Exploring nonequilibrium phases of the generalized Dicke model with a trapped Rydberg-ion quantum simulator

Trapped ions are a versatile platform for the investigation of quantum many-body phenomena, in particular for the study of scenarios where long-range interactions are mediated by phonons. Recent experiments have shown that the trapped ion platform can be augmented by exciting high-lying Rydberg states. This introduces controllable state-dependent interactions that are independent of the phonon structure. However, the many-body physics in this newly accessible regime is largely unexplored. We show that this system grants access to generalized Dicke model physics, where dipolar interactions between ions in Rydberg states drastically alter the collective nonequilibrium behavior. We analyze and classify the emerging dynamical phases and identify a host of nonequilibrium signatures such as multiphase coexistence regions and phonon-lasing regimes. We moreover show how they can be detected and characterized through the fluorescence signal of scattered photons. Our study thus highlights interesting capabilities of trapped Rydberg-ion systems for creating and detecting quantum nonequilibrium phases.

Figure 1

Chain of trapped Rydberg ions. Each ion is modeled as an effective two-level system whose ground state, $\left|\downarrow \right.〉$, is coupled to a Rydberg excited state, $\left|\uparrow \right.〉$, by a laser with Rabi frequency $\mathrm{\Omega }$ and detuning $\mathrm{\Delta }$. The state $\left|\uparrow \right.〉$ spontaneously decays to $\left|\downarrow \right.〉$ with rate $\gamma$. Ions at sites $k$ and $p$ interact through the interaction potential $V_{kp}$ and are subject to a state-dependent trapping potential (with trapping frequency $\omega$ for $\left|\downarrow \right.〉$ and $\omega +{\omega }_a$ for $\left|\uparrow \right.〉$). The internal states of the ions are also coupled to the phononic degrees of freedom of the chain via a far-detuned standing-wave laser. Aspects of the dynamical behavior of the chain can be probed through the fluorescence signal of emitted photons as a function of time.

Figure 2

Interaction-induced (II) coexistence region. (a) Phase diagram in the $g-\gamma$ plane for $\mathcal{V}=5\mathrm{\Omega },\mathrm{\Delta }=0$, and ${\omega }_a=0$. Gray areas denote the coexistence regions [three real solutions to Eq. (3)], while the hatched area signals a PL regime. The II region originates as a consequence of interion interactions and is present for $\mathcal{V}>{\mathcal{V}}_{\mathrm{thr}}$, only. [(b), (d)] Examples of the two possible MF dynamics of $J_z\left(t\right)$ starting from different initial conditions for $\mathcal{V}=5\mathrm{\Omega },\mathrm{\Delta }=0,{\omega }_a=0,N=10$, and (b) $(g,\gamma )=(2.3,0.1)$ and (d) $(g,\gamma )=(1,0.1)$. Here, a PL solution (red curve), with small amplitude oscillations of $J_z\left(t\right)$ around a stationary value, emerges. Phase coexistence is present at small and intermediate times (see blue line and arrow). Inset of panel (d): Long-time limit-cycle dynamics of the c.m. mode position $X\left(t\right)$ and momentum $P\left(t\right)$, shown over a time window $\mathrm{\Delta }T=50{\mathrm{\Omega }}^{-1}$ for the lasing solution. [(c), (e)] Fluorescence photon count as a function of time in QJMC trajectories for $N=3$ ions and $N_{\text{ph}}=400$ with (e) and without (c) PL. Features of the MF solutions are clearly visible.

Figure 3

Effects of a finite detuning. Same as Fig. 2 for (a) $\mathrm{\Delta }=-0.1\mathrm{\Omega }$ and (b) $\mathrm{\Delta }=0.1\mathrm{\Omega }$. In the interaction-induced (II) region, a PL and a stable solution (SS) coexist. (c) Same as Fig. 2 for $\mathrm{\Delta }=0.1\mathrm{\Omega }$. (d) Fluorescence photon count as a function of time in a QJMC trajectory for $N=3$ ions in the II coexistence region with $N_{\mathrm{ph}}=400$.

Figure 4

Emergence of a multicoexistence regime for ${\omega }_a\ne 0$. (a) Same as Fig. 2 for $\mathcal{V}=10\mathrm{\Omega }$ and ${\omega }_a=2\mathrm{\Omega }$. The dark gray area $\mathrm{GDM}+\mathrm{II}$ denotes a multicoexistence region with five real solutions to Eq. (3), while in the interaction-induced (II) region PL and stable solutions coexist. Inset: Distributions of the rate $K$ of emitted photons over a time window of $\mathrm{\Delta }t={10}^3{\mathrm{\Omega }}^{-1}$ for the photon count records of panels (e) (top, yellow) and (c) (bottom, blue). The two histograms show qualitative differences, hinting at a different character of the stationary state (see text). [(b), (d)] Same as Fig. 2 for $\mathcal{V}=10\mathrm{\Omega },{\omega }_a=2\mathrm{\Omega }$, and (b) $(g,\gamma )=(2.5,0.1)$ and (d) $(g,\gamma )=(2,0.1)$. In both panels, two lasing solutions (red and blue lines) coexist, while in the $\mathrm{GDM}+$ II phase, an additional stable solution (green line) is present [panel (b)]. Insets: Long-time limit-cycle dynamics of the phononic position $X\left(t\right)$ and momentum $P\left(t\right)$ over a time window $\mathrm{\Delta }T=50{\mathrm{\Omega }}^{-1}$ for the two lasing solutions of the main panels. [(c), (e)] Fluorescence photon count as a function of time in QJMC trajectories inside the (e) II and (c) $\mathrm{GDM}+\mathrm{II}$ regions for $N_{\mathrm{ions}}=6$ and $N_{\mathrm{ph}}=100$.