Squeezing and adiabaticity breaking in time-dependent quantum harmonic oscillators

TL;DR

This review unifies methods to analyze the dynamics of quantum harmonic oscillators with time-dependent frequencies, highlighting squeezing and adiabaticity breaking, with exact solutions for various protocols.

Contribution

It provides a comprehensive framework connecting invariant methods and squeezing formalism for understanding nonequilibrium quantum dynamics.

Findings

Exact solutions for sudden quenches and smooth ramps.

Connection between invariant methods and squeezing formalism.

Insights into adiabaticity breaking in driven quantum systems.

Abstract

The quantum harmonic oscillator with time-dependent frequency is a paradigmatic model of driven quantum dynamics and one of the few nontrivial systems that admits an exact analytical solution. In this review paper, we present a unified treatment of the time-dependent oscillator based on the Lewis-Riesenfeld invariant method, Bogoliubov transformations and the Ermakov-Pinney equation. We show how these approaches naturally connect to squeezing for the description of excitations production, and to the breakdown of adiabaticity under generic frequency protocols. Exact results for sudden quenches and smooth ramps are discussed in detail. By explicitly bridging invariant methods and squeezing formalism, this review is meant to provide a comprehensive framework for understanding nonequilibrium dynamics in quadratic potentials, with applications ranging from thermodynamics and condensed matter to quantum control theory.