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Journal of the Optical Society of America B

Dynamics and symmetry of a laser with a squeezed reservoir

Christian Ginzel, Rüdiger Schack, Axel Schenzle

Published Aug 1, 1991• 12 citations
Publisher

Abstract

We present a detailed analysis of the stationary and the dynamic behaviors of a single-mode laser coupled to a squeezed reservoir. The eigenfunctions and eigenvalues of the corresponding Fokker–Planck equation are obtained numerically, since the condition of detailed balance is destroyed because of the phase dependence of squeezed noise.

DOI
10.1364/josab.8.001704

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We present a detailed analysis of the stationary and the dynamic behaviors of a single-mode laser coupled to a squeezed reservoir. The eigenfunctions and eigenvalues of the corresponding Fokker–Planck equation are obtained numerically, since the condition of detailed balance is destroyed because of the phase dependence of squeezed noise.
We present a detailed numerical study of the one-atom laser, that is, a single two-level atom interacting with one lasing mode, whereby both the atom and the photon field are coupled to reservoirs. The stationary as well as the dynamical properties of the model are calculated directly from the quantum master equation with the help of two numerical methods. These numerical methods do not need any quasi-probability representation and they do not require approximations. We find that the one-atom laser exhibits most of the typical features of a normal laser. In the region far below threshold some aspects, among them the linewidth, are changed due to eigenvalues of the master equation with imaginary parts. In this regime the complexity of the eigenvalues prominently enters the dynamical behavior.
We investigate the effect of injected squeezed vacuum on the phase and intensity fluctuations of a laser. By using a stochastic simulation of the Langevin equations, we find that the phase diffusion noise of a laser with injected squeezed vacuum can be transiently reduced by a factor of 2, in agreement, with earlier predictions. Further, we find that the laser linewidth is broadened. The key observation is that the phase-diffusion rate is time dependent. We find that the steady-state intensity fluctuations are increased by the injected squeezing, in qualitative (but not quantitative) agreement with an earlier prediction. The problem of locking is examined. We find that the injected squeezed vacuum does not cause the laser to lock to any particular phase. We present an approximate solution of the Fokker-Planck equation in steady state, and a simple geometrical, vector kick model from which most of our results can be obtained.
The influence of injected atomic coherence on a laser's operation is discussed by analyzing the semiclassical equations of motion of this system. Both stationary-steady-state and time-dependent regimes of operation are investigated. We present the dependence of the stationary laser intensity and phase on the external parameters (population inversion, amplitude of atomic coherence, and detuning). For small cavity-field detuning and large atomic coherence, both frequency and phase locking occurs. The laser frequency is locked to the atomic frequency, and the phase is locked to a value determined primarily by the phase of the atomic coherence. Under certain conditions, in the inverted regime, the output intensity is an ssS-shaped function of the atomic coherence, leading to the possibility of a bistable behavior. However, only a portion of the bistable curve gives stable stationary operation, and in critical points (the location of which depends crucially on the external parameters) time-dependent instabilities branch away. When the detuning is larger than a critical value there is neither phase nor frequency locking. We find that the time-dependent behavior of the laser intensity in this case is oscillatory (quasiperiodic), and show that there is a stable limit cycle in the phase plane of the quadratures. There is a small parameter region where a stationary steady state and an oscillatory state may coexist. We also consider the nonlinear quantum theory of a laser with injected atomic coherence, and include the effect of pumping statistics. We derive the Fokker-Planck equation for the P representation, and express the noise in terms of moments. We find that in the steady state the intensity noise can be suppressed below the shot-noise limit but that the phase fluctuations are not affected by pump regularity. For nonzero detuning, we find that transient squeezing of the phase fluctuations is possible.
A two-photon laser is modeled as NNN two-level atoms interacting via a two-photon transition with a single resonant-cavity field mode. The quadratic form of this interaction makes such a system a candidate to exhibit reduced quantum fluctuations (squeezing) in one quadrature of the output field. A Fokker-Planck equation containing all leading quantum noise terms is derived following the method of Haken. A linearized fluctuation analysis reveals that there is no squeezing in a two-photon laser with injected signal. A small amount of squeezing may be present in two-photon optical bistability. Our calculations confirm the results of Lugiato and Strini, who adopted a factorization of moment equations.
We present here a fully quantum-mechanical treatment of the laser using a three-level atom model. The phase-space method employed is identical to that in a previous publication of ours [Phys. Rev. A 38, 4073 (1988)] and again results in a Fokker-Planck description of the laser without any approximations. The third level only provides extra deterministic terms and, if it is regarded as a further reservoir to the lasing levels, then the resultant stochastic equations have much improved stability. These equations are suitable for extensive theoretical analysis of both the laser and optical bistability, although in this paper we have concentrated on the adiabatically eliminated limit and calculated the leading noise terms in various regimes. However, we do use the technique of direct numerical simulation to solve exactly the full system of laser equations. The results display clearly many of the known laser characteristics.