Quantum simulation of the microscopic to macroscopic crossover using superconducting quantum impurities

Despite being a pillar of quantum mechanics, little attention has been paid to the onset of the Fermi golden rule as a discrete microscopic bath of modes approaches the macroscopic thermodynamic limit and forms a continuum. Motivated by recent experiments in circuit quantum electrodynamics, we tackle this question through the lens of single-photon decay in a finite transmission line coupled to a qubit (``quantum impurity''). We consider a single-photon state, coupled via the nonlinear impurity to several baths formed by multiphoton states with different number of photons, which are inherently discrete due to the finite length of the line. We focus on the late-time dynamics of the single-photon, and uncover the conditions under which the photon decoherence rate approaches the decay rate predicted by the Fermi golden rule. We show that it is necessary to keep a small but finite escape rate (unrelated to the impurity) for each single-photon mode to obtain a finite long-time inelastic decay rate. We analyze the contribution of the baths formed by many-body states with different number of photons, and illustrate how the decay rate induced by some bath of $n$ photon states is enhanced by the presence of other baths of $m\ensuremath{\ne}n$ photon states, highlighting the contribution of cascade photon decay processes. Our formalism could be used to analyze recent experiments in superconducting circuits.