How general are time-local master equations?

TL;DR

This paper demonstrates that time-local master equations can sometimes not uniquely determine quantum evolution, especially in non-invertible, non-Markovian cases, using a constructed Jaynes-Cummings model example.

Contribution

It provides an explicit example showing the limitations of time-local master equations in non-invertible, non-Markovian regimes, challenging common assumptions.

Findings

Time-local master equations may not uniquely determine evolution in certain non-invertible cases.

An explicit Jaynes-Cummings model example illustrates non-uniqueness.

Rapid Hamiltonian changes are necessary for the non-uniqueness to occur.

Abstract

Time-local master equations are more generally applicable than is often recognised, but at first sight it would seem that they can only safely be used in time intervals where the time evolution is invertible. Using the Jaynes-Cummings model, we here construct an explicit example where two different Hamiltonians, corresponding to two different non-invertible and non-Markovian time evolutions, will lead to arbitrarily similar time-local master equations. This illustrates how the time-local master equation on its own in this case does not uniquely determine the time evolution. The example is nevertheless artificial in the sense that a rapid change in (at least) one of the Hamiltonians is needed. The change must also occur at a very specific instance in time. If a Hamiltonian is known not to have such very specific behaviour, but is "physically well-behaved", then one may conjecture that a time-local master equation also determines the time evolution when it is not invertible.