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arXiv:quant-ph/0406212 · International Journal of Modern Physics A

A THEOREM ON CYCLIC HARMONIC OSCILLATORS

KENICHI KONISHI, GIAMPIERO PAFFUTI

Published Jun 20, 2006• 1 citation
arXivHTMLPDF

Abstract

It is proven that the energy of a quantum mechanical harmonic oscillator with a generically time-dependent but cyclic frequency, ω(t0) = ω(0), cannot decrease on an average if the system is originally in a stationary state, after the system goes through a full cycle. The energy exchange always takes place in the direction from the macroscopic system (environment) to the quantum microscopic system.

DOI
10.1142/s0217751x06032502

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It is proven that the energy of a quantum mechanical harmonic oscillator with a generically time-dependent but cyclic frequency, ω(t0) = ω(0), cannot decrease on an average if the system is originally in a stationary state, after the system goes through a full cycle. The energy exchange always takes place in the direction from the macroscopic system (environment) to the quantum microscopic system.
In this chapter we introduce the basic concept of a quantum harmonic oscillator, and how such oscillators couple to their environment. Our purpose is not to exhaustively cover all aspects of such systems, but rather to provide the foundation material required to treat the quantum mechanics of mechanical resonators and optical cavities – and their interactions – in later chapters. The quantum fluctuation-dissipation theorem is derived relating the power spectral density of environmental forcing of an oscillator to the rate of energy decay from the oscillator into the environment. This is in direct analogy to the familiar fluctuation-dissipation theorem in classical physics. However, in contrast to the classical case, the noncommutation of position and momentum introduces an asymmetry between the positive and negative frequency components of the power spectrum. This provides a natural definition for the effective temperature of the oscillator. The quantum Langevin approach is introduced as a method to model the open dynamics of quantum systems, both with and without recourse to the rotating wave approximation.
We study the evolution of the energy of a harmonic oscillator when its frequency slowly varies with time and passes through a zero value. We consider both the classical and quantum descriptions of the system. We show that after a single frequency passage through a zero value, the famous adiabatic invariant ratio of energy to frequency (which does not hold for a zero frequency) is reestablished again, but with the proportionality coefficient dependent on the initial state. The dependence on the initial state disappears after averaging over the phases of initial states with the same energy (in particular, for the initial vacuum, the Fock and thermal quantum states). In this case, the mean proportionality coefficient is always greater than unity. The concrete value of the mean proportionality coefficient depends on the power index of the frequency dependence on a time near the zero point. In particular, the mean energy triplicates if the frequency tends to zero linearly. If the frequency attains zero more than once, the adiabatic proportionality coefficient strongly depends on the lengths of time intervals between zero points, so that the mean energy behavior becomes quasi-stochastic after many passages through a zero value. The original Born-Fock theorem does not work after the frequency passes through zero. However, its generalization is found: the initial Fock state becomes a wide superposition of many Fock states, whose weights do not depend on time in the new adiabatic regime. When the mean energy triplicates, the initial Nth Fock state becomes a superposition of, roughly speaking, 6N states, distributed nonuniformly. The initial vacuum and low-order Fock states become squeezed, as well as the initial thermal states with low values of the mean energy.
Abstract We reveal a new face of the old clichéd system: a dissipative quantum harmonic oscillator. We formulate and study a quantum counterpart of the energy equipartition theorem satisfied for classical systems. Both mean kinetic energy Ek and mean potential energy Ep of the oscillator are expressed as Ek = 〈εk〉 and Ep = 〈εp〉, where 〈εk〉 and 〈εp〉 are mean kinetic and potential energies per one degree of freedom of the thermostat which consists of harmonic oscillators too. The symbol 〈...〉 denotes two-fold averaging: (i) over the Gibbs canonical state for the thermostat and (ii) over thermostat oscillators frequencies ω which contribute to Ek and Ep according to the probability distribution Pk(ω){{\mathbb{P}}}_{k}(\omega )Pk​(ω) ℙ k ( ω ) and Pp(ω){{\mathbb{P}}}_{p}(\omega )Pp​(ω) ℙ p ( ω ) , respectively. The role of the system-thermostat coupling strength and the memory time is analysed for the exponentially decaying memory function (Drude dissipation mechanism) and the algebraically decaying damping kernel.
It is pointed out that the property of a constant energy characteristic for the circular motions of macroscopic bodies in classical mechanics does not hold when the quantum conditions for the motion are applied. This is so because any macroscopic body—lo-cated in a high-energy quantum state—is in practice forced to change this state to a state having a lower energy. The rate of the energy decrease is usually extremely small which makes its effect uneasy to detect in course of the observations, or experiments. The energy of the harmonic oscillator is thoroughly examined as an example. Here our point is that not only the energy, but also the oscillator amplitude which depends on energy, are changing with time. In result, no constant positions of the turning points of the oscillator can be specified; consequently the well-known variational procedure concerning the calculation of the action function and its properties cannot be applied.