The solutions of the generalized classical and quantum harmonic oscillators with time dependent mass, frequency, two-photon parameter and external force: The squeezing effects

Abstract

The exact analytical solutions of a generalized classical harmonic oscillator with time dependent mass, frequency, two-photon parameter and external forces are obtained. By using the invariance property of the scaled Wronskian, these solutions are used to obtain the solutions of the quantum mechanical counterpart of the oscillator under Heisenberg picture. In order to discuss the applications of these solutions of the quantum mechanical oscillator, we calculate the exact analytical expressions for the second order variances of both the canonically conjugate quadratures in terms of the time dependent mass, frequency and two-photon parameter. However, these variances calculated in terms of the initial coherent state do not depend on the time dependent driven terms. We argued that the time dependent frequency is on the way of the exact analytical solutions and hence it is kept constant throughout the investigation. We, however, discuss few situations of physical interest where the mass is varying in time. The special circumstance where all the parameters are time independent is used to discuss the squeezing effect in both the quadratures. It is seen that the parameter g involving the two-photon interaction term produces squeezing effects. With the increase of interaction time, the squeezing in both the quadrature components exhibit collapse and revival phenomena for g < ω (frequency of the oscillator). The squeezing of X-quadrature is completely ruled out for g > ω. However, the squeezing for P-quadrature is possible for small interaction time. The squeezing patterns of the X and P-quadratures are also discussed for pulsating mass, and for mass increasing with time. The squeezing is also discussed when the mass is increasing exponentially in time. It is envisage that the solutions could be used in the investigation of quantum statistical properties of the radiation field coupled to the said oscillator.