Linear differential equation approach to the Loschmidt amplitude

The Loschmidt amplitude is a popular quantity that allows making predictions about the stability of quantum states under time evolution. In our work, we present an approach that allows us to find a differential equation that can be used to compute the Loschmidt amplitude. This approach, while in essence perturbative, has the advantage that it converges at finite order. We demonstrate that the approach for generically chosen matrix Hamiltonians often offers advantages over Taylor and cumulant expansions even when we truncate at finite order. We then apply the approach to two ordinary band Hamiltonians (multi-Weyl semimetals and AB bilayer graphene) to obtain the Loschmidt amplitude after a quench for an arbitrary starting state and find that the results readily generalize to find transmission amplitudes and specific contributions to the partition function, too. We then test our methods on many-body spin and fermionic Hamiltonians and find that while the approach still offers advantages, more care has to be taken than in a generic case. We also provide an estimate for a breakdown time of the approximation.