Quantum optics

Contents I MAIN PART OF THE LECTURES 1 1 Classical harmonic oscillator in external field 1.1 Equations of motion . . ...

Author: Kryszewski S.

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Contents I

MAIN PART OF THE LECTURES

1

1 Classical harmonic oscillator in external field 1.1 Equations of motion . . . . . . . . . . . . . . . . . . . . 1.2 General solution to homogeneous equation . . . . . . . . 1.2.1 General solution . . . . . . . . . . . . . . . . . . 1.2.2 Discussion of the roots and solutions . . . . . . . 1.3 Solution to inhomogeneous equation – Green’s function 1.3.1 Requirement of causality . . . . . . . . . . . . . . 1.3.2 Green’s function . . . . . . . . . . . . . . . . . . 1.4 General solution for driven and damped oscillator . . . . 1.5 Harmonic driving force . . . . . . . . . . . . . . . . . . . 1.5.1 Formulation of the problem . . . . . . . . . . . . 1.5.2 Evaluation of the integrals . . . . . . . . . . . . . 1.5.3 General solution for oscillator’s displacement . . 1.5.4 Long–time behaviour . . . . . . . . . . . . . . . . 1.5.5 Resonance approximation for stationary state . .

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1 1 1 1 2 4 4 5 7 8 8 9 9 10 11

2 Classical electrodynamics 2.1 Maxwell’s equations . . . . . . . . . . . . . . . . . . . . 2.1.1 Some general comments . . . . . . . . . . . . . . 2.1.2 Maxwell’s equations in the vacuum . . . . . . . . 2.1.3 Maxwell’s equations in Fourier space . . . . . . . 2.2 Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Introduction and basic definitions . . . . . . . . . 2.2.2 Wave equations for potentials . . . . . . . . . . . 2.2.3 Potentials – gauge invariance . . . . . . . . . . . 2.2.4 Lorentz gauge . . . . . . . . . . . . . . . . . . . . 2.2.5 Coulomb gauge . . . . . . . . . . . . . . . . . . . 2.2.6 Potentials in the Fourier space . . . . . . . . . . 2.3 Longitudinal and transverse fields . . . . . . . . . . . . . 2.3.1 Introduction . . . . . . . . . . . . . . . . . . . . 2.3.2 Transverse and longitudinal Maxwell’s equations 2.3.3 Discussion of the potentials . . . . . . . . . . . . 2.4 Interaction of fields with charged particles . . . . . . . . 2.4.1 Lagrange equations revisited . . . . . . . . . . . 2.4.2 Generalized potential U for a particle in the field 2.4.3 Canonical (Hamilton) formalism . . . . . . . . .

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13 13 13 15 15 16 16 17 17 18 19 19 21 21 22 24 26 26 27 28

3 Quantization of electromagnetic field 3.1 Introductory remarks . . . . . . . . . . . . . . 3.2 Expansion into normal modes . . . . . . . . . 3.2.1 Statement of the problem . . . . . . . 3.2.2 Energy of the field in a cavity . . . . . 3.2.3 Expansion in normal variables . . . . 3.3 Field quantization in a cavity . . . . . . . . . 3.3.1 Field oscillators – harmonic oscillator 3.3.2 Field quantization . . . . . . . . . . . 3.4 Plane wave representation . . . . . . . . . . . 3.4.1 Discussion of our results . . . . . . . . 3.4.2 Introduction of plane waves . . . . . . 3.4.3 Quantization in cubic box of volume V 3.4.4 Density of the modes . . . . . . . . . . 3.4.5 Quantization in free space . . . . . . . 3.5 Equations of motion – Maxwell’s equations .

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30 30 30 30 32 33 34 34 36 37 37 38 39 40 41 42

4 States of quantized electromagnetic fields 4.1 Introduction and general discussion . . . . 4.1.1 Vacuum state . . . . . . . . . . . . 4.1.2 Photon number states . . . . . . . 4.1.3 Single mode field . . . . . . . . . .

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49 49 50 51 51 52 53 54 55 55 57 60 63 64 64 66 66 66 69 71 71 72 73 76

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78 78 78 79 81 81 83 86 87 88 88 89 93 93 94 94

6 Spontaneous emission 6.1 Density operator for open system . . . . . . . . . . . . . . . . . . . . 6.2 Evolution of the reduced density operator . . . . . . . . . . . . . . . 6.2.1 Introductory remarks . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Transformation to interaction picture and formal integration 6.2.3 Assumptions about reservoir . . . . . . . . . . . . . . . . . . 6.2.4 Derivation of the master equation . . . . . . . . . . . . . . . 6.2.5 Transformations of the master equation . . . . . . . . . . . . 6.2.6 Discussion of the approximations . . . . . . . . . . . . . . . . 6.3 Spontaneous emission in two-level atom . . . . . . . . . . . . . . . . 6.3.1 Description of the atom + field system . . . . . . . . . . . . . 6.3.2 Discussion of the properties of the reservoir . . . . . . . . . . 6.3.3 Analysis of the master equation . . . . . . . . . . . . . . . . . 6.3.4 Reservoir in the vacuum state . . . . . . . . . . . . . . . . . . 6.3.5 Physical meaning of the parameter γ . . . . . . . . . . . . . . 6.3.6 Calculation and discussion of γ and δL . . . . . . . . . . . . .

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97 97 99 99 99 101 105 106 107 108 108 108 110 111 112 113

7 Optical Bloch equations 7.1 Introduction. General discussion . . . . . . . . . . 7.2 Derivation of optical Bloch equations . . . . . . . . 7.2.1 Evolution of the atom without damping . . 7.2.2 Addition of radiative and phenomenological 7.2.3 Simple elimination of time dependence . . . 7.3 Stationary optical Bloch equations . . . . . . . . . 7.3.1 Stationary solutions . . . . . . . . . . . . . 7.3.2 Stationary energy balance . . . . . . . . . .

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115 115 117 117 118 119 122 122 124

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4.4

4.5 4.6 4.7

Coherent states (single mode) . . . . . . . . . . . . . . . . . . . . 4.2.1 Expansion in n-photon states . . . . . . . . . . . . . . . . 4.2.2 Scalar product h z | ξ i . . . . . . . . . . . . . . . . . . . . 4.2.3 Completeness of coherent states . . . . . . . . . . . . . . . 4.2.4 Coherent state as a displaced vacuum . . . . . . . . . . . 4.2.5 Minimalization of uncertainty . . . . . . . . . . . . . . . . 4.2.6 Comments on electric field . . . . . . . . . . . . . . . . . . 4.2.7 Time evolution of the coherent state . . . . . . . . . . . . Squeezed states (single mode) . . . . . . . . . . . . . . . . . . . . 4.3.1 Introduction and basic definition . . . . . . . . . . . . . . 4.3.2 Operator approach . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Squeezed vacuum states . . . . . . . . . . . . . . . . . . . 4.3.4 Fluctuations of photon number in squeezed vacuum state 4.3.5 Similarity relations for operators a ˆ 2 , (ˆ a † )2 , a ˆ† a ˆ . . . . . . ~ 2⊥ in squeezed vacuum state . . . 4.3.6 Expectation value for E Squeezed coherent states . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Introductory remarks . . . . . . . . . . . . . . . . . . . . 4.4.2 Expectation values for | z, ξ i . . . . . . . . . . . . . . . . 4.4.3 Expectation values for | α, ξ i . . . . . . . . . . . . . . . . Squeezed photon number states . . . . . . . . . . . . . . . . . . . 4.5.1 New role of squeezed vacuum state . . . . . . . . . . . . . 4.5.2 Squeezed photon number states . . . . . . . . . . . . . . . Expansion of the squeezed vacuum state into n-photon states . . Equivalence of coherent squeezed states | z, ξ i and | α, ξ i . . . . .

5 Atom–field interaction 5.1 Charged particle in electromagnetic field . . . . . . . . . . 5.1.1 Minimum coupling hamiltonian . . . . . . . . . . . 5.1.2 Some additional comments . . . . . . . . . . . . . 5.2 Atom in radiation field . . . . . . . . . . . . . . . . . . . . 5.2.1 Setting up the scene. Free atom hamiltonian . . . 5.2.2 Minimum coupling interaction hamiltonian . . . . 5.2.3 Dipole coupling interaction hamiltonian . . . . . . 5.2.4 Interaction hamiltonian. Summary and discussion 5.3 Hamiltonian for two–level atom in radiation field . . . . . 5.3.1 The two-level atom. Free hamiltonian . . . . . . . 5.3.2 Interaction hamiltonian . . . . . . . . . . . . . . . 5.3.3 Further comments on resonant approximation . . . 5.3.4 Single mode case . . . . . . . . . . . . . . . . . . . 5.4 Semiclassical approximation . . . . . . . . . . . . . . . . . 5.4.1 General discussion . . . . . . . . . . . . . . . . . .

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AUXILIARY CHAPTERS

1

8 Annihilation and creation operators

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8.1 8.2 8.3

General properties . . . . . . . . . . . . . . . . . . . . . Annihilation and creation operators – summary . . . . . Application to harmonic oscillator . . . . . . . . . . . . 8.3.1 Annihilation and creation operators for harmonic 8.3.2 Construction of the vacuum state . . . . . . . . . 8.3.3 Construction of the number states | n i . . . . . .

9 Density operator 9.1 Introductory remarks . . . . . . . . . . . 9.2 The basic concept of density operator . 9.3 Some generalizations . . . . . . . . . . . 9.3.1 Projection operators . . . . . . . 9.3.2 Application to density operator . 9.4 Properties of the density operator . . . . 9.5 Equation of motion for density operator

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APPENDICES

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A Fourier transforms A.1 Time–frequency Fourier transforms . . . . . . . . . . . A.1.1 Definition of the pair of Fourier transforms . . A.1.2 Dirac’s delta function and its Fourier transform A.1.3 Basic properties of Fourier transformation . . . A.1.4 Pseudo-convolution. An auxiliary integral . . . A.1.5 Properties of the Lorentzian curve . . . . . . . A.1.6 Sochocki formulas and related transforms . . . A.2 Three-dimensional Fourier transformation . . . . . . .

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B Useful operator identities B.1 Similarity relations . . . . . . . . . . . . . . . . . . . . . . B.2 Decomposition of the exponential . . . . . . . . . . . . . . B.2.1 General idea of the decomposition . . . . . . . . . B.2.2 The case of [ A, B ] = c . . . . . . . . . . . . . . B.2.3 Special case for annihilation and creation operators B.3 Similarity relation for annihilation operator . . . . . . . . B.3.1 General relation . . . . . . . . . . . . . . . . . . . B.3.2 Some special cases . . . . . . . . . . . . . . . . . . B.3.3 Applications of generalized similarity relation . . . B.4 Squeeze operator . . . . . . . . . . . . . . . . . . . . . . .

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C Certain sum rule for Hermite polynomials D Pseudospin operators D.1 Basic definitions . . . . . . . . . . . . . . D.2 Various products of pseudospin operators D.3 Commutation relations . . . . . . . . . . . D.4 Useful identities and their consequences .

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1. Classical harmonic oscillator in external field

1

Chapter 1

Classical harmonic oscillator in external field 1.1

Equations of motion

We consider a one-dimensional harmonic oscillator with mass m, frequency ω 0 > 0 and charge q. We take the oscillator to be driven by a classical time–dependent electric field E(t). We assume that the oscillator is damped with the rate Γ and the damping force is proportional to velocity. The corresponding equation of motion is of the form m¨ x + mΓx˙ + mω02 x = qE(t),

(1.1)

It is reasonable to assume that Γ ≥ 0. Negative parameter Γ would correspond to a force which increases the oscillator displacement, hence it is unphysical. We rewrite Eq. (1.1) as x ¨ + Γx˙ + ω02 x = f (t),

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f (t) = (q/m)E(t).

(1.2)

Equation (1.2) is an inhomogeneous one. Whatever the method used for finding its solution we need initial conditions. We will assume general initial conditions, that is x(t = t0 ) = x0 ,

x(t ˙ = t0 ) = v(t = t0 ) = v0 .

(1.3)

In some particular cases, we shall use more specific initial conditions.

1.2

General solution to homogeneous equation

Solution to Eq. (1.2) is a sum of a general solution to the homogeneous equation and of a particular solution to the inhomogeneous equation. Thus, we first consider the homogeneous equation.

1.2.1

General solution

So we have to solve the homogeneous equation of the form x ¨ + Γx˙ + ω02 x = 0,

(1.4)

We will seek for the solution in the form e −iωt . It is as well possible to look for a solution in another form, for example eλt . We, however adopt an above given form, which is due to the fact that we will further use the Fourier transforms of the Green’s function. The Fourier S.Kryszewski

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components of the time-dependent functions are taken to have time dependence e −iωt . This explains the adopted form of the solution to the homogeneous equation. Substituting our ansatz into equation (1.4) we find the quadratic characteristic equation −ω 2 − iΓω + ω02 = 0,

(1.5)

The discriminant is equal to 4ω0 − Γ2 . At present, we assume that 12 Γ < ω0 , so the discriminant is positive and the characteristic roots are ω1,2 = −

iΓ ± Ω, 2

(1.6)

where we have introduced a new parameter r 1 Ω = ω02 − Γ2 . 4

(1.7)

We will discuss the roots in more detail later. Then, we will drop the assumption that damping constant is sufficiently small. Having found the characteristic roots, we write the solution to the homogeneous equation as a combination of two exponentials Γt −iΩt −iω1 t −iω2 t + Be = exp − Ae + BeiΩt (1.8) x(t) = Ae 2

with A and B being the constants to be fixed by initial conditions. Differentiating relation (1.8) and using the initial conditions (1.3) we arrive at the set of equations for constants A and B x0 = A + B v0 =

− 12 Γ

(1.9a)

− iΩ A +

− 12 Γ

+ iΩ B.

It is a straightforward matter to solve this set of equations. The solutions are 1 1 1 1 A=− v0 + Γ − iΩ x0 , B= v0 + Γ + iΩ x0 . 2iΩ 2 2iΩ 2

(1.9b)

(1.10)

Substituting the obtained constants into Eq. (1.8), after minor rearrangement we obtain the general solution to the homogeneous equation of motion, which satisfies general initial conditions and is of the form # " v0 + 12 Γx0 1 x(t) = exp − Γt x0 cos (Ωt) + sin (Ωt) . (1.11) 2 Ω The obtained general solution of the homogeneous equation corresponds to simple damped oscillations, provided parameter Ω is real. This is so, when the damping is small, that is when 1 2 Γ < ω0 . Since this is not evident that this requirement is always satisfied we proceed to discuss the characteristic roots as given in (1.6) and (1.7).

1.2.2

Discussion of the roots and solutions

The roots ω1 and ω2 of the characteristic equation are given in (1.6). These roots govern the behaviour of the solution to the homogeneous equation. They are also important for the 2

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solution of the inhomogeneous one, therefore we will discuss their properties. Let us recall that the considered roots are as follows r iΓ 1 ω1,2 = − ± Ω, (1.12) with Ω = ω02 − Γ2 . 2 4 We stress that from the physical point of view, we require that the damping parameter Γ ≥ 0, while the oscillator frequency ω0 > 0. Negative parameters are unphysical, so we do not consider such a case. First of all, we note that for arbitrary values of the parameters Γ and ω 0 , the characteristic roots have the property |ω1,2 | = ω0 .

(1.13)

We start our discussion taking the lowest value of damping parameter, that is Γ = 0. Then Ω = ω 0 and characteristic roots are simply ω 1,2 = ±ω0 , so they lie on the real axis in the plane of complex ω, see figure 1.1. The solution to homogeneous equation follows easily from (1.11), and it is x(t) = x0 cos (Ωt) + vΩ0 sin (Ωt), which are standard undamped oscillations satisfying arbitrary initial conditions. When Γ grows from zero, but satisfies 12 Γ < ω0 , the roots have the form (1.12) and Ω is real. Both roots have nonzero (negative) imaginary part and as Γ grows they move in the complex plane downwards along the arcs of the radius ω0 , as it follows from Fig. 1.1: Behaviour of the characteristic the relation (1.13). The oscillator performs typical roots for fixed frequency ω0 for parameter Γ damped oscillations and the general solution of the varying from zero to infinity. homogeneous equation is as given in Eq. (1.11). The situation changes at the point where 12 Γ = ω0 . Then Ω = 0 and both roots coincide, possessing the values ω 1,2 = ω0 = − 2i Γ. So we do not have two linearly independent solutions as it was in the case of Eq. (1.8). For the case of Ω = 0 we can, however, take the limit in Eq. (1.11). Taking the limit carefully we obtain the following result 1 1 x(t) exp − Γt x0 + v0 + Γx0 t . (1.14) Ω→0 2 2

This solution corresponds to exponential decay without oscillations. When parameter Γ still grows, i.e., 12 Γ > ω0 , then Ω becomes purely imaginary. We can then write r 1 2 ˜ Ω=i Γ − ω02 = iΩ, (1.15) 4 ˜ is again real. The general solution (1.11) changes its character because substitution where Ω (1.15) must be made. Hence, from Eqs. (1.11) and (1.15) we obtain # " v + 1 Γx 1 0 0 2 - exp − Γt ˜ + ˜ x(t) x0 cos iΩt sin iΩt . (1.16) 1 Γ > ω0 ˜ 2 2 iΩ S.Kryszewski

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Then, trigonometric function are transformed into hyperbolic ones, and we obtain # " v + 1 Γx 1 0 0 2 - exp − Γt ˜ + ˜ x(t) x0 ch Ωt sh Ωt . 1 Γ > ω0 ˜ 2 2 Ω

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(1.17)

˜ < 1 Γ and may be called an overdamped one, This solution quickly decays in time because Ω 2 since it corresponds to strong damping. In this case two characteristic roots are purely imaginary and have the property iΓ - 0, ˜ +iΩ (1.18a) Γ→∞ 2 iΓ - − i∞, ˜ ω2 = − − i Ω (1.18b) Γ→∞ 2 Therefore, as presented in Fig. 1.1, we see that when Γ grows ω 1 moves upward the imaginary axis towards zero, while ω2 → i∞ downwards along the imaginary axis. Fig.1.1 illustrates the behaviour of the roots ω 1,2 as functions of the varying parameter Γ. It is important to note that only for Γ = 0 the roots are real, for Γ > 0 the roots are always in the lower half of the complex ω–plane. ω1 = −

1.3

Solution to inhomogeneous equation – Green’s function

Our next aim is to find a particular solution to the inhomogeneous equation x ¨ + Γx˙ + ω02 x = f (t),

with

f (t) = (q/m)E(t).

(1.19)

where we will call f (t) a driving force. We seek a particular solution to the inhomogeneous equation (1.19) in the form Z ∞ x(t) = dt0 g(t − t0 ) f (t0 ), (1.20) −∞

where f (t) is the inhomogeneity, while g(τ ) is an unknown function. We will first discuss the conditions imposed on function g(τ ) which follow for the physics of the problem. Then, we will explicitly construct this function and check that it satisfies all the requirements.

1.3.1

Requirement of causality

We seek the particular solution to the inhomogeneous equation in the form of (1.20). We require that this solution is causal. That means that the force f (t 0 ) can affect the displacement x(t) only at the instants earlier than the current moment. In other words, this means that the displacement x(t) can depend on the force f (t 0 ) only when t0 ≤ t. We may also say that the current state of the oscillator can be influenced by the earlier magnitude of the force, and not by the later ones. Therefore, requirement of causality can be written as g(t − t0 ) 6= 0 0

g(t − t ) = 0

for t0 < t,

(1.21a)

0

(1.21b)

for t ≥ t.

Relation (1.21a) should be understood in the sense that the function g(t − t 0 ) is not identically zero for times t0 earlier than t. Let us note, that since g(t − t 0 ) = 0 for t0 > t the upper limit of the integral in (1.20) is effectively equal to t and not +∞. Hence, instead of (1.20), we can effectively write Z t x(t) = dt0 g(t − t0 ) f (t0 ), (1.22) −∞

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which, in an evident manner, displays the causality requirement. Only moments t 0 earlier than t give nonzero contributions to the current value of the displacement, so that x(t) is determined solely by the earlier magnitudes of the driving force. The condition (1.21) can be put into somewhat more formal way. We write g(t − t0 ) = Θ(t − t0 ) g(t − t0 ),

(1.23)

which is an equality in the sense of generalized functions. Θ(t−t 0 ) denotes the Heaviside function, which is defined as 1 for t0 < t 0 (1.24) Θ(t − t ) = 0 for t0 ≥ t At this stage it is perhaps worth while to give some comments. Searching for the Green’s function in the form of the product Θ(t−t 0 )g(t−t0 ) requires a careful approach within the theory of generalized functions, which is neither easy nor clear. It seems that it is more convenient to seek the Green’s function g(t−t0 ) via the particular solution (1.20) of the inhomogeneous equation with additional conditions summarized by relations (1.21). On the other hand, relation (1.23) may be useful in practical applications, because it automatically restricts the integration domain to times earlier than the current moment. Hence the causality requirement is then explicitly seen. This is put clearly via the equation Z t Z ∞ 0 0 0 0 dt Θ(t − t )g(t − t ) f (t ), = dt0 g(t − t0 ) f (t0 ), (1.25) x(t) = −∞

−∞

Although this relation seems ”tempting”, we will seek the Green’s function via relations (1.20) and (1.21).

1.3.2

Green’s function

One of the ways to construct the particular solution to inhomogeneous equation is to look for the function g(t − t0 ). In order to do so, we substitute Eq. (1.20) into equation (1.2).Thus, we find an equation which must be satisfied by g(t − t 0 ) – the Green’s function 2 Z ∞ 0 dg(t − t0 ) 0 d g(t − t ) 2 0 dt +Γ (1.26) + ω0 g(t − t ) f (t0 ) = f (t). 2 dt dt −∞ Since the right-hand side can be written as Z ∞ dt0 δ(t − t0 ) f (t0 ), f (t) =

(1.27)

−∞

it is straightforward to see that Eq. (1.26) is equivalent to the equation d2 g(τ ) dg(τ ) +Γ + ω02 g(τ ) = δ(τ ), 2 dτ dτ

(1.28)

where we replaced t − t0 = τ . Eq (1.28) is a differential equation for generalized functions of the form typical for the equations determining the Green’s function. Solution to Eq. (1.28) is best sought in the Fourier domain. Hence, we introduce a pair of Fourier transforms Z ∞ dt √ G(ω) = eiωt g(t), (1.29a) 2π −∞ Z ∞ dω √ g(t) = e−iωt G(ω), (1.29b) 2π −∞ S.Kryszewski

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and we recall the relation well-known from distribution theory Z ∞ 1 δ(t) = dω e−iωt . 2π −∞ Transforming equation (1.28) into the Fourier domain we obtain Z ∞ Z ∞ i 1 dω h 2 2 −iωt √ (−iω) − iωΓ + ω0 e G(ω) = dω e−iωt , 2π −∞ 2π −∞

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(1.30)

(1.31)

where we used (1.30) in the right-hand side. Hence, the last relation entails the equality of the Fourier transforms and we get the algebraic equation for the Fourier transform G(ω) of the Green’s function g(τ ). The result is

1 −ω 2 − iωΓ + ω02 G(ω) = √ , 2π

(1.32)

or, equivalently

(−1) 1 (−1) 1 G(ω) = √ =√ 2 2 2π ω + iωΓ − ω0 2π (ω − ω1 )(ω − ω2 )

(1.33)

where ω1,2 = − 2i Γ ± Ω are the previously discussed roots (1.12). This is so, because the denominator in the first equality is exactly the same as the previously discussed characteristic equation of the homogeneous equation. Hence, we have easily found the Fourier transform of the Green’s function for the driven and damped harmonic oscillator. Moreover, due to previous discussion of the roots of characteristic equation we automatically have discussed the poles of the complex valued Fourier transform of the Green’s function. In order to find the Green’s function g(τ ) we must invert the Fourier transform. This is equivalent to compute the integral Z ∞ Z dω −1 ∞ exp(−iωτ ) . √ g(τ ) = dω (1.34) e−iωτ G(ω) = i 2π −∞ ω + 2 Γ − Ω ω + 2i Γ + Ω 2π −∞

Fig. 1.2: Integration contours for evaluation of the Green’s 6 QUANTUM OPTICS function for damped harmonic oscillator according to equation (1.34).

Computation of this integral is simple when one uses the residue theory. For time τ < 0 we close the contour in the upper half–plane of complex ω as indicated in Fig.1.2 by the dashed line. The radius of the semicircle goes to infinity and from the Jordan lemma the integral over the upper semicircle vanishes (because τ < 0 and −iωτ possesses negative real part).The integral reduces to the one over the real axis and since there are no poles within the contour (which has the positive direction) the integral vanishes, yielding g(τ ) = 0 for τ < 0. Similarly, for positive argument, (i.e. for τ > 0) we close the contour in the lower half–plane as indicated by S.Kryszewski

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dotted line. This contour has negative direction. The integral over the semicircle vanishes again due to Jordan lemma for τ > 0 when the radius goes to infinity. Since the poles are within the contour we obtain non-vanishing result which is easily computed via the corresponding residues in the first order poles ω1 and ω2 given in (1.12). The obtained Green’s function follows by evaluation of the residues # " exp −iτ − 2i Γ + Ω exp −iτ − 2i Γ − Ω −1 g(τ ) = −2πi + . (1.35) 2π − 2i Γ + Ω + 2i Γ + Ω − 2i Γ − Ω + 2i Γ − Ω Simple manipulation of this result allows us to write the resulting Green’s function for τ > 0 as g(τ ) =

−i − 1 Γτ iΩτ 1 − 1 Γτ e 2 e − e−iΩτ = e 2 sin (Ωτ ) 2Ω Ω

(1.36)

Summarizing, the Green’s function for the damped harmonic oscillator is given as follows g(τ ) = 0, for τ < 0 1 Γτ = exp − sin(Ωτ ), Ω 2

(1.37a) for τ > 0.

This result clearly satisfies the causality requirement. Moreover, we recall that Ω =

(1.37b) q

ω02 − 14 Γ2

may be real and positive (for ω0 > 12 Γ), zero and purely imaginary (for ω 0 < 12 Γ). All these cases can be investigated with the aid of the Green’s function (1.37). We shall, however, restrict attention to the case of ω0 > 12 Γ, because it corresponds to the physically interesting situation of the damped oscillations. Finally let us note that Green’s function, as given in Eq. (1.37) is continuous at t = 0, but its derivative is not. This is also clear when we use the generalized function approach (1.23) and write 1 0 0 1 0 g(t − t ) = Θ(t − t ) exp − Γ(t − t ) sin Ω(t − t0 ) , (1.38) Ω 2 or equivalently, using full notation, we have g(t − t0 ) = Θ(t − t0 ) q

1 ω02 − 14 Γ2

"r # 1 1 exp − Γ(t − t0 ) sin ω02 − Γ2 (t − t0 ) . 2 4

(1.39)

which is explicitly causal and useful in practical applications.

1.4

General solution for driven and damped oscillator

We can now construct a general solution to the driven oscillator equation (1.2) by summing the general solution to the homogeneous equation (1.11) and the solution to the inhomogeneous equation (1.19) with the Green’s function which follows from Eq.(1.37). We restrict our attention to the case of standard damped oscillations, that is we consider the case in which 0 < 12 Γ < ω0 .

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The obtained solution satisfies arbitrary initial conditions and can be written down for arbitrary driving force f (t). The obtained result is thus, as follows # " v0 + 12 Γx0 Γt x(t) = exp − x0 cos (Ωt) + sin (Ωt) 2 Ω Z t Γ(t − t0 ) 1 0 + dt exp − sin Ω(t − t0 ) f (t0 ). (1.40) Ω −∞ 2

We see that x(t) depends only on the driving force for times earlier than the current moment. This conforms with the causality requirement. It is interesting to note, that when the oscillator is initially at rest x 0 = 0 and v0 = 0, then the first term in (1.40) vanishes. Only the driving force governs its evolution. In the general case (arbitrary initial conditions), if time t is sufficiently long then all the transients depending on initial conditions decay and again the evolution is determined only by the second term in (1.40). Thus, only the influence of the force is of interest. Hence it remains to investigate special cases for which we can take the evolution of the displacement as Z t h i Γ(t − t0 ) 0 1 x(t) dt exp − sin Ω(t − t0 ) f (t0 ). (1.41) no transients Ω 2 −∞ Moreover, in most of the practical cases, the electric field is switched on at the moment t = 0. Thus we take 0 for t < 0, E(t) = (1.42) E(t) for t > 0, with the concrete form of the time dependence of the field to be specified later. If the driving field satisfies the above requirement, then the driving force f (t) = (q/m)E(t) has the same property. In such a case the lower limit of the integral in (1.40) or (1.41) effectively becomes zero instead of minus infinity.

1.5

Harmonic driving force

1.5.1

Formulation of the problem

We shall now assume that the driving force has a simple harmonic character (with frequency ωd ) and that it was switched on at the instant t = 0 (as discussed in connection with (1.42)). Hence, we write f (t) =

qE0 cos(ωd t + φ) = f0 cos(ωd t + φ), m

for t ≥ 0,

(1.43)

while for t < 0, we take f (t) = 0. In the view of these assumptions determination of the inhomogeneous solution for the evolution of the oscillator driven by the force (1.43) requires the integral Z t f0 1 0 0 xinh (t) = dt exp − Γ(t − t ) sin Ω(t − t0 ) cos(ωd t0 + φ). (1.44) Ω 0 2 From elementary trigonometry we get Z t f0 1 0 0 xinh (t) = dt exp − Γ(t − t ) sin Ω(t − t0 ) Ω 0 2 × cos φ cos(ωd t0 ) − sin φ sin(ωd t0 ) .

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Thus the necessary integral splits into two, and we get xinh (t) =

f0 [cos φ I1 (t) − sin φ I2 (t)] . Ω

where the remaining integrals are defined as follows Z t 1 0 0 I1 (t) = dt exp − Γ(t − t ) sin Ω(t − t0 ) cos(ωd t0 ), 2 0 Z t 1 dt0 exp − Γ(t − t0 ) sin Ω(t − t0 ) sin(ωd t0 ), I1 (t) = 2 0

(1.46)

(1.47a) (1.47b)

There is no problem in integral evaluation though it is a bit tedious. Then, the full evolution of the harmonic oscillator driven by the harmonic force (1.43) is the sum of homogeneous and inhomogeneous solutions. Thus, it follows from (1.40) and it is # " v0 + 12 Γx0 1 sin (Ωt) x(t) = exp − Γt x0 cos (Ωt) + 2 Ω +

f0 [cos φ I1 (t) − sin φ I2 (t)] , Ω

(1.48)

where the integrals follow from (1.49). It remains to compute them, then we can construct the full solution corresponding to harmonic driving force (1.43).

1.5.2

Evaluation of the integrals

We present only the final results of the integral computation. I1 (t) = Ω

I2 (t) = Ω

1.5.3

(ω02 − ωd2 ) cos(ωd t) + Γωd sin(ωd t) (ω02 − ωd2 )2 + Γ2 ωd2 2 Γ (ω0 − ωd2 ) cos(Ωt) + 2Ω (ω02 + ωd2 ) sin(Ωt) 1 −Ω exp − Γt , 2 (ω02 − ωd2 )2 + Γ2 ωd2 (ω02 − ωd2 ) sin(ωd t) − Γωd cos(ωd t) (ω02 − ωd2 )2 + Γ2 ωd2 ΓΩ cos(Ωt) + (ωd2 − ω02 + 12 Γ2 ) sin(Ωt) 1 +ωd exp − Γt 2 (ω02 − ωd2 )2 + Γ2 ωd2

(1.49a)

(1.49b)

General solution for oscillator’s displacement

Substituting the integrals (1.49) into the solution (1.48) we regroup the terms and we obtain the following expression for time evolution of the displacement of the harmonic oscillator driven

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by the harmonic force ( v0 + 12 Γx0 1 x0 cos (Ωt) + sin (Ωt) x(t) = exp − Γt 2 Ω Γ (ω02 − ωd2 ) cos(Ωt) + 2Ω (ω02 + ωd2 ) sin(Ωt) (ω02 − ωd2 )2 + Γ2 ωd2 ) ΓΩ cos(Ωt) + (ωd2 − ω02 + 12 Γ2 ) sin(Ωt) ωd f0 sin φ − Ω (ω02 − ωd2 )2 + Γ2 ωd2

− (f0 cos φ)

+ (f0 cos φ)

(ω02 − ωd2 ) cos(ωd t) + Γωd sin(ωd t) (ω02 − ωd2 )2 + Γ2 ωd2

− (f0 sin φ)

(ω02 − ωd2 ) sin(ωd t) − Γωd cos(ωd t) (ω02 − ωd2 )2 + Γ2 ωd2

(1.50)

This general solution may serve as a starting point for discussion of various special cases. We note that it is straightforward to check that the obtained solution satisfies the given initial condition. It is evident that x(0) = x 0 . For the velocity, it requires a little calculation to verify that x(0) ˙ = v0 , as it should be. Since the oscillator frequency Ω may (sometimes considerably) differ from the driving frequency ωd we see that the initial motion of the oscillator may be pretty complicated, especially when damping is very small (Γ ω 0 ). The transients are complicated superposition of the damped oscillations with shifted frequency Ω with the oscillation which follow the driving force with frequency ω d . When the oscillator is initially at rest x 0 = v0 = 0, then the first line of (1.50) vanishes. The transients, however, contain a damped contribution of the oscillation with oscillator frequency Ω – the second and third lines of (1.50). It seems worth giving the expression for oscillator’s displacement for the case when the phase of the driving field vanishes. When φ = 0 the cosine is equal to unity, while the sine is zero. For this case we have ( v0 + 12 Γx0 1 x(t) = exp − Γt x0 cos (Ωt) + sin (Ωt) 2 Ω ) Γ (ω02 − ωd2 ) cos(Ωt) + 2Ω (ω02 + ωd2 ) sin(Ωt) −f0 (ω02 − ωd2 )2 + Γ2 ωd2 + f0

(ω02 − ωd2 ) cos(ωd t) + Γωd sin(ωd t) (ω02 − ωd2 )2 + Γ2 ωd2

(1.51)

The transients are included in the two first lines.

1.5.4

Long–time behaviour

When time t is sufficiently long, i.e., t Γ −1 the motion of the oscillator attains a stationary state. The transients with frequency Ω die out regardless of the initial conditions. First three terms in (1.50) do not contribute any more. Only two last terms survive and the motion of the oscillator occurs with the frequency of the driving field. The motion of the oscillator is then

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completely independent of the initial conditions. Then we have xst (t) = (f0 cos φ)

(ω02 − ωd2 ) cos(ωd t) + Γωd sin(ωd t) (ω02 − ωd2 )2 + Γ2 ωd2 − (f0 sin φ)

(ω02 − ωd2 ) sin(ωd t) − Γωd cos(ωd t) . (ω02 − ωd2 )2 + Γ2 ωd2

(1.52)

Using simple trigonometry we regroup this expression, obtaining the in-phase and out-of-phase contributions xst (t) =

1.5.5

f0 (ω02 − ωd2 ) cos(ωd t + φ) (ω02 − ωd2 )2 + Γ2 ωd2 f0 Γωd sin(ωd t + φ) + (ω02 − ωd2 )2 + Γ2 ωd2

(1.53)

Resonance approximation for stationary state

It is frequently the case that the frequency of the driving field is close to the frequency of the oscillator ωd ' ω0 . Then we can approximate ω02 − ωd2 = (ω0 + ωd ) (ω0 − ωd ) ≈ 2ωd (ω0 − ωd )

(1.54)

Applying this approximation in Eq. (1.53) we obtain xst (t) =

2ωd f0 (ω0 − ωd ) 2 4ωd (ω0 − ωd )2 + Γ2 ωd2 +

cos(ωd t + φ)

4ωd2 (ω0

f0 Γωd sin(ωd t + φ) − ωd )2 + Γ2 ωd2

(1.55)

Some simple rearrangement and substitution of ω 0 instead of ωd in the common factor yields f0 ω0 − ω d xst (t) = cos(ωd t + φ) 2ω0 (ω0 − ωd )2 + (Γ/2)2 Γ/2 sin(ω t + φ) . (1.56) + d (ω0 − ωd )2 + (Γ/2)2 Let us discuss briefly the behaviour of the oscillator within the resonance approximation. We see that the in-phase term (proportional to cos(ω d t + φ), as the driving field) has dispersive character. Its amplitude is Adisp =

f0 ω0 − ω d . 2ω0 (ω0 − ωd )2 + (Γ/2)2

(1.57)

On the other hand, the out-of-phase term [proportional to sin(ω d t + φ)] is absorptive, and its amplitude is Aabs =

f0 Γ/2 . 2ω0 (ω0 − ωd )2 + (Γ/2)2

(1.58)

The figure below illustrates the behaviour of these amplitudes as functions of frequency. When the damping is weak, then apart from the vicinity of the resonance we have Adisp = ω0 − ωd 1. off resonance. (1.59) A Γ/2 abs This allows some interesting conclusions.

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0 -4

-2

0

2

4

Fig. 1.3: Dispersive and absorptive curves.

1. Off resonance, when ωd < ω0 , ie., when the driving force has frequency smaller than the eigenfrequency of the oscillator, the dispersive term dominates, and from (1.56) we get f0 ω0 − ω d xst (t) ≈ cos(ωd t + φ), (1.60) 2ω0 (ω0 − ωd )2 + (Γ/2)2 which means that the oscillator is in phase with the driving force. 2. Off resonance, when ωd > ω0 , ie., when the driving force has frequency larger than the eigenfrequency of the oscillator, the dispersive term dominates again, and from (1.56) we get f0 | ω0 − ωd | xst (t) ≈ − cos(ωd t + φ), (1.61) 2ω0 (ω0 − ωd )2 + (Γ/2)2 so in this case the oscillator is out of phase with the driving force. 3. Finally, exactly on resonance (or very close to it), the absorptive term dominates, and in this case we have f0 Γ/2 xst (t) ≈ sin(ωd t + φ). (1.62) 2ω0 (ω0 − ωd )2 + (Γ/2)2 This indicates, that in resonance the motion of the oscillator is shifted in phase (with respect to the driving force) exactly by a factor of π/2. The discussed features of the driven oscillator are useful in the discussion of some atomic or molecular phenomena.

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2. Classical electrodynamics

Chapter 2

Classical electrodynamics In this chapter we briefly review the fundamentals of classical electrodynamics. Our main aim is to formulate classical electrodynamics in such a manner that the subsequent quantization can be achieved in a simple and convenient way. This chapter summarizes only the main aspects of classical electrodynamics, some other ones, such that are useful in more restricted applications to some specific problems will be dealt with when such a need arises.

2.1

Maxwell’s equations

2.1.1

Some general comments

We will use the SI system of units. In this system of units, Maxwell’s equations in the presence of free charges and currents read ~ = ρ, div D ~ = 0, div B

(2.1a) (2.1b)

∂ ~ B, ∂t ~ = ~j + ∂ D, ~ rot H ∂t ~ = − rot E

(2.1c) (2.1d)

~ D ~ and B, ~ H ~ where ρ is the charge density and ~j the current density. The pairs of the fields E, are connected by the material relations, which may be written as Di = 0 ij Ej ,

Bi = µ0 µij Hj ,

with

1 = c2 . µ0 0

(2.2)

with o the permeability and µo the permeability of the vacuum. The tensors ij and µij are the dielectric and magnetic susceptibilities of the medium in which the fields propagate. For linear and isotropic media these tensors reduce to corresponding constants. In general, susceptibilities ~ ij and µij may be position and time dependent, they may also be the functions of the fields E ~ In the latter case we arrive at the problems of nonlinear optics, which in itself, can be and H. a subject of a separate lecture. In this notes we will not address such questions. ~ and B ~ in media can be expressed via the electric Moreover we note that the fields D polarization and magnetization ~ = 0 E ~ = 0 E ~ + P, ~ D

~ = µµ0 H ~ = µ 0 (H ~ + M), ~ B

(2.3)

or, equivalently ~ = ( − 1)0 E, ~ P S.Kryszewski

~ = (µ − 1)H ~ = µ − 1B ~ = 1B ~ − H. ~ M µµ0 µ0 QUANTUM OPTICS

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~ B ~ and polarization and It is straightforward to write Maxwell’s equations in terms of fields E, magnetization. Then we have ~ + P) ~ div(0 E = ρ, ~ div B = 0,

(2.5a) (2.5b)

~ ~ = −∂ B rot E ∂t ∂ ~ − µ0 M) ~ ~ + µ0 P) ~ rot (B = µ0~j + (0 µ0 E ∂t

(2.5c) (2.5d)

We shall probably return to these expressions when discussing the effects due to the propagation of the fields in material media. It is worth remarking that Maxwell’s equations automatically account for charge conservation. Taking the time derivative of the equation (2.1a) we assume that spatial and temporal ~ by Eq.(2.1d) obtaining derivatives commute. Then we express the time derivative of D ∂ ~ − ~j . ρ = div rot H ∂t

(2.6)

∂ ρ + div ~j = 0. ∂t

(2.7)

Since we have the vector identity div rot ≡ 0, it follows that

which is the charge conservation requirement. It is a local law. Its integral counterpart reads Z Z Z 3 ∂ 3 ~ d r d r div j = − d~S · ~j, (2.8) ρ=− ∂t V V ∂V which means that the charge within certain volume may change only due to the current through its surface. Electric charges create electromagnetic fields, and simultaneously, their motion is influenced by existing fields. We shall consider exclusively point particles with mass m α and charge qα , which contribute delta functions to ρ and ~j, so that X X ~j(~r, t) = ρ(~r, t) = qα δ(~r − ~rα (t)), qα~r˙ α δ(˜ r −˜ rα (t)). (2.9) α

α

The charge and current density defined as above satisfy the continuity equation (2.7). The motion of the particles is governed – in the nonrelativistic case – by Newton’s equations. The force acting on the particles is the Lorentz one, which gives mα

h i d2 ~ rα ) + ~r˙ α × B(˜ ˜ rα ) ~ r = q E(~ α α dt2

(2.10)

The time dependence of the fields and the particle positions and velocities are not explicitly indicated, but in practical applications must be properly accounted for. The set of Maxwell’s equations (2.1), (2.2), and (2.10) give a full description of the dynamics of fields and matter, and form the basis of classical electrodynamics. These equations are strongly coupled so their solution in a general case is impossible. Hence, we usually consider only some simplified cases, which are nevertheless, based on the given set of equations.

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2.1.2

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15

Maxwell’s equations in the vacuum

In vacuum there is no medium, hence polarization and magnetization vanish. There is no need ~ and D, ~ B ~ and H, ~ so we write Maxwell’s equation for the free space as to distinguish fields D ρ , o ~ = 0, div B ~ ~ = −∂ B rot E ∂t 1 ~ 1 ∂ ~ ~ = rot B j+ 2 E 0 c2 c ∂t ~ = div E

2.1.3

(2.11a) (2.11b) (2.11c) (2.11d)

Maxwell’s equations in Fourier space

In some applications it is convenient to express Maxwell’s equations (2.11)in Fourier space. The basic review of the properties of Fourier transforms is given in the appendix A.2. For vector fields ~ or B ~ the Fourier transform can be defined for each Cartesian component separately, so such as E ~ r), ρ(~r), B(~ ~ r) and ~j(~r) that again a vector field results. We denote the Fourier transforms of E(~ ~ ~ ~ ~ ~ ~ ~ as E(k), ρ˜(k), B(k) and J (k) respectively. For example, electric field an its Fourier transform are related as Z 1 ~ ~ ~ ~ r), E(k) = d~r e−ik·~r E(~ (2.12a) 3/2 (2π) Z 1 ~ ~ r) = ~ ~k). (2.12b) E(~ d~k eik·~r E( (2π)3/2 Applying Fourier transformation to Maxwell’s equations (2.1) we obtain their counterparts in the Fourier domain. We list Maxwell’s equations both in the coordinate and Fourier spaces. ~ = ρ, div D ~ = 0, div B ∂ ~ B, ∂t ~ = ~j + ∂ D, ~ rot H ∂t ~ = − rot E

(2.13a) (2.13b) (2.13c) (2.13d)

~ = ρ˜, i~k · D ~ = 0, i~k · B

∂ ~ i~k × E~ = − B, ∂t ~ = J~ + ∂ D, ~ i~k × H ∂t

(2.14a) (2.14b) (2.14c) (2.14d)

It should be, however, noted that in case of dispersive media [that is, media for which the dielectric and magnetic susceptibilities depend on frequency (absolute value of wave vector ~k)] and/or are position dependent, the material relations lead to serious complications. The same applies to nonlinear media (when susceptibilities depend on fields, usually in a nonlinear manner). The problem of the electromagnetic fields in the media is still not fully understood. Therefore, we will mainly focus our attention on the case of the fields in vacuum. In such a case the corresponding Maxwell’s equations in coordinate space and in the Fourier space are simpler, and are of the form

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1 ρ, 0 ~ = 0, div B ~ ~ = − ∂ B, rot E ∂t 1 ~ 1 ∂ ~ ~ = j+ 2 E, rot B 0 c2 c ∂t ~ = div E

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1 ρ˜, (2.16a) 0 ~ = 0, i~k · B (2.16b) ∂ ~ i~k × E~ = − B, (2.16c) ∂t 1 ~ 1 ∂ ~ ~ = E, (2.16d) i~k × B J + 2 0 c2 c ∂t i~k · E~ =

(2.15a) (2.15b) (2.15c) (2.15d)

It is perhaps worth noting that the time derivatives of the Fourier transforms of the fields depend on the values of the transforms taken at the same point ~k of the Fourier space. Hence, Maxwell’s equation in the Fourier domain are local.

