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 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) 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.
We detail the derivation of nanolaser models that include coherent and incoherent variables and predict the existence of a laser threshold, irrespective of cavity size and emitter number, for both single- and multielectron systems. The growth in photon number in the lasing mode is driven by an increase in correlation between absorption and emission processes, leading to the onset of self-sustained stimulated emission (laser threshold), followed in turn by a correlation decrease and ending with the dominance of coherent emission. The first-order coherence
g(1) steadily increases, as the pump grows towards the laser threshold value, and reaches unity at or beyond threshold. The transition toward coherent emission becomes increasingly sharp as the numbers of emitters and coupled electromagnetic cavity modes increase, continuously connecting, in the thermodynamic limit, the physics of nano- and macroscopic lasers at threshold. Our predictions are in remarkable agreement with experiments whose first-order coherence measurements have so far been explained only phenomenologically. A consistent evaluation of different threshold indicators provides a tool for a correct interpretation of experimental measurements at the onset of laser action.
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.
We develop a birth-death model for a cavity QED laser and analyze the dependence of the lasing threshold on the fraction, \ensuremath{\beta}, of spontaneous emission directed into the laser mode. We define threshold in terms of the Fano factor for the intracavity phonon number. We emphasize the role of \ensuremath{\beta} as a parameter characterizing the ``system size'' and show that the concept of laser threshold is well-defined in the ``thermodynamic'' limit, ${\mathrm{\ensuremath{\beta}}}^{\mathrm{\ensuremath{-}}1}$\ensuremath{\rightarrow}\ensuremath{\infty}. An ideal cavity-QED laser operates far outside this limit and therefore, in contrast to a conventional laser, is not a threshold device. We present numerical results showing how this difference affects the noise properties of the device.
The quantum theory of the three-energy-level two-mode laser in which all the levels are being pumped is investigated and the master equation of the laser operation is obtained. It contains two variables,
n1, the photon number of the first mode, and
n2, the photon number of the second mode. There appear four terms in the master equation which represent two-photon processes between the two modes, the absorption of a photon from one mode, and emission of a photon to the other. They are absent in the single-mode case and in the two-mode case with pumping only to the upper level. The master equation can be represented by a diagram of probability in two dimensions which can be extended to infinity, and each arrow of it represents a term on the right-hand side of the master equation. By summing each variable (
n1 or
n2) of the master equation, two equations can be obtained. Each equation contains only one variable, and can be expressed by a diagram of probability flow in one dimension. By considering the correspondence between macroscopic equilibrium and microscopic detailed balance the equations of motion in the steady state are obtained. By introducing a parameter
H, the equation of motion is simplified and a formal solution can be deduced which is dependent on
H. It appears that the parameter
H cannot be fully determined, but some properties of it can be deduced which prove that the introduction of
H is reasonable. The operation characteristics of the laser under different conditions are discussed on the basis of the formal solution: among them are the threshold condition, the condition for one-mode operation, and the change of the photon-statistical distribution. We especially discuss the laser output power curves under two-mode operation above threshold. The curve of one mode, which oscillates first, shows a bending-down phenomenon as the excitation increases when the other mode goes beyond its threshold as demonstrated experimentally by Otsuka. This phenomenon is explained qualitatively and is attributed to the two-photon processes between the two modes.
The general time-dependent problem associated with a two-mode-laser Fokker-Planck equation has been considered. Analytic expressions for the eigenvalues and the eigenfunctions have been derived for the below- and above-threshold operation of the system. It is shown that depending on the values of the pump parameters
a1,a2 and the coupling constant $\ensuremath{\xi}$, a wide variety of dynamical behavior may be encountered. Thus for $\ensuremath{\xi}<1$ interesting threshold effects may appear in the behavior of correlation times and linewidths as a function of excitation which may change their behavior whenever a mode passes through its threshold of oscillation. For $\ensuremath{\xi}=1$ the correlation time of the stronger mode may pass through a maximum before it starts to decrease with increasing excitation. The weaker mode may show particularly interesting behavior for $\ensuremath{\xi}=1$. Both the correlation time and the linewidth may approach a constant value independent of the excitation. For $\ensuremath{\xi}>1$ the system may show bistability. Our analysis provides a deeper mathematical understanding of the onset of bistability and the analogy to the thermodynamic phase transition. Expressions for the switching time have been derived using several different procedures. We have also been able to derive closed-form expressions for the two-time joint probability densities. It should be possible to test some of these predictions in photon-counting experiments. Finally, we mention that our results and method should also be applicable in other similar problems, e.g., to the classical theory of coupled fields.