Time-dependent quantum harmonic oscillator: a continuous route from adiabatic to sudden changes

TL;DR

This paper explores how the rate of frequency change in a quantum harmonic oscillator affects its quantum state, providing a continuous perspective from sudden to adiabatic transitions and clarifying related misconceptions.

Contribution

It introduces an analytical approximation linking squeezing to transition rate and clarifies the interpretation of the adiabatic theorem in this context.

Findings

The final state is a vacuum squeezed state characterized by Bogoliubov transformations.

The squeezing parameter's evolution depends on the transition rate.

An analytical expression relates squeezing to the transition speed and frequencies.

Abstract

In this work, we provide an answer to the question: how sudden or adiabatic is a change in the frequency of a quantum harmonic oscillator (HO)? To do this, we investigate the behavior of a HO, initially in its fundamental state, by making a frequency transition that we can control how fast it occurs. The resulting state of the system is shown to be a vacuum squeezed state in two bases related by Bogoliubov transformations. We characterize the time evolution of the squeezing parameter in both bases and discuss its relation with adiabaticity by changing the rate of the frequency transition from sudden to adiabatic. Finally, we obtain an analytical approximate expression that relates squeezing to the transition rate as well as the initial and final frequencies. Our results shed some light on subtleties and common inaccuracies in the literature related to the interpretation of the adiabatic theorem for this system.