromwich path B being a horizontal line I m E =constant> 0 in the half plane of convergence of the Fourier-Laplace transform (upper half plane). By performing Dyson’s resummation, the propagator reads
Ga(E)==
1
E - w, - C,(E)’
228
where the self-energy function
consists only of a second order contribution and is related to the form factor gn by the equation
n
n
A comment is now in order. If one is only interested in the survival amplitude [or, equivalently, in the expression of the propagator (19)] and not in the details of the interaction gn between la) and different states In) with the same energy wn = w , one can simply replace this set of states with a single, representative state lw), by replacing the Hamiltonian (17) with the following equivalent one
In terms of the Hamiltonian (22) the self-energy function simply reads
5
Unstable systems
We consider now the case of an unstable system. The initial state has energy > wg (w, being the lower bound of the continuous spectrum of the Hamiltonian H ) and is therefore embedded in the continuous spectrum of H . If -C,(w,) < w, (which happens for sufficiently smooth form factors and small coupling), the resolvent is analytic in the whole complex plane cut along the real axis (continuous spectrum of H ) . On the other hand, there exists a pole Epolelocated just below the branch cut in the second Riemann sheet, solution of the equation
w,
229 Ca11 being the determination of the self-energy function in the second sheet. The pole has a real and imaginary part Epole = wa
+ 6wa - i7/2
(26)
given by 6w, = ReCaII(Epole)N ReCa(w,
+ )'0i
y = -21m,Xa~~(Ep01e) N -21mCa(w,
S dw-
=P
+ )'0i
g2(w) , wa-w = 27rg2(w,),
(27) (28)
up to fourth order in the coupling constant. One recognizes the second-order energy shift bw, and the celebrated Fermi "golden" rule y.16 The survival amplitude has the general form A(t) = Apole(t)
+ &ut(t)>
(29)
where
Acut being the branch-cut contribution. At intermediate times, the pole contribution dominates the evolution and
P(t)
IApo1e(t)l2 = 2 e P Y t ,
2
2 = 11 - ~ ~ I I ( E p o 1 e ) l -,
(31)
where 2,the intersection of the asymptotic exponential with the t = 0 axis, is the wave function renormalization. We have found (11) and explicitly determined 2. Notice that, in order to obtain a purely exponential decay, one neglects all branch cut and/or other contributions from distant poles and considers only the contribution of the dominant pole. In other words, one does not look at the rich analytical structure of the propagator and retains only its dominant polar singularity. In this case the self-energy function becomes a constant (equal to its value at the pole), namely
where in the last equality we used the pole equation (25). This is the celebrated Weisskopf-Wigner appr~ximation'~ and yields a purely exponential behavior, A(t) = exp(-iEpolet), without short- and long-time corrections. As is well known, the exponential law is corrected by the cut contribution, which is responsible for a quadratic behavior at short times and a power law
230
at long times. In particular, at short times, by plugging (19) into (18) and changing the integration variable 77 = Et, Eq. (18) becomes
The self-energy function (24) has the following behavior at large energies
where we used Eq. (4) (and assumed the existence of the second moment of the Hamiltonian Hint). Therefore, the survival amplitude at small times has the asymptotic expansion
where
By closing the Bromwich path in Eq. (35) with a large semicircle in the lower half plane, the integral reduces to the sum of the residues at the real poles t771,2and the survival probability at small times reads
in agreement with the expansion (2). Notice that at short times the behavior is governed by two “effective” poles, that replace the global contribution of the cut and the pole on the second sheet. We will come back to this important point in the following sections.
6 Two-pole model and two-pole reduction We consider now a particular solvable model: let the form factor in (22) be Lorentzian
This describes, for instance, an atom-field coupling in a cavity with high finesse mirrors.28 (Notice that the Hamiltonian in this case is not lower bounded and
231
LE
E2X
C,.
-2A
W,
-iA
J
I
0
211
Figure 4. (a) Form factor g2(w)and initial state energy wa. (b) Poles of the propagator in the complex E-plane.
we expect no deviations from exponential behavior at very large timeszg) In this case one easily obtains (for ImE > 0 )
whence the propagator
E+iA Ga(E)= ( E - w a ) ( E i A ) - X2
+
has two poles in the lower half energy plane (see Fig. 4). Their values are
E1=wa+6wa-i- Y 2'
E2=-6wa-i
(A - - 3,
(41)
where
,
with
v2 = w: +4X2 - A 2 .
(42)
(Notice that v2 can be negative.) The propagator and the survival amplitude read
232
1 - R = Res[G,(E1)] =
1
- wa -
1 - EL(&)
+ SW, + i ( A y/2) + 2Sw, + i ( A y) -
w,
-
(45)
is the residue of the pole El of the propagator. The survival probability reads
P ( t )= Id(t)I2= 2exp(-yt)
+ 2Re[R*(1- R)e-z(wa+26wa)t 1 exp(-At)
+ P I 2 e x P [ - W - r)tl,
(46)
where 2 = 11 - RI2 is the wave function renormalization
2=
(Wa
(w,
+
+ (A - 7/2)’
+ 26wa)’ + (A
-
’
7)’
(47)
All the above formulas are exact. We now analyze some interesting limits of the model investigated. 6.1
Weak coupling
In the weak coupling limit X << w,,A, one obtains from Eq. (42)
Notice that the latter formula is the Fermi Golden Rule and El in (41) is the “dominant” pole. Indeed, the second exponential in Eq. (44) is damped very quickly, on a time scale A-l much faster than y-l, whence, after a short initial quadratic (Zeno) region of duration A-l, the decay becomes purely exponential with decay rate y. Notice that the corrections are of order X2
and the Zen0 time is r z = X-’ >> A - l , i.e. the initial quadratic (Zeno) region is much shorter than the Zen0 time: in general, the Zen0 time does not yield a correct estimate of the duration of the Zen0 region.9>22,30 (Beware of many erroneous claims in the literature!) The approximation P ( t ) CY 1- t2/rgholds for times t < A-’ << r z .
233
6.2 Large bandwidth In the limit of large bandwidth A > wa,A, from Eq. (42) one gets y = 2A2/A + O(Ap2) and in order to have a non trivial result with a finite decay rate, we let
A+m,
X+m,
x2
with
-=
A
Y = const. 2
(50)
In this limit the continuum has a flat band, g ( w ) = = const, and we expect to recover a purely exponential decay. Indeed, in this case one gets R = 0 and 6w, = 0, whence
so that the survival amplitude and probability read
A(t)= exp (-iw,t
Y -t)
and P ( t ) = exp(-yt), (52) 2 respectively. In this case the propagator (51) has only a simple pole and the survival probability (52) is purely exponential. -
6.3 Narrow bandwidth In the limit of narrow bandwidth A << w,, A, the form factor becomes &w)
(53)
= A26(w)
and the continuum is “concentrated” in w = 0. Therefore the continuum as a whole behaves as another discrete level and one obtains Rabi oscillations between the initial state) . 1 in ( 2 2 ) and this “collective” level (of energy w = 0). Indeed one gets y=o,
1
wa
R=2
6w,=--++,2
where
R=
J : A2+-
(54)
(55)
is the usual Rabi frequency of a two-level system with energy difference w, (= - 0) and coupling A. By (54) the survival amplitude and probability read
w,
234 If wa = 0, the survival probability (56) oscillates between 1 and 0. On the other hand, if w, # 0 the initial state never “decays” completely. Incidentally, notice that the Zen0 time is still TZ = X-’ and yields now a good estimate of the duration of the Zen0 region. This is, so to say, a “coincidence” due to the oscillatory features of the system. 6.4
Strong coupling
Another interesting case is that of strong coupling, X N A. This is a typical case in which the strong coupling provokes violent oscillations before the system reaches the asymptotic regime. In the limit X >> A, w,, we get
whence the survival amplitude reads
d(t)N exp (-i$t
-tt)
[(t + 7) w, + + (5 7) , i A e-ZXt
1 - w,+iA
.zit]
(581 >
,
which yields fast oscillations of frequency X damped at a rate A << A. 6.5 Two-pole reduction
We now show that the two-pole model introduced in this section is the first improvement, after the Weisskopf-Wigner pole, in the approximation of a generic quantum field model. First note that, according to the WeisskopfWigner approximation (32), an exponential decay is obtained by considering a constant self-energy function C, = - i y / 2 , i.e. a resolvent with a single pole with negative imaginary part (El in Figure 4). On the other hand, as we noted in Sec. 5 , the initial quadratic behavior of the survival amplitude is governed by two effective poles of the resolvent, which ultimately derive from the behavior (34) of the self-energy function at infinity (59) If one wants to capture this short-time behavior while keeping the exponential law at later times, and is not interested in the long-time power-law deviations, one can proceed in the following way. The requirement for having an exponential decay, with decay rate y for t + 00, translates into the behavior of the self-energy function for E + 0, namely in the requirement of having a
235 Weisskopf-Wigner constant self-energy function with negative imaginary part C,(O) = -ib.
( b > 0)
(60)
The simplest form of the self-energy function satisfying both requirements (59) and (60) is
By letting rZ = 1 / X and l/bri = A, this exactly becomes the self-energy function of the twc-pole model (39). Therefore the two-pole model is the simplest approximation that yields the short time quadratic behavior together with the long time exponential one. We call the technique outlined in this subsection “two-pole reduction.” It is useful if one wants to get a first idea of the temporal behavior of a quantum field, keeping information on the lifetime (Fermi golden rule), but also on the short-time Zen0 region. Note that the process outlined above can be iterated to find better a p proximations of the real self-energy function C , ( E ) by adding other poles and/or zeros. But notice also that this approach does not yield the inverse power-law tail. Indeed the latter is essentially due to the nonanalytic behavior of the self-energy function at the branching point, a feature that cannot be captured by a meromorphic function.
7 Modification of the Lee Hamiltonian We now introduce an interesting modification of the Lee Hamiltonian (22), that enables us to look at the Zen0 region from a different perspective. The Hamiltonian (22) describes the decay of a discrete state la) into a continuum of states lw) with a given form factor g(w). According to Eqs. (4) and (22), the Zen0 time is related to the integral of the squared form factor by the simple relation 1
=
J dw g2(w).
TZ
On the other hand, for a two-level system { l a ) , Ib)} with Hamiltonian
H
= X(la)(bl
+ lb)(aI),
TZ
= 1/X
(63)
the Zen0 time TZ is just the inverse off-diagonal element X of the Hamiltonian [and, of course, this is in agreement with Eq. (6?2), as shown by Eq. (53)]. This is therefore a simple system in which the Zen0 time is manifest in
236 the Hamiltonian itself. We seek now an equivalent decay model, that shares with the two-level model this nice property. To this end, let us add a new “intermediate” discrete state Ib) to the Lee model. Consider then the Rabi oscillation X of the two-level system la), Ib) and let the initial state la) decay only through state Ib), i.e. couple Ib) to a continuum with form factor g b ( W ) . In other words, the Hamiltonian (22) is substituted by the following one
H = wala)(al +WbIb)(bl
+A (Ia)(bl
+
+ Ib)(al) +
s1..
dw wlw)(wl gb(w)(lb)(wl
+ b)(bl).
(64)
We require that this Hamiltonian is equivalent to the original one in describing the decay of the initial state la). To this end, notice that the part of Hamiltonian describing the decay of state ( b ) (and neglecting the coupling with la)) is just a Lee Hamiltonian and yields
On the other hand, state la) couples only to state Ib) with a coupling A. Therefore the evolution of state la) is just a Rabi oscillation between state Ib) dressed by the continuum Iw) and state la), namely
+
Ga = GO, GO,AGbXGa ,
(66)
whence
Therefore, in the modified model, the self-energy function of the initial state la) is nothing but the coupling X2 times the dressed propagator Gb(E)
Equation (68) is the sought equivalence relation. One has to choose the auxiliary form factor g b ( W ) in Eq. (64) as a function of the original one g ( w ) , in order to satisfy this relation and get an equivalent description of the decay. Our interest in this equivalence is due to the asymptotic behavior of formula (68)
Ca(E)
N
A2
1
=-
7;E’
for E
+
co ,
(69)
237
Y
I4-
Figure 5. The decay of state la) into a Lorentzian continuum Iw) (a) is equivalent to a Rabi coupling of la) with a state Ib) that in turn exponentially decays into a flat continuum Iw) (b).
which displays the relation between the coupling X and the Zen0 time rz. Thus the Hamiltonian (64) explicitly reads
In the equivalent model, therefore, the initial quadratic behavior is singled out from the remaining part of the decay: the Zen0 region, i.e. the first oscillation, is nothing but the initial unperturbed Rabi oscillation between states la) and Ib) (which initially “represents” the original continuum as a whole). After the initial stage of the decay, the coupling g b ( W ) between Ib) and Iw) (namely the details of the original continuum) comes into play and modifies the initial Rabi oscillation with its characteristic time scale. This explains from a different perspective the difference, already stressed in previous sections, between the Zen0 time and the duration of the initial quadratic region. As an example, we recover the self-energy function (39) of the two-pole model, by requiring that Cb be constant
which implies gb(W)
=
and
Wb
=o.
In other words, the auxiliary state Ib) is placed at the mean energy of the original continuum g ( w ) and decays into a flat-band continuum with decay
rate T b = 2A: the decay into a Lorentzian continuum [Fig. 5(a)] is exactly equivalent to a Rabi coupling with a level that in turn exponentially decays into a flat continuum [Fig. 5(b)]:
Moreover, if one restricts one's attention to the subspace spanned by {la),Ib)}, it is easy to show22 that this Hermitian Hamiltonian reduces to the effective non-Hermitian one
=
(y
a>
721
=
2+
* 2
1 1+46W,yY
.,)
(74)
Therefore, if one is interested only in the decay of the initial state la), the study of the two-pole model reduces to the study of this simple non-Hermitian 2-dimensional matrix. One can reexamine all the results of previous sections just by looking at this matrix. We will not elaborate on this here. A final comment is now in order. One can draw a clear picture of the two-pole reduction, discussed in the previous section, just by looking at the construction of the equivalent model. The first approximation of a real decay, the Weisskopf-Wigner approximation, is represented by the simple exponential decay of level Ib) with its time scale 7;'. The two-pole approximation superimposes an oscillating dynamics with a timescale X - l to the latter, yielding the initial Zen0 region. By complicating the model with the addition of other dynamical elements with their characteristic scales, one can construct a better approximation of the real decay law. 8
ZeneHeraclitus transition
We will now study the Zeno-inverse Zen0 transition in greater detail, by making use of a quantum field theoretical framework, and discuss the primary role played by the form factors of the interaction. The reader should refer to the discussion of Secs. 2-3, where we introduced the effective decay rate 1 2 2 - y e ~ (=r )-- l ogP( r ) = -- log [A(.)[= --Re[ l o g A ( ~ ),] 7
7
7-
(75)
239
which is a linear function of Zen0 region)
T
for sufficiently small values of for
T
5 1/A,
T
(inside the (76)
and becomes, with excellent approximation, a constant equal to the natural decay rate at intermediate times yeff(7)= y
for
T
>> 1/A.
(77)
The transition between Zen0 and Heraclitus occurs at the geometrical intersection T * between the curves P ( t ) and e - T t , solution of the equation yeff(T*) = Y
>
(78)
as shown in Fig. 2. Let us corroborate these general findings by considering for example the
two-pole model studied in detail in Sec. 6, whose survival amplitude is given by Eq. (44)
with 6wa and y given by Eq. ( 4 2 ) . By plugging (79) into (75) one obtains the effective decay rate, whose behavior is displayed in Fig. 6 for different values of the ratio IwaI/A. These curves show that for large values of IwaI (in units A) there is indeed a transition from a Zen0 to an inverse Zen0 (Heraclitus) behavior: such a transition occurs at T = T * , solution of Eq. (78). However, for small values of ( w a ( ,such a solution ceases to exist. The determination of the critical value of lwaI for which the Zeno-inverse Zen0 transition ceases to take place discloses an interesting aspect of this issue. The problem can be discussed in general, but for the sake of simplicity we consider the weak coupling limit (small A) considered in Eqs. (48)-(49). According to the geometrical theorem proved in Sec. 3, a sufficient condition for the system to exhibit an Zeno-inverse Zen0 transition is that the wave function renormalization 2 < 1. In our case, by making use of Eq. (49), this condition reads
240
Figure 6. Effective decay rate yeff(r)for the two-pole model (79), for X = 0.1 and different values of the ratio / w a ( / A(indicated). The horizontal line shows the “natural” decay rate ) the solution T * of ECq. (78). The asymptotic value of y: its intersection with y e ~ ( 7yields all curves is 7,as expected. A Zen0 (inverse Zeno) effect is obtained for T < T * (T > T * ) . Notice the presence of a linear region for small values of T and observe that T * does not belong to such linear region as the ratio ( w , ( / A decreases. Above a certain threshold, given by Eq. (81) in the weak coupling limit of the model (and in general by the condition 2 = l ) , Eq. (78) has no finite solutions: only a Zeno effect is realizable in such a case.
namely
> A’
+ O(X2).
(81) The meaning of this relation is the following: a sufficient condition to obtain a Zeno-inverse Zen0 transition is that the energy of the decaying state be placed asymmetrically with respect to the peak of the form factor (bandwidth) (see Fig. 4). If, on the other hand, w a N 0 (center of the bandwidth), no transition time T * exists (see Fig. 6) and only a QZE is possible: this is the case analyzed in Fig. 3(a). There is more: Equation (79) yields a time scale. Indeed, from the definitions of the quantities in (42) one gets y/2 < A - y/2, so that the second exponential in (79) vanishes more quickly than the first one. (The two time scales become comparable only in the strong coupling regime: y -+ A as X -+ 00.) If the coupling is weak, since y = O(X2), the second term is very rapidly damped so that, after a short initial quadratic region of duration A-’, the decay becomes purely exponential with decay rate y. For r 5 l / A (which is, by definition, the duration of the quadratic Zen0 region), we can use the linear approximation (76). When it applies up to the intersection (i.e., OJ?
24 1
Iw,I >> A) one gets
r* pv yrg.
(82)
When w, gets closer to the peak of the form factor, the linear approximation does not hold and the r.h.s. of the above expression yields only a lower bound to the transition time T * . In this case the solution r* of Eq. (78) becomes larger than the approximation (82), eventually going to infinity when condition (81) is no longer valid. In such a case, only a QZE is possible and no IZE is attainable. This is in contrast with recent claims.26 The conclusions obtained for the two-pole model (79) are of general validity. In general, in the Lee Hamiltonian (22), for any g(w),we assume that w, > w g (the lower bound of the continuous spectrum), in order that the system be unstable. The matrix elements of the interaction Hamiltonian depend of course on the physical model considered. However, for physically relevant situations, the interaction smoothly vanishes for small values of w - wg and quickly drops to zero for w > A, a frequency cutoff related to the size of the decaying system and the characteristics of the environment. This is true both for cavities, as well as for typical EM decay processes in vacuum, where the bandwidth A ”/ is given by an inverse characteristic length12,22 (say, of the order of Bohr radius) and is much larger than the natural decay rate y N lo7 - 1ogS-l. For roughly bell-shaped form factors all the conclusions drawn for the Lorentzian model remain valid. The main role is played by the ratio wag/A, where wag = w, - wg is the available energy. In general, the asymmetry condition wag < A is satisfied if the energy w, of the unstable state is sufficiently close to the threshold. In fact, from Eq. (62) one has = Ti
] dw g2(w)
= g2(w)A,
(83)
where is defined by this relation and is of order w,,,, the energy at which g(w) takes the maximum value. For w, sufficiently close to the threshold w g one has g(w) >> g(wa),the time scale 7 2 .2 is well within the short-time regime, namely
where the Fermi golden rule y = 27rg2(w,) has been used, and therefore the estimate (82) is valid. On the other hand, for a system such that wag pv A (or, better, w, N center of the bandwidth), r* does not necessarily exist and usually only a Zen0
242
effect can occur. In this context, it is useful and interesting to remember that, as shown in Sec. 6.3, the Lorentzian form factor (38) yields, in the limit g 2 ( w ) = X26(w - wa), the physics of a two level system. This is also true in the general case, for a roughly symmetric form factor, when the bandwidth A --t 0. In such a case, if w, = 0 (energy of the initial state at the center of the form factor), the survival probability oscillates between 1 and 0 and only a QZE is possible. On the other hand, if wa # 0 (initial state energy strongly asymmetric with respect to the form factor of “width” A = 0) the initial state never decays completely. By measuring the system, the survival probability will vanish exponentially, independently of the strength of observation, whence only an IZE is possible. If one consider the large bandwidth limit of the two-pole model, which is equivalent to a Weisskopf-Wigner approximation, the propagator (51) has only a simple pole and the survival probability (52) is purely exponential. Therefore the measurements cannot modify the free behavior. Indeed, the conditions for the occurrence of Zen0 effects are always ascribable to the presence of an initial non-exponential behavior of the survival probability, which is caused by a propagator with a richer structure than a simple pole in the complex energy plane. 9
Continuous measurements
Most of our examples dealt with instantaneous measurements, according to the Copenhagen prescription. Our starting point was indeed Eq. (6). However, it is always possible to mimic the effect of pulsed measurements in terms of the coupling to a suitable system, performing a continuous measurement process. This issue has been discussed in other p a p e ~ s ,so ~ let ~ , us ~ ~only give here an example. Let us add to (22) the following Hamiltonian
as soon as a photon is emitted, it is coupled to another boson of frequency w‘ (notice that the coupling has no form factor). One can show that the dynamics of the Hamiltonian (22) and (85), in the relevant subspace, is generated by
and an effective continuous observation on the system is obtained by increasing Indeed, it is easy to see that the only effect due to I? in Eq. (86) is the
r.
