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Journal of Physics A Mathematical and General

Dynamics of the quantized radiation field in an oscillating cavity in the harmonic resonance case

Xiaoxue Yang, Ying Wu

Published Oct 13, 1999• 14 citations
Publisher

Abstract

We investigate the dynamics of the quantized radiation field in an oscillating cavity described by the effective resonance Hamiltonians + where characterizes the oscillating amplitude of the moving boundary of the cavity, the boundary oscillates with the j th unperturbed eigenfrequency ( j = 1,2,3, ... ) and the parametric oscillator part H ( p ) contains only a few terms to be specified in the main text. We present the exact diagonalization forms of the effective Hamiltonians for j = 1,2,3, ... in the absence of the parametric oscillator part H ( p ) . A systematic procedure is then developed to obtain the analytical solutions of the Hamiltonians in the presence of the part H ( p ) accurate up to order k for an arbitrary positive integer k . In this way, we can investigate the dynamics of the corresponding quantized radiative field by explicitly presenting the analytical expressions of the diagonalized Hamiltonians, the time-varying annihilation and creation operators as well as photon number operators for the radiative field, accurate up to any desired order of the dimensionless oscillating amplitude. The analytical expressions of these quantities up to order 4 are also explicitly presented.

DOI
10.1088/0305-4470/32/42/310

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We investigate the dynamics of the quantized radiation field in an oscillating cavity described by the effective resonance Hamiltonians + where characterizes the oscillating amplitude of the moving boundary of the cavity, the boundary oscillates with the j th unperturbed eigenfrequency ( j = 1,2,3, ... ) and the parametric oscillator part H ( p ) contains only a few terms to be specified in the main text. We present the exact diagonalization forms of the effective Hamiltonians for j = 1,2,3, ... in the absence of the parametric oscillator part H ( p ) . A systematic procedure is then developed to obtain the analytical solutions of the Hamiltonians in the presence of the part H ( p ) accurate up to order k for an arbitrary positive integer k . In this way, we can investigate the dynamics of the corresponding quantized radiative field by explicitly presenting the analytical expressions of the diagonalized Hamiltonians, the time-varying annihilation and creation operators as well as photon number operators for the radiative field, accurate up to any desired order of the dimensionless oscillating amplitude. The analytical expressions of these quantities up to order 4 are also explicitly presented.
We study the quantized field in a one-dimensional electromagnetic resonant cavity. The cavity contains a linear and lossless dielectric medium with frequency-independent polarizability. The dielectric permittivity is an externally prescribed function of both the space and the time. We also allow one of the cavity's mirrors to move in a given trajectory. Unlike other previous studies on the same system, we formulate an effective Hamiltonian so that the dynamics of the cavity field can be described in the Schr\"odinger picture. The effective Hamiltonian is quadratic in structure, therefore two-photon generation from the vacuum state can occur. We also discuss the case of resonant behavior of the system.
A quantum optical model for high-order harmonic generation is presented in which both the exciting field and the high-order harmonic modes are quantized, while the target material appears only via parameters. As a consequence, the model is independent of the excited material system to a large extent and allows us to focus on the properties of the electromagnetic fields. Technically, the Hamiltonian known for parametric down-conversion is adopted, where photons in the nthn\mathrm{th}nth harmonic mode are created during the annihilation of nnn photons from the fundamental mode. In our treatment, initially, the fundamental mode is in a coherent state corresponding to large photon numbers, while the high-order harmonic modes are in vacuum state. Due to the interaction, the latter modes get populated while the fundamental one loses photons. Analytical approximations are presented for the time evolution that are verified by numerically exact calculations. For multimode, finite-bandwidth excitation, the time dependence of the high-order harmonic radiation is also given.
We investigate the quantum dynamics of a particle subjected to a linear potential using the Lewis--Riesenfeld invariant operator method. Starting from the time-dependent Schrödinger equation associated with a constant external force, we construct the most general Hermitian quadratic invariant and derive the corresponding coupled differential equations for its time-dependent coefficients. By means of an appropriate sequence of unitary transformations, the invariant operator is reduced to the form of a harmonic oscillator Hamiltonian. This reduction enables a clear classification of the system according to the sign of the conserved quantity ω2. Particular attention is devoted to the physically relevant case ω2 >0, which yields a discrete eigenspectrum. Explicit analytical expressions for the invariant coefficients, the displacement parameters, and the transformed wave functions are obtained. The resulting formalism provides an exact quantum description of a particle under a constant force and establishes a direct connection between invariant theory and harmonic oscillator quantization.
We consider the general open system problem of a charged quantum oscillator confined in a harmonic trap, whose frequency can be arbitrarily modulated in time, that interacts with both an incoherent quantized (blackbody) radiation field and with an arbitrary coherent laser field. We assume that the oscillator is initially in thermodynamic equilibrium with its environment, a non-factorized initial density matrix of the system and the environment, and that at t = 0 the modulation of the frequency, the coupling to the incoherent and the coherent radiation are switched on. The subsequent dynamics, induced by the presence of the blackbody radiation, the laser field, and the frequency modulation, is studied in the framework of the influence functional approach. This approach allows incorporating, in analytic closed formulae, the non-Markovian character of the oscillator-environment interaction at any temperature as well the non-Markovian character of the blackbody radiation and its zero-point fluctuations. Expressions for the time evolution of the covariance matrix elements of the quantum fluctuations and the reduced density-operator are obtained.
The present paper is devoted to the detailed study of quantization and evolution of the point limit of the Pauli-Fierz model for a charged oscillator interacting with the electromagnetic field in dipole approximation. In particular, a well defined dynamics is constructed for the classical model, which is subsequently quantized according to the Segal scheme. To this end, the classical model in the point limit, already obtained by Noja and Posilicano [Ann. I.H.P. Phys. Theor. 71, 425 (1999)], is reformulated as a second order abstract wave equation, and a consistent quantum evolution is given. This allows a study of the behavior of the survival and transition amplitudes for the process of decay of the excited states of the charged particle, and the emission of photons in the decay process. In particular, for the survival amplitude the exact time behavior is found. This is completely determined by the resonances of the systems plus a tail term prevailing in the asymptotic, long time regime. Moreover, the survival amplitude exhibits in a fairly clear way the Lamb shift correction to the unperturbed frequencies of the oscillator.