Wigner Distribution of the Quantized Lorenz Model

Abstract

The quantum-mechanical master equation of a homogeneously broadened single-mode laser is solved in the regime of the Lorenz strange attractor. The quantum Wigner distribution is obtained, replacing the classical invariant measure on the attractor, and reducing to it in the classical limit.PACS numbers: 42.55. Bi, 03.65.Bz, 42.50. The Lorenz model of thermal convection 1 is one of the earliest and simplest examples of the appearance of chaos and strange attractors in continuous dynamical systems. 2 The model is therefore of fundamental importance and has been studied extensively. 3 Physical systems described by this model are not limited to hydrodynamics but occur in other fields as well, most notably in quantum optics. 4 ' 5 The equations of motion of a homogeneously broadened single-mode laser 6 reduce to the Lorenz model in the classical limit. 4 Because of these latter applications of the model and in view of its fundamental importance, it is interesting to ask how quantum theory modifies the Lorenz attractor. It is my purpose here to address this question. The quantum-mechanical equations of motion of laser theory 6 provide a convenient starting point. In an earlier Letter, 7 concerned with the same problem, the master equation of the single-mode laser was used to derive a hierarchy of moment equations. The infinite hierarchy was closed by truncation and solved numerically. In the steady state, the moments obtained in this way exhibited chaotic time dependence for weak quantum noise. However, there remained some doubt 8 as to how the time dependence of the moments could be reconciled with the existence of a time-independent statistical operator in the steady state.In the present note I also use the master equation of the laser as a starting point. The time-independent statistical operator is directly constructed in its representation by the Wigner quasiprobability density. The method employed consists of a combination of general procedures of quantum optics 6 with a special technique devised earlier for an analysis of the Lorenz model under the influence of . + q external classical noise. 9 The analysis also allows identification of three different time scales associated with (i) relaxation of fluctuations towards the attractor, (ii) quantum-mechanically enhanced mixing on the attractor, and (iii) loss of coherence by phase diffusion. The second of these mechanisms is likely to be missed by the factorization scheme of Elgin and Sarkar, 7 which inevitably suppresses at least some correlations due to quantum fluctuations, and thus prohibits the quantum ensemble to spread over the attractor.In the following I describe the essential steps of the analysis, which are applicable to other models of quantum optics with chaotic behavior, and present the results for the single-mode laser in the regime of the Lorenz attractor.The master equation p = Lp which governs the time evolution of the statistical operator p of a single-mode laser is well known, and the form of the Liouville operator L acting on p has been exhibite...

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