Visualizing the Quantum Interaction Picture in Phase Space

TL;DR

This paper demonstrates how the quantum Interaction Picture's evolution can be visualized as local and global transformations of the Wigner function in phase space, using a driven harmonic oscillator model.

Contribution

It provides a novel phase space visualization of the quantum Interaction Picture and derives exact time-evolution operators using the Magnus series.

Findings

Interaction Picture corresponds to local phase space transformations.

Exact time-evolution operator derived for a driven harmonic oscillator.

Phase space visualization clarifies quantum dynamics in the Interaction Picture.

Abstract

We illustrate the correspondence between the quantum Interaction Picture-evolution of the state of a quantum system in Hilbert space and a combination of local and global transformations of its Wigner function in phase space. To this aim, we consider the time-evolution of a quantized harmonic oscillator driven by both a linear and a quadratic (in terms of bosonic creation and annihilation operators) potentials and employ the Magnus series to derive the exact form of the time-evolution operator. In this case, the Interaction Picture corresponds to a local transformation of phase space-reference frame into the one that is co-moving with the Wigner function.