243 substitut,ion of C,(E) with C,(E
+ ir/2) in Eq. (19), namely,
For large values of I’, i.e., for a very quick response of the apparatus, the self-energy function (34) has the asymptotic behavior
[Notice that r + co in (88) means r >> A, the frequency cutoff of the form factor.] In this case the propagator (87) reads
and the survival probability decays with the effective exponential rate (valid for r >> A)
Notice the similarity of this result with (8): r, the strength of the coupling to the (continuously) measuring system, plays the same role as r - l , the frequency of measurements in the pulsed formulation. This is a general result.22,21More to this, we have here a scale for the validity of the linear approximation (90) for ye^: the linear term in the asymptotic expansion (88) approximates well the self-energy function only for values of r that are larger than the bandwidth A. For smaller values of r one has to take into account the nonlinearities arising from the successive terms in the expansion. Note that the flat-band case (51), yielding a purely exponential decay, is also unaffected by a continuous measurement. Indeed in that case C,(E) = -iy/2 is a constant independent of E , whence C,(E+ir/2) = C,(E) is independent of I?. The same happens if one considers the Weisskopf-Wigner approximation (32): in this case one neglects the whole structure of the propagator in the complex energy plane and retains only the dominant pole near the real axis. This yields, as we have seen, a self-energy function which does not depends on energy and a purely exponential decay (without any deviations), that cannot be modified by any observations.
10
Conclusions
The form factors of the interaction Hamiltonian play a fundamental role when the quantum system is “unstable,” not only because of the very formulation
244
of the Fermi golden rule, but also because they may govern the transition from a Zen0 to an inverse Zen0 (Heraclitus) regime. The inverse quantum Zen0 effect has interesting applications and turns out to be relevant also in the context of quantum chaos and Anderson l o ~ a l i z a t i o n . ~ ~ Although the usual formulation of QZE in terms of repeated “pulsed” measurements ci la von Neumann is a very effective one and motivated quite a few theoretical proposals and experiments, we cannot help feeling that the use of continuous measurements (coupling with an external apparatus that gets entangled with the system) is advantageous. Both quantum Zen0 and inverse quantum Zen0 effects have been experimentally confirmed. It is probably time to refrain from academic discussions and look for possible applications.
Acknowledgments We thank Hiromichi Nakazato, Antonello Scardicchio and Larry Schulman for interesting discussions.
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Opt. 43, 165 (1996); M.P. Plenio, P.L. Knight, and R.C. Thompson, Opt. Comm. 123, 278 (1996); E. Mihokova, S. Pascazio, and L. S. Schulman, Phys. Rev. A 56, 25 (1997); B. Militello, A. Messina, and A. Napoli, Phys. Lett. A 286, 369 (2001); S. Maniscalco and A. Messina, Fortschr. Phys. 49, 1027 (2001). L.S. Schulman, Phys. Rev. A 57, 1509 (1998). P. Facchi and S. Pascazio, “Quantum Zen0 and inverse quantum Zen0 effects,” in Progress in Optics 42, edited by E. Wolf (Elsevier, Amsterdam, 2001), Ch. 3, p. 147. P. Facchi and S. Pascazio, “Quantum Zen0 effects with “pulsed” and “continuous” measurements”, in Time ’s arrows, quantum measurements and superluminal behavior, edited by D. Mugnai, A. Ranfagni, and L. S. Schulman (CNR, Rome, 2001) p. 139; Fortsch. Phys. 49, 941 (2001). L.S. Schulman, J. Phys. A 30, L293 (1997); L.S. Schulman, A. Ranfagni, and D. Mugnai, Phys. Scr. 49, 536 (1994). M. Namiki, S. Pascazio and H. Nakazato, Decoherence and Quantum Measurements (World Scientific, Singapore, 1997). A. G. Kofman, G. Kurizki, Phys. Rev. Lett. 87, 270405 (2001); Nature 405, 546 (2000). T.D. Lee, Phys Rev. 95, 1329 (1954). R. Lang, M.O. Scully, and W.E. Lamb, Jr., Phys Rev. A 7,1788 (1973); M. Ley and R. Loudon, J. Mod. Opt. 34, 227 (1987); J. Gea-Banacloche, N. Lu, L.M. Pedrotti, S. Prasad, M.O. Scully, and K. Wodkiewicz, 1990, Phys. Rev. A 41, 381. L.A. KhaEn, Dokl. Acad. Nauk USSR 115,277 (1957) [Sov. Phys. Dokl. 2, 340 (1957)]; Zh. Eksp. Teor. Fiz. 33, 1371 (1958) [Sou. Phys. JET 6 , 1053 (1958)]. I. Antoniou, E. Karpov and G. Pronko, Phys. Rev. A 63,062110 (2001). B. Kaulakys and V. Gontis, Phys. Rev. A 56, 1131 (1997); P. Facchi, S. Pascazio and A. Scardicchio, Phys. Rev. Lett. 83, 61 (1999); J.C. Flores, Phys. Rev. B 60, 30 (1999); B 62, R16291 (2000); S.A. Gurvitz, Phys. Rev. Lett. 85, 812 (2000); J. Gong and P. Brumer, Phys. Rev. Lett. 86, 1741 (2001); M.V. Berry, ‘‘ Chaos and the semiclassical limit of quantum mechanics (is the moon there when somebody looks?)” in Quantum Mechanics: Scientific perspectives on divine action, edited by Robert John Russell, Philip Clayton, Kirk Wegter-McNelly and John Polkinghorne, Vatican Observatory CTNS publications, p. 41.
247
THE TUNNELING TIME PROBLEM AND THE CONSISTENT HISTORY APPROACH TO QUANTUM MECHANICS N. YAMADA Department of Information Science, Fukui University, 3-9-1 Bunkyou, Fukui, Fukui 910-8507, Japan E-mail: [email protected]. ac.jp We review some of the well-known theories of tunneling time and introduce a new theory based on the consistent history approach to quantum mechanics (CHA). The CHA gives a striking result that a probability distribution of tunneling time is not definable. This forces us to reflect what is meaningful about tunneling time. Even in the absence of the probability distribution, the range of values of tunneling time is definable and well-known tunneling times are found to fall within the range for typical opaque barriers. By focusing our attention to the range, we would be able to develop constructive discussions about the tunneling time problem.
1 1.1
Tunneling Time Problem Introduction
More than 70 years after the successful explanation of alpha decay as a tunneling phenomenon by Gamow, Condon, and Gurney, tunneling has now become the most important quantum phenomenon with wide range of applications. Scanning tunneling microscope (STM) is a powerful tool in surface science and various types of semiconductor devices have been developed that utilize tunneling effect. In spite of these remarkable success, there still remains a fundamental question about tunneling. That is, we still do not have a good understanding about the time taken by a particle to tunnel through the potential barrier [l-91. This time, called the tunneling time, is the subject of the present paper. Correct understanding of tunneling time would be important for the developments of both basic and applied research of tunneling. Let us consider a tunneling phenomenon from a spacetime point of view. Fkom a spatial point of view, one would be interested in how many of the incident particles eventually appear on the other side of the barrier. The ratio of the number of the particles that have moved to the other side of the barrier to the total number of the incident particles is nothing but the tunneling probability. Thus we may say that the tunneling probability is related to the spatial aspects of tunneling. On the other hand, an idea of tunneling time arises when one tries to understand tunneling from a temporal point of view. Since the successful explanation of alpha decay as a tunneling phenomenon
248
in 1928, people’s interest has always been in theoretical calculations or experimental estimations of tunneling probabilities. “Applications of tunneling” have actually been the applications of tunneling probabilities. For example, STM and semiconductor tunneling devices utilize tunneling currents, and the currents are deeply related to tunneling probabilities. It is thus the spatial aspects of tunneling that have been studied and applied for decades with great success. By contrast, the temporal aspects of tunneling have not received much attention. There are two major reasons for this. One is that in practical applications it was almost sufficient to treat tunneling as a stationary process, and another reason is that there were no experimental means to observe such phenomena in solids whose time scales are of the order of to s, which are considered to be the typical values of tunneling time in semiconductor heterostructures. It must also be pointed out that the theories gave tunneling times that are qualitatively different. With conflicting theories and without experimental meansa to test the theories, the problem of tunneling time has not been realistic and only the spatial aspects of tunneling have received much attention. Today, the situation is changing due to the progress of time-resolved spectroscopy and the fabrication technology for nanometer scale structures. It is thus timely to reinvestigate the problem of tunneling time. A clear concept of tunneling time with experimental means to measure the time will undoubtedly turn the subject of tunneling time into a truly interesting area of research; for example, the correct understanding of tunneling time could help control the operation speed of quantum devices. The organization of the present paper is as follows. In the rest of this section, we deal with some important questions about the tunneling time problem. In Sec. 2, we review some of the well-known theories of tunneling time. Section 3 presents a new approach proposed by the present author. It leads us t o the conclusion that a probability distribution of tunneling time is not definable but we can still define the range of values of tunneling time. The present author claims that we should shift our focus to the range of tunneling time for constructive discussions. Concluding remarks are given in Sec. 4. 1.2 Some Remarks
Due to the controversy surrounding the problem over the past 70 years, people sometimes argue that it would not make sense to talk about tunneling time. It should make sense, however, simply because tunneling is a physical ‘Tunneling phenomena also exist in optics. Optical tunneling times are experimentally accessible [lo-121. We do not discuss optical tunneling times in the present paper.
249 phenomenon that occurs, as all physical phenomena do, in space-time. At the same time, however, the long-time controversy indicates that we have missed something important in earlier discussions. The theories of tunneling time proposed in the past were often based on implicit assumptions. Sometimes it was assumed that a unique tunneling time should be definable. Sometimes it was assumed that there should be a probability distribution of tunneling time. The present author considers that it is these implicit assumptions that must be critically reconsidered. The implicit assumptions look, at first sight, physically plausible; in fact they often give reasonable answers when the theories are applied to non-tunneling regimes (such as free propagations, the transmissions over a barrier or well), where our classical intuitions based on particle picture work. This means that the implicit assumptions are more or less based on the classical pictures of a particle. Tunneling is a purely quantum mechanical phenomenon, for which the validity of classical intuitions is not warranted. The present author considers that to put an end to the controversy it is necessary to remove classical mechanical intuitions from the discussions and to rely solely on the principles of quantum mechanics. The discussion given in Sec. 3 is securely rooted in the foundations of quantum mechanics. It is based on the real time Feynman’s path integral and on a quantum mechanical rule of constructing probabilities from amplitudes. The dynamics of a tunneling particle is completely described by the time dependent wave function. Does this mean that we do not have to bother to introduce the concept of tunneling time? To answer this question, it is helpful to consider the case of the life time of a particle tunneling out of a metastable state. The time dependent wave function contains all the information about the dynamics of the particle. How the particle escapes from the well can be completely understood by following the time evolution of the wave function. Normally, however, we are not interested in all the details of the decay process and thus not in all the details of the wave function. Instead, we have a special interest in how the probability of finding a particle in the well changes with time. The probability decreases exponentially with time except for very short and very long times, so that we may identify the time constant of the exponentially decaying factor as the quantity that characterizes the decay process in the intermediate time domain. In this way, from the complexvalued time dependent wave function that describes the full dynamics, we extract a single real number that is less informative than the wave function but is sufficiently useful to understand the decay process in the intermediate time domain. No one claims that the concept of life time needs not be introduced because the time dependent wave function tells us the full dynamics. In the same way, the fact that the full dynamics is described by the time dependent
250 wave function does not reduce our motivation of introducing the concept of tunneling time. Of course, as in the case of the life time, the tunneling time, if definable, should be extractable from the time dependent wave function. Thus appropriate questions would be as follows: Is it also possible to extract such quantities from the wave function that describe the amount of time the particle spends in the barrier region? If it is possible, what are the quantities? The tunneling time problem should be set up in this way. It is often argued that the absence of time operator in quantum mechanics is deeply related to the difficulties of the tunneling time problem. The author considers that this is off the point. Remember that the absence of time operator does not bring any difficulties in the study of life time. There seems to be no reason that the absence of time operator is a serious problem only in the case of tunneling time. Also, it must be added that the measurement of life time does not suffer from any sort of “measurement problem of time observables” in quantum mechanics. The quantum mechanical observable in life time measurements is the particle’s position inside (or outside) the well; we observe, at various times, whether the particle is inside or outside the well. It is not a novel “time observable” but the usual position observable that is measured and the time variable simply plays the usual role, that is, it is used to describe the times at which we make the measurements. Similarly, the measurements of tunneling time (although we do not know what they are now) would consist of appropriate measurements of ordinary observables and the time variable would not play more than the usual role.
2
Ideas Proposed in the Past
Here we will review some of the well-known ideas of defining tunneling times. A detailed review is not the purpose here, so that intermediate equations will be omitted as much as possible. To avoid complications, the descriptions will be made for a square barrier in one dimension. For detailed reviews of the tunneling time problem, see [ 1- 91. 2.1
Notation
We consider the square barrier of height VOthat covers the spatial region a < x < b. The Hamiltonian of the particle is
251
where m is the mass of the particle and 1 fora b,
{
{
+
where k > 0, and the complex coefficients R and T are the reflection amplitude and the transmission amplitude, respectively, satisfying \RI2 \TI2= 1. The transmission amplitude is given by
+
with kv, = m / h and d E b-a. The decomposition of T into its modulus and phase is often useful:
T = IT1 eis = IT1 eide-ikd
(4 = B + kd).
(5)
For a particle described by a wave packet Q(z,t ) (initially localized in z < a and moving towards the barrier), the tunneling (or transmission) probability P is given by
-+ IT(kO)l2
in the limit l+(k)12 + S(k - ko),
(8)
where +(k) is the k-space wave function, which is supposed to be vanishing for k < 0 and also for k > m / h (otherwise the particle can go over the barrier). Equation (8) gives the tunneling probability for the stationary case where the wave number is ko. The reflection probability P is given by P=1-P. 2.2 Phase Time by Following the Motion of Wave Packet This is probably the simplest idea, which goes back to Eisenbud [13],Bohm [14], and to Wigner [15]. Let us consider an incident particle described by a wave packet that is narrow in k-space, thus wide in real space. The energy
252
b
U
Figure 1. The phase time from the peak motion of the wave packet
of the particle is nearly monochromatic. The initial wave function can 'be written as
@(x,0 ) =
/$
$ ( k - ko)eikx,
(9)
where $J is such that I$(k - Ic0)l2 is sharply peaked around k = ko > 0 and satisfies Jdk1?,/12 = 1. At t = 0, the incident packet is supposed to be localized in the left side of the barrier and moving towards the barrier with the peak velocity wo = hko/m. At sufficiently later times, the wave function becomes the superposition of a reflected and a transmitted packets, which are substantially non-overlapping. The reflected packet has a clear peak in 3: < < a that leaves the barrier with the velocity -WO, while the transmitted packet has a clear peak in 3: >> b leaving the barrier with the velocity WO. Now, let us focus on the peak position z ( t ) of the transmitted packet and compare it with the peak position q ( t )of the free wave packet that started with the same initial condition. Arguing that the spatial shift of the peak, Az(t) = q ( t )- z ( t ) ,is due to the extra time spent by the particle inside the barrier, we are led to the following expression for the tunneling time, which is called the phase time, Tphase
= Tf
+ AX(t)/vo,
(10)
where q = d/uo is the time that a classical free particle takes to traverse the spatial region a < z < b with the velocity WO. Now, we can calculate ~ ( from t ) the following expression for the transmitted packet:
Because of the peaked nature of $ J ( k- ko), we can evaluate (11) using the stationary phase approximation. We then find that the peak position of the transmitted packet moves as ~ ( t=)wot - O'(ko), where the phase 0 is defined
253 in (5) and ’ = a / a k , while the peak of the free wave packet moves as xf(t) = uot. Using these results in (lo), we have
where E = h 2 k 2 / 2 m , EO = h 2 k i / 2 m , and C#I is defined in ( 5 ) . Essentially the same result can be obtained by following, instead of the peak of the transmitted packet, a suitably defined “center of gravity.” The phase time is very useful in qualitative understanding of the motion of the wave packet. However, it is not appropriate to consider that the phase time represents the time taken by a packet peak to traverse the barrier. In the derivation of phase time, the peak motion of the incident packet in the region 2 << a is linearly extrapolated to 5 = a and the peak motion of the transmitted packet in the region x >> b is linearly extrapolated back to x = b. However, in the actual time evolution of the wave packet, the shape of the packet is quite distorted while the particle is tunneling through the barrier. At both edges of the barrier, we do not find such clear peaks that we get by the extrapolations. Such a peak cannot be found in actual dynamics that takes TpTphme to cross the barrier. 2.3 Larmor Times from Spin Rotations
The derivation of the phase time was based on an equation “x = ut”, while the derivation of the Larmor times to be explained here is based on an analogous equation lLcp = wt” with p being a spin rotation angle and w an angular velocity. It may be noted that “x = ut” was used outside the barrier, while ‘lcp = wt” is going to be used inside the barrier.
Figure 2. The Larmor times from spin rotations
254
Let us consider the tunneling of a particle with spin 1/2. The motion of the particle is supposed to be one dimensional along the y-axis. A small and homogeneous magnetic field Bo2 is applied only in the barrier region ( 2 is the unit vector in the z-direction). The Hamiltonian now has an additional term %ozQ,b(y) on the right-hand side of Eq. (1) (replace z by y and multiply the unit 2 x 2 matrix appropriately), where WL = gqBo/2rn (g is the gyromagnetic ratio and q is the charge of the particle) is the Larmor frequency, and gz is a Pauli spin matrix. The initial spin is supposed to be polarized in the z-direction, and the initial orbital wave function is assumed to be monochromatic with the wave number ko (energy Eo).As the particle enters the barrier, the spin starts to rotate due to the magnetic field. If the particle is simply at rest in the uniform magnetic field Bo2 without the potential barrier, the spin rotates in the xy plane by an angle W L per unit time as known as the Larmor precession, so that the rotation angle divided by W L gives the time the particle is placed in the magnetic field. In the tunneling case, it also seems natural to calculate the tunneling time by dividing the spin rotation angle by W L . However, as pointed out by Buttiker [16], the motion of the spin of the tunneling particle is no longer the usual Larmor precession in the z y plane because of the following reason: The initial spin can be written as the superposition of the spin-up and the spin-down states (“up” and “down” are with respect to the z-direction). Due to the Zeeman splitting, the spin-up component sees a modified barrier of height VO- h w ~ / 2 , while the spin-down component sees a modified barrier of height VO fiw~/2. The spin-up component is preferably transmitted since the barrier is lower, and, as the result, the spin of the transmitted particle has acquired a non-zero z component. Of course, the spin also acquires a non-zero y component due to the conventional Larmor precession. Consequently, the spin of the tunneling particle rotates not only in the z y plane but also in the z z plane, while only the rotation in the zy plane was considered in earlier papers [17,18]. We can obtain two different tunneling times depending on which rotation is used with “cp = wt.” Calculating the expectation values of s, and sy for transmitted particles, we have, to the lowest order in Bo, h alnlTl rnd (s,) = - W L T z 1 Tz = -h(13) 2 avo x -,h K h 84 hkO Ty = -h(Sy) = - - W L T ~ , (14) 2 avo Z -,VOK
+
where K = ,/kb0 - k i , and the last expressions in (13) and (14) are for opaque barriers. The present approach gives the two tunneling times T~ and T ~ One can of course take an average of the two tunneling times to define a
.
255 single quantity such as
7T G
,/r:
+ 7;,
which is Buttiker’s traversal time
(the subscript T means “traversal”). However, there are many ways t o define an average, and there is no reason t o give a special status t o m. We may conclude that the spin rotation method described here does not give a unique answer t o the tunneling time problem. In a narrower sense, 7y is called the Larmor time. In a broader sense, both 7y and 7r are called the Larmor times. Leavens calls 7y the “spin-precession traversal time of Rybachenko” and T~ the “spin-rotation traversal time of Buttiker.” A series of precise measurements of 7y has been reported by a Japanese group (see [19] and references therein). The results are in excellent agreement with theoretical predictions.
