Beyond Fermi's golden rule with the statistical Jacobi approximation

TL;DR

This paper derives an analytic expression for quantum fidelity decay after a quench in ergodic systems, extending the statistical Jacobi approximation's applicability beyond localized regimes and revealing deviations from Fermi's golden rule.

Contribution

It introduces a novel application of the statistical Jacobi approximation to well-thermalizing systems, providing a unified description of fidelity decay for various quench strengths.

Findings

The derived fidelity expression captures initial quadratic and long-time exponential decay.

For strong quenches, the decay rate differs from Fermi's golden rule predictions.

The statistical Jacobi approximation proves effective across different quantum dynamical regimes.

Abstract

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 statistical Jacobi approximation (SJA), which was originally applied in nearly localized systems, and which we here adapt to well-thermalizing systems. Our results demonstrate that the SJA is predictive in disparate regimes of quantum dynamics.