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The Electromagnetic Field and Laser Dynamics in a Cavity Bounded by Moving Mirrors.

Gerald T. Moore

Published Jan 1, 1969• 0 citations
OpenAlex

Abstract

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)

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quant-pharXiv:2610.15312•Oct 13, 2026•0 citations
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)
In this paper the dynamics of wave fields in a cavity with one sinusoidally moving mirror are investigated. The wave equation is solved approximately using the method of multiple time scales. Frequency shifts as well as amplitude and phase modulations of cavity modes are given. In all cases when it is possible, an approximate effective field Hamiltonian operator is derived. In the adiabatic case it is then used to analyze the interaction of radiation with a two-level atom passing through the cavity. Corrections to the Rabi oscillations due to the time-dependent frequency of the cavity photons are investigated. It is shown that adiabatic motion of a mirror can lead to suppression of the Rabi oscillations of atomic inversion.
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.
The dynamics of an ensemble of two-level atoms moving through a single-mode lossless cavity is investigated in the semiclassical and rotating-wave approximations. The dynamical system for the expectation values of the atomic and field observables is considered as a perturbation to one of the following integrable versions: (i) a model with atoms moving through a spatially inhomogeneous resonant field, and (ii) a model with atoms interacting with a nonresonant eigenmode which is assumed to be homogeneous on the cavity size. We find the general exact solutions for both the models and show that they contain special solutions describing a coherent effect of population and radiation trapping. Using the Melnikov method, we prove analytically transverse intersections of stable and unstable manifolds of a hyperbolic fixed point under a small modulation of the vacuum Rabi frequency. These intersections are believed to provide the Smale horseshoe mechanism of Hamiltonian chaos. The analytical results are accompanied with direct computation of topographical maps of maximal Lyapunov exponents that give a representative image of regularity and chaos in the atom-field system in different ranges of its control parameters--the frequency detuning, the number, and the velocity of atoms.
Abstract This present paper is an attempt to describe the operation of a laser by a method patterned closely after the classical theory of dispersion. There, the electromagnetic field is treated as a classical system interacting with a collection of classical harmonic oscillators which are at rest. In the present case the radiation field is still treated classically, but, in accordance with the correspondence principle, the oscillators are now virtual oscillators associated with the up-and-down transitions in effective two-state atoms which are described quantum mechanically. The atoms carrying these virtual oscillators are not at rest but move with different velocities. We show how the Lorentz averaging can be performed for such systems, and derive a closed set of equations linking the average electric field E, with the average polarization density P„ and the average population inversion density Mp associated with atoms moving with the velocity v. Next, we investigate the conditions for the existence of a steady state if pumping is present, and if the electric field is represented by a standing wave, or a travelling wave of a single frequency (single-mode case). It turns out that, notwithstanding the nonlinear nature of the equations, the steady-state conditions give a simple complex dispersion relation; moreover, the real and imaginary part of this dispersion relation are equivalent to the usual heuristic expressions which specify the operating frequency in terms of the index of refraction and cavity length, and which balance the gain against the losses. The single-mode case is analysed in detail without any smallness assumptions for the resultant intensity. The small-intensity case gives the usual results exhibiting the tuning dip. For high intensities the relative depth of the dip tends to zero. At the end of the paper we discuss proposed extensions and additional applications.
We have solved a master equation to obtain the diagonal elements of the density matrix for laser light. The master equation represents a generalization of a master equation derived earlier for a single radiation mode interacting with NNN stationary two-level atoms. The generalization takes into account the pumping scheme which characterizes a three-level laser. For positive temperatures and a lossless cavity, the equilibrium solution of the master equation gives the correct statistical-mechanical description of the atoms and radiation. For negative temperatures, the distribution function for the number of photons in the mode is either exponential or peaked, depending on whether the laser is operating below or above threshold. The calculated intensity fluctuations are in good agreement with semiclassical results for lasers operating slightly above threshold.
We perform a detailed mathematical study of the effective Hamiltonian describing a non-stationary electromagnetic cavity field, introduced by C.~K.~Law, in the presence of uniform linear motion. We find that the corresponding dynamical equations admit a unitary diagonalization and that the normalized velocity of motion satisfies the two-sided bound ∣v/c∣≲10.1716\lvert v/c \rvert \lesssim 10.1716∣v/c∣≲10.1716. Although this result is mathematically consistent, the interval obtained is an order of magnitude larger than the physically admissible range ∣v/c∣<1\lvert v/c \rvert < 1∣v/c∣<1. To address this limitation, we propose a reinterpretation of bosonic coupling terms as generators of the symplectic algebra Sp(4,R)\mathrm{Sp}(4,\mathbb{R})Sp(4,R), identifying them with transformations that can be implemented through linear optical elements. Within this photonic approach, the coupling parameters are not restricted to small values, allowing us to explore regimes of strong effective interactions. In this case, we demonstrate the generation of two-mode squeezed states and derive analytical expressions for the average photon number in each mode. Furthermore, we analyze statistical properties that characterize mixedness and entanglement, including intermode correlations, the Mandel parameter, and the linear entropy. The corresponding Wigner function provides a phase-space visualization of the system's behavior, illustrating the connection between cavity motion, mode coupling, and nonclassical field correlations via the underlying symplectic algebra.