We consider the radiation field operators in a cavity with varying dielectric medium in terms of solutions of Heisenberg's equations of motion for the most general one-dimensional quadratic Hamiltonian. Explicit solutions of these equations are obtained and applications to the radiation field quantization, including randomly varying media, are briefly discussed.

On the Problem of Electromagnetic-Field Quantization

Quasi-coherent states for a photon in time varying dielectric media

Quantum energies for Caldirola–Kanai Hamiltonian systems with and without a singular perturbation are investigated in coherent states on the basis of invariant operator theory. Our results are compared with those of several different approaches from various angles. Further, the advantages of our method over that adopted in the previous researches are addressed in detail.

Analysis of quantum energy for Caldirola–Kanai Hamiltonian systems in coherent states

Nonextensive dynamics for a quantum dissipative system described by a Caldirola-Kanai (CK) Hamiltonian is investigated in SU(1,1) coherent states. To see the effects of nonextensivity, the system is generalized through a modification fulfilled by replacing the ordinary exponential function in the standard CK Hamiltonian with the q-exponential function. We confirmed that the time behavior of the system is somewhat different depending on the value of q which is the degree of nonextensivity. The effects of q on quantum energy dissipation and other parameters are illustrated and discussed in detail.

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The effects of nonextensivity on quantum dissipation

We consider the radiation field operators in a cavity with varying dielectric medium in terms of solutions of Heisenberg’s equations of motion for the most general one-dimensional quadratic Hamiltonian. Explicit solutions of these equations are obtained and applications to the radiation field quantization, including randomly varying media, are briefly discussed.

We realized SU(1,1) Lie algebra in terms of the appropriate SU(1,1) generators for the time-dependent quadratic Hamiltonian system perturbed by a singularity Exact quantum states of the system are investigated using SU(1,1) Lie algebra Various expectation values in two kinds of the generalized SU(1,1) coherent states, that is, BG coherent states and Perelomov coherent states are derived We applied our study to the CKOPS (Caldirola–Kanai oscillator perturbed by a singularity) Due to the damping constant γ, the probability density of the SU(1,1) coherent states for the CKOPS converged to the center with time The time evolution of the probability density in SU(1,1) coherent states for the CKOPS are very similar to the classical trajectory

Su(1,1) lie algebra applied to the time-dependent quadratic hamiltonian system perturbed by a singularity

We obtained the uncertainty relation in squeezed states for a time-dependent oscillator. The uncertainty relation in coherent states is same as that of the number states with n=0. However, the uncertainty relation in squeezed states does not satisfy this property and depends on squeezing parameter c. For instance, the uncertainty relation is ℏ/2 which is the minimum value as far as quantum mechanics permits for c=1, same as that in coherent state for c=±∞, and infinity for c=-1. If the time-dependency of the Hamiltonian for the system vanishes, the uncertainty relation in squeezed states will no longer depend on c and becomes the same as that in number state with n=0, like the uncertainty relation in coherent states.

The dependency on the squeezing parameter for the uncertainty relation in the squeezed states of the time-dependent oscillator

We defined normalized density operator that satisfies Liouville–von Neumann equation in terms of the invariant operator. The energy density inside the cavity decreased exponentially with time due to the conductivity of the media. We also evaluated the total number of photons in the cavity.

Dissipative blackbody radiation: radiation in a lossy cavity

The quantum states with discrete and continuous spectrum for the damped harmonic oscillator perturbed by a singularity have been investigated using invariant operator and unitary operator together. The eigenvalue of the invariant operator for ω0≤β/2 is continuous while for ω0>β/2 is discrete. The wave functions for ω0=β/2 expressed in terms of the Bessel function and for ω0<β/2 in terms of the Kummer confluent hypergeometric function. The convergence of the probability density is more rapid for over-damped harmonic oscillator than that of the other two cases due to the large damping constant.

Erratum: "Wave functions with discrete and with continuous spectrum for quantum damped harmonic oscillator perturbed by a singularity"

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.

Jeong Ryeol Choi

Papers

Su(1,1) lie algebra applied to the time-dependent quadratic hamiltonian system perturbed by a singularity

The dependency on the squeezing parameter for the uncertainty relation in the squeezed states of the time-dependent oscillator

Dissipative blackbody radiation: radiation in a lossy cavity

Erratum: "Wave functions with discrete and with continuous spectrum for quantum damped harmonic oscillator perturbed by a singularity"

Commutation relations and self-consistency in electrodynamics for the fields in time-varying media