2.2

Potentials

2.2.1

Introduction and basic definitions

Maxwell’s equations may be formulated in terms of potentials. To define the potentials, we refer to general Maxwell’s equations (2.1) or (2.13). We also recall to identities known from vector analysis div rot ≡ 0,

rot grad ≡ 0.

(2.17)

~ = 0 is always satisfied, since it signifies that there are no The second Maxwell’s equation div B magnetic monopoles. Hence, due to the first of the identities 2.17) we can always write ~ r, t) = rot A(~ ~ r, t). B(~

(2.18)

~ is specified [the magnetic field B ~ being determined We conclude that when the vector potential A by Eq.(2.18)], then Maxwell’s equation (2.1b) or (2.13b) is automatically satisfied. Introducing ~ = −∂ rot A/∂t, ~ (2.18) into the third Maxwell’s equation (2.1c)or (2.13c) we obtain rot E which ~ ~ suggests that the electric field is given as E = −∂ A/∂t. This is, however not sufficient. Due to the second of relations (2.17) we can always add a gradient of arbitrary function. Thus we write ~ r, t) = −grad φ(~r, t) − ∂ A(~ ~ r, t), E(~ ∂t

(2.19)

and the Maxwell’s equation (2.11c) or (2.15c) is still satisfied. Therefore, we may formulate the problem of specifying the electromagnetic field as follows. We postulate the existence of a scalar ~ r, t) such that the electric field is determined by Eq.(2.19) field φ(~r, t) and of a vector field A(~ and the magnetic field by (2.18). Then, we automatically satisfy two out of four Maxwell’s equations (namely, Eqs. (2.1b) and (2.1c) or (2.13b) and (2.13c). There are still two other Maxwell’s equations to consider, namely ~ = ρ, div D

~ = ~j + rot H

∂ ~ D, ∂t

(2.20)

which also must be satisfied. It remains to check what are the conditions imposed on potentials by equations (2.20).

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2.2.2

2. Classical electrodynamics

17

Wave equations for potentials

We should now look for the restrictions imposed on the potentials by two remaining Maxwell’s equations (2.20). ~ and φ specify the We note that there arises a serious problem. Namely, the potentials A ~ and B, ~ while equations (2.20) contain fields D ~ and H. ~ The latter and the former fields fields E are connected by material relations (2.2). The dielectric and magnetic susceptibilities may be complicated functions of position, time and also of the fields themselves. Thus, equations (2.20) after insertion of material relations and potentials can be expected to be very complicated. To avoid such problems, we shall restrict our attention to fields in vacuum. The relations between ~ B ~ and potentials A, ~ φ remain unchanged, because the second and the third Maxwell’s fields E, equations in vacuum are the same as in media. On the other hand, equations (2.20) become simpler, and in vacuum they are ~ = div E

ρ , 0

~ = rot B

1 ~ 1 ∂ ~ j+ 2 E, 0 c2 c ∂t

(2.21)

Introducing electric field determined by (2.19) into the first of the above equations we get ! ~ 1 ∂ ∂A ρ ~ =⇒ ∇2 φ = − ρ − div −grad φ − = ∇ · A, (2.22) ∂t 0 0 ∂t which is the wave equation (in vacuum) for scalar potential. This wave equation is not, strictly speaking, an equation of motion for φ since it does not include its time derivative. It is rather ~˙ at a certain time moment. a relation between φ and A It remains to make use of the second of equations (2.21). We replace fields by the corresponding expressions with potentials and we obtain 1 ∂ 1 ∂ ~j + ~ = ~ . rot rot A −∇ φ − A (2.23) 0 c2 c2 ∂t ∂t Using the vector analysis identity ~ = ∇(∇ · A) ~ − ∇2 A, ~ ∇ × (∇ × A)

(2.24)

we further obtain 1 ∂2 1 ~ 1 ∂φ 2 ~ ~ −∇ A= 2 j−∇ ∇·A+ 2 . c2 ∂t2 c 0 c ∂t

(2.25)

We may conclude, restricting our considerations to vacuum, that the introduction of potentials by (2.18) and (2.19) guarantees that two of the Maxwell’s equations are satisfied automatically, while the other two are equivalent to wave equations (2.22) and (2.25).

2.2.3

Potentials – gauge invariance

We recall the definition of the fields via potentials, that is ~ r, t) = −grad φ(~r, t) − ∂ A(~ ~ r, t), E(~ ∂t

~ r, t) = rot A(~ ~ r, t). B(~

(2.26)

The vector identity rot grad ≡ 0 holds, hence redefinition (called gauge transformation) of the vector potential ~ r, t) A(~ S.Kryszewski

gauge

- A ~ 0 (~r, t) = A(~ ~ r, t) + ∇ F (~r, t), QUANTUM OPTICS

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~ Such a transformation results in the corresponding change does not change the magnetic field B. of the electric field which becomes i h ~ r, t) = −grad φ(~r, t) − ∂ A ~ 0 (~r, t) − ∇F E(~ ∂t ∂ ∂ ~0 = −grad φ(~r, t) − F (~r, t) − A (~r, t). (2.28) ∂t ∂t Hence we conclude that the joint transformation ~ r, t) A(~ φ(~r, t)

- A ~ 0 (~r, t) = A(~ ~ r, t) + ∇χ(~r, t),

(2.29a)

∂ φ0 (~r, t) = φ(~r, t) − χ(~r, t), ∂t

(2.29b)

gauge

gauge

~ and B, ~ unchanged. This fact is called the gauge leaves the physical quantities, that is the fields E invariance of the fields. Gauge transformations allow us to choose potentials in a convenient way, suited to particular applications. At present, we will not discuss this subject. We will only briefly indicate two most commonly used gauges – methods of choosing the potentials.

2.2.4

Lorentz gauge

Lorentz gauge consists in such a choice of the potentials, that the relation ~ r, t) + div A(~

1 ∂ φ(~r, t) = 0, c2 ∂t

(2.30)

is satisfied. This requirement still leaves some freedom. Namely, let us assume that the potentials satisfy the Lorentz gauge (2.30). Let us make a second gauge transformation by adopting new potentials ~0=A ~ + ∇G, A

φ0 = φ −

∂ G, ∂t

(2.31)

which are still required to satisfy Lorentz gauge. Inserting the gauge (2.31) into the condition (2.30) and using the fact that old potentials also satisfy the Lorentz gauge, we obtain the equation for the gauge function G ∇2 G −

1 ∂2 G = 0, c2 ∂t2

(2.32)

which clearly indicates some arbitrariness in the choice of the gauge function G. Lorentz gauge imposed on the chosen potentials leads to some simplifications of the wave equations (2.22) and (2.25) and allows us to rewrite these wave equations as 1 ∂2 1 ∇2 − 2 2 φ(~r, t) = − ρ(~r, t) (2.33a) c ∂t 0 1 ∂2 ~ 1 2 ∇ − 2 2 A(~r, t) = − 2 ~j(~r, t). (2.33b) c ∂t 0 c As a result, we obtain uncoupled, symmetric wave equations for the potentials. Finally, we note the Lorentz gauge is relativistically invariant, and as such is particularly useful in relativistic considerations. It is easy to recast wave equations (2.33) into four-dimensional (space-time) notation. This, however, is beyond the scope of our present interests.

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2.2.5

2. Classical electrodynamics

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Coulomb gauge

Coulomb gauge is specified by the requirement ~ r, t) = 0. div A(~

(2.34)

So, there is no requirement imposed on the scalar potential. Unfortunately, relativistic invariance is thus lost. Nevertheless this gauge is extremely useful. In this case wave equations are also simplified. From (2.22) and (2.25), for Coulomb gauge, we get the wave equations 1 ∇2 φ(~r, t) = − ρ(~r, t), 0 1 ∂2 ~ 1 ~ 1 ∂ 2 ∇ − 2 2 A(~r, t) = − grad φ(~r, t). j(~r, t) + 2 2 c ∂t 0 c c ∂t

(2.35a) (2.35b)

We lost the symmetry of the wave equations, although the equation for the scalar potential is simpler. We note, that from the Poisson equation (2.35a) it follows that the scalar potential is due to the instantaneous charge distribution. This may lead to the conclusion that there is some kind of an interaction which spreads with infinite velocity. It can be shown, that this is not really a problem. At present, we only state that the discussion of this problem can be found elsewhere.

2.2.6

Potentials in the Fourier space

Similarly as we introduced the Fourier transforms of the fields, we can define Fourier transforms of the potentials Z 1 ~ ~ ~k, t) = ~ r, t), d~r e−ik·~r A(~ (2.36a) A( (2π)3/2 Z 1 ~ ˜ ~ φ(k, t) = d~r e−ik·~r φ(~r, t) (2.36b) 3/2 (2π) where we have included the time dependencies. It is straightforward to write the inverse transforms. Z 1 ~ ~ ~ ~k, t), A(~r, t) = d~r eik·~r A( (2.37a) 3/2 (2π) Z 1 ~ ˜ ~k, t) φ(~r, t) = d~k eik·~r φ( (2.37b) 3/2 (2π) All equations given in previous sections, can easily be translated into Fourier domain ( ~k space). The fields are expressed by potentials via the relations (2.18) and (2.19) which after Fourier transformation yield ˜ ~k, t) ~ ~k, t) − i~k φ( ~ ~k, t) = − ∂ A( E( ∂t

~ ~k, t) = i~k × A( ~ ~k, t). B(

(2.38)

General wave equations (2.22) and (2.25) in Fourier domain read ˜ ~k, t) = k 2 φ( 1 ∂2 ~ ~ k 2 + 2 2 A( k, t) = c ∂t

S.Kryszewski

1 ~ ∂ ~~ ρ˜(k, t) + i~k · A(k, t), 0 ∂t 1 ~ ~ ~k i~k · A( ˜ ~k, t) , ~ ~k, t) + 1 ∂ φ( J ( k, t) − i c2 0 c2 ∂t

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(2.39a) (2.39b)

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Gauge transformation (2.29) in the Fourier domain is of the form ~ ~k, t) A( ˜ ~k, t) φ(

- A ~ 0 (~k, t) = A( ~ ~k, t) + i~k χ( ˜ ~k, t),

(2.40a)

-

(2.40b)

gauge

gauge

˜ ~k, t) − ∂ χ( φ˜0 (~k, t) = φ( ˜ ~k, t), ∂t

Lorentz gauge (2.30) in the Fourier domain reads ˜ ~k, t) = 0, ~ ~k, t) + 1 ∂ φ( i~k · A( c2 ∂t

(2.41)

and the wave equations (2.33) become 2 ~k2 + 1 ∂ φ( ˜ ~k, t) = 1 ρ( ˜ ~k, t) 2 2 c ∂t 0 2 1 ~ ~ ~k2 + 1 ∂ A( ~ ~k, t) = J (k, t). c2 ∂t2 0 c2

(2.42a) (2.42b)

Finally, the Coulomb gauge (2.34) in the Fourier domain attains the form ~k · A( ~ ~k, t) = 0.

(2.43)

This relation means that the Fourier transform of the vector potential is perpendicular to the wave vector ~k, which explains why the Coulomb gauge is sometimes called the transverse gauge. Wave equations (2.35) now become ˜ ~k, t) = ~k2 φ( 2 ~k2 + 1 ∂ A( ~ ~k, t) = c2 ∂t2

1 ~ ρ( ˜ k, t), 0 1 ~ ~ i~k ∂ ˜ ~ J ( k, t) − φ(k, t). 0 c2 c2 ∂t

(2.44a) (2.44b)

It is perhaps worth commenting upon Eq.(2.44a). We will show that it corresponds to the Fourier transform of the standard Coulomb potential. Indeed, (2.44a) may be written as 1 ˜ ~k, t) = 1 ρ( φ( ˜ ~k, t) (2π)3/2 3/2 0 (2π) k 2

(2.45)

We see that φ˜ is a product of two Fourier transforms, so it is a transform of a convolution. It follows that the inverse transform, being the potential φ(~r, t) is a convolution of two functions 1 ρ(~r, t) 0

and

(2π)3/2

1 4π|~r |

(2.46)

where the second one follows from (1.89). Therefore we have the potential φ(~r, t) – the inverse of φ˜ given as the convolution, that is Z Z 1 1 1 ρ(~r, t) 0 1 3/2 φ(~r, t) = d~r ρ(~r, t) (2π) = d~r0 , (2.47) 0 3/2 ~ 0 4π|~r − r | 4π0 |~r − ~r0 | (2π) so we reproduce the well-known Coulomb potential, the Fourier transform of which satisfies equation (2.44a).

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2.3

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Longitudinal and transverse fields

2.3.1

Introduction

In many practical applications it is convenient to split vector fields in a longitudinal part, for which the rotation is zero, and a transverse part, which has a vanishing divergence (it is a sourceless field). For example, for the electric field we write ~ r) = E ~ k (~r) + E ~ ⊥ (~r) E(~

(2.48)

with ~ k (~r) = 0, ∇×E

~ ⊥ (~r) = 0. ∇·E

(2.49)

For any square integrable field this separation is unique when we also require that the transverse and longitudinal parts vanish separately at infinity. This statement is known as Helmholtz’s ~ r) at theorem. The given separation is non-local, in the sense that knowledge of the values of E(~ ~ ~ a certain position is not sufficient to determine the values of E⊥ and Ek at that position. The differential operators do not specify the field in a unique way, some integration is necessary. ~ ⊥ (~r) have ~ k (~r) and E The same separation is more transparent in Fourier space. The fields E E~k (~k) and E~⊥ (~k) as their Fourier transforms.The equivalents of Eqs.(2.48) and (2.49) read now ~ ~k) = E~k (~k) + E~⊥ (~k) E(

(2.50)

with ~k × E~k (~k) = 0,

~k · E~⊥ (~k) = 0.

(2.51)

This implies that E~k (~k) is the component parallel to ~k, whereas E~⊥ (~k) is simply the component ~ ~k) orthogonal to ~k. Introducing a unit vector of E( ~nk =

~k | ~k |

(2.52)

we may also write h i ~ ~k) , E~k (~k) = ~nk ~nk · E(

(2.53a) h

i

~ ~k) − E~k (~k) = E( ~ ~k) − ~nk ~nk · E( ~ ~k) , E~⊥ (~k) = E(

(2.53b)

which clearly indicate that E~k (~k) is indeed parallel to vector ~k, while E~⊥ (~k) is perpendicular to wave vector. Hence, we see that in Fourier space the separation is local. Moreover, it is obviously unique. Here, several comments are necessary. To separate the field into longitudinal and transverse parts, we must satisfy equations (2.48) and (2.49) for all positions ~r. Similarly, we have to satisfy Eqs.(2.50) and (2.51) for all wave vectors ~k. This may be a tricky problem. To clarify it, let us consider an example, a point charge Q located at a position ~r0 . In this case the charge density ~ = Q δ(~r − ~r0 ) and it is zero almost everywhere (except at is ρ(~r) = Q δ(~r − ~r0 ). Hence 0 div E the point at which the charge is located). Transforming into Fourier space we get ~ ~k) = i0 ~k · E( S.Kryszewski

Q ~ e−ik·~r0 . (2π)3/2

(2.54) QUANTUM OPTICS

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~ ~k) and wave vectors ~k are not orthogonal almost everywhere. Although div E ~ =0 The field E( almost everywhere, the field is not transverse. This is clearly seen from Fourier relation (2.54). Secondly, the separation into longitudinal and transverse parts is not relativistically invariant. A vector which is transverse in one reference (coordinate) frame, usually is not transverse in another frame – obtained via Lorentz transformation.

2.3.2

Transverse and longitudinal Maxwell’s equations

We shall again restrict our attention to the vacuum fields. Maxwell’s equations (2.15) in coordinate space, or (2.16) in Fourier space, can now be expressed as separate equations for the longitudinal and transverse parts of the fields and the current density. Firstly, we discuss the longitudinal components. Two first equations (with divergence) Equations (2.15b) or (2.16b) state that the divergence of the magnetic field always vanishes, which means that the magnetic field is sourceless. This also means that the magnetic induction ~ is purely transverse, so that B B~k (~k) = 0,

~ k (~r, t) = 0, B

(2.55)

~ must always be zero. and the longitudinal component of B The other scalar equations (2.15a) or (2.16a) state that i~k · E~ = ρ/ ˜ 0 . This says that the ~ ~ transverse part E⊥ of the electric field (orthogonal to k) does not contribute to the left hand side. So, scalar equation (2.15a) or (2.16a) concern only the longitudinal part of the electric field, and may be expressed as ~ k (~r) = ∇·E

ρ(~r) , 0

or

ρ˜(~k) i~k · E~k (~k) = . 0

(2.56)

We can show that the longitudinal electric field is simply the Coulomb field corresponding to the instantaneous position of the charges. Hence we may write Z X qα ~r − ~r0 ~r − ~rα 1 ~ Ek = d~r0 ρ(~r0 ) = . (2.57) 0 3 4π0 |~r − ~r | 4π0 |~r − ~rα |3 α To prove that this is indeed the case, we use Eq.(2.53a) and we get E~k (~k) =

i ~k h ~k · E~k (~k) + E~⊥ (~k) . | ~k |2

(2.58)

Obviously, ~k · E~⊥ = 0, so that we are left with the relation E~k (~k) =

~k ~k · E~k (~k) . | ~k |2

(2.59)

Replacing the scalar product in the right hand side by Eq.(2.56) we obtain E~k (~k) = −

i ~k ρ( ˜ ~k). 0 | ~k |2

(2.60)

So, the Fourier transform of the longitudinal electric field is a product of two transforms − 22

i ~k k2

F −1-

(2π)3/2

~r , 4πr 3

ρ( ˜ ~k)

F −1-

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ρ(~r).

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The first transform follows from (1.90b). Then, using the properties of the convolution, we arrive at Eq.(2.57), which indeed is the Coulomb field of the given charge distribution. ~ = ρ/0 and div B ~ = 0 specify the longituHence we can say that Maxwell’s equations div E ~ k by Eq.(2.57). ~ k ≡ 0 and E dinal fields: B ~ Third equation (with rot E) By definition, rotation of longitudinal field is identically zero. Therefore, Eq.(2.15c) or its Fourier equivalent (2.16c) ~ r) = − rot E(~

∂ ~ B(~r), ∂t

~ ~k) = − ∂ B( ~ ~k), i~k × E( ∂t

(2.62)

~ or E~ does contain no information on the longitudinal fields because longitudinal component of E ~ is purely transverse so there is no magnetic not contribute to left hand side, and magnetic field B longitudinal contribution. Hence Maxwell’s equations (2.15c) or (2.16c) (as in Eq.(2.62)) are equivalent to ~ ⊥ (~r) = − rot E

∂ ~ B⊥ (~r), ∂t

∂ ~ ~ i~k × E~⊥ (~k) = − B ⊥ (k), ∂t

(2.63)

~ Fourth equation (with rot B) It remains to discuss Eq.(2.15d) or (2.16d), that is 1 ~ 1 ∂~ ~ ~k) = 1 J~ (~k) + 1 ∂ E( ~ ~k). j(~r) + 2 E(~r), i~k × B( (2.64) 0 c2 c ∂t 0 c2 c2 ∂t Taking into account the transversality of the magnetic field we can rewrite these equations as (in the Fourier domain) ~⊥ (~k) = 1 ~⊥ (~k) + J~k (~k) + 1 ∂ ~⊥ (~k) + E~k (~k) . i~k × B (2.65) J E 0 c2 c2 ∂t The last equation splits into the transverse part ~ r) = rot B(~

1 ~ ~ 1 ∂ ~ ~ J⊥ (k) + 2 E⊥ (k), 2 0 c c ∂t ~=B ~⊥ does not contribute to left-hand side and longitudinal part, where B ~⊥ (~k) = i~k × B

0=

1 ~ ~ 1 ∂ ~ ~ Jk (k) + 2 E (k). 2 0 c c ∂t k

(2.66)

(2.67)

We will now argue that equation (2.67) does not bring any new information, and as such can be discarded. Indeed, multiplying both sides by i~k we get i ~ ~ ~ ∂ ~ ~ ~ k · Jk (k) = −i k · Ek (k). 0 ∂t

(2.68)

Due to (2.56) we express right-hand side via charge density, and we have ∂ i ~k · J~k (~k) = − ρ˜(~k). (2.69) ∂t Projection of the part J~k onto ~k is obviously equivalent to the projection of a whole vector. Hence, instead of (2.69) we can write ∂ ~ i ~k · J~ (~k) = − ρ( ˜ k), (2.70) ∂t which is clearly seen to be the charge continuity equation in the Fourier space. Hence we conclude that (2.67) does not bring any new information. S.Kryszewski

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Summary of longitudinal and transverse Maxwell’s equations • Two first (scalar) Maxwell’s equations, which in the Fourier domain read ~ ~k, t) = 1 ρ( i ~k · E( ˜ ~k, t), 0

~ ~k, t) = 0, i ~k · B(

(2.71)

specify the longitudinal fields ~ k (~r, t) = 0, B ~ k (~r, t) = E

(2.72)

1 4π0

Z

d~r0 ρ(~r0 , t)

− ~r0

~r = |~r − ~r0 |3

X α

qα 4π0

~r − ~rα (t) . |~r − ~rα (t)|3

(2.73)

~ k of the electric field is fully determined by the charge In summary, the longitudinal part E ~ k is not an independent dynamical variable. density ρ. This implies that E ~ is purely transverse and it reads • The third Maxwell’s equation (this with rot E) ~ ⊥ (~r, t) = − rot E

∂ ~ B⊥ (~r, t), ∂t

∂ ~ ~ i~k × E~⊥ (~k, t) = − B ⊥ (k, t), ∂t

(2.74)

~ splits into the transverse part • Finally, the fourth Maxwell’s equation (this with rot B) ~ r, t) = rot B(~ ~⊥ (~k, t) = i~k × B

1 ~ 1 ∂ ~ j(~r, t) + 2 E(~r, t), 2 0 c c ∂t 1 ~ ~ 1 ∂ ~ ~ J⊥ (k, t) + 2 E⊥ (k, t). 2 0 c c ∂t

(2.75)

The longitudinal part of Maxwell’s equation (2.15d) or (2.16d) is of the form given in (2.67), which is equivalent to the charge continuity, and hence brings no new information and can be discarded. ~ =B ~ ⊥ are mutually related by (2.74) and (2.75), where ~ ⊥ and B • The transverse fields E the transverse part of the current density ~j⊥ acts as a source of the transverse fields.

2.3.3

Discussion of the potentials

Definitions The electric field in the Fourier domain is given by the Fourier transforms of the potentials by Eqs.(2.38). We have ˜ ~k, t) ~ ~k, t) = − ∂ A( ~ ~k, t) − i~k φ( E( ∂t

(2.76)

where we see that the last term is parallel to wave vector. Hence this term is longitudinal. Thus we arrive at the splitting ∂ ~ ~ E~⊥ (~k, t) = − A ⊥ (k, t), ∂t

∂ ~ ~ ˜ ~k, t), E~k (~k, t) = − A (k, t) − i~k φ( ∂t k

(2.77)

which can easily be transformed to coordinate space, yielding ~ ⊥ (~r, t) = − ∂ A ~ ⊥ (~r, t), E ∂t 24

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Magnetic induction is transverse, so according to (2.38) it is specified only by transverse component of the vector potential. Thus, we can write ~ ⊥ (~k, t), B~⊥ (~k, t) = i~k × A

~ r, t) = ∇ × A ~ ⊥ (~r, t). B(~

(2.79)

We thus see, that the scalar potential φ(~r, t) and the longitudinal part of the vector potential ~ k (~r, t) are necessary only to determine the longitudinal part of the electric field. A Wave equations for potentials General wave equations for potentials in the Fourier domain are given in Eqs(2.39). Equation (2.39a) contains only scalar potential and longitudinal component (parallel to ~k of vector potential. So we have ˜ ~k, t) = k 2 φ(

1 ~ ∂ ~ ~ ρ( ˜ k, t) + i~k · A (k, t), 0 ∂t k

The second wave equation, i.e. (2.39b) can be split into a transverse part 1 ∂2 ~ ~ 1 ~ ~ k2 + 2 2 A ⊥ (k, t) = 2 J⊥ (k, t) c ∂t c 0 and a longitudinal part (parallel to the wave vector ~k) 1 ∂2 ~ ~ 1 ~ ~ 1 ∂ ˜~ ~ ~ ~ ~ k2 + 2 2 A ( k, t) = J ( k, t) − i k i k · A ( k, t) + φ( k, t) . k k c ∂t c2 0 k c2 ∂t

(2.80)

(2.81)

(2.82)

In the longitudinal equation (2.82) we take the scalar product of both sides with the vector i ~k. The first term in the left-hand side cancels with the second one in right-hand side, and we get i ~k ·

∂2 ~ 1 ∂ ˜ Ak = i ~k · J~k + k 2 φ. 2 ∂t 0 ∂t

(2.83)

Differentiating Eq.(2.80) over time, we eliminate the second order time derivative of the longitudinal component of the vector potential. Then, the terms containing the time derivative of scalar potential cancel out and we obtain −

1 ∂ 1 ~ ~ ρ˜ = i k · Jk , 0 ∂t 0

(2.84)

which again reproduces the Fourier domain charge continuity equation (2.70). Hence, we conclude that the longitudinal equation (2.82) do not contribute any significant or new information. Moreover, we conclude that only equations (2.80) and (2.81) play meaningful role in the determination of the potentials. These equations can easily be transformed to usual coordinate space yielding

1 ∂2 c2 ∂t2

−∇2 φ(~r, t) = 2 ~ − ∇ A⊥ (~r, t) =

1 ∂ ~ ρ(~r, t) + div A (~r, t), 0 ∂t k 1 ~ j⊥ (~r, t) 2 c 0

(2.85a) (2.85b)

We see that equation (2.85a) is a scalar one, hence it is insufficient to determine the scalar ~ k . There is no harm in it, because potential and three components of the longitudinal vector A potentials are defined in a redundant manner (freedom of the gauge). Hence, to specify the potentials we need some additional requirements. It means that some gauge should be chosen. We also note that transverse equation (2.85b) has the same form as the wave equation (2.42b). S.Kryszewski

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Choice of the gauge From Eq.(2.40a) we have the gauge transformation for the vector potential in the Fourier domain ~ ~k, t) A(

- A ~ 0 (~k, t) = A( ~ ~k, t) + i~k χ( ˜ ~k, t),

(2.86)

gauge

with arbitrary function χ. Since the second term is parallel to the wave vector ~k, gauge transformation changes only the longitudinal component of vector potential. It means that in any gauge the transverse component ~ ⊥ (~k, t) A

- A ~ 0 (~k, t) = A ~ ⊥ (~k, t), ⊥

(2.87)

gauge

so it is unchanged, hence gauge invariant. Therefore, it is nothing strange, that the wave ~ has the same for as in the Lorentz gauge. It will have equation (2.85b) for transverse part of A such form in any gauge. In view of these remarks it seems reasonable to choose the Coulomb gauge (2.34), which we write as ~ ~ div A⊥ + Ak = 0. (2.88)

~ ⊥ is by definition a sourceless field (i.e. with zero divergence) the Since the transverse part A ~ k = 0. The simplest way to fulfill this condition is to requirement (2.88) simply reduces to div A choose Coulomb gauge ~k = 0 =⇒ A (2.89) ~ =0 div A Then, from (2.78) and (2.79) we get ~ ⊥ (~r, t) = − ∂ A ~ ⊥ (~r, t), E ∂t ~ r, t) = B ~⊥ = ∇×A ~ ⊥ (~r, t), B(~

~ k (~r, t) = − ∇ φ(~r, t), E

(2.90a) (2.90b)

Moreover, with Coulomb gauge wave equations for the potentials (2.85) are of the form 1 ∇2 φ(~r, t) = − ρ(~r, t), 0 2 1 ∂ 1 ~ ~ ⊥ (~r, t) = − ∇2 A j⊥ (~r, t). c2 ∂t2 c2 0

(2.91a) (2.91b)

Finding the scalar potential we obtain longitudinal component of the electric field. The second ~ ⊥ to equation yields transverse component of the vector potential and it is sufficient to know A compute the transverse components of the electric field and of the magnetic induction according to (2.90).

2.4 2.4.1

Interaction of fields with charged particles Lagrange equations revisited

Lagrange equations of motion are of the form d ∂T ∂T − = Qi , dt ∂ q˙i ∂qi 26

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(2.92)

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where T is the kinetic energy of the considered physical system, (q i , q˙i ) are generalized coordinates and velocities and Qi – generalized forces. If the forces are conservative, then they can be expressed via the corresponding potential energy V which depends only on generalized coordinates {qi }, so that Qi = −∂V /∂qi . If such forces are introduced into Lagrange equations (2.92), then we get equations of motion d ∂ ∂ T −V − T −V = 0, (2.93) dt ∂ q˙i ∂qi and we can denote the Lagrangian as L = T − V . However, in the general case it may happen that some of the forces can be described by the potential energy V , while some other ones may still require the generalized expressions, as in (2.92). Some kind of an intermediate case arises, when we can express the force by the so-called generalized potential U which may depend on velocities ∂U d ∂U Qj = − + . (2.94) ∂qi dt ∂ q˙i If such an expression for forces is used in Eq.(2.92), then our equations of motion become d ∂ ∂ T −U − T − U = 0. (2.95) dt ∂ q˙i ∂qi In this case, the Lagrangian is L = T − U , so it is generalized by taking into account forces which may depend on generalized velocities of the constituents of the system (the second term in (2.94)). We will show, that charged particles interacting with electromagnetic field can be described by such a generalized potential, This will allow us to construct the corresponding canonical formalism.

2.4.2

Generalized potential U for a particle in the field

Since we will apply the results of this section to nonrelativistic quantum mechanics, our approach will be strictly nonrelativistic. To keep our derivations as simple as possible, we will consider ~ r, t) and B(~ ~ r, t). just one particle of mass m and charge q moving in the electromagnetic field E(~ In our notation we will nor write the arguments of the fields. The fields are purely classical, thus they may be considered as externally imposed. Hence the energy of the field is constant (depends on the parameters from outside of our system: particle + field), so it will not enter our equations. As we know, the fields can be described by the potentials (see (2.18) and (2.19)). We stress, that we are not adopting any particular potential gauge. Employing the potentials we can write the Lorentz force which acts upon the particle as ! ~ ∂ A ~ = q E ~ + ~v × B ~ ~ . F = q −∇φ − + ~v × rot A (2.96) ∂t Using elementary vector analysis, we can write the last term in the form ~ = ~ei vj ∇i Aj − (~v · ∇)A. ~ ~v × (∇ × A)

(2.97)

Let us note that the full time derivative of the vector potential is expressed by ~ ~ ~ dxk ~ dA ∂A ∂A ∂A ~ = + = + (~v · ∇) A. dt ∂t ∂xk dt ∂t S.Kryszewski

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Furthermore, velocity ~v is, in Lagrange formalism, independent of position, which implies that ~ ~ei vj ∇i Aj = ~ei ∇i (vj Aj ) = ∇(~v · A).

(2.99)

Using two last equations, we reexpress Eq.(2.97) and we obtain ~ = ~v × rot A

~ ~ − dA = ∇(~v · A) dt

+

~ ∂A ∂t

(2.100)

Thus, using this in (2.96), we are able to find the following formula giving the Lorentz force upon the charged particle ! ~ d A ~ = = q −∇φ + ∇(~v · A) ~ − . (2.101) F dt The field is considered as a function only of position and time, it is an external field. Therefore, we have d dAj d ∂ ~ d ∂ (Ak vk ) = (Ak δjk ) = (2.102) A · ~v = dt ∂vj dt ∂vj dt dt Thus, we can write the last term in (2.101) according to (2.102) d ∂ ~ ~ ~ F = = q −∇ φ − ~v · A − (~v · A) , dt ∂~v

(2.103)

where we combined two first terms with gradients. If we note that the scalar potential is also velocity independent, we can finally write d ∂ ~ ~ ~ F = = q −∇ φ − ~v · A + (φ − ~v · A) . (2.104) dt ∂~v Comparing this equation with Eq.(2.94) we see that for the charged particle interacting with electromagnetic field we can introduce the generalized potential U e ~ Ue = qφ − q ~v · A,

(2.105)

so that the Lorentz force is expressed as d ∂Ue ~ F = = − ∇Ue + . dt ∂~v

(2.106)

This allows us to write down the Lagrangian Le = T − U e =

m~v2 ~ − qφ + q ~v · A, 2

(2.107)

and the corresponding equations of motion follow immediately from Eq.(2.95).

2.4.3

Canonical (Hamilton) formalism

We still do not account for the energy of the field, since we take it to be external. Having constructed the Lagrangian (2.107) we find the canonical momentum to be ~p = 28

∂Le ~ = m ~v + q A. ∂~v

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Although we denote it by ~p, we stress that it is a canonical momentum, while the kinetic one is ~pkin = m ~v. Now, we are in the position to construct the Hamiltonian. According to the well-known definition we have He = ~p · ~v − Le =

m~v2 + qφ 2

(2.109)

But in the canonical formalism velocity ~v is not an independent variable. Eliminating velocity by means of the canonical momentum we arrive at He =

2 1 ~ ~p − q A + qφ, 2m

(2.110)

which we will call the minimal coupling hamiltonian for a particle of mass m and charge Q in the external electromagnetic field. The field is described by the scalar potential φ(~r, t) and vector ~ r, t). Having found the Hamiltonian, we stress that the following comments should be one A(~ taken into account. • We considered one particle, but it is easy to generalize the result for many particle case. • Our considerations are nonrelativistic. • We do not include the hamiltonian of the free field. • The gauge in which the potentials are taken is not specified. If some special gauge is adopted, the hamiltonian (2.110) may have somewhat different form.

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Chapter 3

Quantization of electromagnetic field

3.1

Introductory remarks

We will quantize electromagnetic field in a simple, intuitive manner. We consider the sourceless electromagnetic field in a cavity with Coulomb gauge, that is the field, for which the vector potential satisfies the requirement ~ = 0. div A

(3.1)

As it follows from the considerations in the previous chapter, the vector potential is transverse, gauge independent, and since there are no sources, it satisfies the homogeneous wave equation ~⊥ 1 ∂2 A ~ ⊥ (~r, t) = 0. − ∇2 A 2 c ∂t2

(3.2)

Due to the Coulomb gauge and to the absence of the sources we can take the scalar potential to be identically zero. Then the fields are fully specified by the vector potential ~ ⊥ (~r, t) = − ∂ A ~ ⊥ (~r, t), E ∂t ~ r, t) = B ~⊥ = ∇×A ~ ⊥ (~r, t), B(~

~ k (~r, t) = 0, E

(3.3a) (3.3b)

We will not go into the subtleties of the gauge problems, or other mathematical nuances. We will consider the electromagnetic field in the cavity of volume V . The procedure we will describe does not depend on the shape of the cavity, although the proof of this fact is far from trivial. We will also indicate the limiting procedure allowing the description of the fields in all space.

3.2 3.2.1

Expansion into normal modes Statement of the problem

We seek the solution to the wave equation (3.2) in a form with separated variables r 1 X ~ A⊥ (~r, t) = qn (t) ~un (~r). 0 n

(3.4)

The index n may have the meaning of the multiindex (the Laplace operator in the wave equation is usually degenerate). The coefficient in the front is introduced for future convenience. We 30

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will call qn (t) the field amplitudes, while the function ~un (~r) will be called field modes. At present we will assume that the set of field modes is linearly independent, and as such can be orthonormalized. This point will be discussed later. Since the fields are physical observables, we can for present purposes take the field amplitudes and modes to be real. Before analyzing the wave equation let us express the fields via the adopted vector potential. From (3.3) and (3.4) we get r 1 X ~ E⊥ (~r, t) = − q˙n (t) ~un (~r), (3.5a) 0 n r 1 X ~ B(~r, t) = qn (t) rot ~un (~r) (3.5b) 0 n Let us also note, that the Coulomb gauge implies the relation r 1 X ~ 0 = div A(~r, t) = qn (t) div ~un (~r) 0 n

(3.6)

We return to the wave equation. Substituting (3.4) into the wave equation (3.2) we employ the linear independence of the field modes to obtain 1 q¨n (t)~un (~r) − qn (t) ∇2 ~un (~r) = 0. c2 We can add and subtract the same quantity. Then we get i 1 h ωn2 2 2 ~ n (~r) q¨n (t) + ωn qn (t) ~un (~r) − qn (t) ∇ ~un (~r) + 2 u = 0. c2 c

(3.7)

(3.8)

This procedure is fully equivalent to usual variable separation. This equation must be satisfied identically for any time instant and at any point within the cavity, therefore, the coefficients in square brackets must vanish separately. Hence, our wave equation is equivalent to the set of equations q¨n (t) + ωn2 qn (t) = 0, ω2 ∇2 ~un (~r) + 2n ~un (~r) = 0. c

(3.9a) (3.9b)

Equation (3.9a) has dynamical character, while (3.9b) is geometrical. First, we consider the geometrical one with the reasonable assumption that ω m 6= ωn . The field within the cavity must satisfy the boundary conditions at the walls of the cavity. The tangent component of the electric field must vanish, and so must the normal component of the magnetic field. Since the fields are given via Eqs.(3.5), we see that the geometric equation (3.9b) must be solved with three conditions ~un (~r)|tangent = 0

on the boundary ∂V,

(3.10a)

rot ~un (~r)|normal = 0

on the boundary ∂V,

(3.10b)

in all volume V.

(3.10c)

div ~un (~r) = 0

The third condition follows from the Coulomb gauge (3.6) and must be satisfied within all volume of the cavity. It can be shown that relations (3.10a) and (3.10c) imply that the electric field should vanish on the cavity walls. Hence, we can say that ~un |wall = 0. We will not solve Eq.(3.9b) with the above given conditions. We refer to mathematical handbooks, and we will S.Kryszewski

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only state that such a problem can be solved once the shape of the cavity is given. Moreover, the obtained cavity modes ~un (~r) can be shown to characterize the cavity in a unique manner. Such modes are called normal modes of the cavity. Hence, normal modes fully characterize the geometry of the problem. The dynamical behavior of the fields is thus described by the amplitudes qn (t). Determination of amplitudes automatically determines the fields, since the normal (geometrical) modes are fixed once the cavity shape is given. Therefore we proceed to analyze the field amplitudes qn (t).

3.2.2

Energy of the field in a cavity

The field amplitudes qn (t) are best discussed via the field energy. We recall, that in classical electrodynamics the energy of the field in cavity (in vacuum) is given by the integral Z h i 0 ~ 2 + c2 B ~2 E = d~r E (3.11) 2 V Inserting the fields according to relations (3.5) we get " Z X 1 E = d~r q˙m (t)q˙n (t) ~um (~r) · ~un (~r) 2 V n,m

# c2 X qm (t)qn (t) rot ~um (~r) · rot ~un (~r) . + 2 m,n

We proceed with the analysis of the second integral, which we denote as Z Jmn = d~r rot ~um (~r) · rot ~un (~r),

(3.12)

(3.13)

V

and which is obviously symmetric, that is J mn = Jnm . We will now transform this integral so as to make use of the boundary conditions imposed on cavity modes. We use the vector analysis identities (rot rot = grad div − ∇2 ), ~ ~ ~ ~ rot a · rot b = div a × rot b . + ~a · rot rot ~b (3.14) = div ~a × rot ~b . + ~a · grad div ~b − ~a · ∇2 ~b, which allows us to rewrite the integral J mn Z Jmn = d~r div (~um × rot ~un ) + ~um · grad div ~un − ~um · ∇2 ~un .

(3.15)

V

Firstly, we note that the second term in (3.14) does not contribute, because the fields are transverse: div ~un = 0, see (3.10c). Secondly, the functions ~un satisfy Helmholtz equation (3.9b). Hence, from (3.150 we obtain Z ωn2 Jmn = d~r div (~um × rot ~un ) + 2 ~um · ~un . (3.16) c V The symmetry of the integral Jmn gives Jmn − Jnm = 0, and by subtraction of the equations of the type of (3.16) we obtain Z 2 − ω2 Z ωm n d~r [ div (~um × rot ~un ) − div (~un × rot ~um )] = d~r ~um · ~un , (3.17) c2 V V 32

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Now, we consider the integral of the term similar to the ones appearing in the above formula. First we use the Gauss theorem and transform the volume integral into the surface one. Then we argue that, for any m and n, the resulting integral vanishes, that is we have Z I d~r div (~um × rot ~un ) = d~S · (~um × rot ~un ) = 0. (3.18) V

∂V

The scalar product in the surface integral ”selects” the component of ~um ×rot ~un along the vector d~S which is perpendicular to the surface. This (perpendicular) part of the vector ~um × rot ~un arises from the components which are parallel (tangent) to the surface, as it follows from the properties of the vector product of two vectors. But the tangent component of rot ~un vanishes [see the boundary conditions (3.10a)]. Hence, the normal component of the vector product vanishes. The integrand is thus zero, and the relation (3.18) is proved. Left-hand side of (3.17) is zero, and we arrive at the conclusion that 2 − ω2 Z ωm n d~r ~um · ~un = 0. (3.19) c2 V Since parameters ωn = 6 ωm , we may write Z d~r ~um · ~un = δmn ,

(3.20)

V

because orthogonal functions can be normalized. This fact may be explained in a different manner. Namely, we can refer to the wave equation (3.9b) and since the laplacian ∇ 2 is a Hermitian operator, its eigenfunctions belonging to different eigenvalues should be orthonormal, which is reflected by (3.20). We complete our discussion by returning to the integral (3.16) in which we use relations (3.18) and (3.20) to get Z ω2 Jmn = Jnm = d~r rot ~um (~r) · rot ~un (~r) = 2n δmn . (3.21) c V Inserting the obtained results (3.20) and (3.21) into the expression (3.12) for the energy of the field in the cavity, we obtain E

= =

1 X 2 c2 X ω2 q˙n (t) + qm (t)qn (t) 2n δmn 2 n 2 m,n c 1 X 2 q˙n (t) + ωn2 q 2 (t) . 2 n

(3.22)

This result confirms that all dynamical information on the fields is included in the field amplitudes qn (t). The obtained result clearly reminds of the harmonic oscillator. We may interpret (3.22) as the sum of the energies of the so-called field oscillators. This analogy will be very important, therefore, we will devote some attention to the harmonic oscillator.

3.2.3

Expansion in normal variables

To proceed further we introduce new, in this case complex, time dependent functions which reexpress the amplitudes qn and associated momenta pn as r r ~ ~ ωn ∗ qn = ( an + an ) , pn = q˙n = − i ( an − a∗n ) , (3.23) 2ωn 2 S.Kryszewski

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where Planck’s constant is introduced for future convenience. These relations are easily inverted to give an =

1 √ ( ωn qn + ipn ) , 2~ωn

1 a∗n = √ ( ωn qn − ipn ) . 2~ωn

(3.24)

The functions an and a∗n are called ”normal variables” of the fields which, as follows from (3.4) and (3.5) after substitution of (3.23), are now of the form r X ~ ∗ ~ A⊥ (~r, t) = an + an ~un (~r). (3.25a) 2 0 ωn n r X ~ ωn ~ ⊥ (~r, t) = i E an − a∗n ~un (~r), (3.25b) 20 n r X ~ ~ B(~r, t) = an + a∗n rot ~un (~r). (3.25c) 20 ωn n

Energy of the field, given in (3.22), can also be given in terms of the normal variables. It becomes h i X 1 X = ~ωn an a∗n + a∗n an ~ωn a∗n an , (3.26) E = 2 n n

where the second equality follows from the commutation of classical normal variables. We have already mentioned analogies to classical harmonic oscillator. The energy of the field expressed as in (3.26) by normal variables brings further associations, but this time with quantum-mechanical harmonic oscillator.

3.3

Field quantization in a cavity

3.3.1

Field oscillators – harmonic oscillator

We recall some basic facts about quantum mechanical harmonic oscillator. We consider onedimensional harmonic oscillator with unit mass, thus we write its Hamiltonian as 1 2 1 p + ω2 q2 (3.27) 2 2 The position and momentum operators satisfy the well-known canonical commutation relation [q, p] = i~. It is straightforward to find the Heisenberg equations of motion for both operators h i 1 (3.28a) i~ q˙ = q, H = q, p2 = i~p 2 h i 1 i~ p˙ = p, H = p, ω 2 q 2 = −i~ω 2 q (3.28b) 2 We see that quantum-mechanical Heisenberg equations yield the same equations as classical Hamilton equations Hosc =

q˙ = p =

∂H ∂p

p˙ = − ω 2 q = −

∂H ∂p

(3.29)

We know (see the Appendix 8) that the variables of the harmonic oscillator can be reexpressed in terms of the dimensionless annihilation and creation operators. Such a procedure is called second quantization. We assign the following operators to the position and momentum ones r r ~ ~ω † q = a ˆ + a ˆ , p = −i a ˆ − a ˆ† , (3.30) 2ω 2 34

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These relations are easily inverted to give a ˆ =

1 √ ( ω q + ip ) , 2~ω

1 a ˆ† = √ ( ω q − ip ) . 2~ω

(3.31)

So far, we easily see full analogy between quantum-mechanical annihilation and creation operators and normal variables. The essential difference follows from the canonical commutation relation for position and momentum operators. It implies the canonical commutation relation for annihilation and creation operators i h = 1. (3.32) a ˆ, a ˆ†

The Hamiltonian of the oscillator, rewritten in terms of annihilation and creation operators is of the form 1 † Hosc = ~ω a , (3.33) ˆ a ˆ + 2

where the term 1/2 is due to noncommutativity of a ˆ and a ˆ†. It is also important to note, than the states of the harmonic oscillator (ie., eigenstates of the Hamiltonian) are denoted by | n i with n = 0, 1, 2, , . . . . . .. So we have 1 Hosc | n i = ~ω n + |ni (3.34) 2 The state with n = 0 is called the vacuum state and it has the property a ˆ | 0 i = 0.

