Novel quantum description for nonadiabatic evolution of light wave propagation in time-dependent linear media

A simple elegant expression of nonadiabatic light wave evolution is necessary in order to have a deeper insight for complicated optical phenomena in light science as well as in everyday life. Light wave propagation in linear media which have time-dependent electromagnetic parameters is investigated by utilizing a quadratic invariant of the system. The time behavior of the nonadiabatic geometric phase of the waves that yield a cyclic nonadiabatic evolution is analyzed in detail. Various quantum properties of light waves in this situation, such as variances of electric and magnetic fields, uncertainty product, coherent and squeezed states, and their classical limits, are developed. For better understanding of our research, we applied our analysis in a particular case. The variances of the fields D and B are illustrated and their time behaviors are addressed. Equivalent results for the corresponding classical systems are deduced from the study of the time evolution of the appropriate coherent and squeezed states.

If electromagnetic parameters of the media, such as permittivity, magnetic permeability, and conductivity, change as time goes by, the media are classified as time-dependent media. Recently, great attention has been paid to achieving the accurate and efficient description for light wave propagation in time-dependent linear and random media; this is partly due to the concern to quantum optics processes in modern optical materials such as optical fibers [1][2][3][4][5][6][7][8][9] . The light waves in such media are described by a time-dependent Hamiltonian on account of the time-dependence of parameters. It is interesting to note that, even if the time-dependence of parameters disappears, the Hamiltonian of an electromagnetic field is time-dependent as long as the conductivity exists 7,10 .
There exist generally accepted approaches for developing the quantum theory of light waves of which the corresponding Hamiltonian is explicitly time-dependent. Some of us 8,9 , studied the geometric phase of quantized light waves in time-dependent linear media, where the parameters of the system undergo adiabatic change. They considered the eigenstates of a time-dependent Hamiltonian in order to derive the adiabatic geometric phase change under the assumption that the parameters of the system vary sufficiently slowly with time. Their research is principally based on Berry's report appearing in his classic paper 11 , which states that the wave function acquires a geometric phase (now known as Berry's phase) in addition to the usual dynamical phase when a physical system evolves in a cyclic and adiabatic fashion.
In contrast to the adiabatic process of phase change treated in the previous works 8, 9 , we, in this paper, investigate the dynamical properties of nonadiabatic geometric phase accumulated by a somewhat fast change of control parameters of the medium for a light wave propagating in time-dependent linear media. For this reason, we introduce a quadratic invariant of Lewis and Riesenfeld 12 and consider the eigenstates of this invariant (instead of those of the Hamiltonian) in order to study the features of the geometric phase. Then the quantum description of electromagnetic field whose Hamiltonian is time-dependent can be achieved on a fundamental level by taking advantage of the invariant formulation of quantum electrodynamics.
The importance of Berry's finding of the geometric phase and its impact on various areas of physics have naturally led to arousing the interest in the generalizations of geometric phases. One of the most significant contributions in this direction is the nonadiabatic generalization of Berry's phase firstly fulfilled by Aharonov and Anandan 13 . This generalization employs a geometric picture of quantum dynamics and shows that the nonadiabatic geometric phase can be defined for any closed curve in the space of (pure) quantum states. Perhaps one of the most important research studies connected to these is the classic seminal paper of Lewis and Riesenfeld 12 on dynamical invariants. The correspondence of Berry's phase and Lewis's phase has been pointed out by Morales 14 . The classical counterpart of Aharonov-Anandan quantum geometric phase is the nonadiabatic Hannay angle 15,16 . Since the invariant action proposed by Lewis 12 exists independently of whether the Hamiltonian is changed slowly or not, the geometric angle can be defined on constant-action tori for a cyclic evolution, independently of whether the evolution is adiabatic or not. The appropriate interpretation of the angle obtained in this way is the classical counterpart (Hannay angle) of the geometric phase of the Aharonov and Anandan 13 . For light waves described by the generalized harmonic oscillator, the adiabatic-approximation-results can be obtained from an asymptotic theory of nonadiabatic process.
To investigate the problem of the nonadiabatic geometric phase for quantized light waves propagating through homogeneous conducting linear media, we use Coulomb's gauge and assume that the medium has no free charge for simplicity. We reduce the problem to that of a generalized time-dependent harmonic oscillator Hamiltonian and we examine the geometric character of light waves with the help of the dynamical invariant. From Maxwell's equations, we obtain the classical Hamiltonian for the light waves propagating through homogeneous conducting linear media without charge density and we survey the basic results of the dynamical invariants and their relationship with geometric angles. In particular, the expression of electromagnetic fields in terms of the nonadiabatic geometric angle will be derived. We will address the characteristics of the geometrically equivalent quantum systems and, through a construction of annihilation and creation operators, we will investigate the quantum properties of light in time-dependent linear media. The coherent and squeezed states of the generally described light wave will be investigated and the adiabatic limit of our results will be compared to the previously known ones.

