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arXiv:quant-ph/9611044 · Physical Review A

Quantum-state diffusion model and the driven damped nonlinear oscillator

M. Rigo, G. Alber, Francisca Mota-Furtado, Patrick F. O’Mahony

Published Mar 1, 1997• 44 citations
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Abstract

We consider a driven damped anharmonic oscillator that classically leads to a bistable steady state and to hysteresis. The quantum counterpart for this system has an exact analytical solution in the steady state that does not display any bistability or hysteresis. We use quantum-state diffusion theory to describe this system and to provide a new perspective on the lack of hysteresis in the quantum regime so as to study in detail the quantum to classical transition. The analysis is also relevant to measurements of a single periodically driven electron in a Penning trap where hysteresis has been observed.

DOI
10.1103/physreva.55.1665

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We consider a driven damped anharmonic oscillator that classically leads to a bistable steady state and to hysteresis. The quantum counterpart for this system has an exact analytical solution in the steady state that does not display any bistability or hysteresis. We use quantum-state diffusion theory to describe this system and to provide a new perspective on the lack of hysteresis in the quantum regime so as to study in detail the quantum to classical transition. The analysis is also relevant to measurements of a single periodically driven electron in a Penning trap where hysteresis has been observed.
Recent numerical work has demonstrated how one of the bistable states of a driven, damped, anharmonic classical oscillator is metastable in the corresponding quantum oscillator [D. Enzer and G. Gabrielse, Phys. Rev. Lett. 78, 1211 (1997)]. Based on an analysis of the classical oscillator, we present a simple analytical treatment of the quantum oscillator. Calculated fluctuations in the quantum observables and the estimated decay rate of the metastable state are in an excellent agreement with the numerical results of Enzer and Gabrielse.
We consider the quantum model of a driven anharmonic oscillator, in the presence of dissipation, and present an exact analytic solution for the corresponding Wigner function in the steady-state regime. This provides explicit phase-space images of the resulting state of the cavity mode, and allows us to understand how the quantum interference is built up into it. The photon number probability distribution is calculated and analysed as well. We monitor the transition from the semiclassical to extreme quantum regime of operation, and identify qualitative changes, where the conventional characteristics of the model, such as bistability or turning points, become meaningless.
The dynamics of hybrid systems -- i.e. ones in which classical and quantum degrees of freedom co-exist and interact -- feature both diffusion in the classical sector and decoherence in the quantum state. In this article, we will consider the simple setup of a classical damped oscillator interacting with its quantum counterpart and show that, for any initial state, the dynamics flows to a unique (non-equilibrium) steady state, which we compute explicitly. To do so, we make use of a useful mapping between hybrid and classical diffusive dynamics, which we characterise in detail in the master equation formalism.
For the model of a linearly driven quantum anharmonic oscillator, the role of damping is investigated. We compare the position of the stable points in phase space obtained from a classical analysis to the result of a quantum mechanical analysis. The solution of the full master equation shows that the stable points behave qualitatively similar to the classical solution but with small modifications. Both the quantum effects and additional effects of temperature can be described by renormalizing the damping.
Driving a damped, anharmonic oscillator produces stable, steady-state excitations according to the familiar classical description. A quantum mechanical description, however, shows the surprising result that such excitations are intrinsically unstable, and can be much smaller than classical expectations. We illustrate by calculating the motion of one electron in a magnetic field. This physically realizable system is among the simplest of nonlinear systems, being anharmonic because of special relativity and being damped by the spontaneous emission of synchrotron radiation.