Bistability Versus Metastability in Driven Dissipative Rydberg Gases

We investigate the possibility of a bistable phase in an open many-body system. To this end, we discuss the microscopic dynamics of a continuously off-resonantly driven Rydberg lattice gas in the regime of strong decoherence. Our experimental results reveal a prolongation of the temporal correlations exceeding the lifetime of a single Rydberg excitation and show strong evidence for the formation of finite-sized Rydberg excitation clusters in the steady state. We simulate the dynamics of the system using a simplified and a full many-body rate-equation model. The results are compatible with the formation of metastable states associated with a bimodal counting distribution as well as dynamic hysteresis. However, a scaling analysis reveals that the correlation times remain finite for all relevant system parameters, which suggests the formation of many small Rydberg clusters and finite correlation lengths of Rydberg excitations. These results constitute strong evidence against the presence of a global bistable phase previously suggested to exist in this system.

Popular Summary

The quantum world behaves very differently from our everyday experience. Particles can exist in many states simultaneously; an electron, for example, can spin in more than one direction at once. This “superposition of states” gets lost on the way from the microscopic world to the macroscopic one. One way to handle this conceptual difficulty is to separate the problem into a quantum system and its environment, which are coupled to each other. The environment, however, can do more than destroy superpositions—entirely new phenomena may emerge. In particular, we are interested in whether two steady states can coexist in a system that is constantly interacting with its environment, a phenomenon known as bistability.

We investigated a sample of Rydberg atoms, where each atom can be either in the excited (or Rydberg) state or in the ground state. Because the Rydberg states have a finite lifetime, they continuously decay back to the ground state. Under off-resonant excitation, the interaction between the Rydberg states facilitates the excitation of a ground-state atom, provided there is already a Rydberg state in the vicinity, initiating an excitation cascade. We show experimentally and theoretically that in a small system, the metastable character leads to configurations where zero or many Rydberg states are excited. However, as we approach larger systems, we identify a global configuration with many Rydberg excitations, which is incompatible with a bistable state.

Our results solve a long-lasting debate and highlight new concepts for the emerging research field of open quantum systems. The understanding of phase transitions in open many-body systems could offer a new route to prepare and stabilize interesting phases of matter.

Figure 1

(a) Schematic sketch of the experimental measurement. Atoms confined in a 3D optical lattice (only one plane is shown) are continuously excited to the Rydberg state with Rabi frequency $\mathrm{\Omega }$ and detuning $\mathrm{\Delta }>0$. The Rydberg atoms are ionized at the rate ${\mathrm{\Gamma }}_{\mathrm{ion}}$, which is much smaller than the natural decay rate. Therefore, the ions serve as a probe of the excitation dynamics. The color scheme of the atoms is chosen according to the rates in (c): Rydberg excitations are red, ground-state atoms are gray, and atoms that can be facilitated are blue. (b) Pair-correlation function $g^{(2)}(\tau )$ deduced from the ion signal ($\mathrm{\Omega }=2\pi \times 77\mathrm{kHz}$, $\mathrm{\Delta }=2\pi \times 13\mathrm{MHz}$). The correlation function shows super-Poissonian fluctuations with large bunching amplitude $g^{(2)}(0)\gg 1$ and relaxation time ${\tau }^{(2)}\gg {\tau }_{\mathrm{sp}}$, where ${\tau }_{\mathrm{sp}}=20\mu \mathrm{s}$ is the natural lifetime of the excited Rydberg state $25{\mathrm{P}}_{1/2}$. Inset: Measured ion signal for single (blue) and averaged-over-40 (gray) experimental realizations. The shaded area around the gray line denotes Poissonian fluctuations. (c) Effective cluster model: A single seed excitation with rate $N_{\text{at}}{\mathrm{\Gamma }}_{\mathrm{seed}}$ initializes the growth of a cluster. While a facilitation rate ${\mathrm{\Gamma }}_{\uparrow }\gg {\mathrm{\Gamma }}_{\mathrm{seed}}$ leads to an increase of cluster size $m$, the spontaneous decay rate ${\mathrm{\Gamma }}_{\mathrm{sp}}$ limits the size of a cluster.

Figure 2

Generic spectrum of relaxation (damping) rates as a function of system size $L$ for (a) a metastable system and (b) a truly bistable system.

