Quadratic Time-dependent Quantum Harmonic Oscillator

TL;DR

This paper introduces a Lie algebraic method for solving a broad class of time-dependent quantum harmonic oscillators, providing analytic solutions and unifying various models including driven, parametric, and historically significant cases.

Contribution

The authors develop a unitary transformation approach that analytically solves general quadratic time-dependent quantum harmonic oscillators, including models previously solved only numerically.

Findings

Analytic solutions for periodically driven quantum harmonic oscillators without rotating wave approximation.

Unification of the Caldirola--Kanai oscillator with the Paul trap Hamiltonian via unitary transformations.

Enhanced understanding of dynamics in models with numerically unstable Schrödinger equations.

Abstract

We present a Lie algebraic approach to a Hamiltonian class covering driven, parametric quantum harmonic oscillators where the parameter set -- mass, frequency, driving strength, and parametric pumping -- is time-dependent. Our unitary-transformation-based approach provides a solution to our general quadratic time-dependent quantum harmonic model. As an example, we show an analytic solution to the periodically driven quantum harmonic oscillator without the rotating wave approximation; it works for any given detuning and coupling strength regime. For the sake of validation, we provide an analytic solution to the historical Caldirola--Kanai quantum harmonic oscillator and show that there exists a unitary transformation within our framework that takes a generalized version of it onto the Paul trap Hamiltonian. In addition, we show how our approach provides the dynamics of generalized models whose Schr\"odinger equation becomes numerically unstable in the laboratory frame.