(3.35)

It is useful to remind that given the vacuum state, we can construct all states | n i by successive application of the creation operator (ˆ a † )n |ni = √ | 0 i. n!

(3.36)

Obviously, we can also construct the wave functions of the oscillator. For example, in the position representation we can find the eigenfunctions ψ n (q) = h q | n i. We refer the reader to the Appendix 8. We can also easily derive the Heisenberg equations of motion for annihilation and creation operators. This can be done directly by the differentiation of the definitions (3.31) or in a standard way. We get i h i h i 1 h a ˆ˙ = a ˆ, Hosc = − iω a ˆ, a ˆ† a ˆ = − iω a ˆ, a ˆ† a ˆ = − iωˆ a, (3.37a) i~ i h i h i 1 h † ˆ˙a† = a ˆ , Hosc = − iω a ˆ† , a ˆ† a ˆ = − iωˆ a† a ˆ, a ˆ = iωˆ a† . (3.37b) i~ This summarizes all the information on the quantum mechanical harmonic oscillator. This is relevant for the context of electromagnetic fields in the cavity.

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Field quantization

We proceed with the intuitively simple quantization of the electromagnetic field in the cavity. With each of the field oscillators we associate corresponding annihilation and creation operators which satisfy the commutation relation h i a ˆm , a ˆ†n (3.38) = δmn ,

with all other commutators vanishing. This commutation relation reflects the independence of the field modes. Quantization of the fields consists in replacing the normal variables by corresponding operators a ˆ and a ˆ† . Adopting such an equivalence we now have the quantized fields in the form identical to (3.25) only with normal variables replaced by annihilation and creation operators. We thus obtain the fields as r X ~ ~ ⊥ (~r, t) = A a ˆn + a ˆ†n ~un (~r). (3.39a) 2 0 ωn n r X ~ ωn ~ E⊥ (~r, t) = i a ˆn − a ˆ†n ~un (~r), (3.39b) 20 n r X ~ ~ B(~r, t) = a ˆn + a ˆ†n rot ~un (~r) (3.39c) 20 ωn n

Thus, instead of classical functions describing the electromagnetic field we now have quantummechanical field operators. These operators do not commute, since the annihilation and creation operators do not commute. It is important to understand that the time dependence (or, dynamical behavior) of the fields is hidden in the annihilation and creation operators. This is evident, if we take into account that in the classical case, the dynamics was hidden in the field amplitudes q(t), as it can be seen from Eq.(3.22). Since the amplitudes are, in the quantum-mechanical case, replaced by annihilation and creation operators, they must account for the dynamics of the fields. Applying the same procedure to the field energy, we reexpress it in terms of the annihilation and creation operators. Thus, we replace the classical energy of the field by the quantum mechanical operator X 1 † Hf ield = ~ωn a ˆn a ˆn + . (3.40) 2 n The term 1/2, absent in the classical case, now follows from the noncommutativity of the annihilation and creation operators. To complete the field quantization we must specify the Hilbert space of the field eigenstates. We again employ the analogy with the harmonic oscillator. First we define the vacuum state by the requirement a ˆn | Ω i = 0,

for any mode n.

(3.41)

Since the field modes are independent we construct other eigenstates of the field as a tensor product | n(1) , n(2) , . . . , n(k) , . . . i = | n(1) i ⊗ | n(2) i ⊗ . . . ⊗ | n(k) i ⊗ . . . O (ˆ a†n(i) )n(i) p = |Ωi n(i) !

(3.42)

(i)

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where numbers n(i) are nonnegative integers and index (i) numbers all modes. The states defined above are called states with n(i) photons in the mode number (i). We note, that in many practical cases the number of modes in the cavity is infinite. In such a case the expectation value of the energy of the vacuum state follows form (3.40) h Ω | Hf ield | Ω i =

X 1 ~ωn 2 n

(3.43)

and is infinite. We renormalize the energy by dropping the term 1/2. Thus the free field hamiltonian is taken to be X Hf ield = ~ωn a ˆ†n a ˆn . (3.44) n

Omission of the 1/2 term does not change the equation of motion for field operators. They follow again by the analogy to harmonic oscillator, and we have a ˆ˙ n = −iωˆ an , ˙ † † an . ˆan = iωˆ

(3.45a) (3.45b)

Hence the time dependence of the field annihilation and creation operators for the case of free field is simple. By direct integration we get a ˆn (t) = a ˆn (t0 ) e−iωn (t−t0 ) ,

a ˆ†n (t) = a ˆ†n (t0 ) eiωn (t−t0 ) ,

(3.46)

with a ˆn (t0 ) and a ˆ†n (t0 ) being the initial values. As we already stressed, the dynamics of the fields is ”hidden” in the annihilation and creation operators. If we insert relations (3.46) into the fields (3.39), we see that their time dependence is sinusoidal, as might be expected due to the oscillator analogy.

3.4 3.4.1

Plane wave representation Discussion of our results

As we already mentioned, the index n numbering field modes should be understood as a multiindex. We generalize our results by using an additional index α explicitly. Moreover, the wave equation (3.9b) is real, but in general, it allows complex valued solutions. Therefore, we generalize the fields (3.39) by writing r X ~ ∗ ~ r, t) = A(~ a ˆnα ~fnα (~r) + a ˆ†nα ~fnβ (~r) , (3.47a) 20 ωn n,α r X ~ ωn ∗ ~ ⊥ (~r, t) = i E a ˆnα ~fnα (~r) − a ˆ†nα ~fnα (~r) , (3.47b) 2 0 n,α r X ~ ∗ ~ r, t) = B(~ a ˆnα rot ~fnα (~r) + a ˆ†nα rot ~fnα (~r) (3.47c) 20 ωn n,α The functions ~fnα (~r) possess similar properties as initial functions ~un . They satisfy the boundary conditions, are orthonormal, etc. We stress that the form of the fields as above ensures hermiticity of the field operators, as it should be, because the fields are most certainly the physically observable quantities. S.Kryszewski

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The annihilation and creation operators satisfy the commutation relation which is an obvious generalization of (3.38), and which reads h i a ˆmα , a ˆ†nβ = δmn δαβ . (3.48)

Other expressions, as for example the free field Hamiltonian (3.44), are also suitably generalized in an obvious manner. It may be worth noting that our generalization can be viewed as a unitary transformation of the previous results. Unitary transformation preserve commutation relations, orthonormality etc., thus, there is no need to discuss this point in more detail. As we also mentioned, the specific form of cavity modes ~fnα depends on the geometry of the cavity. It is possible to discuss cavities of various shapes, symmetries, but we will focus attention on the simplest, but most widely used case – the plane waves.

3.4.2

Introduction of plane waves

We will now discuss the quantization of fields in a cubic box of volume V . The simplest set of orthonormal eigenfunctions of such a cavity consists of plane waves ~f~ (~x) = kλ

1 ~ √ ~e~kλ eik·~x , V

(3.49)

which are labelled by the wave vector ~k and by an additional index λ (which replace our multiindex n, α). The wave vector satisfies the dispersion relation ωk = kc,

with

k = | ~k |,

(3.50)

which is a consequence of the requirement that functions ~f~kλ (~x) satisfy the Helmholtz equation (3.9b). The vectors ~e~kλ called polarization vectors are in general complex and normalized to unity || ~e~kλ || = 1.

(3.51)

In order to discuss these functions we expand the vector potential in terms of them. Adjusting the summation indices according to the present situation, from (3.47a) we get s h i X ~ ~ ~ ∗ † ~ ⊥ (~x, t) = ~e~kλ a~kλ eik·~x + ~e~kλ A a~ e−ik·~x , (3.52) kλ 20 ωk V ~ kλ

Vector potential must satisfy the Coulomb gauge (3.1), that is we consider a transverse field ~ ≡A ~ ⊥ . This requirement applied to expansion (3.52) yields A ∗ ~ ~e~kλ · ~k = ~e~kλ · k = 0.

(3.53)

So, polarization vectors are orthogonal to wave vector, which explains their name. By analogy to classical electrodynamics we conclude that there are two vectors orthogonal to the given ~k. Hence the index λ takes on two possible values λ = 1, 2. Two real vectors ~e~kλ correspond to two linear polarizations. When polarization is circular polarization vectors are complex. Moreover, it is convenient to assume that two polarization vectors are mutually orthogonal, that is ~e~kλ · ~e~kµ = δλµ .

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(3.54)

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The boundary conditions imposed on the fields result in the requirements imposed upon functions ~f~kλ (~x). These requirements may be phrased as periodic boundary conditions ~f~ (~x + ~ei L) = ~f~ (~x), kλ kλ

(3.55)

where L is the length of the edge of the cubic cavity and ~ei is a unit vector directed along one of the three orthogonal edges. Imposing this condition on the plane waves (3.49) we arrive at the quantization of the wave vector ~k = 2π nx~ex + ny~ey + nz~ez , (3.56) L where nx , ny , nz are triples of integers. Hence numbering of the plane wave modes by wave vector ~k is fully equivalent to numbering by triples of integers. It is, however, important that the numbering of the modes, as discussed here, should not be mixed with photon numbers n ~kλ which are nonnegative integers and they number the states of the quantized field in the abstract Hilbert space of states of the type indicated in (3.42). Finally we note that (3.57) rot ~f~kλ (~x) = i ~k × ~f~kλ (~x) . which is necessary to express the magnetic induction according to Eq. (3.47c).

3.4.3

Quantization in cubic box of volume V

Having discussed the main features of the plane wave representation of modes in the cubic cavity we can express the fields in this representation. Although we have already considered the vector potential (see (3.52) we collect all the results which follow from Eqs.(3.47) and from the above given discussion. Vector potential quantized in the cubic box of volume V is s h i X ~ ~ ~ ∗ † ~ ⊥ (~x, t) = ~e~kλ a~kλ (t) eik·~x + ~e~kλ A a~ (t) e−ik·~x , (3.58a) kλ 20 ωk V ~ kλ i X Ek h ~ ~ ∗ † ~e~kλ a~kλ (t) eik·~x + ~e~kλ = a~ (t) e−ik·~x . (3.58b) kλ ωk ~ kλ

Transverse electric field (in Coulomb gauge) in the cubic box of volume V , r i X ~ωk h ~ ~ ∗ † ~ ~e~kλ a~kλ (t) eik·~x − ~e~kλ E⊥ (~x, t) = i a~ (t) e−ik·~x , kλ 20 V ~ kλ h i X ~ ~ ∗ † = i Ek ~e~kλ a~kλ (t) eik·~x − ~e~kλ a~ (t) e−ik·~x . kλ

(3.59a) (3.59b)

~ kλ

The corresponding magnetic field is of the form s h i X ~ i~ k·~ x ~k × ~e~ ~k × ~e ∗ a† (t) e−i~k·~x(3.60a) ~ x, t) = i B(~ (t) e − a , ~ ~ ~ kλ kλ kλ kλ 20 ωk V ~ kλ i X Ek h i~ k·~ x ~k × ~e~ ~k × ~e ∗ a† (t) e−i~k·~x . = i a (t) e − (3.60b) ~ ~ ~ kλ kλ kλ kλ ωk ~ kλ

In these equations we have introduced a useful and convenient notation r ~ωk Ek = . 20 V S.Kryszewski

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It is worth noting that operators of the electric field and magnetic induction are related as ~ = B

X 1 ~k × E ~~ kλ ωk

(3.62)

~ kλ

The Hamiltonian of the field in the plane wave representation has an obvious form X ~ ωk a~† a~kλ . HF = kλ

(3.63)

~ kλ

The eigenstates of this Hamiltonian are denoted similarly as in (3.42) the only difference being in numbering of the modes | { n~kλ } i = | . . . , n~kλ , . . . i =

O ~ kλ

(ˆ a†n~ )n~kλ q kλ |Ωi (n~kλ ) !

(3.64)

For sake of completeness let us write the commutation relation for field operators. It obviously follows from (3.38) and now is of the form h i a~kλ , a~† 0 0 (3.65) = δ~k~k0 δλλ0 , kλ

where the first Kronecker delta is understood as a product of three deltas with indices following from the allowed values of the wave vector as in Eq.(3.56). The Hamiltonian (3.63) and commutation relations are sufficient to derive the Heisenberg equations of motion for field operators i d ih a~kλ = − a~kλ , HF = − iωk a~kλ . dt ~

(3.66)

a~kλ (t) = a~kλ (t0 ) e−iωk (t−t0 ) .

(3.67)

This equation of motion obviously yields the solution

Since annihilation and creation operators determine the dynamics (time evolution) of the fields, we see that after inserting (3.67) into expansions (3.58)–(3.60) we obtain the fields as the combinations of plane waves.

3.4.4

Density of the modes

In many practical applications need to perform summations over allowed wave vectors and polarizations. We will consider this problem in more detail. Let us assume that we have to compute the sum XX ( . . . ), (3.68) ~ k

λ

of some function of summation variables. Summation is usually difficult, therefore we will show how to replace summation by integration. In the ~k-space the allowed wave vectors are specified by points with integer coordinates (see (3.56). The region of the volume of (2π/L) 3 around such a point is inaccessible for other wave vectors. Thus, the given volume determines the elementary cell in the ~k-space. Hence summation as in (3.68) corresponds to counting the number of points in ~k-space with weights

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specified by the summed function. The number of such points is equal to the volume in ~k space divided by the volume of the elementary cell. Thus we can write Z X XX X V Z 1 ~ ( ... ) = dk ( . . . ) = d~k ( . . . ) (3.69a) (2π/L)3 (2π)3 λ λ λ ~k Z ∞ Z V X 2 = k dk dΩ~k ( . . . ) (3.69b) 8π 3 0 λ

where we expressed the last integral in spherical coordinates. We will apply this result to some practically important cases. As a example, which can be easily adopted to practical problems, we consider the function Gλ (ωk ) which depends only on polarizations and the length of the wave vector k = ω k /c. Then in (3.69b) we can easily integrate over the angles. Changing the integration variable to ω k we get X X Z ∞ V Gλ (ωk ) −→ ωk2 dωk Gλ (ωk ) (3.70) 2π 2 c3 0 λ

~ k,λ

As the second example, we go further and take the function G(ω k ) which depends only on the field frequency. The sum over polarizations in (3.70) can be performed and yield a factor 2 because there are two polarizations. Hence, we obtain X Z ∞ X V G(ωk ) −→ ωk2 dωk Gλ (ωk ), (3.71) π 2 c3 0 λ

~ k,λ

which is sometimes written as Z ∞ X G(ωk ) −→ dωk V ρ(ωk ) G(ωk ).

(3.72)

0

~ k,λ

The introduced quantity ρ(ωk ) is given as ρ(ωk ) =

ωk2 , π 2 c3

(3.73)

an is called the density of the modes. It gives the number of modes of any polarization which lie within the frequency interval (ω k , ωk + dωk ) per unit volume of the cavity. The notion of mode density is useful when we have to evaluate the integrals of the type given in (3.72).

3.4.5

Quantization in free space

Fields given above in terms of the plane waves in a cubic box by Eqs.(3.58)–(3.60), are in fact Fourier series. There is no difficulty in transforming Fourier series in the bounded region into Fourier integrals in the whole space. The quantity (2π/L) 3 = 8π 3 /V determines the elementary cell in the ~k-space (see the discussion in the previous section). This √ implies that the functions orthonormalized to Kronecker delta in the box by a constant 1/ V and summed over discrete ~k will in the whole space be normalized by a factor 1/(2π) 3/2 to Dirac delta and integrated over whole space of wave vectors. This procedure allows us to rewrite the above discrete expansions of the fields into Fourier integrals over whole ~k-space.

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Vector potential quantized in free space with plane waves s h i 3 XZ ~ d k ~ ~ ∗ † ~ ⊥ (~x, t) = p ~e~kλ a~kλ (t) eik·~x + ~e~kλ A a~ (t) e−ik·~x , kλ 20 ωk (2π)3 λ Z i X Ek0 h d3 k ~ ~ ∗ † p ~e~kλ a~kλ (t) eik·~x + ~e~kλ = a~ (t) e−ik·~x . kλ (2π)3 ωk

(3.74)

λ

Electric field (in Coulomb gauge) in this case is r i XZ ~ωk h d3 k ~ ~ ∗ † ~ p ~e~kλ a~kλ (t) eik·~x − ~e~kλ E⊥ (~x, t) = i a~ (t) e−ik·~x , kλ 20 (2π)3 λ Z h i X d3 k ~ ~ ∗ † p = i Ek0 ~e~kλ a~kλ (t) eik·~x − ~e~kλ a~ (t) e−ik·~x . kλ (2π)3 λ

(3.75)

Corresponding magnetic field ~ x, t) = i B(~

XZ λ

= i

XZ λ

d3 k p (2π)3 d3 k p (2π)3

s

h ~ ~ ~k × ~e~ a~kλ (t) eik·~x kλ 20 ωk i ~ ∗ − ~k × ~e~kλ a~† (t) e−ik·~x , kλ h 0 Ek ~ ~k × ~e~ a~kλ (t)eik·~x kλ ωk i ~ ∗ − ~k × ~e~kλ a~† (t)e−ik·~x , kλ

We have also introduced a useful notation r ~ωk 0 Ek = . 20

(3.76)

(3.77)

The relation between the electric field and magnetic induction (3.62) still holds, but the sum is replaced by the integral. Similarly the commutation relation for annihilation and creation operators now reads h i (3.78) a~kλ , a~† 0 0 = δ(~k − ~k0 ) δλλ0 , kλ

The Hamiltonian of the field in the plane wave representation has now the integral form Z Hf ield = d~k ~ ωk a~† a~kλ . (3.79) kλ

The eigenstates of this Hamiltonian are denoted as in (3.64) only the tensor product is now performed over the continuous variable.

3.5

Equations of motion – Maxwell’s equations

~ = 0. This ensures that two homogeneous Maxwell’s We work in the Coulomb gauge, that is div A equations (for vacuum, in absence of sources) ~ = 0, div B

~ = 0, div E

(3.80)

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~ = −∂ B/∂t. ~ Let us first check that rot E We first calculate rotation of the electric field. Taking the field as in (3.59b), using relation (3.57) and its complex conjugate, we get h i X i~ k·~ x ~k × ~e~ ~k × ~e ∗ a† e−i~k·~x . ~ ⊥ (~x, t) = − rot E Ek a e + (3.81) ~ ~ ~ kλ kλ kλ kλ

~ kλ

~ we take Eq.(3.60) and we see that the On the other hand, computing the time derivative of B only time dependence in the right-hand side may enter via annihilation and creation operators. Then using Eq.(3.66) and its hermitian conjugate we obtain i X Ek h ~ i~k·~x ∂B † ∗ −i~ k·~ x ~k × ~e~ ~ ~ = i −i ω a e − k × e i ω a e , k ~ k ~ ~ kλ kλ kλ kλ ∂t ωk ~ kλ h i X † i~ k·~ x ∗ −i~ k·~ x ~k × ~e~ ~ ~ = a a . (3.82) Ek e + k × e e ~ ~ ~ kλ kλ kλ kλ

~ kλ

~ ⊥ = −∂ B/∂t ~ Comparing rhs of the last two equations we see that the Maxwell’s equation rot E is indeed satisfied. Thus it remains to check the fourth Maxwell’s equation, namely ~⊥ ∂E ~ = c2 rot B ∂t

(3.83)

We compute left-hand side by differentiating Eq.(3.59) over time and taking into account relations (3.66). We get, similarly as in (3.82): h i X Ek ~ ⊥ (~x, t) ∂E ~ ~ ∗ † = c2 k 2 ~e~kλ a~kλ eik·~x + ~e~kλ a~ e−ik·~x , kλ ∂t ωk

(3.84)

~ kλ

where we have used the dispersion relation: ω k = ck. Now we proceed to compute the rotation of the magnetic induction. h i X Ek i~ k·~ x ~k × ~e~ ~k × ~e ∗ a† e−i~k·~x , ~ x, t) = i rot B(~ rot e − a (3.85) ~ ~ ~ kλ kλ kλ kλ ωk ~ kλ

i h ~k × ~e~ ei~k·~x . So we compute it. To find the rotation we need the expression rot kλ h i h i i~ k·~ x ~k × ~e~ ei~k·~x ~ ~ rot = ε ∂ k × e e abc b ~ kλ kλ c a h i ~ = εabc εcmn ∂b km ~e~kλ n eik·~x ~ = δam δbn − δan δbm i km ~e~kλ n kb eik·~x h i ~ = i ka ~k · ~e~kλ − ~e~kλ a ~k · ~k eik·~x .

(3.86)

Since polarization vectors and wave vector are orthogonal, the first term vanishes, and we finally obtain h i ~ ~k × ~e~ ei~k·~x rot = −i ~e~kλ k 2 eik·~x (3.87) kλ a

We use the obtained relation and its complex conjugate in (3.85), this yields rotation of the magnetic induction h i X Ek ~ ~ ∗ † ~ x, t) = rot B(~ k 2 ~e~kλ a~kλ eik·~x + ~e~kλ a~ e−ik·~x (3.88) kλ ωk ~ kλ

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Comparing Eqs.(3.84) and the last one we see that the Maxwell’s equation (3.83) is indeed satisfied. We conclude this section by stating that the Maxwell’s equations are satisfied by the quantized field (in the cubic cavity). Checking that this is so also in a general case is much more tedious, but nevertheless can be done along the same lines. On the other hand Maxwell’s equations are equivalent to Heisenberg equations of motion for field operators. This follows, since we used (3.66) in the derivation, and the latter equations are just the Heisenberg ones for annihilation and creation operators.

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Chapter 4

States of quantized electromagnetic fields Quantization of the electromagnetic field has led us to the so-called Fock space, that is to the states | { n~kλ } i = | . . . , n~kλ , . . . i =

O ~kλ

(ˆ a†n~ )n~kλ q kλ |Ωi (n~kλ ) !

(4.1)

where | Ω i is a vacuum state specified by a sequence of zeroes – no photons in any of the allowed modes, which are numbered by the wave vector ~k and polarization index λ = 1, 2. Each of such states is specified by a sequence of nonnegative integers. Obviously, these photon number states are the eigenstates of the Hamiltonian H F given in (3.63).

4.1

Introduction and general discussion

4.1.1

Vacuum state

It is natural to consider the vacuum state | Ω i as the first one. Since vacuum state can be defined by the relation a~kλ | Ω i = 0 or h Ω |a~† = 0, we easily see that the vacuum expectation values of kλ ~ (3.58a), electric field E ~ ⊥ and magnetic field B ~ (3.60a) all vanish the vector potential A⊥ ~ ⊥ | Ω i = 0, hΩ|A

~ ⊥ | Ω i = 0, hΩ|E

~ | Ω i = 0. hΩ|B

(4.2)

On the other hand, expectation values of the intensities, that is of the squares of the fields do not vanish in the vacuum state. Let us compute the expectation value of the square of the electric field (which is a Hermitian operator). Using (3.59a) we get X X ~√ωk ωk0 † i~ k·~ x ∗ −i~ k·~ x ~2 |Ωi = − ~ ~ hΩ|E h Ω | e a ˆ e − e a ˆ e ~ ⊥ ~ kλ ~ kλ kλ ~ kλ 20 V ~ kλ ~ k 0 λ0 ~0 ~0 ~e~k0 λ0 a ˆ~k0 λ0 eik ·~x − ~e~∗k0 λ0 a ˆ~† 0 0 e−ik ·~x | Ω i. (4.3) kλ

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~ ⊥ are both taken at the same space-time point (~x, t). By straightforWe stress that the fields E ward multiplication we get n X X ~√ωk ωk0 ~ ~0 ~2 |Ωi = − ~e~kλ · ~e~k0 λ0 a hΩ|E h Ω | ˆ~kλ a ˆ~k0 λ0 ei(k+k )·~x ⊥ 20 V ~ kλ ~ k 0 λ0

− ~e~kλ · ~e~∗k0 λ0 a ˆ~kλ a ˆ~† 0

k λ0

~ ~0

ei(k−k )·~x ~ ~0

− ~e~∗kλ · ~e~k0 λ0 a ˆ~† a ˆ~k0 λ0 e−i(k−k )·~x +

~e~∗kλ

· ~e~∗k0 λ0

kλ a ˆ~† kλ

a ˆ~† 0

k λ0

o ~ ~0 e−i(k+k )·~x | Ω i.

(4.4)

The first term vanishes, since each of the annihilation operators acting on | Ω i yields zero. The same applies to the last (fourth) term (only it acts on the right, that is on h Ω |). So the nonzero contribution is at most due to two terms only, and we get n X X ~√ωk ωk0 ~ ~0 2 ~ h Ω | E⊥ | Ω i = h Ω | ~e~kλ · ~e~∗k0 λ0 a ˆ~kλ a ˆ~† 0 0 ei(k−k )·~x kλ 20 V ~ kλ ~ k 0 λ0 o ~ ~0 (4.5) + ~e~∗kλ · ~e~k0 λ0 a ˆ~† a ˆ~k0 λ0 e−i(k−k )·~x | Ω i. kλ

The diagonal terms behave differently than off-diagonal. Let us discuss the latter ones first. If ~k 6= ~k0 and/or λ 6= λ0 then operators a ˆ~kλ and a ˆ~† 0 0 commute. So we can move annihilation kλ operators to the right and then these operators act on vacuum state giving zeroes. The conclusion is that all off-diagonal terms vanish, do not contribute. Non-zero contribution may arise only due to diagonal terms with ~k = ~k0 and λ = λ0 . The double sum reduces to a single one. Thus (4.5) reduces to o n X ~ωk ~2 |Ωi = (4.6) hΩ|E h Ω | a~kλ a~† + a~† a~kλ | Ω i, ⊥ kλ kλ 20 V ~ kλ

because exponential factors give unity and so do the products of polarization vectors (see (3.54). Due to canonical commutation relation we finally arrive at an expression n o X ~ωk X ~ωk † ~2 |Ωi = hΩ|E h Ω | 2a a + 1 | Ω i, = . (4.7) ~ ⊥ ~ kλ kλ 20 V 20 V ~ kλ

~ kλ

~ specified in (3.60a). The only Very similar calculation can be performed for the magnetic field B difference consists in different vectorial factors. The argument about annihilation and creation operators in diagonal and off-diagonal terms remains unchanged. The sum reduces to a single one and we have to consider the vector products. It is easy to show that (~k × ~e~kλ )2 = ~k2 ~e~2kλ − (~k · ~e~kλ )2 = ~k2 ,

(4.8)

where the second term vanishes due to transversality of the field (3.53), so the vectorial products reduces to ~k2 . Therefore, the expectation value of the square of the magnetic field follows in exactly the same manner as that for electric field, yielding ~ 2 |Ωi = hΩ|B

X ~ kλ

~~k2 20 ωk V

=

X ~ kλ

~ωk , 20 V c2

(4.9)

because (see (3.50)) we have the dispersion relation ω k = |~k| c. Due to dispersion relation, both ~ 2 | Ω i and h Ω | B ~ 2 | Ω i, in principle, diverge with growing ~k. quantities h Ω | E ⊥ 46

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4. States of quantized electromagnetic fields

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In our considerations we use nonrelativistic approach, so we do not allow for creation or destruction of material particles. Hence the energy range in which we work must be restricted to energies less than me c2 , where me is the electron rest mass. Or, in other words, nonrelativistic approach becomes invalid for frequencies approaching ω M = me c2 /~. Hence we can limit the energies by introducing the frequency cut-off equal to ω M . Adopting such a limit we see that our expressions (4.7) and (4.9) contain summations over all modes but concern the functions of frequency only. Thus we can use the summation prescription (3.72) and we can expressed the obtained expectation values as Z ωM ω 2 ~ω 2 2 2 ~ ~ h Ω | E⊥ | Ω i = c h Ω | B | Ω i = dω 2 3 . (4.10) π c 20 0 4 . Since the averages (4.2) vanish, the correHence, the these expectation values diverge as ω M sponding variances are equal to the expectation values (4.10), so the variances are also divergent 4 . This is typical (purely quantum) problem with vacuum fields. The procedure of renoras ωM malization is aimed at removal of the divergencies, but it is a subject which we will not consider here.

4.1.2

Photon number states

The states (4.1) introduced previously are called photon number states. So they are the eigenP states of the hamiltonian (3.63) with eigenvalues E ~kλ = ~kλ ~ωk n~kλ . It is easy to argue that when the field is in the photon number state the expectation values of the field operators (3.58a)– (3.60a) are ~ | {n~ } i = h {n~ } | E ~ ⊥ | {n~ } i = h {n~ } | B ~ | {n~ } i = 0, h {n~kλ } | A kλ kλ kλ kλ kλ

(4.11)

which follows directly from the fact that a ˆ ~kλ lowers and a~† raises the number of photons, while kλ states with different photon numbers (different eigenstates of H F ) are orthogonal. The expectation values of the squares of the fields can be computed in an exactly the same ~ 2 in the vacuum state was, up manner as for vacuum state. Computation for the average of E ⊥ to the first part of eq.(4.7), done in a quite a general manner. Thus, it is sufficient to replace | Ω i by | {n~kλ } i, and we get n o X ~ωk † ~ 2 | {n~ } i = h {n~kλ } | E h {n } | 2a a + 1 | {n~kλ } i, ~ ⊥ ~ kλ kλ kλ kλ ~ 20 V ~ kλ

X ~ωk = (2n~kλ + 1). 20 V

(4.12)

~ kλ

We note that terms such as a~kλ a~kλ do not contribute in this case due to orthonormality of the ~2 states with unequal photon numbers. Obviously the expression for the expectation value of B is the same as (4.12) only divided by an additional factor c 2 . We shall return to the discussion of this results in a more specific case of a single mode field. At present we will add only several comments. The variances of the fields follow immediately from two previous formulas. We obtain ~ ⊥) = σ 2 (E

X ~ωk ~ (2n~kλ + 1) = c2 σ 2 (B) 20 V

(4.13)

~ kλ

Certainly, we encounter here the same problems with divergencies, so again we can consider the cut-off frequency ωM . Finally, we note that photon number states (eigenstates of H F ) S.Kryszewski

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are stationary ones. Therefore, all expectation values are time independent. This is strictly nonclassical, because in the classical picture the fields are quantities oscillating in time. We can view the photon number states as states of well-defined amplitude of the oscillations, but with completely undetermined phase, hence the fields average out to zero as in (4.11).

4.1.3

Single mode field

It is possible to consider a multimode field, just as we have done in the previous section, that is a field in which many modes specified by (~k, λ) are occupied – many numbers n~k,λ are nonzero. However, the field may be viewed as a Fourier series, or linear combination of many modes. Thus, it is frequently sufficient to consider only one (single) mode, while the generalizations to many modes usually poses no difficulties. For these reason, and also for reason of simplicity, we will now consider only a single mode of the quantized electromagnetic field. Hence, we drop the indices denoting the modes and from (3.58a)–(3.60a) we have the fields given by single terms, as r h i ~ ~ ~ ~ ~e a ˆ eik·~x + a A⊥ (~x, t) = ˆ† e−ik·~x , (4.14a) 20 ωV r i h ~ω ~ ~ ~ ~e a (4.14b) E⊥ (~x, t) = i ˆ eik·~x − a ˆ† e−ik·~x , 2 V r 0 h i ~ ~ ~ ~ x, t) = i B(~ (~k × ~e) a ˆ eik·~x − a ˆ† e−ik·~x , (4.14c) 20 ωV where we assumed the polarization vector ~e to be real. The physical state of the field is specified by the n-photon state | n i. We note that the vacuum state | Ω i corresponds to n = 0. The results of previous section can be specified to fit the present needs. In particular, the expectation values for the field follow from (4.11) and in this special case they are ~ ⊥ |ni = hn|E ~⊥ |ni = hn|B ~ | n i = 0, hn|A

(4.15)

where n = 0, corresponding to vacuum, is also allowed. Similarly, for the expectation value of the square of the field, from (4.12) we get ~2 |ni = hn|E ⊥

~ω 2n + 1 20 V

(4.16)

~ 2 i/2 gives half of the energy density of the field. This is not unexpected. The expression 0 h E Since we use oscillator analogy, we see that result (4.16) is indeed proportional to energy density. However, for the field in the n-state the expectation value of the field amplitude is zero (see (4.15)). This may be explained by saying that photons can have any phases, so that the field averages out to zero. Energy is phase independent, hence the non-zero result (4.16). Before proceeding, let us recall the identifications (3.30), namely r r ~ ~ω a ˆ + a ˆ† , p = −i a ˆ − a ˆ† , (4.17) q = 2ω 2 which immediately yield expectation values in n-state h q in = h n | q | n i = 0,

48

h p in = h n | p | n i = 0.

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(4.18)

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The expectation values of the squares are also easy to compute, and we get 2 ~ ~ h q 2 in = h n | q 2 | n i = hn| a ˆ+a ˆ† | n i = (2n + 1) 2ω 2ω 2 ~ω ~ω hn| a ˆ−a ˆ† | n i = (2n + 1) h p 2 in = h n | p 2 | n i = − 2 2

49

(4.19a) (4.19b)

The quantum averages (4.18) vanish, so the above expectations are equal to variances, eg. σn2 (q) = h q 2 in , and similarly for p. Therefore the product of variances becomes σn2 (q) σn2 (p) =

~2 ~2 (2n + 1)2 ≥ 4 4

(4.20)

Since [ q, p ] = i~, the last inequality follows from Heisenberg uncertainty relation for noncommuting observables. We see that even for the vacuum state (n = 0) the product of variances satisfies the uncertainty principle, which for n ≥ 1 is satisfied as a ”real–sharp” inequality. Therefore, an important question arises: can we construct fields such, that the uncertainty principle is minimized ?

4.2

Coherent states (single mode)

Coherent states are the states which answer to the given question. We shall introduce these states in a formal manner and we will investigate their properties. The coherent state | z i is defined as the normalized eigenstate of the annihilation operator a ˆ| z i = z| z i,

z ∈ C,

h z | z i = 1.

(4.21)

Annihilation operator is not hermitian, so we do not a priori know whether states | z i and | ξ i are orthogonal.

4.2.1

Expansion in n-photon states

Fock states, that is n-photon states, are complete and orthonormal (they are eigenstates of ˆ). Thus, any state can be expanded as hermitian operator n ˆ=a ˆ†a |z i =

∞ X

n=0

| n ih n | z i =

∞ X

n=0

| n i Cn (z).

(4.22)

Cn (z) are probability amplitudes, that for the field in coherent state | z i we will find it in the n-photon state. Applying the annihilation operator to both sides we get ∞ ∞ X X √ √ a ˆ|z i = n | n − 1 i Cn (z) = n + 1 | n i Cn+1 (z), n=0

(4.23)

n=0

where in the second equality we have renumbered the series. On the other hand, from (4.22) we obtain a ˆ|z i = z|z i =

∞ X

n=0

z | n i Cn (z).

(4.24)

Comparing rhs of two last formulas we arrive at the recurrence relation z Cn+1 (z) = √ Cn (z), n+1 S.Kryszewski

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which easily gives the probability amplitude zn Cn (z) = √ C0 (z), n!

(4.26)

so it remains to compute C0 (z). This is done by invoking the normalization requirement. From expansion (4.22) after insertion of (4.26) we have ∞ X (z ∗ )m z n √ 1 = h z | z i = |C0 (z)| hm|ni n! m! m,n=0 2

= |C0 (z)|2

∞ X |z|2n

n=0

n!

2

= |C0 (z)|2 e|z| ,

(4.27)

where we used orthonormality of the n-states. Adopting zero phase we get C 0 (z) = exp(−|z|2 /2). Hence, the final form of the expansion of the coherent state | z i in the n-states becomes | z i = exp(−|z|2 /2)

∞ X zn √ | n i. n! n=0

(4.28)

We note that the vacuum state | Ω i is the coherent state corresponding to z = 0. The expansion coefficients give probability Pn (z) = |Cn |2 = exp(−|z|2 )

|z|2n , n!

(4.29)

which is the Poisson distribution with mean h n i z = |z|2 . Indeed, it is straightforward to check that the average number of photons for the field in the coherent state | z i is given as h n iz = h z | a ˆ† a ˆ|z i = ka ˆ| z i k2 = |z|2 ,

(4.30)

as it follows from the definition (4.21).

4.2.2

Scalar product h z | ξ i

Employing expansions (4.28) for two coherent states we can write |z|2 |ξ|2 h z | ξ i = exp − − 2 2

X ∞ (z ∗ )n ξ m √ h n | m i, m! n! m,n=0

and due to orthonormality of n-states we get |z|2 |ξ|2 ∗ h z | ξ i = exp − − +z ξ . 2 2

(4.31)

(4.32)

The obtained relation immediately entails |h z | ξ i|2 = exp −|z|2 − |ξ|2 + z ∗ ξ + zξ ∗

= exp −|z − ξ|2 .

(4.33)

The bigger the difference between two complex numbers z and ξ, two coherent states become ”more orthogonal”.

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4. States of quantized electromagnetic fields

Completeness of coherent states

Complex numbers which parameterize coherent states span a two-dimensional space with continuous variable. Thus it seems natural to investigate the operator Z Z ∞ X 2 (z ∗ )n z m 2 √ d z | z ih z | = d2 z e−|z| | m ih n | m! n! m,n=0 Z ∞ X | m ih n | 2 √ = d2 z e−|z| (z ∗ )n z m (4.34) m! n! m,n=0 Taking the polar coordinates in the complex plane, we transform the integral and we obtain Z Z ∞ Z 2π 2 −|z|2 ∗ n m m+n+1 −r 2 d z e (z ) z = dr r e dϕ ei(m−n)ϕ 0 0 Z ∞ m+n+1 −r 2 = 2π δmn dr r e = π δmn n! (4.35) 0

Using this result in the operator (4.34) we express it as Z

2

d z | z ih z | = π

∞ X

n=0

ˆ | n ih n | = π 1.

(4.36)

Hence, we arrived at the completeness relation for coherent states, which can be written as Z 1 ˆ d2 z | z ih z | = 1. (4.37) π Since coherent states are not orthogonal (see (4.32)) but complete, it is possible to expand one coherent state | α i in terms of all other ones. This means, that the coherent states constitute, the so called, overcomplete set. The obtained relation allows us to write for an arbitrary (single mode) state | ψ i of the radiation field Z 1 |ψ i = d2 z | z ih z | ψ i. (4.38) π It seems tempting to call h z | ψ i the wave function of state | ψ i in the coherent state representation. This, is, however, incorrect, because h z | ψ i is a function of two real variables which have the sense of phase space variables, so can be interpreted (considered) as position and momentum.

4.2.4

Coherent state as a displaced vacuum

Let us combine expression (4.28) with that for the n state, that is with (3.42). As a result we can write X ∞ |z|2 z n (ˆ a † )n |z|2 √ √ | z i = exp − | Ω i = exp − exp zˆ a† | Ω i (4.39) 2 2 n! n! n=0 ∗

Since there holds the relation a ˆ| Ω i = 0, then it is obvious that e −z aˆ | Ω i = | Ω i. Therefore, we can recast the above relation as |z|2 | z i = exp − exp zˆ a† exp (−z ∗ a ˆ) | Ω i. (4.40) 2 S.Kryszewski

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†

Using relation (2.27b), ie.: eαˆa+βˆa = eβˆa eαˆa eαβ/2 with α = −z ∗ and β = z we rewrite the above formula as | z i = exp zˆ a† − z ∗ a ˆ | Ω i = D(z)| Ω i, (4.41)

where we have introduced an operator, which we will call the displacement operator D(z) = exp zˆ a† − z ∗ a ˆ . The displacement operator is unitary, indeed we have h i D † (z) = exp z ∗ a ˆ − zˆ a† = exp −(zˆ a† − z ∗ a ˆ† ) = D(−z),

(4.42)

(4.43)

and it is evident that

D † (z) = D(−z) = D −1 (z),

(4.44)

which proves that it is unitary, while the operator acting on the vacuum in (4.39) does not possess such a property. Unitarity is the reason why we have introduced D(z). † † ˆ eαˆa+βˆa = a ˆ + β, which allows us to write Let us also recall relation (2.37), that is e −αˆa−βˆa a D † (z) a ˆ D(z) = ez

∗a ˆ −zˆ a†

a ˆ ezˆa

† −z ∗ a ˆ

= a ˆ + z,

(4.45)

which explains the name ”displacement” operator. This is also the reason why we call the coherent state (see (4.41)) ”the displaced vacuum” state. It may be worth discussing some analogy with quantum mechanical harmonic oscillator. The hamiltonian of the oscillator is quadratic in momentum and position. If we add an additional potential energy term (for example coupling of the charged oscillator with an external, uniform electric field) linear in position, it is then easy to show that the wave functions and energies will be displaced. This follows by expressing the hamiltonian in a canonical form: ax 2 + bx = a(x+b/2a)2 −b2 /4a. Similar considerations could also be done for linear shift in momentum. This is so, because position and momentum of the oscillator both enter the hamiltonian quadratically (they are canonically equivalent). The given arguments suggest that a displaced state of the oscillator is generated by an operator proportional to position (or momentum), that is by a combination of annihilation and creation operators. Requirement of unitarity then leads to the displacement operator D(z) as given in (4.43). So we see, that the discussed analogy could be used to define a coherent state via the relation (4.41), which could be then used to prove other properties of coherent states, including the fact that | z i is an eigenvector of the annihilation operator. Moreover, this analogy gives additional clarification to the notion of coherent state as a displaced vacuum state.

4.2.5

Minimalization of uncertainty

In Eqs.(4.19) we have computed the expectation values of operators q and p for the field in the n-photon state. Here, we shall repeat these calculations for the field in the coherent state | z i. First we compute the corresponding expectation values r r ~ ~ ∗ † h q iz = h z | a ˆ + a ˆ |z i = z +z (4.46a) 2ω 2ω r r ~ω † ~ω ∗ h p iz = h z | i a ˆ − a ˆ |z i = i z −z . (4.46b) 2 2 52

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These relations follow from definition of the coherent state (4.21) and its hermitian conjugate: h z |ˆ a† = z ∗ h z |. Next we proceed to find the expectation values of the squares q 2 and p2 . In the calculation we use the canonical commutation relation for annihilation and creation operators and we obtain 2 i ~ ~ h ∗ 2 h q 2 iz = h z | a ˆ + a ˆ† (z ) + 2|z|2 + z 2 + 1 |z i = (4.47a) 2ω 2ω h i 2 ~ω ~ω a ˆ† − a (z ∗ )2 − 2|z|2 + z 2 − 1 h p 2 iz = − h z | ˆ |z i = − (4.47b) 2 2 Variances follow immediately, we easily get the following expressions σz2 (q) = h z | q 2 | z i − h z | q | z i2 = σz2 (p) = h z | p2 | z i − h z | p | z i2 =

~ 2ω ~ω . 2

(4.48a) (4.48b)

Hence, the product of the variances, for the field in the coherent state | z i is given as σz2 (q)σz2 (p) =

~2 ~2 ≥ , 4 4

(4.49)

so indeed the uncertainty relation is satisfied, but it is minimized, the product of variances attains the minimum allowed value. Thus, we can say that coherent states minimize the uncertainty, and as such can be considered to be as close to classical states as it is allowed by the principles of quantum mechanics. From general course of quantum mechanics we, for example, know how to construct the minimum uncertainty wave packet. Replacing the averages of q and p by the expectation values in the coherent state, we may construct the coherent state wave packet.

4.2.6

Comments on electric field

The electric field for one mode is given by Eq.(4.14b). Its expectation value in the coherent state reads r ~ω ~ ~ ~ ~e z eik·~x − z ∗ e−ik·~x . h z | E⊥ | z i = i (4.50) 20 V We already know that this field corresponds to the state with minimum uncertainty and as we see it, is of the form of classical plane wave. This is another argument, why the coherent states are considered to be the closest to the classical ones. Let us analyze the variance of the photon number for the field in the coherent state. We have the obvious relations (in the second one we use commutation relation) h n iz = h z | a ˆ† a ˆ | z i = |z|2 2 h n 2 iz = h z | a ˆ† a ˆ |z i = hz |a ˆ† (ˆ a† a ˆ + 1)ˆ a | z i = |z|4 + |z|2 .

(4.51a) (4.51b)

Thus, the variance of the photon number in this case is given as σz2 (n) = h n2 iz − h n i2z = |z|2 = h n iz .

