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Physical Review Letters

Dressed Coherent States of the Anharmonic Oscillator

Daphna G. Enzer, G. Gabrielse

Published Feb 17, 1997• 16 citations
Publisher

Abstract

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.

DOI
10.1103/physrevlett.78.1211

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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.
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.
In terms of excitation creation and annihilation operators of the Jaynes-Cummings model, acting in the representation of dressed states, the Hamiltonian is written which describes the character of the spectrum of excitations of two modes, representing a quantum analog of the classical behavior of two interacting one-dimensional anharmonic oscillators, namely, the field and atomic oscillators. The anharmonicity is caused by the nonlinearity of the oscillator interaction and manifests itself in the dependence of the frequencies of both modes on the number of excitations, i.e., on the energy. It is shown that an external deterministic force, acting on the system during a certain time t 0, transfers it from a vacuum state to a coherent state or from one of the coherent states to another coherent state. The probability of the transition from the vacuum state to the coherent state with a given number of excitations represents the Poissonian distribution for the number of excitations formed in the (atom + field) system by the end of action of the external force. It was found to be proportional to the excitation time t 0.
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.
Abstract We study various techniques to characterize the quantum states of a strongly dissipative system, namely, the an‐harmonic oscillator. One cannot directly access the quantum states themselves but only transitions between them. Various methods bring various advantages and inconvenients. We compare several observables under coherent and incoherent excitation. Incoherent excitation leads to huge broadening while coherent excitation quickly dresses the system at the energies required to access the excited states. Less perturbative methods such as four wave mixing appear to be superior but are, by construction, only usable for the lowest rungs. (© 2012 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
The present review is devoted to the study of certain aspects of anharmonic, time-dependent and damped oscillator(s) system using different theoretical techniques. A theoretical understanding of these systems is important for application in many problems in physics, mechanics and other fields. We discuss in detail the difficulties in the theoretical analysis of the problem. In particular we discuss here the regular, well-behaved perturbative solution, the large quantum number behaviour of anharmonic oscillator(s) using the technique of coherent states, exact solution of quantum anharmonic oscillators, the electromagnetic radiation emitted by a charged particle executing damped anharmonic oscillator motion using Krylov-Bogoliubov approximation method, use of invariants to obtain solution and coherent states of time-dependent oscillator(s), the derivation of perturbative frequencies of a damped coupled anharmonic oscillators system using suitable canonical transformation in the framework of Hamilton-Jacobi formalism and the quantisation and construction of coherent states of a damped oscillator using time-dependent operators.