Spontaneous avalanche dephasing in large Rydberg ensembles

Strong dipole-exchange interactions due to spontaneously produced contaminant states can trigger rapid dephasing in many-body Rydberg ensembles [E. A. Goldschmidt et al., Phys. Rev. Lett. 116, 113001 (2016)]. Such broadening has serious implications for many proposals to coherently use Rydberg interactions, particularly Rydberg dressing proposals. The dephasing arises as a runaway process where the production of the first contaminant atoms facilitates the creation of more contaminant atoms. Here we study the time dependence of this process with stroboscopic approaches. Using a pump-probe technique, we create an excess “pump” Rydberg population and probe its effect with a different “probe” Rydberg transition. We observe a reduced resonant pumping rate and an enhancement of the excitation on both sides of the transition as atoms are added to the pump state. We also observe a time scale for population growth that is significantly shorter than predicted by homogeneous mean-field models, as expected from a clustered growth mechanism where high-order correlations dominate the dynamics. These results support earlier works and confirm that the time scale for the onset of dephasing is reduced by a factor which scales as the inverse of the atom number. In addition, we discuss several approaches to minimize these effects of spontaneous broadening, including stroboscopic techniques and operating at cryogenic temperatures. It is challenging to avoid the unwanted broadening effects, but under some conditions they can be mitigated.

Figure 1

Interaction mechanism: Blackbody-dominated decay from the Rydberg state produces nearby $p$ states ($|p\rangle$), which trigger an off-diagonal dipole-exchange interaction of the form $|p,s^{\prime }\rangle \to |s^{\prime \prime },p^{\prime }\rangle$. This creates a dipole-dipole exchange interaction between two different Rydberg $s$ states. For some orientation and distance r between the dipoles, this brings atom 2 into resonance and the excitation is enhanced (antiblockade). This energy shift inhomogeneously broadens the transition in the sample, as represented by the gray shading on $|s^{\prime }\rangle$ and $|s^{\prime \prime }\rangle$. If $|s\rangle =|s^{\prime }\rangle$, we are in the “self-broadening” situation where a single $s$-state transition is broadened via the $p$ states created from it.

Figure 2

Cross-state broadening. (a) The ground state is divided into a fraction $f$ in the “probe state” $|g^{\prime }\rangle$ (highlighted by a green square), leaving $1-f$ in the “pump state” $|g\rangle$. Here $f=0.25$, so that contaminant $p$ atoms come predominantly from the pump while leaving enough atoms participating in the probe transition for a sufficient signal to noise ratio. The excitation to $18S$ is done via a two-photon transition with the $5P_{1/2}$ intermediate state: the intermediate detuning $\mathrm{\Delta }$ is much greater than the single-photon Rabi frequencies and than the two-photon detuning ${\delta }_{\mathrm{probe}}$ (not to scale on this figure). The pump population is driven resonantly to $|s\rangle =|18S,2,-2\rangle$ with two-photon Rabi frequency ${\mathrm{\Omega }}_{\mathrm{pump}}$. (b) Example of broadening due to the presence of pump-induced Rydberg population. The blue (red) curve is the probe spectrum with the pump turned off (on). The fits are Lorentzian and are used to estimate the resonant rates $R_0$, the widths $\mathrm{\Gamma }$, and their uncertainties throughout this work. (c) Probe transition widths and resonant excitation rates as a function of the fraction of $|g\rangle$ that was pumped to other states during the $300\mu \mathrm{s}$ excitation. We observe a doubling of $\mathrm{\Gamma }$ and a significant decrease of $R_0$ between no pump and the maximum pumped population.