Relative fluctuations of the mean photon number can thus be estimated as p σz2 (n) 1 = p , h n iz h n iz S.Kryszewski

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(4.52)

(4.53)

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so it becomes very small when the field is strong (with large mean photon number h n i z = |z|2 ). So, if the mode contains (on average) many photons, the fluctuations of h n i z are small and the field approaches the classical one. We continue our discussion of the field in the coherent state and we compute the variance of the electric field intensity. The average is already given in Eq.(4.50) so we proceed to find the average of the square † −i~ k·~ x i~ k·~ x i~ k·~ x † −i~ k·~ x ~ 2 | z i = ~ω h z | a hz |E ˆ e − a ˆ e a ˆ e − a ˆ e |z i ⊥ 20 V ~ω ~ ~ = 2|z|2 + 1 − (z ∗ )2 e−2ik·~x − z 2 e2ik·~x . (4.54) 20 V Calculation of the variance is now easy, and we get ~2 |z i − hz |E ~ ⊥ | z i2 = hz |E ⊥

~ω 20 V

- 0,

~→0

(4.55)

because classical limit corresponds to ~ → 0. This limit for the variance is independent of the field intensity (its energy), in contrast to the n-photon states (see Eqs.(4.16) and (4.15)). This is an additional argument explaining why coherent states of the field are closely related to classical ones.

4.2.7

Time evolution of the coherent state

We still consider one mode field, hence its Hamiltonian is simply H F = ~ω a ˆ† a ˆ. Let us assume that the mode was initially in the coherent state | ψ(t0 ) i = | z i.

(4.56)

We want to find what happens with the mode when time t > t 0 . Since Hamiltonian is explicitly time independent, we can invoke general rules of quantum mechanics to write iHF (t − t0 ) | ψ(t0 ) i = exp − iωˆ a† a ˆ(t − t0 ) | z i. (4.57) | ψ(t) i = exp − ~ Expanding the coherent state | z i in n-photon states, as in (4.28) we obtain ∞ X z n −iωn(t−t0 ) √ | ψ(t) i = exp(−|z| /2) e | n i. n! n=0 2

(4.58)

Additional phase factor e−iω(t−t0 ) does not change the modulus of number z, thus we can write rhs of (4.58) as | ψ(t) i = | z(t) i = | z e−iω(t−t0 ) i, (4.59)

which still is a coherent state only with ξ = e −iω(t−t0 ) z, that is with time dependent phase. So, free evolution of the coherent state produces a new coherent state, or we may say equivalently, that an evolving coherent state remains coherent. Let us also note that this result may be written as (4.60) | ψ(t) i = | z(t) i = D z e−iω(t−t0 ) | Ω i, which agrees with the property (2.43) of the displacement operator.

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For some further discussion, let us denote z(t 0 ) = x0 + iy0 . Thus, the time evolution of z can be written as z(t) = x0 cos ω(t − t0 ) + y0 sin ω(t − t0 ) − ix0 sin ω(t − t0 ) + iy0 cos ω(t − t0 ) Then, the time dependent expectation values of phase space variables are r ~ h q(t) i = h z(t) | a ˆ+a ˆ† | z(t) i 2ω r r ~ ~ ∗ = z(t) + z (t) = 2 Re[z(t)] 2ω 2ω r i 2~ h = x0 cos ω(t − t0 ) + y0 sin ω(t − t0 ) . ω

(4.61)

(4.62)

And similarly, we obtain

r

~ω h p(t) i = −i h z(t) | a ˆ−a ˆ† | z(t) i 2 r r ~ω ~ω ∗ = −i z(t) − z (t) = 2 Im[z(t)] 2h 2 i √ = 2~ω − x0 sin ω(t − t0 ) + y0 cos ω(t − t0 ) .

(4.63)

These relations reproduce the evolution of classical harmonic oscillator. We conclude that a coherent state during its time (free) evolution remains coherent and follows the trajectory of the classical orbit in the phase space. The uncertainties (variances) σ z (q) and σz (p) remain constant and minimal.

4.3

Squeezed states (single mode)

4.3.1

Introduction and basic definition

We know that a quantized mode of the radiation field can be expressed via fields (4.14), that is via annihilation and creation operators. We will concentrate our attention on the electric field, r h i ~ω ~ ~ ~ ~e a E⊥ (~x, t) = i ˆ eik·~x − a ˆ† e−ik·~x . (4.64) 20 V Discussing the field oscillators and coherent states we have used the phase space operators q and p (position and momentum operators of the oscillator with unit mass). We will now introduce two operators 1 X = √ (ˆ a+a ˆ† ) 2

−i Y = √ (ˆ a−a ˆ† ), 2

(4.65)

which, by comparison with (4.17) may be viewed as dimensionless (rescaled) position and momentum. We shall discuss the importance of the absence of dimensionality later. At present we note that both operators X and Y are hermitian, and as such, can be considered observables. The relations (4.65) imply the inverse ones 1 a ˆ = √ (X + i Y ) 2

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(4.66)

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in the essentially the same manner as it was done for harmonic oscillator. Moreover, we note that operators X and Y satisfy an obvious commutation relation h i X, Y = i, (4.67)

which follows immediately from the properties of annihilation and creation operators. To clarify the physical meaning of the operators X and Y , let us consider the electric field (4.64). We recall that in the Heisenberg picture a ˆ(t) = a ˆe −iωt (as in (3.67)), so we can write r h i ~ω ~ ⊥ (~x, t) = i ~e a ˆ eiφ − a E ˆ† e−iφ , (4.68) 20 V

where, for brevity, we have denoted φ = ~k · ~r − ωt. Introducing relations (4.66), after simple manipulations we get r h i √ ~ω ~ ~e X sin φ + Y cos φ . E⊥ (~x, t) = − 2 (4.69) 20 V Hence, X and Y are amplitudes of the two quadratures of the electric field (4.64) having a phase difference of π/2. The commutation relation (4.67) implies that the variances of observables X and Y satisfy the uncertainty relation 1 σ 2 (X) σ 2 (Y ) ≥ . 4

(4.70)

Before continuing our discussion, it may be worth recalling that the variances of quadratures are: • for n − photon state (see(4.19)) : • for coherent state (see(4.48)) :

σ n2 (X) = σn2 (Y ) = σz2 (X) = σz2 (Y ) =

1 (2n + 1) 2

1 2

(4.71a) (4.71b)

These relations show that we reach the minimum uncertainty (the product of variances equal to 1/4) for the vacuum state | Ω i = | n = 0 i and for a coherent state | z i. For future reference, we also recall that according to Eq.(4.18), for n-photon state we have the expectation values h X in = h n | X | n i = 0,

h Y in = h n | Y | n i = 0.

(4.72)

For the coherent state, these expectations become (see (4.46) √ 1 h X iz = h z | X | z i = √ (z + z ∗ ) = 2 Re(z), 2 √ −i h Y iz = h z | Y | z i = √ (z − z ∗ ) = 2 Im(z), 2

(4.73a) (4.73b)

We are now in position to define a squeezed state of the radiation field as such, for which one of the variances of quadrature operators goes below the vacuum limit 1/2. That is: Definition: State | ψ i of the radiation field is a squeezed state if either σ 2 (X) < 1/2 or σ 2 (Y ) < 1/2. The product of the variances, however, must still satisfy the uncertainty relation (4.70). Sometimes an additional requirement is imposed, namely, that state | ψ i is also a minimum uncertainty state, but this is not necessary. We shall call state | ψ i a squeezed state when one of 56

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the quadratures has less fluctuations than in vacuum or coherent state. The squeezed state can be called ideal if it is also a state with minimum uncertainty (when in (4.70) we have equality). The fact that both quadratures have the same dimension is essential, because then scaling the dimensions does not change the ratio of the variances. The feature of quantum fluctuations below vacuum value is most important and it implies the quantum-mechanical nature of squeezed states.

4.3.2

Operator approach

Squeeze operator. Introduction The wave function of the ground state of the harmonic oscillator is (see (8.57)): mω 1/4 mωq 2 , ϕ0 (q) = exp − π~ 2~

(4.74)

so it becomes narrower when ω gets larger. The width of this function is governed by the strength of the potential energy which is quadratic in position. Frequency can be enlarged by the additional term in the hamiltonian which must be quadratic in position. Hence, a transformation generated by q 2 is expected to narrow the oscillator’s wave function. This suggests that an operator quadratic in annihilation and creation operators should ”narrow”, or squeeze the states of the radiation field. This reasoning is similar to that concerning the displacement operator leading to coherent states. Following this argument, we postulate that an operator 1 ∗ 2 1 † 2 S(ξ) = exp ξ a ˆ − ξ a ˆ , with ξ = ρ eiθ ∈ C, (4.75) 2 2 generates states which can be justified to be called squeezed states of the radiation field. The simplest candidate for a squeezed state would then be | 0, ξ i = S(ξ) | Ω i,

(4.76)

where | Ω i = | n = 0 i is a usual vacuum state. Obviously, we have to prove that the state (4.76) indeed is a squeezed state, according to the above given definition. Before we do so, we will study some basic properties of squeeze operator (4.75) which is also investigated in the Appendix B. We have shown that S(ξ) is unitary and that it transforms the annihilation and creation operators as S † (ξ) a ˆ S(ξ) = a ˆ cosh(ρ) − a ˆ † eiθ sinh(ρ), †

†

†

S (ξ) a ˆ S(ξ) = a ˆ cosh(ρ) − a ˆe

−iθ

sinh(ρ).

(4.77a) (4.77b)

Since squeezing of the states is defined by quadrature operators, we should devote some time to study them together with operator S(ξ). But before we do so, let us note that relations (4.77) immediately imply that h 0, ξ | a ˆ | 0, ξ i = h Ω | S † (ξ) a ˆS(ξ) | Ω i = hΩ| a ˆ cosh(ρ) − a ˆ † eiθ sinh(ρ) | Ω i = 0,

h 0, ξ | a ˆ† | 0, ξ i = h Ω | S † (ξ) a ˆ† S(ξ) | Ω i = hΩ| a ˆ† cosh(ρ) − a ˆ e−iθ sinh(ρ) | Ω i = 0,

(4.78a)

(4.78b)

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Similarity relations for quadratures To investigate the expectation values of quadrature operators in state | 0, ξ i we obviously need expressions of the type of S † (ξ)XS(ξ). Hence, we proceed to find such expressions. From the definition (4.65) of the quadratures and from (4.77) we have i 1 h S † (ξ) XS(ξ) = √ a ˆ cosh(ρ) − a ˆ † eiθ sinh(ρ) + a ˆ † cosh(ρ) − a ˆ e−iθ sinh(ρ) 2 1 † iθ = X cosh(ρ) − √ a ˆ e +a ˆ e−iθ sinh(ρ). (4.79) 2

Expressing the term in brackets by (4.66) we get √ ˆ e−iθ = 2 X cos θ + Y sin θ . a ˆ† eiθ + a

(4.80)

Combining the two last equations yields S † (ξ) XS(ξ) = X cosh(ρ) − cos θ sinh(ρ) − Y sin θ sinh(ρ).

(4.81)

In fully analogous manner we easily derive the similarity relation for the second quadrature operator (4.82) S † (ξ) Y S(ξ) = Y cosh(ρ) + cos θ sinh(ρ) − X sin θ sinh(ρ). These relations, though correct are neither illuminating nor convenient. Nevertheless, we can find the expectation values of the quadratures for the field in the state (4.76): h X i = h 0, ξ | X | 0, ξ i = h Ω | S † (ξ) XS(ξ) | Ω i = 0,

(4.83a)

†

h Y i = h 0, ξ | Y | 0, ξ i = h Ω | S (ξ) Y S(ξ) | Ω i = 0.

(4.83b)

This is so, because operator S † (ξ) XS(ξ) is expressed by (4.81), i.e., by a combination of a ˆ † and a ˆ . The expectation values of the latter vanish for the field in the vacuum state. Hence our result (4.83). We can also refer to Eqs.(4.78), quadratures are combinations of a ˆ and a ˆ † , so mentioned equations imply the obtained ones. Computation of the expectation values of the squares of the quadratures is greatly inconvenient. It appears that to consider the quadratures further it is useful to introduce some additional transformations. Rotated operators To simplify the above obtained relations it is convenient to introduce new operators ˆb = a ˆ e−iθ/2 ,

ˆb† = a ˆ† eiθ/2 ,

which, obviously, satisfy the commutation rule: h i ˆb, ˆb† = 1,

(4.84)

(4.85)

and which means that they are also annihilation and creation operators. Analyzing the similarity relation (4.77), we almost automatically obtain S † (ξ) ˆb S(ξ) = ˆb cosh(ρ) − ˆb† sinh(ρ), S † (ξ) ˆb† S(ξ) = ˆb† cosh(ρ) − ˆb sinh(ρ).

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Since ˆb and ˆb† are annihilation and creation operators we can construct new (hermitian) quadrature operators in an exactly the same manner as previously. Namely, we introduce ˆb + ˆb† , ˜ = √1 X 2 so the inverse relations read ˆb = √1 ˜ + iY˜ , X 2

−i ˆ Y˜ = √ b − ˆb† , 2 ˆb† = √1 ˜ − iY˜ . X 2

Moreover, we note that new quadratures satisfy the commutation relation h i ˜ Y˜ X, = i,

(4.87)

(4.87)

(4.88)

which is the same one as for old quadratures (4.67). This implies that new quadratures also satisfy the same uncertainty relation ˜ σ 2 (Y˜ ) ≥ 1 , σ 2 (X) 4

(4.89)

where the variances are taken in the arbitrary state of the field. This suggests that we can look for squeezed state in terms of new quadratures. Before we do so, let us look at the properties of new quadratures. First we seek the connection between old and new quadratures. We insert operators ˆb, ˆb† given according to (4.84) into Eqs.(4.87), secondly we express old annihilation and creation operators by old quadratures as in (4.65). Simple manipulation yields ˜ X Y˜

= X cos(θ/2) + Y sin(θ/2),

(4.90a)

= −X sin(θ/2) + Y cos(θ/2),

(4.90b)

which is a simple rotation by an angle θ/2. Rotation is an orthogonal transformation, so automatically unitary. Hence, it is not surprising that new quadratures satisfy commutation relation (4.88), the same as old quadratures. Moreover, rotational character of transformation (4.90) enables us to write an inverse one X Y

˜ cos(θ/2) − Y˜ sin(θ/2) = X ˜ sin(θ/2) + Y˜ cos(θ/2), = X

(4.91a) (4.91b)

which is a rotation by a negative angle −θ/2. Finally, having specified new quadrature operators, we look for the similarity relations. We ˜ and Y˜ . Then we use equations apply operator S † (ξ) on the left and S(ξ) on the right of X (4.87). Applying similarity relations (4.86) and using the properties of hyperbolic functions, we arrive at ˜ S(ξ) = √1 ˆb + ˆb† (cosh ρ − sinh ρ) = X ˜ e−ρ S † (ξ) X (4.92a) 2 −i ˆ ˆ† S † (ξ) Y˜ S(ξ) = √ (4.92b) b − b (cosh ρ + sinh ρ) = Y˜ eρ 2 It is possible to cross-check the consistency of the theory, for example by obtaining equations (4.81) and (4.82) from relations (4.92) by using suitable correspondence between new and old operators.

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Moreover, having specified new quadrature operators we look for their expectation values in n-photon and coherent states. Thus, from (4.90) and (4.72) we easily obtain ˜ | n i = h n | X cos(θ/2) + Y sin(θ/2) | n i = 0, ˜ in = h n | X (4.93a) hX h Y˜ in = h n | Y˜ | n i = h n | − X sin(θ/2) + Y cos(θ/2) | n i = 0. (4.93b) Similarly, for the field in the coherent state | z i, using (4.73) we obtain ˜ | z i = h z | X cos(θ/2) + Y sin(θ/2) | z i ˜ iz = h z | X hX √ = 2 cos(θ/2)Re(z) + sin(θ/2)Im(z) √ = 2 Re z e−iθ/2 h Y˜ iz = h z | Y˜ | z i = h z | − X sin(θ/2) + Y cos(θ/2) | z i √ = 2 − sin(θ/2)Re(z) + cos(θ/2)Im(z) √ = 2 Im z e−iθ/2

(4.94a)

(4.94b)

Having specified some auxiliary quantities, we proceed to investigate the single mode squeezed state (4.76).

4.3.3

Squeezed vacuum states

Calculation of expectation values for quadratures As we already mentioned the state (4.76), that is | 0, ξ i = S(ξ) | Ω i is suspected to be a squeezed state of the electromagnetic single mode field. We suggested the construction of this state by reasoning stemming from harmonic oscillator, expecting that squeezing occurs due to displacement quadratic in annihilation and creation operators. We recall that the state | ψ i is called squeezed if the variance of one of the quadratures goes below 1/2 – its vacuum value. So we have to check whether the discussed state | 0, ξ i satisfies this definition. We need to find the variances of the quadratures, and in Eqs.(4.81) and (4.82) we found that old quadratures are inconvenient. Therefore, we will consider the expectation values and variances ˜ and Y˜ . of new quadratures X First we compute simple averages – expectation values. From similarities (4.92) we obtain ˜ i = h 0, ξ | X ˜ | 0, ξ i = h Ω | S † (ξ)XS(ξ) ˜ ˜ | Ω i e−ρ , hX |Ωi = hΩ|X

(4.95)

and similarly for the second quadrature Y˜ . Next, we express new quadratures by the old ones, as in (4.90) and since the averages of old quadratures vanish in vacuum state, [see (4.72)], we finally obtain ˜ i = h 0, ξ | X ˜ | 0, ξ i = 0, hX

h Y˜ i = h 0, ξ | Y˜ | 0, ξ i = 0,

(4.96)

Computation of the expectation values of the squares of the quadratures is a bit lengthy, ˜ 2 . By definition, in but rather straightforward. We illustrate the procedure by considering X state | 0, ξ i we have ˜ 2 i = h 0, ξ | X ˜ 2 | 0, ξ i = h Ω | S † (ξ)XS(ξ) ˜ ˜ hX S † (ξ)XS(ξ) | Ω i,

(4.97)

due to unitarity of S(ξ).Next, from (4.92a) we get ˜2 i = h Ω | X ˜ 2 | Ω i e−2ρ , hX 60

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Expressing new quadratures by the old ones, according to (4.90) (remembering that quadratures do not commute) we have h ˜ 2 i = e−2ρ h Ω | X 2 | Ω i cos2 (θ/2) + h Ω | Y 2 | Ω i sin2 (θ/2) hX + h Ω | (XY + Y X) | Ω i sin(θ/2) cos(θ/2)] .

(4.99)

The first two matrix elements correspond to vacuum expectations, each of which due to (4.71a) gives 1/2. Thus, 2 −2ρ 1 ˜ hX i = e + h Ω | (XY + Y X) | Ω i sin(θ/2) cos(θ/2) . (4.100) 2

The remaining matrix element can be simplified with the aid of the commutation relation (4.88), which yields 1 2 −2ρ ˜ + i sin(θ/2) cos(θ/2) hX i = e 2 + 2 h Ω | Y X | Ω i sin(θ/2) cos(θ/2) . (4.101)

So we have to calculate the matrix element h Ω | Y X | Ω i. We invoke definitions (4.65) and we write i h Ω | Y X | Ω i = − h Ω | (ˆ a−a ˆ † )(ˆ a+a ˆ† ) | Ω i 2 h i i = − hΩ| a ˆ2 + 1 + a ˆ† a ˆ −a ˆ† a ˆ − (ˆ a† )2 | Ω i, (4.102) 2 where in the second term we used the commutation relation for a ˆ and a ˆ † . We easily see that operator terms do not contribute, and only term with unity survives, so we have i i hΩ|Y X |Ωi = − hΩ|1|Ωi = − . (4.103) 2 2 Finally, we see that two last terms in (4.101) cancel out and we have for the expectation value of the square of the first quadrature ˜ 2 | 0, ξ i = h Ω | S † (ξ) X ˜ 2 S(ξ) | Ω i = 1 e−2ρ . ˜ 2 i = h 0, ξ | X (4.104) hX 2 Calculation of the expectation value for Y˜ 2 goes along exactly the same lines, so we give the final result, which is of the form 1 2ρ h Y˜ 2 i = h 0, ξ | Y˜ 2 | 0, ξ i = h Ω | S † (ξ) Y˜ 2 S(ξ) | Ω i = e . (4.105) 2 Thus, summarizing our results we can say that for the state of the radiation field which is generated by the operator S(ξ) | 0, ξ i = S(ξ) | Ω i

(4.106)

the expectation values of quadratures vanish (see Eqs.(4.96)), while the expectation values of the squares of quadratures are given by (4.104) and (4.105). This means that the corresponding variances are 1 2ρ ˜ = 1 e−2ρ , σ 2 (X) σ 2 (Y˜ ) = e , (4.107) 2 2 so that the product of variance follows as ˜ σ 2 (Y˜ ) = 1 . σ 2 (X) (4.108) 4 ˜ is We see that the uncertainty relation (4.89) is minimalized and variance of quadrature X reduced with respect to its vacuum value, while the variance of Y˜ is correspondingly enhanced. S.Kryszewski

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Squeezed vacuum states The state | 0, ξ i = S(ξ) | Ω i is a minimum uncertainty state, for which variance of one quadrature is enhanced, while variance of the second one is reduced below the so-called vacuum limit (equal to 1/2). By definition, this state is a squeezed state, which we can call a vacuum squeezed state. We also note, that the expectation values of the quadratures themselves give zero (see (4.96), ~ ⊥ (~x, t) is linear which is characteristic for a vacuum state. As a consequence, since the field E in quadratures (see (4.69)), relations (4.96) imply that the expectation value of the field in the squeezed vacuum state vanishes ~ ⊥ i = h 0, ξ | E ~ ⊥ | 0, ξ i = h Ω | S † (ξ) E ~ ⊥ S(ξ) | Ω i = 0, hE

(4.109)

similarly as in the n-photon state, but differently from a coherent state. Next, let us compute the expectation value of the number of photons in the squeezed vacuum state. We then have ha ˆ† a ˆ i = h 0, ξ | a ˆ† a ˆ | 0, ξ i = h Ω | S † (ξ) a ˆ† a ˆ S(ξ) | Ω i = h Ω | S † (ξ) a ˆ† S(ξ)S † a ˆ S(ξ) | Ω i.

Using similarity relations (4.77) we further get ˆi = hΩ| a ˆ† a ˆ cosh2 (ρ) − (ˆ a† )2 eiθ sinh(ρ) cosh(ρ) ha ˆ† a

−ˆ a2 e−iθ sinh(ρ) cosh(ρ) + a ˆa ˆ † sinh2 (ρ) | Ω i.

(4.110)

(4.111)

It is obvious that three first terms do no contribute (give zeroes), while due to commutation rule, the term a ˆa ˆ† = a ˆ† a ˆ + 1 contributes unity, so we have ha ˆ† a ˆ i = h 0, ξ | a ˆ† a ˆ | 0, ξ i = sinh2 (ρ).

(4.112)

So, the squeezed vacuum state has nonzero expectation value of photon number. The averages of the quadratures, however, remain zero, as it is in a ”normal vacuum”. This explains why we keep the word ”vacuum”, calling the state | 0, ξ i – squeezed vacuum. The states | 0, ξ i are made out of ”real vacuum”, but may be arbitrarily intense. Time evolution of squeezed vacuum state The squeezed vacuum state | 0, ξ i evolves freely according to usual rules of quantum mechanics, that is i † | ψsq (t) i = exp − HF t | 0, ξ i = e−iωˆa aˆt S(ξ)| Ω i. (4.113) ~ Obviously we can write †

†

†

| ψsq (t) i = e−iωˆa aˆt S(ξ)eiωˆa aˆt e−iωˆa aˆt | Ω i = S ξe−2iωt | Ω i = | 0, ξe−2iωt i,

(4.114)

θ(t) = θ0 − 2ωt,

(4.115)

where in the second line we have used similarity relation(2.54) from the appendix, while in the last step we applied a definition of the squeezed vacuum state with time-shifted (time-dependent) argument. Since the number ξ = ρ eiθ , we may say that the angle θ is a time dependent function with

which corresponds to a clockwise rotation with angular frequency 2ω, We shall return to the discussion of this point in more geometric context in next sections. 62

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Fluctuations of photon number in squeezed vacuum state

To investigate the fluctuations of photon number we need the variance of photon number and thus we need h (ˆ a† a ˆ )2 i = h n 2 i h n2 i = h 0, ξ | (ˆ a† a ˆ)2 | 0, ξ i = h Ω | S † (ξ) a ˆ† a ˆ S(ξ) S † (ξ) a ˆ† a ˆ S(ξ) | Ω i. We can use the operator appearing in (4.111), so we can write h h n2 i = h Ω | a ˆ† a ˆ cosh2 (ρ) − (ˆ a† )2 eiθ sinh(ρ) cosh(ρ)

ˆa ˆ † sinh2 (ρ) −ˆ a2 e−iθ sinh(ρ) cosh(ρ) + a

h a ˆ† a ˆ cosh2 (ρ) − (ˆ a† )2 eiθ sinh(ρ) cosh(ρ)

(4.116)

i

i −ˆ a2 e−iθ sinh(ρ) cosh(ρ) + a ˆa ˆ † sinh2 (ρ) | Ω i.

(4.117)

In general, multiplication gives sixteen terms, but those which contain unequal numbers of a ˆ † ˆ as the rightmost operator also vanish. and a ˆ do not contribute. Moreover, the terms having a So, out of sixteen terms, the nonzero ones may arise only from three terms. We thus have h n2 i = cosh2 (ρ) sinh2 (ρ) h Ω | a ˆ† a ˆa ˆa ˆ† | Ω i

+ cosh2 (ρ) sinh2 (ρ) h Ω | a ˆ2 (ˆ a † )2 | Ω i ˆa ˆ† a ˆa ˆ† | Ω i. + sinh4 (ρ) h Ω | a

(4.118)

Calculations of the remaining three matrix elements is very simple if we take into account the canonical commutation relation a ˆa ˆ† = 1 + a ˆ† a ˆ and the fact that a ˆ| Ω i = 0. We just state the results hΩ|a ˆ† a ˆa ˆa ˆ† | Ω i = 0, 2

(4.119a)

† 2

(4.119b)

†

(4.119c)

a ) | Ω i = 2, h Ω | aˆ (ˆ †

hΩ|a ˆa ˆ a ˆa ˆ | Ω i = 1.

Inserting results (4.119) into (4.118) we obtain h n2 i = h (ˆ a† a ˆ)2 i = 2 cosh2 (ρ) sinh2 (ρ) + sinh4 (ρ) 1 = sinh4 (ρ) + sinh2 (2ρ). (4.120) 2 The variance of the photon number in the vacuum squeezed state follows from (4.120) and (4.112). It is 1 sinh2 (2ρ). (4.121) 2 Let us now discuss relative fluctuations, as we did it for the field in the coherent state as in (4.53). In the present case of squeezed vacuum we have p √ σ 2 (n) sinh(2ρ) cosh(ρ) =√ = 2 . (4.122) 2 hni sinh(ρ) 2 sinh (ρ) p Since h n i = sinh(ρ) according to (4.112), we can write the relative fluctuations as p σ 2 (n) √ cosh(ρ) 1 = 2 p >p . (4.123) hni hni hni σ 2 (n) = h n2 i − h n i2 =

Comparing this result with (4.53) we can say that relative photon number fluctuations in vacuum squeezed state are larger than in the coherent state, because cosh(ρ) ≥ 1. S.Kryszewski

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Similarity relations for operators a ˆ2 , (ˆ a † )2 , a ˆ† aˆ

4.3.5

In the main text (see (4.77) and in the appendix (2.53) we have already used the following similarity relations for annihilation and creation operators ˆ S(ξ) = a ˆ cosh(ρ) − a ˆ † eiθ sinh(ρ), S † (ξ) a †

†

†

S (ξ) a ˆ S(ξ) = a ˆ cosh(ρ) − a ˆe

−iθ

sinh(ρ).

(4.124a) (4.124b)

Since operator S(ξ) is a unitary one, it is straightforward to utilize these formulas to compute other similarity relations. Therefore, we will only give the results, without a detailed derivation, which in fact, reduces to performing some operator multiplication. We also note that, when necessary, we use the canonical commutation relation a ˆa ˆ† = 1 + a ˆ† a ˆ. The similarities which we will employ in further developments are as follows. For the square of the annihilation operator S † (ξ) a ˆ2 S(ξ) = S † (ξ) a ˆ S(ξ) S † (ξ) a ˆ S(ξ) = a ˆ2 cosh2 (ρ) − (2ˆ a† a ˆ + 1) eiθ sinh(ρ) cosh(ρ) + (ˆ a† )2 e2iθ sinh2 (ρ).

(4.125)

Analogous similarity relation for the square of the creation operator follows by hermitian conjugation and yields a† )2 S(ξ) = S † (ξ) a ˆ† S(ξ) S † (ξ) a ˆ† S(ξ) S † (ξ) (ˆ a† a ˆ + 1) e−iθ sinh(ρ) cosh(ρ) = (ˆ a† )2 cosh2 (ρ) − (2ˆ + (ˆ a)2 e−2iθ sinh2 (ρ).

(4.126)

Finally, for photon number operator we get S † (ξ) a ˆ† a ˆ S(ξ) = S † (ξ) a ˆ† S(ξ) S † (ξ) a ˆ S(ξ) ˆ cosh2 (ρ) + (ˆ a† a ˆ + 1) sinh2 (ρ) = a ˆ† a − [ˆ a2 e−iθ + (ˆ a† )2 eiθ ] sinh(ρ) cosh(ρ).

(4.127)

We shall need these relation in our next steps.

4.3.6

~ 2 in squeezed vacuum state Expectation value for E ⊥

Using notation introduced earlier in (4.68) we investigate the expectation value of the square of the electric field which is in the squeezed vacuum state. Thus, we write ~ 2 i = h 0, ξ | E ~ 2 | 0, ξ i hE ⊥ ⊥ 2 ~ω iφ = − h a ˆe −a ˆ† e−iφ i 20 V ~ω † = h 2ˆ aa ˆ+1−a ˆ2 e2iφ − (ˆ a† )2 e−2iφ i 20 V

from the definition (4.76) of the squeezed vacuum state we have n ~ 2 i = ~ω 1 + 2h Ω | S † (ξ) a hE ˆ† a ˆ S(ξ) | Ω i ⊥ 20 V − h Ω | S † (ξ) a ˆ2 S(ξ) | Ω i e2iφ o − h Ω | S † (ξ) (ˆ a† )2 S(ξ) | Ω i e−2iφ . 64

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We have to consider each of the three matrix elements using similarity relations (4.125)–(4.127). For the term involving the photon number operator we get n h Ω | S † (ξ) a ˆ† a ˆ S(ξ) | Ω i = h Ω | a ˆ† a ˆ cosh2 (ρ) + (ˆ a† a ˆ + 1) sinh2 (ρ) o − [ˆ a2 e−iθ + (ˆ a)2 eiθ ] sinh(ρ) cosh(ρ) | Ω i = sinh2 (ρ).

(4.130)

The next term includes the square of the annihilation operator. For this term we get n h Ω | S † (ξ) a ˆ2 S(ξ) | Ω i = h Ω | a ˆ2 cosh2 (ρ) − (2ˆ a† a ˆ + 1) eiθ sinh(ρ) cosh(ρ) o + (ˆ a† )2 e2iθ sinh2 (ρ) | Ω i = − eiθ sinh(ρ) cosh(ρ).

(4.131)

The last term with the square of creation operator follows by hermitian conjugation of the previous one h Ω | S † (ξ) (ˆ a† )2 S(ξ) | Ω i = − e−iθ sinh(ρ) cosh(ρ).

(4.132)

Inserting the obtained matrix elements into Eq.(4.129) we express the expectation value of the square of the field as ~2 i = hE ⊥

~ω 1 + 2 sinh2 (ρ) 20 V

+ eiθ sinh(ρ) cosh(ρ) e2iφ + e−iθ sinh(ρ) cosh(ρ) e−2iφ

=

~ω 1 + 2 sinh2 (ρ) + 2 sinh(ρ) cosh(ρ) cos(θ + 2φ) . 20 V

o

(4.133)

Due to well-known properties of the hyperbolic functions we have 2 sinh(ρ) cosh(ρ) = sinh(2ρ) and 1 + 2 sinh2 (ρ) = cosh(2ρ). Therefore we obtain h i ~ 2 (~x, t) i = ~ω cosh(2ρ) + sinh(2ρ) cos(θ + 2φ) . hE ⊥ 20 V i ~ω h −2ρ = e + sinh(2ρ) (1 + cos(θ + 2φ)) . (4.134) 20 V

First of all, we note that in the absence of squeezing, that is when ρ = 0, formula (4.134) exactly reproduces the expectation value (4.7) which is characteristic for the pure vacuum state | Ω i. Next, we recall that φ = ~k · ~x − ωt. Therefore, we conclude that there exist such space-time points in which the cosine in (4.134) equals minus unity. Then, we have ~2 i = hE ⊥

i ~ω h ~ω −2ρ cosh(2ρ) − sinh(2ρ) = e . 20 V 20 V

(4.135)

This relation may be interpreted as showing that in certain regions of space-time the field is squeezed, while in some other ones it is not necessarily the case. We will not present a detailed investigation of the temporal and/or spatial dependencies of squeezing. It suffices to realize, that squeezing may occur only in some regions of space-time, not necessarily everywhere.

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Squeezed coherent states

4.4.1

Introductory remarks

We have already investigated the squeezed vacuum states (4.76) defined as | 0, ξ i = S(ξ)| Ω i. We also know that a displacement operator D(z) = exp(zˆ a† − z∗a ˆ) when applied to the vacuum state | Ω i produces a coherent state | z i = D(z)| Ω i. Thus we see that we have two alternatives to construct new quantum-mechanical states of the radiation field, namely | α, ξ i = D(α)S(ξ)| Ω i,

(4.136a)

| z, ξ i = S(ξ)D(z)| Ω i.

(4.136b)

The question is whether these two possible definitions are equivalent or not. To answer this question we refer to Appendix. In eq.(2.57) we have shown that D(α) S(ξ) = S(ξ) D(z),

(4.137)

where ξ = ρ eiθ as previously, while α and z are complex numbers connected by the relations z = α cosh(ρ) + α∗ eiθ sinh(ρ)

(4.138a)

∗ iθ

α = z cosh(ρ) − z e sinh(ρ),

(4.138b)

which can be written equivalently, as Re z e−iθ/2 = eρ Re α e−iθ/2 , Im z e−iθ/2 = e−ρ Im α e−iθ/2 .

(4.139a) (4.139b)

These relations allow discussion of two possible definitions of the coherent squeezed states (4.136). Therefore we conclude that the states (4.136a) and (4.136b) are the same, provided relations (4.138) or, equivalently, (4.139) are met. We can say that squeezing of vacuum (done by S(ξ)) followed by displacement (performed by D(α)) leads to a squeezed coherent state | α, ξ i = D(α)S(ξ)| Ω i. This has the same effect as a displacement (by D(z)) of the vacuum state followed by squeezing S(ξ), provided the parameters α and z are connected by relations (4.138) or (4.139). So we can consider either the state | α, ξ i, or the | z, ξ i. Due to relations (4.138)– (4.139) we can easily transform results concerning | α, ξ i into those corresponding to | z, ξ i or vice versa. The choice between the two states is rather a matter of convenience, and less of physics. We note, that we have used the name ”squeezed coherent state” not really knowing whether the discussed states are indeed squeezed or not. Hence we proceed to investigate the properties of these states and to validate the name associated with them.

4.4.2

Expectation values for | z, ξ i

Here, we choose to investigate the expectation values for various operators, for the case when the radiation field is in the state | z, ξ i = S(ξ)D(z)| Ω i = S(ξ)| z i.

(4.140)

According to our discussion above, we shall then transform our results to describe the other state: | α, ξ i = D(α)S(ξ)| Ω i. The subsequent calculations are rather straightforward, so we only indicate main steps. 66

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First, we compute the expectation value of the annihilation operator. ha ˆ i = h z, ξ | a ˆ | z, ξ i = h Ω | D † (z)S † (ξ) a ˆ S(ξ)D(z) | Ω i = h z | S † (ξ) a ˆ S(ξ) | z i. (4.141) By similarity relation (4.77a) we get ha ˆi = hz | a ˆ cosh(ρ) − a ˆ† eiθ sinh(ρ) | z i = z cosh(ρ) − z ∗ eiθ sinh(ρ).

(4.142)

As the second expectation value, we consider the one for photon number operator. Thus, we calculate as follows ha ˆ† a ˆ i = h z, ξ | a ˆ† a ˆ | z, ξ i

= h Ω | D † (z)S † (ξ) a ˆ† a ˆ S(ξ)D(z) | Ω i

= h z | S † (ξ) a ˆ† S(ξ) S † (ξ) a ˆ S(ξ) | z i † −iθ = hz | a ˆ cosh(ρ) − a ˆ e sinh(ρ) a ˆ cosh(ρ) − a ˆ † eiθ sinh(ρ) | z i,

(4.143)

where we used the similarity relations (4.77). Performing the multiplications and using the commutation relation for annihilation and creation operators, we get h ha ˆ† a ˆi = hz | a ˆ† a ˆ cosh2 (ρ) + a ˆ† a ˆ + 1 sinh2 (ρ) i a† )2 eiθ sinh(ρ) cosh(ρ) | z i −ˆ a2 e−iθ sinh(ρ) cosh(ρ) − (ˆ = |z|2 cosh2 (ρ) + sinh2 (ρ) + sinh2 (ρ)

− z 2 e−iθ sinh(ρ) cosh(ρ) − (z ∗ )2 eiθ sinh(ρ) cosh(ρ) = z cosh(ρ) − z ∗ eiθ sinh(ρ) z ∗ cosh(ρ) − z e−iθ sinh(ρ) + sinh2 (ρ) 2 = z cosh(ρ) − z ∗ eiθ sinh(ρ) + sinh2 (ρ) (4.144)

˜ as the third one. In the We calculate the expectation value of the quadrature operator X similar manner we obtain ˜ i = h z, ξ | X ˜ | z, ξ i hX

˜ S(ξ)D(z) | Ω i = h Ω | D † (z)S † (ξ) X ˜ e−ρ | z i, = hz |X

(4.145)

where we used similarity relation (4.92a). Employing also the connection between old and new quadratures (4.90a) we have ˜ i = e−ρ h z | X cos(θ/2) + Y sin(θ/2) | z i. hX (4.146)

Next, taking into account expectation values (4.73) taken in the coherent state | z i we get i √ h ˜ i = e−ρ 2 Re(z) cos(θ/2) + Im(z) sin(θ/2) hX √ = e−ρ 2 Re z e−iθ/2 (4.147) The fourth expectation value to consider is the one for the second quadrature. In this case ˜ but we will present a little different approach. we can calculate along the same lines as for X, ˜ At first, we start as for X, and using (4.92b) we get h Y˜ i = h z, ξ | Y˜ | z, ξ i = h Ω | D † (z)S † (ξ) Y˜ S(ξ)D(z) | Ω i = h z | Y˜ eρ | z i,

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Next, we express the quadrature Y˜ via operators ˆb and ˆb† according to (4.87), then we use (4.84) to arrive at the expression with old annihilation and creation operators. This yields −i h Y˜ i = eρ h z | √ ˆb − ˆb† | z i 2 −i = √ eρ h z | a ˆ e−iθ/2 − a ˆ† eiθ/2 | z i, (4.149) 2 Then, by simple properties of usual coherent states we obtain −i h Y˜ i = eρ √ z e−iθ/2 − z ∗ eiθ/2 2 −i = eρ √ Im z e−iθ/2 2

(4.150)

In order to estimate the uncertainties we need also the expectation values of the squares of ˜ 2 , obtaining the quadratures. So we compute them. First we consider X ˜ 2 i = h z, ξ | X ˜ 2 | z, ξ i = h Ω | D † (z)S † (ξ) X ˜ 2 S(ξ)D(z) | Ω i hX ˜ S(ξ) S † (ξ) X ˜ S(ξ) | z i = e−2ρ h z | X ˜ 2 | z i, = h z | S † (ξ) X where we used similarity (4.92a). Expressing the quadrature via ˆb and ˆb† we get 2 ˜ 2 i = 1 e−ρ h z | ˆb + ˆb† | z i hX 2 1 −ρ = e h z | ˆb2 + (ˆb† )2 + 2ˆb†ˆb + 1 | z i 2

(4.151)

(4.152)

where we used the commutation relation (4.85). Going to old annihilation and creation operators as in (4.84) we get ˜ 2 i = 1 e−2ρ h z | a hX ˆ2 e−iθ + (ˆ a† )2 eiθ + 2 a ˆ† a ˆ + 1 |zi 2 1 −2ρ 2 −iθ e z e = + (z ∗ )2 eiθ + 2|z|2 + 1 2 2 1 −2ρ −iθ/2 ∗ iθ/2 e ze + (z ) e + 1 = 2 2 1 −2ρ = e 1 + 4 Re(z e−iθ/2 ) . (4.153) 2

Finally, we need the expectation value of the second quadrature. Since the calculation goes ˜ 2 we now give only the final result along exactly the same lines as for X h Y˜ 2 i = h z, ξ | Y˜ 2 | z, ξ i = h Ω | D † (z)S † (ξ) Y˜ 2 S(ξ)D(z) | Ω i 2 1 2ρ −iθ/2 = e 1 + 4 Im(z e ) . 2

(4.154)

Having computed the necessary expectation values we are in position to write down the corresponding variances. From (4.147) and (4.153) for the first quadrature, from (4.150) and (4.154) for the second one we get the variances ˜ = hX ˜2 i − h X ˜ i2 = e−2ρ /2, σ 2 (X) σ 2 (Y˜ ) = h Y˜ 2 i − h Y˜ i2 = e2ρ /2.

(4.155a) (4.155b)

To summarize, we have considered states of the radiation field which arise by first displacing and then squeezing of the vacuum state | z, ξ i = S(ξ)D(z)| Ω i. We may also view this state 68

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as squeezing of the coherent state | z, ξ i = S(ξ)| z i. The expectation values of the quadratures (4.147) and (4.150) can be written as √ √ ˜ i = 2 Re e−ρ z e−iθ/2 , hX h Y˜ i = 2 Im eρ z e−iθ/2 . (4.156)

Comparing these results with Eqs.(4.94) we can say that they indeed correspond to some specific coherent state. Moreover, we see that the considered state is a minimum uncertainty state, since the product of variances equals 1/4. On the other hand, one of the variances is reduced below 1/2 – the vacuum limit, and the second is correspondingly enhanced. Thus, we see that we indeed can call the state | z, ξ i = S(ξ)D(z)| Ω i a coherent squeezed state.

4.4.3

Expectation values for | α, ξ i

Now, we proceed to investigate the similar expectation values, but for the state of the radiation field defined as | α, ξ i = D(α)S(ξ)| Ω i,

(4.157)

which is first squeezed and then displaced. Due to relations (4.138) and (4.139) we have an easy connection between two types of squeezed coherent states. So in principle we can just rewrite previous results with suitable replacements of z by α, as indicated in (4.138) and (4.139). However, the direct computation may be of interest. The reason is as follows. Calculation with | z, ξ i = S(ξ)D(z)| Ω i = S(ξ)| z i, was simplified since in fact we have dealt with the coherent state | z i. We only had to consider similarity transformations induced by squeeze operator. In the present case we also need the similarities induced by a displacement operator D(α). Therefore, we first consider such similarities. First we recall already used similarity relations D † (α) a ˆ D(α) = a ˆ+α †

†

†

(4.158a) ∗

ˆ D(α) = a ˆ +α . D (α) a

(4.158b)

Other similarity relations for combinations of annihilation and creation operators follow from the two given above. For example, we evidently have for the photon number operator D † (α) a ˆ† a ˆ D(α) = a ˆ† a ˆ + αˆ a † + α∗ a ˆ + |α|2 .

(4.159)

Since old quadratures X and Y are combinations of a ˆ and a ˆ † (4.65) we easily find that 1 1 D † (α) X D(α) = √ D † (α)(ˆ a+a ˆ † )D(α) = √ a ˆ+α+a ˆ † + α∗ 2 2 √ = X + 2 Re(α). (4.160) In completely analogous manner we get √ D † (α) Y D(α) = Y + 2 Im(α). Since new quadratures are linear combinations of the old ones (4.90) we obtain √ ˜ D(α) = X ˜ + 2 Re α e−iθ/2 , D † (α) X √ D † (α) Y˜ D(α) = Y˜ + 2 Im α e−iθ/2 . S.Kryszewski

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(4.161)

(4.162a) (4.162b)

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Having collected auxiliary relations we can proceed to calculations of various expectation values for the state | α, ξ i = D(α)S(ξ)| Ω i. As previously, we start with the annihilation operator. Using similarity relation (4.158a) we simply get ha ˆ i = h α, ξ | a ˆ | α, ξ i = h Ω | S † (ξ)D † (α) a ˆ D(α)S(ξ) | Ω i

= h Ω | S † (ξ)(ˆ a + α)S(ξ) | Ω i = α + h Ω | S † (ξ) a ˆ S(ξ) | Ω i = α.

(4.163)

The remaining matrix element vanished due to (4.78a) – it reproduces the corresponding expectation value for the squeezed vacuum. If we take into account the relation (4.138b) then we see that (4.163) reproduces exactly (4.142), as it should. The next expectation value concerns photon number operator. Employing (4.159) we get ha ˆ† a ˆ i = h α, ξ | a ˆ† a ˆ | α, ξ i = h Ω | S † (ξ)D † (α) a ˆ† a ˆ D(α)S(ξ) | Ω i ˆ† a ˆ + αˆ a † + α∗ a ˆ + |α|2 S(ξ) | Ω i = h Ω | S † (ξ) a = |α|2 + h Ω | S † (ξ) a ˆ† a ˆ S(ξ) | Ω i

(4.164)

because terms linear in annihilation and creation operators do not contribute, as it was the case with squeezed vacuum state [see (4.78). The last term is the same as for squeezed vacuum, hence from (4.112) we get ha ˆ† a ˆ i = |α|2 + h 0, ξ | a ˆ† a ˆ | 0, ξ i = |α|2 + sinh2 (ρ).

