Beyond Fermi's golden rule with the Jacobi method

Many problems in quantum dynamics can be cast as the decay of a single quantum state into a continuum. The time dependent overlap with the initial state, called the fidelity, characterizes this decay. We derive an analytic expression for the fidelity after a quench to an ergodic Hamiltonian. The expression is valid for both weak and strong quenches, and timescales before finiteness of the Hilbert space limits the fidelity. It reproduces initial quadratic decay and asymptotic exponential decay with a rate which, for strong quenches, differs from Fermi's golden rule. The analysis relies on the Jacobi method, which was originally applied in nearly localized systems, and which we here adapt to well-thermalizing systems. Our results demonstrate that the Jacobi method is predictive in disparate regimes of quantum dynamics.