Figure 3

Experimentally measured correlation time ${\tau }^{(2)}={\tau }_{\mathrm{cl}}$ (left scale), corresponding to the lifetime of a cluster in the full antiblockade regime $\mathrm{\Delta }>0$ for different Rabi frequencies $\mathrm{\Omega }$. The cluster lifetime increases with decreasing detuning for fixed Rabi frequency. Using the cluster model, we extract a typical cluster size $\overline{m}$ (right bar). The gray dashed line indicates the spontaneous decay time of a single atom ${\tau }_{\mathrm{sp}}=20\mu \mathrm{s}$. Inset: The bunching amplitude $g^{(2)}(0)-1$ shows peaks at detunings, which correspond to a cluster size of approximately two. Error bars correspond to the statistical uncertainty from fitting an exponential decay to the experimental retrieved correlation function. For simplicity, they are not shown in the inset.

Figure 4

Number of clusters $N_{\mathrm{cl}}$ calculated from the experimental data using Eq. (15) for the different detunings and Rabi frequencies. For decreasing detuning, the number of clusters increases steadily, exceeding the size of a cluster by roughly 1 order of magnitude. The color code corresponds to the same Rabi frequencies as in Fig. 3. The error bars correspond to the statistical uncertainties from fitting the cluster lifetime ${\tau }_{\mathrm{cl}}$ as well as the multiparticle seed rate $N_{\text{at}}{\mathrm{\Gamma }}_{\mathrm{seed}}$.

Figure 5

Comparison between the mean excitation fraction ${\rho }_e$ extracted from the maximal detected ion rate $R_{\mathrm{ion}}(t)$ and the excitation fraction ${\rho }_{\mathrm{cl}}=N_{\mathrm{cl}}\overline{m}/N_{\text{at}}$ calculated using the effective cluster model. The color code corresponds to the same Rabi frequencies as in Fig. 3. The gray dashed line is a linear curve with a slope of 1.

Figure 6

Finite-size extrapolation of the cluster lifetime ${\tau }_{\mathrm{cl}}$ for Rabi frequency $\mathrm{\Omega }=2\pi \times 160\mathrm{kHz}$ and different detuning $\mathrm{\Delta }$. The dashed lines indicate an exponential saturation fit of about $[1-c\exp (-L/L_0)]$ with a constant $c$. Here, the characteristic length scale $L_0=6.2,3.2$, and 2.1 is comparable to the cluster size $\overline{m}$.

Figure 7

(a) Excitation number probability $p_{N_{\mathrm{Ry}}}$ for $\mathrm{\Omega }=2\pi \times 185\mathrm{kHz}$ and $\mathrm{\Delta }=2\pi \times 15\mathrm{MHz}$ for three different linear system sizes $L$ after 1-ms propagation. For intermediate system size ($L=6$), the distribution shows a bimodal behavior. (b) Hysteresis scan in the detuning parameter $\mathrm{\Delta }$ for $\mathrm{\Omega }=2\pi \times 160\mathrm{kHz}$ with linear system size $L=10$. The sweep times in the forward and backward directions are $\tau =0.88\mathrm{ms}$ and $\tau =88\mathrm{ms}$. In the case of a large sweep time $\tau =88\mathrm{ms}$, the dynamic hysteresis disappears. The mean Rydberg excitation number in the backward and forward scans are averaged over 500 trajectories. The standard deviation is indicated by the shaded area.

Figure 8

Numerical results of the cluster lifetime ${\tau }_{\mathrm{cl}}$ and the bunching amplitude $g^{(2)}(0)-1$ (inset) using a full many-body rate-equation model with parameters estimated from the experiment. The color code corresponds to the same Rabi frequencies as in Fig. 3. The system contains $N_{\text{at}}=1000\mathrm{atoms}$. We use an interaction potential $V(r)=C_9/r^9$ with $C_9=2\pi \times 2.1\mathrm{kHz}\mu {\mathrm{m}}^9$. Other parameters are ${\mathrm{\Gamma }}_{\mathrm{ion}}=1\mathrm{kHz}$, ${\gamma }_0=300\mathrm{kHz}$, ${\mathrm{\Gamma }}_n=1\mathrm{kHz}$, and $\sigma =0.06\mu \mathrm{m}$. The gray shaded area corresponds to the area where the decoherence rate ${\gamma }_{\mathrm{inh}}$ is not a good approximation anymore (see Appendix pp4).