Materials and Methods
In this section, we investigate the nonadiabatic geometric phase of light waves in time-dependent linear media using invariant theory developed for time-dependent Hamiltonian systems. Since the Hamiltonian is nonconservative in this situation, the invariant related approach for the geometric phase is useful now.
In time-dependent linear media, the relations between fields and current are given by and σ ( ) t are time-dependent electric permittivity, magnetic permeability, and conductivity, respectively. Due to the time-dependence of the parameters, the speed of a light wave varies with time and is given by In the Coulomb gauge, Maxwell's equations in time-dependent linear media give a damped wave equation for the vector potential such that 4,8 µ ε σ ε µ To separate ( , ) and performing some algebra for the angle with Eq. (7) leads to Then, the exact solution for q l is given by This oscillates in time with a time-dependent phase increment.

Effects of Geometric Phase on Radiation Fields. Now let us see the effects of the nonadiabatic geomet-
ric phase for the light wave propagating under a periodic boundary condition. If we replace the classical variables , it is possible to investigate this problem in view of quantum mechanics. Each mode of the electromagnetic field behaves like a time-dependent harmonic oscillator whose quantum features are clearly understood through the introduction of annihilation and creation operators associated with the invariant presented in the previous section. If we consider an annihilation operator and a creation operator that are given by  . From the result of the direct differentiation of ( ) a t l with respect to time: we see that the time evolution of the annihilation operator is given by In terms of the canonical variable obtained from the inverse representation of â l and ˆ † a l (given in Eqs. (12) and (13)), the vector potential given in Eq. (2) can be completely described as Then, from the basic relations given in Eq. (5) and the first term of Eq. (1), we see that the electric displacement and the magnetic field operators are represented in the form in these equations, both the electric and the magnetic fields decrease with time according to the electromagnetic energy dissipation produced by the conductivity σ ( ) t in media. We can confirm from this that the time behavior of the amplitude of the fields are the same as that of the classical fields. Of course, similar behavior appears in the case of adiabatic change of geometric phase 8,9 . In the limit σ ( ) → t 0, the fields no longer dissipate with time as expected. We are able to understand the geometric character of the light wave from the phase factor As you can see, the light waves undergo geometric phase change θ ∆ l g as well as familiar change of dynamical phase, θ ∆ l d . The interference fringe, produced when two or more light waves with different modes meet from different paths, would be altered more or less by the existence of the geometric phase. This concept is important for accurate prediction of interference pattern in interferometers.
Adiabatic Limit: Phase Splitting into Dynamical and Geometrical Parts. One can find the adiabatic limit of the expressions presented up to now by using the following argument. In the adiabatic regime, the differentiation of ρ l with respect to time induces a multiplication by a small adiabatic parameter , i.e., ρ l . Thus, concerning the expression of the invariant given in Eq. , which is valid up to the first order in ; one then recovers the well known relation 8 The time derivative of the angle, θ ( )  t l , consists of two parts. The first one, θ ϖ =  l l , which exists even for a system with fixed parameters, corresponds to the so-called dynamical component of the angle, while the second one, l H l is the (time derivative of the) geometrical Hannay's phase. For dielectric materials σ ( ( ) = ) t 0 , the geometric phase vanishes. Therefore, we see that the geometric properties of E and B fields vary depending on the characteristics of the medium. Moreover, the geometric phase appears when the medium becomes a conducting one with a slowly time-dependent conductivity while the electric permittivity is finite.

