Squeezing an harmonic oscillator with a sawtooth pulse

Abstract

In recent years there has been a marked interest in quantum-system states called squeezed and correlated (in particular, electromagnetic-field modes), differently named in [1-5]. The history of the question and its present status are treated in various surveys and conference proceedings (we mention here only the latest ones [6, 7]). Squeezed states of an harmonic cavity oscillator (which include in the final analysis practically all the physical systems investigated in this connection) are the natural result of parametric excitation. The actual excitation mechanisms can vary. Quantum optics deals most frequently with various nonlinear interactions of several modes in crystals (see, e.g., the articles in [8-10]). Other possible mechanisms, for example, are variation of the boundary conditions in the cavities [11, 12] or collisions [13]. The main features of the phenomenon, however, are best understood and analyzed within the framework of a model of an oscillator with a specified time dependence of the natural frequency. The parametric quantum oscillator problem was first solved by Husimi [14] who devised Gaussian wave packets and an analog of Fock states for an oscillator with a time-dependent frequency, and also found the amplitudes of transitions between energy levels corresponding to a stationary oscillator (albeit only in series form). His results were obtained and generalized in [15-18] by a simpler procedure, based on the method of time dependent operator integrals of motion, an expanded exposition of which can be found in [19]. It turns out that all the quantities indicative of the behavior of a quantum oscillator (transition probabilities, variances, etc.) can be expressed in terms of the trajectory parameters of a corresponding classical oscillator. The explicit forms of classical trajectories are known, however, only for certain special time dependences of the oscillator frequency. It is easy to obtain solutions for a jump change of the frequency [20-22] and also for a delta-function pulsed frequency perturbation [23]. Exact solutions for a frequency that varies exponentially with time were obtained in [24]. Some other dependences of the frequency on the time, which admit of both exact and approximate solutions, were investigated in [6]. The aim of the present article is an examination of the case of a sawtooth time dependence of an oscillator frequency so as to obtain an exact solution. A sawtooth pulse simulates an experimental situation of interest and makes it possible to consider, in limiting cases, both a delta-function excitation and a slow variation of the frequency. For completeness, we shall repeat in the next section the approach based on the method of timedependent integrals of motion [15-19]. In Sec. 3 we solve the classical equations of motion for a parametric oscillator with a sawtooth time dependence of the frequency, in a form convenient for finding the transition amplitudes of a quantum oscillator. In Sec. 4 we find in explicit form, in the framework of the approach developed in [15:19], the transition amplitudes in the case of a sawtooth pulse. In Sec. 5. we investigate the excitation of an oscillator from an initial thermal-equilibrium state. The analysis of the results is presented in Sec. 6.