(4.165)

Noting that (4.138b) holds, we see that the above relation exactly reproduces (4.144), as it should. To study squeezed states we must investigate the expectation values of quadratures. So we proceed to do that. The expectation value of the first of the new quadratures is found by using similarity relation (4.162a) and it is ˜ i = h α, ξ | X ˜ | α, ξ i = h Ω | S † (ξ)D † (α) X ˜ D(α)S(ξ) | Ω i hX √ ˜ + 2 Re(α e−iθ/2 ) S(ξ) | Ω i = h Ω | S † (ξ) X √ ˜ S(ξ) | Ω i = 2 Re(α e−iθ/2 ) + h Ω | S † (ξ) X √ = 2 Re(α e−iθ/2 ),

(4.166)

where the last result follows from (4.93a) or from (4.96) for the squeezed vacuum state. Due to connection (4.139a) we see that the obtained formula (4.166) reproduces (4.147). In the exactly the same manner we obtain the expectation value of the second quadrature. The result is √ (4.167) h Y˜ i = h α, ξ | Y˜ | α, ξ i = 2 Im(α e−iθ/2 ), which in turn, reproduces (4.150) when we take into account (4.139b). Comparing the obtained expectation values with (4.94) we see that we indeed have the expectations for a coherent state. To estimate the variances we also need the expectation values of the squares of the quadratures. Hence, using the square of the similarity (4.162a) we get ˜ 2 i = h α, ξ | X ˜ 2 | α, ξ i = h Ω | S † (ξ)D † (α) X ˜ 2 D(α)S(ξ) | Ω i hX 2 √ ˜ + 2 Re(α e−iθ/2 ) S(ξ) | Ω i = h Ω | S † (ξ) X h i2 ˜ 2 S(ξ) | Ω i, = 2 Re(α e−iθ/2 ) + h Ω | S † (ξ) X 70

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(4.168)

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˜ does not contribute. The remaining matrix element because due to (4.96), the term linear in X is identical to the one for squeezed vacuum (4.104), so we obtain h i2 ˜ 2 | α, ξ i = 1 e−2ρ + 2 Re(α e−iθ/2 ) , ˜ 2 i = h α, ξ | X (4.169) hX 2 which, together with (4.139a) is clearly identical to (4.153), as it is expected to be. Computation of the expectation value of Y˜ 2 is evidently similar and it yields h i2 1 2ρ h Y˜ 2 i = h α, ξ | Y˜ 2 | α, ξ i = e + 2 Im(α e−iθ/2 ) , (4.170) 2

which coincides with (4.154), when we replace α by z according to (4.139b). Corresponding variances follow immediately from Eqs.(4.169), (4.166), (4.170) and (4.167). We easily obtain ˜ = hX ˜2 i − h X ˜ i2 = σ 2 (X) σ 2 (Y˜ ) = h Y˜ 2 i − h Y˜ i2 = We conclude that we can in fact | z, ξ i = S(ξ)D(z)| Ω i and | α, ξ i expectation values of quadratures squeezed states.

4.5

1 −2ρ (4.171a) e , 2 1 2ρ (4.171b) e , 2 repeat the comments given after Eqs.(4.156). Both states = D(α)S(ξ)| Ω i are indeed squeezed states and since the correspond to coherent state, they can be called coherent

Squeezed photon number states

4.5.1

New role of squeezed vacuum state

The vacuum state | Ω i is an eigenstate of the annihilation operator belonging to the eigenvalue zero. On the other hand, the squeeze operator S(ξ) is a unitary one. Therefore we can write 0=a ˆ| Ω i = a ˆS † (ξ)S(ξ)| Ω i = a ˆS † (ξ)| 0, ξ i,

(4.172)

where | 0, ξ i is the squeezed vacuum state defined in (4.76). The obvious conclusion from this relation is S(ξ)ˆ aS † (ξ)| 0, ξ i = 0.

(4.173)

The operators on the left do not have a typical form of the similarity relation, so we may ask a question: what kind of an operator is the one appearing in the lhs of (4.173). The answer is simple, if we notice that S(ξ) is unitary and S † (ξ) = S −1 (ξ) = S(−ξ). Thus, we can define the operator cˆ = S(ξ)ˆ aS † (ξ) = S † (−ξ)ˆ aS(−ξ).

(4.174)

Recalling similarity relation (4.77a) we note that −ξ = −ρ e iθ and that sinh is an odd function. Therefore we get cˆ = S † (−ξ)ˆ aS † (−ξ) = a ˆ cosh(ρ) + a ˆ † eiθ sinh(ρ),

(4.175a)

†

(4.175b)

cˆ

†

† †

†

= S (−ξ)ˆ a S (−ξ) = a ˆ cosh(ρ) + a ˆe

−iθ

sinh(ρ),

Definition of cˆ operators allows us to check the commutation relation h i h i cˆ, cˆ† = a ˆ cosh(ρ) + a ˆ † eiθ sinh(ρ), a ˆ† cosh(ρ) + a ˆ e−iθ sinh(ρ) = cosh2 (ρ) − sinh2 (ρ) = 1,

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which shows that operators cˆ and cˆ† are also annihilation and creation operators. Thus relation (4.173) is equivalent to cˆ| 0, ξ i = 0.

(4.177)

We can interpret this expression as follows. The vacuum squeezed state | 0, ξ i plays the role of the vacuum state for the annihilation operator cˆ. This clearly suggest the following definition of the new states of a single mode electromagnetic field | n, ξ i =

(ˆ c † )n √ | 0, ξ i, n!

(4.178)

and tempts us to call these states – squeezed photon number states. This, however, requires some discussion, we have to see if these states have the necessary properties. Obviously they are number states for operators cˆ and cˆ† , so we can write √ cˆ | n, ξ i = n | n − 1, ξ i (4.179a) √ † cˆ | n, ξ i = n + 1 | n + 1, ξ i (4.179b) cˆ† cˆ | n, ξ i = n | n, ξ i

(4.179c)

So our next steps should be aimed at investigation of the properties of the newly introduced quantum states of the field.

4.5.2

Squeezed photon number states

Firstly, we study the definition (4.178). Expressing cˆ† according to the hermitian conjugate of (4.174) and using the definition of the vacuum squeezed state we get n S(ξ) a ˆ† S † (ξ) (ˆ c † )n √ √ | n, ξ i = | 0, ξ i = S(ξ) | Ω i n! n! 1 (ˆ a † )n = √ S(ξ) (ˆ a† )n S † (ξ)S(ξ) | Ω i = S(ξ) √ | Ω i = S(ξ)| n i, (4.180) n! n! which, at least partly, justifies the name ”squeezed photon number state”. Nevertheless, it remains to check whether this state is indeed squeezed, in the sense defined earlier. To do so we need to study the variances of quadratures of the field. ˜ and Y˜ . The simple So we investigate the expectation values of the new quadratures X averages follow from (4.180) and from (4.83) ˜ i = h n, ξ | X ˜ | n, ξ i = h n | S † (ξ) X ˜ S(ξ) | n i = 0, hX h Y˜ i = h n, ξ | Y˜ | n, ξ i = h n | S † (ξ) Y˜ S(ξ) | n i = 0.

(4.181a) (4.181b)

The expectation values of the quadratures vanish, as it is the case for a usual photon number state | n i. So we proceed to compute the expectation values of the squares of the quadratures. ˜ quadrature We start with the X 2 ˜ 2 i = h n, ξ | X ˜ 2 | n, ξ i = h n | S † (ξ) X ˜ 2 S(ξ) | n i = h n | S † (ξ) X ˜ S(ξ) | n i hX (4.182) In this case we have to be more careful, since we deal with product of operators. We shall proceed similarly as in (4.151)-(4.153). The present case differs from the previous one only by the presence of photon number states instead of coherent ones. So we can write ˜ 2 i = 1 e−2ρ h n | a hX ˆ2 e−iθ + (ˆ a† )2 eiθ + 2 a ˆ† a ˆ + 1 | n i. (4.183) 2 72

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The two first terms do not contribute, hence we have ˜ 2 i = 1 e−2ρ (2n + 1) . hX 2

(4.184)

Clearly, the similar calculation for the second quadrature yields h Y˜ 2 i = h n, ξ | Y˜ 2 | n, ξ i =

1 2ρ e (2n + 1) . 2

(4.185)

The averages (4.181) and the expectation values of the squares of quadratures allow us to write the corresponding variances ˜ = 1 e−2ρ (2n + 1), σ 2 (X) 2

1 2ρ σ 2 (Y˜ ) = e (2n + 1). 2

(4.186)

The product of these variances is ˜ σ 2 (Y˜ ) = σ 2 (X)

1 1 (2n + 1)2 ≥ , 4 4

(4.187)

which is characteristic for the photon number states. State | n, ξ i is not minimalizing the uncertainty relation. Nevertheless, for given (but arbitrary) n we can choose the real number ρ in such a way that one of the variances will attain a value less than 1/2, that is below the magnitude which is characteristic for the vacuum state. Thus, we see that the name squeezed number state, for the state | n, ξ i, may be considered justified.

4.6

Expansion of the squeezed vacuum state into n-photon states

Considering the coherent states, we have expanded the coherent state | z i into n-photon ones in Eq.(4.28). We now intend to find a similar expansion for the squeezed vacuum state, that is we look for series | 0, ξ i =

∞ X

n=0

An (ξ) | n i.

(4.188)

Finding this expansion means finding the explicit expression for coefficients A n (ξ). In order to do so, we recall that the squeezed vacuum state | 0, ξ i is a vacuum for the operator cˆ as in (4.177), with annihilation operator cˆ related to ”usual” ones by (4.175a). Now, we apply operator cˆ to both sides of (4.188) obtaining 0 =

∞ X

n=0

An (ξ) cˆ | n i.

(4.189)

Expressing operator cˆ via (4.175a) we get 0 = cosh(ρ)

∞ X

n=0

An (ξ) a ˆ | n i + eiθ sinh(ρ)

∞ X

n=0

An (ξ) a ˆ† | n i.

(4.190)

We know how operators a ˆ and a ˆ † act on n-photon states. We also note that the term n = 0 in the first sum vanishes, hence we have 0 = cosh(ρ)

∞ X

n=1

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∞ X √ √ iθ An (ξ) n | n − 1 i + e sinh(ρ) An (ξ) n + 1 | n + 1 i.

(4.191)

n=0

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In the first sum we change the summation index n → m = n − 1, with m = 0, 1, 2, . . ., and we take the term m = 0 out of the sum. In the second sum we also introduce a new summation index n → m = n + 1, with m = 1, 2, 3, . . .. Then from (4.191) we get 0 = cosh(ρ)A1 (ξ) | 0 i + cosh(ρ) + eiθ sinh(ρ)

∞ X

∞ X

Am+1 (ξ)

m=1

Am−1 (ξ)

m=1

√ m + 1 |mi

√ n | m i.

(4.192)

This relation obviously implies that A1 ≡ 0.

(4.193)

Moreover, the kets | n i are the basis of the space of state vectors of the field. Thus, all the coefficients must be equal to zero. Therefore, we arrive at the relation √ √ cosh(ρ) m + 1 Am+1 (ξ) = − eiθ sinh(ρ) m Am−1 (ξ), (4.194) which is equivalent to the recurrence relation r m iθ Am+1 (ξ) = − e tanh(ρ) Am−1 (ξ), m+1

for

m = 1, 2, 3, . . .

(4.195)

This recurrence is valid for m > 1, so we conclude that A 0 is the first unknown coefficient. Since A1 = 0, we see that only coefficients with even index are nonzero. In other words, all coefficients with odd index are equal to zero. Putting m = 2k + 1 in recurrence relation (4.195) we rewrite it as r 2k + 1 iθ for k = 0, 1, 2, . . . (4.196) A2k+2 (ξ) = − e tanh(ρ) A2k (ξ), 2k + 2 Writing down several first coefficients, we can easily generalize the recurrence relation, which enables us to write s (2k − 1)!! A2k (ξ) = (−1)k eikθ tanhk (ρ) A0 (ξ), for k = 1, 2, 3, . . . (4.197) (2k)!! It is straightforward to check (by induction) that the expression (4.197) agrees with the recurrence relation (4.195). The coefficient A 0 (ξ) is unknown, and must be determined from the requirement of normalization of the vacuum squeezed state | 0, ξ i. Before we do so, let us consider the term with factorials. It is evident that (2k)!! = 2k k!,

and

(2k)! = (2k)!!(2k − 1)!!,

(4.198)

Which implies that (2k − 1)!! (2k)! (2k)! = = , 2 (2k)!! [(2k)!!] (2k k!)2

(4.199)

which, after inserting into (4.197) gives eikθ A2k (ξ) = (−1) k! k

74

tanh(ρ) 2

k

p (2k)! A0 (ξ),

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k = 1, 2, 3, . . .

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We use the obtained coefficients in the expansion (4.188), we also account for the fact that odd terms are absent, and we have ∞ ikθ X tanh(ρ) k p k e | 0, ξ i = A0 (ξ) (−1) (2k)! | 2k i. (4.201) k! 2 k=0

It remains to find the coefficient A0 (ξ) from the normalization requirement. Since the n-photon states are orthonormal, we easily obtain ∞ X (2k)! tanh(ρ) 2k 2 1 = h 0, ξ | 0, ξ i = |A0 (ξ)| . (4.202) (k!)2 2 k=0

So it is necessary to perform the remaining summation. This not an easy task. To do so, we first note that from the definition of Hermite polynomials we have for ones of even order H2k (x) = (2k)!

k X

(−1)m

m=0

(2x)2k−2m , m! (2k − 2m)!

(4.203)

so, for x = 0 only the term with m = k does not vanish. Hence, we have H2k (x = 0) = (2k)!

(−1)k . k!

(4.204)

This allows us to reexpress the terms under summation in (4.202), which is therefore rewritten as k ∞ X 1 tanh2 (ρ) 2 1 = |A0 (ξ)| − H2k (x = 0). (4.205) (k!) 4 k=0

Since tanh2 (ρ) < 1, the sum rule (3.1) applies. The summation is, thus, performed and we arrive at the formula 1 1 = |A0 (ξ)|2 q 1 − tanh2 (ρ)

= |A0 (ξ)|2 cosh(ρ),

(4.206)

because cosh(ρ) is always positive. Denoting an arbitrary phase by ϕ, we express the last expansion coefficient as eiϕ . A0 (ξ) = p cosh(ρ)

(4.207)

We adopt the overall phase ϕ = 0, then we insert A 0 into expansion (4.201) and we get ∞ ikθ X 1 tanh(ρ) k p k e | 0, ξ i = p (−1) (2k)! | 2k i. (4.208) k! 2 cosh(ρ) k=0

If we employ the expression (4.204) we can transform the obtained formula into q ∞ X (−1)k eikθ tanh(ρ) k p | 0, ξ i = (−1)k H2k (0) | 2k i. 2 k! cosh(ρ)

(4.209)

k=0

Combining the powers of (−1) into exponential phase we have ∞ X eik(θ+3π/2) tanh(ρ) k p p | 0, ξ i = H2k (0) | 2k i. 2 Γ(k + 1) cosh(ρ) k=0 S.Kryszewski

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(4.210)

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At this point we can generalize this result by noting that Hermite polynomials of odd order taken at x = 0 are equal to zero, i.e., H 2k+1 (x = 0) = 0. Therefore, we can include the odd terms in the sum (4.210) and we can write ∞ X tanh(ρ) n/2 p ei(θ+3π/2)n/2 p | 0, ξ i = Hn (0) | n i, (4.211) 2 Γ(n/2 + 1) cosh(ρ) n=0

where the odd terms (with odd n) give the zero contribution, while the even terms (in which n = 2k) reproduce the sum (4.210). The obtained expression is the sought expansion of the vacuum squeezed state in the n-photon states. The coefficients of the expansion are the probability amplitudes. Therefore, we can say that n 1 |Hn (0)| Pn (ξ) = tanh(ρ) , (4.212) 2 Γ(n/2 + 1) cosh(ρ)

is the probability that for the field in the vacuum squeezed state | 0, ξ i we find n photons. We note that this probability is zero for n odd, while for n even, that is for n = 2k it reads 2k 2k 1 1 |H2k (0)| 1 (2k)! P2k (ξ) = tanh(ρ) = tanh(ρ) , (4.213) 2 Γ(k + 1) cosh(ρ) 2 cosh(ρ) (k!)2

where we used relation (4.204). The squeezed vacuum state | 0, ξ i = S(ξ)| Ω i, and P n is the probability that for the field in this state, we find n photons. The fact that P 2k+1 (ξ) ≡ 0 seems ˆ2 /2 − ξ(ˆ a† )2 /2)], while operators a ˆ 2 and (co)2 not to be very surprising. Operator S(ξ) = exp[ξ ∗ a correspond either to destruction or to creation of two photons. Hence S(ξ)| Ω i is composed of n-photon states in which photons are created (or annihilated) in pairs. Therefore, only P 2k )(ξ) may be expected to be nonzero.

4.7

Equivalence of coherent squeezed states | z, ξ i and | α, ξ i

Let us consider the coherent squeezed state (4.136b): | z, ξ i = S(ξ)D(z)| Ω i = S(ξ)| 0, ξ i, and expand the coherent state | z i into photon number states according to (4.28). Then we get | z, ξ i = S(ξ) exp

− 12 |z|2

∞ X zn √ |ni n! n=0

∞ X zn √ S(ξ) | n i = exp − 12 |z|2 n! n=0 ∞ X zn √ | n, ξ i, = exp − 12 |z|2 n! n=0

(4.214)

where we used the definition (4.180) of the squeezed photon number states. Next we employ (4.178) to write ∞ X z n (ˆ c † )2 | z, ξ i = exp − 12 |z|2 | 0, ξ i, n! n=0 † 2 1 = exp − 2 |z| ezˆc | 0, ξ i.

(4.215)

The state | 0, ξ i is an eigenstate of the annihilation operator cˆ. Thus The above formula is equivalent to † ∗ | z, ξ i = exp − 12 |z|2 ezˆc e−z cˆ | 0, ξ i. (4.216) 76

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Operators cˆ† and cˆ are creation and annihilation operators, thus relation (2.27b) (with α = −z ∗ and β = z) applies, and we get † ∗ exp − 12 |z|2 ezˆc e−z cˆ = exp zˆ c† − z ∗ cˆ . (4.217) We combine two last equations, and we obtain c† − z ∗ cˆ S(ξ)| Ω i. | z, ξ i = exp zˆ c† − z ∗ cˆ | 0, ξ i = exp zˆ

(4.218)

Next we need to consider the exponent in the leftmost operator. Due to relations (4.175) zˆ c† − z ∗ cˆ = z a ˆ† cosh(ρ) + a ˆ e−iθ sinh(ρ) − z ∗ a ˆ cosh(ρ) + a ˆ † eiθ sinh(ρ) ˆ, = αa ˆ† − α ∗ ∗ a

(4.219)

where we have denoted α = z cosh(ρ) − z ∗ eiθ sinh(ρ). Eq.(4.2180 we have | z, ξ i = exp αˆ a † − α∗ a ˆ S(ξ)| Ω i.

Introducing this new variable into

| z, ξ i = S(ξ)D(z)| Ω i = D(α)S(ξ)| Ω i = | α, ξ i

(4.221)

(4.220)

Recognizing the displacement operator we summarize our calculations by

provided the complex numbers z and α are connected by the relation α = z cosh(ρ) − z ∗ eiθ sinh(ρ).

(4.222)

This result is in full agreement with the discussion of two possible squeezed coherent states. This fact elucidates the sense of the squeezed photon number states. They have the same relation to squeezed coherent states as the corresponding usual number states to usual coherent ones. Obviously, the derivation presented here could have been done in the reversed direction, leading to the same final conclusion.

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Chapter 5

Atom–field interaction 5.1

Charged particle in electromagnetic field

5.1.1

Minimum coupling hamiltonian

Discussing classical electrodynamics we have derived the so-called minimum coupling hamiltonian 2 1 ~ ~p − q A H= + qφ + V (~r), (5.1) 2µ for a particle of mass µ and charge q. In this equation we have added the term V (~r) which accounts for the additional potential energy due to interactions other than coupling to the ~ = A(~ ~ r, t) external electromagnetic field which is characterized by vector and scalar potentials A and φ = φ(~r, t). The potentials specify the fields ~ r, t) = rot A(~ ~ r, t), B(~

and

~ r, t) = −∇φ(~r, t) − ∂ A(~ ~ r, t) E(~ ∂t

(5.2)

In the classical approach, all quantities appearing in Hamiltonian 5.1) are certainly classical and the fields (or potentials) are treated as prescribed functions of position and time. In quantummechanical context the situation changes radically. First of all, both the charged particle and the fields are described by corresponding operators. The field is a dynamical quantity, not a set of given functions. Thus the hamiltonian (5.1) must be augmented by H F – the hamiltonian of the field. Therefore our hamiltonian becomes H = HA + H1 + H2 + qφ + HF

(5.3)

where we have denoted the terms ~p2 + V (~r), 2µ q ~ ~ · ~p , ~p · A + A H1 = − 2µ q2 ~ 2 H2 = A , 2µ X 1 † HF = ~ωk a~ a~kλ + . kλ 2

HA =

(5.4a) (5.4b) (5.4c) (5.4d)

~ kλ

We note, that keeping the sequence of operators, we account for their possible noncommutativity. 78

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Having outlined the general scope, we shall devote some time to its discussion. Hamiltonian HA is frequently called atomic hamiltonian (we shall fully justify this name later). The terms H1 and H2 depend on the charge of the particle and on the field, thus are responsible for the interaction of the field with the charged particle. Sometimes, though typically in different contexts, H1 is called a paramagnetic term, while H 2 – diamagnetic one. As we already mentioned the particle and field operators usually do not commute. This fact is summarized by the following theorem. Theorem 5.1 Cartesian components of the momentum of the particle and of the vector potential operator obey the commutation rule h i ∂Aj pk , Aj = −i~ . (5.5) ∂xk

Proof. Component Aj of vector potential depends on creation and annihilation operators and on some functions of position. The first ones do commute with momentum, but the position dependent functions do not. Thus, if ψ(~r) is an arbitrary wave function of the particle (in position representation), then i h pk , Aj ψ(~r) = − i~ ∂k Aj − Aj ∂k ψ(~r) = − i~∂k Aj ψ + i~Aj ∂k ψ = − i~ ∂k Aj ψ − i~Aj ∂k ψ + i~Aj ∂k ψ = − i~ ∂k Aj ψ. (5.6) Arbitrariness of the function ψ(~r) implies that (5.5) holds true.

In the view of this theorem we can rewrite hamiltonian H 1 as i q q q h H1 = − pk Ak + A k pk = − pk , A k − Ak pk 2µ 2µ µ iq~ q ~ − A ~ · ~p. = divA 2µ µ

(5.7)

Therefore, the full hamiltonian (5.3) reads H = HA +

2 iq~ ~ −q A ~ · ~p + q A ~ 2 + qφ + HF . divA 2µ µ 2µ

(5.8)

~ r, t) (and so of divA) ~ and φ(~r, t), depends on the particular Concrete form of the potentials A(~ physical problem we intend to consider. This hamiltonian can be used for analysis of the phenomena in the static fields (what is usually done in standard course of quantum mechanics) and for the case in which an atom is irradiated by the time-dependent field, that is by an electromagnetic wave – light. We also stress that the discussed hamiltonian (5.8) does not include the spin of the particle, so it does not account for any effects in which the couplings of the spin are of importance. We will not consider such effects, but it may be worth remembering that the presented theory can be generalized in a manner allowing for the description of the effects connected with particle spin.

5.1.2

Some additional comments

In the strict sense, the system we consider is a system consisting of charged particles(s) and of the electromagnetic field. Thus, the state vector of the system is a joint one | ϕ a , {n~kλ } i, where | ϕa i denotes quantum state of the particle and {n~kλ } is a set of the photon numbers occupying modes designated by (~k, λ). It is perhaps worth recalling that in many applications it S.Kryszewski

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is sufficient to employ a semiclassical approach. This approach consists in treating the particle as a quantum-mechanical object, while the fields are treated classically, that is as prescribed functions of ~r and t. Such an approach is presented in standard course of quantum mechanics. It is quite well suited to description of such phenomena as Zeeman or static Stark effects. Some of the results of semiclassical theory are also useful in more general context. For example, it is possible to argue that typical energies ∆E 1 connected with hamiltonian H1 are of the order of ∆E1 ∼ ~

~ q|B| , 2µ

(5.9)

~ is the value of the corresponding magnetic field ( B ~ = rot A). ~ where |B| Numerical estimate easily shows that ∆E1 ∆EA ,

(5.10)

with ∆EA being a typical energy (eigenvalue) arising from the atomic hamiltonian H A . Similarly, one can show, that hamiltonian H2 corresponds to typical energies of the order of ∆E2 ∼

~ 2 q 2 a2B |B| , µ

(5.11)

where aB is the Bohr radius. This allows the comparison, which yields ∆E2 ∆E1 ∼ 1. ∆E1 ∆EA

(5.12)

So the effects connected with H2 are typically of several orders of magnitude smaller than those due to H1 . One has to be aware, that the given estimates are just rough ones, and usually not valid in more extreme conditions. For example,, for laser intensities of order of GW/cm 2 , the fields are so strong that the above estimates must be treated with suspicion, as probably being wrong. Nevertheless, in typical spectroscopic conditions, the discussed approximate estimates give a good feeling of relative magnitudes. The reader interested in more details of semiclassical approach is encouraged to consult a variety of handbooks on standard quantum mechanics. Finally, let us comment on the problem of gauge invariance. In the Schr¨odinger equation for considered system i~

∂ψ(~r, t) = H ψ(~r, t) ∂t

(5.13)

with hamiltonian (5.1) or (5.8), we perform the gauge transformation of the potentials ~ r, t) A(~ φ(~r, t)

~ 0 (~r, t) = A(~ ~ r, t) + ∇χ(~r, t) A ∂ gauge φ0 (~r, t) = φ(~r, t) − χ(~r, t), ∂t gauge -

(5.14a) (5.14b)

thus leaving the fields unchanged. If simultaneously we transform the wave function according to the rule iq gauge 0 ψ(~r, t) ψ (~r, t) = exp χ(~r, t) ψ(~r, t) (5.15) ~ then, the Schr¨odinger equation for the ”new” wave function ψ 0 is of the form i~ 80

∂ψ 0 (~r, t) = H 0 ψ 0 (~r, t) ∂t

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with ”new” hamiltonian H 0 is of the form identical to (5.1), but with new – transformed potentials. Hence the Schr¨odinger equation is invariant with respect to gauge transformation of the potentials, provided the new choice of the potentials is accompanied by the transformation (5.15) of the system wave function. It should be stressed that the gauge transformation of the wave function is not just a change of the (otherwise arbitrary) overall phase. Change of the overall phase is a global transformation, in the sense that overall phase does not depend on the position and time. On the contrary, the gauge transformation introduced in (5.15) changes the phase in a manner which depends on space-time coordinates, and as such is local. Gauge transformations are of great importance in the quantum field theory. The invariance of the matter-field interaction may be studied from such a field-theoretic point of view, which allows a rigorous derivation. This subjects, however, go beyond the scope of these lectures, so we will just stop here, remembering that the theory is indeed gauge invariant.

5.2

Atom in radiation field

5.2.1

Setting up the scene. Free atom hamiltonian

We consider a simple one-electron atom. The center of an atom (its nucleus) is positioned at the point denoted by the ~ The electron, with respect to the nucleus position vector R. has radius vector ~r, while with respect to the point S – the center of the coordinate system, electron’s position is given by a vector ~x. In our description of the atom we will neglect the spin of the electron. The atom as a single whole may perform a uniform motion. Such a case of a moving atom, if time permits, will be considered later (in next chapters). ~ will be a time-dependent function varyThen its position R ing (usually linearly) with time. The kinetic energy of the atom may be separated in the center-of mass frame and it is just a constant term in the Hamiltonian. So we will consider the electron motion in the center-of-mass frame and we will omit the kinetic energy of the uniform motion of the atom. Fig. 5.1: Positions of atomic nucleus Therefore, we will denote by m the reduced mass, which is and its electron in arbitrary coordi- very close to the well-known electron mass (the nucleus is nate frame. much heavier than the electron). We assume that in the center-of-mass frame the Hamiltonian H A of the atom may be diagonalized, that is the necessary stationary Schr¨odinger equation can be solved. We denote the eigenstates of HA by | a i and the corresponding energies by ~ω a . The standard hamiltonian of a free atom can be written as

HA =

X ~p2 + V (r) = ~ ωa | a ih a |. 2m a

(5.17)

By definition | a i are the eigenstates of the atomic hamiltonian: HA | a i = ~ωa | a i.

(5.18)

Atomic eigenstates are assumed to be orthonormal and complete: X h a | b i = δab , and | a ih a | = 1.

(5.19)

a

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The atom, description of which is given above, interacts with electromagnetic wave. The sources of radiation are considered to be far away, thus the atom interacts with a free field. We will consider a general case in which the radiation is treated quantum-mechanically. First, we will derive the necessary expressions for the description of the interaction. Then, we will discuss some approximations which still retain the quantum description of the field. We will also consider approximations in which radiation is classical (or in the coherent state). The hamiltonian of the quantized free radiation field is of the form X 1 † HF = . (5.20) ~ωk a~ a~kλ + kλ 2 ~ kλ

We also recall that the vector potential of the electromagnetic field quantized in the cubic box of volume V , evaluated at the position ~x where the atomic electron is situated, is given as ~ x, t) = A(~

i X Ek h ~ ~ ∗ † ~e~kλ a~kλ (t) eik·~x + ~e~kλ a~ (t) e−ik·~x , kλ ωk

(5.21)

~ kλ

while the corresponding electric field (in Coulomb gauge) is h i X ~ ~ ∗ † ~ x, t) = i E(~ Ek ~e~kλ a~kλ (t) eik·~x − ~e~kλ a~ (t) e−ik·~x , kλ

(5.22)

~ kλ

where ~e~kλ are the polarization vectors, assumed to be in general complex. We have also introduced a useful notation r ~ωk Ek = . (5.23) 20 V Finally, ~k and ωk are wave vector and frequency of the mode of the field, we note that they satisfy the dispersion relation for each mode, which reads ωk = c |~k|.

(5.24)

The interaction of an atom with the incident electromagnetic field is described by the total hamiltonian of the system atom + field H = HA + HF + HAF ,

(5.25)

with HA given by (5.17) and HF by (5.20). The main purpose of this chapter is to discuss the interaction hamiltonian HAF . There are two basic formulations of the problem. • We already discussed the minimum coupling hamiltonian. By the standard correspondence principle, the quantities appearing there are quantum-mechanical operators. Thus we may take for the atom (atomic electron) in the field described by a vector and scalar potentials ~ and φ, the total hamiltonian in the minimum coupling in the form A H min =

2 1 ~ ~p − q A + V (r) + qφ + HF , 2m

(5.26)

where V (r) is the internal (Coulomb) term responsible for interaction between the electron and the atomic nucleus. Thus, in this case it remains to construct and discuss the interaction term HAF . We return to this question in next sections. 82

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• The minimum coupling hamiltonian have been derived by correspondence principle from the classical one. Another possibility also arises from classical notions. It is possible to expand the charge density of the atom into multipolar series.Then, an expression for the energy of the atom in the electromagnetic field can be derived. Invoking the correspondence principle, one can then construct a quantum-mechanical multipolar hamiltonian. An atom is electrically neutral, so the first term is an electric dipole one. Next terms are electric quadrupole and magnetic dipole. An estimation of relative energies connected with these terms shows that the leading one – electric dipole – is roughly by three orders of magnitude larger than the two next ones, which in turn are larger than the third order multipoles, and so on. Therefore, it is reasonable to truncate the multipolar expansion and to retain only the leading term. This allows us to represent the atom-field interaction by the expression dip ~ HAF = − ~d · E,

(5.27)

where ~d = e~r denotes the atomic dipole moment operator, while the electric field is given by (5.22). This interaction hamiltonian enters the total one (5.25 giving the full quantum mechanical hamiltonian of the atom in the radiation field. Obviously, there arises the question whether the two given descriptions are equivalent. We shall not discuss this question in great detail. At present we will adopt these two hamiltonians as given. We shall give some additional comments and remarks (also concerning their equivalence), but more detailed analysis of the two hamiltonians can be found in more specialized literature.

5.2.2

Minimum coupling interaction hamiltonian

General discussion and basic simplifying assumptions Taking the hamiltonian (5.26) we combine the scalar (internal) potential V (r) with the kinetic energy term into the atomic hamiltonian. Thus, we write q ~ q2 ~ 2 ~ H min = HA + HF − A · ~p + ~p · A + A + q φ. (5.28) 2m 2m

~ is a function of position it does not have to commute with the Since the vector potential A momentum operator. Employing the arguments of the previous section we can rewrite H min as H min = HA + HF −

2 iq~ ~ − q A ~ · ~p + q A ~ 2 + q φ. div A 2m m 2m

(5.29)

Now, we discuss some additional assumptions allowing considerable simplifications. ~ 2 was already briefly discussed. We recall Assumption 1. The diamagnetic term containing A that it leads to small corrections to the energy, and as such, can usually be neglected. Moreover, the diamagnetic term contains the square of the vector potential. Therefore it contains products of annihilation and creation operators which obviously correspond to two-photon processes. Such processes are much less probable than the one-photon ones. This gives an additional justification ~ 2 term. for neglecting the A Assumption 2. Our prior quantization of the electromagnetic field was performed within the Coulomb gauge. As a consequence, we adopt this gauge in the present considerations. Thus, we ~ is transverse, and that it satisfies the Coulomb gauge recall that the vector potential A ~ = 0, divA

(5.30)

while the scalar potential of the radiation field can be taken as zero. We note, that in Coulomb gauge the momentum and vector potential operators commute (see (5.6)). S.Kryszewski

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Under these assumptions the hamiltonian (5.26) assumes the form: H min = HA + HF −

q ~ ~p · A. m

(5.31)

We will name the last term of Eq.(5.31) minimal coupling interaction hamiltonian min HAF =−

q ~ =−q A ~ · ~p. ~p · A m m

(5.32)

Taking the explicit expression (5.21) for the vector potential we write the minimal coupling interaction hamiltonian as i q X Ek h ~ ~ min ∗ ~p · ~e~kλ a~kλ (t) eik·~x + ~p · ~e~kλ HAF = − a~† (t) e−ik·~x , (5.33) kλ m ωk ~ kλ

with Ek given in (5.23). The obtained expression (5.33) will be subject of further discussion. Dipole approximation The size of an atom is of the order of 10 −10 m while the wavelength of the electromagnetic radiation in the optical region is by three orders of magnitude larger. Therefore, the field intensity at the point where the atomic electron is situated is, within very good approximation, the same as in the center of the atom (that is, at the atomic nucleus). Hence we can neglect the ~ which determines variation of the field within an atom, and we can take the field at the point R ~ as the external variable which the position of the atom as a whole. Moreover, we may treat R commutes with the momentum operator (the latter being the internal variable). Thus, we rewrite minimal coupling interaction hamiltonian (5.33) as min HAF =−

i q X Ek h ~ ~ ~ ~ ∗ ~p · ~e~kλ a~kλ (t) eik·R + ~p · ~e~kλ a~† (t) e−ik·R , kλ m ωk

(5.34)

~ kλ

Next, we employ the completeness relation (5.19) at the left and at the right of the momentum operators, and from (5.34) we have min HAF =−

q X X Ek ~ ~ | a i~pab h b | · ~e~kλ a~kλ (t) eik·R + H.C., m ωk

(5.35)

~ kλ a,b

where ~pab = h a | ~p | b i denotes the matrix element of the electron’s momentum operator. We now intend to reexpress the matrix element of the momentum operator. To this end we consider the commutation relations for the electron’s position operator. We easily obtain i~ ~p2 1 ~r, ~p2 = ~p. [ ~r, HA ] = ~r, + V (~r) = (5.36) 2m 2m m Then we compute the matrix element h a | ~p | b i = −

h i im im h a | ~r, HA | b i = − ωba ~dab , ~ q

(5.37)

where ~dab = h a | ~d | b i = h a | q ~r | b i is the matrix element of the atomic dipole moment. The frequency ωba is the usual Bohr frequency between the two atomic levels | b i and | a i, that is

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ωba = (Eb − Ea )/~ = ωb − ωa . Using the above obtained relations we can reexpress the minimal coupling interaction hamiltonian as   X Ek X ~ ~ min  HAF = i ωba | a i ~dab h b |  · ~e~kλ a~kλ (t) eik·R ωk a,b ~ kλ    X ~ ~ ∗ † −  ωba | b i ~dba h a |  · ~e~kλ a~ (t) e−ik·R  (5.38) kλ

a,b

New notation scheme We consider the last relation and in the second sum we interchange the indices a ↔ b. This can be done because the sum runs over a complete set of atomic states. So we get X X i Ek h ~ ~ min HAF = ωba | a i ~dab h b | · ~e~kλ a~kλ (t) eik·R ωk ~ kλ a,b i ~ ~ ∗ † − ωab | a i ~dab h b | · ~e~kλ a~ (t) e−ik·R . (5.39) kλ

Since ωab = −ωba we can rewrite the above expression as XX ωab h~ ~ ~ min = | a ih b | (− i Ek ) dab · ~e~kλ a~kλ (t) eik·R HAF ωk ~ kλ a,b i ~ ~ ∗ + ~dab · ~e~kλ a~† (t) e−ik·R . kλ

(5.40)

Now we introduce the following notation ~ ~ ~ = − i Ek ωab ~dab · ~e~ gab,~kλ (R) eik·R . kλ ωk

(5.41)

From this definition, by interchange of the indices a and b, it follows that ~ ~ ~ = − i Ek ωba ~dba · ~e~ gba,~kλ (R) eik·R , kλ ωk

(5.42)

and further, by hermitian conjugation, we also have ~ ∗ ~ = − i Ek ωab ~dab · ~e ∗ e−i~k·R gba, ( R) , ~ ~ kλ kλ ωk

(5.43)

~ ∗ = ~dab . We note, that due to ωaa = 0 the coefficients are characterbecause ωab = −ωba and d ba ized by the property ~ = 0. gaa,~kλ (R)

(5.44)

We conclude that the minimum coupling is not suitable for the description of the atoms which possess the permanent dipole moment (ie. atoms for which ~daa 6= 0). Now, we introduce coupling coefficients (5.41) and (5.42) into the interaction hamiltonian (5.40), and we obtain h i XX min ~ a~ (t) + g ∗ ~ a† (t) HAF = | a ih b | gab,~kλ (R) ( R) (5.45) ~ kλ ba,~ kλ kλ

~ kλ a,b

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Dipole coupling interaction hamiltonian

This case seems to be easier, because we simply have, as it follows from (5.27), the interaction dip ~ We stress that at the present stage of hamiltonian for the dipole coupling as H AF = − ~d · E. our considerations, this Hamiltonian is not derived but treated as given. We may say that it simply follows from classical associations, and as such, may be treated as an assumption. Moreover, we assume that the electric field entering the expression (5.27) is evaluated at the atomic center of mass (this is justified by the same argument which we used to explain the dipole approximation in the previous section). Due to this assumption we take the electric field ~ and we get as in (5.22) with ~x replaced by R h i X ~ ~ ~ ~ dip ∗ † HAF = − i Ek ~d · ~e~kλ a~kλ (t) eik·R − ~d · ~e~kλ a~ (t) e−ik·R . (5.46) kλ

~kλ

We employ the completeness of the atomic states once more, using this at the both sides of the atomic dipole moment operator and we find the following expression X X ~ ~ dip HAF = −i Ek | a i ~dab h b | · ~e~kλ a~kλ (t) eik·R ~ kλ

a,b

~ ~ ∗ † − | a i ~dab h b | · ~e~kλ a~ (t) e−ik·R kλ

.

(5.47)

It may seem that the hermiticity is lost. This not the case, when we interchange the atomic indices in the second term we can rewrite Eq.(5.47) in the form analogous to Eqs.(5.38). Thus, we have  X X ~ ~ dip HAF = −i Ek  | a i ~dab h b | · ~e~kλ a~kλ (t)eik·R ~ kλ

a,b

−

X a,b



~ ~ ∗ † | b i ~dba h a | · ~e~kλ a~ (t)e−ik·R  , kλ

(5.48)

which is evidently Hermitian. Next, we proceed along the lines very much similar to those in the previous case. Transforming eq.(5.47) we arrive at the expression h X X ~ ~ dip HAF = | a ih b | −i Ek ~dab · ~e~kλ eik·R a~kλ (t) ~ kλ

a,b

i ~ ~ ∗ + i Ek ~dab · ~e~kλ e−ik·R a~† (t) . kλ

In this case the new notation is as follows ~ ~ ~ = − i Ek ~dab · ~e~ Gab,~kλ (R) eik·R , kλ

(5.49)

(5.50)

which in an obvious manner, implies that ~ ~ ~ = − i Ek ~dba · ~e~ Gba,~kλ (R) eik·R , kλ

(5.51)

and

~ ~ = i Ek ~dab · ~e ∗ e−i~k·R G∗ba,~kλ (R) . ~ kλ

(5.52)

Using these coupling coefficients in the interaction hamiltonian (5.49) we rewrite it in the form h i X X dip ~ a~ (t) + G∗ ~ a† (t) HAF = | a ih b | Gba,~kλ (R) ( R) (5.53) ~ ~ kλ ba,kλ kλ

~kλ

a,b

This hamiltonian is a counterpart of (5.45) and also will be extensively used in next sections. 86

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Interaction hamiltonian. Summary and discussion

First we summarize the obtained results • For the minimum coupling scheme, the interaction hamiltonian is of the form h i ~ a~ (t) + g ∗ ~ a† (t) | a ih b | gab,~kλ (R) ( R) ~ kλ ba,~ kλ

XX

min = HAF

kλ

~ kλ a,b

~ i~ k·R ~ = − i Ek ωab ~dab · ~e~ gab,~kλ (R) . kλ e ωk

with

(5.54a) (5.54b)

• For the dipole coupling scheme, the interaction hamiltonian is of the form dip HAF =

XX ~ kλ a,b

with

h i ~ a~ (t) + G∗ ~ a† (t) | a ih b | Gab,~kλ (R) ( R) ~ kλ ba,~ kλ

~ = − i Ek Gab,~kλ (R)

kλ

~ ~ ~dab · ~e~ eik·R . kλ

(5.55a) (5.55b)

• We note the formal similarity of both interaction hamiltonians. The corresponding coupling coefficients are obviously related ~ = ωab G ~ (R), ~ gab,~kλ (R) ωk ab,kλ

(5.56)

which simplifies going from one coupling to the other. • It is straightforward to check that both interaction hamiltonians are Hermitian (as they should be). For example, from (5.54a), by Hermitian conjugation we get

min HAF

†

=

XX ~ kλ a,b

| b ih a |

h

i † ∗ gab, a (t) + g a (t) . ~ ba,~ kλ ~ kλ kλ ~

(5.57)

kλ

Since it is possible to interchange the indices a ↔ b, we get next

min HAF

†

=

XX ~ kλ a,b

| a ih b |

h

∗ gba, a† (t) + gab,~kλ a~kλ (t) ~ kλ ~ kλ

i

min = HAF .

(5.58)

dip The proof for HAF is certainly identical.

• The total hamiltonian of the atom-field system follows from Eqs.(5.25), (5.17) and (5.20). It is X X 1 † H = ~ ωa | a ih a | + ~ωk a~ a~kλ + + HAF , (5.59) kλ 2 a ~ kλ

with HAF given either by (5.54a) or by (5.55a). Having summarized the main results we also give several additional comments on the validity and applications of the discussed formalism.

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• In expressions (5.54a), (5.55a) and (5.59) we sum over all modes of the radiation field, while the dipole approximation is valid for radiation wavelengths much larger than the size of an atom. This may seem to be contradictory. The contradiction, however, disappears when we consider realistic light sources. Laser radiation can frequently be considered monochromatic and in such a case only one mode remains with nonzero photon number. In more general situation the wavelength range is restricted to optical spectrum, so modes with higher frequencies (smaller wavelength) are absent, and we sum the modes only within some (usually small) range. Hence the dipole approximation is indeed valid even in the ultraviolet range of the radiation spectrum. The unoccupied modes may, however, contribute to some expectation values (as it is the case, for example, with energy of the field). Therefore, it may seem dangerous to omit these empty modes from the summations. We will consider the coupling of the atom to the vacuum field and we will show that such coupling results in spontaneous emission effect. Thus, empty modes and their influence upon the atom are accounted for in the proper description of spontaneous emission. We shall examine this problem in further sections. • The obtained interaction hamiltonians include atomic states | a i, which in general, are fairly complex, since they depend on several quantum numbers (index a is, in fact, a multiindex). This usually complicates practical calculations. Various techniques are devised to deal with realistic atoms which usually possess degeneracy with respect to angular momentum quantum numbers. At this stage it is also possible to account for the electron spin. This can be done by considering the basis of atomic states spanned by the eigenstates of ~ + ~S. Then, many useful computational tools, known from total angular momentum ~J = L the general theory of angular momentum, can be employed. For example, it is possible to construct the atomic basis with the aid of spherical tensors. This approach is frequently used in many spectroscopic investigations. This is, however, a problem beyond the scope of these lectures. We will focus our attention on the very simple model, namely we shall adopt the two-level approximation for the description of the atomic states.

