Frequency Switching of Quantum Harmonic Oscillator with time-dependent frequency

An explicit solution of the equation for the classical harmonic oscillator with smooth switching of the frequency has been found . A detailed analysis of a quantum harmonic oscillator with such frequency has been done on the base of the method of linear invariants. It has been shown that such oscillator possesses cofluctuant states, different from widely studied Glauber’s coherent and ”ideal” squeezed states. Oscillator models are widely used in many branches of physics, such as quantum optics, atomic, molecular, and solid state physics. Small vibration of dynamic system can be describe in terms of harmonic oscillators in both quantum and classical mechanics. To include surrounding influences on the vibration, or to simulate the coupling of the vibration with other degree of freedom, one can consider time-dependent parameters specifying the Hamiltonian of a harmonic oscillator, for example, mass and frequency. Besides, nonstationary oscillator models show essentially nonclassical effects, such as squeezing and covariance of their quantum fluctuations. Some examples for such phenomena are: motion of one ion in Paul trap, which is precisely described by harmonic oscillator, with periodically time-dependent frequency [1], also Berry phase can be achieved when parameters of oscillator undergoes a cyclic change [2]-[7]. Agarwal and Kumar have shown that a nonstationary oscillator with linear sweep of the restoring force owns nonclassical states [8]. In the present Letter we study the switching of the frequency of a quantum nonstationary oscillator by using the method suggested in [11, 12]. The Hamiltonian of a harmonic oscillator is given by Ĥ = 1 2m p̂ + mΩ(t) 2 q̂, (1) where the constants m and Ω(t) are mass and the frequency of the quantum harmonic oscillator. The case M = M(t) can be reduced to the case M = m ( see for example [11], eq. 122 ). We recall the method of linear invariants, developed in series of papers [9]-[12], which we apply to one dimensional case: For each quantum system, described by a quadratic Hamiltonian, there is a classical twodimensional isotropic nonstationary harmonic oscillator with a Lagrangian (classical) L = m 2 (ǫ̇1 + ǫ̇ 2 2)− m 2 Ω(t)(ǫ1 + ǫ 2 2). (2) 1 and equations of motion d dt ∂L ∂ǫ̇k − ∂L ∂ǫk = 0, k = 1, 2. (3) As it is shown in [15], these two real equations are equivalent to one complex classical equation of nonstationary harmonic oscillator ǫ̈(t) + Ω(t)ǫ(t) = 0, (4) where complex function ǫ(t) = ǫ1(t) + iǫ2(t) completely describes the quantum evolution of the system [12], in particular with Hamiltonian (1). Despite of various investigations of nonstationary harmonic oscillator, smooth switching of the frequency for finite interval of time, was not presented in the literature. Here we study the behaviour of the nonstationary harmonic oscillator with varying frequency and constant mass for finite interval of time. We will show that the found classical solution for this case completely determines the quantum evolution of the corresponding quantum oscillator too. Let we consider switching of the frequency Ω(t) in the form;