Quantum simulation of thermalization dynamics of a nonuniform Dicke model

Abstract

Previous experimental realizations of Dicke model in atomic or ionic systems are based on global observables assuming uniform spin-boson coupling, while inevitable experimental nonuniformity on the one hand requires site-resolved measurement of spin states, and on the other hand provides potential quantum advantage on the simulation of multi-spin distributions. Here we report the quantum simulation of a nonuniform Dicke-like model in a two-dimensional (2D) crystal of up to 200 ions. We explicitly demonstrate the sensitivity of few-spin observables and multi-spin distributions to the spatial inhomogeneity of the model, and examine the thermalization dynamics of the nonuniform model by measuring the subsystem entropies of selected ion groups. Our work enables the study of Dicke-like models beyond the symmetric subspace, paving the way toward understanding the role of disorder in its thermalization and quantum chaos behavior.

Figures (4)

• Figure 1: Experimental scheme. (a) Schematic of the experimental setup. We use counter-propagating $411\,$nm laser beams to couple the spin states of a two-dimensional crystal of up to $N=200$${}^{171}\mathrm{Yb}^+$ ions to a shared collective phonon mode. (b) Relevant energy levels. The qubits are encoded in the clock states of $S_{1/2}$ hyperfine levels, which can be controlled globally by resonant $12.642\,$GHz microwave. We further use counter-propagating global $411\,$nm laser beams to generate a spin-dependent force on the ions. We set two pairs of frequency components on the two sides of the $S$-$D$ transition to compensate the time-averaged AC Stark shift. (c) Experimental sequence. After laser cooling and optical pumping, we initialize all the spins by a global microwave SK1 composite pulse. Then we turn on the Dicke-like Hamiltonian $H_0$ for a controllable time $t$ with an SK1 spin echo inserted in the middle. Finally, we can use another SK1 composite pulse to rotate the spins globally into any desired basis (not shown), and measure their individual states by electron shelving. (d) and (e) We verify the simulated Dicke-like model by dynamics of average single-spin observables $\sigma_x$ (red), $\sigma_y$ (green) and $\sigma_z$ (blue) from the initial spin states of $N=60$ ions in (d) $|-x\rangle^{\otimes N}$ or (e) $|-y\rangle^{\otimes N}$. The dashed lines are numerical simulation results for a homogeneous model, and the solid lines are computed by dividing the spins into two subensembles to account for the nonuniformity of the spin-phonon coupling. Here we plot the cumulative time-averaged dynamics for robustness against slow drift in the experimental parameters.
• Figure 2: Thermalization dynamics of a Dicke-like model. We couple the spins dominantly to the center-of-mass phonon mode, and measure the time-averaged distributions of total spins $S_x$ (red), $S_y$ (green) and $S_z$ (blue) for different evolution times. (a)-(c) We initialize $N=10$ ions in $|+x\rangle^{\otimes N}$ and evolve the system for $t=1,\,2,\,4\,$ms. The bars represent the experimental data, the dashed lines are numerical simulation results for an ideal homogeneous mode, and the solid lines further consider the nonuniformity of the spin-phonon coupling. (d)-(f) Similar plots for $N=10$ ions initialized in $|+y\rangle^{\otimes N}$. (g) and (h) Similar experimental results for $N=60$ ions initialized in the $x$ and $y$ basis, respectively.
• Figure 3: Evolution of subsystem entropy. We couple $N=200$ spins dominantly to the center-of-mass phonon mode, and measure the dynamics of the subsystem entropy under (a, b) a weak or (c, d) a strong transverse field, starting from the initial spin state (a, c) $|+x\rangle^{\otimes N}$ or (b, d) $|+y\rangle^{\otimes N}$. Here we choose a typical group of four neighboring ions (see Supplementary Material) and measure the subsystem entropy $S_k$ for the $k=1,\,2,\,3,\,4$ spins assuming symmetric states as the circles connected by solid lines. We also compute the $k=2$ subsystem entropy by reconstructing the complete density matrices as the squares connected by the dashed lines. (e) Subsystem entropy $S_k$ versus subsystem size $k$ at the evolution time $t=4.5\,$ms in the weak-$B$ regime. (f) Similar plot in the strong-$B$ regime.
• Figure 4: Thermalization dynamics under nonuniform spin-phonon coupling. We couple $N=200$ spins dominantly to the third highest phonon mode, and similarly measure the dynamics of the subsystem entropy starting from the initial spin state (a) $|+x\rangle^{\otimes N}$ or (b) $|+y\rangle^{\otimes N}$. We observe faster dynamics for ions with a stronger spin-phonon coupling (blue dots for pair 1 and green dots for pair 3) than the ions with a weaker spin-phonon coupling (orange dots for pair 2 and red dots for pair 4). (c) Experimentally calibrated spin-phonon coupling strength for individual ions and the locations of the ion pairs used for entropy measurement.