5.3 5.3.1

Hamiltonian for two–level atom in radiation field The two-level atom. Free hamiltonian

In many practical cases the electromagnetic field irradiating an atom is closely tuned to one of the atomic res onances and has a relatively narrow spectral bandwidth equal to ∆ωR . In such a case only two atomic levels with energy separation ~ω21 close to the central frequency ωR

of the field, are strongly coupled to the incoming radiation. Moreover, if the other levels are separated by much more than ∆ωR , we can safely disregard all other levels, except those two coupled to the field. Then we may say that we deal with a two-level approximation to the real atom. We shall still make an additional simplifying assumption. Namely, we will assume that the considered Fig. 5.2: The level scheme of a two levels do not exhibit spatial degeneracy. This means, two-level atom. that we do not consider the angular momentum issues, which reduces the atomic basis just to two states, and two quantum numbers (numbering the levels) are sufficient to fully describe the state of the atom. The Hilbert space of these states is, 88

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thus, two-dimensional. It may be worth noting, that it is possible to prepare such states of real atoms that are indeed well described by a two-level model. Moreover, investigations of two-level atom give excellent insights in the phenomena occurring in real atoms. So, we restrict our attention to the simplest two-level atom (TLA) model, and we denote upper (excited) state of a TLA by | 2 i and lower (ground) state by | 1 i. Thus, the hamiltonian of the free TLA may be then written as it follows from (5.17), that is HA = ~ω1 | 1 ih 1 | + ~ω2 | 2 ih 2 |.

(5.60)

The form (5.60) is fairly self-evident. We can, however, select other forms of the Hamiltonian for TLA, depending on the choice of the zero on the energy scale. When we take zero energy at the ground level, then we can write HA = ~ω21 | 2 ih 2 |.

(5.61)

Another possibility consists in choosing the zero of energy midway between the levels. Then, we can rewrite hamiltonian (5.60) as h i 1 (5.62) HA = ~ω21 | 2 ih 2 | − | 1 ih 1 | = ~ω21 S3 , 2

where S3 is the third component of the quasi–spin operator (for the details see the appendix D). Since the space spanned by the states | 1 i and | 2 i is two-dimensional, it is isomorphic to the space of the eigenstates of spin 1/2 operators. The corresponding operators can be associated with 2×2 matrices, and are expressible by Pauli matrices. These operators can also be expressed by the so-called pseudo-spin operators introduced and discussed in appendix D. We will use pseudo-spin operators without reference to particular formulas given in that appendix, which should be consulted if necessary. Obviously, the total hamiltonian of TLA-light system contains three terms as in (5.59). The interaction hamiltonian should be, however, transformed to suit our current needs.

5.3.2

Interaction hamiltonian

Minimum coupling case We adjust the general expression (5.54a) for the TLA. We assume that the atomic dipole moment, due to parity considerations has the property ~d11 = ~d22 = 0 (moreover, coupling coefficients possess property (5.44). Therefore the sum in (5.54a) reduces to two terms only. Using the pseudospin operators we obtain Xn min ~ S− a~ (t) + g ∗ ~ S− a† (t) HAF = g12,~kλ (R) (R) ~ kλ 21,~ kλ kλ

~ kλ

~ S+ a~ (t) + g ∗ ~ S+ a† (t) + g21,~kλ (R) (R) ~ kλ 12,~ kλ kλ

where, as it follows from (5.54b) ~ ~ ~ = − i Ek ωab ~dab · ~e~ gab,~kλ (R) eik·R . kλ ωk

o

,

(5.63)

(5.64)

This hamiltonian stems from a more general one (5.54a), hence it can be generalized to describe levels | 1 i and | 2 i with spatial degeneracy (nonzero angular momentum). Since polarization of radiation is a vector, it couples to the vector characteristics of the atom (this follows from the general theory of quantum angular momentum). As we do not consider angular momentum S.Kryszewski

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issues, we can take the polarization in its simplest form. Therefore, here and in future we will assume that polarization vectors are real ∗ ~e~kλ = ~e~kλ

(5.65)

This assumption allows us to simplify the notation. According to (5.64) we have ∗ ω12 ~ ω21 ~ ~ ~ ~ ~ g12,~kλ = − i Ek d12 · ~e~kλ eik·R = i Ek d21 · ~e~kλ eik·R , ωk ωk ω21 ~ ~ ∗ −i~ k·R g12,~kλ = − i Ek d21 · ~e~kλ e , ωk ω21 ~ ~ ~ g21,~kλ = − i Ek d21 · ~e~kλ eik·R , ωk ∗ ω21 ~ ~ ~ ∗ g21,~kλ = i Ek d21 · ~e~kλ e−ik·R . ωk where we used (5.65) and ~d∗ab = ~dba . We now introduce (see (5.23)) s 2 ω21 ~ ~ω21 ~d21 · ~e~ , d21 · ~e~kλ = − i ~g~kλ = − i Ek kλ ωk 20 V ωk

(5.66a) (5.66b) (5.66c) (5.66d)

(5.67)

which allows us to ”forget” the signs of atomic frequencies ω ab . Thus, from (5.66) we get ~ ~

∗ g12,~kλ = ~ g~kλ eik·R , ~ ~

∗ g12, = ~ g~kλ e−ik·R , ~ kλ

~ ~

g21,~kλ = ~ g~kλ eik·R , ~ ~

∗ ∗ g21, = ~ g~kλ e−ik·R , ~ kλ

(5.68)

Using this notation in the hamiltonian (5.63) we obtain X n ~ ~ min ∗ i~ ∗ −i~ HAF = ~ g~kλ e k·R S− a~kλ (t) + g~kλ e k·R S− a~† (t) kλ

~ kλ

o ~ ~ ~ ~ + g~kλ eik·R S+ a~kλ (t) + g~kλ e−ik·R S+ a~† (t) . kλ

(5.69)

Since S− = | 1 ih 2 | and S+ = | 2 ih 1 | are the lowering and raising operators, we can interpret the physical effects described by each of the terms in the above expression. They are as follows: i) S− a~kλ (t) – atom deexcitation and photon annihilation; ii) S− a~† (t) – atom deexcitation and photon creation; kλ iii) S+ a~kλ (t) – atom excitation and photon annihilation; iv) S+ a~† (t) – atom excitation and photon creation. kλ

In standard time-dependent perturbation theory we distinguish between resonant and nonresonant processes, and find that nonresonant ones are much less probable that the resonant ones. Absorption process consists in annihilation of a photon, energy of which is transferred to an atom, which is then excited to a higher level (up-transition). Emission process is the inverse one, the atom undergoes the transition from upper to a lower state (becomes deexcited) and its excess energy is given to the photon which is created and escapes from the atom. In view of these remarks we may call processes (ii) and (iii) resonant, while the first and the fourth are nonresonant. The standard practice in perturbation theory is to omit (neglect) the nonresonant terms. We shall do so at later stages of our discussion, At present we shall mark these terms by a

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control parameter ξ = 1, while the resonant approximation consists in taking ξ = 0. Therefore, we rewrite (5.69) as X n ~ ~ ~ min ∗ −i~ = ~ e k·R S− a~† (t) HAF g~kλ eik·R S+ a~kλ (t) + g~kλ kλ

~ kλ

h io ~ ~ ~ ∗ i~ + ξ g~kλ e k·R S− a~kλ (t) + g~kλ e−ik·R S+ a~† (t) kλ

X n

= ~

h

~ ~

~ ~

g~kλ S+ eik·R a~kλ (t) + ξ e−ik·R a~† (t)

~ kλ

kλ

i

h io ~ ~ ~ ~ ∗ + g~kλ S− ξ eik·R a~kλ (t) + e−ik·R a~† (t) .

(5.70)

kλ

This form of the minimum coupling hamiltonian for the TLA in radiation field will be useful in future applications. We shall, however write it in one more form, namely we will explicitly put ξ = 1, and then from (5.70) we have ~ X ~ ~ ~ min ∗ HAF = ~ g~kλ S+ + g~kλ S− eik·R a~kλ (t) + e−ik·R a~† (t) , (5.71) kλ

~ kλ

~ = 0. This concludes the general aspects which attains especially simple form when we can put R of the discussion of the interaction minimum coupling hamiltonian for the TLA irradiated by quantum-mechanical electromagnetic radiation. We warn the reader that in other literature sources the notation may be different. So our formulas must be ”translated” into the notation specific to particular needs. Dipole coupling case In this case we work upon general expression (5.55a). We take into account that ~d11 = ~d22 = 0 (as it follows from parity considerations), which implies that the sum over the atomic states includes only two terms. Thus, similarly as in the case of minimum coupling we obtain X n dip ~ a~ (t) + G∗ ~ a† (t) = | 1 ih 2 | G12,~kλ (R) ( R) HAF ~ ~ kλ 21,kλ kλ

~ kλ

io ~ a~ (t) + G∗ ~ a† (t) . + | 2 ih 1 | G21,~kλ (R) ( R) ~ kλ 12,~ kλ kλ

We recall that ~ = − i Ek Gab,~kλ (R)

~dab · ~e~ kλ

~ ~

eik·R .

(5.72)

(5.73)

By the similar argument as in the case of minimum coupling we take the polarization vectors to be real. Then we have ~ ∗ ~ ~ ~ G12,~kλ = − i Ek ~d12 · ~e~kλ eik·R = − i Ek ~d21 · ~e~kλ eik·R , (5.74a) ~ ~ G∗12,~kλ = i Ek ~d21 · ~e~kλ e−ik·R , (5.74b) ~ ~ G21,~kλ = − i Ek ~d21 · ~e~kλ eik·R , (5.74c) ∗ ~ ~ G∗21,~kλ = i Ek ~d21 · ~e~kλ e−ik·R . (5.74d) We now adopt a new notation (see (5.23)) ~G~kλ S.Kryszewski

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which, similarly as previously, simplifies the employed notation. Thus, we get ~ ~

~ ~

G12,~kλ = − ~ G~∗kλ eik·R ,

G21,~kλ = ~ G~kλ eik·R ,

G∗12,~kλ = − ~ G~kλ e−ik·R ,

G∗21,~kλ = ~ G~∗kλ e−ik·R ,

~ ~

~ ~

(5.76)

Using this notation in the hamiltonian (5.72) we obtain X n ~ ~ ~ ~ dip HAF = ~ − G~∗kλ eik·R S− a~kλ (t) + G~∗kλ e−ik·R S− a~† (t) kλ

~ kλ

o ~ ~ ~ ~ + G~kλ eik·R S+ a~kλ (t) − G~kλ e−ik·R S+ a~† (t) .

(5.77)

kλ

Inspecting the obtained expression we see that we have four terms which have the same interpretation as that given after Eq.(5.69). As previously we recognize that the second and the third terms are resonant while the remaining two (the first and the fourth) are nonresonant. In the present case we distinguish the nonresonant terms by the control parameter ξ¯ = −1, and directly by rearrangement of (5.77) we have X n ~ ~ ~ ~ dip HAF = ~ G~kλ eik·R S+ a~kλ (t) + G~∗kλ e−ik·R S− a~† (t) kλ

~ kλ

+ ξ¯ = ~

X n ~ kλ

G~kλ S+

h

~ ~

~ ~

G~∗kλ eik·R S− a~kλ (t) + G~kλ e−ik·R S+ a~† (t) kλ

~ ~ ~ ~ eik·R a~kλ (t) + ξ¯ e−ik·R a~† (t) kλ

i

o

h io ~ ~ ~ ~ + G~∗kλ S− ξ¯ eik·R a~kλ (t) + e−ik·R a~† (t) kλ

If we explicitly put ξ¯ = −1 in (5.78), then we can write X ~ ~ ~ ~ dip G~kλ S+ − G~∗kλ S− eik·R a~kλ (t) − e−ik·R a~† (t) , HAF = ~ kλ

(5.78)

(5.79)

~ kλ

~ = 0. We remind, that the adopted notation scheme is not which is especially simple when R unique. In literature other schemes are employed, so necessary care must be exercised to find the connections between various ways of notation. We note the formal similarity of equations (5.70) and (5.71) for the minimum coupling with corresponding equations (5.78) and (5.79) for the dipole coupling. This similarity also explains, why in the present case we have chosen the control parameter ξ¯ = −1. We will use these formulas in future applications. Summary Having introduced the notation r Ek ~ ωk ~d21 · ~e~ , G~kλ = − i d21 · ~e~kλ = − i kλ ~ 2~ 0 V

(5.80)

we have obtained two forms of the hamiltonian describing the interaction between a two-level atom and the electromagnetic field. • For the minimal coupling scheme we have X |ω21 | n ~ ~ ~ ~ min HAF = ~ G~kλ eik·R S+ a~kλ (t) + G~∗kλ e−ik·R S− a~† (t) kλ ωk ~kλ h io ~ ~ ~ ~ + ξ G~∗kλ eik·R S− a~kλ (t) + G~kλ e−ik·R S+ a~† (t) , kλ

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with ξ = 1, or with ξ = 0 which corresponds to the resonant approximation. • On the other hand, for the dipole coupling scheme we have X n ~ ~ ~ ~ dip HAF = ~ G~kλ eik·R S+ a~kλ (t) + G~∗kλ e−ik·R S− a~† (t) kλ

~kλ

h io ~ ~ ~ ~ + ξ¯ G~∗kλ eik·R S− a~kλ (t) + G~kλ e−ik·R S+ a~† (t) , kλ

(5.82)

where ξ¯ = −1, or ξ¯ = 0 for the resonant approximation.

5.3.3

Further comments on resonant approximation

Resonant approximation already discussed in the previous section is sometimes called rotating wave approximation (RWA, this name will become clear later on). Such an approximation consists in neglecting the nonresonant terms which appear in the hamiltonian. As already mentioned, it is equivalent to putting ξ = ξ¯ = 0 in Eqs.(5.81) or (5.82). In such a case, we have o X n ~ ~ ~ ~ dip HAF = ~ G~kλ eik·R S+ a~kλ (t) + G~∗kλ e−ik·R S− a~† (t) (5.83a) kλ

~ kλ

min HAF

o X |ω21 | n ~ ~ ~ ~ = ~ G~kλ eik·R S+ a~kλ (t) + G~∗kλ e−ik·R S− a~† (t) kλ ωk

(5.83b)

~kλ

These expressions differ only by the ratio of frequencies appearing in the minimal coupling hamiltonian. If this factor can be approximated by the unity, then (5.83a) and (5.83b) coincide. Indeed, the two-level model is valid provided the central frequency of the field is close to the atomic one. So, in the sum in (5.83b) only those ω k contribute significantly for which ω k ≈ ω21 . Hence, the mentioned factor can be approximated by a unity. Therefore, with the resonant approximation, we arrive at a single expression for the interaction Hamiltonian for two-level atom coupled to the electromagnetic field. It reads o X n ~ ~ ~ ~ HAF = ~ G~kλ eik·R S+ a~kλ (t) + G~∗kλ e−ik·R S− a~† (t) (5.84) kλ

~ kλ

with coupling coefficients given in (5.80). In most cases of physical interest it is reasonable to assume that the resonant (rotating wave) approximation holds very well. Thus, the interaction Hamiltonian (5.84)) will be usually employed to describe the discussed interaction.

5.3.4

Single mode case

In many practical cases the field illuminating an atom consists of a single mode (eg., laser radiation) with specified characteristics: wave vector ~k and frequency ω. In such a case, the sum over the field modes is trivial, it contains just one term. Then, the interaction Hamiltonian is simply n ~ o ~ ~ ~ ˆ + e−ik·R S− a ˆ† (5.85) HAF = ~ G eik·R S+ a

where we omitted the indices, since they are not necessary as we deal with only one, well specified mode. The coupling constant follows from (5.80). It may be assumed real, because the phases may be incorporated into atomic lowering and raising operators. Hence it is r ω ~d21 · ~e = E ~d21 · ~e . G = (5.86) 2~ 0 V ~ S.Kryszewski

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5.4

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Semiclassical approximation

5.4.1

General discussion

We start the discussion of the two-level atom interacting with electromagnetic field by invoking a general Hamiltonian (5.27) corresponding to the dipole coupling. We shall focus our attention on this type of coupling and we will omit the superscript ”dip”. So we have ~ R, ~ t) = − | 2 ih 1 | ~d21 + | 1 ih 2 | ~d12 · E( ~ R, ~ t), HAF = − ~d · E( (5.87)

where we treat the field quantum-mechanically. At present we will adopt a slightly different approach. Firstly, we write the field as the sum of the positive- and negative-frequency parts: ~ =E ~ (+) + E ~ (−) . Thus, the hamiltonian (5.87) is written as the sum of four terms E ~ (+) − | 1 ih 2 | ~d12 · E ~ (+) HAF = − | 2 ih 1 | ~d21 · E ~ 21 · E ~ (−) − | 1 ih 2 | ~d12 · E ~ (−) , −| 2 ih 1 | d

(5.88)

where, for sake of brevity, we have omitted the arguments of the fields. Since the positive~ (+) contains annihilation operators, while the negative-frequency part E ~ (−) frequency field E creation ones, we see that the second and third terms are nonresonant. Performing rotating wave approximation (RWA) we can neglect those terms. Therefore, we have the interaction Hamiltonian of the form ~ (+) − | 1 ih 2 | ~d12 · E ~ (−) . HAF = − | 2 ih 1 | ~d21 · E

(5.89)

~ (±) (R, ~ t) quantum mechanically. Although the RWA is already made, we can still treat the fields E However, the aim of this section is to deal with semiclassical fields. Within this intuitive approach we can treat the fields appearing in Eq.(5.89) as the corresponding amplitudes of the classical electric field. Such amplitudes are usual (vector) functions of the position of the atomic ~ and time t. Adopting this simplification we can introduce the quantity which center of mass R is called ”Rabi frequency” and defined as ~ t) = Ω ≡ Ω(R,

1~ ~ (+) (R, ~ t). d21 · E ~

(5.90)

Since we have made a semiclassical approximation, the Rabi frequency is not an operator but a classical function of its arguments. With the aid of the defined Rabi frequency we can write the Hamiltonian (5.89) as HAF = − ~Ω | 2 ih 1 | − ~Ω∗ | 1 ih 2 | = − ~Ω S+ − ~Ω∗ S− .

(5.91)

We stress that Ω is a numeric (scalar) function, perhaps quite complicated, of position and time, so that all operator character of the fields is lost. Fields are no more quantum-mechanical but simply classical. As we have discussed, the minimal coupling Hamiltonian (5.81) can be shown to lead to the same semiclassical Hamiltonian. Indeed, when we take (5.81), employ RWA and plug in G ~kλ from (5.80), then we get o X |ω21 | n ~ ~ ~ ~ min HAF = ~ G~kλ eik·R S+ a~kλ (t) + G~∗kλ e−ik·R S− a~† (t) kλ ωk ~ kλ X ω21 n ~ ~ = | 2 ih 1 | ~d21 · − iEk ~e~kλ eik·R a~kλ (t) ωk ~kλ o ~ ∗ −i~ + | 1 ih 2 | ~d12 · iEk ~e~kλ e k·R a~† (t) (5.92) kλ

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The factors in normal brackets are (up to the sign) nothing else but the amplitudes of the fields ~ (+) and E ~ (−) , respectively, as seen by comparing with Eq.(3.59b). Therefore, instead of (5.92) E we can write o ω21 n min ~ (+) + | 1 ih 2 | ~d12 · E ~ (−) , HAF = − | 2 ih 1 | ~d21 · E (5.93) ωk which, as previously, is almost of the same form as H AF in (5.91). The only difference consists in the appearance of the additional factor ω 21 /ωk which, as discussed under RWA, can be approximated by the unity. Then (5.93) and (5.91) coincide exactly. Thus, we see that in the RWA and in the semiclassical approximation we can deal with the Hamiltonian (5.91) regardless of the adopted coupling scheme, either dipole or the minimal one. Single-mode field ~ and In general, the Rabi frequency defined in Eq.(5.90) is a complicated function of position R time t. The situation is much simpler if we restrict our attention to a single mode field with wave vector ~k and frequency ω. For such a case, the sum over the modes (implicit in the field ~ (+) ) contains effectively only one term. The Rabi frequency can then be written as follows E ~ ~ t) = 1 ~d21 · E ~ 0 ei~k·R Ω(R, e−iωt , (5.94) ~

~ 0 which is assumed to be a constant, specifies the amplitude and polarization where the vector E of the given mode. Eq.(5.94) follows from the classical expression for the positive frequency part of the single mode of the electric field. Generally, the quantity in brackets in Eq.(5.94) is complex and so is the Rabi frequency. There is, therefore, some flexibility of notation. We shall write ~ t) = χ(R) ~ e−iφ−iωt , Ω(R,

(5.95)

~ being complex, with where, despite the separation of the phase factor e −iφ , we allow for χ(R) ~ ~ = 1 ~d21 · E ~ 0 ei~k·R χ(R) (5.96) ~

Two last definitions require some explanation. We have always assumed the phase of the field to be zero. Allowing for nonzero φ in (5.95) generalizes the case. Moreover, the origin of the factor e−iφ can be given another interpretation. For example, the frequency of the field may be slowly varying (chirped pulse). Then the phase φ will be a slowly varying function of time. Such a case requires care, since there is an additional source of time variation. Nevertheless, it is convenient to adopt a general scheme, which can be easily adapted to various physical situations. ~ depends on the position of the atomic center We also stress that the new quantity χ( R) ~ enters the relevant of mass. This dependence is, however, parametrical. This means that R equations of motion as a parameter. In most of the physically important cases such equations of motion can be considered local, that is depending on the position parametrically. Hence, in most cases we will not indicate that χ is position dependent, but this fact should not be forgotten. In the view of the above remarks we redefine the Rabi frequency according to (5.95) and we finally adopt the interaction Hamiltonian in the form following from (5.91) with (5.95) taken into account. Thus, we have HAF = − ~χe−iφ−iωt | 2 ih 1 | − ~χ∗ eiφ+iωt | 1 ih 2 | c = − ~χe−iφ−iωt S+ − ~χ∗ eiφ+iωt S− ,

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with χ defined in (5.96) and called Rabi frequency. We remind that this is a semiclassical Hamiltonian with RWA for a two-level atom interacting with a (classical) single mode field of frequency ω and wave vector ~k. We will show, that consistent quantum-mechanical description of the field as the coherent state, leads to the same result.

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Spontaneous emission 6.1

Density operator for open system

The formalism of the density operator is especially useful for description of the open system. By an open system we understand a system consisting of two parts. The first part is labelled by S and the observations (measurements) we make concern only this part of the total system. The second part, labelled by B is the surrounding which may exchange energy with the subsystem S. Thus, the subsysten B is a heat reservoir, such as frequently discussed in the context of statistical mechanics. System S may, for example, correspond to a vapor cell with some gas, while system B is the surroundings which may exchange the heat with the gas (across the cell walls). Another example is an atom immersed in the radiation field. The heat bath B is considered to be much larger that the system of interest S. That is, the number of the degrees of freedom of the bath is much larger. In many cases it is possible to assume that the bath is always in thermal equilibrium. The state vector | Ψ i of the joint system S + B can always be expanded on an orthonormal basis of states {| ξn i} of B X |Ψi = Cn | ψ n i | ξ n i (6.1) n

where the | ψn i are normalized (but not necessarily orthogonal) states of subsystem S. The coefficients Cn are the probability amplitudes that the bath B is in its basis state | ξ n i, which correlates with the normalized state | ψ n i of the subsystem S. The normalization of the joint state | Ψ i is as follows XX ∗ 1 = hΨ|Ψi = h ξm | h ψ m | Cm Cn | ψ n i | ξ n i m

=

XX m

=

X m

∗ Cm

n

2

n

Cn h ξ m | ξ n i h ψ m | ψ n i =

| Cm | h ψm | ψm i =

X m

2

XX m

n

| Cm | .

∗ Cm Cn δmn h ψm | ψn i

(6.2)

Performing this computation we note that the states of the bath, that is states | ξ n i and the states | ψn i of subsystem S are independent. Hence, we take separate scalar products, corresponding to two different Hilbert spaces. We again stress, that states | ψ n i of S are only normalized, no assumption is made on their orthogonality. Finally, the result (6.2) indicates that interpretation of coefficients Cn as probability amplitudes is indeed correct. Let us now consider a measurement of an observable A performed on subsystem S only. By measurement on S we understand that the operator A operates only on states of S. The S.Kryszewski

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expectation value of A, due to orthonormality of the basis states { | ξ n i }, can be then found to be X hAi = hΨ|A|Ψi = |Cn |2 h ψn | A | ψn i. (6.3) n

This relation is obviously equivalent to h A | = iTrs+b { %s+b A }

(6.4)

with %s+b = | Ψ ih Ψ | being the density operator for the joint (total) system S + B, while Tr s+b indicates that we take the trace with respect to the states of both subsystems. The density operator %s+b includes information on both parts of the whole system. However, we are not interested in the heat reservoir. Therefore, we define the reduced dendity operator % s which contains data only on the relevant subsystem S. We define % s as %s = Trb { %s+b } = Trb { | Ψ ih Ψ | } .

(6.5)

Then, using (6.1) we calculate the trace with respect to the basis vectors { | ξ k i } of the subsystem B. We get ( ) X ∗ %s = Trb Cn | ψn i| ξn ih ξm |h ψm |Cm m,n

=

X

k,m,n

=

X

k,m,n

=

X m,n

=

X n

∗ Cm Cn h ξ k |

h

| ψn i| ξn ih ξm |h ψm |

i

| ξk i

∗ Cm Cn | ψn ih ξk | ξn ih ξm | ξk ih ψm | =

∗ Cm Cn | ψn i δmn h ψm |

| ψn i | Cn |2 h ψn |.

X

k,m,n

∗ Cm Cn | ψn i δkn δmk h ψm |

(6.6)

The advantage of the reduced density operator is that in order to find expectation value of the observable A concerning the subsystem S only we can write h A | = iTrs { %s A },

(6.7)

where Trs indicates that we compute the trace only with respect to the states of the subsystem S. To find the trace in (6.7) we introduce a basis { | ϕ a i } in the Hilbert space of states of the subsystem S. Then, we have X h A | i = Trs { %s A } = h ϕa | A % s | ϕa i a

=

XX a

=

a

=

n

XX X n

n

h ϕa | A | ψ n i | C n | 2 h ψn | ϕa i | C n |2 h ψ n | ϕ a i h ϕ a | A | ψ n i

| C n |2 h ψ n | A | ψ n i

(6.8)

Comparison of (6.8) and (6.3) shows that the reduced density operator is indeed sufficient to find necessary information about measurements on subsystem S only. The structure of the 98

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density operator %s as in (6.6) is similar to that given in Eq.(9.15). The difference is that the statistical probabilities are now replaced by quantum-mechanical probabilities | C n |2 . We recall that the amplitudes Cn are the probability amplitudes that the reservoir B is in its basis state | ξn i, which correlates with the normalized state | ψ n i of the observed subsystem S which is of interest. The discussed case concerns a pure state | Ψ i of the joint system. Similar considerations can be also done for a more general case, when the joint system is in the mixed state. It is, however, worth noting that even in this simple case the description of the relevant subsystem S must be done with density operator. Although the joint system is in the pure state, the subsystem S is in the mixed state described by the reduced density operator % s which is defined in eq.(6.6).

6.2

Evolution of the reduced density operator

6.2.1

Introductory remarks

We are interested in the evolution of the subsystem S which is coupled to the heat reservoir B. This evolution cannot be described by the von Neumann equation for the subsystem S alone due to the influence of the interaction with B. Therefore, we must start studying the evolution of %s with the evolution of the whole system S + B, which is assumed to be closed. The total density operator %s+b obeys the von Neumann equation (9.41), where the total Hamiltonian of the combined system can be separated as HT = H0 + V,

with

H 0 = Hs + Hb ,

(6.9)

where Hs and Hb describe the free evolution of subsystems S and B, whereas V describes their interaction. We note that the Hamiltonian H s operates only on states | ψn i of the subsystem S, while Hb only on the state vectors | ξn i of the reservoir B. The von Neumann equation for the total system reads i~

h i d %s+b = HT , %s+b dt

(6.10)

This is our starting point for finding the evolution equation for the reduced density operator for the system of interest %s = Trb {%s+b }. Since the reservoir may safely be assumed to be large, its state is very weakly influenced by its interaction with subsystem S. The reservoir is also expected to be incoherent in the sense, that its correlations decay very quickly. We will express the evolution of the subsystem S by these correlations and employ the expected short correlations.

6.2.2

Transformation to interaction picture and formal integration

The main advantage of the interaction picture is that the free time evolution (due to free Hamiltonian H0 ) is transformed away. The remaining time evolution is entirely due to the interaction. Therefore we take the full density operator % s+b (t) in the Schr¨odinger picture and we transform it to the interaction picture T rs+b (t) = eiH0 t/~ %s+b (t) e−iH0 t/~ ,

(6.11)

or equivalently %s+b (t) = e−iH0 t/~ T rs+b (t) eiH0 t/~ , S.Kryszewski

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where tilde denotes the interaction picture. Before proceeding furter with the problem of evolution, let us note that the definition (6.5) of the reduced density operator implies (in the Schr¨odinger picture) n o %s (t) = Trb e−i(Hs +Hb )t/~ T rs+b (t) ei(Hs +Hb )t/~ . (6.13)

It is not obvious that we can simply use the cyclic property of the trace. The reason is that S variables are not affected by Trb . Free Hamiltonians Hs and Hb act in two different Hilbert spaces, so they commute, hence we can write e±i(Hs +Hb )t/~ = e±iHs t/~ e±Hb t/~ .

(6.14)

In (6.13) we compute the trace with respect to reservour variables only, hence due to (6.14) we can write n o %s (t) = e−iHs t/~ Trb e−iHb t/~ T rs+b (t) eiHb t/~ eiHs t/~ . (6.15) Using the cyclic property of the trace, we obtain eiHs t/~ %s (t) e−iHs t/~ = Trb { T rs+b (t) } .

(6.16)

We easily see that the left-hand side represents the reduced density operator in the interaction picture (since it depends solely on the variable of the subsystem S it is not influenced by the exponential term including Hb ). So we have T rs (t) = Trb { T rs+b (t) } .

(6.17)

Hence the relation (6.5) is formally indentical to (6.17). The connection between the reduced density operator and the total one is the same in both pictures. Now, we transform the von Neumann equation (6.10) into the interaction picture. We insert (6.12) into the lhs, and we use (6.9) in the rhs. By differentiation we get d i −iH0 t/~ iH0 t/~ −iH0 t/~ T rs+b (t) e + e i~ − H0 e T rs+b (t) eiH0 t/~ ~ dt i −iH0 t/~ iH0 t/~ +e T rs+b (t) H0 e ~ = ( H0 + V ) e−iH0 t/~ T rs+b (t) eiH0 t/~ − e−iH0 t/~ T rs+b (t) eiH0 t/~ ( H0 + V )

(6.18)

We multiply this equation by eiH0 t/~ at the left and by e−iH0 t/~ at the right. We also notice that the Hamiltonian Ho commutes with the exponential operators. Hence, we get d Ho T rs+b (t) + i~ T rs+b (t) − T rs+b (t) H0 dt = Ho T rs+b (t) + eiH0 t/~ V e−iH0 t/~ T rs+b (t)

− T rs+b (t) Ho − T rs+b (t) eiH0 t/~ V e−iH0 t/~ .

(6.19)

As it could be expected when working in the interaction picture, the terms containing H o cancel out, and we arrive at i d 1 h˜ T rs+b (t) = V (t), T rs+b (t) , dt i~ 100

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where V˜ (t) is the interaction hamiltonian in the interaction picture V˜ (t) = eiH0 t/~ V e−iH0 t/~ .

(6.21)

Equation (6.20) is the interaction picture version of the von Neumann equation for the density operator of the full system S + B. We shall investigate it further to extract evolution equation for the reduced density operator %s fot the subsystem of interest. In the next step we formally integrate the von Neumann equation (6.20). This yields 1 T rs+b (t) = T rs+b (t0 ) + i~

Z

t

dt1 t0

h

i V˜ (t1 ), T rs+b (t1 ) ,

(6.22)

where T rs+b (t0 ) is the initial condition. The result of the iteration is the substituted into the von Neumann equation, and we get Z i h h ii d 1 h˜ 1 2 t T rs+b (t) = V (t), T rs+b (t0 ) + dt1 V˜ (t), V˜ (t1 ), T rs+b (t1 ) . (6.23) dt i~ i~ t0 We will analyze this equation in great detail. Taking the trace Tr b which commutes with time derivative, we obtain io nh 1 d T rs (t) = Tr b V˜ (t), T rs+b (t0 ) dt i~ Z nh h iio 1 2 t + dt1 Tr b V˜ (t), V˜ (t1 ), T rs+b (t1 ) . (6.24) i~ t0 This is still an exact equation. However, to get useful information we must introduce several simplifying assumptions which will yield a tractable and closed equation for % s (t). This is so, because right-hand side of (6.24) still contains the full density operator T r s+b (t).

6.2.3

Assumptions about reservoir

To use equation (6.24) practically, we need some additional assumptions concerning the reservoir. These assumptions will enable us to simplify the discussed equation and to make and justify some approximations needed to obtain a closed equation for the reduced density operator % s (t) of the system of interest. Reduced density operator of the reservoir In analogy to (6.5) we define the reduced density operator of the reservoir %b (t) = Trs { %s+b (t) }.

(6.25)

Since the reservoir is very large, we may assume that it depends very weakly on the intetaction with subsystem S. We will neglect the influence on B of the perturbation due to the coupling between S + B. Therefore, we will assume that %b (t) ≈ %b (t0 ) = σ ¯B .

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Stationarity of the reduced density operator of the reservoir We will also assume that σ ¯ B – reduced density operator of the reservoir is stationary, that is time-independent. It is equivalent to assume that h i σ ¯ B , Hb = 0. (6.27) Since σ ¯B and Hb commute, they posses a common set of eigenstates. Hence we can write Hb | b i = Eb | b i,

(6.28)

and simultaneously X σ ¯B = p(a)| a ih a |.

(6.29)

a

The eigenstates | a i of the reservoir Hamiltonian are orthonormal and form a basis in the corresponding Hilbert space. It is worth noting that in many practical cases we can take X Ea Ea 1 exp − , with Z= exp − . (6.30) p(a) = Z kB T kB T a This obviously corresponds to the case of thermodynamic equlibrium. Special form of the interaction hamiltonian Our third assumption consists in adopting a special for of the interaction hamiltonian. Namely we will take X V = Ai Xi , (6.31) i

where Ai are the operators acting in the Hilbert space of the states of subsystem S, while X i act in the space of the states of the reservoir. All operators in (6.31) are taken to be hermitian, because they are observables pertaining to two systems. Since these operators act in different spaces, they are independent and they commute. Obviously, in the interaction picture we have X ˜ i (t), V˜ (t) = A˜i (t) X (6.32) i

with A˜i (t) = eiHs t/~ Ai e−iHs t/~

and

˜ i (t) = eiHb t/~ Xi e−iHb t/~ . X

(6.33)

Zero average value of the interaction Our fourth assumption concerning the reservoir is summarized be the requirement Trb { Xi %b (t) } = Trb { Xi σ ¯ B } = 0.

(6.34)

Since σ ¯B commutes with Hb is poses no problems to transform this requirement to the interaction picture. So we have n o ˜ i (t) σ Trb X ¯B = 0. (6.35) 102

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As a consequence, the average value of the interaction calculated with respect to the state of the reservoir vanishes. Indeed, we have ( ) n o X ˜ i (t) σ A˜i (t) X h V˜ (t) ib = Trb V˜ (t) σ ¯B = Trb ¯B i

=

X

A˜i (t) Trb

i

n

˜ i (t) σ X ¯B

o

= 0,

(6.36)

because A˜i (t) does not depend on the variables over which we compute the trace. Hence, the expectation value of the interaction with respect to the state of the reservoir vanishes. Two-time correlations The previous assumption concerns the one-time average of the interaction. We consider now the two-time averages, or in other words, the correlation function     X X ˜ i (t1 ) ˜ j (t2 ) h V˜ (t1 ) V˜ (t2 ) ib = Trb σ ¯B A˜i (t1 ) X A˜j (t2 ) X   i j n o X ˜ i (t1 ) X ˜ j (t2 ) A˜i (t1 ) A˜j (t2 ) Trb σ ¯B X = i,j

=

X

A˜i (t1 ) A˜j (t2 ) Gij (t1 , t2 ).

(6.37)

i,j

˜ The last equality defines the two-time correlation function of the operators X(t) with respect to the state of the reservoir n o ˜ i (t1 ) X ˜ j (t2 ) Gij (t1 , t2 ) = Tr b σ ¯B X (6.38) We show one of the basic properties of the introduced correlation function, namely o∗ † n ∗ ˜ ˜ ˜ ˜ = Tr b Gij (t1 , t2 ) = Tr b σ ¯B Xi (t1 ) Xj (t2 ) σ ¯B Xi (t1 ) Xj (t2 ) n o ˜ j (t2 ) X ˜ i (t1 ) σ = Tr b X ¯B = Gji (t2 , t1 ).

(6.39)

Although the one-time average (6.35) vanishes by definition, it is not apriori clear that the two-time average Gx also vanishes. Just the opposite, the correlation function usually is nonzero. It can be shown that such a two-time correlation function is connected with dynamics of the fluctuations in the reservoir and with linear susceptibility of the system with respect to the external perturbations. These facts are, however, important in other contexts. At present we will not study the properties of these two-time correlations. Stationarity of two-time correlation functions The reduced density operator of the reservoir is assumed to be stationary. This fact has an important consequence regarding the two-time correlation function. From the definition, we have n o Gij (t1 , t2 ) = Trb σ ¯B eiHb t1 /~ Xi e−iHb t1 /~ eiHb t2 /~ Xj e−iHb t2 /~ . (6.40)

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Employing the cyclic property of trace and commutation of σ ¯ B with Hb we get n o Gij (t1 , t2 ) = Trb σ ¯B eiHb (t1 −t2 )/~ Xi e−iHb (t1 −t2 )/~ Xj n o ˜ j (0) ¯ ij (t1 − t2 ). ˜ i (t1 − t2 ) X ¯B X = G = Trb σ

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(6.41)

¯ ij depends effectively only on one time argument. Hence we write The correlation function G n o n o ¯ ij (τ ) = Trb σ ˜ i (τ ) Xj G ¯B X = Trb σ ¯B eiHb τ /~ Xi e−iHb τ /~ Xj (6.42)

We call such a correlation function a stationary one. Its form is a result of the stationarity of the density operator σ ¯B . Using the obtained results in (6.37) we express the two-time average of the interaction as X ˜ 1 ) A(t ˜ 2) G ¯ ij (t1 − t2 ), A(t h V˜ (t1 ) V˜ (t2 ) ib = (6.43) i,j

which still is a two-time function. For sake of completeness, let us compute the explicit form of the stationary correlation ¯ ij (τ ) in the basis common to the reservoir Hamiltonian and density operator. From function G (6.41), (6.33) and (6.29) we obtain ( ) X iH τ /~ −iH τ /~ b ¯ ij (τ ) = Tr b G p(a)| a ih a |e b Xi e Xj a

=

X a

=

X a,b

=

X a,b

p(a) h a | eiHb τ /~ Xi e−iHb τ /~ Xj | a i

p(a) eiωa τ h a | Xi | b i h b | Xj | a ie−iωb τ /~ p(a) eiωab τ h a | Xi | b i h b | Xj | a i.

(6.44)

Writing these relations we employed standard and self-explanatory notation. We see that the stationary correlation function of the reservoir operators X i is a superposition of oscillations with Bohr frequencies. The reservoir is very large, so that the Bohr frequncies are very densely packed (with very small spacing). Therefore, if the time τ is large enough, these oscillations will interfere destructively. So, for τ sufficiently large, the correlation function is essentially zero. We denote by τc the characteristic time of the reservoir correlations. We may say that for times τ τc the correlations in the reservoir vanish. Summary of the properties of the reservoir The assumptions made may be summarized as follows. • Reservoir is large and it is in the stationary state which is described by the reduced density operator σ ¯ B which commutes with Hamiltonian Hb . See eqs.(6.26) and (6.27). • We have assumed the interaction between S and B to be in the special factorized form (6.31). • The reservoir exerts a ”force” on the system S. This force fluctuates around zero average value. See eqs.(6.32) and (6.36). • The correlations of this ”force” are given in (6.43). They decay with time scale given by τc which may be assumed to be short so the correlations decay very rapidly. 104

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Derivation of the master equation

We return to the analysis of our main, and still exact eqaution (6.24) of motion for the reduced dendity operator of the subsystem S. It is nh io 1 d V˜ (t), T rs+b (t0 ) T rs (t) = Tr b dt i~ Z iio nh h 1 2 t + dt1 Tr b . (6.45) V˜ (t), V˜ (t1 ), T rs+b (t1 ) i~ t0 As the first step we make the assumptions central to further developments. We assume that at any time instant the density operator of the whole system can be written as T rs+b (t) = T rs (t) T rb (t) + T rcorel (t)

(6.46)

where the last term T rcorel (t) describes the correlations between subsystems S and B. It reasonable to think that such an assumption is exact. But now comes the main approximation. We will neglect the correlation term and we will adopt assumption (6.26) about density operator od the reservoir. Hence, we approximately write ¯B T rs+b (t) ≈ T rs (t) σ

(6.47)

We will use this approximation and its discussion and justification will be done later. At present we insert (6.47) into (6.45) and we get o n d 1 T rs (t) = Tr b V˜ (t) T rs (t0 ) σ ¯B − T rs (t0 ) σ ¯B V˜ (t) dt i~ Z nh h iio 1 2 t + dt1 Tr b ¯B V˜ (t), V˜ (t1 ), T rs (t1 ) σ . (6.48) i~ t0 Due to our assumption (6.36) about vanishing average of the interaction we see that the first term does not contribute. Only the second term remains, and we have Z nh h iio d 1 2 t T rs (t) = dt1 Tr b V˜ (t), V˜ (t1 ), T rs (t1 ) σ ¯B . (6.49) dt i~ t0 The integral contains the correlations between subsystems. But the time τ c is very short, hence these correlations are significantly different from zero only for times t 1 which are close to the instant t, the initial correlations arising at t 0 decay quickly. Hence the integral does not vanish only for times close to the final moment t. Therefore, if we assume that the time interval ∆t = t − t0 is large as compared to τc , we can replace the argument of T rs (t1 ) by the final moment t. This is also an approximation which should be discussed and justified. Now, we write Z nh h iio d 1 2 t T rs (t) = dt1 Tr b V˜ (t), V˜ (t1 ), T rs (t) σ ¯B . (6.50) dt i~ t0 We change the integration variable from t 1 to τ = t − t1 and we obtain Z nh h iio d 1 2 t−t0 T rs (t) = dτ Tr b V˜ (t), V˜ (t − τ ), T rs (t) σ ¯B . dt i~ 0

(6.51)

The correlation are important only when times t and t − τ differ slightly (approximately by τ c ). Since ∆t = t − t0 is large, the main contribution to the integral comes from the vicinity of τ S.Kryszewski

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close to zero. Hence, we make a small error when we push the upper limit of integration to infinity. In such a manner we have d T rs (t) = dt

1 i~

2 Z

∞

dτ Tr b

0

nh

V˜ (t),

h

V˜ (t − τ ), T rs (t) σ ¯B

iio

.

(6.52)

This is our main result. What remains is to discuss the approximations made and to transform this result to more tractable form.

