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Journal of the Korean Physical Society

Quantum optical electromagnetic field and Rabi oscillation in time-varying medium

Eun Ji Jang, Min Jung, Ji Hoon Cha, Young Kyu Lee, Won Sang Chung

Published Sep 1, 2016• 0 citations
Publisher

Abstract

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.

DOI
10.3938/jkps.69.708

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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.
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.
After the birth of quantum mechanics, the notion in physics that the frequency of light is the only factor that determines the energy of a single photon has played a fundamental role. However, under the assumption that the theory of Lewis–Riesenfeld invariants is applicable in quantum optics, it is shown in the present work that this widely accepted notion is valid only for light described by a time-independent Hamiltonian, i.e., for light in media satisfying the conditions, ε(i) = ε(0), μ( t ) = μ(0), and σ( t ) = 0 simultaneously. The use of the Lewis–Riesenfeld invariant operator method in quantum optics leads to a marvelous result: the energy of a single photon propagating through time-varying linear media exhibits nontrivial time dependence without a change of frequency.
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.
We investigated coherent and squeezed states of light in linear media whose parameters are explicitly dependent on time by making use of the Lewis–Riesenfeld invariant operator method. Not only the field strengths but also the fluctuations of the fields both in coherent and in squeezed states are decayed with time. The relative noise of the field strengths are calculated in coherent state. Quantum statistical properties of the chaotic field are investigated. We applied our theory to a phenomenological model of the biophoton system and compared the corresponding result of the uncertainty product with that obtained from a previous report.
The time-dependent Schrödinger equation describing a generalized two-dimensional quantum parametric oscillator in the presence of time-variable external fields is solved using the evolution operator method. For this, the evolution operator is found as a product of exponential operators through the Wei–Norman Lie algebraic approach. Then, the propagator and time-evolution of eigenstates and coherent states are derived explicitly in terms of solutions to the corresponding system of coupled classical equations of motion. In addition, using the evolution operator formalism, we construct linear and quadratic quantum dynamical invariants that provide connection of the present results with those obtained via the Malkin–Man’ko–Trifonov and the Lewis–Riesenfeld approaches. Finally, as an exactly solvable model, we introduce a Cauchy–Euler type quantum oscillator with increasing mass and decreasing frequency in time-dependent magnetic and electric fields. Based on the explicit results for the uncertainties and expectations, squeezing properties of the wave packets and their trajectories in the two-dimensional configuration space are discussed according to the influence of the time-variable parameters and external fields.