Nonequilibrium Phase Transition in a Dilute Rydberg Ensemble

We demonstrate a nonequilibrium phase transition in a dilute thermal atomic gas. The phase transition, between states of low and high Rydberg occupancy, is induced by resonant dipole-dipole interactions between Rydberg atoms. The gas can be considered as dilute as the atoms are separated by distances much greater than the wavelength of the optical transitions used to excite them. In the frequency domain, we observe a mean-field shift of the Rydberg state which results in intrinsic optical bistability above a critical Rydberg number density. In the time domain, we observe critical slowing down where the recovery time to system perturbations diverges with critical exponent $\alpha =-0.53\pm 0.10$. The atomic emission spectrum of the phase with high Rydberg occupancy provides evidence for a superradiant cascade.

Figure 1

Theoretical model for the cooperative optical response of the ensemble. (a) Simplified two-level model with ground state $|\mathrm{g}⟩$ and Rydberg state $|\mathrm{R}⟩$. The levels are coupled by a laser with Rabi frequency $\Omega$ and detuning from resonance $\Delta$. (b) Rydberg state population ${\rho }_{\mathrm{RR}}$ as a function of laser detuning $\Delta$ for increasing Rabi frequency $\Omega$. As a result of the cooperative excitation-dependent shift, the response exhibits intrinsic optical bistability with hysteresis dependent upon the history of the ensemble. Theoretical parameters: interaction strength $V/\Gamma =-11$ and Rabi frequency $\Omega /\Gamma =(0.15,0.3,0.5)$.

Figure 2

(a) Three-photon excitation scheme to Rydberg states in cesium. (b) Schematic of the experimental setup. The three excitation lasers copropagate through a 2 mm vapor cell. The nonequilibrium dynamics is probed by measuring the transmission of the probe laser or analyzing the emitted fluorescence. (c) Experimental optical response $\Delta T$ as a function of Rydberg laser detuning ${\Delta }_{\mathrm{R}}$ for Rydberg Rabi frequency ${\Omega }_{\mathrm{R}}$ increasing from (i) to (iii). Experimental parameters: ground-state density $\mathcal{N}=4.3\times {10}^{12}{\mathrm{cm}}^{-3}$, probe Rabi frequency ${\Omega }_{\mathrm{p}}=2\pi \times 37\mathrm{MHz}$, coupling Rabi frequency ${\Omega }_{\mathrm{c}}=2\pi \times 77\mathrm{MHz}$, and Rydberg Rabi frequency ${\Omega }_{\mathrm{R}}=2\pi \times (14,36,74)\mathrm{MHz}$.

Figure 3

Atomic emission spectra for low and high Rydberg state occupancy. The visible fluorescence spectrum is shown for (a) $\mathcal{N}=3.1\times {10}^{11}{\mathrm{cm}}^{-3}$ and (c) $\mathcal{N}=4.3\times {10}^{12}{\mathrm{cm}}^{-3}$. In the low Rydberg occupancy phase, the spontaneous emission originates from high-lying Rydberg states as illustrated in (b). However, in the high Rydberg occupancy phase, the spontaneous emission originates from low-lying Rydberg states, as illustrated in (d), due to a superradiant cascade between high-lying Rydberg states. The ionization limits from $6s_{1/2}$, $6p_{1/2}$, and $6p_{3/2}$ are shown by thick red vertical lines. The blue shaded regions highlight the absence of spontaneous emission between $26p_{3/2}$ and ionization which would occur due to a blackbody or collisional excitation process. The thin cyan vertical lines indicate the dipole-allowed transitions. Probe Rabi frequency ${\Omega }_{\mathrm{p}}=2\pi \times 41\mathrm{MHz}$, coupling Rabi frequency ${\Omega }_{\mathrm{c}}=2\pi \times 74\mathrm{MHz}$, and Rydberg Rabi frequency ${\Omega }_{\mathrm{R}}=2\pi \times 122\mathrm{MHz}$.

Figure 4

Critical slowing down as the temporal signature of a phase transition. (a) Continuous Rydberg laser intensity $I_{\mathrm{R}}$ scan showing bistability and hysteresis in the optical response $\Delta T$. (b) Discrete Rydberg laser intensity $I_{\mathrm{R}}$ scan showing the divergence of the switching time to steady-state $\tau$ around the critical transition intensity $I_{\mathrm{R},\mathrm{crit}}\approx 17.5\mathrm{W}/{\mathrm{mm}}^2$. The switching time diverges as $(I_{\mathrm{R}}-I_{\mathrm{R},\mathrm{crit}}{)}^{\alpha }$ with critical exponent $\alpha =-0.53\pm 0.10$ (standard deviation error) shown by the dashed line of best fit. Ground-state density $\mathcal{N}=4.3\times {10}^{12}{\mathrm{cm}}^{-3}$, probe Rabi frequency ${\Omega }_{\mathrm{p}}=2\pi \times 57\mathrm{MHz}$, coupling Rabi frequency ${\Omega }_{\mathrm{c}}=2\pi \times 116\mathrm{MHz}$, and Rydberg detuning ${\Delta }_{\mathrm{R}}=2\pi \times -220\mathrm{MHz}$. The error bars represent the standard deviation error on the determination of the laser intensity and switching time.