Interacting spins in a cavity: Finite-size effects and symmetry-breaking dynamics

We calculate the ground state and simulate the dynamics of a finite chain of spins with Ising nearest-neighbor interactions and a Dicke collective spin interaction with a single-mode cavity field. We recover the signatures of first- and second-order phase transitions predicted by mean-field theory, and for small chains, we find significant and nontrivial finite-size effects. Below the first-order phase transition, even quite large spin chains of 30–40 spins give rise to a mean photon number and number fluctuations significantly above the mean-field vacuum result. Near the second-order phase critical point, our calculations reveal photon number fluctuations that grow beyond Poisson statistics with the size of the spin chain. We simulate the stochastic evolution of the system when the cavity output field is subject to homodyne detection. For an initial state close to the first-order phase-transition the random character of the measurement process causes a measurement-induced symmetry breaking in the system. This symmetry breaking occurs on the time scale needed for an observer to gather sufficient information to distinguish between the two possible (mean-field) symmetry-broken states.

Figure 1

(a) Expected photon density $\langle n\rangle /L$, (b) energy gap to the excited state, (c) energy gap to the second excited state, and (d) the correlation length for ${\sigma }^y$. Values are for 32 spins with $\omega =1$ and $h+J+g=1$.

Figure 2

Photon number (a) mean and (b) variance as functions of $h$ for $\omega =1$, $g=0.4$, and $J=0.1$. Note the peak in $\mathrm{Var}n/n$ indicating a super-Poissonian distribution at the phase transition. As a function of $L$ the peak height scales as $\mathrm{Var}n/n\propto L^{0.19}$, and the peak location scales as $\left|h-h_c\right|\propto L^{-0.5}$, where $h_c$ is the location of the critical point.

Figure 3

Parameters are $\omega =1$, $g=0.49$, and $J=0.3$. (a) The mean photon number as a function of $h$ near the first-order transition. (b) The photon number variance in the region from $h=0.1$ to $h=0.25$. The dashed curve in (a) shows the mean photon number for a low, but finite, temperature.

Figure 4

Time evolution of a system being monitored by homodyne detection. (a) The conditioned expectation value of the $a+a^{†}$ quadrature of the cavity field. The dashed lines are the values of $q$ for the two symmetry-broken states in the thermodynamic limit. The thick line is the ensemble average of $q$. (b) The thick line is the von Neumann entropy of the average time-evolved oscillator state, and the thin lines are the entropy of the oscillator state in a few sample trajectories. (c) The expectation value of the symmetry operator $P$ for individual trajectories (thin lines) and the ensemble average (thick line).