Full Counting Statistics and Phase Diagram of a Dissipative Rydberg Gas

Ultracold gases excited to strongly interacting Rydberg states are a promising system for quantum simulations of many-body systems. For off-resonant excitation of such systems in the dissipative regime, highly correlated many-body states exhibiting, among other characteristics, intermittency and multimodal counting distributions are expected to be created. Here we report on the realization of a dissipative gas of rubidium Rydberg atoms and on the measurement of its full counting statistics and phase diagram for both resonant and off-resonant excitation. We find strongly bimodal counting distributions in the off-resonant regime that are compatible with intermittency due to the coexistence of dynamical phases. Our results pave the way towards detailed studies of many-body effects in Rydberg gases.

Figure 1

(a) Schematic representation of the experimental procedure. (b)–(d) Resonant and off-resonant excitation processes for interacting Rydberg atoms. (vdW indicates the energy shift due to the van-der-Waals interaction). (e) Mean number of Rydberg excitations as a function of detuning. Here and throughout the Letter we omit the subscript “obs” indicating the observed quantities. The excitation durations are $1\mu \mathrm{s}$ (gray symbols, right-hand vertical axis) and $20\mu \mathrm{s}$ (black symbols, left-hand vertical axis). (f)–(h) Histograms of the counting distributions in the resonant and off-resonant regimes. The detunings are $\mathrm{\Delta }/2\pi =-3.5\mathrm{MHz}$ (f), $\mathrm{\Delta }/2\pi =0$ (g), and $\mathrm{\Delta }/2\pi =+3.5\mathrm{MHz}$ (h). The dashed vertical lines indicate the mean number of excitations. The Rabi frequency is $2\pi \times 400\mathrm{kHz}$, the interaction volume $10^{-7}{\mathrm{cm}}^3$, and the density $1.8\times 10^{11}{\mathrm{cm}}^{-3}$.

Figure 2

Experimental results showing the crossover to the dissipative regime in off-resonant excitation ($\mathrm{\Delta }=11.5\mathrm{MHz}$). As the duration of the excitation is increased from $20\mu \mathrm{s}$ (dotted line) to $950\mu \mathrm{s}$ (solid line), the counting distributions become more strongly bimodal. The inset shows the bimodality parameter as a function of the excitation duration.

Figure 3

(a) Off-resonant ($\mathrm{\Delta }/2\pi =11.5\mathrm{MHz}$) and (b) resonant counting distributions in the dissipative regime, for equal mean numbers 6 (dashed lines) and 23 (solid lines), respectively [indicated by the arrows in the inset of (a), which shows the mean number of excitations as a function of the Rabi frequency]. The dotted red lines are the results of the bimodal model in (a) and the perfectly monomodal model in (b), whereas the solid light gray lines show the distributions expected for Poissonian distribution with the same mean values. (c)–(e) The second, third, and fourth central moments of the off-resonant (blue) and on-resonant (gray) counting distributions. Panels (f)–(h) show the associated normalized quantities (see text). The lines in (c)–(h) are the results of the simple bimodal (dashed, with $x_1=1$ and $x_2=65$), Poissonian monomodal (dotted), and perfectly monomodal (dot-dashed) models. The detection efficiency of 40% was taken into account. For the bimodal model, the theoretical values were scaled by a factor of $\approx 0.5$ in order to facilitate a qualitative comparison with the experimental data. The interaction volume is $3.6\times 10^{-6}{\mathrm{cm}}^3$, the density $1\times 10^{11}{\mathrm{cm}}^{-3}$, and the excitation duration $950\mu \mathrm{s}$.

Figure 4

Phase diagram in the dissipative regime. Panel (a) shows the mean number of Rydberg excitations and (b) the Mandel $Q$ factor as a function of Rabi frequency and detuning. In (a) and (b) the red dashed lines indicate the transition to $Q>1$ and $Q>7$. The interaction volume is $3\times 10^{-7}{\mathrm{cm}}^3$ and the density $1.6\times 10^{11}{\mathrm{cm}}^{-3}$.

Figure 5

(a)–(c) Numerical simulations of the counting distributions for $\mathrm{\Delta }/2\pi =-5,0,5\mathrm{MHz}$, respectively. The finite detection efficiency is taken into account through a convolution with a binomial function. In (d) the mean values are plotted as a function of detuning. Comparison with a Lorentzian fit (dashed line) centered at $\mathrm{\Delta }=0$ highlights the asymmetry of the line shape (simulation parameters: Rabi frequency 130 kHz, excitation duration $4\mu \mathrm{s}$, density $9\times 10^{10}{\mathrm{cm}}^{-3}$, and interaction volume $8.9\times 10^{-8}{\mathrm{cm}}^3$). (e) Energy levels in the Dicke model of Rydberg excitations (for $\mathrm{\Delta }>0$). The laser excitation is blocked at the state $N_{\text{Ryd}}$ within a dark state superposition unbalanced towards the ground state (schematically represented here for the case $N_{\text{Ryd}}=6$). In the mean-field approximation, the resonance condition $W_{ss}^{(N_{\text{Ryd}}/2)+1}-W_{ss}^{(N_{\text{Ryd}}/2)-1}=\hslash \mathrm{\Delta }$ leads to several multiphoton resonances between the intermediate Dicke states. Those excitations produce multilambda-level configurations within the $0\to N_{\text{Ryd}}$ ladder.