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Physical Review A

Semiclassical analysis of the metastable driven and damped quantized anharmonic oscillator

Bjarne Vestergaard, Juha Javanainen

Published Aug 1, 1998• 3 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.58.1537

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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 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.
The analytic solution of the time evolution of a dissipative nonlinear quantum oscillator in the generalized s-ordered representation is derived. The analytical solution allows for comparsion of the various representations with the classical and semiclassical dynamics. It is shown by the eigenvalue analysis of the evolution operators that in the case of a small nonlinearity a semiclassical analysis is an excellent approximation to the exact quantum dynamics. This is further demonstrated by using the Green's function of the anharmonic oscillator to solve the initial-value problem for a superposition of coherent states.
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.
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.
We study quantum mechanical behavior of a Morse oscillator coupled to a bath of harmonic oscillators and driven by an infrared laser. The purpose is to compare quantum behavior of such a system with the classical and semiclassical solutions of a driven damped Morse oscillator. The general picture coming out of classical and semiclassical studies is that the total energy of the Morse oscillator in steady states or steady oscillations show jump and hysteretic behavior, as expected in a cusp catastrophe, when laser intensity or frequency is varied. Along with the bistable behavior we have also found period-doubling bifurcations leading to chaos on one or both branches. In the quantum treatment a generalized master equation is derived from the Liouville equation, in which the Hamiltonian of the Morse oscillator is expressed in terms of generators of an SU(2) algebra. This algebra and the Markoffian approximation allows us to derive the equation of motion of the reduced density matrix, which is solved numerically. Bistable behavior does not exist in this quantum treatment, but solutions do show jump behavior and bifurcation phenomenon. Experimental implications of these studies are also discussed.