Invalidation of the Bloch-Redfield Equation in Sub-Ohmic Regime via a Practical Time-Convolutionless Fourth-Order Master Equation

TL;DR

This paper develops an optimized fourth-order time-convolutionless quantum master equation that accurately describes open quantum system dynamics in sub-Ohmic environments, revealing limitations of the Bloch-Redfield equation in such regimes.

Contribution

It introduces a computationally efficient fourth-order master equation that captures relaxation and dephasing, addressing divergence issues in sub-Ohmic environments and improving accuracy over second-order methods.

Findings

The new master equation accounts for relaxation and dephasing simultaneously.

It invalidates the Bloch-Redfield equation in sub-Ohmic regimes due to infrared divergence.

Ground-state approach is reliably computed to second order in bath coupling.

Abstract

Despite recent advances in quantum sciences, a quantum master equation that accurately and simply characterizes open quantum dynamics across extremely long timescales and in dispersive environments is still needed. In this study, we optimize the computation of the fourth-order time-convolutionless master equation to meet this need. Early versions of this master equation required computing a multidimensional integral, limiting its use. Our master equation accounts for simultaneous relaxation and dephasing, resulting in coefficients proportional to the system's spectral density over frequency derivative. In sub-Ohmic environments, this derivative induces infrared divergence in the master equation, invalidating the second-order Bloch-Redfield master equation findings. We analyze the approach to a ground state in a generic open quantum system and demonstrate that it is not reliably computed by the Bloch-Redfield equation alone. The optimized fourth-order equation shows that the ground-state approach is accurate to second order in bath coupling regardless of the dispersion, even though it can diverge in the fourth order at zero temperature.