Explicit Computation of the Quantum Phase of the Time Perturbed Harmonic Oscillator

Abstract

We consider a simple harmonic oscillator subject to a time-dependent, spatially homogeneous perturbation and show that it is canonically equivalent to a non-perturbed oscillator. The canonical transformation that relates both systems can be implemented as a unitary transformation mapping the perturbed system and the non-perturbed systems onto each other. This unitary transformation allows us to explicitly compute the time dependence of the states of the perturbed system, and in particular to compute the phase that affects all states. The phase turns out to be a classical action along the classical trajectory of the origin (in phase space), along its motion under the action of perturbation. In this simple system, the transition probabilities due to the perturbation can also be computed explicitly. Our approach is independent of the magnitude of the perturbation, and does not require an adiabaticity assumption.