2.4 Buttiker-Landauer Time and Takagi Time from Time-Modulated Barriers The previous two approaches attempt to measure the tunneling time directly. The peak shift of the wave packet or the spin rotation angles are supposed to be measured and translated into the information of time through the classical equation “x = vt” or “cp = wt.” By contrast, the approach presented here attempts t o measure the tunneling time indirectly. The basic idea is simple. Suppose that the barrier is perturbed with a time scale T . If there is a well defined tunneling time 7 , then the effect of the perturbation would be different depending on whether 7/T is greater or smaller than one. This in turn means that it would be possible t o estimate 7 by monitoring the effect of the perturbation as the function of T . Buttiker and Landauer [20,21] investigated an oscillatory barrier, where a small perturbation Vl coswt is added t o the original static barrier. The Hamiltonian is therefore given by Eq. (1) with V, replaced by V, Vl coswt. Since the potential is time dependent, the energy of the transmitted particle
+
Figure 3. Oscillating barrier of Buttiker and Landauer
256 can differ from that of the incident particle. The particle can emit or absorb energy quanta n h w (n’s are positive integers) through the interaction with the barrier. Treating Vl as a perturbation, they analyzed the case of emission or absorption of a single quantum iiw and calculate the tunneling probabilities P+ and P- that the particle is transmitted at the sideband energies EO+ Aw and EO- hw, respectively. Assuming iiw << EO and EO iiw << V , they showed, for opaque barriers (i.e., nd >> 1 with K = d2rn(Vo - Eo)/h),
+
where P is the tunneling probability at the energy Eo and T = 2 a / w (here, T represents the modulation period, not the transmission amplitude). It may be noted that W E is the absolute value of the imaginary velocity J2(Eo - Vo)/rn under the barrier (here the subscript E in W E means “Euclidean”). We are now interested in how P* change as T is varied. Equation (15) shows that the values of P+/P are negligible for ~ T T B L /<< T 1, while they are not for ~TTBL/T > 1. Because of this crossover behavior at T ” r r B L , one would identify, apart from the numerical factor 2a, T B L as the tunneling time. For opaque barriers, using the transmission amplitude T , N
As pointed out by Buttiker [2], it was difficult to extend the original crossover argument to general barriers beyond the WKB regime. Thus a different approach was taken which essentially uses the low frequency behavior of P+ to extract a characteristic time scale of tunneling [2]. For opaque rectangular barriers, this method also gives Eq. (17) as the characteristic time. The time T B L is normally called the Buttiker-Landauer time, while Hauge and Stmvneng [l]called TT = r2 r2 the Biittiker-Landauer time. For opaque barriers, TT M T B L since rt = T B L >> ry. Note that T B L coincides with rt obtained in the spin rotation method. Also, the so called bounce time, which appears in the instanton technique, is essentially the same as TBL for opaque barriers. Does this popularity of T B L mean that TBL is the tunneling time? Before giving TBL a special status, it must be checked whether different modulations of the barrier also lead us to T B L as the time scale that char-
+ 6
257
acterizes the crossover behaviors. Indeed, Takagi [22] studied another timemodulated barrier, where not only the height but also the width of the barrier changes with time. In particular, the tunneling of a nearly monochromatic particle thraugh a squeezing potential barrier was studied; the height V ( t ) and the width d ( t ) of the potential change as
V ( t )=
VO
+
d ( t ) = (1 t / T ) d , (1 t / T ) 2 ' where T is now a constant representing the time scale of modulation. A remarkable feature of this model is that the exact expression for an evolving wave packet can be obtained. Since the potential changes with time, the energy of the transmitted particle can differ from the incident energy. Note that the energy shift is not n 2 d i l T because the barrier modulation is not periodic in time. Thus, instead of P J P , we need a new measure that quantifies how the tunneling is affected by the barrier modulation. Focusing attention to the barrier effect on the transmitted wave function in k space, rather than in real space, Takagi employed Apout/Apinas the measure, where Apin is the momentum width (uncertainty) of the incident packet and Apout is that of the transmitted packet. In the limit of slow modulation, he showed, to the first order in 1 / T , Apout/Apin= 1 - TTAKAGI/T for opaque barriers, where
+
If we follow the spirit of Biittiker and Landauer, TTAKAGI should now be identified a s the tunneling time, which differs from 7BL qualitatively and quantitatively. The characteristic time scales extracted from time modulated barriers thus depend on the manner of time modulations. The time modulated barrier methods, a s well as the spin rotation method, do not give a unique answer to the tunneling time problem. 3 3.1
Consistent History Approach
Doubt the Probabilities!
Although the question of determining the time of tunneling may sound simple, the subject has so far remained obscure because the notion of the purported tunneling time has been introduced in an ad hoc manner not securely rooted in the principles of quantum mechanics. For constructive discussions of the tunneling time problem, the author considers t,hat it is absolutely necessary to clarify, on the basis of the foundations of quantum mechanics, what is in principle speakable about tunneling time.
258 As we partially saw in Sec. 2, many different tunneling times were proposed in the past. This raises a question “Which is the right tunneling time?” From a quantum mechanical point of view, however, it is more likely that the tunneling time is not unique. In fact, we often encounter the phrase “average tunneling time” in literature. This phrase implicitly assumes the existence of a probability distribution of tunneling time. However, it is not necessarily clear in literature what the probability distribution is. Although some authors proposed the distributions, the most important point was not questioned in such attempts. The question that must be answered first is not “What is the probability distribution of tunneling time?”, but rather “Is it possible to define a probability distribution of tunneling time?” This would need further explanations. In quantum mechanics, probabilities are constructed from amplitudes. The amplitudes obey the superposition principle and the probabilities satisfy a set of probability axioms. As is well known in the two-slit experiments, a consistent construction of probabilities from amplitudes is not always possible due to the quantum mechanical interferences between alternatives (events). If the interferences do not vanish, the sum rule for probabilities, one of the probability axioms, fails to hold; this means that it is not possible to define probabilities in a mathematically consistent way. In quantum mechanics, the definability of probabilities is not always guaranteed and is thus a subject of investigation. It is only recently that a theoretical framework appeared in which one can discuss the definability of probabilities with a rigorous measure of interference between alternatives. It is the consistent history approach to quantum mechanics (CHA for short) constructed by Griffiths 1231, O m n k [24,25], GellMann and Hartle [26,27], and also by Yamada and Takagi [28,29]. In [30,31], the present author applied CHA to the tunneling time problem to investigate the definability of a probability distribution of tunneling time. For the tunneling time of resident time type (resident time for short),bit was concluded that a probability distribution is not definable but the range of values of the time is definable [31]. The rest of this section provides an overview of this argument. 3.2 Decoherence Functional and Weak Decoherence Condition In CHA, the interferences between alternatives are quantified by the decoherence functionals. In the tunneling time problem, the decoherence functional bThere is another type of tunneling time which the author calls the passage time. The origin of these different types of tunneling time will be explained briefly in Sec. 3.2.
259
Figure 4. The resident time of a Feynman path inside the barrier region
O [ T ; T ’(to ] be precise, its real part) describes the interference between the
alternative that the tunneling time is 7 and another alternative that the tunneling time is 7’. In this subsection, we derive a formal expression for O[T;7’1 and write down the condition for vanishing interference. a Feynman path We begin by considering the amount of time spends in the barrier region. For a Feynman path that crosses the barrier many times, we do not count the time while it is outside the barrier.c In the of the illustrated Feynman path is the case of Fig. 4, the resident time sum A, A2 As. Generally,
I).(.[.
+
+
I).(.[.
where eab(x) is unity for a<x
.I).(.[.
s
Q ( x ,t ) = d7 @(x,t17)
(21)
with @(x, t(7)= J d x o K ( z ,t;zo,to(7)\k(zo, t o ) , where @(Q, t o ) is the initial CIf we include the contributions from outside the barrier, the resultant time is longer than the time of residence inside the barrier, for which the phrase “resident time” is not apprcpriate. However, this longer time is also interesting in its own right and the present author calls it the passage time since it is the time taken by the Feynman path to pass through the barrier region. The first analysis of this passage time was done by Schulman and Ziolkowski [32]. The first CAH analysis of the passage time was done by the present author [30].
260
packet and K ( z ,t ;20,t01r) is defined as the sum over those paths whose resident time in the barrier is r. Equation (21) is an analogue of
in the two-slit experiment, where Q(z,t ) is the total wave function and Qj(z,t ) is the partial wave function that comes from slit j ( j = 1, 2). Equation (21) tells us that a t the level of amplitudes it is possible to consider a sort of distribution (“amplitude distribution”) of resident time. What we want to know is, however, the definability of a distribution at the level of probabilities. Let us substitute Eq. (21) into Eq. (6). We obtain
P=
J J dr
dr’D[r;r’],
1
(23)
00
D[r;r’]E t-00 lim
da: Q’*(z,tlr)Q(z,t1r1),
(24)
where note that only the real part of D contributes to the double r integrals since D*[r;7’1 = D[r’;7-1. ReD[r;7’1 is the measure of the interference between the two alternatives of a particle having different resident times r and 7’. Now, CHA claims the following: If the proportionality relationship ReD[r;r’] c( S(r - 7’) holds (i.e., if different values of resident time are not interfering), then a probability distribution of resident time is definable and it is given b y the proportionality coeficient; otherwise, the probability distribution of resident time is considered undefinable. Denoting the proportionality coefficient by P ( r ) ,we may write the condition of no interference as Re D[r;7’1 = 6(r - r ’ ) P ( r ) ,
(25)
which is called the weak decoherence condition (WDC for short) in CHA. We can easily check that, if WDC holds, then the proportionality constant P ( r ) has such properties that a probability distribution of resident time must have. To see this, substitute Eq. (25) into Eq. (23), noting that only the real part of D contributes to the double r integrals. We then have
P=
s
drP(7).
(26)
We can also prove that P ( r ) 2 0 if WDC holds. A probability distribution of resident time must have these properties.
261
3.3 Function G ( r )
To see whether WDC [Eq. (25)] holds or not, it is convenient to introduce function G ( r )defined by
from which we immediately have G(0) = 0 and G(m) = 1. Note that G ( r ) is a real-valued function of one variable, while D[r’;r”]is a complex-valued function of two variables. To see how G ( r ) is useful, let us assume that WDC holds. Using Eq. (25) on the right-hand side of Eq. (27) (noting again that only the real part of D contributes to the double r integrals), we have
1
l G ( r )= p
T dT‘P(r’).
(28)
Since we have assumed that WDC holds, P ( r ) is positive definite, so that G(7) is an increasing function of r (physically, G ( r )is the accumulated distribution of r if WDC holds). This means the following: If G ( r ) is not an increasing function of r , then we can immediately conclude that WDC does not hold. In Sec. 3.4, we will see in a specific example that G ( r )is not an increasing function of r. In the rest of this subsection, we will see that G ( 7 ) is also useful to estimate the range of resident time. Let us again assume that WDC holds. Let r1 < r < r2 be the “approximate support” of the resultant probability distribution, i.e.,
l>
P(T) % p,
(29)
which means that the probability that the particle spends less than r1 or greater then 72 inside the barrier is negligible. It is then natural to identify r1 < r < 7 2 as the range of the values of resident time. We can express Eq. (29) in terms of the decoherence functional as
If WDC holds, Eq. (29) and Eq. (30) are the same thing. We can use either of them to determine r1 and 7 2 . Now, note that, irrespective of whether WDC holds or not, it is always possible to find such r1 and 7 2 that make Eq. (30) hold. The meaning of the resultant 71 and 72 is that those Feynman paths whose resident time in the barrier region is less than r1 or greater than 72 do not contribute to the tunneling probability substantially. It seems natural
262
to expect that such r1 and 7 2 would have some physical meaning even when WDC does not hold; for example, when the tunneling particle interacts with environment such as phonon, it could be that the tunneling particle does not feel such modes of phonon whose frequency is lower than 1/72 or higher than 1/71. The present author considers that, even when WDC does not hold, the time interval 71 < r < r2 [TI and 7 2 are determined from Eq. (30)] would still make some physical sense as the range of values of resident time. Now, the determination of r1 and r 2 is easier if we use G(T). Let E be a small positive number, 0 < E << 1. Then, as shown in Fig. 5, let us find such r1 and 7 2 that
G ( r )< E for ‘r < 71, 11 - G(r)l< E for ‘7 > 7-2. It is not difficult to prove that (31) and (32) guarantee
so that Eq. (30) holds with the values of r1 and 7 2 that we determined from (31) and (32). As we will see in a specific example in Sec. 3.4, some of the well-known tunneling times proposed in the past fall within the range of resident time which we have just proposed. This suggests that the range of resident time is indeed an important quantity that deserves further studies; unknown time scales could also be found in the range, especially when the particle interacts with other degrees of freedom. To obtain an explicit expression for G ( r ) , we must first calculate the right-hand side of Eq. (24) using a concrete expression for @(x, tlr),which
Figure 5. Estimation of the range of tunneling time with G ( T )
263 can be obtained by using Eq. ( 3 . 3 ) in [33]. We then use Eq. ( 2 7 ) to have [31]
with P = J d k l$(k)12 ITk(V0)l2,where Tk(V) is the transmission amplitude when the wave number of the incident particle is k and the barrier height is V . Note that the initial state of the incident particle and the transmission amplitudes determine G ( T )and thus the range of resident time. 3.4
Example of G(T)in Monochromatic Limit
In the monochromatic limit G(T)
+
14(k)I2 + S(k - ko), we have
ICko(T)I2 7
where cko( T ) is Fertig’s C function [34,35].Figure 6 shows a typical example of G(T)for opaque barriers in the monochromatic limit, where the free passage time 7-f = rnd/tiko is used as the time unit; the parameter values are kod = 1 and V o / E o = 5 [Eo= ( h k 0 ) ~ / 2 m for ] , which P = (Tko(vo)1*= 4.64 x lo-’. It is clear that G ( T )is not an increasing function of T . We thus conclude that the probability distribution is not definable. If we put E = 0.01 in Eqs. (31) and (32) to estimate the range of resident time from Fig. 6 , we find T I / q M 0.0335 and Q / T ~M 5.95. In Fig. 6 , LM and BL indicate the Larmor time [TLM = h / ( V o , / m ) ] and the Buttiker-Landauer time [TBL = d , / m / ( 2 ( & - E o ) ) ]respectively. , Note that 71 << RM and 7-2 >> TBL, which is true for wide range of parameters in the opaque barrier regime. If both TI and 7 2 were close to TBL (or T L M ) , then we would be able to regard TBL (or T L M ) as the unique tunneling time, but that is not the case. In addition , can find (though only loosely) another characteristic time to 71 and ~ 2 we 7,; it is the time at which G(T),when its noisy behavior is ignored, takes its maximum. Figure 6 shows that T, T B L , which seems also true for wide range of parameters in the opaque barrier regime. Although G(T)takes significantly large values around T,, the class of Feynman paths whose resident time is T~ cannot be considered as the dominantly contributing class of paths, because, as the decreasing behavior of G(7) at larger T shows, the contribution from the class is mostly canceled by the contributions from other classes of paths. Since we are working in the tunneling regime, it is not surprising that we do not have a sharply defined “dominantly contributing class of paths.”
-
264
4
5
6
7
8
9
10
Figure 6. G(7) for monochromatic case
3.5 SummamJ
We have defined the range of resident time and found three characteristic times associated with tunneling: 71, 7 2 , and rc. In the present context, the meanings of these times are rather abstract because they just characterize how Feynman paths add up in r space. At the same time, however, it is also sure that Feynman paths determine the dynamics of the tunneling particle, so that we should be able to extract important time scales of tunneling from Feynman paths. What we saw in Sec. 3 is a demonstration of such extractions. Further studies of the problem in the presence of time dependent environments would help clarify the physical status of the three times, and such studies might even lead us to find other characteristic time scales in the range. Feynman’s path integral is suitable for such studies. To analyze real time Feynman paths in r space by using decoherence functionals and function G ( r ) seems to be a general and successful way of finding important time scales of tunneling. 4
Concluding Remarks
To settle the controversy, it is important to recognize what is in principle speakable about tunneling time. The results of Sec. 3 tell us that the probability distribution of resident time is not speakable. We should accordingly shift our focus to the range of values of resident time, which is speakable, for
265
constructive discussions of the tunneling time problem. Finally, it may be noted that the arguments of Sec. 3 do not exclude the possibility of such a probability distribution that is valid only when the resident time is measured in a specific way. However, such a distribution is not intrinsic to the scattering event itself and hence not of our primary interest.
Acknowledgments The author thanks Professor H. Yamamoto and Professor S. Tanaka for valuable discussions. He thanks Professor S. Takagi for discussions and encouragement over a long period of time. The author acknowledges partial financial support of Grant-in-Aids for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology, Japan. The author thanks Information Synergy Center at Tohoku University for CPU time on the supercomputer SX4 for numerical computations.
References 1. E.H. Hauge and J. A. Stgvneng, Rev. Mod. Phys. 61, 917 (1989). 2. M. Buttiker, in Electronic Properties of Multilayers and low Dimensional Semiconductors, ed. J. M. Chamberlain, L. Eaves, and J. C. Portal, (Plenum, New York, 1990), p. 297. 3. C.R. Leavens and G. C. Aers, in Scanning Tunneling Microscopy and Related Methods, ed. R. J. Behm and N. Garcia and H. Rohrer (Kluwer, Dordrecht, 1990), p. 59. 4. M. Jonson, in Quantum Transport in Semiconductors, ed. D. K. Ferry and C. Jacoboni (Plenum, New York, 1991), p. 193. 5 . A.P. Jauho, in Hot Carriers i n Semiconductor Nanostructures: Physics and Applications, ed. J. Shah (Academic, Boston, 1992), p. 121. 6. V.S. Olkhovsky and E. Recami, Phys. Rep. 214,339 (1992). 7. R.Landauer and Th. Martin, Rev. Mod. Phys. 66, 217 (1994). 8. Proceedings of the Adriataco research conference on Tunneling and its Implications, ed. D. Mugnai, A. Ranfagni, L. S. Schulman (World Scientific, Singapore, 1997). 9. V. Gasparian, M. Ortuiio, G. Schon, and U. Simon, in Handbook of Nanostructured Materials and Nanotechnology, ed. H. S . Nalwa (Academic Press, San Diego, 1999), Vol. 2, p. 514. 10. A. Enders and G. Nimtz, J. Phys. I France 2, 1693 (1992). 11. A.M. Steinberg, P.G. Kwiat, and R.Y. Chiao, Phys. Rev. Lett. 71,708 (1993).
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12. A. Ranfagni, P. Fabeni, G.P. Pazzi, and D. Mugnai, Phys. Rev. E 48, 1453 (1993). 13. L. Eisenbud, Princeton thesis (unpublished) (1948). 14. D. Bohm, in Quantum Theory (New York, Prentice-hall, 1951), p. 257. 15. E. P. Wigner, Phys. Rev. 98, 145 (1955). 16. M. Biittiker, Phys. Rev. B 27,6178 (1983). 17. A.I. Baz’, Yad. Fiz. 4, 252 (1966) [Sov. J. Nucl. Phys. 4, 182 (1967)]. 18. V.F. Rybachenko, Yad. Fiz. 5, 895 (1966) [Sov. J. Nucl. Phys. 5, 635 (1967)]. 19. M. Hino, N. Achiwa, S. Tasaki, T. Ebisawa, T. Kawai, and D. Yamazaki, Phys. Rev. A 61,013607 (1999). 20. M. Biittiker and R. Landauer, Phys. Rev. Lett. 49, 1739 (1982). 21. M. Biittiker and R. Landauer, Phys. Scr. 32,429 (1985). 22. S. Takagi, in Proceedings of the 4th International Symposium on the Foundations of Quantum Mechanics i n the Light of New Technology, ed. M. Tsukada et al. (JJAP, Tokyo 1993), p. 82. 23. R.B. Griffiths, J. Stat. Phys. 36,219 (1984). 24. R. Omnbs, J. Stat. Phys. 53,893 (1988). 25. R. Omnbs, The interpretation of Quantum Mechanics (Princeton University Press, Princeton, NJ, 1994). 26. M. Gell-Mann and J.B. Hartle, in Proceedings of the 3rd International Symposium on the Foundations of Quantum Mechanics in the Light of New Technology, edited by S . Kobayashi, H. Ezawa, Y. Murayama, and S. Nomura (Physical Society of Japan, Tokyo, 1990), p. 321. 27. J.B. Hartle, Phys. Rev. D 44, 3173 (1991). 28. N. Yamada and S. Takagi, Prog. Theor. Phys. 85, 985 (1991). 29. N. Yamada and S. Takagi, Prog. Theor. Phys. 86,599 (1991). 30. N. Yamada, Phys. Rev. A 54,182, (1996). 31. N. Yamada, Phys. Rev. Lett. 83,3350, (1999). 32. L.S. Schulman and R.W. Ziolkowski, in Proceedings of Third International Conference on Path Integrals from m e V to MeV, edited by V. Sa-yakanit et al. (World Scientific, Singapore, 1989), p. 253. 33. D. Sokolovski, Phys. Rev. A 52,R5 (1995). 34. H.A. Fertig, Phys. Rev. Lett. 65,2321 (1990). 35. H.A. Fertig, Phys. Rev. B 47, 1346 (1993).
267
TO DECAY OR NOT TO DECAY? TEMPORAL BEHAVIOR OF A QUANTUM SYSTEM -ANALYSIS BASED ON A SOLVABLE M O D E L H. NAKAZATO Department of Physics, Waseda University 3-4-1 Okubo, Shinjuku, Tokyo 169-8555, Japan E-mail: hiromiciOwaseda.jp The exact expression of the survival amplitude of a model system, which may describe photodetachment of a negative ion, is derived and analyzed in detail with an attention on its behavior as a function of time. The latter is governed by the location of poles of the amplitude in the complex energy plane. Poles located on the real energy axis can prevent the system from decay (known as plasmon poles), while other poles on the second Riemannian sheet give rise to exponentially decaying terms. The solvability of the model enables us to see explicitly the analytic structure of the amplitude, e.g., the movement of the poles in the complex energy plane as functions of parameters of the model. The behavior both at short and long times is also derived exactly and its peculiarity is pointed out and discussed.
1
Introduction
It is well known that a quantum system, interacting with another quantum system of (infinitely) many degrees of freedom and being supposed to be unstable, generally exhibits three different types of behavior for short, intermediate and long times.’ The situation would be contrasted with our naive expectation: we naturally expect the familiar exponential decay form for unstable systems. The above behavior is most easily seen in the so-called survival probability, a probability of finding a system in its initial state at later times. Let I$) be the initial state of the system and H the total Hamiltonian which includes the interaction between the quantum system and its surrounding large quantum system (usually called environment). Then the survival probability P ( t ) is nothing but the absolute value squared of the survival amplitude
~ ( t=)1 (+le-iHtl+)I2 . (1) The temporal behavior of P ( t ) is characterized in each of the three different time regions in the following way. 1) “Short” times: If the system of interest is initially prepared not in an eigenstate of the total Hamiltonian, the transition to other states occurs at a rate of order At for the amplitude, while the change of probability is of the order of (At).. Here the exponent cy is shown to be greater than
268 one, which implies that the system does not start to decay exponentially: The survival probability has a flat derivative at t = 0 and behaves like
P(t)
-+ 1
O(t‘l),
CY
>1
This is considered to be rooted to the unitary evolution of quantum systems, reflecting the time-reversal symmetry at short times and this peculiar behavior of quantum systems leads to the so-called quantum Zen0 effect (QZE)2 (and sometimes also to the inverse quantum Zen0 effect (IQZE)3,4,5),which states that repeated inspections of the system whether it is still in the initial state a t later times would result in the dramatic changes of its evolution, i.e. hindrance for QZE and acceleration for IQZE of decay.6
2) “Intermediate” times: The exponential decay, familiar, e.g. for unstable radioactive atoms, is expected to show up at later times and the survival probability decays exponentially
-
P ( t ) ePrt,
r > 0.
