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Physica A Statistical Mechanics and its Applications

Rigorous solution to a quantum statistical mechanical laser model

Fumiaki Shibata, Chikako Uchiyama

Published Mar 1, 1995• 3 citations
Publisher
DOI
10.1016/0378-4371(94)00218-i

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quant-pharXiv:2511.13622•Physical Review A•Nov 18, 2025•0 citations
A quantum statistical procedure for studying interacting radiation fields is developed and applied to a model laser operating as a linear amplifier. The method utilizes an expansion of the density matrix in terms of a complete set of orthogonal operators originally used by Weyl. One mode of the laser field is considered to interact with a pumping mechanism and with a loss mechanism. A general solution for the density matrix is obtained and then applied to various initial conditions. It is found that a classical statistical interpretation of the results is frequently appropriate.
Recently, a rigorous foundation of several aspects of the theory of universality for statistical mechanics models with continuously varying exponents (among which are interacting planar Ising models, quantum spin chains, and one-dimensional Fermi systems) has been reached; it has its root in the mapping of such systems into fermionic interacting theories and uses the modern renormalization group methods developed in the context of constructive quantum field theory. No use of exact solutions is done and the analysis applies either to solvable or not solvable models. A review of such developments will be given here.
The random nature of spontaneous emission leads to unavoidable fluctuations in a laser's output. This is often included through random Langevin forces in laser rate equations, but this approach falls short for nanolasers. In this paper, we show that the laser quantum noise can be quantitatively computed for a very broad class of lasers by starting from simple and intuitive rate equations and merely assuming that the number of photons and excited electrons only takes discrete values. While the approach has seen previous success, we here derive it rigorously from an open quantum system master equation, whereas it was previously introduced only on phenomenological grounds. We further show that in the many-photon limit, the model simplifies to Langevin equations. We perform an extensive comparison of different approaches for computing quantum noise in lasers, identifying the best approach for different system sizes, ranging from nanolasers to macroscopic lasers, and different levels of excitation, i.e., cavity photon number. In particular, we show that below the laser threshold, stochastic fluctuations in the numerical solution to the Langevin equations can drive populations to unphysical negative values, requiring the introduction of population bounds, which in turn skew the noise statistics, leading to inaccuracies. The Laser Markov Chain model, on the other hand, is accurate for all pump values and laser sizes when collective emitter effects are excluded.