Modifying the Time-Convolutionless Master Equation via the Moore-Penrose Pseudoinverse

TL;DR

This paper explores modifying the time-convolutionless master equation by using the Moore-Penrose pseudoinverse to improve its robustness, but finds that the modification introduces convergence issues and fails to accurately capture dynamics.

Contribution

It introduces a novel modification of the TCL master equation using the Moore-Penrose inverse and analyzes its limitations through specific quantum models.

Findings

Modified equation scales exponentially with bath dimension

Fails to match exact dynamics in tested models

Loss of convergence in perturbative expansion

Abstract

We attempt to modify the time-convolutionless master equation (TCL-ME) to be more resistant to breakdown. We remove the standard assumption that a portion of the generator is invertible by instead taking the Moore-Penrose inverse. We rederive the perturbative expansion using Israel and Charnes' result, and test the equation up to sixth and fifth orders on the Jaynes-Cummings and Ising models, respectively. We find that in both cases, the modified equation fails to capture the dynamics of the exact solution compared to the standard TCL due to the terms of the modified equation scaling exponentially with the dimension of the bath, and connect this failure to a loss of convergence of the perturbative expansion.