Quantum speed of evolution in a Markovian bosonic environment

TL;DR

This paper evaluates quantum speed limits for a bosonic mode in a Markovian environment, providing explicit formulas and comparing different indicators to determine the tightest bounds for state evolution times.

Contribution

It derives explicit quantum speed limit formulas for a cavity mode coupled to a thermal reservoir, comparing fidelity and Hilbert-Schmidt metrics for the first time in this setting.

Findings

Hilbert-Schmidt metric yields a tighter speed limit than fidelity.

Explicit formulas for speed limits for coherent and Fock states.

Analysis of damping effects on quantum states using derived bounds.

Abstract

We present explicit evaluations of quantum speed limit times pertinent to the Markovian dynamics of an open continuous-variable system. Specifically, we consider the standard setting of a cavity mode of the quantum radiation field weakly coupled to a thermal bosonic reservoir. The evolution of the field state is ruled by the quantum optical master equation, which is known to have an exact analytic solution. Starting from a pure input state, we employ two indicators of how different the initial and evolved states are, namely, the fidelity of evolution and the Hilbert-Schmidt distance of evolution. The former was introduced by del Campo {\em et al.} who derived a time-independent speed limit for the evolution of a Markovian open system. We evaluate it for this field-reservoir setting, with an arbitrary input pure state of the field mode. The resultant formula is then specialized to the coherent and Fock states. On the other hand, we exploit an alternative approach that employs both indicators of evolution mentioned above. Their rates of change have the same upper bound, and consequently provide a unique time-dependent quantum speed limit. It turns out that the associate quantum speed limit time built with the Hilbert-Schmidt metric is tighter than the fidelity-based one. As apposite applications, we investigate the damping of the coherent and Fock states by using the characteristic functions of the corresponding evolved states. General expressions of both the fidelity and the Hilbert-Schmidt distance of evolution are obtained and analyzed for these two classes of input states. In the case of a coherent state, we derive accurate formulas for their common speed limit and the pair of associate limit times.