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Physical Review A

Effective Hamiltonian for the radiation in a cavity with a moving mirror and a time-varying dielectric medium

C. K. Law

Published Jan 1, 1994• 327 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.49.433

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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.
We study the quantum properties of the electromagnetic field in optical cavities coupled to an arbitrary number of escape channels. We consider both inhomogeneous dielectric resonators with a scalar dielectric constant $\ensuremath{\epsilon}(\mathbf{r})$ and cavities defined by mirrors of arbitrary shape. Using the Feshbach projector technique we quantize the field in terms of a set of resonator and bath modes. We rigorously show that the field Hamiltonian reduces to the system-and-bath Hamiltonian of quantum optics. The field dynamics is investigated using the input-output theory of Gardiner and Collet. In the case of strong coupling to the external radiation field we find spectrally overlapping resonator modes. The mode dynamics is coupled due to the damping and noise inflicted by the external field. For wave chaotic resonators the mode dynamics is determined by a non-Hermitean random matrix. Upon including an amplifying medium, our dynamics of open-resonator modes may serve as a starting point for a quantum theory of random lasing.
The propagation of light in a one-dimensional cavity bounded by moving mirrors is studied in the context of the theory of gas lasers. A canonical quantization of the free field is carried out by utilizing the symplectic structure of the space of solutions of the wave equation. It is shown that no Hamiltonian and no Schrodinger picture exists and that the photon number operator is in general not conserved. Changes in the number operator are, however, ordinarily too small to be observed experimentally. Automorphic coordinate transformations are used to develop adiabatic expansions for the mode functions of cavities with one fixed and one moving mirror. The coupling of the field to two-level atoms is done both quantum-mechanically and semi-classically. For the interacting quantum field an interaction Hamiltonian does exist, even though there is no total Hamiltonian. The semi-classical equations treat the atomic density matrix as a distribution in phase space. They possess traveling wave solutions which are expected to be important in understanding the behavior of mode-locked lasers. (Author)
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.
The spectral distribution of light generated by a finite cavity with one moving mirror is compared with that produced by a fixed cavity containing a time-varying dielectric. In both cases a motion over a finite time interval is considered. Although the moving mirror is usually considered to be an idealization for the time-varying dielectric, there are qualitative differences in the spectra produced. The spectral distribution for the moving mirror case behaves as 1/n3{\mathit{n}}^{3}n3, while that for the time-varying dielectric behaves on average as 1/n4{\mathit{n}}^{4}n4 but is rapidly oscillating.