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

TL;DR

This paper investigates how a finite quantum bath transitions to a continuum, analyzing single-photon decay in superconducting circuits and identifying conditions for Fermi's golden rule to hold in the microscopic to macroscopic crossover.

Contribution

It introduces a formalism for understanding photon decay in finite quantum baths, bridging microscopic discrete modes and macroscopic continuum behavior, relevant for superconducting circuit experiments.

Findings

Decay rate approaches Fermi's golden rule with finite escape rate.

Bath interactions enhance decay rates through cascade processes.

Formalism applicable to recent superconducting circuit experiments.

Abstract

Despite being a pillar of quantum mechanics, little attention has been paid to the onset of Fermi's 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 multi-photon states with different number of photons, which are inherently discrete due to the finite size of the line. We focus on the late-time dynamics of the single-photon, and uncover the conditions under which the photon's decoherence rate approaches the decay rate predicted by Fermi's 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 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 $m \neq n$ photon states, highlighting the contribution of cascade photon decay processes. Our formalism could be used to analyze recent experiments in superconducting circuits.