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Physical Review Letters

Exact quantum-statistical dynamics of an oscillator with time-dependent frequency and generation of nonclassical states

G. S. Agarwal, Satish Kumar

Published Dec 23, 1991• 151 citations
Publisher

Abstract

Exact results on the time evolution of the density matrix and the Wigner function for an oscillator with time-dependent frequency are given. Explicit time dependence is given for the case of linear sweep of the restoring force. Such a sweep is shown to generate states of the oscillator with remarkable nonclassical properties such as squeezing and sub-Poissonian character.

DOI
10.1103/physrevlett.67.3665

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Exact results on the time evolution of the density matrix and the Wigner function for an oscillator with time-dependent frequency are given. Explicit time dependence is given for the case of linear sweep of the restoring force. Such a sweep is shown to generate states of the oscillator with remarkable nonclassical properties such as squeezing and sub-Poissonian character.
A quantum system described by a two-dimensional harmonic oscillator with a varying mass parameter plus the projection of the angular momentum in the z direction with a time-dependent mode coupling constant is considered to study the two-mode squeezing and photon generation phenomena. For some specific time dependences of the oscillator mass and an arbitrary function of time for the coupling constant exact results on the time evolution of the density matrix and the Wigner function are presented. An analytical expression of the photon distribution is obtained explicitly in terms of Hermite polynomials of two variables, and it shows the effect of oscillations characteristic of squeezing and correlation of the photon quadratures.
We study the phenomena at the overlap of quantum chaos and nonclassical statistics for the time-dependent model of nonlinear oscillator. It is shown in the framework of Mandel Q parameter and Wigner function that the statistics of oscillatory excitation numbers is drastically changed in the order-to-chaos transition. The essential improvement of sub-Poissonian statistics in comparison with an analogous one for the standard model of driven anharmonic oscillator is observed for the regular operational regime. It is shown that in the chaotic regime, the system exhibits the range of sub-Poissonian and super-Poissonian statistics which alternate one to other depending on time intervals. Unusual dependence of the variance of oscillatory number on the external noise level for the chaotic dynamics is observed. The scaling invariance of the quantum statistics is demonstrated and its relation to dissipation and decoherence is studied.
Photon distribution function, means and dispersions are found explicitly for the nonclassical state of light which is created from the photon--added coherent state ∣α,m⟩\vert α,m \rangle∣α,m⟩ due to a time--dependence of the frequency of the electromagnetic field oscillator. Generating function for factorial momenta is obtained. The Wigner function and Q--function are constructed explicitly for the excited photon--added coherent state of light. Influence of added photons on known oscillations of photon distribution function for squeezed light is demonstrated. oscillator are considered.
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.