# Critical phenomena and nonlinear dynamics in a spin ensemble strongly coupled to a cavity. II. Semiclassical-to-quantum boundary

## TL;DR

This paper investigates the quantum-to-classical transition in a driven-dissipative spin ensemble coupled to a cavity, revealing significant quantum effects near critical points even in large ensembles of up to 10,000 spins.

## Contribution

It introduces a generalized cumulant expansion method to analyze finite-size quantum corrections and the validity of semiclassical equations in a driven spin-cavity system.

## Key findings

- Quantum corrections remain significant near critical points.
- Semiclassical equations are invalid close to phase transition thresholds.
- Finite-size effects are crucial for understanding the quantum-classical boundary.

## Abstract

We numerically study the dynamics and stationary states of a spin ensemble strongly coupled to a single-mode resonator subjected to loss and external driving. Employing a generalized cumulant expansion approach we analyze finite-size corrections to a semiclassical description of amplitude bistability, which is a paradigm example of a driven-dissipative phase transition. Our theoretical model allows us to include inhomogeneous broadening of the spin ensemble and to capture in which way the quantum corrections approach the semiclassical limit for increasing ensemble size $N$. We set up a criterion for the validity of the Maxwell-Bloch equations and show that close to the critical point of amplitude bistability even very large spin ensembles consisting of up to $10^4$ spins feature significant deviations from the semiclassical theory.