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Journal of Physics A Mathematical Nuclear and General

Approach to equilibrium of single mode radiation in a cavity

A. J. Schell, Richard G. Barakat

Published Jun 1, 1973• 33 citations
Publisher

Abstract

The master equation formed from the diagonal elements of the density matrix is solved in general for arbitrary initial photon probability distributions. Several examples are studied including the decay of an initial Poisson distribution (coherent state) to the equilibrium Bose-Einstein distribution. The concept of a mixed Poisson process is introduced and its physical implications examined in the context of the present problem. A general expression for the nonstationary correlation function of the photon field is also obtained.

DOI
10.1088/0305-4470/6/6/011

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The master equation formed from the diagonal elements of the density matrix is solved in general for arbitrary initial photon probability distributions. Several examples are studied including the decay of an initial Poisson distribution (coherent state) to the equilibrium Bose-Einstein distribution. The concept of a mixed Poisson process is introduced and its physical implications examined in the context of the present problem. A general expression for the nonstationary correlation function of the photon field is also obtained.
The transient behavior of the density operator for radiation in a single-mode cavity at a finite temperature is considered. Any initial state will evolve toward thermal equilibrium because of the interaction with the mirrors. This steady state is determined uniquely by the temperature, but the transient state depends on the initial conditions. The equation of motion for the matrix elements of the density operator is solved analytically, given an arbitrary initial state. The factorial moments, the generating function, and the time-dependent spectral distribution are also obtained. The results yield known expressions in the appropriate limits.
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 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.
The interaction is considered between NNN stationary two-level atoms and a radiation field described by a single cavity mode. The state vector for the complete system of atoms plus radiation is expressed as a linear superposition of states constructed from a product of photon states in the nnn representation and products of Pauli spin eigenstates describing all combinations of atoms in the lower and upper energy levels. Equations for the corresponding probability amplitudes are derived by substituting this superposition into the Schr\"odinger equation. The resulting equations are combined into bilinear form and phenomenological damping contributions are added. After the neglect of certain of the bilinear quantities, a master equation is derived which governs the probability Pmn{{P}_{m}}^{n}Pm​n of having mmm atoms in the lower level and nnn photons in the mode. This master equation takes account of multiple single-quantum absorption and emission processes but not of simultaneous multiple processes involving two or more atoms at a time. The equation which governs the expected number of photons 〈n〉〈n〉〈n〉 derived from the master equation bears a close resemblance to a rate equation. The effect of radiation loss from the cavity is incorporated into the master equation. Numerical calculations for a QQQ-spoiled laser show that the statistics of the number of photons in the mode bear a qualitative resemblance to Poisson statistics.
Any initial state of the radiation field in a finite-Q cavity relaxes towards thermal equilibrium. For the case of a zero-temperature cavity, it is shown that the factorial moments of any photon distribution evolve independently, and have the same time evolution. This is applied to evaluate the transient behaviour of the photon statistics for a variety of initial states of practical interest. The coherences of the density operator obey the same equation of motion as the populations (the photon probability distribution), after a simple transformation. This is used to recover the result that a coherent state remains a pure and coherent state at all times. Also the time-dependent frequency spectrum has been evaluated, and it appeared that this spectrum is the same for any state of the radiation field. The spectral distribution is determined entirely by the cavity damping rate, the spectral width of the detector, and the observation time.