Steady State Convergence Conditions for the Fourth Order Time-Convolutionless Master Equation

Accurately describing the long time behavior of open quantum systems is crucial for modelling how any real system is affected by its environment. Second order (in interaction strength, λ) master equations such as the Redfield equation are limited in their ability to describe reduced systems at long times, as they can only determine diagonal elements of the density matrix to order O(λ​​​​​​^​0). A fourth order master equation is necessary to determine all elements of the asymptotic state to the first nonzero order (λ²) precision. We establish a simplified form of the fourth order generator of the Non-Markovian time-convolutionless master equation (TCL4), which is then numerically implemented to efficiently find the steady state for an arbitrary finite system weakly coupled to a reservoir of linear oscillators at some finite temperature. With our optimal representation of the TCL4 generator, we investigate the requirements for return to equilibrium, and at zero temperature, approach to the global ground state. For systems such as the spin-boson model at zero temperature, where the Hamiltonian is known to not always admit a global ground state, we see that the TCL4 generator replicates this behavior, whereas the Redfield equation relaxes to a nonphysical ground state. We find that for the TCL4 master equation to have a convergent steady state, it must be asymptotically complete in the Hilbert Space.