(3)
Even though such a deviation from the exponential decay form at short times has never been observed experimentally until very r e ~ e n t l ywe , ~ have to admit its existence: The deviation a t short times is due to the finiteness of the energy expectation value in the initial state.’ Furthermore, even though it would be rather difficult to observe experimentally, a theorem (the Paley-Wiener theorem’) strictly prescribes the behavior of the survival probability at very long times.
3) “Long” times: At very long times, the system decays more slowly than exponentially, usually like (power decay)
-
P(t) tPb, b > 0,
(4)
only if the total Hamiltonian is bounded from below. The temporal behavior of quantum systems stated above is understood in general on the basis of the analyticity of the survival amplitude in the complex energy plane: Possible poles on the second Riemannian sheet contribute to the exponentially decaying terms, while the contour integrals give dominant contributions both at short and long times.’ It is, however, still not clear, for example, when the exponential decay manifests itself and is overridden by the power decay. Furthermore, it is known that there are cases where a system shows an oscillating behavior, even if it interacts with another system with infinitely many degrees of freedom, which actually prevents it from decay at
269
t =
00. This behavior is due to a pole appearing on the real energy axis, known a s a p l a s m ~ n . ~ These issues are surely related to a profound theme in quantum mechanics, that is, the reason why and the condition under which a system is destined to or not to decay. In order to find a clue to this question, a detailed and explicit study of a solvable model would be of great help and give us deeper understanding of the quantum dynamics. In this paper, according to this line of thought, the temporal behavior of a model system, which is sometimes used to describe the photodetachment phenomenon of a negative ion, is considered. The model is introduced and ~ i m p l i f i e d ’ ~for , ~later convenience in the next section, and then in Sec. 3, the survival amplitude is examined analytically and its explicit expression is given in a closed form. The locations of poles are explicitly written down as functions of the parameters in the model, that is, energy detuning, strength and energy scale of interaction in Sec. 4,through which we can see quantitatively a non-trivial and quite intriguing parameter dependence of the survival amplitude. On the basis of the exact expression, the typical behavior at short and long times is extracted and discussed in Sec. 5 . Section 6 is devoted to discussions.
2
Model: photodetachment of a negative ion
Let us consider a system described by the total Hamiltonian H 00
H=wol+)(+l+J’
0
d w w I w ) ( w l + R a t a + ~ ~ d w g ( w[Iw)(+b+I+)(wlat]. )
(5) The Hamiltonian models a photodetachment process where a negative-ion (bound electron) state I+) with an energy wo < 0 can be ionized to the continuum (free electron state) Iw)through the interaction with a single mode laser field a of frequency 0, in the rotating-wave approximation. The form factor g(w) characterizes the strength of the interaction responsible for bound-free transition of the electron. If the total system is initially prepared in the bound electron state with N single-mode photons I+, N ) , the system remains in the Tamm-Duncoff sector spanned by the initial state I+, N ) and the continuum with N - 1 photons Iw, N - 1). As far as its dynamics is concerned, we can confine ourselves to this sector and consider a simpler H a m i l t ~ n i a n , ’ ~ , ~
by suppressing the photon degrees of freedom. Clearly, parameters here are
270 properly modified from the original ones:
+
A = wo 0, d R g ( w ) + g ( w ) . (7) The discrete level I$) and continuous ones Iw) are mutually orthogonal and normalized. Two parameters A and g(w) represent the energy level of I$) relative to the continuum (i.e., the detuning parameter) and the strength of interaction, respectively. Notice that the value of parameter A is controlled by the frequency of the laser field and takes both signs.
3 Survival a m p l i t u d e We prepare the system in the discrete level I$) at time t = 0 and let it evolve. The dynamics is governed by the above Hamiltonian (6) and we are interested in the temporal behavior of the system, which can be read off from the behavior of the survival amplitude, an amplitude of the system to be in the initial state,
($le-iHtl+) = z ( t ) , (t 2 0 ) . The standard manipulation4 yields the following expression of z(t):
(8)
where the contour C runs from --oo to -oo, just above the real E-axis. The self-energy part
usually has a branch point at E = 0, a cut and possible poles on the second Riemannian sheet. It is known that a specific choice of the form f a ~ t o r , ~
where positive parameters A and ,O represent the strength of interaction (proportional to the laser intensity), and the peak position and width of g2(w), respectively, enables us to calculate explicitly the self-energy part, leading to a closed form of the survival amplitude z ( t ) .It is actually an elementary task to integrate (10) to obtain
27 1
with being defined as m e i V P l 2 . It is manifest that the self-energy part has a cut from the origin E = 0 and we need two Riemannian sheets. There is no logarithmic branch point and no infinite number of sheets is needed either in this case. By inserting the above expression of C ( E )into (9) and changing the integration variable E -+C = z ( t ) is now expressed as
a,
The contour C1 +C2 is composed of two segments C1 and C2: the former (C,) runs infinitesimally right of the imaginary C-axis from +im to 0, while the latter (C2) just above the positive real <-axis from 0 to +w. These contours C1 and C2 are rotated around the origin by angles n/4 and -n/4, respectively, to form a straight-line contour C3: w we^^^/^, + ~ e - z . " /1.~ In this way, we arrive at the following expression of z ( t ) ,
where the last term represents residues of the integrant in (13) at poles located in the region between the two contours C1 Cz and C3. Notice that this region, subdivided into two regions with ReCzO, just corresponds, in the original complex E-plane, to the region with ImE < 0, and therefore the last term in (14) represents the exponential decay if it exists, while the first term corresponds to the contour integration along the cut in the complex E-plane. It will be shown later that poles contributing exponential decay only appear in one of the above two regions, i.e. in the region between C2 and C3 where R e 5 > 0 and this region does correspond to that on the second Riemannian sheet with I mE < 0. The evaluation of each term in (14) is not difficult, only if one notices that the denominator of the integrant is the third-order polynomial function. Indeed, if we put
+
E = ei7r/4 y,
or [ = -i<, the denominators in (14) and (13) are written respectively as
(15)
272
Let the three roots of this function be
&,El
and Ez. They satisfy
EoEi+EiEz+EzEo=A,
EoEiEz=-fi(A-A). (17) It is now an easy exercise to evaluate the first term in (14), which is rewritten in the following form after a simple manipulation: Eo+El+Ez=-fi,
-i
c[
k=0,1,2
+ f l ) Jm + 2 d @ k + A -m
Ek(Ek
3t:
dy
e-y2t
r Y - e - i K / 4Ek
This type of integration can be performed with the help of the integral formula for the Error function
On the other hand, the integrant in (13) is easily rearranged to be
and we finally arrive at the analytic expression of the survival amplitude
where, as already stated, the residues are evaluated at those poles located in the region between two contours C1 CZ and CJ, i.e. at such poles satisfying 0 5 Arg(&) < ~ / or 4 5 r / 4 < Arg(&) 5 3 r / 2 .
+
4
Behavior of survival amplitude
In order to clarify the behavior of the survival amplitude, especially the existence or absence of the exponential decay, we have only to investigate the second term in ( 2 1 ) , responsible for it. That is, we need to know the three roots & explicitly. For later convenience, let us introduce c by
273 It is a tedious but elementary task to write down the three roots &. Notice that when the condition A > ,813 is satisfied or both conditions A < ,813 and Ic( > are satisfied, we have one real (Q) and two complex (El = 6;) roots conjugate to each other, and otherwise, we have three real roots.
4.1
Dependence on the interaction strength A
Let us first investigate the behavior of the survival amplitude as a function of the strength of interaction A , with the remaining parameters A and ,8 being fixed. Case I: A > 813 We have one real root and two complex roots = [; in this case and their explicit expressions read (see (22) above)
z1=--+ zo 2i
/ I + ;z; = z;.
(23)
+
Observe that (a) when the interaction is weak, i.e. 0 < A < 28/27 2A/3, c is positive and zo < 0, while for strong interaction 2,8127 2A/3 < A, c < 0 and zo > 0 and (b) the complex quantities z1 and z2 lie on a hyperbola y2 - 322 = 1. It is not difficult to find the locations of these poles as functions of A. As A is increased from zero (free case), the real pole 60 shifts right from -flalong the real <-axis, crosses the origin at A = A and becomes large indefinitely. This pole can contribute only when it appears on the positive real E- axis, for it is then included in the area 0 5 Arg(&) < ~ / 4 .Remark that this pole is responsible for an oscillating behavior of the survival amplitude and if it appears, the system ceases to decay out completely even for t -i co. Two poles and J2 locate at and when A = 0, representing free oscillation of state I$), and start to move on each leaf of a hyperbola y2 - 3(2 f l / 3 ) 2 = A - 813 > 0 toward its center (-fl / 3, 0) and move away it as A becomes stronger. Since only one of the hyperbola's asymptotic G / 3 ) can have intersections with the boundary of the area lines y = &(x 5 ~ / 4< Arg(&) 5 3 ~ 1 2 that , is, with the straight line y = x, only (2 can contribute to give the residue in (21) when it stays in this area, while has no contribution at all.
+
ifi
+
+
-ia
274 Thus we have the following classification of the behavior of the survival amplitude: (i) In the perturbative regime, no matter how small the interaction is, the exponentially decaying pole is always present and the system exhibits exponential decay together with the power behavior arising from cut contribution (the first term in (21)) and dominating the former both at short and long times. (ii) When the interaction becomes strong and (813 <)A < ,012 is satisfied (then the hyperbola has intersections with the boundary line y = x), this pole suddenly disappears at a critical value of A and reappears for even stronger A , while for the system with A > 812, the decaying pole is always present for any value of A . (iii) If the value of A exceeds the threshold A, the system never decays completely and the oscillating behavior survives at t = co.See Fig. 1. Case 11: 0 < A < 813 In this case, we obtain the following expressions of [ k : for weak and one real pole [O and two complex strong interaction, where IcI >
m,
poles
= (1 exist. That is, for 0
or 28/27
+ 2A/3 + d m d
< A < 28/27+2A/3 m 3< A ,
-
d m , / w 3
f l and we have d m d m 3 < A < 28/27 +
On the other hand, for intermediate cases, IcI is less than
+ 2A/3 2A13 + d m d m 3 , &=-y+ t/p J s z k , (k=O,1,2),
three real roots; for 28/27
2
zo = -cos
J3
({ *) , -
-
2 z1= -cos
43
n (?e + ?),
2 z2 = -cos
43
(; ); -
,
Notice that if the interaction is sufficiently weak or strong, the two complex poles [I and (2 lie on a hyperbola 3(2 + f i / 3 ) 2 - y2 = ,013 - A > 0.
275
A = 0.35 p
A = 0.7p
1.5
15 /
1.0
/
1.0
/
/
/
/
/ /
/ /
/
0.5
0.5
// /
/ /
/
0.5
1.0
-
-1.0
-0.5
//
/
/.
,/
0.5
1.0
/
/
/
/
-0.5
/
-0.5
-1.0
-1.0
-1.5
-1.5
Figure 1. Locations of poles as functions of the coupling constant A, in cases of 8/3 < A = 0.358 < 8 / 2 (left) and 8 /2 < A = 0.708 (right). There are one real pole [o and two complex poles €1 = c;. The complex poles lie on each leaf of hyperbola y2 - 3 ( ~ + f i / 3 ) ~=
-fi
2 < 0. Notice that at A = 0 (free case), [ O = and = +(-)Z Asa the.coupling A is increased, (0 moves right on the real axis to become a plasmon pole when it crosses the origin, which occurs at A = A, while the two complex poles move left on the hyperbola. The poles contribute t o z ( t ) if they locate in the two regions which are both below the line y = z (broken line) and either above the positive z-axis or left of the negative y-axis. When p / 3 < A < P / 2 is satisfied, there are two critical values of A between which the exponentially decaying term does not exist. If A is greater than 8/2, the exponentially decaying term is always present, irrespectively of the value of A. Here the poles c k are measured in units of
A
-
p / 3 in the left-half plane
fi,
We summarize the behavior of the survival amplitude in this case: (i) In the perturbative regime, the system always exhibits the exponential decay (at intermediate times). (ii) When A exceeds a critical value, it shows no exponential decay and the behavior is governed solely by the cut contribution. (iii) When the interaction reaches and exceeds the threshold A = A, an oscillating behavior appears. (iv) If the interaction becomes even stronger, the
276
A
= 0.27B
-1.5 Figure 2. Locations of poles as functions of A, same as in Fig. 1, but in the case of 0 < A = 0.270 < 0/3. Both for small and large couplings, two complex poles are present on each leaf of the hyperbola 3(2+ &/3)z - y2 = p/3- A > 0, one of which can contribute to give the exponential decay if it lies below y = 2 (broken line), while at intermediate couplings, they are both real and have no contributions to z ( t ) . The exponentially decaying term appears both at small coupling and at very large coupling. The oscillation pole is present for large coupling, i.e., A 2 A.
exponentially decaying term appears again, however, the system’s behavior is dominated by the above oscillation and it never decays out. See Fig. 2. Case III:
A <0
Since A is negative, there are three real roots existing for small A . For sufficiently large A , c becomes less than and only one root remains to be real and two others become complex. Though the actual critical values of A are different depending on whether A is bigger or smaller than -,0/9, the expressions of the poles are the same as in (25) for small A and in (24) for A exceeding the critical value, respectively. See Fig. 3.
-m
We can conclude, in this case, that the exponentially decaying term is
277
/
1.0
/
/ /
/ /
0.5
// /
/ /
,/
’-
0.5
1.0
-0.5
-1.0
-1.5 Figure 3. Locations of poles as functions of A, same as in Fig. 1, but in the case of A = -0.OlP < 0. At A = 0 (free case), all three poles are real, to = El(z) = The hyperbola is the same as in Fig. 2. For a sufficiently large coupling, an exponentially decaying term appears, while an oscillation pole is always present and the system never decays out.
-&,
zkm.
present if the interaction is weak, disappears for stronger interaction and reappears for even stronger interaction, however, an oscillating term is always present and the system never decays completely. 4.2 Dependence o n the detuning parameter A
So far, we have treated A as a given, fixed parameter and examined the behavior of the survival amplitude (actually, the locations of poles) as a function of parameter A , the strength of interaction. Since we can also change the value of A by adjusting the frequency of the laser field R (see (7)), the locations of the poles & can be seen as functions of A, given the strength of the laser field A > 0. It turns out that (i) the complex poles are allowed to exist only for those A larger than a critical value and if they exist, they lie on each leaf of a
278 2.5
2 -
1.5
-
1 -.........................................
0.5 -
0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
P-A
d m )
Figure 4. The curve y = ( p - A)(l f as a function of fi - A. The intersection with y = A gives the critical value of A below which no exponentially decaying term is present. Both axes are in units of fi and the case A = p(= 1) is drawn here.
+
curve y2 = -(z J p ) 2- JPA/x, (z < 0), and (ii) an exponentially decaying term appears only if A exceeds a threshold value A,(< p/2), given by
In other words, if A is smaller than A,, no exponential decay can be realized, while for those A greater than p/2(> A,), (26) has no solution and the exponential decay appears for all values of A > 0 and p > 0. See Fig. 4. 5
Behavior at short and long times
We can draw the survival probability P ( t ) = Iz(t)I2as a function of time for various cases, some of which are seen in Fig. 5, and extract the behavior of z ( t )both at short and long times from its exact and explicit expression (21).
279
0
10
20
30
40
t Figure 5 . The survival probability P ( t ) in Case I (A = 0.35p > 8/3). Here time t is measured in units of p-l. The four curves (a)N(d) represent the typical cases: (a) A = 0.4A (weak coupling and exponentially decaying pole present), (b) A = 0.9A (no pole contribution present), (c) A = 1.7A (only plasmon pole present) and (d) A = 4.04 (both plasmon pole and exponentially decaying pole present). Every curve has been confirmed to have a flat derivative at t = 0.
5.1
Short-time behavior
First, remember the following expansion of the Error function erfc(6) for small E
It is then easy to see that, independently of the presence or absence of pole contributions, the short-time behavior of the survival amplitude always reads
which, after some manipulation, is reduced to
280
Notice that the term of order & has disappeared from the final expression (29), which is in accord with the finiteness of the derivative of the survival amplitude a t the origin IS(0)l < 00 and that the linear (in t ) term is purely imaginary. Therefore we arrive at the following short-time expansion of the survival probability
which is valid for all cases. Observe that there is no linear term and the first 3 term is of order & : The system does not start to decay exponentially. It does not start to decay quadratically either, however, the survival probability has a vanishing derivative at the origin P(0) = 0. Deviation from the quadratic decay is due to the infiniteness of the variance of our interaction Hamiltonian in the initial state, which is a result of our choice of the form factor g(w) in (11): The usual naive expansion of the evolution operator for small t is not meaningful in this case, however the flatness of the derivative of the survival probability is maintained even in such a case. 5.2 Long-time behavior We know that the Error function has the following expansion for large value of its argument 1 - -+... 1 223
-
t
,
fort
-+
00,
JArg(t)J<
37r -. 4
(31)
We understand that for large t , the exponentially decaying term, if it exists, is to be overwhelmed by other terms decaying more slowly like an inverse power of t , which come from the above expansion of the Error function originated from the contour integrals. Inserting the above expansion (31) into s(t)in ( 2 l ) , we find that the leading term is, not of order l/&, but of order l/A3
irrespectively of the presence or absence of the exponentially decaying term. There are exceptional cases, for which the above expression is not valid. Such an exception occurs for the threshold case A = A. In this case, the
281
amplitude decays much more slowly5
The exception also occurs if the coupling A exceeds the threshold value A and we have a plasmon pole. Let 60 be positive and responsible for the plasmon. Then the survival amplitude never decays out even at t = 00, instead it keeps to oscillate:
It is interesting to see that, while (0 -i00 as the coupling A goes to infinity, the survival probability remains at a value of 4/9, and never becomes unity.
6
Discussions
The behavior of a quantum system which is in interaction with another large quantum system and is therefore supposed to be unstable, is not simple and is characterized by different functions in each time region, as stated in 1)-3) in Introduction. The present analysis of a solvable model explicitly shows that there is more: In some cases, a system which is supposed to be unstable owing to its coupling to the environmental degrees of freedom can survive and never decays out completely even in the t co limit. This is due to the presence of an oscillation pole, known as a plasmon and the existence of such a pole implies that the spectrum of the total Hamiltonian includes a discrete level, represented by a delta-function. It is easily understood that if the spectrum of the total Hamiltonian is continuous and has no discrete levels, the system has to vanish in the t -i co limit: More precisely, the survival amplitude (and therefore the survival probability) decays to 0 as t + m, since -+
where the energy density of the initial state p ( E ) = l(E1$)1’ vanishes at the boundaries of the spectrum p ( 0 ) = p ( m ) = 0 and is assumed to have ith As long as the left most quantity is local maximum (minimum) at E,?ix(min). finite, survival amplitude decays to 0 as t -+ 00. It is also evident that if the
282
spectrum contains a discrete level, represented by a delta-function, then the above estimation loses its meaning, for p(E) is not bounded any longer. It would easily be understood that for a quantum system of finite degrees of freedom t o decay out, it is necessary that it is coupled t o another system of infinitely many degrees of freedom (environment) and that the energy spectrum of the total system is continuous and does not contain any discrete levels. It is, however, still not clear, at least for the present author, the physical reason why and the condition under which most of possible process are to be overwhelmed by a certain decay process, even though the dynamics is governed by a Hamiltonian which is hermitian. The issue clearly needs much deeper analysis of temporal behaviors of quantum systems, which would result in better understanding of quantum dynamics.
Acknowledgments The author would like to thank those participants of Japan-Italy Joint Waseda Workshop 2001 who contributed to the discussion during and after his talk, according to which the manuscript has been improved. In particular, helpful discussions with Paolo Facchi, Dr. K. Imafuku, Saverio Pascazio, Antonello Scardicchio and Dr. K. Yuasa are greatly appreciated.
References 1. H. Nakazato, M. Namiki and S. Pascazio, Int. J . Mod. Phys. B 10,247
(1996) and references therein. 2. A. Beskow and J. Nilsson, Arkiv fiir Fysik 34, 561 (1967); L.A. Khalfin, Zh. Eksp. Teor. Fiz. Pis. Red. 8, 106 (1968) [ J E T P Letters 8, 65 (1968)l; B. Misra and E. C. G. Sudarshan, J. Math. Phys. 18, 758 (1977); D.Home and M.A.B. Whitaker, Ann. Phys. 258,237 (1997). 3. A.G. Kofman and G. Kurizki, Acta Physica Slovaca 49, 541 (1999); S. Pascazio and P. Facchi, Acta Physica Slovaca 49, 557 (1999); A.G. Kofman and G. Kurizki, Nature 405, 546 (2000); P. Facchi and S. Pascazio, Phys. Rev. A 62,023804 (2000). 4. P. Facchi, H. Nakazato and S. Pascazio, Phys. Rev. Lett. 86, 2699 (2001). 5. M.Lewenstein and K. Rzqzewski, Phys. Rev. A 61,022105 (2000). 6. M.C. Fischer, B. Gutikrrez-Medina and M.G. Raizen, Phys. Rev. Lett. 87,040402 (2001). 7. S.R. Wilkinson et al, Nature 387,575 (1997).