6.2.5

Transformations of the master equation

Obtaining the master equations we used only one assumption concerning the reservoir, namely stationarity of its density operator. We have not employed the assumptions on the form (6.31) of the interaction. Using this assumption and expanding the commutators we get Z n X d 1 2 ∞ ˜ i (t) A˜j (t − τ ) X ˜ j (t − τ ) T rs (t) σ T rs (t) = dτ Tr b A˜i (t) X ¯B dt i~ 0 i,j

˜ i (t) T rs (t) σ ˜ j (t − τ ) − A˜i (t) X ¯B A˜j (t − τ ) X ˜ j (t − τ ) T rs (t) σ ˜ i (t) − A˜j (t − τ ) X ¯B A˜i (t) X ˜ j (t − τ ) A˜i (t) X ˜ i (t) + T rs (t) σ ¯B A˜j (t − τ ) X

o

(6.53)

Operators of two subsystems are independent and commute, so we have Z n o X h d 1 2 ∞ ˜ ˜ ˜ ˜ T rs (t) = dτ Ai (t) Aj (t − τ ) T rs (t) Tr b Xi (t) Xj (t − τ ) σ ¯B dt i~ 0 i,j n o ˜ i (t) σ ˜ j (t − τ ) − A˜i (t) T rs (t) A˜j (t − τ ) Tr b X ¯B X n o ˜ j (t − τ ) σ ˜ i (t) − A˜j (t − τ ) T rs (t) A˜i (t) Tr b X ¯B X n oi ˜ j (t − τ ) X ˜ i (t) + T rs (t) A˜j (t − τ ) A˜i (t) Tr b σ ¯B X (6.54)

The quantities involving Tr b are easily recognized as the stationary correlation functions of the reservoir. From (6.40) and (6.41), using the cyclic property of the trace (where necessary) and relation (6.39) we transform the above equation into Z X h d 1 2 ∞ ¯ ij (τ ) T rs (t) = dτ A˜i (t) A˜j (t − τ ) T rs (t) G dt i~ 0 i,j

¯ ∗ (τ ) − A˜i (t) T rs (t) A˜j (t − τ ) G ij ¯ ij (τ ) − A˜j (t − τ ) T rs (t) A˜i (t) G ¯ ∗ij (τ ) + T rs (t) A˜j (t − τ ) A˜i (t) G

i

(6.55)

All operators are Hermitian, hence we see that the first and the fourth terms are hermitian conjugates of each other, so are the second and the third terms. We retain the first and the third terms, and we write Z h X d 1 2 ∞ ¯ ij (τ ) A˜i (t) A˜j (t − τ ) T rs (t) T rs (t) = dτ G dt i~ 0 i,j i − A˜j (t − τ ) T rs (t) A˜i (t) + H.C. (6.56) 106

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The operators in this equation are expressed in the interaction picture. We are used to working in the Schr¨odinger one, hence we will transform back to the latter picture. We must perform the transformation inverse to (6.12). It is obvious that the term with H b can be omitted and transformation of the derivative will reproduce the free evolution, due to H s . Thus, we get i d 1 h %s (t) − Hs , % s dt i~ Z h X 1 2 ∞ ¯ ij (τ ) e−iHs t/~ A˜i (t) A˜j (t − τ ) T rs (t) = dτ G i~ 0 i,j i − A˜j (t − τ ) T rs (t) A˜i (t) eiHs t/~ + H.C. (6.57) Transformation of the operators back to Schr¨odinger picture is straighforward. The resulting Schr¨odinger picture master equation is of the form i d 1 h %s (t) − Hs , % s dt i~ Z h X 1 2 ∞ ¯ ij (τ ) Ai e−iHs τ /~ Aj eiHs τ /~ %s (t) = dτ G i~ 0 i,j i − e−iHs τ /~ Aj eiHs τ /~ %s (t) Ai + H.C. (6.58) This is the final form of the master equation, which we will exploit in practical cases.

6.2.6

Discussion of the approximations

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Spontaneous emission in two-level atom

6.3.1

Description of the atom + field system

We consider a two-level atom in the quantized electromagnetic field. We may adopt Hs = HA = ~ ω21 S3 .

(6.59)

Hamiltonian of the field which plays the role of the reservoir is X Hb = H F = ~ωk a~† a~kλ .

(6.60)

As the interaction hamiltonian in the RWA we take (5.84) i X h ∗ † g~kλ a~kλ S+ + g~kλ a~ S− , V = HAF = − ~

(6.61)

kλ

~ kλ

kλ

~ kλ

with the coupling coefficient r ωk ~d21 · ~e~ , g~kλ = i kλ 2 V ~ 0

(6.62)

where we have assumed that the atom is placed at the origin of the coordinate system and that ~ = 0). Hence, we see that the interaction hamiltonian is given in the it is at rest (we have put R form (6.31) with identifications X X1 = − ~ g~kλ a~kλ , (6.63a) A1 = S + , ~ kλ

A2 = S − ,

X2 = − ~

X

∗ † g~kλ a~ . kλ

(6.63b)

~ kλ

We note that each of the operators is nonhermitian, but they are hermitian conjugates of each other: A†1 = A2 and X1† = X2 . Deriving the master equation we have reffered to hermiticity, but it can be readily checked, that the pairwise hermitian conjugation of the interaction operators ensures that all the properties of the master equation are retained.

6.3.2

Discussion of the properties of the reservoir

We must check whether the assumuptions concerning the resrvoir made in the general context in the previous section are satisfied. Obviously, we assume that reservoir – the electromagnetic field – is in the state described by the stationary density operator σ ¯F = σ ¯B , such that h i σ ¯F , HF = 0. (6.64) We do not (at present) specify σ ¯ F any further. We know, however, that since H F and σ ¯F commute, they posses common set of eigenstates. These eigenstates are the photon number states | { n~kλ } i. Therefore we can the reduced density operator of the field as σ ¯F =

X ~ kλ

p~kλ | { n~kλ } ih { n~kλ } |,

(6.65)

which is in analogy to (6.29). 108

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The second point to be checked is whether the assumption (6.34) is satisfied. That is, whether the average value of the interaction vanishes. Firstly, we consider i = 1. Thus, we analyze     n o X X ˜ 1 (t) σ TrF X ¯F = −~ ¯F a~kλ . (6.66) ¯F = − ~ TrF σ g~kλ a~kλ g~kλ TrF σ   ~ kλ

~kλ

Annihilation operator decreases the number of photons in the mode, in other words it is nondiagonal in photon numbers. The density operator σ ¯ F is diagonal in these states. Hence the average value of a product of the nondiagonal operator with the diagonal one gives zero average value TrF σ ¯F a~kλ = 0. (6.67) In a fully analogous manner we check for i = 2 that n o n o X ∗ ˜ 2 (t) σ TrF X ¯F = −~ g~kλ TrF σ ¯F a~† = 0,

(6.68)

kλ

~kλ

because creation operator is also nondigonal since it increases the photon number. From relations (6.66)–(6.68) we conclude that the average value of the interaction indeed vanishes, so this assumption about the reservoir–system interaction is satisfied. The next step in our analysis consists in the discussion of the correlation functions of the resrevoir. We investigate the stationary correlation functions of the type of n o ¯ ij (τ ) = TrF σ ˜ i (τ ) Xj . G ¯F X (6.69)

To discuss such functions we need the operators in the interaction picture. According to general ˜ i (τ ) = eiHF τ /~ Xi e−iHF τ /~ . Substituting our identifications we rules we have (as in (6.33)) X have      X X X ˜ 1 (τ ) = −~ exp iτ X ωk a~† a~kλ   g~k0 λ0 a~k0 λ0  exp −iτ ωk00 a~† 00 00 a~k00 λ00  , (6.70) kλ

~ kλ

~ k 0 λ0

~k00 λ00

k λ

and analogously for X2 (τ ) which contains the creation operators in the middle term. Considering expression (6.70) we note that annihilation and creation operators with different indices commute. Therefore, only the terms with the same indices contribute to the final expression. In this manner, eq(6.70) reduces to X iτ ω a† a −iτ ωk a~† a~kλ ˜ 1 (τ ) = −~ kλ X g~kλ e k ~kλ ~kλ a~kλ e . (6.71) ~kλ

Explicit expression for the transformation of the annihilation operator follows from the properties of the free quantized electromagnetic field. The term in (6.71) can be viewed as the free time evolution of the annihilation operator. Then, from (2.41) we get e

iτ ωk a~† a~kλ kλ

a~kλ e

−iτ ωk a~† a~kλ kλ

= a~kλ e−iωk τ .

Using this result in (6.71) we finally obtain X ˜ 1 (τ ) = −~ X g~kλ a~kλ e−iωk τ .

(6.72)

(6.73a)

~ kλ

˜ 2 (τ ) = −~ X

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X

∗ † g~kλ a~ eiωk τ ,

(6.73b)

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~ kλ

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where the second equality follows by hermitian conjugation of the first one. Having made the transformation to the interaction picture we can return to investigating the stationary correlation function of the reservoir – the electromagnetic field. Since the are two ¯ 11 , G ¯ 12 , G ¯ 21 and G ¯ 22 . It operators X1 and X2 we can conctruct four correlation functions G is not difficult to see that n o XX ¯ 11 (τ ) = TrF σ ˜ 1 (τ ) X1 = ~2 G ¯F X g~kλ g~k0 λ0 e−iωk τ TrF σ ¯F a~kλ a~k0 λ0 = 0, (6.74) ~ kλ ~ k 0 λ0

due to nondiagonal character of the annihlation operators, similarly as in (6.68). Analogously ¯ 22 (τ ) = 0, since it contains two nondiagonal creation operators. Hence, there we can show that G ¯ 12 and G ¯ 21 . We will consider them separately. For the first of the nonvanishing remain only G correlation functions we have n o n o XX ˜ 1 (τ ) X2 = ~2 ¯ 12 (τ ) = TrF σ G ¯F X g~kλ g~k∗0 λ0 e−iωk τ TrF σ ¯F a~kλ a~† 0 0 . (6.75) kλ

~ kλ ~ k 0 λ0

The terms with two different indices do not contribute again, due to nondigonal character. Only single sum survives and we have n o X † ¯ 12 (τ ) = ~2 g~ 2 TrF σ G ¯ a a e−iωk τ F ~ kλ kλ ~ kλ

~ kλ

= ~2

o n X † g~ 2 TrF σ a + 1 e−iωk τ , ¯ a F ~ ~ kλ kλ

(6.76)

kλ

~ kλ

where the last step follows from canonical commutation relations for annihilation and creation operators. The density operator σ ¯ F is normalized, while using (6.65) we get n o TrF σ ¯F a~† a~kλ = p~kλ n~kλ = h n~kλ i (6.77) kλ

is the average number of photons in the mode ~kλ. Inserting (6.77) into (6.76) we get X ¯ 12 (τ ) = ~2 g~ 2 h n~ | + i1 e−iωk τ G kλ kλ

(6.78)

~ kλ

In a fully analogous manner we compute the second nonvanishing correlation function of the field n o ¯ 21 (τ ) = TrF σ ˜ 2 (τ ) X1 G ¯F X n o XX ∗ = ~2 g~kλ g~k0 λ0 eiωk τ TrF σ ¯F a~† a~k0 λ0 kλ

~ kλ ~ k 0 λ0

= ~2

X g~ 2 h n~ | ieiωk τ . kλ kλ

(6.79)

~ kλ

Hence the stationary correlation functions of the reservoir – electromagnetic field are found and can be used in our master equation.

6.3.3

Analysis of the master equation

We now move further to analyze the master equation (6.58). In our case indices i, j taken on ¯ ij vanish. Thus the only two values. Moreover two out of four possible correlation functions G 110

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sum over i and j in (6.58) effectively contains only two terms. Hence, from (6.58) we obtain the equation of motion for the atomic density operator (we will omit index s) i d 1 h %(t) − HA , % dt i~ Z n h ∞ 1 ¯ 12 (τ ) A1 e−iHA τ /~ A2 eiHA τ /~ %(t) = − 2 dτ G ~ 0 i − e−iHA τ /~ A2 eiHA τ /~ %(t) A1 h ¯ 21 (τ ) A2 e−iHA τ /~ A1 eiHA τ /~ %(t) +G io − e−iHA τ /~ A1 eiHA τ /~ %(t) A2 + H.C.

(6.80)

Since we have A1 = S+ and A2 = S− , we easily obtain e−iHA τ /~ S± eiHA τ /~ = S± e−iω21 τ .

(6.81)

Using this in master equation (6.80) we get Z ∞ i d 1 h 1 ¯ 12 (τ ) eiω21 τ [ S+ S− %(t) − S− %(t) S+ ] %(t) − HA , % = − 2 dτ G dt i~ ~ 0 ¯ 21 (τ ) e−iω21 τ [ S− S+ %(t) − S+ %(t) S− ] + H.C. +G (6.82) Finally, we substitute the stationary correlation functions of the reservoir. From (6.78), (6.79) and (6.82) we arrive at the equation of motion of the atomic density operator in the form i 1 h d %(t) − HA , % dt i~  Z ∞  X g~ 2 ei(ω21 −ωk )τ h n~ | + i1 [ S+ S− %(t) − S− %(t) S+ ] dτ = − kλ kλ  0 ~ kλ   X 2 −i(ω −ω )τ 21 k g~ e + h n i [ S S %(t) − S %(t) S ] − + + − ~ kλ kλ  ~ kλ

+ H.C.

6.3.4

(6.83)

Reservoir in the vacuum state

Reservoir in the vacuum state corresponds to the situation when h n~kλ | = i0,

for all modes ~k, λ.

(6.84)

Then, the second term in the master equation (6.83) vanishes, while the first one gives only one contribution. So, for such a case we have i d 1 h %(t) − HA , % dt i~ Z ∞ X g~ 2 ei(ω21 −ωk )τ [ S+ S− %(t) − S− %(t) S+ ] + H.C. = − dτ (6.85) kλ 0

~ kλ

Let us introduce a special notation for the summation over the modes and integral over dτ Z ∞ X g~ 2 ei(ω21 −ωk )τ = γ + iδL , dτ (6.86) kλ 2 0 ~kλ

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which we will compute and discuss later on. Inserting this notation and writting the hermitian conjugate term explicitly, we arrive at the equation i γ h i d 1 h S+ S− %(t) − S− %(t) S+ %(t) − HA , % = − + iδL dt i~ 2 h i γ − %(t) S+ S− − S− %(t) S+ , (6.87) − iδL 2 because S±† = S∓ . Combining the terms we further get i i d 1 h γ h %(t) − HA , % = − S+ S− %(t) + %(t) S+ S− − 2 S− %(t) S+ dt i~ 2 h i − iδL S+ S− %(t) − %(t) S+ S− ,

(6.88)

Let us consider the last term in this equation. We transform it as follows h i 1 1 − iδL S+ S− %(t) − %(t) S+ S− = − iδL S3 + %(t) − %(t) S3 + 2 2 h i i 1 h = − iδL S3 , %(t) = ~δL S3 , %(t) . (6.89) i~

This term can, therefore, be combined with the free evolution term in the master equation (6.88), so we get i d 1 h %(t) − HA + ~δL S3 , % dt i~ i γ h = − S+ S− %(t) + %(t) S+ S− − 2 S− %(t) S+ . (6.90) 2

Since the Hamiltonia HA of the free atom is HA = ~ ω21 S3 as in (6.59), we can combine two terms in the commutator in the left hand side of (6.90). We see that the imaginary part of the sum (6.86) gives a shift of the atomic eigenfrequency. So we can redefine the atomic frequency 0 = ω + δ , thus accounting for the frequency shift which is due to the coupling to the as ω21 21 L reservoir. So will omit the prime in atomic frequency, but we will always consider the frequency shift to included in the atomic frequency appearing in the atomic Hamiltonian H A . Hence, there remains the term depending on the parameter γ, and the resulting master equation becomes i i d 1 h γ h %(t) − HA , % = − S+ S− %(t) + %(t) S+ S− − 2 S− %(t) S+ . dt i~ 2

(6.91)

This is the final form of the master equation for a two-level atom immersed in the electromagnetic radiation. We stress that this equation refers to the situation when the field is in its vacuum state and that the frequency shift is included in the redefined atomic Hamiltonian. Equation (6.91) gives the free evolution of the atom (the second term on left-hand side) and the influence of the field is accounted for by the right-hand side. We may say that the right-hand side of this equation describes the reaction of the field on the atom.

6.3.5

Physical meaning of the parameter γ

Let us consider the term describing the reaction of the vacuum-state radiation on the two-level atom. This reaction is given by i d γ h %(t) = − S+ S− %(t) + %(t) S+ S− − 2 S− %(t) S+ dt 2 AF i γ h = − | 2 ih 2 | %(t) + %(t) | 2 ih 2 | − 2| 1 i %22 (t) h 1 | , (6.92) 2 112

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where in the second equality we have used the definition and the properties of the pseudospin operators expressed via the projectors constructed from the two-level atom states (see the appendix). Moreover, we have used an obvious notation for the two-level atom density operator matrix elements: %ij = h i | % | j i. Taking matrix elements of the both sides of the above equation we get d d γ = γ %22 (t), = − %12 (t), %11 (t) %12 (t) dt dt 2 AF

AF

d γ %21 (t) = − %21 (t), dt 2 AF

d %22 (t) = − γ %22 (t). dt AF

(6.93)

It is worth noting that these relations are nonhermitian. On the other hand, we see that the field reaction preserves the trace of the atomic dendity operator, that is d d d %11 (t) + %22 (t) Tr {%(t)} = = 0. (6.94) dt dt dt AF AF So the value of the trace Tr {%(t)} is constant, and thus, can be taken equal to the initial value, which is unity due to normalization condition. Integration of the last one of equations (6.93) for the upper state population is straightforward and it gives %22 (t) = %22 (0) e−γt .

(6.95)

The equation for lower state population can be also integrated, but it is simpler to use trace conservation condition (6.94), which easily yields %11 (t) = 1 − %22 (t) = 1 − %22 (0) e−γt .

(6.96)

From equations (6.95) and (6.96) we see that the inverse of the parameter γ may be inerpreted as a certain characteristic time, called the atomic lifetime τA =

1 . γ

(6.97)

Conversely, we can interpret γ as the probability per unit time of the atom to decay from the upper (excited state) and to make a transition to the lower (ground) state.

6.3.6

Calculation and discussion of γ and δL

We have introduced parameters γ and δ L as a shorthand for the summation and integration Z ∞ X γ g~ 2 + iδL = dτ ei(ω21 −ωk )τ . (6.98) kλ 2 0 ~ kλ

The integral over dτ can be analyzed in the following manner Z ∞ Z ∞ √ 1 i(ω21 −ωk )τ dτ e = 2π √ dτ θ(t) ei(ω21 −ωk )τ 2π −∞ 0 √ = 2π F[θ(t)](ω21 −ωk ) .

(6.99)

We see, that apart from a factor, the considered integral is the Fourier transform of the Heavisise θ(t) function. Thus we obtain Z ∞ √ 1 1 i(ω21 −ωk )τ √ dτ e = 2π πδ(ω21 − ωk ) + iP (6.100) ω21 − ωk 2π 0 S.Kryszewski

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Therefore, we have X 2 1 γ + iδL = πδ(ω21 − ωk ) + iP , g~kλ 2 ω21 − ωk

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(6.101)

~kλ

from which it follows that X 2 γ = π g~kλ δ(ω21 − ωk ) 2

(6.102a)

~kλ

δL =

X g~ 2 P kλ ~kλ

1 . ω21 − ωk

(6.102b)

The computation of the frequency shift δ L leads to divergent integrals which must be renormalized. We will not investigate this question. The main reason is that δ L is included in the free atom Hamiltonian with the shifted frequency. We will always think about the atomic frequencies as the quantities with Lamb shift already included. So it remains to compute the atomic lifetime τA or its inverse. We proceed to find γ. From (6.102a) and (6.62) we obtain, after some minor simplifications 2 X π ωk γ = (6.103) ~d21 · ~e~kλ δ(ω21 − ωk ) V ~ 0 ~kλ

The summation over the modes can be performed with the aid of relation (3.70), because we need to sum the function of polarisations and mode frequency. Using this relation we can write 2 X Z ∞ V π ωk ~ 2 ~ γ = ω dω d · e δ(ω21 − ωk ) (6.104) 21 k ~ k kλ 2π 2 c3 V ~ 0 0 λ

The integration is trivial due to the presence of the Dirac delta function. After some simplifications we arrive at 2 3 X ω21 ~d21 · ~e~ γ = (6.105) kλ 2π~ 0 c3 λ

It remains to sum over polarizations. But let us note that we know nothing on the orientation of the atomic dipole moment. In other words, we can say that the dipole moment is oriented arbitrarily. Hence we can write 2 2 ~ (6.106) d21 · ~e~kλ = ~d21 h cos2 θ | , i

where θ is the angle between the atomic dipole moment and one of the polarization vectors. The angular brackets denote the averaging over this angle, that is over the orientation of the dipole moment with respect to the polarization vector. In an obvious manner we get Z Z 1 1 π 1 2 2 h cos θ | i = dΩ cos θ = dθ sin θ cos2 θ = . (6.107) 4π 2 0 3

This reasoning can be repeated with the same result for the second polarization vector. Since there are two polarizations, we may say that 2 X 2 ~ 2 ~ ~ (6.108) d21 · e~kλ = d21 3 λ

Substituting the last result into (6.105) we obtain 2 3 ω21 ~ γ = d21 3 3π~0 c 114

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(6.109)

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7. Optical Bloch equations

Chapter 7

Optical Bloch equations 7.1

Introduction. General discussion

We return to the more detailed analysis of the interaction of a two-level atom with electromagnetic field. The problem was already main discussed in chapter 5, hence we can just briefly recall some basic facts. • The atom–field system has the hamiltonian (see (5.25)) H = HA + HF + HAF .

(7.1)

• The atomic Hamiltonian HA is taken to be (see (5.62)) h i 1 HA = ~ω21 | 2 ih 2 | − | 1 ih 1 | = ~ω21 S3 . (7.2) 2 • The free-field Hamiltonian HF is (in the representation of plane waves) given by (3.63), that is X HF = ~ ωk a~† a~kλ . (7.3) kλ

~kλ

• The Hamiltonian of the atom–field interaction H AF is taken in the resonant (rotating wave) approximation as in eq.(5.84) o X n ~ ~ ~ ~ HAF = ~ G~kλ eik·R S+ a~kλ + G~∗kλ eik·R S− a~† , (7.4) kλ

~kλ

with the coupling coefficients defined in (5.80), i.e., r ωk ~d21 · ~e~ G~kλ = − i kλ 2~ 0 V

(7.5)

Despite its superficial simplicity, the problem, in its full generality very complicated. The discussed model may describe a great variety of physical problems. Moreover, allowing for the ~ = R(t) ~ time dependence of the atomic position R leads to many phenomena connected with atomic motion. Hence it seems necessary to restrict our attention to a more simple model. We will consider a simpler situation. We will assume that the two-level atom interacts with the field, in which at the initial moment t = 0, only a single mode with well-specified wave vector ~k0 and frequency ω0 is excited. We will also assume that the state of this mode (at the initial instant) is a coherent state. All other modes are assumed to be at t = 0 in the vacuum state. Therefore, the initial state of the whole system can be written as | Ψ(t = 0) i = | ψat i | (α)~k0 λ0 i, S.Kryszewski

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where | ψat i is some atomic state. In order to consider the discussed simplified problem we introduce the unitary transformation h i Tˆ (t) = U † α e−iω0 t = exp α∗ eiω0 t a0 − α e−iω0 t a†0 , (7.7)

where a0 and a†0 are the annihilation and creation operators for the selected – excited mode of the field. The operator U transforms the vacuum state into a coherent one, while U † reduces the coherent state to the vacuum one. When this transformation is applied to the considered initial state, we see that the new initial state becomes | Φ(t = 0) i = | ψat i| 0 i,

(7.8)

that is the atomic state remains unaffected, while the state of the field is transformed into the vacuum one. The transformation of the hamiltonian (7.1) and all its terms leads to the following expression ˆ† ˜ = Tˆ (t)H Tˆ † (t) − i~Tˆ (t) d T (t) H dt = HA + HF + HAF − ~ΩS+ − ~Ω∗ S− . The quantity Ω called Rabi frequency is defined as r ~ω0 ~ ~ ~Ω = i α eik0 ·R−iω0 t ~d21 · ~e~kλ 2 0 V

(7.9)

(7.10)

The obtained results can be interpreted as follows. Due to the elimination of the coherent state, the atom interacts with the field as in the semiclassical picture. This was already discussed earlier and the conclusions of that discussion can be used in the present context. The last two terms in the Hamiltonian (7.9) are exactly the same as those in Eq.(5.97). Hence, similarly as before, we can write the transformed Hamiltonian as ˜ = HA + HF + HAF − ~χe−iφ−iωt S+ − ~χ∗ eiφ+iωt S− , H

(7.11)

After the transformation the atom is coupled to the field in the vacuum state. Hence, the spontaneous emission is possible. The description of the atom must be done by means of the reduced density operator. Therefore, the evolution of the atomic density operator is given by the equation following from Eq.(6.91) which accounts for the coupling with the vacuum. In the present case we must take into account the last two terms of (7.11). In such a manner, the atom interacting with the field in a coherent state, is described by the evolution equation i i d 1 h γ h %(t) = HA , % − S+ S− %(t) + %(t) S+ S− − 2 S− %(t) S+ dt i~ 2 i 1 h + − ~χe−iφ−iωt S+ − ~χ∗ eiφ+iωt S− , % (7.12) i~

The first term gives the free evolution of the atom, the second one describes spontaneous emission (coupling of the atom to the vacuum field), while the last term has semiclassical character and describes the interaction of the field with the single mode of the field. The Rabi frequency χ follows from (7.10) and can be shown to be χ = 116

1 ~ ~ ~ (+) ei~k·R d21 · E 0 ~

(7.13)

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where ω and ~k are the wave vector and frequency of the semiclassical field. Equation of motion (7.12) can thus be written as i i h γ h d i~ %(t) = Hs , % − i~ S+ S− %(t) + %(t) S+ S− − 2 S− %(t) S+ , dt 2

117

(7.14)

with effective hamiltonian

Hs = ~ω21 S3 − ~χ e−i(ωt+φ) S+ − ~χ∗ ei(ωt+φ) S− .

7.2

(7.15)

Derivation of optical Bloch equations

7.2.1

Evolution of the atom without damping

We continue our analysis of the two-level atom in the framework presented in the previous section. The evolution without damping is given by the first term in (7.14) in a manner similar to the von Neuman equation h i d i~ %(t) = Hs , %(t) . (7.16) dt

The density operator can obviously be expanded by employing the completeness relation for the atomic states % = %11 | 1 ih 1 | + %22 | 2 ih 2 | + %21 | 2 ih 1 | + %12 | 1 ih 2 |,

(7.17)

where %ab are the matrix elements of the density operator. Derivation of the equations of motion for matrix elements which follow form equation (7.16) can be done in many ways. We will briefly illustrate one of the possible methods. First we note that eq.(7.16) implies that d h a | % | b i = h a | Hs % − %Hs | b i i~ dt = h a | Hs %(t) | b i − h a | %(t)Hs | b i = h a | Hs | 1 ih 1 | % | b i + h a | Hs | 2 ih 2 | % | b i

−h a | % | 1 ih 1 | Hs | b i − h a | % | 2 ih 2 | Hs | b i

= (Hs )a1 %1b + (Hs )a2 %2b − (Hs )1b %a1 − (Hs )2b %a2 .

(7.18)

In the third equality we have employed the completeness relation for the atomic states, while in the fourth one we simply adopted a simpler notation. It is straightforward to find the matrix elements of the Hamiltonian (7.15), we get 1 (Hs )11 = − ~ω21 , 2

(Hs )12 = −~χ∗ ei(ωt+φ) ,

(Hs )21 = −~χ e−i(ωt+φ) ,

(Hs )22 =

1 ~ω21 . 2

(7.19)

Having the matrix elements of the Hamiltonian (7.19) it is an easy matter to construct the equations of motion for the matrix elements of the density operator which follow from the last of Eqs.(7.18). We obtain the set of equations %˙ 11 = iχ∗ ei(ωt+φ) %21 − iχ e−i(ωt+φ) %12 ∗

%˙ 22 = −iχ e

%˙ 21 = −iχ e

i(ωt+φ)

−i(ωt+φ)

%21 + iχ e

−i(ωt+φ)

%12

(%22 − %11 ) − i ω21 %21

%˙ 12 = iχ∗ e−i(ωt+φ) (%22 − %11 ) + i ω21 %12

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This set of equations constitutes the optical Bloch equations (OBE) for a two-level atom interacting with the single-mode (monochromatic) electromagnetic field which is in the coherent state. This set of equations can also be viewed as corresponding to the classical treatment of the mode which interacts with the atom. The corresponding Hamiltonian was constructed with RWA and the dipole approximation. This result, as it was discussed above follows from the minimal and the dipole coupling schemes. Since OBE play an extremely important role in quantum optics we summarize the notation. • ω21 denotes the atomic frequency; • ω is the frequency of the incoming electromagnetic field; • χ is the Rabi frequency, defined in (7.13). We stress that χ depends parametrically on the ~ of the atomic center of mass. position R Discussion of the OBE will be given in next sections. At present we note that OBE preserve the trace of the atomic density matrix. The trace Tr % = %11 + %22 = 1

(7.21)

is conserved for an arbitrary moment of time as it follows from the first two equation of the set (7.20). The requirement (which follows from the Hermiticity of the density operator) that %12 = %21∗ is also clearly satisfied by the OBE. It is also evident that the optical Bloch equations are not independent. The first two equations are actually the same, while two last equation are complex conjugates of each other. Therefore, the trace conservation requirement (7.21) plays an essential role in any attempts to find the solution to Eqs.(7.20). Before proceeding further we note the presence of the term e ±(iφ+iωt) in equations (7.20). Certainly the presence of a time dependent factor makes the solution to the set of equations more difficult. Therefore, it is desirable to eliminate the time dependent factor. We will do this in the further sections, transforming the OBE to such a form, that right-hand sides of the equations will not include any time dependencies.

7.2.2

Addition of radiative and phenomenological damping

Radiative damping (spontaneous emission) is accounted for by the last term in eq.(7.14). The contributions due to this term were already discussed in eqs.(6.93). This results in the appearance of additional terms in eqs. (7.20). We obtain %˙ 11 = A%22 + iχ∗ ei(ωt+φ) %21 − iχ e−i(ωt+φ) %12 ∗

%˙ 22 = −A%22 − iχ e

i(ωt+φ)

%21 + iχ e

−i(ωt+φ)

%12

%˙ 21 = −iχ e−i(ωt+φ) (%22 − %11 ) − (A/2 + i ω21 ) %21 ∗ −i(ωt+φ)

%˙ 12 = iχ e

(%22 − %11 ) − (A/2 − i ω21 ) %12

(7.22a) (7.22b) (7.22c) (7.22d)

where we denoted A = γ – the Einstein A-coefficient for spontaneous emission. In many practical cases there are some other mechanisms which lead to the damping of the atomic dipole moment. They can be included in our picture by introducing a new damping rate of the coherences, Γc =

118

A + γph , 2

(7.23)

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119

where γph describes the dephasing of the atomic dipole moment. The physical reasons for the dephasing will be discussed elsewhere. At present, we will simply include it into the optical Bloch equations. We arrive at the following set of equations %˙ 11 = A%22 + iχ∗ ei(ωt+φ) %21 − iχ e−i(ωt+φ) %12 ∗

%˙ 22 = −A%22 − iχ e

i(ωt+φ)

(7.24a)

%12

(7.24b)

%˙ 21 = −iχ e−i(ωt+φ) (%22 − %11 ) − (Γc + i ω21 ) %21

(7.24c)

∗

%˙ 12 = iχ e

7.2.3

−i(ωt+φ)

%21 + iχ e

−i(ωt+φ)

(%22 − %11 ) − (Γc − i ω21 ) %12

(7.24d)

Simple elimination of time dependence

Elimination procedure Equations (7.24) have right-hand sides explicitly time-dependent. This is very inconvenient when seeking practical solutions. It is desirable to transform out the unnecessary time dependence. To achieve this end we introduce new auxiliary variables ρ˜11 = %11 ,

ρ˜12 = %12 e−i(ωt+φ) ,

ρ˜21 = %21 ei(ωt+φ) ,

ρ˜22 = %22 .

(7.25)

The set of equations (7.24) can now be reexpressed in terms of new variables denoted by the tilde. It is an easy matter with equations (7.24a) and (7.24b). The exponential factors are simply incorporated in the quantities with tilde. Before dealing with two remaining equations we first write down the relations inverse to (7.25) %11 = ρ˜11 ,

%12 = ρ˜12 ei(ωt+φ) ,

%21 = ρ˜21 e−i(ωt+φ) ,

%22 = ρ˜22 .

(7.26)

We proceed to transform eq.(7.24c). We use relations (7.26) to obtain ˙ ρ21 e−i(ωt+φ) ρ˜˙ 21 e−i(ωt+φ) − i(ω + φ)˜

= − iχ e−i(ωt+φ) (˜ ρ22 − ρ˜11 ) − (Γc + iω21 ) e−i(ωt+φ) ρ˜21 ,

(7.27)

The exponential term cancels out. Upon some manipulation we arrive at an equation ρ˜˙ 21 = − iχ (˜ ρ22 − ρ˜11 ) − (Γc − i∆) ρ˜21 ,

(7.28)

where we introduced the generalized detuning defined as ∆ = ω + φ˙ − ω21 .

(7.29)

Although in the general context the derivative φ˙ appears in the definition of the detuning, we will not consider such case. In our approach we will consequently assume either φ = const or even simply φ = 0. Transformation of (7.24d) follows easily by complex conjugation. As a result of transformation (7.25) or (7.26) we obtain the following set of equations, fully equivalent to Eqs.(7.24) ρ˜˙ 11 = A˜ ρ22 + iχ∗ ρ˜21 − iχ ρ˜12 , ρ˜˙ 22 = −A˜ ρ22 − iχ∗ ρ˜21 + iχ ρ˜12 , ρ˜˙ 21 = −iχ (˜ ρ22 − ρ˜11 ) − (Γc − i∆) ρ˜21 , ρ˜˙ 12 = iχ∗ (˜ ρ22 − ρ˜11 ) − (Γc + i∆) ρ˜12 ,

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7. Optical Bloch equations

It is convenient to   ρ˜11   ρ˜  d   22  =  dt   ρ˜21  ρ˜12

write this  0   0   iχ  −iχ∗

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set of equations in the matrix form  A iχ∗ −iχ ρ˜11  ∗   ρ˜22 iχ −A −iχ    ρ˜ −iχ −Γc + i∆ 0   21 iχ∗ 0 −Γc − i∆ ρ˜12

     

(7.31)

We note that the matrix in this relation is explicitly time independent, (since we take φ = 0) which greatly facilitates the practical computations. As it was in the case of untransformed equations (7.20), also here the equations in the set (7.30) are linearly dependent. The trace conservation ρ˜11 + ρ˜22 = 1 is still very important. Finally, we stress that the variables ρ˜ab have only the auxiliary character. Physical significance is associated with the old ones (i.e., those without a tilde) given by eqs(7.26) via the new ones (those with tilde). We have to remember that the physically relevant variables are expressed via the old elements of the atomic density operator. Discussion of the new variables Having relations (7.26), let us use them to find the expectation values of the pseudospin operators. h S+ i = Tr{ S+ % } = Tr{ | 2 ih 1 |% } = %12 = ρ˜12 ei(ωt+φ) ,

(7.32a)

h S− i = Tr{ S− % } = Tr{ | 1 ih 2 |% } = %21 = ρ˜21 e−i(ωt+φ) ,

(7.32b)

1 (| 2 ih 2 | − | 1 ih 1 |) % } 2 1 1 (%22 − %11 ) = (˜ ρ22 − ρ˜11 ) . = 2 2 From these equations we derive the following ones h S3 i = Tr{S3 %} = Tr{

h S+ i = %12 ,

h S− i = %21 ,

2 h S3 i = %22 − %11 ,

(7.32c)

ρ˜12 = h S+ i e−i(ωt+φ) , ρ˜21 = h S− i e

i(ωt+φ)

,

ρ˜22 − ρ˜11 = 2 h S3 i.

(7.33a) (7.33b) (7.33c)

Comparing both columns we see that when applying the new variables (with tilde), it is convenient to introduce ”new” operators S˜+ = S+ e−i(ωt+φ) ,

S˜− = S− ei(ωt+φ) ,

S˜3 = S3 .

(7.34)

From the above given relations it is easy to derive the following expressions for the expectation values of the ”new” operators h S˜+ i = Tr{S˜+ %} = ei(ωt+φ) h S+ i = ρ˜12

(7.35a)

h S˜− i = Tr{S˜− %} = e−i(ωt+φ) h S− i = ρ˜21

(7.35b)

1 h S˜3 i = Tr{S˜3 %} = h S3 i = (˜ ρ22 − ρ˜11 ) . (7.35c) 2 Physical sense is assigned to operators S ± and S3 . Nevertheless, new operators (with tilde) are useful. We invert Eq.(7.34) and express old ones via the new operators S± = S˜± exp [±i (ωt + φ)] , 120

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7. Optical Bloch equations

121

From the above given arguments we can draw the following conclusion. When working with the transformed Bloch equations (7.30) we eliminate the time dependence via substitutions (7.26). Solution to these equations yields the time evolution of matrix elements ρ˜ab . To compute the time-dependent expectation values of various observables we need the averages of pseudospin operators S± , S3 . To do so, we express these operators via the ones with tilde (according to relations (7.36)). Calculation of the expectation values for S˜ operators is easy, due to relations (7.35). Additional exponential factor which follows from substitutions (7.36) ensures the correct factors in the physically significant quantities, i.e., as in Eqs.(7.32). This seems a little complicated, but we will give an example illustrating this point quite clearly. Atomic dipole moment Atomic dipole moment is defined earlier, but for clarity, we repeat it again. We have ~d = ~d21 | 2 ih 1 | + ~d12 | 1 ih 2 | = ~d21 S+ + ~d12 S− .

(7.37)

Therefore expectation value is as follows from relations (7.32), namely h ~d i = ~d21 h S+ i + ~d12 h S− i = ~d21 %12 + ~d12 %21 ,

(7.38)

where we have used parts of Eqs.(7.32). Expression (7.38) gives the expectation value of the atomic dipole moment in terms of the physically significant quantities. However, Bloch equations (7.20) yielding the elements %ab are inconvenient since they have time dependent right-hand sides. It is much easier to work with Eqs.(7.30) since they are not time-dependent. There is no problem in changing the variables in (7.38) to the new ones – with tilde. According to (7.26) we rewrite (7.38) as h ~d i = ~d21 ρ˜12 ei(ωt+φ) + ~d12 ρ˜21 e−i(ωt+φ) .

(7.39)

To find this average, we need the solution to Eqs.(7.30), which is not a very difficult task. We notice, that due to the change of variables, Eq.(7.39) contains an explicit time dependence, which was absent in (7.38). We can, however, adopt a slightly different approach. In the operator definition of the atomic dipole (7.37) we change pseudospin operators to the new ones, according to (7.36). Thus, we express the dipole moment in new variables as ~˜ = d ˜ 21 S ˜ 12 S ˜ + ei(ωt+φ) + d ˜ − e−i(ωt+φ) . d

(7.40)

Computing the expectation value from Eq.(7.40) we immediately obtain ˜ h ~d i = ~d21 h S˜+ i ei(ωt+φ) + ~d12 h S˜− i e−i(ωt+φ) = ~d21 ρ˜12 ei(ωt+φ) + ~d12 ρ˜21 e−i(ωt+φ) ,

(7.41)

where the last line follows from Eqs.(7.35). We see that we have recovered Eq.(7.39). Therefore, we have two equivalent approaches to computation of the expectation values. We may reformulate the conclusion given at the end of previous section. Working with Bloch equations (7.30) with new variables ρ˜ab , we obtain the same physical predictions if we replace operators S± , S3 by the ones with tilde according to Eqs.(7.36). The mentioned replacement is done in the expressions for observables. The example of such a replacement is given by a transition from Eq.(7.37) to (7.40). S.Kryszewski

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We can rephrase this conclusion by saying that the physical predictions following from optical Bloch equations (7.30) (with variables ρ˜ab )are correct, provided in all observables we make the replacement (7.36). Then we compute expectation values of S˜ operators with matrix elements ρ˜ab . The correct time dependent factors appear due to relations (7.36). We also note one more consequence of the change of variables. The atomic dipole moment expressed via the operators with tilde (new ones) may be written as ˜ ˜ (+) ~d ˜ (−) , =d + d

(7.42)

where we have the positive and negative frequency parts ~d(+) = ~d12 S˜− e−i(ωt+φ) ,

~d(−) = ~d21 S˜+ ei(ωt+φ) .

(7.43)

The assignment of the positive- and negative-frequency parts follows the same rules as that ~ (±) . concerning the electric fields E A note of caution In some literature it is said that the change of variables, as in Eqs.(7.26), can be done with the aid of the operator transformation given by the relation ρ˜ = ei(ωt+φ) S3 % e−i(ωt+φ) S3 .

(7.44)

This is indeed so. Nevertheless, this manner of approach is not advised. The reason is that relation (7.44) gives associations with unitary transformation, given by an operator Tˆ = e−i(ωt+φ) S3 . What we do is just a change of variables in the Bloch equations (7.20) leading to equations (7.30). We do not use all the mathematical apparatus of unitary transformations. Change of variables induced by relation (7.44) may be viewed as a partial unitary transformation. Partial – in the sense that only the density matrix is transformed and not the atomic states. This clearly warns against using the transformation as in (7.44). We will not use such an approach, and we will not discuss it any more, because it does not seem useful.

7.3

Stationary optical Bloch equations

7.3.1

Stationary solutions

Stationary optical Bloch equations follow, when we take the left-hand sides of equations (7.30) to be equal zero. Then we have ¯21 − iχ σ ¯12 , 0 = A¯ σ22 + iχ∗ σ

0 = −A¯ σ22 − iχ∗ σ ¯21 + iχ σ ¯12 ,

0 = −iχ ( σ ¯22 − σ ¯11 ) − (Γc − i ∆) σ ¯21 , 0 = iχ ( σ ¯22 − σ ¯11 ) − (Γc + i ∆) σ ¯12 .

(7.45)

The bar indicates that we deal with stationary solutions, in the sense σ ¯ab = lim ρ˜ab (t).

(7.46)

t→∞

The obtained set is homogeneous, moreover, it is straightforward to see that the first two equations differ only by a sign, so they are linearly dependent. It may seem that there are no

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7. Optical Bloch equations

nontrivial solution. This not the case since we must account for the trace conservation. We discard the first equation of the above set, and instead, we adopt the trace conservation requirement as the first equation. As a result we arrive at the set of equations 1 = σ ¯11 + σ ¯22 ,

(7.47a)

0 = −A¯ σ22 − iχ∗ σ ¯21 + iχ σ ¯12 ,

(7.47b)

0 = iχ (¯ σ22 − σ ¯11 ) − (Γc + i∆) σ ¯12 .

(7.47d)

0 = −iχ (¯ σ22 − σ ¯11 ) − (Γc − i∆) σ ¯ 21 ,

(7.47c)

From two last equations we express coherences as functions of populations σ ¯21 =

−iχ (¯ σ22 − σ ¯ 11 ) , Γc − i∆

(7.48)

while the second coherence σ ¯ 12 follows by complex conjugation. Inserting these expressions into the first two equations of the set (7.47) we get two, closed equations for populations only ¯22 , σ ¯11 = 1 − σ 2Γc |χ|2 A¯ σ22 = − 2 (¯ σ22 − σ ¯11 ) , Γc + ∆ 2

(7.49a) (7.49b)

Solution to Eqs.(7.49) poses no difficulties. The obtained populations are then substituted to Eq.(7.48) which yield the coherences. Straightforward algebra leads to the following results for stationary solutions to optical Bloch equations with phenomenologically included damping A(Γ2c + ∆2 ) + 2Γc |χ|2 , A(Γ2c + ∆2 ) + 4Γc |χ|2 2Γc |χ|2 = , A(Γ2c + ∆2 ) + 4Γc |χ|2 χA(iΓc − ∆) = σ ¯12∗ = , 2 A(Γc + ∆2 ) + 4Γc |χ|2

σ ¯11 =

(7.50a)

σ ¯22

(7.50b)

σ ¯21

(7.50c)

The trace conservation requirement is obviously satisfied. Moreover, we note the inequality σ ¯11 > σ ¯22

(7.51)

It is sometimes convenient to introduce the parameter s=

Γc |χ|2 , A(Γ2c + ∆2 )

(7.52)

which is called the saturation parameter. With its aid we can transform stationary solutions (7.50) into σ ¯11 =

1+s 1 + 2s

σ ¯21 =

iχ Γc − i∆

S.Kryszewski

1 1 + 2s

σ ¯12 =

−iχ Γc + i∆

σ ¯22 =

s 1 + 2s

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1 1 + 2s (7.53)

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7.3.2

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Stationary energy balance

As we have discussed above our problem of atom-light interaction can be given semiclassical interpretation, due to which the atom is coupled to the electric field ~ R, ~ t) = E ~ (+) (R, ~ t) + E ~ (−) (R, ~ t) E( ~ ~ ~ (−) e−i~k·R+iωt+iφ ~ (+) ei~k·R−iωt−iφ +E . = E 0

0

(7.54)

This field acts upon the atom and within the time interval it performs the elementary work ~ R, ~ t) · d~r, dW = q E(

(7.55)

where d~r is the displacement of the electron. Thus, the power absorbed by the atom is given as d ~ R, ~ t) · d (q~r) , = E( ~ R, ~ t) · d ~d(t) , W = E( (7.56) dt dt dt

with ~d(t) being the operator of the atomic dipole moment. We average the obtained relation both quantum-mechanically and over time (we denote the latter averaging by the bar) h

d ~ R, ~ t) · d h ~d(t) i. W i = E( dt dt

(7.57)

Quantum-mechanical averaging refers only to the atomic dipole, since the field is treated classically. In this section we are interested only in the stationary regime, hence we can use Eq.(7.41) with matrix elements of the atomic density operator replaced by their stationary values, so we get h ~d(t) i = ~d21 σ ¯12 ei(ωt+φ) + ~d12 σ ¯21 e−i(ωt+φ) .

