Genuine Quantum effects in Dicke-type Models at large atom numbers

TL;DR

The paper shows that genuine quantum effects can persist in Dicke-type models at large but finite $N$, using the nuHOPS method to obtain numerically exact dynamics up to $N\approx 1000$. In the balanced model, dynamics converge to mean-field predictions for large $N$, while in the unbalanced model, finite-size tunneling between normal and superradiant states leads to steady-state mixtures and a shifted quantum phase transition. The study reveals noncommutativity between the steady-state limit and the thermodynamic limit, with substantial implications for understanding quantum-to-classical crossovers in driven-dissipative many-body systems and for guiding future experimental explorations of mesoscopic quantum effects in cavity QED.

Abstract

We investigate the occurrence of genuine quantum effects and beyond mean-field physics in the balanced and unbalanced open Dicke model with a large, yet finite number of atoms $N$. Such driven and dissipative quantum many-body systems have recently been realized in experiments involving ultracold gases inside optical cavities and are known to obey mean-field predictions in the thermodynamic limit $N\to\infty$. Here we show quantum effects that survive for large but finite $N$, by employing a novel open-system dynamics method that allows us to obtain numerically exact quantum dynamical results for atom numbers up to a mesoscopic $N\approx 1000$. While we find that beyond-mean-field effects vanish quickly with increasing $N$ in the balanced Dicke model, we are able to identify parameter regimes in the unbalanced Dicke model that allow genuine quantum effects to persist even for mesoscopic $N$. They manifest themselves in a strong squeezing of the steady state and a modification of the steady-state phase diagram that cannot be seen in a mean-field description. This is due to the fact that the steady-state limit $t\rightarrow \infty$ and thermodynamic limit $N\rightarrow \infty$ do not commute.

Figures (4)

• Figure 1: Numerically exact dynamics of the spin state in the (balanced) open Dicke model. Plots (a1) to (d1) show the quantum dynamics of the spin expectation values on a generalized Bloch sphere (thick blue line) compared to the dynamics obtained from the mean-field equations \ref{['eq:meanfieldDicke']} (black line). Plots (a1) to (c1) show the dynamics for $N = 100, 500, 1000$ respectively with the initial state $\ket{\Psi (0)} = \ket{\theta=\pi/4, \phi=\pi}\otimes\ket{0}$, where $\ket{\theta, \phi}$ denotes a spin coherent state. (d1) shows the dynamics for the initial state $\ket{\Psi (0)} = \ket{\theta=\pi, \phi=0}\otimes\ket{0}$ and $N=1000$, corresponding to a quench from the free spin and field Hamiltonian. Selected trajectories are shown as thin purple lines and the mean-field steady-state predictions are marked by orange crosses. Curves inside or on the backside of the sphere are shown increasingly transparent. Plots (a2) - (d2) show the spin-Q function $Q(\theta, \varphi) = \bra{\theta, \varphi}\rho\ket{\theta, \varphi}$ at a fixed time $t_1 = 7.5/\omega_a$, marked by a green dot in the upper plots.
• Figure 2: Time evolution of the covariance between the atoms and the field mode for different $N$ (solid) as well as the result of a second order cumulant expansion for $N=500$ (black dashed). For the inset we divide the $N$ spins in two groups of equal size and quantify the entanglement between the two groups with the help of the negativity of the partial transpose. This supports the view that the steady state correlations are of classical origin and true quantum correlations are only present during transient dynamics.
• Figure 3: Finite size effects on the bistable phase of the unbalanced Dicke model. (a) Example of a spin-Q function for $N=500$ in the bistable phase, showing the possibility of quantum tunneling. (b) Occupation of the superradiant state after starting in the normal state and evolving for $\omega_a t = 50$ for different parameters and $N=500$. Dashed lines show the mean-field boundaries of the superradiant (SR), bistable (BI) and normal (NP) phases. Inset shows a fit of the rate exponents $\gamma_i \propto \exp\left( r_i N \right)$ for $i\in \{\mathrm{ns},\mathrm{sn}\}$ and their intersection which marks the sharp transition in the quantum phase diagram and is shown as a green cross. (c)/(d) Occupation of the superradiant/normal state after starting in the normal/superradiant state and evolving for $\omega_a t= 150$ for different $N$ along a cut through the phase diagram. Color of the data points encodes the corresponding number of atoms and shape encodes the position along the cut. Colored dashed lines are obtained from a cubic spline fit and serve merely as a guide to the eye. Black dashed line corresponds to the mean-field prediction. Insets show the same data points as a plot with respect to $N$ together with the fit of the rate exponents $r_i$.
• Figure S4: Example data for how the tunneling rates were obtained for the parameters $g_- = 1.8$, $g_+ = 0.782$ and $\omega_a = \omega_c = \kappa = 1$, corresponding to the data points at $s=0.69$ in Fig. \ref{['fig:pdGenDicke']} (marked by circles, close to the crossing of the exponents). The upper left figure shows some individual trajectories obtained from a simulation with $N=100$ atoms starting in the normal state. The dashed lines mark the steady state expectation values in the superradiant phase. While most trajectories remain in the normal phase some tunnel into the superradiant phase marked by the blue background. The upper right figure shows trajectories starting in the superradiant state, with tunneling events to the normal phase marked in blue. The lower figure shows the probability to find the state in the initially unoccupied phase over time for different $N$ and both initial states. A fit to the solution of the rate equation \ref{['eq:app_rate_eq']} shows very good agreement and is used to extract the tunneling rates from the numerical data. Note that for this combination of $g_+$, $g_-$ no tunneling to the superradiant state was observed for $N=300$.