283 8. R.E.A.C. Paley and N. Wiener, Fourier Transforms in the Complex Domain (American Mathematical Society Colloquium Pub. XIX, New York, 1934). 9. G. Gaveau and L.S. Schulamn, J . Phys. A 28, 7359 (1995). 10. K. Rzqzewski, M. Lewenstein and J.H. Eberly, J . Phys. B 15, L661 (1982); J. Javanainen, J. Phys. B 16,1343 (1983); J. Javanainen, Optics Comm. 46,175 (1983); S.L. Haan and J. Cooper, J. Phys. B 17,3481 (1984).
284
QUANTUM DECOMPOSITION AND QUANTUM CENTRAL LIMIT THEOREM AKIHITO HORA Department of Environmental and Mathematical Sciences Faculty of Environmental Science and Technology Okayama University Okayama 700-8530 Japan E-mail: [email protected] NOBUAKI OBATA Graduate School of Information Sciences T o h o b University Sendai 980-8579 Japan E-mail: [email protected] On the basis of the canonical relation between an interacting Fock space and a system of orthogonal polynomials we introduce the notion of quantum decompcsition of a real random variable in an algebraic probability space. To understand prototypes we review some basic examples appearing from the Boson, Fermion, and free Fock spaces. We then prove quantum central limit theorems for the quantum components of the adjacency matrices (combinatorial Laplacians) of a growing family of regular connected graphs. As a corollary, asymptotic properties of the adjacency matrix are obtained. Concrete examples include lattices, homogeneous trees, Cayley graphs of the Coxeter groups, Hamming graphs and Johnson graphs. In particular, asymptotic spectral distribution of the adjacency matrix of a Johnson graph is described by an interacting Fock space corresponding to the Meixner polynomials which are one-parameter deformation of the Laguerre polynomials.
Introduction Motivated by the canonical relation between an interacting Fock space and a system of orthogonal polynomials established by Accardi-Bozejko' , we have explicitly introduced the notion of quantum decomposition of a real random variable in an algebraic probability space (including a classical case in principle) in the series of works10>11>12 to study the adjacency matrix (combinatorial Laplacian) of a large graph. This idea together with quantum central limit theorems provides a new method of obtaining approximation of the spectral distribution of the adjacency matrix. The main purpose of this paper is to give a self-contained account of the idea of quantum decomposition together with concrete examples and applications. In particular, we shall unify results obtained in the previous works into
285 a general statement applicable to a wider class of graphs. Thus, the mechanism of our method has become clearer and one may expect many potential applications in geometry, combinatorics, physics and so on. The paper is organized as follows. In Section 1 we review shortly some notions in algebraic probability theory, interacting Fock space and its relation to orthogonal polynomials. Then we introduce the quantum decomposition of a classical random variable and some basic examples. In Section 2 we recall some notions in graph theory. We introduce a stratification of a graph by means of the natural distance function and an orientation compatible with the stratification. Then a quantum decomposition of the adjacency matrix is defined and the main question is formulated. In Section 3 we prove a quantum central limit theorem (Therem 3.1) for the quantum components under three essential conditions (Al)-(A3) and discuss lattices, homogeneous trees and the Cayley graphs of the Coxeter groups as concrete examples. In Section 4 we discuss a possibility of removing condition ( A l ) by illustration of distanceregular graphs. In fact, condition ( A l ) is replaced with (A4) for a certain class of distance-regular graphs and we obtain a quantum central limit theorem (Theorem 4.1). Concrete examples are Hamming graphs and Johnson graphs. It is noteworthy that the Meixner polynomials, which is a one-parameter deformation of the Laguerre polynomials, play a role in the study of Johnson graphs. Finally, introducing the weak quantum decomposition for a general distance-regular graph, we show the quantum central limit theorem for its quantum components. 1 1.1
Notation and Background
Algebraic Probability Space
Let A be a *-algebra with unit 1 = Id, i.e., a unital *-algebra. A linear function 4 : A + C is called a state if $(a*a) 2 0 for all a E A and 4(1) = 1. The pair (A,4) is called an algebraic probability space. Each a E A is called an algebraic random variable or a random variable for short. Consider two algebraic probability spaces (A,4) and (a,$). Algebraic random variables a E A and b E U are called stochastically equivalent if
4(at1at2. . . a E m )= $(bE1bE2. . . b E m ) for any choice of m = 1 , 2 , . . . and €1,. . . ,,t E {I, *}. For a real random variable a = a* E A it is sufficient to consider the moment sequence 4 ( a m ) , m = 0 , 1 , 2 , . . . . Let (A,, &) be a sequence of algebraic probability spaces and {a,} a sequence of algebraic random variables such that a , E A,. Then
286 an algebraic random variable b in another algebraic probability space is called a stochastic limit of {a,} if lim &(a:az
n-cc
(a,4)
. . . akm) = +(b'l be2 . . . P )
for any choice of m = 1 , 2 , . . . and c l , . . . ,en. E (1, *}. In that case we also say that {a,} converges stochastically to b.
1.2 Interacting Fock Space We refer to Accardi-Bozejko' for more details. Let A0 = 1, A1, Az, . . . 2 0 be a sequence of nonnegative numbers and assume that if A, = 0 occurs for some m 2 1 then A, = 0 for all n 2 m. According as A, > 0 for all n or A, = 0 occurs for some m 2 1, we define a Hilbert space of infinite dimension or of finite dimension: mn-1
n=O
n=O
where mo is the first number such that, ,A = 0, and {@,} is an orthonormal basis. We then define the creation operator B+ and the annihilation operator B- by
B-@o=O,
B-@,=
@,-I,
n 2 1.
In the case when r is of finite dimension we tacitly understand that B+@,,-l = 0. Equipped with the natural domains, B* become closed oper= (I?, {An}, B+,B - ) is called ators which are mutually adjoint. Then r({A,}) an interacting Fock space associated with {An}. By simple computation we have
B+B-@o = 0,
An B+B-@, = -@n, An-1
An+l B-B+@, = -@n, n 2 0, An
72
2 1,
287
The number operator N is defined as usual by
n 2 0.
N Q n = nQ,,
Proposition 1.1 Let I' = (r,{A,], B+,B - ) be the (one-mode) Boson Fock space, i.e., the interacting Fock space with An = n!. Then the Boson commutation relation holds: B-B+ - B+B- = 1. Moreover,
where the probability distribution in the right hand side is the standard Gaussian distribution, i.e., with mean 0 and variance 1. Proposition 1.2 Let r = (r,{An}, B+,B - ) be the (one-mode) Fermion Fock space, i.e., the interacting Fock space with A0 = A1 = 1 and A, = 0 for n 2 2. Then the Fermion commutation relation holds: B-B+ + B+B- = 1. Moreover, +oo
(@o, (B'
+ B-)mQpo)r= 1 J 2 -cO
z"(S-~+
&+I)(&),
m = 0 , 1 , 2 , .. . .
Proposition 1.3 Let r = (r,{An}, B+,B - ) be the (one-mode) free Fock space, i.e., the interacting Fock space with A, = 1 for all n 2 0. Then the free commutation relation holds: B-B+ = 1. Moreover,
1
+2
+ B-)mQo)r = !27r -2
(Qo, (B+
x m d s d z , m = 0,1,2,. . . ,
where the probability distribution in the right hand side is called the Wigner semi-circle law. Remark 1.4 The above commutation relations are special cases of the socalled q-commutation relation: B-B+ - qB+B- = l, where -1 5 q l. This is realized by the interacting Fock space r = (r,{An}, B+,B - ) with
<
A, = [n],!= [n],[n - 11, . . . [I],,
[n],= 1+ q
+ q2 + . . . + qn-l.
For more information on this topic, see e.g., Bozejko-Kummerer-Speicher5 and references cited therein. 1.3 Orthogonal Polynomials Let p be a probability measure on R with finite moments of all orders, i.e., F
IR
( z ( " p ( d x ) < 00,
m = 0,1,2,. . . ,
288 and {P,} the associated orthogonal polynomials normalized in such a way that P,(3:) = zn . . . . Then there exists uniquely a pair of sequences a1, a2, . . . E R a n d w 1 , w 2 , . . . 2 0 such that
+
Po(.)
= 1,
P l ( 2 ) = 3: - a1,
xPn(z) = Pn+1(2)
+ an+lPn(z) +wnPn-l(s),
n2 I
(1)
The pair {a,}, {w,} is called the Szego-Jacobi parameter. When the probability measure p is supported by a finite set of exactly mo points, the orthogonal polynomials {P,} terminate at n = mo - 1 and the Szegij-Jacobi parameter becomes a pair of finite sequences a1, . . . ,amo and w 1 , . . . , w m o - l , where the last numbers are determined by (1) with P,+1 = 0. Note also that p is symmetric if and only if a, = 0 for all n 2 1. Theorem 1.5 (Accardi-Boiejko') Let {P,} be the orthogonal polynomials with respect to p with Szego-Jacobi parameters {a,}, {w,}. Let I'({A,}) be an interacting Foclc space associated with A0 =
1,
A,
= ~ 1 ~ 2. w,, . .
Then there exists an isometry U from I'({A,}) mined by
UQo = PO, UB+U*Pn = Pn+l,
n 2 1.
(2)
into L 2 ( R , p ) uniquely deter-
Q = U(B+
+ B- + a N + l ) U * ,
where Q is the multiplication operator by x densely defined in L 2 ( R , p ) and operator defined by a ~ + l @ = , cr,+l@,. The proof is straightforward. In fact, the isometry U is uniquely specified by &an H P,. A question of when U is a unitary, or equivalently when the polynomials span a dense subspace in L 2 ( R ,p ) is related to the so-called determinate moment problem, see e.g., Deift'. CXN+ is~the
1.4
Quantum Decomposition of Position Variable
We keep the same notation as in the previous subsection. In quantum mechanics Q stands for the position variable. Define an algebraic probability space (A,d),where A is the unital *algebra generated by Q and 6 is the state defined by
6(a) = (P0,aPO)L2(R,p), a
E A.
Another algebraic probability space is naturally constructed from the interacting Fock space (I?, {Am}, B+,B - ) . Let B be the unital *-algebra generated
289
by B+,B - , a ~ + and 1 $ the vacuum state defined by
$(b) = (@o, b@o)r,
b E B.
It tthen follows from Theorem 1.5 that
4(Q") = $ ((B ++ B-
+a~+l)"),
m = 0,1,2,.. .,
or equivalently
+ B- + aN+l)"@O)r,
m = 0,1,2,.. . .
zmp(dz) = (GO,(B+
(3)
/Fl
+ +a~ +1
In other words, two real random variables Q and B+ Btically equivalent. In this sense we have
are stochas-
Q=B++B-+cxN+~, which is a prototype of the quantum decomposition. Note that the quantum components are no longer commuting each other.
1.5 Quantum Decomposition of a Classical Random Variable Let X be a real random variable defined on a classical probability space. The distribution p of X is not necessarily uniquely specified by the moment sequence E(Xm)due to the famous determinate moment problem. If it is the 1 also referred to case, X and Q are identified so that X = B+ B- a ~ + is as the quantum decomposition. Example 1.6 Let X be a Bernoulli random variable such that P ( X = + I ) = P ( X = -1) = l / 2 . Then X = B+ B - , where B* are the creation and annihilation operators of the Fermion Fock space defined in Proposition 1.2. In an explicit form we have
+
+
+
which is sometimes called a quantum coin-tossing. Example 1.7 Let X be a real random variable obeying the standard Gaussian distribution. Then X = B+ B - , where B* are the creation and annihilation operators of the Boson Fock space defined in Proposition 1.1. Example 1.8 Let X be a real random variable obeying the Poisson measure with parameter X > 0. Then the quantum decomposition is given by
+
X
= (I?++
&)(B-+
A)= &(El++
B-) + N + X ,
where B* and N are the creation, annihilation, and number operators of the Boson Fock space defined in Proposition 1.1.
290
Quantum decomposition of a stochastic process is a question of interest. For example, the Brownian motion is decomposed into a sum of the creation and annihilation processes defined in the Boson Fock space over L2(R) (hence this Boson Fock space is of infinite mode). In fact, this is a root of the quantum It6 theory initiated by Hudson-Parthasarathy16. Moreover, the quantum decomposition of the white noise is also performed within the framework of white noise distribution theory and is a clue to study white noise differential equations, see e.g., Chung-Ji-Obata'.
Quantum Decomposition of Adjacency Matrices
2
2.1 Basic Notions in Graph Theorg
A non-empty set V equipped with E c {{x,y } ; x,y E V, x # y } is called a graph and denoted by 6 = ( Y E ) . Elements of V and of E are called a vertex and an edge, respectively. If {x,y } E E, we say that x and y are adjacent and write x N y . The degree (or valency) of x E V is defined by K(X) = I{y E V ;y A graph is called regular if K(X) = K is a finite
- .}I.
constant independent of x E V. A finite sequence X O , X ~. .,. ,x, E V is called a path of length n if xi xi+l for all i = 0 , 1 , . . . ,n - 1. A graph is called connected if any pair of points x,y E V are connected by a path. For x,y E V the distance 8(x,y) is by definition the length of the shortest path connecting x and y . Obviously, x y if and only if 8(x,y ) = 1. The diameter of a graph is defined to be SUP{8(X, Y ) ; 5,Y E V ) . For a graph 6 = ( yE ) the i-th adjacency matrix Ai = (Ai)zy, where x,y run over V, is defined by N
-
We set A = A1 which is called the adjacency matrix for simplicity. 2.2
Stratification of a Graph and Associated Hilbert Space
LFrom now on let 6 = (V,E ) be a regular connected graph with a fixed origin xo E V. The degree is denoted by K . Then, the graph is stratified into a disjoint union of strata:
u v,, 00
v=
n=O
vn = { x E v ;d(zI),x) = n ) .
(5)
29 1
Obviously, lVol = 1, IVll = n, and IV,l 5 n(n - l),-' for n 2 2. The next result is immediate by the triangle inequality, see Figure 1. Lemma 2.1 Let x, y E V . I f x E V , and x N y, then y E Vn-l U Vn U Vn+l. For x E V we denote by 6, the indicator function of the singlet {x}. The collection (6, ; x E V } forms a complete orthonormal basis of C2(V). According to ( 5 ) , we define a Hilbert space:
r(q = C @can, an = I
C
V ~ ~ - ~6,,' ~ XEV,
n=O
where
{a,}
becomes an orthonormal basis of
r(6).
Vntl
Vn
Vn-1
.. .. .. . . .
Figure 1. Stratification of 0 = (V, E ) with n = 4
2.3 Adjacency Matrix as Algebraic Random Variable First note that An is well defined for all n 2 1. In fact, the ( 5 ,y)-component of An coincides with the number of paths of length n that connect x and y. Obviously, 0 5 (A"),, 5 I C ~ . Hence we can define the unital *-algebra generated by A, which will be denoted by A. The adjacency matrix A acts on the Hilbert space t 2 ( V )in a natural manner: Af(x) =
c
C f(Y),
YEV
YWX
AXYf(Y) =
f E t2(V)
292
Note that IlAll = K and A is injectively imbedded in B(12(V)). In general, a sta,te q5 on A is chosen by a question. In this paper we consider the vacuum state defined by
4(a)
=
(Sxo,a6x,),
a E A.
Thus the adjacency matrix A is regarded as a real random variable of the algebraic probability space (A,4). 2.4
Quantum Decomposition of Adjacency Matrix
By virtue of Lemma 2.1 we assign to each edge x y of the graph Q = (V,E ) an orientation compatible with the stratification, i.e., in such a way that x i y if x E V, and y E V,+l. For an edge x y lying in a stratum V, there are two ways of assigning an orientation and, as a result, there are many ways of giving an orientation to the graph Q, see Figure 2. Then we define N
N
=
{tYx =
1 if y >- x, otherwise,
otherwise,
or equivalently,
As is easily verified: (A+)* = A- and
A
= A+
+ A-,
(7)
which is called a quantum decomposition. We keep in mind that the above quantum decomposition depends on an orientation introduced into Q. Remark 2.2 There is a slightly different idea of quantum decomposition based on a Fock space structure of r(Q).Namely, to an edge x y lying in a stratum V, we do not give an orientation, we define instead a new operator A" to have A = A+ A- A". Obviously, these operators are more like the creation, annihilation, and number operators. This decomposition will be discussed elsewhere. N
+
+
2.5 Fundamental Question Given a "growing" family of graphs {Qx = (V('),Idx)) ; A E A}, where A is an infinite directed set, we consider the adjacency matrix Ax of Qx with its
293
Figure 2. Quantum decomposition: A = A+
+A-
+
quantum decomposition Ax = A;t' A , as in (7). A general question is to B+, B - ) in which the limits construct an interacting Fock space (r,{A,},
are described, where ZX is a normalizing constant. A rigorous statement will be naturally formulated as a quantum central limit theorem. When C* admit simple expressions such as linear combinations of El* and the number operator N on r', the distribution of C+ C- is easily computed with the help of Theorem 1.5, see also Subsection 1.4. Note that the distribution of C+ C- reflects asymptotic spectral properties of Ax.
+
+
3 3.1
Quantum Central Limit Theorem Conditions (AI)-(A3) and Main Statement
Let {GA = ( V ( X ) , E ( X )X) ; E A} be a family of regular connected graphs, where A is an infinite directed set, with stratification:
n=O
The degree of GX is denoted by K ( X ) and assume limx %(A) = 00. The associated Hilbert space is denoted as usual by
294
where {@,( A ) } forms an orthonormal basis. Keeping Lemma 2.1 in mind, for
x
E
v,"'
we put
,(A' +
(x)= { y E
v;:;
;y
N
x}, w'x'(z)= { y
E
v;i\
;y
N
z}.
With these notation we consider the following conditions:
( A l ) there is no edge lying in a common stratum ViX', i.e., Iw?'(5)1
+ Iw'X'(x)I = .(A),
zE
v('),
E A;
(9)
(A2) for each n = 1 , 2 , .. . there exist an integer w, 2 1 and a constant C, 2 0 such that
I{.
E
vix); I W ' x ' ( 2 ) I # W,}l 5 C n K ( A ) " - '
for all A E A;
(A3) for any n = 1 , 2 , .. . ,
By ( A l ) there is only one orientation which is compatible with the stratification, and therefore a quantum decomposition Ax = A: A, of the adjacency matrix of BX is unique. Condition (A2) means that for any X E A and n = 1 , 2 , .. . , a generic x E V,'" is connected by edges with exactly w, vertices in the lower stratum V;i\. As for condition (A3), we note that supxEvp' (w-(A) )I(. 5 .(A), which follows from (9). Hence (A3) ensures uni-
+
form boundedness. Existence of the stochastic limits (8) is now claimed in the following
Theorem 3.1 Let {GA = ( V ( X ) , E ( XX) )E; A} be a family of regular connected graphs such that limxtc(X) = 00. Assume conditions (Al)-(A3) are satisfied and let (I', {A,}, B+,B - ) be the interacting Fock space with A, = w1.. . w,. Then,
for any choice of j , k = 0 , 1 , 2 , .. . , m = 1 , 2 , . . . and € 1 , . . . E , E {k}. In particular, taking the vacuum states (i.e.l j = k = 0) we come to the following
295
Corollary 3.2 The normalized quantum components A : / m of the adjacency matrix of 6~ converges stochastically t o the annihilation and creation operators on the interacting Foclc space (r,{An}, B+,B - ) , where A n = w1 . . .W n . The classical reduction is now immediate. Corollary 3.3 It holds that lim XEA
(a?), (A)
m
m = 0 ,1 ,2 ,. . . ,
zmp(dx),
=
P(V(X))
where p is a probability measure corresponding to the interacting Foclc space (r,{ A n } , B+,B-1. 3.2 Proof of Theorem 3.1 Theorem 3.1 was first proved by Hashimoto'' for a class of Cayley graphs and the proof therein is easily adapted to our case. Here we only mention a sketch. For simplicity we drop the suffix A. By condition (A2) we split Vn into a disjoint union of two subsets:
v, = v y u vex n ,
v y = {x E v, ; W-(.)
VF = {x E vn; W-(.)
= w,},
# w,}.
Lemma 3.4 IVnl=
Kn
n 2 1.
+O(K~-'),
~
wn...wl
(10)
PROOF.By counting the number of edges whose endpoints are in V, we have KlKl
=
c
lw-(Y)l+
YEVn+1
=wn+llvn+ll+
c
Iw+(z)I
zEV,-1
C
( ~ w - ( Y )-wn+1) I
Y€v:'n";l
+ ( ~ . - w n - 1 ) 1 v n - 1-/
C
(IW-(Z)I
ZEV,e?',
For n 2 1 we put
Pwn-1).
296
where ug = 0. Applying two inequalities: IV,""l 5 C,K,-' and Iw-(x)I 5 Wn for all x E V,, which follow respectively from (A2) and (A3), we come to IS,(
I Wn+lCn+lKn f Wn-1Cn-1Kn-2
= O(Kn).
(11)
Then, by repeated application of (11) we obtain (10) with no difficulty.
I
Lemma 3.5 n = 0,1,2, . . .
PROOF. By definition we have
X€V,
Y€V*+1
+
= wn+11Vn+l11/2Qn+1
C
(w-(Y) -un+1)6,.
