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Generalized Coherent States for Quantized Electromagnetic Fields in Time-varying Linear Media

Jeong Ryeol Choi, Mustapha Maamache

Published Dec 25, 2009• 4 citations

Abstract

If the parameters of the materials, such as electric permittivity, magnetic perme- ability, and conductivity, vary explicitly with time, the corresponding Hamiltonian is a time- dependent form and such materials are classifled as the time-varying media of light. The quan- tization of electromagnetic flelds in time-varying linear media is well described by means of the invariant operator theory of Lewis-Riesenfeld. The Coulomb gauge is chosen in the development of theory, which allows us to evaluate quantized electric and magnetic flelds by expanding only the vector potential in charge free space, since the scalar potential in this situation vanishes. The generalized coherent state, i.e., Gaussian-Klauder state, of electromagnetic flelds conflned inside a cavity fllled with a nontrivial conductive medium is investigated. The Gaussian-Klauder state provides a general means of construction for Husimi-Wigner distributions and can be used to show the quantum and classical correspondence. We conflrmed that the uncertainty product in Gaussian-Klauder state is the same as the minimum uncertainty product in number state. This property of the generalized coherent state is well agree with that of the well known Glauber coherent state.

DOI: 10.2529/piers091004232119

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We investigated exact quantum states of the light confined in cubes filled with conductive media whose parameters are explicitly dependent on time and the light propagating under periodic boundary condition by making use of the LR (Lewis–Riesenfeld) invariant operator method. The choice of Coulomb gauge in the charge free space allowed us to evaluate quantized electric and magnetic fields by expanding only the vector potential, since the scalar potential is zero. We also described the fields with a spectrum of continuous mode, which can be obtained by setting the side L to infinity.

Integrals of motion and statistical properties of quantized electromagnetic field (e.-m. field) in time-dependent linear dielectric and conductive media are considered, using Choi-Yeon quantization, based on Caldirola-Kanai type Hamiltonian. Eigenstates of quadratic and linear invariants are constructed, the solutions being expressed in terms of a complex parametric function that obeys classical oscillator equation with time-varying frequency. The time evolutions of initial Glauber coherent states and Fock states are considered. The medium conductivity and the time-dependent electric permeability are shown to generate squeezing and non-vanishing covariances. In the time-evolved coherent and squeezed states all the second statistical moments of the electric and magnetic field components are calculated and shown to mminimize the Robertson-Schrödinger uncertainty relation.

In this paper, we consider the quantum optical electromagnetic wave in time-varying media, where the electric permittivity ϵ(t) and the magnetic permeability μ(t) depend on time explicitly as ϵ(t) = ϵ 0(1+qt) and μ(t) = μ 0(1+qt). For these time-varying parameters, we solve Maxwell’s equations. To construct time-dependent coherent states for this light, we adopt the invariant method proposed by Lewis and Riesenfeld and express the fields in terms of the time-dependent step operators for the invariant. We also discuss the deformed Rabi oscillation for both on-resonance and off-resonance.

Dynamical invariants and statistical properties of the quantized electromagnetic field in nonstationary linear media (dielectric and/or conductive) are considered in the framework of Choi‐Yeon quantization scheme. It is shown that in the eigenstates of linear dynamical invariant the Robertson‐Schrödinger uncertainty relation is minimized, both for the photon annihilation operator quadratures and for the electric and magnetic field components. The time evolution of initial Glauber coherent states and Fock (photon number) states is considered. On an initial coherent state the medium conductivity and the time‐dependent electric permeability both are shown to act as squeezing and correlating factors.

Dynamical invariants and statistical properties of the quantized electromagnetic field in nonstationary linear media (dielectric and/or conductive) are considered in the framework of Choi‐Yeon quantization scheme. It is shown that in the eigenstates of linear dynamical invariant the Robertson‐Schrödinger uncertainty relation is minimized, both for the photon annihilation operator quadratures and for the electric and magnetic field components. The time evolution of initial Glauber coherent states and Fock (photon number) states is considered. On an initial coherent state the medium conductivity and the time‐dependent electric permeability both are shown to act as squeezing and correlating factors.

We present a quantum description of electromagnetic waves propagating through time-dependent homogeneous nondispersive conducting and nonconducting linear media without charge sources. Based on the Coulomb gauge and the quantum invariant method, we find the exact wave functions for this problem. In addition, we construct coherent and squeezed states for the quantized electromagnetic waves and evaluate the quantum fluctuations in coordinate and momentum space as well as the uncertainty product for each mode of the electromagnetic field.