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arXiv:1812.07428 · Physical review. B./Physical review. B

Spontaneous symmetry breaking in the laser transition

P. Gärtner

Published Mar 22, 2019• 4 citations
arXivHTMLPDF

Abstract

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 $\text{U}(1)$ invariance with respect to a phase factor, and the perturbation is a coherent field $E$, 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 ${g}^{(2)}(0)=1$, and the advantages of the former are discussed.

DOI
10.1103/physrevb.99.115313

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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, \ensuremathκ\ensuremath→0\ensuremath{\kappa}\ensuremath{\rightarrow}0\ensuremathκ\ensuremath→0, g\ensuremath→0g\ensuremath{\rightarrow}0g\ensuremath→0, so that g2/\ensuremathκ{g}^{2}/\ensuremath{\kappa}g2/\ensuremathκ remains finite. We show that in the lasing regime, anomalous averages persist in the E\ensuremath→0E\ensuremath{\rightarrow}0E\ensuremath→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.
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.
The problem of phase transitions in the system of high-density gas of two-level atoms non-resonantly interacting with a pump field in the presence of optical collisions (OCs) and placed in the cavity is studied. The OCs are considered in the framework of thermalization of atomic dressed state (DS) population. For the case of a strong atom-field coupling condition we analyze the problem of thermodynamic equilibrium superradiant phase transition for the order parameter representing real amplitude of cavity mode and taking place as a result of the atomic DS thermalization process. At the same time such a transition can be connected with condensed (coherent) properties of low branch DS-polaritons occurring in the cavity. Various aspects of non-equilibrium phase transition to laser field generation by using atomic DS levels are studied in detail. It is important that for positive atom-light detuning DS lasing can be obtained in the presence of quasi-equilibrium DS population that corresponds to a true two-level atomic system with inversion in perturbative limit.
After reviewing the interpretation of laser operation as a nonequilibrium Bose-Einstein condensation phase transition, we illustrate the novel features arising from the nonequilibrium nature of photon and polariton Bose-Einstein condensates recently observed in experiments. We then propose a quantitative criterion to experimentally assess the equilibrium versus nonequilibrium nature of a specific condensation process, based on fluctuation-dissipation relations. The power of this criterion is illustrated on two models which show very different behaviors. Historical and Conceptual Introduction The first introduction of concepts of nonequilibrium statistical mechanics into the realm of optics dates back to the early 1970s with pioneering works by Graham and Haken [1] and by DeGiorgio and Scully [2], who proposed a very insightful interpretation of the laser threshold in terms of a spontaneous breaking of the U( 1 ) symmetry associated with the phase of the emitted light. Similar to what happens to the order parameter at a second-order phase transition, such an optical phase is randomly chosen every time the device is switched on and remains constant for macroscopic times. Moreover, a long-range spatial order is established, as light emitted by a laser device above threshold is phase-coherent on macroscopic distances. While textbooks typically discuss this interpretation of laser operation in terms of a phase transition for the simplest case of a single-mode laser cavity, rigorously speaking this is valid only in spatially infinite systems. In fact, only in this case one can observe nonanalytic behaviors of the physical quantities at the transition point. In particular, the long-range order is typically assessed by looking at the long-distance behavior of the correlation function of the order parameter, which, for a laser, corresponds to the first-order spatial coherence of the emitted electric field, the spontaneous symmetry breaking is signaled by this quantity becoming nonzero (see Chapter 5). The average is taken on the stationary density matrix of the system. In order to be able to probe this long-distance behavior, experimental studies need devices with a spatially extended active region.
Liouvillian spectral collapse in Scully-Lamb lasing: Non-equilibrium second-order phase transition with or without U(1)U(1)U(1) symmetry breaking
The paradigm of second-order phase transition and symmetry breaking for thermal and quantum systems is a pillar of modern physics that has been fruitfully applied to out-of-equilibrium open quantum systems. Despite the non-equilibrium nature of lasing, the laser threshold was compared with thermal phase transitions systems more than forty years ago. For instance, the Landau theory -- based on thermodynamical consideration -- has been used to characterize the Scully-Lamb laser model (SLLM), describing a single-mode laser without saturable absorbers. Accordingly, the laser threshold is a second-order phase transition, associated with a spontaneous symmetry breaking (SSB) of the U(1)U(1)U(1) symmetry of the model. To capture a genuine non-equilibrium, quantum, and dissipative phase transition of the SLLM, here we employ a Liouvillian superoperator formalism derived from a Lindblad master equation. Our results confirm that the U(1)U(1)U(1) SSB can occur at the lasing threshold but, in contrast to the Landau theory and semiclassical approximation, they signal that the SLLM fundamental transition is a different phenomenon, which we call Liouvillian spectral collapse; that is, the emergence of diabolic points of infinite degeneracy. By considering a generalized SLLM with additional dephasing, we witness a second-order phase transition, with a Liouvillian spectral collapse, but in the absence of symmetry breaking. This demonstrates the difference between the Liouvillian spectral collapse and a standard second-order phase transition with symmetry breaking. We characterize the new phenomena associated with the spectral collapse, such as the onset of a rather unusual multistability enabled (and not suppressed) by the very presence of fluctuations. This multistability, characterizing the dissipative and quantum origin of lasing, can be experimentally revealed by the presence of a dynamical hysteresis.