YGV,.;,
Then, (12) follows by a direct computation with the help of Lemma 3.4. The proof of (13) is similar. I Consider an interacting Fock space (r,{An}, B+,B - ) with A n = u1. . .W n . Then, at a formal level we see immediately from (12) and (13) that
In fact, the proof of Theorem 3.1 is a direct computation using Lemma 3.5. 3.3 Cayley Graphs
Consider a discrete group G with the identity e and a set of generators C c G satisfying (i) CT E C + CTE' C, i.e., C-l = C; and (ii) e @ C. Then G becomes a graph, where a pair x , y E G satisfying y x - l E C constitutes an edge. This is called a Cayley graph and denoted by (G, C). A Cayley graph is regular with degree IE = 1x1. We consider e as the origin of the Cayley graph and introduce a stratification as usual. Example 3.6 (Lattice) The additive group Z N furnished with the standard generators g+1 = ( + l , O , . . . , O ) , . . . ,Q+N = (0,... , O , f l ) , is the N dimensional lattice. Conditions (Al)-(A3) are easily verified with K ( N )= 2N, w, = n and W, = n. Hence An = n! and the limit is described by the Boson
297
Fock space. By Proposition 1.1 for the normalized adjacency matrix we have
AN/^
This reproduces the central limit theorem in the classical probability theory. Example 3.7 (Homogeneous tree) Let FN be the free group on N free generators 91,. . . ,gN. Then the Cayley graph ( F N ,{g*1,. . . , g + ~ } ) where , g-% = gZp1 for simplicity, becomes a homogeneous tree. Conditions (Al)-(A3) are easily verified with K ( N )= 2 N , w, = 1 and W, = 1. Hence A, = 1 for all n 2 0 so that the limit is described by the free Fock space. Moreover, by Proposition 1.3 for the normalized adjacency matrix A N / mwe have
This is a prototype of the central limit theorem in the free probability theory, see e.g., Hiai-Petz13, Voiculescu-Dykema-Nica19. Example 3.8 (Coxeter group) We refer to Humphreys17 for generalities. Let C = (91,g2,.. . } be a countable infinite set and consider a Coxeter matrix m(i,j) E { 1 , 2 , . . . ,ca}l where i, j run over { 1 , 2 , .. . }, such that m(i,i) = 1 and m ( i , j ) = m ( j , i ) 2 2 for i # j . For each N 2 1 let G ( N )be the group generated by C“) = (91,g2,. . . ,gN} subject only to the relations: (g.z 93.)+>A = e,
i , j E { 1 , 2, . . . ,N } .
In case of m ( i , j )= 00 we understand that gigj is of infinite order. Note that any gi E C is of order two and hence K ( N )= N . Condition ( A l ) is satisfied by every Coxeter group G ( N ) . Furthermore, (A2) and (A3) are satisfied if m ( i , j ) 2 3 for any pair of distinct i , j . In that case wn = 1 and W, = 2 for all n 2 1. Hence, for a growing family of Coxeter groups (G(N)lC(N)) with Coxeter matrix satisfying m ( i l j ) 2 3 for i # j , the situation falls into the same as in Example 3.7. The distribution of AN/^ was obtained by Fendler’ with a different method. Detailed discussion is found in HashimotoHora-Obata”. 4
4.1
More on Quantum Central Limit Theorem
Motivation
In the previous section we proved a quantum central limit theorem for adjacency matrices (Theorem 3.1) under the three assumptions (Al)-(A3). However, there are many interesting graphs which do not satisfy condition ( A l ) .
298
As is discussed in Subsection 2.4, a quantum decomposition of the adjacency matrix is always possible since it is induced from an orientation compatible with the stratification, where condition ( A l ) is not necessary. Hence our question is how to introduce an orientation good enough to obtain quantum central limit theorems for the quantum components of the adjacency matrix. We shall illustrate this problem with distance-regular graphs.
4.2 Distance-Regular Graphs
A finite connected graph 6 = (V, E ) is called distance-regular if for any choice of h, i , j E {0,1,. . . ,d } , d being the diameter of the graph, ( { Z E V ; a ( x , z ) = i , d ( z , y ) = j } ( ~ ph i j does not depend on the choice of x , y E V such that a ( x ,y ) = h. We call { p b } the intersection numbers of 6. For simplicity we set Kn
0
= Pan,
0 n = K.1 = p11.
Obviously, the distance-regular graph is regular with degree n.
4.3 Q u a n t u m Decomposition Induced f r o m Euler Paths Let 4 = (V,E ) be a distance-regular graph with intersection numbers {&}. We fix an arbitrary xo E V as the origin and introduce the stratification:
u v,, d
v=
v, = {z E v ;d ( x 0 , x ) = n}.
n=O
-
Note that (V,( = K,. For a quantum decomposition of the adjacency matrix A it is sufficient to specify an orientation of an edge x y lying in V,. Let X , be a subgraph of 6, where the set of vertices is V, and the edges are those of lying in V,. Then X , is a regular graph with degree p;",. We now consider another condition:
(A4) for each n
=
1 , 2 , . . . one of the two cases occurs:
(Case 1) p;", is even; (Case 2) p?, is odd and X , admits a perfect matching. Here we recall definition: in general, a graph (V,E ) is said to admit a perfect matching if there is a subset M c E such that each x E V is an endpoint of just one edge of M .
299
When (Case 1) occurs, by Euler's unicursal theorem there is an Euler path for X , along which each edge is given an orientation. When (Case 2) occurs, deleting M from the subgraph X , we obtain another subgraph 2, which is a regular graph with even degree. Then, taking an Euler path for X,, we give an orientation to each edge of X,. An edge of M is given an arbitrary orientation. Thus, a distance-regular graph E satisfying condition (A4) is given an orientation. Let A = A+ + A - be the quantum decomposition induced from the above orientation. Put X€V,
where {@,} is an orthonormal basis of
r(6).Then, by a direct computation,
(Case 1)
n E {0,1,. . . , d } .
(15)
(Case 2) Let V; (resp. V Z ) be the set of all x E V, which are initial (terminal) vertex of an edge of M . Then Vn = V', U V; and
Hence the action of A* is defined on r(E) when (Case 1) occurs. This is not true for (Case 2) as in the case discussed in Section 3.
4.4
Quantum Central Limit Theorem
We consider a growing family of distance-regular graphs {EX; X E A} such 00 and .(A) 00, where d ( X ) and .(A) are the diameter and that d(X) the degree of G, respectively. The adjacency matrix and the intersection numbers of 6, are denoted by Ax and {P(X);~}, respectively. By definition --f
--f
n=O
Theorem 4.1 Assume that every E, satisfies condition ( A d ) and let Ax = A I + A , be the adjacency matrix of G, with its quantum decomposition defined as an Subsection 4.3. Assume that the limits
300
exist f o r all n = 0 , 1 , 2 , .. . , and let (I?, {A,}, B+,B - ) be the interacting Foclc space, where A0 = 1, A, = w1. . .w, f o r n 2 1. Define
C*=B+++N, where N is the number operator of r. Then f o r any choice of j , lc = 0,1,2,.. . , m = 1 , 2 , . . . and € 1 , . . . ,E , E {k},it holds that
This is proved by employing a similar argumant as in HashimoteObataTabei12, see HashimoteHora-Obatall for details. The classical reduction is immediate. Corollary 4.2 Under the same assumptions as in Theorem 4.1 it holds that
f o r all m = 0 , 1 , 2 , .. . . Remark 4.3 As we shall see later in Lemma 4.6, the inner product in the ) , Tr stands for left hand side of (19) coincides with T r ( ( A ~ / r n ) ~where the normalized trace. Therefore a probability measure p on R such that
gives an approximation of the eigenvalue distribution of A x / m . For some classes of distance-regular graphs including the Hamming and Johnson graphs the distributions p were first obtained by Hora14 with a classical method.
4 . 5 Hamming Graph
+
Let F be a finite set of n 1 points and d 2 1 an integer. For x = ((1, and y = (71,. . . ,q d ) in V = F d we put
. . ' ,&)
a(x,y) = H i ; ti # %}I. A pair x,y E V is by definition an edge if a(x,y) = 1. Then V becomes a distance-regular graph called a Hamming graph and is denoted by H ( d , n+ 1). As is easily verified, p:;'=k+l, k
pk. = k ( n -
lc=O,l;..,d-l, I),
k = 0 , 1 , ..
pkT1 = n ( d - lc+ l), 0
p , , = nd = IE.
'
, d,
k = 1,2,... ,d,
301
Moreover, it was proved in Hashimoto-Obata-Tabei12 that condition (A4) is satisfied by H ( d ,n 1) for any choice of d and n. Let A(++l) = A&,n+l)+AG,n+l)be the adjacency matrix of H ( d ,n+ 1) and its quantum decomposition. Then the conditions in Theorem 4.1 are fulfilled with w k = lc and '& = l c f i / 2 , and hence the limit is described by the Boson Fock space as follows. Theorem 4.4 It holds that
+
lim d,n+m,nld+r
= Blk ~
J;'N ,
+ -
2
in the sense of quantum random variables (see Theorem 4.1), where B' and N = B+B- are the creation, annihilation and number operators on the Boson Foclc space. By the classical reduction we have
=
(Qo, ( f i B + B -
+ B+ + B-)mQ.o),
rn = 0 , 1 , 2 , . . . . (20)
The unique probability distribution v, whose rn-th moment is given by (20) is the standard Gaussian distribution for r = 0 and by the image of the Poisson distribution of parameter l / r under the map x H f i x - (l/J;') for r > 0, see Examples 1.7 and 1.8. 4.6
Johnson Graph
Let v , d be a pair of positive integers such that d 5 v . Put S = {I, 2 , . . . ,v } and V = {z c S ; 1x1 = d } . We say that z, y E V are adjacent if d - lxnyl = 1. Thus a graph structure is introduced in V, which is called a Johnson graph and denoted by J ( v ,d). By symmetry we may assume that 2d 5 v . The Johnson graph J(w, d ) is distance-regular with intersection numbers /c
= d(v
-
d),
p;", = n ( v
-
2 n ) , p;",+' = ( n
+ v,
(21)
where n = 0,1, . . . ,d. It was proved by Hashirnoto-Hora-Obatal1 that every Johnson graph J ( v ,d ) fulfills condition (A4). Consider the growing family of Johnson graphs J ( v , d ) , where d + co and 2d/v -+ p E (0, I]. Condition (17) in Theorem 4.1 is satisfied with
302
Then for the quantum decomposition matrix of J ( u , d ) we have the following Theorem 4.5 Let 0 < p 5 1. Then
= A:,,)
+AG,,) of the adjacency
in the sense of Theorem 4.1, where B', N are respectively the creation, annihilation, and number operators on the interacting Fock space r = (r,{(n!)'},B+,B-1. The classical reduction is immediate:
= (Qo,
2N ),.> (B' + B- + diF3
, m = 0 , 1 , 2, . . . . r
The probability measure vp on R whose m-th moment is given as above is obtained by observing the associated orthogonal polynomials. For p = 1, the Laguerre polynomials L,(x) = 2, . . . satisfying the recurrence formula:
+
Lo(.) = 1, Ll(2) = 5 - 1,
+
+
+
zL,(x) = L,+l(X) (an l)L,(X) n2L,-1(x), n 2 1, play a role. By using the fact that the Laguerre polynomials form the orthogonal polynomials with respect to the probability measure e-"dx on the half line [O, m), i.e.,
Lrn
xme-"d2 = (Qo, (I?+
r
+ B- + 2N + l)"Qo)r,
(I',{(n!)'},B+,B-) and N is the number operator, we see that on [-1,m). For 0 < p < 1 the Meixner polynomials (see Schoutens18 for definition) play a role. By modification the polynomilas M,(z) = 2, + . . . defined by where
=
vl(d5) = e-("+l)da:concentrated
Mo(x) = 1,
303 form the orthogonal polynomials with respect to the probability measure
that is,
where I? = (I?,
TI!)^}, B+,B-). Thus, by translation of pp we obtain
4.7’ Bose-Mesner Algebra and Weak Quantum Decomposition Let = ( V , E ) be a general distance-regular graph for which (A4) is not necessarily fulfilled. As usual let A be the unital *-algebra generated by the adjacency matrix A. This is called the Bose-Mesner algebra of 6. It is known (see e.g., Bannai-Ito4) that the adjacency matrices 1 = Ao, A = Al, . .. ,Ad are linearly independent and satisfy the relation:
Hence A is a vector space with linear basis 1 = Ao, A = Al, . . . ,Ad. The Bose-Mesner algebra A becomes an algebraic probability space equipped with the normalized trace Tr. The GNS-representation of A is realized on the Hilbert space V ( A )obtained from A equipped with an inner product:
( a ,b ) A = Tr (a*b),
a, b E A.
We see from an obvious relation:
( A ~ , A ~=)6Ai j ~ = i (wi,t.j),
~i =
C 6, E 12(V), X€Vd
that the correspondence Ai ++ wi yields a unitary isomorphism between V ( A ) and r(6)c 12(V).Moreover, since d
h=O
304
which is easily verified by definition, we see from (22) that the above unitary isomorphism intertwines the action of the Bose-Mesner algebra A. In particular , Lemma 4.6 T h e action of the adjacency m a t r i x A o n r(Q) i s unitarily equivalent t o that induced f r o m t h e GNS-representation of (d,Tr)and (Qo, A”Q0) = Tr(Am),
m = O,1,2,. . . .
It follows from (22) with a simple triangle inequality that
AAn = $l1A,-1
+ pYnAn + p1,n+l An+l,
n = 0,1,. . . , d.
We then define
n = 0, 1,. . . ,d. + P7 L A,, and A = A+ + A - , which is referred to as weak quan-
A*An = pY:’An*1
Obviously (A+)* = Atum decomposition. This is equivalent to adopt (15) ignoring the orientation of the graph. Thus the weak quantum decomposition does not reflect an orientation of the graph though the actions of the quantum components are well defined on r(Q). Theorem 4.1 remains valid if the quantum decomposition is replaced with the weak one. In fact, the proof is almost similar.
Acknowledgments This work is supported by JSPS-PAN Joint Research Project “Infinite Dimensional Harmonic Analysis.”
References 1. L. Accardi and M. Bozejko: Interacting Fock spaces and Gaussianization of probability measures, Infin. Dimen. Anal. Quantum Probab. Relat. Top. 1 (1998), 663470. 2. L. Accardi, Y. Hashimoto and N. Obata: Notions of independence related t o the free group, Infin. Dimen. Anal. Quantum Probab. Relat. Top. 1 (1998), 201-220. 3. L. Accardi, Y. Hashimoto and N. Obata: Singleton independence, Banach Center Publ. 43 (1998), 9-24. 4. E. Bannai and T. Ito: “Algebraic Combinatorics I, Association Schemes,” Benjamin, 1984.
305 5. M. Bozejko, B. Kummerer and R. Speicher: q-Gaussian processes: Noncommutative and classical aspects, Commun. Math. Phys. 185 (1997), 129-1 54. 6. D. M. Chung, U. C. Ji and N. Obata: Quantum stochastic analysis via white noise operators in weighted Fock space, Rev. Math. Phys. 14 (2002), 241-272. 7. P. Deift: “Orthogonal Polynomials and Random Matrices: A RiemannHilbert Approach,” Courant Lect. Notes Vol. 3, Amer. Math. SOC., 1998. 8. G. Fendler: Central limit theorems f o r Coxeter systems and Artin systems of extra large type, preprint 2000. 9. Y. Hashimoto: Deformations of the semicircle law derived f r o m random walks o n free groups, Prob. Math. Stat. 18 (1998), 399-410. 10. Y. Hashimoto: Q u a n t u m decomposition in discrete groups and interacting Foclc spaces, Infin. Dimen. Anal. Quantum Probab. Relat. Top. 4 (2001), 277-287. 11. Y. Hashimoto, A. Hora and N. Obata: Central limit theorems for large graphs: Method of quantum decomposition, J. Math. Phys. in press. 12. Y. Hashimoto, N. Obata and N. Tabei: A quantum aspect of asymptotic spectral analysis of large H a m m i n g graphs, in “Quantum Information I11 (T. Hida and K. Sait6, Eds.),” pp. 45-57, World Scientific, 2001. 13. F. Hiai and D. Petz: “The Semicircle Law, Free Random Variables and Entropy,” Amer. Math. SOC.,2000. 14. A. Hora: Central limit theorems and asymptotic spectral analysis o n large graphs, Infin. Dimen. Anal. Quantum Probab. Relat. Top. 1 (1998), 221-246. 15. A. Hora: Gibbs state o n a distance-regular graph and its application t o a scaling limit of the spectral distributions of discrete Laplacians, Probab. Theory Relat. Fields 118 (2000), 115-130. 16. R. L. Hudson and K. R. Parthasarathy: Q u a n t u m A6’s formula and stochastic evolutions, Commun. Math. Phys. 93 (1984), 301-323. 17. J. E. Humphreys: “Reflection Groups and Coxeter Groups,” Cambridge UP, 1990. 18. W. Schoutens: “Stochastic Processes and Orthogonal Polynomials,” Lect. Notes in Stat. Vol. 146, Springer-Verlag, 2000. 19. D. Voiculescu, K. Dykema and A. Nica: “Free Random Variables,” CRM Monograph Series, Amer. Math. SOC.,1992.
306
AN INTRODUCTION TO THE EPR-CHAMELEON EXPERIMENT LUIGI ACCARDI, KENTARO IMAFUKU, MASSIMO REGOLI Centro Vito Volterra, Universitd di Roma “Tor Vergata” E-mail: [email protected], WEB page: http://volterra. mat.uniroma2.it On September 27 (2001), as a side activity to the ”Japan-Italy Joint workshop on: Quantum open systems and quantum measurement”, the first public demonstration of the dynamical EPR-chameleon experiment was performed at Waseda University in order to give an experimental answer to a long standing question in the foundations of quantum theory: do there exist classical macroscopic systems which, by local independent choices, produce sequences of data which reproduce the singlet correlations, hence violating Bell’s inequality? The EPR-chameleon experiment gives an affirmative answer to this question by concretely producing an example of such systems in the form of three personal computers which realize a local deterministic dynamical evolution whose m a t h e matical structure is very simple and transparent. In the experiment performed on September 27 the local dynamics used was not a reversible one because the interaction with the degrees of freedom of the apparatus was integrated out giving rise to an effective Markovian dynamics which, although mapping probability measures into probability measures, did not preserve the fl-values of the spin (or polarization) observables. This feature was criticized by some of the partecipants and the following two questions arose: i) is it possible to prove that the Markovian evolution, used in the experiment, is indeed the reduced evolution of a bona fide reversible evolution? ii) if the answer to question (i) is affirmative, is it possible to reproduce the EPR correlations by simply considering empirical averages of f-1-values, as one does in usual EPR type experiments? An affirmative answer to these questions was given in the paper [AcImReOl]and it is briefly reviewed in what follows.
1
Introduction
In the past 20 years quantum probability has challanged the widespread belief that classical macroscopic systems cannot, by local independent choices, produce sequences of data whose correlations violate Bell’s inequality. The possibility of such a violation is not a matter of interpretation, but of fact: ”local independent choices” means that two separated and non communicating experimenters make measurements but one does not know what the other measures (or even if the other one measures something); correlations
307 are evaluated by means of standard procedures. There is no space for verbal compromise in this question: either the s u p porters of Bell’s thesis or those of quantum probability are wrong. Tertium non datur. The goal of the present experiment is to prove that the point of view of quantum probability is correct. On the other hand, since the Bell thesis is widely known and the quantum probability thesis - widely unknown, it might be useful to quickly pinpoint the main critiques quantum probability moves to Bell’s analysis. It is interesting to notice that while, in the past years, a multiplicity of books and papers have appeared, which reproduce modulo inessential variants, the original Bell argument, the problem to reply to the quantum probability critiques has not been tackled. In absence of such a reply, the only reasonable conclusion is that the quantum probability critiques are sound and that the theoretical analysis, from which the contradiction between quantum theory, locality and reality is deduced, is effectively a weak one. The present experiment brings this theoretical analysis to its logical conclusion by producing a concrete counterexample. The reason why the interpretation of quantum mechanics has been one of the most fundamental conceptual problems of modern science and practically all the best physicists, philosophers of science and mathematicians of the past century confronted themselves with this problem, is probably the uneasy feeling that there might be a deeply rooted contradiction between two of the basic pillars of contemporary physics: quantum theory and relativity theory. For almost 40 years arguments based on violation of Bell’s inequality have been deemed to provide the basis for an experimental proof of such contradiction (with the meager consolation that this contradiction might not be exploitable for human telecommunications). The present experiment shows that this is not the case: in no way the EPR correlations and related experiments can be considered as a support of the incompatibility of quantum theory with local realistic theories, in particular relativity. 2
The Bell inequality
In this section we briefly survey the quantum probabilistic analysis of Bell’s inequality. Bell’s inequality was proved in the paper [Be64]. In this paper, while the thesis, i.e. the inequality itself, is clearly stated and correctly proved, the mathematical assumptions from which the thesis follows (and without which
308 the thesis cannot be proved) were not formulated. This opened a debate whose goal was to try and establish which these assumptions effectively were. The reader interested in having an idea of the arguments used before quantum probability may consult the famous [Wig701 or [Stap82] or, for the connections with probability [SuZa76]. The mathematical formulation, now commonly adopted, of the Bell inequality was first given in [Ac81]. The main result of this paper consists in having realized that the mathematical assumptions on which the validity of the inequality depends are only the following two ones: (i) that the random variables take values in the interval [-1,1] (originally Bell considered only the set { -1,1} but shortly after he extended his result to the full interval) (ii) that all the random variables are defined on a single probability space. More precisely: Lemma (1) Let A, B , C be random variables defined on the same probability space (0,F,P ) and with values in the set { -1,l). Denote
(AB) :=
/
A(w)B(w)P(dw)
R
the correlations (mean zero can be assumed without loss of generality). Then the following inequality holds:
\ ( A @ - ( C B ) \5 1 - (AC)
(1)
Remark The proof below is, modulo notational variations, Bell’s original one. For generalizations and variants, cf. [AcReOOb]. Proof. Since the expectation value is linear
[ ( A B )- (CB)I = I(AB - CB)I (2) Now we use I(X)l 5 (1x1) and the fact that A , B , C are fl-valued to deduce that
5 (IAB-CBI)
=
( I A B I . \ l - A C ( )= (Il-ACI) = ((1-AC))
=
l-(AC) (3)
In the original Bell’s inequality (1) the observable C is measured by experimenter 1 in the second experiment and by experimenter 2 in the third. Thus, in order to verify violation of this inequality there must be a preliminary agreement between the two experimenters (or recourse to random choices with a posteriori matching). The following corollary of (1) (which is an equivalence for fl-values observables), due to Clauser, Home, Shimony, Holt (CHSH), allows to perform the violation experiments without exchange of informations on the random
309
variable C (typically this represents a spin or polarization direction). For this reason, even if mathematically it does not add anything to Bell’s argument, it is widely used in the literature. Corollary (2) Let A , B , A’, B’ be random variables defined on the same probability space ( O , F , P ) and with values in the set {-ill}. Then the following inequality holds:
I(AB) - (A’B)+ (AB’)+ (A’B’)I 5 2 Proof. With the replacements B
+ B’,
(4)
C + -C, (1) becomes
+
I(AB’) (CB’))= 1
+ (AC)
(5)
Adding (1) to ( 5 ) and replacing C by A‘ we get
I(AB) - (A’B)I
+ I(AB‘) + (A’B’)I I 2
which implies (4). The following rephrasing of Corollary (2) is used to establish a connection between Bell’s inequality and the predictions of quantum mechanics. Corollary (3). Let S S be fl-valued random variables defined on a probability space (0,F,P ) . Then
s?),A’)’ i2)
I5
1 - (spsp)
l(sps:2)) - (sps:2))
(6)
Proof. (6) is obtained by replacing, in (1):
A+,”?)
