One-Parameter Squeezed Gaussian States of Time-Dependent Harmonic Oscillator

We show that the nonlinear auxiliary equation for the invariant introduced by Lewis and Riesenfeld for a time-dependent harmonic oscillator is satisfied by the amplitude of a complex soltuion to the classical equation of motion. One-parameter squeezed Gaussian states are found whose parameter denotes the mixing of positive and negative frequency solutions. The minimization of the energy expectation value is shown to provide a criterion to select the vacuum state. PACS number(s): 42.50.D; 03.65.G; 03.65.-w Typeset using REVTEX 1 Harmonic oscillators have played many important roles in quantum physics, partly because they are exactly solvable quantum mechanically and partly because any system around an equilibrium can be approximated as a harmonic oscillator system. As a non-stationary system, a time-dependent quantum harmonic oscillator can also be exactly solved. One can find typical time-dependent harmonic oscillators in a system of harmonic oscillators interacting with an environment or evolving in an expanding universe. In the former case, the harmonic oscillator system depends on time through parametric coupling to the environment. In the latter case, for instance, a massive scalar field, as a collection of harmonic oscillators when appropriately decomposed into modes, gains time-dependence from a time-dependent spacetime background. As a method to find the exact quantum states of a time-dependent harmonic oscillator, Lewis and Riesenfeld [1] have introduced an invariant, quadratic in momentum and position, which satisfies the quantum Liouville-Neumann equation. The exact quantum states are given by the eigenstates of this invariant up to some time-dependent phase factors. Since then there have been many variants and applications of the invariants [2–5]. The coherent states are constructed [2] and shown to be squeezed states [3] of those states at an earlier time. In this paper, by finding a pair of first order invariants in terms of a complex solution to the classical equation of motion [5], we circumvent the task of solving a time-dependent nonlinear auxiliary equation in terms of which the quadratic invariant was expressed and show that the amplitude of the complex solution satisfies the auxiliary equation. Furthermore, by making use of these invariants we find one-parameter squeezed Gaussian states whose parameter denotes the mixing of positive and negative frequency solutions, an analog of quantum field theory. It is also found that the minimization of the energy expectation value gives a criterion to select the vacuum state. Let us consider a time-dependent harmonic oscillator studied in Refs. [1–3] of the form Ĥ = 1 2m0 p̂ + m0ω (t) 2 q̂. (1)