Figure 3

Delayed pulses experiment. (a) To observe cross-broadening effects, we chop both 795 nm beams (probe and pump) into pulses, which can be delayed relative to one another. The 485 nm light is kept on. (b) Pump-induced slowdown of resonant excitation, visible in the number of atoms remaining in the probe ground state as a function of excitation time $N_{p}T$. The two examples here, fitted with exponentials, are for $\mathrm{\Delta }t=0\mu \mathrm{s}$ (red) and $\mathrm{\Delta }t=40\mu \mathrm{s}>t_p$ (blue). We observe rates of $R_{\mathrm{\Delta }t=0\mu \mathrm{s}}=1.8\pm 0.13{\mathrm{ms}}^{-1}$ and $R_{\mathrm{\Delta }t=40\mu \mathrm{s}}=3.3\pm 0.14{\mathrm{ms}}^{-1}$. (c) Following the sequence in (a), we observe a broadening that depends on the overlap of the pulses. We focus on the turn on time (probe before pump, corresponding to negative $\delta$ for a pump pulse at $\mathrm{\Delta }t=0\mu \mathrm{s}$). The red zones represent the pump pulse, while the green bands show the probe pulse width. The gray dashed line is the (self-broadened) width obtained when the pump is turned off. Spectra corresponding to a high $p$ population are obtained for $\mathrm{\Delta }t\simeq 0\mu \mathrm{s}$ and spectra corresponding to no pump population (pump turned off, marked by the dashed line) are recovered for $\left|\mathrm{\Delta }t\right|\gtrsim t_p$. This is visible around $\mathrm{\Delta }t=-40\mu \mathrm{s}$, when the probe is delayed well beyond the lifetime of the Rydberg states. The red line is the result of the nonlinear mean-field theory.

Figure 4

The $\mathrm{\Delta }{m}_F=3$ case. (a) Pumping scheme for two states separated by $\mathrm{\Delta }{m}_F=3$. The pump population is driven resonantly to $|s\rangle =|18S,2,-2\rangle$, while the probe transition is driven to $|s^{\prime }\rangle =|18S,1,+1\rangle$. (b) Time dependence of the pump-probe experiment. The red zones represent the times the pump is on, while the green bands show the probe pulse width. The gray dashed line is the (self-broadened) width obtained when the pump is turned off. Again we observe a spectral width increase when in phase, relative to out of phase. This plot spans the full period $T$ and shows an asymmetry between having the probe after (0–40 $\mu \mathrm{s}$) or before (40–80 $\mu \mathrm{s}$) the pump. A rate-equation theory [see Eqs. (5, 6, 7, 8)] shows (red line) the correct shape, including asymmetry, but consistently has a delay of $\sim 10\mu \mathrm{s}$.

Figure 5

Stroboscopic excitation on a single transition. Observed self-broadened width (blue) of the pump transition as a function of the individual pulse width. Here a single $|18S,2,-2\rangle$ population is excited with a Rabi frequency $\mathrm{\Omega }=2\pi \times 66\mathrm{kHz}$. Below $t_p\approx 3{\tau }_0$ (where ${\tau }_0$ is the $18S$ lifetime), the broadening decreases, with a reduction of the width by a factor of $\sim 3$. However, the natural linewidth (black dashed line) cannot be recovered via this scheme due to the fundamental Fourier limit (red line, delimiting the forbidden region in gray). Here too the homogeneous mean-field prediction (green line) is slower than observed. The dashed green line shows the prediction for homogeneous mean-field plus Fourier broadening.

Figure 6

Photon counting on a single transition. Observed $5P_{3/2}-5S$ fluorescence (blue trace) vs time. The detected fluorescence is proportional to the $18S$ population. Here a single $|18S,2,-2\rangle$ population is excited with a Rabi frequency $\mathrm{\Omega }=2\pi \times 140\mathrm{kHz}$ (light red areas). The red line is the model. (a) Resonant case: an overshoot is visible in both the data and the model, signaling the presence of the $nP$ population. Since the model is slower to produce pollutants, the overshoot in the model has more time to grow at the full single-particle rate than in reality. (b),(c) Detuned case: since the facilitated excitation relies on a pollutant population, the $18S$ population is slower to rise in the model than experimentally observed, confirming the faster-than-homogeneous mean-field property of the avalanche dephasing. Since the experimentally observed broadening is symmetric, the sign of the detuning does not change these observations.

Figure 7

Temperature scaling. (a) Temperature dependence of the sum of branching ratios $b$ from the different $nS$ states to the contaminant $nP$ states playing a significant role in the avalanche dephasing process. (b) Effective lifetime ${\tau }_0$ (determined by spontaneous emission and blackbody radiation-induced transitions) for different $nS{}^{87}\mathrm{Rb}$ Rydberg states as a function of the ambient temperature. (c) Temperature $T_N^{*}$ needed to compensate for the dephasing effect as a function of $N$. The sharp drop stems from the intrinsic branching ratio to contaminant states: even at zero temperature, there is a nonzero chance to produce a contaminant atom.