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

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.