The classical and quantum dynamics of the inhomogeneous Dicke model and its Ehrenfest time

TL;DR

This paper compares classical and quantum dynamics of the inhomogeneous Dicke model, revealing agreement in large N regimes and identifying an Ehrenfest time scale where quantum effects diverge.

Contribution

It provides an exact analysis of the inhomogeneous Dicke model's dynamics, introducing the concept of Ehrenfest time for the first time in this context.

Findings

Classical and quantum dynamics agree for large N in the few-excitation regime.

1/N-expansion reproduces quantum results for single excitation.

Ehrenfest time scale tau_E = sqrt{N<g^2>} marks divergence between classical and quantum solutions.

Abstract

We show that in the few-excitation regime the classical and quantum time-evolution of the inhomogeneous Dicke model for N two-level systems coupled to a single boson mode agree for N>>1. In the presence of a single excitation only, the leading term in an 1/N-expansion of the classical equations of motion reproduces the result of the Schroedinger equation. For a small number of excitations, the numerical solutions of the classical and quantum problems become equal for N sufficiently large. By solving the Schroedinger equation exactly for two excitations and a particular inhomogeneity we obtain 1/N-corrections which lead to a significant difference between the classical and quantum solutions at a new time scale which we identify as an Ehrenferst time, given by tau_E=sqrt{N<g^2>}, where sqrt{<g^2>} is an effective coupling strength between the two-level systems and the boson.