Figure 9

We retrieve the seed rate ${\mathrm{\Gamma }}_{\mathrm{seed}}$ by fitting an exponential decay $\exp (-{\mathrm{\Gamma }}_{\mathrm{seed}}t)$ to the probability of detecting a first ion after the time of flight $t$ to our ion detector (inset). From a $\sim {\mathrm{\Delta }}^2$ fit to the extracted seed rates, we can calculate the Rabi frequency $\mathrm{\Omega }$, taking into account the finite detection and ionization efficiency, as well as the bare decoherence rate.

Figure 10

(a) Different configurations during the time evolution of a single cluster starting from an initial seed excitation. In the last configuration, the two excited atoms prevent the excitation of the atom in the middle. A cluster, which is split apart, cannot merge. (b) The configuration space of the cluster dynamics is spanned by the cluster size $m$ and the number of splittings $n$.

Figure 11

Numerical simulation of the dynamics using a many-body rate-equation method in a large 1D lattice system with size $L\gg 100$ for different detunings $\mathrm{\Delta }$ over the line width $\omega ={\gamma }_0\sqrt{{\mathrm{\Omega }}^2/{\gamma }_0{\mathrm{\Gamma }}_{\mathrm{sp}}+1}$ and fixed facilitation radius $r_{\mathrm{fac}}=a$. The decoherence rate is ${\gamma }_0/\mathrm{\Omega }=2.05$ and decay rate is ${\mathrm{\Gamma }}_{\mathrm{sp}}/\mathrm{\Omega }=0.1$. (a) Cluster lifetime ${\tau }_{\mathrm{cl}}$ and (b) bunching parameter $g^{(2)}(0)-1$. The dashed lines are the results from the cluster model using $z_{\mathrm{eff}}\simeq 2+(\mathrm{\Omega }/{\mathrm{\Gamma }}_{\mathrm{sp}}{)}^2/8$ as the effective coordination number.

Figure 12

Interaction potentials of the pair state $|25{\mathrm{P}}_{1/2},25{\mathrm{P}}_{1/2}⟩$ obtained from a diagonalization of the interaction Hamiltonian including up to quadrupole-quadrupole interactions. The different potentials correspond to different superpositions of the $m_j$ states. The green dashed line is a fit to the only repulsive potential with $V(r)=C_9/r^9$ and $C_9=2\pi \times 2.1\mathrm{kHz}\mu {\mathrm{m}}^9$.

Figure 13

Excitation rate ${\mathrm{\Gamma }}_{\mathrm{ex}}/{\mathrm{\Gamma }}_{\mathrm{sp}}$ for three different interparticle distances $r_0$ using the parameters from the experiment and $\mathrm{\Omega }/2\pi =500\mathrm{kHz}$. The excitation rate for $r_0=2a_x$ already agrees well with the uncorrelated excitation rate $r_0\to \infty$. The inset shows the same rates with linear scale.

Figure 14

Comparison of the bare excitation rate ${\mathrm{\Gamma }}_{\mathrm{ex}}^0/{\mathrm{\Gamma }}_{\mathrm{sp}}$ without an additional broadening mechanism, the full excitation rate ${\mathrm{\Gamma }}_{\mathrm{ex}}/{\mathrm{\Gamma }}_{\mathrm{sp}}$, Eq. (d3), and the approximate excitation rate ${\mathrm{\Gamma }}_{\mathrm{ex}}^{\mathrm{L}}/{\mathrm{\Gamma }}_{\mathrm{sp}}$. Here, we use the parameters from the experiment; the interparticle distance is $r_0=a_x$, and Rabi frequency $\mathrm{\Omega }/2\pi =500\mathrm{kHz}$. The inset shows the same rates with linear scale.

Figure 15

Mismatch between the equilibrium Rydberg excitation number ${\overline{N}}_{\mathrm{Ry}}$ and the Rydberg excitation number $N_{\mathrm{Ry}}({\tau }_{\text{sweep}})$ after a finite sweep time ${\tau }_{\text{sweep}}$. We use the same parameters as in Fig. 7. The hysteresis scan starts at $\mathrm{\Delta }/2\pi =40\mathrm{MHz}$ and stops at $\mathrm{\Delta }/2\pi =10\mathrm{MHz}$. The dashed line indicates a scaling $\propto {\tau }_{\text{sweep}}^{-1}$.