(7.58)

Now, we substitute (7.54) and the time derivative of (7.58) into (7.57) to get h

h i d ~ ~ ~ (+) ei~k·R−iωt−iφ ~ (−) e−i~k·R+iωt+iφ Wi = E + E 0 0 dt h i · iω ~d21 σ ¯12 ei(ωt+φ) − iω ~d12 σ ¯21 e−i(ωt+φ) ,

(7.59)

where, for simplicity, we have again assumed φ˙ = 0. We are averaging over time, therefore we can neglect the quickly oscillating terms. We arrive at the expression h

d ~ ~ (+) ~ −i~ k·R ~ (+) · ~d21 ei~k·R ~ W i = iω E σ ¯ − iω E · d σ ¯21 . 12 12 e 0 0 dt

(7.60)

d W i = iω~χ¯ σ12 − iω~χ∗ σ ¯21 = 2~ωIm (χ∗ σ ¯21 ) dt

(7.61)

According to eq.(7.13) we recognize the Rabi frequency χ and its complex conjugate. We obtain h

We now take σ ¯ 21 from eq.(7.50c) and we obtain h

d 2|χ|2 Γc W i = ~ωA . dt A(Γ2c + ∆2 ) + 4Γc |χ|2

(7.62)

Finally we see that the fraction reproduces the stationary-state upper state population σ ¯ 22 . Therefore we have h 124

d W i = ~ωA σ ¯22 . dt

(7.63) QUANTUM OPTICS

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125

We have calculated the average power absorbed by the atom from the incident field. Let us ¯abs the average number of the photons absorbed per unit time, we can also call N ¯abs denote ny N the photon absorption rate. Then we can write h

d ¯abs = ~ωA σ W i = ~ω N ¯22 . dt

(7.64)

Obviously, this entails ¯abs = A σ N ¯22 .

(7.65)

On the other hand, the term A σ ¯ 22 is the emission rate, because it appears in the right-hand side of eq.(7.30b) as the rate of the decay of the upper state population. And the upper state decays due to photon emission. Therefore, we can say that eq (7.65) informs us that in the stationary state the rate of emission is equal to the rate of absorption. This seems to be an intuitively plausible conclusion.

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1

Extras. 8. Annihilation and creation operators

Chapter 8

Annihilation and creation operators 8.1

General properties

We introduce two nonhermitian operators which, by definition, satisfy the canonical commutation relation: h i a ˆ, a ˆ† = 1. (8.1)

ˆ =a By | z i we denote a normalized eigenstate of the operator N ˆ† a ˆ. We assume that such states ˆ are orthogonal, since operator N is hermitian. So we have ˆ| z i = a N ˆ† a ˆ| z i = z | z i,

h z | z 0 i = δzz 0 .

(8.2)

ˆ is real and nonnegative: z ∈ R+ . Lemma 8.1 Eigenvalue of the operator N ˆ , we have Proof. Since | z i denotes the normalized eigenvector of N z = z hz |z i = hz | z |z i = hz | a ˆ† a ˆ |z i = hz | a ˆ† ( a ˆ |z i ) ˆ | z i ) = || a ˆ | z i ||2 . = (a ˆ | z i )† ( a

(8.3)

So we see that z is equal to a norm of a certain vector, and as such is real and nonnegative. Lemma 8.2 The following commutation relations hold h i a ˆ† a ˆ, a ˆ = −ˆ a, h i a ˆ† a ˆ, a ˆ† = a ˆ† .

Proof. By simple calculation, we get from the canonical relation (8.1): h i h i a ˆ† a ˆ, a ˆ = a ˆ† [ a ˆ, a ˆ] + a ˆ† , a ˆ a ˆ = a ˆ† · 0 + (−1)ˆ a. i h i h i h a ˆ† a ˆ, a ˆ† = a ˆ† a ˆ, a ˆ† + a ˆ† , a ˆ† a ˆ = a ˆ† + 0 · a ˆ,

(8.4a) (8.4b)

(8.5)

which completes the proof.

ˆ = a Lemma 8.3 The ket a ˆ | z i is an eigenstate of the operator N ˆ† a ˆ, and it belongs to an eigenvalue (z − 1), that is ˆa N ˆ | z i = (z − 1) a ˆ | z i. S.Kryszewski

(8.6) QUANTUM OPTICS

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Proof. If a ˆ | z i 6= 0, then we have ˆa N ˆ |z i = a ˆ† a ˆa ˆ | z i.

(8.7)

ˆa ˆ=a ˆa ˆ† a ˆ−a ˆ, and hence Due to commutation relation (8.4a) we can write a ˆ† a ˆa N ˆ |z i = a ˆ (ˆ a† a ˆ − 1) | z i = a ˆ z |z i − a ˆ | z i = (z − 1) a ˆ | z i.

(8.8)

ˆ with an eigenvalue (z − 1). This shows that vector a ˆ | z i is an eigenstate of N ˆ = a Lemma 8.4 The ket a ˆ † | z i is an eigenstate of the operator N ˆ† a ˆ, and it belongs to an eigenvalue (z + 1), that is ˆa N ˆ† | z i = (z + 1) a ˆ | z i.

(8.9)

Proof. The proof is analogous to that of the previous lemma, only we use commutation relation (8.4b) instead of (8.4a). Lemma 8.5 Norms of the vectors a ˆ | z i and a ˆ † | z i are given as √ √ || a ˆ† | z i || = z + 1 . || a ˆ | z i || = z ,

(8.10)

Proof. The first norm follows automatically from the proof of the first lemma, see relation (8.3). The second relation is proved similarly. We have ||ˆ a† | z i||2 =

a ˆ† | z i

†

a ˆ† | z i

= hz|a ˆa ˆ† | z i.

(8.11)

Using the canonical commutation relation we have a ˆa ˆ† = a ˆ† a ˆ + 1, thus, we get ||ˆ a† | z i||2 = h z | a ˆ† a ˆ +1|z i = hz|a ˆ† a ˆ | z i + h z | z i = || a ˆ | z i ||2 + 1 = z + 1, (8.12) since vector | z i is normalized and || a ˆ | z i || 2 = z. Second relation (8.10) follows immediately. ˆ belonging to the eigenvalue Lemma 8.6 If a vector a ˆ n | z i 6= 0, then it is an eigenvector of N (z − n): ˆa N ˆn | z i = (z − n) a ˆn | z i

(8.13)

Proof. The proof follows by mathematical induction. The case n = 1 was already shown in ˆa ˆa (8.6). In the proof essential role is played by the relation N ˆ = N ˆ−a ˆ, which follows from (8.4a). We easily have n+1 ˆ a ˆa ˆ −a ˆ [ˆ N ˆ |z i = N ˆ [ˆ an | z i] = (ˆ aN ˆ) [ˆ an | z i] = a ˆN an | z i] − a ˆn+1 | z i (8.14) By induction assumption, we further get n+1 ˆ a N ˆ |z i = a ˆ(z − n)ˆ an | z i − a ˆn+1 | z i = (z − n − 1)ˆ an+1 | z i.

(8.15)

and the lemma follows.

Lemma 8.7 There exists such an integer n, that a ˆn | z i 6= 0, 2

but

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ˆ and Proof. From the previous lemma it follows that a ˆ n | z i is an eigenvector of the operator N ˆ it belongs to the eigenvalue (z − n). Lemma (8.1) states that eigenvalues of N are nonnegative. For n sufficiently large we would have (z − n) < 0. This contradicts lemma (8.1). Hence, there must exist an integer n such that relations (8.16) are satisfied. This completes the proof. ˆ defined in Eq.(8.2) are nonnegative integers. Theorem 8.1 The eigenvalues z of the operator N ˆ that Moreover, there exists such a normalized eigenvector | 0 i of N a ˆ |0i = 0

(8.17)

which will be called the vacuum state. ˆ belonging to the eigenvalue z − n, we can Proof. Since a vector a ˆ n | z i is an eigenvector of N normalize it and write it as |z − ni =

a ˆn | z i . ||ˆ an | z i||

(8.18)

Let the integer n be such, that Eq.(8.16) is satisfied. This means that a ˆ | z − n i = 0,

(8.19)

and the norm of the obtained vector is || a ˆ | z − n i || = 0.

(8.20)

Now, from the first of relations (8.10) it follows that √ || a ˆ | z − n i || = z − n = 0.

(8.21)

ˆ = a This implies that z = n. Hence the eigenvalues z of the operator N ˆ† a ˆ are nonnegative integers. We also conclude that there exists a normalized vector | 0 i for which eq.(8.16) is satisfied for n = 0. Theorem 8.2 According to the previous theorem, we denote by | n i the normalized eigenstate ˆ belonging to the eigenvalue n – nonnegative integer. Then, the vectors of the operator N |n − 1i =

a ˆ |ni √ , n

and

a ˆ† | n i |n + 1i = √ , n+1

(8.22)

ˆ . These relations enable us to construct all the eigenstates of operator are the eigenstates of N ˆ N , provided one of the states | n i is given. ˆ belonging Proof. In lemma (8.3) we have shown that the vector a ˆ | n i is an eigenstate of N to the eigenvalue (n − 1). This means (according to the introduced notation), that a ˆ | n i is proportional to the vector | n − 1 i. It remains to find the coefficient of proportionality. From √ lemma (8.5) we have the norm || a ˆ | n i || = n . Thus the vector a ˆ |ni a ˆ |ni = √ , || a ˆ | n i || n

(8.23)

ˆ with eigenvalue (n − 1). Hence it is equal to | n − 1 i. So the is a normalized eigenvector of N first part of the theorem is proved. The second part can be shown in the same manner. Let us note that relations (8.22) can be rewritten as √ a ˆ |ni = n |n − 1i (8.24a) √ † a ˆ |ni = n + 1 |n + 1i (8.24b) S.Kryszewski

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ˆ =a Lemma 8.8 The eigenstate | n i of the operator N ˆ† a ˆ can be constructed as n 1 |ni = √ | 0 i, a ˆ† n! if the vacuum state | 0 i defined in eq.(8.17) is given.

(8.25)

Proof. The proof follows by induction from relation (8.24b). For n = 1 we have 1 1 √ |1i = √ a ˆ† | 0 i = √ 1 | 1 i = | 1 i, (8.26) 1! 1! as it should be. Now, we have 1 1 1 √ |n + 1i = p (ˆ a† )n+1 | 0 i = √ a ˆ† (ˆ a † )n | 0 i n + 1 n! (n + 1)! √ a ˆ† |n + 1i = √ |ni = n + 1 √ = | n + 1 i. (8.27) n+1 n+1 Going from the first to the second line we have employed the principle of mathematical induction, and thus the proof is completed. This lemma clearly indicates the manner of construction of the eigenstates of the operator ˆ N =a ˆ† a ˆ. We must find the ground state – the vacuum one | 0 i which should be unique. If this is not the case, we must find a complete set of commuting observables and classify the vacuum states with the aid of additional quantum numbers. Normalizing the vacuum state we apply the creation operators to construct the eigenstates | n i. Lemma 8.9 The eigenstates | n i specified in (8.25) are orthonormal, that is h n | m i = δnm .

(8.28)

ˆ =a Orthogonality follows from the fact that | n i are eigenstates of the hermitian operator N ˆ† a ˆ, so it is sufficient to prove that the are normalized. Proof. Without loss of generality we can assume n ≥ m. Then from (8.25) we have 1 hn|mi = √ h 0 | aˆn (ˆ a† )m | 0 i. n! m! But h i h i h i a ˆ (ˆ a† )m − (ˆ a † )m a ˆ = a ˆ, (ˆ a † )m = a ˆ† a ˆ, (ˆ a† )m−1 + a ˆ, a ˆ† (ˆ a† )m−1 h i = a ˆ† a ˆ, (ˆ a† )m−1 + (ˆ a† )m−1 .

(8.29)

(8.30)

Continuing such a reasoning we finally obtain a ˆ (ˆ a† )m − (ˆ a † )m a ˆ = m (ˆ a† )m−1 ,

(8.31)

which can easily be verified by mathematical induction. Therefore, we obtain h i 1 hn|mi = √ h 0 | aˆn−1 m(ˆ a† )m−1 + (ˆ a † )m a ˆ |0i n! m! 1 = √ m h 0 | aˆn−1 (ˆ a† )m−1 | 0 i, (8.32) n! m! because a ˆ | 0 i = 0. Repeating such a procedure m times we will arrive at the relation r m! hn|mi = h 0 | aˆn−m | 0 i. (8.33) n! For n > m we have a ˆ n−m | 0 i = 0, which follows from the definition of the vacuum state. When n = m we get h n | m i = h 0 | 0 i = 1. So the states | n i are orthogonal (which is not unexpected) and normalized, as it should be. 4

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Annihilation and creation operators – summary

Annihilation and creation operators (non-hermitian) are specified by the commutation relation h i a ˆ, a ˆ† = 1. (8.34) ˆ =a The number states | n i are the eigenstates of the number operator N ˆ† a ˆ, that is ˆ |ni = a N ˆ† a ˆ| n i = n | n i,

with

n = 0, 1, 2, . . . . . .

(8.35)

The state | 0 i is called a vacuum state and it satisfies the condition a ˆ | 0 i = 0.

(8.36)

ˆ) Number states | n i are orthonormal (eigenstates of the Hermitian operator N h m | n i = δmn .

(8.37)

Annihilation and creation are sometimes called ladder operators. This follows from the properties of lowering and raising the number of the state √ a ˆ |ni = n | n − 1 i, (8.38a) √ † n + 1 | n + 1 i. (8.38b) a ˆ |ni = Let us note that these relations are fully consistent with the previous ones. Relation (8.38a) agrees with the definition (8.36) of the vacuum state. Moreover, we have √ √ a ˆ† a ˆ |ni = a ˆ† n | n − 1 i = n a ˆ† | n − 1 i √ p n (n − 1) + 1 | n i = n | n i, (8.39) =

as it should be, when compared to definition (8.35). Matrix elements of the annihilation and creation operators follow immediately from Eqs.(8.38)and from orthonormality requirement. We have √ √ hm|a ˆ|ni = n h m | n − 1 i = n δm,n−1 , (8.40a) √ √ † n + 1 h m | n + 1 i = n + 1 δm,n+1 . (8.40b) h m | aˆ | n i = Finally, practical construction goes along the following way • Construct annihilation and creation operators a ˆ and a ˆ † , check their commutation relation (to reproduce the canonical one (8.34)). • Find (construct) the vacuum state | 0 i. • Construct the number states by using the relation |ni =

S.Kryszewski

(ˆ a † )n √ | 0 i. n!

(8.41)

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8.3

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Application to harmonic oscillator

8.3.1

Annihilation and creation operators for harmonic oscillator

Hamiltonian of the quantum–mechanical harmonic oscillator is of the form 2 ˆ = pˆ + 1 mω 2 x H ˆ2 , 2m 2

(8.42)

and the momentum and position operators satisfy the canonical commutation relation h i x ˆ, pˆ = i~.

It is an easy matter to check that two operators r mω pˆ √ x ˆ and , ~ mω~

(8.43)

(8.44)

are dimensionless. Theorem 8.3 Two dimensionless, nonhermitian operators a ˆ and a ˆ † defined as r 1 mω iˆ p 1 ˆb = √ x ˆ + √ = √ ( mω x ˆ + iˆ p ), ~ 2 mω~ 2mω~ r mω iˆ p 1 ˆb† = √1 x ˆ − √ = √ ( mω x ˆ − iˆ p ), ~ 2 mω~ 2mω~ satisfy the commutation relation h i ˆb, ˆb† = 1.

(8.45a) (8.45b)

(8.46)

Hence we may identify: ˆb – annihilation, and ˆb† – creation operators. Proof. The facts that these operators are nonhermitian and dimensionless are evident. We show the commutation relation. h i h i 1 ˆb, ˆb† mω x ˆ + iˆ p, mω x ˆ − iˆ p = 2mω~ n h i h i h i h io 1 = m2 ω 2 x ˆ, x ˆ − imω x ˆ, pˆ + imω pˆ, x ˆ + pˆ, pˆ 2mω~ h i h io imω n i − x ˆ, pˆ + pˆ, x ˆ = { − i~ + (−i~) } = 1. (8.47) = 2mω~ 2~

Since operators ˆb and ˆb† satisfy commutation relation typical for annihilation and creation operators, they posses all the necessary properties and the identification made in the theorem is fully justified and correct. Relations (8.45) can easily be inverted, and we can express the position and momentum operators via annihilation and creation ones r ~ ˆ x ˆ = b + ˆb† , (8.48a) 2mω r mω~ ˆ pˆ = −i b − ˆb† , (8.48b) 2 6

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Having expressions (8.48) we can now express the Hamiltonian of the oscillator in terms of the annihilation and creation operators. We obtain # "r # " r 2 2 1 1 mω~ ~ † 2 † ˆb − ˆb ˆb + ˆb ˆ = H −i + mω 2m 2 2 2mω 2 2 ~ω ˆ ~ω ˆ b − ˆb† b + ˆb† = − + 4 4 ~ω ˆˆ ˆˆ† ˆ†ˆ ˆ†ˆ† ~ω ˆˆ ˆˆ† ˆ†ˆ ˆ†ˆ† = − + bb − bb − b b + b b bb + bb + b b + b b 4 4 ~ω ˆ ˆ† ˆ† ˆ = bb +b b (8.49) 2 Using the commutation relation (8.46) we have ˆb ˆb† = 1 + ˆb†ˆb, thus from the above we finally get 1 ˆ = ~ω 2 ˆb† ˆb + 1 ˆ+1 H = ~ω ˆb† ˆb + = ~ω N (8.50) 2 2 2

ˆ = ˆb† ˆb. where, as previously, we introduced the number operator N Theorem 8.4 Energy eigenstates of the quantum-mechanical harmonic oscillator are the numˆ = ˆb† ˆb. The energy eigenvalues are ber states | n i – the eigenstates of the number operator N 1 En = ~ω n + . (8.51) 2 Proof. The proof follows immediately from relation (8.50) and from the properties of the number operator, as discussed in the previous section.

8.3.2

Construction of the vacuum state

Construction of the vacuum state is the first step in building the energy eigenstates of the harmonic oscillator. We will do this in the position representation, that is we are looking for the wave function ϕ0 (x) = h x | 0 i. The vacuum state is defined by eq.(8.17), so using the annihilation operator ˆb as given in (8.45a), we get 1 0 = ˆb | 0 i = √ ( mω x ˆ + iˆ p ) | 0 i. 2mω~

(8.52)

In position representation, this equation reads 1 0 = hx| √ ( mω x ˆ + iˆ p ) |0i 2mω~ 1 d = √ mω x + i −i~ ϕ0 (x). dx 2mω~ The latter relation is a simple differential equation of the first order d mω 0 = λx + ϕ0 (x), with λ = . dx ~ Solution to this equation is very simple. It is λx2 ϕ0 (x) = Ao exp − , 2 S.Kryszewski

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(8.53)

(8.54)

(8.55) 7

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Extras. 8. Annihilation and creation operators

where Ao is a normalization constant. Computation of this constant yields r Z ∞ λx2 π 2 2 dx exp − = | Ao | . 1 = | Ao | 2 λ −∞

2.02.2004

(8.56)

Choosing the arbitrary phase of the constant A o to be zero we obtain the wave function of the ground state of the oscillator, or in other words, the vacuum state in the position representation 1/4 λ λx2 ϕ0 (x) = , (8.57) exp − π 2 which is properly normalized.

8.3.3

Construction of the number states | n i

Having constructed the vacuum state in the position representation, we proceed to construct further states. To do so, we use relation (8.41) in position representation 1 h x | (ˆb† )n | 0 i. (8.58) ϕn (x) = h x | n i = √ n! In order to deal with this expression let us consider a bra (dual form) h x | ˆb† . Using Eq.(8.45b) we get

r 1 mω i hx|b = hx| √ ( mω~ x ˆ − iˆ p) = hx| x ˆ − pˆ 2~ mω 2mω~ r r † † λ i λ ~ d = x ˆ + pˆ | x i = x + |xi . 2 mω 2 mω dx Since the differential operator d/dx is antihermitian, we get r λ 1 d † ˆ hx|b = x − h x |. 2 λ dx ˆ†

(8.59)

Using this relation n times in (8.58), we get n/2 1 1 d n λ √ ϕn (x) = x − h x | 0 i. (8.60) 2 λ dx n! Inserting the wave function (8.57), we obtain the differential relation specifying the n-th eigenstate of the harmonic oscillator 1/4 r 1 λ 1 d n λx2 n/2 ϕn (x) = λ x − exp − . (8.61) π 2n n! λ dx 2

This is a functional equation similar to the Rodrigues formula for Hermite polynomials. This is clarified by the following theorem Theorem 8.5 Hermite polynomials can be expressed as follows 2 y d n y2 Hn (y) = exp y − exp − . 2 dy 2

(8.62)

We accept this theorem without proof (which is not difficult, when √ one uses the Rodrigues formula for Hermite polynomials). Changing the variable y = x λ, we can easily show that eq.(8.61 leads to the expression 1/4 r √ λ 1 λx2 ϕn (x) = exp − H (x λ), (8.63) n π 2n n! 2

which, together with notation introduced in (8.54) exactly reproduces the standard wave functions of the n-th energy eigenstate of the quantum-mechanical harmonic oscillator. 8

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Appendix B

Useful operator identities B.1

Similarity relations

Theorem B.1 Let A and B be operators. Let ξ be a parameter which may be complex or real. Then, the following identity holds eξA Be−ξA = B +

i ii iii ξ h ξ2 h h ξ3 h h h A, B + A, A, B + A, A, A, B + ... 1! 2! 3!

(2.1)

ˆ For arbitrary operator B we define Before proving this theorem let us specify a superoperator A. h i ˆ = A, B . AB (2.2) Formally we can also write h i Aˆ = A, ,

(2.3)

where an empty place at the second position within a commutator is understood as a place where ˆ should be inserted. Having the the operator B, which is acted upon by the superoperator A, ˆ definition of the superoperator A we can rewrite the theorem (2.1) equivalently as ˆ B. exp(ξA) B exp(−ξA) = exp(ξ A)

(2.4)

Proof. We introduce an operator-valued function g(ξ) = eξA Be−ξA ,

with initial condition :

g(0) = B.

(2.5)

Next, we expand g(ξ) in Taylor series ∞ X ξ n dn g(ξ) g(ξ) = g(0) + n! dξ n ξ=0 n=1

(2.6)

It remains to compute explicitly the coefficients of the expansion, that is the derivatives evaluated at ξ = 0. The first derivative is as follows d d ξA −ξA ξA −ξA ξA −ξA g(ξ) = e Be = Ae Be − e BAe dξ dξ ξ=0 ξ=0 ξ=0 h i h i = Ag(ξ) − g(ξ)A = A, g(ξ) = A, B (2.7) ξ=0

14

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where the first equality in the second line follows from the fact that operator A commutes with the exponential eξA . The last equality follows from initial condition for the function g(ξ). Substituting (2.7) into expansion (2.6) we obtain ∞ i X ξ h ξ n dn g(ξ) g(ξ) = g(0) + A, B + (2.8) 1! n! dξ n ξ=0 n=2

In the similar manner we calculate next terms of the expansion. i d2 d h dg(ξ) A, g(ξ) g(ξ) = = A, dξ 2 dξ dξ ξ=0 ξ=0 ξ=0 h h ii h h ii = A, A, g(ξ) = A, A, B ξ=0

where the last steps follow from Eq.(2.7). Thus, (2.8) transforms into the relation ∞ i ξ2 h h ii X ξ n dn g(ξ) ξ h g(ξ) = g(0) + A, B + A, A, B + 1! 2! n! dξ n ξ=0 n=3

(2.9)

(2.10)

Further derivatives may be found in the same manner. It is also possible to employ the method of mathematical induction to show that the theorem (2.1) indeed holds. Finally, we note that the relation (2.4) follows from (2.1) simply by expansion of the exponential in the right-hand-side of (2.4). Comparing the obtained expansion we easily see that right-hand-side of (2.1) is reproduced. This completes the proof of the theorem. The previous theorem can easily be generalized. We shall now formulate a generalized similarity relation. Theorem B.2 Let g(B1 , . . . , Bk ) be a function of k different operators. We assume that this function can be expanded into series X g(B1 , B2 , . . . , Bk ) = gn1 n2 ...nk B1n1 B2n2 . . . Bknk . (2.11) {nk }

If it is necessary, commutation relations can be used to rearrange the operators {B j } in the power series. Then, the following similarity relation holds for operator A and a complex number ξ eξA g(B1 , B2 , . . . , Bk ) e−ξA = g eξA B1 e−ξA , eξA B2 e−ξA , . . . , eξA Bk e−ξA , (2.12)

that is, the function g is unchanged, only each of its arguments is transformed according to the given similarity.

Proof. Applying similarity operator e ξA on the left of the expansion, and e−ξA on the right, ˆ = e−ξA eξA between all factors in each term of the series. Then we can also introduce the 1 each of the operators undergoes the similarity transformation, and the series coefficients remain unchanged. Resummation yields rhs of the theorem.

B.2

Decomposition of the exponential

B.2.1

General idea of the decomposition

In many practical applications we need to express the operator exp[ξ(A + B)], where A and B are also operators, as a product of separate exponentials, that is eξ(A+B) = ef1 (ξ)A ef2 (ξ)B ef3 (ξ) . S.Kryszewski

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where fk (ξ) are ordinary (c-numbered, complex) functions of the parameter ξ ∈ C. These functions must satisfy an obvious boundary condition f k (0) = 0. The whole problem is to determine these functions. Before we start looking for the solution, let us note that we can easily write the relation inverse to (2.13), namely e−ξ(A+B) = e−f3 (ξ) e−f2 (ξ)B e−f1 (ξ)A .

(2.14)

Surely, exp[−f3 (ξ)] is a number so it commutes with all operators. To find functions fk (ξ) let us differentiate both sides of )(2.13), thus obtaining 0

0

(A + B) eξ(A+B) = f1 (ξ)A ef1 (ξ)A ef2 (ξ)B ef3 (ξ) + ef1 (ξ)A f2 (ξ)B ef2 (ξ)B ef3 (ξ) 0

+ ef1 (ξ)A ef2 (ξ)B ef3 (ξ) f3 (ξ),

(2.15)

where the prime denotes the derivative with respect to ξ. Next, we multiply both sides of (2.15) on the right by both sides of the inverse relation (2.14). We get 0

0

0

A + B = f1 (ξ) A + f2 (ξ) ef1 (ξ)A B e−f1 (ξ)A + f3 (ξ).

(2.16)

Equating the coefficients multiplying operator A, we see that 0

f1 (ξ) = 1,

=⇒

f1 (ξ) = ξ,

(2.17)

which satisfies the boundary condition. Using (2.17) in (2.16) we reduce it to 0

0

B = f2 (ξ) eξA B e−ξA + f3 (ξ).

(2.18)

Now, we employ the similarity expansion (2.1) to write i ii ξ h ξ2 h h 0 0 B = f2 (ξ) B + A, B + A, A, B + . . . + f3 (ξ). 1! 2!

(2.19)

Further steps obviously depend on the shape of the commutators which appear within the curly brackets. If we know the commutators, we can try to find the remaining functions of the parameter ξ.

B.2.2

The case of [ A, B ] = c

Let us now assume that the commutator [ A, B ] = c, where c ∈ C. In such a case, all terms in (2.19), except the first two ones, vanish and we have 0 0 B = f2 (ξ) B + ξ c + f3 (ξ). (2.20) Hence, we arrive at the equations 0

f2 (ξ) = 1,

0

and

0

f2 (ξ)ξc + f3 (ξ) = 0.

(2.21)

These equations are immediately integrated, and taking into account the boundary conditions we get f2 (ξ) = ξ,

and

f3 (ξ) = −

1 2 ξ c. 2

(2.22)

Thus we can state the following Theorem B.3 If two operators A and B have the commutator [ A, B ] = c ∈ C, then eξ(A+B) = eξA eξB e−cξ

2 /2

,

(2.23)

for any complex parameter ξ. Equivalently we can write eξ(A+B) = eξB eξA ecξ 16

2 /2

,

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Math. App. B. Useful operator identities

Special case for annihilation and creation operators

We employ the theorem (2.23) taking ξ = 1 and specifying the operators as A = αa ˆ,

B = βˆ a† ,

and

(2.25)

where a ˆ and a ˆ† are annihilation and creation operators. Since h i αˆ a, βˆ a† = α β,

(2.26)

from Eqs.(2.23) and (2.24) we obtain eαˆa+βˆa

†

†

= eαˆa eβˆa e−α β/2 = e

βˆ a†

(2.27a)

eαˆa eα β/2 .

(2.27b)

These relations are very useful in many practical cases.

B.3

Similarity relation for annihilation operator

B.3.1

General relation

Let a ˆ and a ˆ † be the annihilation and creation operators, which satisfy the canonical commutation relation a ˆ, a ˆ† = 1. Let us, moreover, define an operator Z = αˆ a + βˆ a† + γˆ a† a ˆ,

(2.28)

with α, β and γ being complex parameters (numbers). We consider the similarity relation a ˆ(ξ) = e−ξZ a ˆ eξZ ,

(2.29)

with an obvious boundary condition a ˆ(0) = a ˆ. We can, in principle, use general expression ˆ, the commutator series do (2.1). This is, however, inconvenient because due to the term γˆ a†a not truncate. Therefore, we employ a different approach. We differentiate Eq.(2.29) with respect to parameter ξ, obtaining h i d a ˆ(ξ) = e−ξZ − Zˆ a+a ˆ† Z eξZ = e−ξZ a ˆ, Z eξZ . (2.30) dξ It is straightforward to compute the commutator h i h i a ˆ, Z = a ˆ, αˆ a + βˆ a† + γˆ a† a ˆ = β + γˆ a.

(2.31)

Thus we have the differential equation d a ˆ(ξ) = e−ξZ β + γˆ a eξZ = β + γˆ a(ξ). dξ

(2.32) 0

a has an This is an inhomogeneous differential equation. The homogeneous one: a ˆ (ξ) = γˆ γξ obvious solution a ˆ(ξ) = a ˆ(0)e . Hence, we look for the solution of (2.32) in the form a ˆ(ξ) = eγξ b(ξ),

(2.33)

with boundary condition b(0) = a ˆ(0). Inserting (2.33) into (2.32) we obtain an equation for b(ξ) 0

b (ξ) = e−γξ β S.Kryszewski

which yields

b(ξ) = b0 −

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β −γξ e , γ

(2.34) 17

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where the constant b0 has to be fixed. From boundary condition we get a ˆ(0) = b(0) = b 0 − β/γ. Therefore, the sought solution to Eq.(2.32) follows as β β −γξ γξ a ˆ(ξ) = e a ˆ(0) + − e . (2.35) γ γ This completes our derivation and we can finally write a ˆ(ξ) = e−ξ(αˆa+βˆa

B.3.2

† +γˆ a† a ˆ)

a ˆ eξ(αˆa+βˆa

† +γˆ a† a ˆ)

= a ˆ eγξ +

β γξ e − 1 . γ

(2.36)

Some special cases

Let us take γ = 0, ξ = 1. Then, relation (2.36) reads †

e−αˆa−βˆa a ˆ eαˆa+βˆa

†

= a ˆ + β,

(2.37)

and it can be rewritten as a ˆ eαˆa+βˆa

†

†

†

= eαˆa+βˆa a ˆ + eαˆa+βˆa β,

(2.38)

which yields the commutation relation i h † † = eαˆa+βˆa β, a ˆ, eαˆa+βˆa

(2.39)

Let us note, that we can apply relations (2.27) to formula (2.37). This gives †

†

†

e−βˆa e−αˆa eαβ/2 a ˆ eαˆa eβˆa e−αβ/2 = e−βˆa a ˆ eβˆa

†

= a ˆ + β,

(2.40)

because e±αˆa commutes with a ˆ. Another special case follows easily, when we put α = β = 0, and γ = 1. Then (2.36) yields a ˆ(ξ) = e−ξˆa

B.3.3

†a ˆ

a ˆ eξˆa

†a ˆ

= a ˆ eξ .

(2.41)

Applications of generalized similarity relation

The generalized similarity theorem (2.12) has several immediate applications. The first one is for arbitrary function g(ˆ a, a ˆ † ) which can be expanded into power series of annihilation and creation operators, namely we have † † † † † † e−ξˆa aˆ g(ˆ a, a ˆ† ) eξˆa aˆ = g e−ξˆa aˆ a ˆ eξˆa aˆ , e−ξˆa aˆ a ˆ† eξˆa aˆ = g a ˆ eξ , a ˆ† e−ξ (2.42)

where we used relation (2.41). Let us note, that (2.42 implies for y ∈ R that † † † † e−iyˆa aˆ D(z) eiyˆa aˆ = e−iyˆa aˆ exp zˆ a† − z ∗ a ˆ eiyˆa aˆ = exp ze−iy a ˆ† − z ∗ eiy a ˆ = D(ze−iy )

18

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19

Math. App. B. Useful operator identities

Squeeze operator

We define squeezing operator, for a complex parameter ξ ∈ C, as 1 ∗ 2 1 † 2 S(ξ) = exp ξ a ˆ − ξ (ˆ a ) . 2 2

(2.44)

We easily see that †

S (ξ) = exp

1 1 ∗ 2 † 2 ξ (ˆ a ) − ξ a ˆ , = S(−ξ) = S −1 (ξ), 2 2

(2.45)

which indicates that operator S(ξ) is a unitary one. In the spirit of previous sections, we intend to investigate the similarity transformation of the annihilation operator induced by the squeezing operator. That is, we are interested in the expression 1 1 1 ∗ 2 1 aS (ξ) = S † (ξ) a ˆ S(ξ) = exp ξ(ˆ a † )2 − ξ ∗ a ˆ2 a ˆ exp ξ a ˆ − ξ(ˆ a † )2 . (2.46) 2 2 2 2 To consider this relation it is convenient to write the complex parameter in polar coordinates, ξ = ρ eiθ .

(2.47)

Then, we can rewrite (2.46) in the form 1 † 2 iθ 1 2 −iθ 1 † 2 iθ 1 2 −iθ (ˆ a) e − a ˆ e (ˆ a) e − a ˆ e aS (ξ) = exp ρ a ˆ exp − ρ . 2 2 2 2

(2.48)

We analyze this expression by means of formula (2.1), in which we make the identifications ξ → ρ,

A→

1 † 2 iθ 1 2 −iθ (ˆ a) e − a ˆ e , 2 2

B→a ˆ.

(2.49)

We see that we have to consider the commutators of operator A with B which, due to the introduced identifications, reads h i h i 1 † 2 iθ 1 2 −iθ 1 A, B = (ˆ a) e − a ˆ e , a ˆ = eiθ a ˆ† a ˆ† , a ˆ = − eiθ a ˆ† . (2.50) 2 2 2 Using the obtained commutator, we compute the next one, as it follows from the general expansion (2.1). We get h h ii i 1 † 2 iθ 1 2 −iθ 1h A, A, B = (ˆ a) e − a ˆ e , − e−iθ a ˆ† = a ˆa ˆ, a ˆ† = a ˆ. (2.51) 2 2 2 By careful inspection of the obtained commutators we conclude that: • when operator A identified according to (2.49) occurs even number of times, the result of such a multiple commutator will always be equal to the annihilation operator a ˆ; • when operator A occurs odd number of times, the result of a corresponding multiple commutator will always be equal to the − e iθ a ˆ† .

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Math. App. B. Useful operator identities

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Therefore, the general similarity expansion (2.1) splits into two series: with odd and even terms, and we get ρ2 ρ4 ρ3 ρ5 † iθ aS (ξ) = a + + ...... − a ˆ e + + ...... ˆ 1+ ρ+ 2! 4! 3! 5! = a ˆ cosh(ρ) − a ˆ † eiθ sinh(ρ).

(2.52)

Summarizing we write aS (ξ) = S † (ξ) a ˆ S(ξ) = a ˆ cosh(ρ) − a ˆ † eiθ sinh(ρ),

(2.53)

The generalized similarity theorem (2.12) can be applied to find a transformation of the squeeze operator. Since the exponential function is expandable into the power series, we get for y ∈ R: 1 ∗ 2 1 † −iyˆ a† a ˆ iyˆ a† a ˆ −iyˆ a† a ˆ † 2 ξ a a ) e S(ξ) e = e exp ˆ − ξ(ˆ eiyˆa aˆ 2 2 2 1 2 1 † † † † = exp ξ ∗ e−iyˆa aˆ a ˆ eiyˆa aˆ − ξ e−iyˆa aˆ a ˆ† eiyˆa aˆ 2 2 1 ∗ 2 2iy 1 † 2 −2iy = exp ξ a a ) e ˆ e − ξ(ˆ 2 2 (2.54) = S ξ e−2iy

where in the third line we have used (2.41), while in the fourth we used the definition of the squeeze operator with shifted argument. Next, we note that by means of the general similarity theorem (2.12) and using (2.52) and its hermitian conjugate, we can write S † (ξ) g(ˆ a, a ˆ † ) S(ξ) = = g a ˆ cosh(ρ) − a ˆ † eiθ sinh(ρ), a ˆ† cosh(ρ) − a ˆ e−iθ sinh(ρ)

(2.55)

where g(., .) is a function, which can be expanded into power series. In particular, taking function g as the displacement operator g(ˆ a, a ˆ † ) = D(α) = exp(αˆ a † − α∗ a ˆ) we obtain h S † (ξ) D(α) S(ξ) = exp α a ˆ† cosh(ρ) − a ˆ e−iθ sinh(ρ) i −α∗ a ˆ cosh(ρ) − a ˆ † eiθ sinh(ρ) h = exp a ˆ† α cosh(ρ) + α∗ eiθ sinh(ρ) i −a ˆ α∗ cosh(ρ) + αe−iθ sinh(ρ) = D α cosh(ρ) + α∗ eiθ sinh(ρ) = D(z).

(2.56)

which defines new argument z = α cosh(ρ) + α ∗ eiθ sinh(ρ) of the transformed displacement operator. Relation (2.56) can be written as D(α) S(ξ) = S(ξ) D(z),

(2.57)

with z given via α and ξ = ρeiθ as above. Taking the expression for z and its complex conjugate, we multiply the first one by cosh(ρ) and the second one by − sinh(ρ) e iθ . Then we add both equations, and using the hyperbolic unity we express α as α = z cosh(ρ) − z ∗ eiθ sinh(ρ) 20

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Math. App. B. Useful operator identities

21

which, together with z = α cosh(ρ) + α∗ eiθ sinh(ρ)

(2.59)

allows us to use Eq.(2.57) in an effective manner. From the last relation we see that z e−iθ/2 = α e−iθ/2 cosh(ρ) + α∗ eiθ/2 sinh(ρ) 1 1 ρ −iθ/2 e αe = + α∗ eiθ/2 + e−ρ α e−iθ/2 − α∗ eiθ/2 2 2

This allows us to derive a useful relation between parameters α and z, namely Re z e−iθ/2 = eρ Re α e−iθ/2 −iθ/2 −ρ −iθ/2 Im z e = e Im α e

S.Kryszewski

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(2.60)

(2.61)

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Math. App. C. Certain sum rule for Hermite polynomials

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Appendix C

Certain sum rule for Hermite polynomials Here we will prove the following summation rule involving even Hermite polynomials ∞ X 1 tk 4tx2 H2k (x) = √ exp , k! 1 + 4t 1 + 4t

(3.1)

k=0

To prove this rule, we recall the generating function of Hermite polynomials e

−s2 +2sx

∞ X sn Hn (x). = n! n=0

(3.2) 2

We multiply both sides of (3.2) by e−as , with a being a real positive parameter, and then we integrate both sides over s ∈ R1 . Thus, we get Z ∞ Z ∞ X Hn (x) ∞ 2 −(a+1)s2 +2sx ds e = ds sn e−as . (3.3) n! −∞ −∞ n=0

Both integrals appearing in (3.3) are simple. The one in the lhs we compute according to 2 r Z ∞ π q −py 2 −qy dy e = exp , (3.4) p 4p −∞ where in our case p = a + 1 and q = −2x. The integral in the rhs of (3.3) vanishes for n = 2k + 1, that is for odd n. Thus we have Z ∞ Γ(k + 1/2) 2 ds s2k e−as = , for n = 2k. (3.5) ak+1/2 −∞ Using (3.4) and (3.5) in (3.3) we get r 2 ∞ X aπ x H2k (x) Γ(k + 1/2) exp = a+1 a+1 ak (2k)!

(3.6)

k=0

Next we consider the combinatorial term in the rhs. We know that Γ(k + 1/2) =

√ (2k − 1)!! π . 2k

(3.7)

Moreover, we have (2k)! = (2k)!!(2k − 1)!! = 2 k k!(2k − 1)!!, so by combining these relations √ Γ(k + 1/2) 1 = π 2k . (2k)! 2 k! 22

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23

Using (3.8) in the summation rule (3.6) we obtain r 2 ∞ X 1 k H2k (x) a x = exp . k 4a a a+1 a+1

(3.9)

k=0

We see that substitution a = 1/4t yields r ∞ X H2k (x) k 1 4tx2 t = exp , k! 1 + 4t 1 + 4t

(3.10)

k=0

which is the sum rule (3.1) which we intended to prove, so the proof is therefore completed. Finally we note that the obtained expression is well defined for t > −1/4.

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Math. App. D. Pseudospin operators

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Appendix D

Pseudospin operators D.1

Basic definitions

Identifications |1i =

0 1

,

|2i =

1 0

Construction of the pseudospin matrices 0 0 0 | 1 ih 1 | = 0, 1 = 1 0 1 0 0 0 | 1 ih 2 | = 1, 0 = 1 1 0 1 0 1 | 2 ih 1 | = 0, 1 = 0 0 0 1 1 0 | 2 ih 2 | = 1, 0 = 0 0 0 Pauli matrices 0 1 σ1 = , 1 0 It is convenient to denote 0 S− = | 1 ih 2 | = 1 0 S+ = | 2 ih 1 | = 0

σ2 =

0 0 1 0

0 −i i 0

.

(4.1)

,

(4.2a)

,

(4.2b)

,

(4.2c)

.

(4.2d)

,

σ3 =

1 0 0 −1

.

(4.3)

,

(4.4a)

.

(4.4b)

We call these operators lowering and raising, respectively. We now define some more operators via their matrices. 1 1 0 1 S1 = (S+ + S− ) = , (4.5a) 1 0 2 2 i 1 0 −i S2 = − (S+ − S− ) = , (4.5b) i 0 2 2 1 1 1 0 S3 = | 2 ih 2 | − | 1 ih 1 | = . (4.5c) 0 −1 2 2 24

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Math. App. D. Pseudospin operators

25

An obvious connection with Pauli matrices 1 Sj = σj (4.6) 2 explains why we call Sj operators the pseudospin. Before discussion of the properties of the pseudospin operators we make two additional comments. From Eqs.(4.5a) and (4.5b) it follows that S+ = S1 + iS2 ,

S− = S1 − iS2 .

(4.7)

We also note the Hermiticity relations Sj† = Sj − Hermitian,

† S± = S∓ − Hermitian conjugates,

(4.8)

what follows by inspection of the matrix representation.

D.2

Various products of pseudospin operators

The products of pseudospin operators follow: • from their ket-bra definitions; • from their matrix representations; • from the fundamental property of Pauli operators: σj σk = ijkm σm

σj2 = 1

for j 6= k,

for j = 1, 2, 3.

(4.9)

All this sources are in fact equivalent. The proofs of the given below relations are omitted since such proofs are very easy to do. Before we give many particular examples, we note that Eqs.(4.9) and (4.6) imply i jkm Sm for j 6= k, 2 For raising and lowering operators we have Sj Sk =

Sj2 =

S+ S+ = 0

S+ S− = 1/2 + S3 ,

S− S− = 0

S− S+ = 1/2 − S3 .

D.3

1 4

for j = 1, 2, 3.

(4.10)

(4.11)

Commutation relations

For Pauli operators we have h i σj , σk = 2ijkm σm ,

(4.12)

which, together with Eq.(4.6), yields the commutation relation for the pseudospin operators h i Sj , Sk = ijkm Sm , (4.13) For raising and lowering operators we have h i h i S± , S1 = ±S3 , S± , S 2 = i S 3 , and

h

i S+ , S− = 2S3 ,

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h

i S± , S3 = ∓S± ,

(4.14)

(4.15)

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Math. App. D. Pseudospin operators

D.4

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Useful identities and their consequences

Theorem D.1 For numbers α and β real or complex, there holds an identity exp [ iα S+ + iβ S− ] = cos

p i αβ + √ αβ

( α S+ + β S− ) sin

p αβ .

(4.16)

This identity has several interesting and useful consequences. Putting α = β, we get eiα(S+ +S− ) = cos α + (S+ + S− ) sin α.

(4.17)

Since 2S1 = S+ + S− , we also get e2iαS1 = cos α + 2S1 sin α.

(4.18)

If we take a limit β → 0 in (4.16), we get eiαS+ = 1 + iα S+ .

(4.19)

Similar procedure, but with α → 0 yields eiβS− = 1 + iβ S− .

(4.20)

Combining two last relations we have eiαS± = 1 + iα S± .

(4.21)

If we put α = −iξ, β = iξ, then from (4.16) we can derive ei(αS+ +βS− ) = e2iξS2 = cos ξ + 2iS2 sin ξ,

(4.22)

which should be compared to (4.18). Theorem D.2 For numbers α and β real or complex, there holds an identity exp [ iα S3 + iβ S1 ] = ! p α2 + β 2 2i = cos + p ( α S3 + β S1 ) sin 2 2 α + β2

! p α2 + β 2 2

(4.23)

This theorem also leads to many useful specific cases. Putting β = 0 we get α α eiαS3 = cos + 2iS3 sin 2 2 Combining (4.18), (4.22) and (4.24) we can write a useful relation α α eiαSj = cos + 2iSj sin , for j = 1, 2, 3. 2 2

(4.24)

(4.25)

As a conclusion from the above derived relations we get the third useful theorem.

Theorem D.3 For any number α real or complex there holds an identity ( Sk for j = k, iαSk −iαSk e Sj e = Sj cos α + jkm Sm sin α for j 6= k.

(4.26)

From this theorem it follows that eiαS3 S± e−iαS3 = S± e±iα

26

(4.27)

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