Quantum predator-prey cycles in dissipative Rydberg lattices

Abstract

The Lotka-Volterra model is a paradigm for self-organized predator-prey oscillations in far-from-equilibrium systems, yet testing it in real-world ecosystems is hindered by uncontrollable microscopic parameters. Here, we propose a quantum analogue of predator-prey dynamics using a tunable two-dimensional Rydberg atom array. Through mean-field analysis and numerical simulations based on the open-system discrete truncated Wigner approximation, we demonstrate that Rydberg excitations exhibit predator-prey cycles on microsecond timescales. We show that quantum coherence drives spontaneous symmetry breaking, while long-range interactions stabilize global oscillations against quantum-noise-induced desynchronization. We further reveal that quantum jump induce quasicycles whose amplitude scales inversely with the square root of the system size. Our work extends the study of predator-prey models to the quantum realm and advances quantum simulation stratagies that leverage engineered many-body nonequilibrium effects.

Figures (4)

• Figure 1: (a) The three-level scheme. The ground state $\ket{g}$ is coupled to the Rydberg states $\ket{s}$ and $\ket{r}$ by a laser field. Excited atoms in the state $\ket{s}$ ($\ket{r}$) decay to the ground state at rates $\Gamma_r$ ($\Gamma_s$). Mean-field analysis reveals (b) predator-prey dynamics with alternating dominance between $\ket{r}$ (preys) and $\ket{s}$ (predators) populations characterized with period $T \approx 2/\Gamma_r$ and relative time advance $dT_{rs}$ of peaks in $n_r$ (yellow dotted lines) over peaks in $n_s$ (purple dotted lines), manifesting a (c) limit cycle (LC) in the $n_s$-$n_r$ phase space with (d) three distinct dynamical phases emerging subsequently: In phase I, low excited populations favor $\ket{g}\to\ket{r}$ transition, while interaction-induced level shifts suppress $\ket{g}\to\ket{s}$ transition. In phase II, high $n_r$ enhances $\ket{g}\to\ket{s}$ transition and suppress $\ket{g}\to\ket{r}$ transition. In phase III, both excitation processes are blocked and the system returns to the low-excitation phase I.
• Figure 2: Mean-field analysis. (a) Real and (b) imaginary parts of the Jacobian eigenvalues $\lambda_l$. Quasicycles occur when the ratio (color-coded) $-\left\vert {\Im[\lambda_0]} \right\vert/\Re[\lambda_0]\gg 0$. In panel (b), the blue dashed lines mark a quasicycle at $\Delta_r=2.1$ with frequency $\approx 3.7$. Eigenvalues are conjugate pairs sorted by real parts in decending order. (c) Phase diagram in $\Omega$-$\Delta_r$ parameter space showing monostable stationary (STA), purely LC, STA/LC coexistence, and bistable STA phases. (d) Normalized time advance $t_{rs}=dT_{rs}/T$ of LC, where $dT_{rs}$ is the time advance of $n_r$ peaks relative to $n_s$ and $T$ is the period. Parameters for (a)(b): $\Omega=2$.
• Figure 3: Results from OSDTWA with all-to-all coupling on a 2D lattice. (a) Time series of average Rydberg population $n_s,n_r$ for both components, and the relative faction $f_{rs}=(n_r-n_s)/(n_r+n_s)$ for LC. Time is rescaled by the period $T$ measured from the time series, and the vertical dashed gray lines stand for half-period intervals, with the corresponding spatial profile of $f_{rs}^l$ shown in (b). $f_{rs}= 1$ stands for $n_s=0,n_r=1$, while $f_{rs}= -1$ corresponds to $n_s=1,n_r=0$. The auto-correlation function $G_{rr}(t)=\braket{\delta n_r(t')\delta n_r(t'+t)}_{t'}$, and the cross-correlation function $G_{rs}(t)=\braket{\delta n_r(t')\delta n_s(t'+t)}_{t'}$ with $\delta n_{r,s}=n_{r,s}-\braket{n_{r,s}}_t$ for $N=256$ for (c) LC and (d) quasicycle. The characteristic lifetime $\tau$ is obtained from fitting the peak-envelope to $A e^{-t/\tau}$ (gray dashed lines). The Fourier spectra for (e) LC and (f) quasicycles (rescaled by system size $N$). The intrinsic frequency is marked by gray dotted lines. Parameters for (a)-(c)(e): $\Delta_r=3$; parameters for (d)(f): $\Delta_r=2.1$. Correlation functions are averaged over trajectories of duration $\geq 10^4$ for well-converged statistics.
• Figure 4: Results from OSDTWA with vdW coupling on a 2D lattice of edge length $L$. (a) Time series of average Rydberg population $n_s$ and $n_r$ for a small system ($L=8$). Left: full time series, right: zoom-in view, with corresponding spatial profile of the relative fraction $f_{rs}^l$ shown in (c) and dashed lines marking the snapshots times. (b) Time series for a large system ($L=32$). Left: whole system, right: a subsystem with edge length $9$. (d) Fourier components of $G_{rr}$ (rescaled by system size $N=L^2$) for small and large systems. Dashed line indicates oscillation frequency. (e) Two-time auto-correlation $G_{rr}$ and cross-correlation $G_{rs}$. Parameters for (a)-(g): $\Delta_r=3$.