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arXiv:1203.2324 · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

A scaling-limit approach to the theory of laser transition

P. Gärtner

Published Apr 10, 2012• 1 citation
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Abstract

The conditions for the appearance of a sharp laser transition are formulated in terms of a scaling limit, involving vanishing cavity loss and light-matter coupling, k → 0, g → 0, such that g2/k stays finite. It is shown analytically that in this asymptotic parameter domain, and for pump rates above the threshold value, the photon output becomes large in a sense that is specified, and the photon statistics becomes strictly Poissonian. Numerical examples for the case of a two-level and a three-level emitter are presented and discussed in relation to the analytic result.

DOI
10.1117/12.921772

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The conditions for the appearance of a sharp laser transition are formulated in terms of a scaling limit, involving vanishing cavity loss and light-matter coupling, k → 0, g → 0, such that g2/k stays finite. It is shown analytically that in this asymptotic parameter domain, and for pump rates above the threshold value, the photon output becomes large in a sense that is specified, and the photon statistics becomes strictly Poissonian. Numerical examples for the case of a two-level and a three-level emitter are presented and discussed in relation to the analytic result.
The problem of the two-level laser is studied analytically. The steady-state solution is expressed as a continued fraction and allows for accurate approximation by rational functions. Moreover, we show that the abrupt change observed in the pump dependence of the steady-state population is directly connected to the transition to the lasing regime. The condition for a sharp transition to Poissonian statistics is expressed as a scaling limit of vanishing cavity loss and light-matter coupling, $\ensuremath{\kappa}\ensuremath{\rightarrow}0$, $g\ensuremath{\rightarrow}0$, such that ${g}^{2}/\ensuremath{\kappa}$ stays finite and ${g}^{2}/\ensuremath{\kappa}>2\ensuremath{\gamma}$, where $\ensuremath{\gamma}$ is the rate of nonradiative losses. The same scaling procedure is also shown to describe a similar change to the Poisson distribution in the Scully-Lamb laser model, suggesting that the low-$\ensuremath{\kappa}$, low-ggg asymptotics is of more general significance for the laser transition.
In analogy with equilibrium phase transitions, we address the problem of the instability to symmetry-breaking perturbations of systems undergoing a laser transition. The symmetry in question is the U(1)\text{U}(1)U(1) invariance with respect to a phase factor, and the perturbation is a coherent field EEE, coupled to the exciton. At the rate-equation level we analyze first the case of a cavity containing a single, two-level emitter, and then a chain of such cavities interacting by photon-hopping processes. In both cases, spontaneous symmetry breaking takes place when the system is in the lasing phase. For the laser transition, the analog of the thermodynamic limit is the scaling limit of vanishing cavity loss and light-matter coupling, $\ensuremath{\kappa}\ensuremath{\rightarrow}0$, $g\ensuremath{\rightarrow}0$, so that ${g}^{2}/\ensuremath{\kappa}$ remains finite. We show that in the lasing regime, anomalous averages persist in the $E\ensuremath{\rightarrow}0$ limit, provided that the scaling limit is performed first. Lasing diagnosis based on robust anomalous averages is compared numerically with the familiar coherence criterion g(2)(0)=1{g}^{(2)}(0)=1g(2)(0)=1, and the advantages of the former are discussed.
The laser--phase-transition analogy is considered for a system of identical emitters interacting with a cavity mode. As in the case of equilibrium phase transitions, the thermodynamic limit is shown to be an essential condition for the appearance of an abrupt transition, with a well-defined threshold. The proof is discussed first in the simpler formalism of the rate equation and then at the higher level of the master equation for probabilities. The latter is derived from the von Neumann--Lindblad evolution of the density operator. In the thermodynamic limit and for two-level emitters the master equation becomes a two-dimensional dynamic flow with two possible fixed points, related to the normal and the lasing regimes, respectively. The master equation analysis confirms the threshold value deduced from the rate equation, but also provides further evidence of lasing by showing that all photonic autocorrelation functions converge to unity.
In the present work the model of a single-atom laser generating in the regime when incoherent pumping rate coincides with the cavity decay rate is theoretically investigated. Using the stationary equation for phase averaged Glauber P function the analytical expressions for mean number of photons and its dispersion are obtained. In the limiting case of the strong-coupling regime the exact expression for the photon number distribution function is derived. Obtained results describe the sub-Poissonian photon statistics and show an increase in mean number of photons to non-zero value when the pumping tends to zero.
A recent analysis of the influence of the pump statistics on the laser output fluctuations is applied to the experimentally feasible case of a laser being pumped by light coming from a separate pump field cavity that is driven by a broad-band squeezed vacuum plus a coherent driving field. In contrast to the 'squeezed-reservoir laser', where a reservoir of squeezed modes coming from all directions coupled to the dipole moment of the lasing transition and leads to atomic phase decay inhibition and phase locking of the laser field, here the squeezed light merely pumps the gain atoms, that is gives rise to inversion of the laser transition. In analogy to photodetection theory the photon statistics of the laser output is calculated as a function of the photon statistics of the absorbed squeezed pump light. For the case of a squeezed pump field chosen to display sub-Poissonian statistics, it is demonstrated that the Mandel Q-parameter of the laser cavity output can theoretically reach the optimum value of Q=- 1 / 2 , similar to the case of a deterministic pumping mechanism.
A fully quantum treatment of the non-degenerate optical parametric oscillator in the extreme quantum limit of small photon numbers and very high nonlinear coupling strength is presented. When the nonlinear coupling constant becomes comparable with the cavity decay rate, the sharp threshold that is usually encountered disappears, a situation reminiscent of microlasers. Furthermore the output light exhibits unusual statistical properties such as strong super-Poissonian behaviour of the pump mode for (very) large coupling constants.