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International Journal of Modern Physics B

COMMUTATION RELATIONS AND SELF-CONSISTENCY IN ELECTRODYNAMICS FOR THE FIELDS IN TIME-VARYING MEDIA

Jeong Ryeol Choi, Jun‐Young Oh

Published Jan 30, 2008• 2 citations
Publisher

Abstract

In linear media with time-dependent parameters, various commutation relations for the field operators obtained from the Lewis–Riesenfeld invariant operator method are calculated. We investigated whether our development is self-consistent or not by evaluating the Heisenberg equation of motion for field operators using the associated commutation relation.

DOI
10.1142/s0217979208038491

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In linear media with time-dependent parameters, various commutation relations for the field operators obtained from the Lewis–Riesenfeld invariant operator method are calculated. We investigated whether our development is self-consistent or not by evaluating the Heisenberg equation of motion for field operators using the associated commutation relation.
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 investigate the relation between the invariant operators satisfying the quantum Liouville-von Neumann and the Heisenberg operators satisfying the Heisenberg equation. For time-dependent generalized oscillators we find the invariant operators, known as the Ermakov-Lewis invariants, in terms of a complex classical solution, from which the evolution operator is derived, and obtain the Heisenberg position and momentum operators. Physical quantities such as correlation functions are calculated using both the invariant operators and Heisenberg operators.
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.
Consideration of the asteroid belt (Kuiper belt) as a jammed-granular media establishes a bridge between condensed matter physics and astrophysics. It opens up an experimental possibility to determine the deformation parameters for noncommutative space-time. Dynamics of the Kuiper belt can be simplified as dynamics of a dynamical effective mass for a jammed granular media under a gravitational well. Alongside, if one considers the space-time to be noncommutative, then an experimental model for the determination of the deformation parameters for noncommutative space-time can be done. The construction of eigenfunctions and invariance for this model is in general a tricky problem. We have utilized the Lewis-Riesenfeld invariant method to determine the invariance for this time-dependent quantum system. In this article, we have shown that a class of generalized time-dependent Lewis-Riesenfeld invariant operators exist for the system with dynamical effective mass in jammed granular media under a potential well in noncommutative space. To keep the discussion fairly general, we have considered both position-position and momentum-momentum noncommutativity. Since, up to a time-dependent phase-factor, the eigenfunctions of the invariant operator will satisfy the time-dependent Schrödinger equation for the time-dependent Hamiltonian of the system, the construction of the invariant operator fairly solve the problem mathematically, the results of which can be utilized to demonstrate an experiment.