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arXiv:1908.08679 · Journal of the Physical Society of Japan

Probing the Ginzburg–Landau Potential for Lasers Using Higher-order Photon Correlations

Naotomo Takemura, Masato Takiguchi, Masaya Notomi

Published Jul 1, 2022• 1 citation
arXivHTMLPDF

Abstract

Lasing transition is known to be analogous to the second-order phase transition. Furthermore, for some cases, it is possible to define the Ginzburg-Landau (GL) potential, and the GL theory predicts the photon statistical properties of lasers. However, the GL potential for lasers is surprising, because lasers are operating far from equilibrium. In this paper, we theoretically examine the validity of the GL theory for lasers in terms of various parameters, particularly, the ratio between photon and carrier lifetimes. For this purpose, we use stochastic rate equations and higher-order photon correlation functions. With higher-order photon correlation measurements, we can check whether or not laser dynamics are described by the GL theory. We demonstrate that, for low-$β$ lasers, the GL theory is applicable even when the photon lifetime is comparable to the carrier lifetime and that photon-carrier relaxation oscillation is the fundamental origin of the breakdown of the GL theory, which can be understood in the framework of center manifold reduction.

DOI
10.7566/jpsj.91.074401

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Lasing transition is known to be analogous to the second-order phase transition. Furthermore, for some cases, it is possible to define the Ginzburg-Landau (GL) potential, and the GL theory predicts the photon statistical properties of lasers. However, the GL potential for lasers is surprising, because lasers are operating far from equilibrium. In this paper, we theoretically examine the validity of the GL theory for lasers in terms of various parameters, particularly, the ratio between photon and carrier lifetimes. For this purpose, we use stochastic rate equations and higher-order photon correlation functions. With higher-order photon correlation measurements, we can check whether or not laser dynamics are described by the GL theory. We demonstrate that, for low-βββ lasers, the GL theory is applicable even when the photon lifetime is comparable to the carrier lifetime and that photon-carrier relaxation oscillation is the fundamental origin of the breakdown of the GL theory, which can be understood in the framework of center manifold reduction.
Nanocavity lasers are commonly characterized by the spontaneous coupling coefficient β that represents the fraction of photons emitted into the lasing mode. While β is conventionally discussed in semiconductor lasers where the photon lifetime is much shorter than the carrier lifetime (class-B lasers), little is known about β in atomic lasers where the photon lifetime is much longer than the other lifetimes and only the photon degree of freedom exists (class-A lasers). We investigate the impact of the spontaneous coupling coefficient β on lasing properties in the class-A limit by extending the well-known Scully–Lamb master equation. We demonstrate that in the class-A limit all the photon statistics are uniquely characterized by β and that the laser phase transition-like analogy becomes transparent. In fact, β perfectly represents the “system size” in phase transition. Finally, we investigate the laser-phase transition analogy from the standpoint of a quantum dissipative system. Calculating a Liouvillian gap, we clarify the relation between β and the continuous phase symmetry breaking.
We examine higher-order photon statisitics of laser diodes (LDs) and microchip solid-state lasers (SSLs). When the lasers are operated above lasing threshold, significantly different features of fluctuations are exhibited, which are the result of the asymmetric non-Gaussian distribution characteristics that are always present in SSLs, whereas, in contrast, Gaussian fluctuation persists in LDs. It is found that the deterministic dynamic trajectory in phase space, which is controlled by the ratio between the fluorescence and the photon lifetimes, is crucial to the corresponding behavior of higher-order statistics.
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.
A simple analytic expression for the ground state of a dilute gas of NNN ideal bosons in a 3D harmonic potential at temperature TTT is derived from the steady state solutions of nonequilibrium equations of motion. The NNN particle constraint plays the important role of introducing the essential nonlinearity yielding a Ginzburg-Landau free energy. The present analysis has much in common with the quantum theory of the laser, and with the laser phase transition analogy.
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.
Abstract Despite the enormous technological interest in micro and nanolasers, surprisingly, no class‐B quantum density‐matrix model is available to date, capable of accurately describing coherence and photon correlations within a unified theory. In class‐B lasers—applicable for most solid‐state lasers at room temperature—, the macroscopic polarization decay rate is larger than the cavity damping rate which, in turn, exceeds the upper level population decay rate. Here, a density‐matrix theoretical approach for generic class‐B lasers is carried out, and closed equations are provided for the photonic and atomic reduced density matrix in the Fock basis of photons. Such a relatively simple model can be numerically integrated in a straightforward way, and exhibits all the expected phenomena, from one‐atom photon antibunching, to the well‐known S‐shaped input–output laser emission and super‐Poissonian autocorrelation for many atoms (), and from few photons (large spontaneous emission factors, ) to the thermodynamic limit ( and ). Based on the analysis of , it is concluded that super‐Poissonian fluctuations are clearly related to relaxation oscillations in the photon number. A strong damping of relaxation oscillations with an atom number as small as is predicted. This model enables the study of few‐photon bifurcations and nonclassical photon correlations in class‐B laser devices, also leveraging quantum descriptions of coherently coupled nanolaser arrays.