Galilean boost invariance does not survive the trace: symmetry breaking in open quantum systems

TL;DR

Tracing out a Galilean-invariant environment in open quantum systems breaks boost symmetry at the operator level, with implications for fluctuation-dissipation relations and potential control via parametric driving.

Contribution

The paper reveals how environment tracing breaks Galilean boost invariance in open quantum systems and analyzes the implications for fluctuation-dissipation and symmetry preservation.

Findings

Boost invariance is broken in the dissipative anticommutator term.

Fluctuation-dissipation theorem and boost covariance cannot hold simultaneously for non-trivial baths.

Parametric driving can suppress boost-breaking effects during a cycle.

Abstract

Tracing out a Galilean-invariant Caldeira-Leggett environment breaks Galilean boost covariance of the reduced dynamics, while spatial translations and rotations survive intact. An operator-level analysis of the exact Hu-Paz-Zhang master equation localizes the violation entirely in the dissipative anticommutator term, scaling with the damping coefficient $\Gamma(t)f(t)$. The fluctuation-dissipation theorem ties this coefficient to the absorptive bath response that drives equilibrium momentum diffusion, so for any non-trivial bath spectral density bilinear-coupled Galilean invariance, the fluctuation-dissipation theorem, and reduced boost covariance cannot hold simultaneously. The stochastic decomposition of the influence functional extends the mechanism beyond the quadratic regime. The dimensionless ratio $\hbar\gamma/k_\mathrm{B} T$ delineates the crossover: cold atoms in dissipative optical lattices and ultracold molecules sit at its edge. Parametric driving offers a one-directional escape: the squeezing rate that protects nonequilibrium entanglement above the standard quantum limit also suppresses boost-breaking over a driving cycle.