Classical Limit: Coherent and Squeezed States for the Generalized GHO Model. The classical
counterparts of the results developed up to now, i.e., the identity of the phase properties for the classical GHO and the expression of the classical GHO invariant (which is a generalized one of the results of refs 8,9 established under the restrictive adiabatic hypothesis), can be derived from classical limit of the quantum analysis. Indeed, the classical invariant and angle variables can be obtained by utilizing the quantum evolution of coherent states and squeezed states 21,22 . These states play the same role as the ordinary coherent states in the harmonic oscillator when considering the quantum-classical correspondence 23,24 .
The coherent state is an eigenstate of the annihilation operator â l : ( ) a 24 l l l l We can confirm from Eq. (16) that the eigenvalue is given by In some cases, it may be useful to recast the initial eigenstate in the form α α ( ) = Then, by taking into account Eqs. (17) and (24)(25)(26), the expectation values of the canonical variables in the coherent state are given by where μ and ν are complex parameters that follow µ ν − =1 2 2 . We can easily check that b l and its hermitian l l . If we represent the eigenvalue equation for b l in the form l l l l then β l is the squeezed state. In general, the squeezed state is obtained by first squeezing the vacuum state and then displacing it 25 . The expectation value of canonical variables in the squeezed state, β q l and β p l , are identical to those in the coherent state given in Eqs. (27) and (28). Hence, the corresponding characteristics of the geometric phase for the canonical variables are very much the same as those of the coherent state.
However, the variances are different and their straightforward evaluations yield It is well known that one can make the size of (∆ ) β q l 2 sufficiently small at the expense of a relative increase of the value (∆ ) β p l 2 or vice versa. Consequently, the uncertainty product is If we consider long time behavior (  t 1, τ )  1 , we easily have τ τ τ ( + )/ /  1 1 . Then, the solution of Eq. (49) in this approximation is given by where ν J and ν N are Bessel functions of the first and the second kinds, respectively, while ν = / G b 2 l 2 . This equation can be rewritten in terms of t as

Discussion
Starting from the basic Maxwell's equations, the GHO model of quantized light wave in time-dependent linear media was investigated with emphasis on its nonadiabatic evolution and accompanying properties of geometric phase. For convenience, we have considered charge free space and took advantage of the Coulomb gauge. Various quantum properties associated with light wave propagation in such situations and their adiabatic and classical limits were analyzed.
The vector potential ( , ) t A r was separated into position function ( ) u r l and time function ( ) q t l as shown in Eq. (2). In terms of ( ) q t l and its conjugate variable ( ) p t l , the time-dependent Hamiltonian of the system was con- . All values are taken to be dimensionless for the sake of convenience. This convention will also be used in the subsequent figure. structed. To derive quantum solutions of the system, a Hamiltonian-like quadratic invariant was introduced. We can see from Eq. (14) that this invariant is represented as a simple form in terms of the annihilation and the creation operators. If we consider Eq. (16), the absolute value of the annihilation operator does not vary with time even if ( ) a t l is an explicit function of time. With the use of the annihilation and the creation operators, full expressions of the field operators ( , ) t D r and ( , ) t B r are derived and presented in Eqs. (20) and (21), respectively. The expectation values of q l and p l and their variances in the coherent and the squeezed states have been investigated. In the limit µ = 1 and ν = 0, the corresponding uncertainty product in the squeezed state reduces to that of the coherent state. The expectation values of the field operators ( , ) D t r l and ( , ) B t r l have also been derived. We have confirmed that such expectation values in the coherent state oscillate sinusoidally according to the increment of phase angle Θ ( , ) t r l . This time behavior is exactly the same as that of the classical fields. Not only the field operators but also their variances decrease with time due to the existence of the conductivity. This analysis agrees with the characteristics of classical fields.
We have studied the effects of time-varying parameters of the medium in nonadiabatic evolution of the geometric phase of a light wave. The geometric phase exhibits gauge structure relevant to a phase shift in nonadiabatic processes. There appears a classical analogue of this phase, which is the Hannay angle that can be formulated using the theory of action variable in the canonical structure of light wave phenomena. The adiabatic limit of the phase factor was investigated and it is shown in this limit that the angle θ l exactly recovers Eq. (23), where the first term is an ordinary dynamical component of the angle and the second one is the geometrical Hannay angle. We can see from the second term that the geometric phase change takes place when the conductivity varies slowly with time.
Finally, we note that there are several potential scientific applications of the results of this work, relevant to nonadiabatic geometric phase. One is a technique for quantum computation that can be carried out by using superconducting nanocircuits 27 or nuclear magnetic resonance (NMR) 28 . For instance, a method to implement the Deutsch-Jozsa algorithm and Grover's search algorithm using the nonadiabatic geometric phase in a two-qubit system has been suggested 28 . Our results can also be applied to the problem of shortcuts to adiabaticity 29 that are introduced for speeding up quantum adiabatic processes. If we consider the fact that adiabatic processes are ubiquitous, the theory of shortcuts to adiabaticity can be employed to various fields in physics relevant to dynamical systems, ranging from the population inversion in two-level quantum systems 30 to the trapping and control of Bose gases 31 . Another branch applicable to this work is to investigate the characteristics of quantized light waves in oscillating turbulent plasma, which is important for diagnosing and controlling the plasma state in a tokamak where nuclear fusion takes place 32 .