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arXiv:1105.2189 · Physical Review A

Two-level laser: Analytical results and the laser transition

P. Gärtner

Published Nov 2, 2011• 35 citations

Abstract

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-$g$ asymptotics is of more general significance for the laser transition.

DOI: 10.1103/physreva.84.053804

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•Jan 1, 2002•2 citations

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-

g

asymptotics is of more general significance for the laser transition.

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.

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.

The quantum statistical properties of the radiation of a one-atom two-level laser with incoherent pumping are analyzed. Solution of the Liouville equation for the density operator in the basis of Fock states shows that stationary radiation from a single-mode laser with incoherent pumping can be in a squeezed (sub-Poissonian) stationary state if the rate of spontaneous decay is lower than the rate of cavity losses and the pump rate. Inside the cavity the Fano factor reaches F =0.85 (15% squeezing). Multiple squeezing ( F =0.19) is possible in the transient lasing regime. Significant squeezing obtains at the cavity output; the spectral Fano factor at zero frequency is 0.36 under optimal conditions.

The quantum-statistical properties of the optical field of a two-mode laser with two coupled transitions have been studied by means of a generalization of the Scully and Lamb theory. The photon number distribution and the mode intensity distribution are obtained in the steady state for a system of homogeneously broadened atoms in resonance with the laser field. It is shown that the mode-coupling constant $\ensuremath{\xi}$ is unity and that near threshold the results of earlier treatments are recovered. Furthermore, it is shown that certain limit measures for the relative intensity fluctuations predicted by the semiclassical Fokker-Planck treatments based on the third-order theory are valid even in the limit of arbitrarily high gain levels.

The statistics of the light emitted by two-level lasers is evaluated on the basis of generalized rate equations. According to that approach, all fluctuations are interpreted as being caused by the jumps that occur in active and detecting atoms. The intra-cavity Fano factor and the photo-current spectral density are obtained analytically for Poissonian and quiet pumps. The algebra is simple and the formulas hold for small as well as large pumping rates. Lasers exhibit excess noise at low pumping levels.