Asymptotic expansion of the solution of the master equation and its application to the speed limit

TL;DR

This paper develops an asymptotic expansion method for the master equation's solution under control modulation, demonstrating its convergence and application to heat, activity, and speed limits in quantum systems.

Contribution

It introduces an asymptotic series expansion of the master equation solution, analyzes its convergence, and applies it to derive heat, activity, and speed limit relations in two-level quantum systems.

Findings

Borel summation matches the exact solution in the relaxation time approximation.

The series expansion provides an analytic expression for heat and activity.

The speed limit relation holds at lowest order of energy modulation frequency.

Abstract

We investigate an asymptotic expansion of the solution of the master equation under the modulation of control parameters. In this case, the non-decaying part of the solution becomes the dynamical steady state expressed as an infinite series using the pseudo-inverse of the Liouvillian, whose convergence is not granted in general. We demonstrate that for the relaxation time approximation model, the Borel summation of the infinite series is compatible with the exact solution. By exploiting the series expansion, we obtain the analytic expression of the heat and the activity. In the two-level system coupled to a single bath, under the linear modulation of the energy as a function of time, we demonstrate that the infinite series expression is the asymptotic expansion of the exact solution. The equality of a trade-off relation between the speed of the state transformation and the entropy production (Shiraishi, Funo, and Saito, Phys. Rev. Lett. ${\bf 121}$, 070601 (2018)) holds in the lowest order of the frequency of the energy modulation in the two-level system. To obtain this result, the heat emission and absorption at edges (the initial and end times) or the differences of the Shannon entropy between the instantaneous steady state and the dynamical steady state at edges are essential: If we ignore these effects, the trade-off relation can be violated.