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Correlated states of a quantum oscillator acted by short pulses

Olga V. Man’ko

Published Jan 1, 1993• 0 citations
OpenAlex

Abstract

Correlated squeezed states for a quantum oscillator are constructed based on the method of quantum integrals of motion. The quantum oscillator is acted upon by short duration pulses. Three delta-kickings of frequency are used to model the pulses' dependence upon the time aspects of the frequency of the oscillator. Additionally, the correlation coefficient and quantum variances of operations of coordinates and momenta are written in explicit form.

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Correlated squeezed states for a quantum oscillator are constructed based on the method of quantum integrals of motion. The quantum oscillator is acted upon by short duration pulses. Three delta-kickings of frequency are used to model the pulses' dependence upon the time aspects of the frequency of the oscillator. Additionally, the correlation coefficient and quantum variances of operations of coordinates and momenta are written in explicit form.
The behaviour of an oscillator may be controlled by the frequency time-dependence. For example, one can kick the oscillator frequency by short pulses and this kicking produces an excitation of the parametric oscillator. The amplitude of the oscillator vibrations and its energy may increase due to the external influence expressed as the frequency time-dependence. Also the statistical properties of the oscillator state may be changed due to the action of external forces. The aim of the talk is to discuss the exact solution of the time-dependent Schrodinger equation for a damped quantum oscillator subject to a periodical frequency delta-kicks describing squeezed states which are expressed in terms of Chebyshev polynomials. The cases of strong and weak damping are investigated in the frame of Caldirola-Kanai model [1], [2].
In a squeezed state, the variance in one canonical variable may be suppressed below that normally associated with either the ground state or a coherent state, at the expense of an expansion in the variance of the conjugate variable. Squeezed states are usually discussed in the context of quantized light fields. The principal properties of squeezed states are demonstrated using the motion of quantum mechanical simple harmonic oscillators. The motion of a single oscillator is discussed to introduce the key concepts of squeezing, including quadrature operators, and the error contours of the Wigner function describing the quantum quasiprobability distributions in phase space of the oscillator motion. The main topic of two-mode squeezing is then addressed, in which the fluctuations in a system of two oscillators are tightly correlated. The existence of squeezing is demonstrated in normal mode coordinates representing motion of superpositions of the motion of the two oscillators. The fluctuations within a single oscillator are shown to increase when squeezing increases in the normal modes, generating thermal noise in individual mode subsystems.
New generalized squeezed states for the time-dependent harmonic oscillator are found through the theory of invariants. Our method gives a comparatively clearer picture than methods using evolution operators because we can establish a direct connection between the classical and quantum solutions. The additional significance of our method is that it is possible to find new generalized quantum squeezed states from just one particular solution to the classical time-dependent oscillator. Accordingly, more general results for the variance of x (squared uncertainty) are found for the cases of linear sweep of the restoring force and compared to recent results encountered in the literature.