Systems of Quantum Oscillators

I. H. Umirzakov

As known all physical properties of solids are described well by the system of quantum linear harmonic oscillators. It is shown in the present paper that the system consisting of classical linear harmonic oscillators having temperature dependent masses or (and) frequencies has the same partition function as the system consisting of quantum linear harmonic oscillators having temperature independent masses and frequencies while the means of the square displacements of the positions of the oscillators from their mean positions for the system consisting of classical linear harmonic oscillators having: the temperature dependent masses; temperature dependent frequencies; and temperature dependent masses and frequencies differ from each other and from that of the system consisting of quantum linear harmonic oscillators, and hence, the system consisting of classical linear harmonic oscillators describes well the thermodynamic properties of the system consisting of quantum linear harmonic oscillators and solids.

A. M. Perelomov

Olivier Henri‐Rousseau, Paul Blaise

Biswabibek Bandyopadhyay, Tanmoy Banerjee

Restoration of oscillations from an oscillation suppressed state in coupled oscillators is an important topic of research and has been studied widely in recent years. However, the same in the quantum regime has not been explored yet. Recent works established that under certain coupling conditions, coupled quantum oscillators are susceptible to suppression of oscillations, such as amplitude death and oscillation death. In this paper, for the first time, we demonstrate that quantum oscillation suppression states can be revoked and rhythmogenesis can be established in coupled quantum oscillators by controlling a feedback parameter in the coupling path. However, in sharp contrast to the classical system, we show that in the deep quantum regime, the feedback parameter fails to revive oscillations, and rather results in a transition from a quantum amplitude death state to the recently discovered quantum oscillation death state. We use the formalism of an open quantum system and a phase space representation of quantum mechanics to establish our results. Therefore, our study establishes that the revival scheme proposed for classical systems does not always result in restoration of oscillations in quantum systems, but in the deep quantum regime, it may give counterintuitive behaviors that are of a pure quantum mechanical origin.

Michael B. Mensky

This chapter considers continuous measurements of harmonic oscillators. As has been mentioned, in many situations real systems may actually be treated as harmonic oscillators. Many real systems may be treated approximately as harmonic oscillators or systems of harmonic oscillators. For example, measurement of an oscillator is often performed in a narrow frequency interval near the resonance frequency. The chapter investigates a measurement of this type for a harmonic oscillator as the measured system. Measurement of the coordinate frequency component allows one to estimate the frequency component of the force acting on the oscillator during the measurement. The system of connected harmonic oscillators is a universal model for any oscillating system in a state close to its equilibrium state. All of two or more connected oscillators may display quantum properties. Therefore the problem of measurements in a system of connected quantum oscillators naturally arises.