.,
B
+
SL2)
C
;
+ S:’)
Remark. Notice that, in a quantum mechanical context, (6) would not be measurable for a # c. Remark. The CHSH analogue of (6) is
I(spsj2)) - (Sa, (1) s,( 2 ) ) + (spsp)+ (sps:f’)/ 52
( 7)
With these notations the main mathematical conclusion, used in the comparison of the predictions of Bell’s inequality with those of quantum theory is the following Theorem (4). There cannot exist a stochastic process
Sp) , S:’)
a, b E [0,27r]
defined on a probability space (O,F,P ) and with values in the set {+I}, whose correlations are given by:
(S?)S:2 ) ) = -cos(a
-
b)
;
a,b E [0,27r]
(8)
310
Remark. According to quantum theory the expression in the right-hand side of (8) is the correlation of two spin or polarization observables, along directions a, b, of two quantum particles in singlet state. These correlations have been experimentally confirmed by many experiments since the early days of quantum mechanics. Proof. Suppose, by contradiction, that such a process exists. Then, by Corollary (3)’ it must satisfy (6). A corollary of (8) is the singlet condition:
which is equivalent to
s:’) = -sL2)
P
-
a.e.
(9)
The singlet condition (9) implies that (6) is equivalent to
But, choosing a=O
;
b=r/2
;
c=r/4
one sees that (10) leads to the contradiction 4 5 1. Remark. The deduction of (7) (i.e. CHSH) does not require the singlet condition (9). However once a single probability space is postulated, the quantum mechanical condition (8) implies (9).
3
The implications of Bell’s inequality according to Bell
Theorem (4) above synthetizes the conclusions of the quantum probabilistic analysis of the implications of Bell’s inequality. Bell’s conclusions about these implications were quite different! They are clearly stated both in the Introduction and in the Conclusions of [Be641 and can be summarized by Bell’s own words: ” ... In a theory in which parameters are added to quantum mechanics to determine the results of individual measurements, without changing the statistical predictions, there must be a mechanism whereby the setting of one measuring device can influence the reading of another instrument, however remote. Moreover the signal involved must propagate instantaneously so that the theory could not be Lorentz invariant. ...” (cf. the beginning of Section VI of [Be64]). A theory ” ... in which parameters are added to quantum mechanics to determine the results of individual measurements, without changing the
31 1
statistical predictions ...” is called a ”hidden variable theory”. Such a theory is also called ”realistic” because in it observables have always a definite value, as opposed to the orthodox interpretation of quantum mechanics in which observables have virtual values which become actual by virtue of the measurement. To ” ... add parameters to quantum mechanics ...” means that the quantum observables are represented as functions on a single space 0 (the space of hidden parameters). ”... without changing the statistical predictions ...” means (in the EPR context) that the hidden parameters are distributed according to a (possibly unknown) probability measure P on 0 and that the experimentally measurable correlations of any pair of observables A , B are those predicted by classical probability, i.e.
The locality condition, according to Bell, is that: ... the result B for particle 2 does not depend on the setting a, of the magnet for particle 1, nor A on b. ...” (cf. Section I1 of [Be64]. In fact at this point Bell quotes EPR.). A hidden variable theory, satisfying the above conditions is called a local realistic model for the EPR (singlet) correlations. For these reasons Bell’s conclusion on the implications of his inequality is often synthetized in the statement: ” ... A local realistic model for the EPR correlations cannot exist. ...” Bell’s stat,ement reported above has given rise to a huge literature turning around the apparent dichotomy pointed out in this statement: if we insist on a realistic point of view, then we have to give up locality and conversely, if we insist on locality, then we have to give up a realistic point of view. Both choices would be heavy for a physicist. The quantum probabilistic approach offers to the physicists a way out by criticizing Bell’s analysis and proving that: i) the contradiction, pointed out by Bell, arises only from his implicit postulate that 3 statistical correlations, coming from 3 mutually incompatible experiments, can be described within a single classical probabilistic model ii) that this implicit postulate is by no means a consequence of locality and reality. If this implicit postulate is not assumed, then Bell’s proof is at fault already in its first step, i.e. (2), because if we write explicitly the identity ”
( A B )- (CB)= (AB - C B )
(1)
312
we find
and, while the pair joint probabilities Pa$, P c , b, ... are experimentally observable, there is no reason to postulate, as Bell implicitly does when using this formula, that the, experimentally unobservable, triple joint probabilities Pa,c,b exist. It is well known from classical probability that there are constraints, i.e. compatibility conditions, which relate the pair with the triple joint probabilities and which are necessary conditions for the existence of the latter ones. Since the pair correlations are deduced from the pair probabilities and since, when using ( 2 ) 1 Bell is postulating a priori the existence of these (experimentally unobservable) triple joint probabilities, the only rational conclusion he can draw from his argument is that the inequality (1) (Bell's inequality) is one of these necessary conditions. This was the critique that, starting from 1981 [Ac81],quantum probability opposed to Bell's argument. One might try to counter this critique by arguing that the existence of the triple probabilities is a consequence of the "realism" assumption. For example suppose that in a box there are many pairs of balls whose color can be either green or brown. Moreover each ball is either made of glass or of wood and it weights either 10 or 20 grams. The rules of the game are such that you can only measure one observable at the time on each ball (color, weight, material). Thus on each pair we can simultaneously measure at most two observables and we can make an experimental analysis of the joint statistics of all possible pairs of observables ("color-material" "color-weight",
...). Because of the rules of the game the triple joint probabilities "colormaterial-weight" are not accessible to experiment. However the "realism assumption" tells us that any one of the possible combinations (color, material, weight) has a definite relative frequency in the box and that the pair statistics we observe, is a consequence of this (unobservable) triple statistics. Arguments of this kind are quite reasonable: for example they are at the basis of classical statistical mechanics and it is probable that Einstein had in mind something of this kind when speaking of "objective reality". According to quantum probability there is a more subtle notion of "objec-
313
tive reality” which gives a better intuition of the behavior of quantum systems (but it by no means restricted to them). We call the corresponding realism ”chameleon realism” as opposed to the ”ballot box realism” of classical statistical mechanics. Suppose that, in the above example, you leave the rules of the game unaltered, but you replace the pairs of balls by pairs of chameleons and the observables (color, material, weight) by (color on the leaf, color on the wood, weight). Is it still reasonable to believe that the pair statistics you observe, is a consequence of some (unobservable) triple statistics? A little thought shows that the answer is: No! According to quantum probability, quantum systems are much more similar to chameleons (adaptive: we measure the response to an interaction) than to balls (passive: we read what was in the box). 4
The chameleon effect
The attempt to translate in a precise mathematical and physical language the intuitive difference between ”ballot boxes” and ”chameleons” leads to a natural generalization of von Neumann’s measurement theory. The generalization consists in introducing, in this theory, the notions of locality and causality. It is widely accepted, since von Neumann’s original analysis, that a qualitative analysis of the measurement process should start from the joint (unitary) evolution U S , A (system, apparatus): for simplicity we consider discrete time. Thus, if $o is the initial state of the system, its state at the time of measurement is
Now suppose that we want to measure the observable S, = S, 8 1~ of the system (say: spin in direction u ) . Then the apparatus must be prepared to measure S, (say: by orienting a magnetic field in direction u ) . Therefore the interaction Hamiltonian between system and apparatus, hence also the joint dynamics, will depend on a: U S , A := U S , A ( a ) := u a
(4) In other words: the dynamics of a system depends on the observable we want to measure: this is the chameleon effect. As anybody can see, it is a simple
314
corollary of the standard ideas on measurement theory. Now suppose that the system, hence the apparatus, is made up of two spatially separated parts: (1,2, A l , Az) and that we measure independently
sL1)= (s,8 1 2 ) 8 1 M
(resp. 5L2)= (11 8 s b ) 8 IM)
on particle 1 (resp. particle 2). Then, according to the chameleon effect, we will have
(5)
U S , A := u a , b
and, according to quantum (or classical) mechanics, the pair correlations will be
(s2)sf)):=
$0
0
Ua,b
(s2)sf)) = $a,b
(s2)sL2))
(6)
This shows that the pair joint probability Pa$, corresponding to these correlations, depends on a, b, hence the application of Bell’s inequality is impossible. However, by considering the mean value of a single particle observable, say
~2):
we see that, for a general dynamics, the mean value of an observable of particle 1 will depend on the measurement we do on particle 2: this means that the EPR locality condition is not satisfied. If we want it to be satisfied, we have to restrict the class of allowed dynamics and also the class of allowed initial states. The physical arguments which allow to define such restrictions have been discussed in previous papers of the authors (cf. [AcReOla] for bibliography). The EPR locality condition is mathematically expressed by
and the causality condition by $0
:= $1,2 8 $A1 8 $A2
(9)
where both $ A ~ $A* , may depend on state of the system 1 (resp.2). With these restrictions one easily computes that the EPR locality condition is satisfied. However (8) and (6) show that the pair joint probabilities, corresponding to pair correlations, still depend on a, b, hence the application of Bell’s inequality is still impossible.
315
This extension of the standard quantum theory of measurement was first proposed in [Ac93]. The experiment discussed in the present conference is a concrete realization of this abstract scheme. 5
Description of the dynamical model
In the present section we construct a dynamical system which simulates locally the EPR correlation (8). In the idealized dynamical system considered in our experiment we consider only two time instants 0 (initial) and 1 (final) so, in our case, a “trajectory” consists of a single jump. We do not describe the spacetime details of the trajectory because we are only interested in distinguishing 2 cases: -
at time 1 the particle is in the apparatus (and in this case it is detected with certainty)
- at time 1 the particle is not in the apparatus (and in this case it makes no
sense to speak of detection) Thus our “configuration space” for the single particle will be made of 3 points: s (source), 1 (inside apparatus), 0 (outside apparatus). Since at time 0 the “position” of both particles is always s, because of the chameleon effect, the position qj,l of particle j = (1,2) at time 1 will depend on the polarization a j , on the inner state o and on the state X j of the apparatus M j ( j = 1,2): 4j,l = e , l ( ” j , a , X j )
;
j = 1,2,
The local, deterministic dynamical law of this dependence is described as followed. 1. The state space of the composite system (particles, apparatus) is {position space} x {inner state space} x {apparatus space} = {s, 0,1} x
[O, 27f
x [O, 112
2. The initial state is always of the form (s,s,01,o2,X1,X2) E {s}2x [0,27T]2x [0,1l2 i.e. the initial position of both particles is always s.
316 3. To speak of correlations only makes sense if the deterministic trajectories of both particles end up in the detectors (pre-determination). This means that the statistics is conditioned to the subset
{ 1}2 x [O, 2 4 2 x [O,
112
of the state space. 4. Just by changing the order of the factors the state space can be realized as {(Sl
,c1,X1; s 2 1 0 2 , Xz))
E ({S,O, 1) x [0,2Tl x [0111) x ({s,O, 1) x [O, 2
4 x [O, 11)
Therefore a local deterministic dynamics is uniquly determined by the assignment of two functions Tl,a,T 2 , b : (s1,cl, X1; SZ,oZ,
XZ)
(Tl,a(sl,01, Xl),TZ,b(SZ,0 2 ,
A,))
=: (q1,a(sl,o1? ~ ~ ) , ~ l , a ( ~ l ~ ~ l ~ ~ ~ ) , m l , a q Z , b ( S Z , 0 2 , XZ)rSZ,b(SZ,oZ, X 2 ) , m Z , b ( S Z ? f f 2 ,X 2 ) )
Moreover it is convenient to identify the endpoints of both intervals [0,27r] and [O, I], i.e. to identify these intervals to circles so that the functions q j , x ,~ j ,m ~j , , ,( j = 1,2,x = a , b ) , as functions of the variables o,X can be extended by periodicity to the whole real line (period 27r in o,period 1 in A). This allows to give a meaning to formula (17) of [AcImReOl] in full generality, i.e. without appealing to special choice (11) of [AcImReOl]. 5 . With these conventions, for every a, b E [0,27r],a deterministic dynamics as follows (10) i.e. the inner state of the particle and of the apparatus do not vary under the evolution, but the position varies according to the law: ( S l , o l ,X 1 ; S 2 , o Z r X 2 )
(ql,a(sli 0 1 ,Xl), 01; q 2 , b ( S 2 , o Z , XZ), 0 2 , x 2 )
Remember that the initial position of both particles is always s. Therefore it is sufficient to define the dynamics only in this case.
317 6. For every setting ( a ,b) E [0,27rI2 of the apparata, the initial probability distribution of our deterministic dynamical system is given by:
where ma,mb are fixed numbers in [0,1] and
(2) . Finally the random variables S a(1),Sb .
{S}Z
x [O, 27ry x [O, 112+ {fl}
are defined by
It is now a matter if simple calculations (cf. section (2) of [AcImReOl]) to verify that the correlations
XI; s , 0 2 , X2)) are precisely the EPR correlations. Finally notice that the dynamics (10) is slightly simplified with respect to the one described in [AcImReOl]. However, due to the choice (11) of IAcImReOl] this simplification does not change the calculations in the specific case under consideration. For more general classes of models the simplification (10) is convenient because with this choice the state space is mapped into itself by the dynamics and no additional identifications are required. There is no conceptual difficulty to include in our model the consideration of the space-time trajectory of the particle. This surely would improve the present model, however the main conclusion of our experiment, i.e. the reproducibility of the EPR correlations by a classical, deterministic, local dynamical system, will not change. (W
= (s,01,
318
5.1 Description of the experiment
1. Let N 5 Ntot be natural integers and let {oj : j = l , ..., N }
be the sequence of numbers either deterministically or pseudo-randomly distributed in [0,27~]with good equidistribution properties. (cf. the o p tion D (deterministic) or R (random) that has been inserted in the program of the experiment.) Let N ( o j ) ( j = 1,.. . , N ) denote a sequence of natural integers such that N “Cj)
=N o t
j=1
Remark. Ntot represents the (physically unobservable) “total number” of entangled pairs emitted by the source. N ( o j ) is the number of times that the imput aj is produced in the sequence (13). 2. For each j from 1 to N , repeat the following 3 operations (a), (b), (c), N ( o j ) times (a) The central computer sends oj to the computers 1 and 2. (b) Computer 1 computes the position of particle 1 using the deterministic dynamics and sends back S c ) ( o j )(= 1 or -1) if the particle is inside the apparatus. It sends back nothing if the particle is outside the apparatus. Computer 2 does the same thing. The deterministic dynamics is such that S c ) ( o j )is sent back with probability p l , , ( a j ) and Sb(2) (oj)is sent back with probability pz,b(.j) where ~ l , ~ , p 2 , b are sufficiently regular probability densities (say piecewise smooth with a finite number of discontinuities in [0,an]. Remark. This corresponds in the real experiments, to labeling the local detection time of the photon. When both computers send back a value 51, then we say that a coincidence occurs. The emergency of these probability in a deterministic context is due to the fact that the dynamics has strong chaotic properties. (c) Only in case of a coincidence, i.e. when the central computer receives the value fl f r o m both computers, the central computer computes the “correlation product” S~)(c~j)S~~)(oj).
319
3. The central computer computes the correlation as Sum of all correlation products The total number of coincidences
(14)
Remark. This is what is done in all experiments and a corresponds to the statement of the problem because, up to now, the EPR correlations have always been integrated as equal time correlations. 5.2
Computation of the correlations
Introducing
the expected number of coincidences ^'coincidences and the sum of all correlation products Correlations Decome respectively N
•^coincidences = j=i N
OPLafaKftfo)
(16)
j=i N
orrelations
Thus the correlation defined by (14) is Correlations
=
•^coincidences
EJLiP^Qpi.a^Qpa^^)^^^)^^) E^li P(< 7 j)Pi,a(CTj)p 2 ,6(crj)
and therefore Correlations ^ J027r da •^coincidences
p(a)Plta((r)p^b((T)s(a\a)Sf\a) /0 ^ df
p(ff)pi,a(^)p2,b(<^)
where p(a) is a probability density with properties analogue to pi >a and
(lg)
320
5.9 The EPR correlations In our experiment the deterministic dynamics is chosen in such a way that the emerging probabilities have the form :
With these choices we obtain 1 numerator of (19) = -- cos(a - b) 2.rr
(22)
1 denominator of (19) = 2T
Therefore for large N the correlation (14) or equivalently (18), (19) is well approximated by - COS(U -
b)
(24)
which is exactly the EPR correlation. We underline that, as shown by ( 2 3 ) , even if the mechanism of coincidences depends on the setting of the apparatus, the expected number of coincidences (23) is independent of it, in agreement with the experimental results [Aspects82].
References
[Ad11 Luigi Accardi: Topics in quantum probability, Phys. Rep. 77 (1981) 169-192 [Ac93] L. Accardi:Einstein-Bohr: one all, in: The interpretation of quantum theory: where do we stand?, Acta Enciclopedica, Istituto dell’Enciclopedia Italiana (1994) 95-115; Volterra preprint N. 174 (1993). [AcReOOb] Luigi Accardi, Massimo Regoli: Locality and Bell’s inequality. Preprint Volterra, N . 427 (2000) quant-ph/0007005; an extended version of this paper, including the description of the present experimen will appear in the proceedings of the conference “Foundations of Probability and Physics”, Vaxjo University, Sweden, November 27 - December 1 (ZOOO), A. Khrennikov (ed.), World Scientific (2001)
321
[AcReOla] Luigi Accardi, Massimo Regoli: The E P R correlations and the chameleon effect, discussed during the ” Japan-Italy Joint workshop on Quantum open systems and quantum measurement”, Waseda University, 27-29 September 2001; Preprint Volterra, N. 487 (2001) [AcImReOl] L.Accardi, K.Imafuku, M.Regoli: O n the physical meaning of the EPR-chameleon experiment, Infinite dimensional analysis, quantum probability and related topics, 5 N. 1 (2002) 1-20; quant-ph/0112067; Volterra Preprint December (2001) N. 494 [Be641 Bell J.S: O n the Einstein Podolsky Rosen Paradox, Physics 1 no.3. 195-200 1964. [Stap82] Stapp H. P.:Bell’s Theorem as a Nonlocality Property of Quantum Theory, Physical Review Letters, vol. 49, no. 20 (1982) [SuZa76] Suppes P., Zanotti M.:On the Determinism of Hidden Variable Theories with Strict Correlation and Conditional Statistical Independence of Obseruables, Logic and Prob. in Quant. Mech., 445-455 (1976) [Wig701 E.P. Wigner: Amer Journ. of Phys. 38 (1970) 1005 [Aspects821 Alain Aspect, Philippe Grangier, and Gerard Roger, Phys. Rev. Lett. 49, 91 (1982)
322
DESCRIPTION OF THE DAMPED OSCILLATOR BY A SINGULAR FRIEDRICHS OPERATOR
WILHELM VON WALDENFELS Institut fur Angewandte Mathematik der Universitat Heidelberg, I m Neuenheimer Feld 294, 69120 Heidelberg, Germany The damped oscillator is described by coupling to a continuum of bath oscillators. In square integrable coupling the total Hamiltonian is an operator considered already by Friedrichs [4].Exponential damping is achieved in the singular coupling limit. The resolvent of the Hamiltonian converges. The limiting Hamiltonian and its spectrum are calculated explicitely.
1. Introduction The damped oscillator is one of the basic examples of classical mechanics.
It is, however, difficult to translate into quantum mechanics, as the energy of the oscillator is not conserved, but decreases to zero. In order to obtain energy conservation, one has to enlarge the system. One couples the oscillator, now called main oscillator to a system of bath oscillators and considers the total system. If the number of bath oscillators is infinite, the main oscillator will eventually loose all its energy to the bath, whereas the total energy is conserved. A well known example in classical electrodynamics is an infinite telegraph line behaving as an ohmic resistance . Another example in classical mechanics is the oscillator coupled to an infinite string. The energy of the oscillator is transferred to the string and travels to infinity. So, whereas the oscillator constantly looses energy, the total system consisting of oscillator and string conserves energy. In quantum mechanics the infinite string is represented by an infinite set of osillators. One of the easiest models for a damped oscillator goes back to the Wigner Weisskopf theory of emission of spectral lines. Neclecting rapid fluctuations they arrived at the Hamiltonian extensively studied by Friedrichs [4]
323
The operator lives on the Hilbert space
4 = c e3 L2(R), its elements being described in the form
t=(t).
(3)
R is the multiplication operator given by
(Rf)(w) = w f ( 4 We denote by (91 the functional f H (glf).
(4)
We assume at first, that g is square integrable.The time development operator is
Uoo(t) describes the motion of the main oscillator, if a time 0 all bath oscillators are at rest, and the main oscillator is at position 1. We have the differential equation
with
k(t) =
s
e-i"tlg(w)12dW
Now Uoo(t)behaves asympotically like an exponential function. In order t o obtain exponential decay not only asymptotically, but for all t one has to perform the socalled singular limit, i.e. to replace g by a constant. Let gn be a sequence of square integrable functions converging to 1 in a symmetric way, then
k n ( t ) 4 27rA(t) and
~ ~ o (4t e--?rltl. ) ~ The resolvent can be calculated explicitely and, as we will see in section 4 that it converges to a function R ( z ) in operator norm. Of course, R ( z ) fulfills the resolvent equation. Formally R ( z ) = ( z - H ) - l with
324
but what does that mean, as 1 is not a square integrable function. One object of this paper is to define H in a rigorous way, such that the limiting resolvent R ( z ) is the resolvent of H . This will be done in section 4. Our problem is a special case of an old problem in quantum probability, which we want to describe without going too much into details. The Fockspace is defined by 00
r ( L 2 ( R ) )= @rn(L2(R)) 0
with ro = C , r l = L2(W) and ,?I = Lz(R"), whereLz(R") is the space of all square integrable symmetric functions. Let I be a Hilbert space,B(I) the space of all bounded operators on t and = I @ r(L2(R)). A quantum stochastic differential equation with constant coefficients in B(I) defines a strongly continuos,unitary one parameter group W ( t ) on A, provided that the coeffients obey certain algebraic relations. This fact is known for long time. But the explicit form of the generator or Hamiltonian was not known. Chebotarev[3] gave in 1998 an explicit formula for normal ,commuting coefficients. Gregoratti [6]was able to calculate in the general case an explicit generator as an essentially self adjoint operator on a dense domain.His proof , however, is very long and difficult and his final formula is not apparently symmetric. In order to get some insight into the structure of such an involved, but important problem it seems advantageous to study a special case, which is easier to treat. We obtain the Hamiltonian in a different, symmetric form, which,is equivalent to what would be in the special case Gregoratti's formula. In section 4 you may find our form (34), and the same Fourier transformed (39) and Gregoratti's formula (40). The use of the resolvent facilitates the proofs very much. A worthful hint to find the explicit form of the Hamiltonian were the eigenvectors . We sketch now the special case considered here. In the quantum stochastic differential equation related to the Wigner Weisskopf problem of spontaneous emission we have I = C2 and the subspace
AO with eo =
(i)
and el =
fio and our space
= (el @
(;)
ro) CB (eo @ rl)
stays invariant under W(t).The subspace
sj see (2) are just the same Hilbert spaces. Then the
restriction of W ( t )to % , which can be easily calculated explicitely [13], coincides (after Fourier transform) with the one parameter group V ( t )on
325
U ( t )is determined by the resolvent R(z), and both determine the generator H defined in section 4, we have solved the problem of the generator in a different way. In order to determine the resolvent of H, we use the singular coupling limit. If one approximates quantum white noise by coloured noise, then in the Wigner Weisskopf case the approximating one parameter group leaves again -Ro invariant. The generator of the restriction is (after Fourier transform) our operator Hg.Going from coloured noise to white noise, means going with g to 1in a symmetric way. This is the socalled singular coupling limit [12]. It is related to Accardi's stochastic limit [l]. By the singular coupling limit the unitary one parameter group converges to W ( t )[12]. This implies convergence of the resolvents. As the resolvent of Hgcan be easily calculated, we can perform the singular coupling limit in the resolvent and establish then the selfadjoint generator H belonging to that resolvent. fj calculated in proposition 7. As
A large part of the paper consists in calculating the spectral decomposition of H and Hg.We have here before us a special case, analogous to the case of finite dimensional hermitian matrices with only single eigenvalues. In order to determine the eigen projectors, we use the continous analogon of the finite dimensionsal method of calculating the residues of the resolvent at its poles, which are the eigenvalues. This theory is presented in section 2 under strong, surely not optimal assumptions.It is then applied in sections 3 and 4. The author wants t o thank Luigi Accardi and the Volterra Center in Rome, for a nice stay and stimulating discussions. There much of the work has been done. He acknowledges fruitful discussions with Belavkin, Chebotarev and Gregoratti.
2. A generalized eigenvalue problem We have in our paper three examples of a special eigenvalue problem, which we discuss without attempting generality. If fi is a Hilbert space with scalar product (.I.) we denote by f or the elements and by ( f l the linear functional g +-+ (f1s). IfYjo is a dense subset of fj , we imbed intofib, the space of all antilinear functionals on 40by adjoining to f E s j the antilinear functional g ++ (glf). We denote the action of an antilinear functional 'p in the same way
If>
326
we define
(‘plf) = (fl’p)
and denote by(cp1 the linear functional f
H (‘plf).
If H is a selfadjoint operator in some Hilbert space 3 , then there exists a spectral measure ((dz) on R such that for any measurable function f on the real line
f(H) =
1
f(.)t(dz),
(5)
where the domain of the operator f ( H ) consists of all u E 3 , such that
1
< 0.
lf(.)l2(.IE(d.)I4
In order to motivate, what we are doing, we define
4)
(6)
1 P - -7 .rriA(a: - y), (xkiO)-y 2-y
(7)
A(2 - H ) = (d/dz)t(] - 0
1
and recall the equation
where P denotes the principal value.So, if
R(2)= ( 2 - H y , 1 -(R(z 2ri
- i0) - R ( x + i0)) = A(z - H ) = la,)(a,I.
(8) (9)
we call a, a generalized eigenvector for the eigenvalue x. We expect orthonormality (azlay)
= A(%-
(10)
and completeness
I
1a2)(az1d.= 1.
(11)
We go to make our statments more precise. Assume a dense linear subset .Cj, of such that 2
(‘plR(z)I$J);‘p, 1c, E 30
327
is holomorphic in the upper and the lower halfplane and the restrictions to the halfplanes are continous at the bounderies. Furthermore there exist antilinear functionals a, : 40-+CCI such that ax
#0
(12)
(‘plaz)
(13)
and
x is continous for
‘p
E Ej0,We assume, that
1 -(4l(fqx 21ri
- i0) - R(.
+ iO))l+) = (’plaz)(azld4
(14)
for ‘p,+ E 30,where (‘pla,) = (a,l‘p).This is the exact formulation of (9). If H is bounded, then at infinity the resolvent has a simple pole with a Laurent expansion starting with l/z.Here we assume instead SUP IzI=r
l4‘plWZ)l+)
- (‘pl+)l
-
0 for r
-+
(15)
0.
Consider an integral over a circle of radius r
or
(16) That means that the set of eigenvectors is complete and is the rigorous statement of (ll).By a similar argument
So for the spectral measure one obtains
(‘plJ(dx)I+) = (‘plaz)(azl+)d.
(18)
Proposition 2.1. The spectrum of H is purely continuous and consists of the whole of the real line. Proof. If zo E R then there exits a neighborhood I such that (a,l‘p) # 0 for z E I . Hence by (18)
cR
and a
‘p
E 40
328
So the spectrum of H is the whole real line. On the other hand
-
If p is an antilinear functional onfjo, such that ((PIP)= (PI'p) is bounded for [[cpII 5 1 then there exists a unique element E(P) E rj, such that (VIP) = ('plE(P)) for all 'p E r j o . The following proposition shows, that the eigenvectors are orthonormal in a generalized sense. Proposition 2.2. port, then
Iff is a bounded measurable function with compact sup-
Proof. Let h be a bounded measurable function, and
.1c, E 4 0
+-+
(.1c,lh(H)I'p)=
/
'p
E 30.Then
h(.)(.1c,laz)(azl'p)d.
is bounded by \ ~ h ~ ~ m ~ ~so~it\ can ~ \ ~be. lextended c , ~ ~ , to sj. We have
Assume, that f is bounded of compact support and that the support is so small, that (possibly after multiplication with a factor), there exists 'p such thatl('pIaz)I 2 1 for z in the support o f f and put h ( z ) = f(z)/(a,lp). Applying the last equation we establish the existence of
329
By adding up we arrive to arbitrary measurable bounded functions with compact support. 0 Define a mapping TOfrom0 into L2 by ( T o p ) ( x )= ( a z l p ) .
Proposition 2.3. The mapping To can be extended in way to a unitary operator from 4 to L 2 . Proof. By equation (16) TOis an isometrie. As 40is dense in 4,it can be extended to an isometry from fj into L2 in a unique way. Define by SO the mapping f H E ( J f ( x ) a , d x ) from all bounded measurable functions of compact support into 4.As these functions are dense in L 2 , by the last proposition SOcan be extended to an isometry S from L2 into 40. If f is measurable , bounded and of compact support and 'p E 40 ,then
(Sof I4 =
s
f (z)(azldda: = (f FOP).
This shows, that S is the adjoint of T and the unitarity.
0
3. Square integrable coupling The first example of last section's theory is the multiplication operator R with (Rf)(w) = wf(w) in 4 = L2(R) .It has the domain
Da = { f
E &(R) : /(1
+ w2)1f (w)I2dw < m}.
The operator is selfadjoint and its spectrum is purely continuous and consists of the real 1ine.We start by a lemma.
Lemma 3.1. Let f be a complex valued function o n the real line, such that f is C2 , i.e. f ,f', f" exist and are continuous, and assume, that their products with 1 w2 are integrable. Then the function 1 F ( 2 ) = dwf (w)
+
s
2-w
330
exists for imz # 0 and is holomorphic. For real x the limits z -+ exist and equal F ( x f i0) =
J
P
dw-f x-w
(w)
i~f(x).
where P denotes the principal value. For r -+
x f i0
(19)
03
Proof. Define the Fourier transform g(t) = /dwf(w)e'"' then
F ( z )=
{
sooog(t)eiztdt for imz > 0 i sooo g(t)eiztdt for imz < 0
-i
The assumptions imply, that g is C2 and that g, g', g" are O(t-2) for t Hence for z = x iy and y --+ f O , we have
+
F(a:f iO) =
{
-i
(21)
-,
03.
g(t)e'"'dt (22)
i J_", g(t)eiZtdt
Equation (20) follows from (21) by partial integration.
0
Choose 30= Cp the space of all infinitely differentiable functions with compact support. Call Rn = ( z - 52)-l. The function
is holomorphic in the upper and in the lower halfplane and continuous at the boundaries and approaches for real x the limits by equation (19)
1 -(R(x 27~i
- i0) - R(x
+ i0)) = A(x - 52) = IE,)(E,~ P
R ( x f i 0 ) = -FiTIEz)(Ezl, X-R
where E, is the functional ( € , I f ) = f(z).Othonormality and completeness of the generalized eigenvalues are trivial.
331
The unperturbed Hamiltonian
is an operator in fi = C @ L2(R) It is selfadjoint on the domain C @ Vn. It is similar to the cases considered in section 2. We choose fi0 = C @ Cp The spectrum consists of the continuous spectrum R and the point (0). Its (generalized) eigenvectors are eo =
(i)
and
=
(O) for all real x.
The spectral measure is given by
A(z - Ho) = A ( z ) l e o ) ( e o l + I&)(Gl Inserting a square integrable coupling function g one obtains the Hamiltonian already considered by F'riedrichs
This Hamiltonian is defined in the same Hilbertspace fi. It is selfadjoint on One checks immediately the same domain C @ Dn.We choose the same 90. that its resolvent is given by the matrix
with
The function Cg(z) is holomorphic for imz imz # 0, as
# 0 . It does not vanish for
We calculate in the next proposition the generalized eigenvectors, the form of which coincides with the one, Friedrich has given. Proposition 3.1. Assume that g i s C2 and that g , g', g" themselves and multiplied with w are sqare integrable, and that g ( w ) # 0 f o r all w . T h e n C g ( z ) does not vanish neither in the interior nor o n the boundaries. W e have
cg(zfio) = c9R(z)fi7rlg(z)I2
(24)
332
SUP ICg(Z)/Z Itl=T
/
IS(W)I2dWI
+
0
(26)
f o r r t 00. Furthermore f o r f o r cp,+ E fio the limits of (cplRg(z)l+)f o r z 4 x f i0 exist and 1 A ( x - Hg)= G ( R g ( x- i0) - R g ( x i0)) = ~ a , ) ( ~ , ~ (27)
+
with
Conditions (12),(13),(15) of section 2 are fulfilled. The spectrum of Hg is purely continous and the la,) f o r x E R f o r m a complete orthonormal system of generalized eigenvectors. Proof. : Lemma 1 applies and from there follows (24),(25) and (26). Applying again lemma 1, the rest of the proposition follows. The calculations 0 to determine (28) are lengthy, but straight forward.
4. Singular coupling We denote the constant function 1 again by 1. The functional
is defined for integrable f . The function 1 1 -* i+w
i+R1=
so exists for square integrable f . In the last section the coupling g was done by a square integrable function. We want to replace g by the constant 1. We perform the so-called singular coupling limit[l2],mentioned in the introduction. We consider a sequence gn of square integrable functions, converging to 1 pointwise , uniformely bounded by some constant function,with the property
gn (w ) = gn ( -w).
333
# 0 the resolvents Rgn(z)converge in operator
Then for fixed z with imz norm to
with again Rn(z) = (z - R)-l. The function
C ( z )= z
+ irg(z)
with
1 for imz -1 for imz
=
.(z)
>0 <0
is holomorphic in the upper and lower halfplane and continous at the boundaries. The function R ( z ) fullfills the resolvent equation
(z2 - Zl)R(Zl)R(Z2) = R(z1) - R(z2). We define the the subspace L
c L2 of all f
1
+.f)
f = -(cl 1+R with
.f E L2 and 4jl (ilf)=
= C €3 L
lim
r+w
(30)
c 4.Define the functional i on C by
f(w)dw = -irc+
/ r
such that
-r
/
&.f(w)clw.
Define the operator
A : L L2@Ci Af = cli) + f - if. 4
Compair it to the equation
~f
=cl+
f- if.
Define the operator
A : C €3 L + C a3 (P(IW€3) Cli))
(33) (34)
-
Lemma 4.1. The sesquilinear f o m E,rl E
is symmetric.
31
(EIAlrl)
334
Proof. The only non trivial part of the proof is to show that
is real. Now this expression equals
An easy calculation leads to
Lemma 4.2. The operator H maps
<=
1
with
E
= -(-to1
1+R
(t)
E 41 into r j i f f
+f)
(35)
L2.
Definition 4.1. Denote by DH the subspace of all and by H the restriction of H to DH.
< E 31 fulfilling (4.2)
Proposition 4.1. The subspace DH is dense in rj, H is selfadjoint and R(z) is the resolvent of H. Proof. If
< E 3 , then (R:
and
(&)
are in
D H ,hence R(z)< E
DH.One obtains HR(z)< =
-< + z R ( z ) ( .
(36)
If 6 is a bounded function of compact support, then for y -i f c o one obtains iyR(iy)< -+ in norm. So DH is dense in rj. Lemma 2 and 3 imply, that H is symmetric and the domain of H is contained in the domain of the adjoint of H. We have still to prove the inverse inclusion. Introduce the matrix
<
Then by the lemma 3 T maps 4 onto
V H one to one. We have for t = T i
H< = HTi with
335
considering that A and coincide with R and 1 on functions of the form (l/(i+R)) f with f E L2. An elements 5 E 3 is in the domain of the adjoint iff there exists an element 77 E 3 such that
or
(J'IHTIV)= (771TlV) for all Y E
DH or V E 3. Denoting by t the usual adjoint (HT>+ It)= Tt177)
and taking the second component of this vector 1
R
1
-11) + --y&)= - i + R
50 - i + R
~
1771).
Considering both sides as functions of w and multiplying with -i obtains
+ w one
and
So H is densely defined and selfadjoint and equation (36) shows, that R ( z ) is the resolvent of H . 0 Similar calculations as for (28) show, assuming again that 30= C @ CF, Proposition 4.2. The operator H has only a continous spectrum consisting of the real line. It has the generalized eigenvectors
They f o r m an orthonormal complete system in the generalized sense. Conditions (12), (15') and (15) are fulfilled.
Remark 4.1. By RE, = ZE, the functional E, appears as an eigenvector of R in the proper sense. Generalize A andi to act on some Schwartz distributions in a natural way by
336
h&,= XE,; R-- X -PR
1=-1+x-
P l 2-R
(ile,) = 2;(i 1-1)P
2-R
=0
Then
Ha, = xa,. In this sense a, is a proper eigenvector
Remark 4.2. The orthogonality = A(x
(azlay)
- Y)
can be checked directly by using the equation
P
P
x-wy-w
-
--
-)P
+ 7r2A(x- w ) A ( y - w ) .
We establish now the time evolution of the operator H . Proposition 4.3. Define for t E
R Ol (Ul0 UUl1) uoo
u(t) =
ull= ,-ifit
t2
-ltdt2
dtle-i"(t-t2)ll)e-?'It2-tll
(Ile-iW
The U ( t )f o r m a one parameter unitary strongly continuous group generated -iH. That means
U ( s ) U ( t )= U ( s + t ) U ( 0 )= 1
337
U(-t) For t
+0
= U@)t=
and E E fi
U(t)14) and for
v(t)-'
+
14)
5 E VH U ( t )- 1 It) --iHlE). i t
Proof. We have to prove that
0
~ ( z=)iJ, U(t)eiztdt for imz < o
(38)
The unitarity can be checked either directly or from the equation
I?(.)+
= R(Z).
The rest follows then from the general theory [7]. We use the formalism of convolution and write
Where
e ( t )=
For imz
1 for t 0 for t
>0 <0
> 0 we have the Laplace transforms
J
i dteiztUS2(t)O(t)= z-R
] dteiztA(t)
=1
338
As Laplace transform maps convolutions into ordinary products we obtain
On the other hand performing the convolutions one obtains for t U ( t )= U ( t ) . That proves (37). For (38) use a similar method.
> 0 that 0
It is interesting to calculate the Fouriertransform of I?. Consider the Sobolev space H1(R) of all square integrable functions on the real line , whose derivatives in the sense of Schwartz distributions are square intergrable as well. Their Fourier transforms are of the form (l/(i R))f with F E L2. Call
+
As the Fourier transform i 1 F‘c(w) = - eiwtC(t)dt= -&i+w’ we see that
,c = F(Hl(1W)@a=<). As the elements of H1(R) are continous, we observe that the functions of H1(R) @ CC are continous in R \ (0) and that their limits in f O exist. Define
(4f ) = (1/2)(f(O+) - f(0-1) hf
= dcf
+ (f(o+) - f ( O - ) A
where dcf is the restriction of df to R \ {O}.One obtains
We have still to recover Gregoratti’s formula. The domain of the limiting operator ?i is the subspace (see definition 1)
One has then f(O+)
= f(0-)
x(;)
+ i&c
= (i;?f-)
and arrives at
(;)
witha-(f) = f(0-). This is exactly Gregoratti’s formula restricted to our special case.
339
References 1. L. Accardi and Lu Yun Gang, Quantum Mechanics and its Stochastic Limit Springer 2002 2. V.P. Belavkin, Lecture in the Conference on Infinite Dimensional Analysis and Quantum Probability, Levico 2001. 3. A.M. Chebotarev, Quantum stochastic differential equation is unitary equivalent to a symmetric boundary problem in Fock space. 1nf.Dim.Anal.Quantum Prob. l(1998). 4. K.O. Friedrichs, On the permutation of continuous spectra. Comm.Pure Applied Math 1(1948), 361-406. 5. I. Gelfand and N. Vilenkin, Generalized functions, vo1.4. Academic Press, New York 1964 6. M. Gregoratti, The Hamiltonian associated to some quantum stochastic differential equations.Thesis. Milano 2000. 7. E. Hille, R.S. Phillips, Functional Analysis and Semigroups. AMS Coll.Pub1. Vol.XXX1, Providence, R.I.1957. 8. P.A. Meyer, Quantum Probability for Probabilists Lecture Notes in Mathematics 1538(Springer, Berlin, Heidelberg 1993). 9. K.R. Parthasarathy, A n Introduction to Quantum Stochastic Calculus (Birkhauser, Basel, Boston, Berlin 1992). 10. W. von Waldenfels, Spontaneous Light Emission described by Quantum Stochastic Differential Equation. Lecture Notes in Mathematics 1136, p.515534 (Springer, Berlin, Heidelberg 1985) 11. W. von Waldenfels, Continuous Maassen Kernels and the Inverse Oscillator. Sem. de Probabilites XXX, Springer Lecture Notes im Mathematics. 12. W. von Waldenfels, An Example of the Singular Coupling Limit. In: Stochastic Analysis and Mathematical Physics.Anestoc '98, p.155-166. Birkhaeuser Boston 2000. 13. W. von Waldenfels, The Method of Quantum Jumps and Quantum White Noise. In: Modern Challenges in Quantum Optics, p. 279-293.Springer 2001.
