Emergence of Synchronization in a Driven-Dissipative Hot Rydberg Vapor

We observe synchronization in a thermal ($35–60°\mathrm{C}$) atomic (Rb) ensemble driven to a highly excited Rydberg state (principle quantum number $n$ ranging from 43 to 79). Synchronization in this system is unexpected due to the atomic motion; however, we show theoretically that sufficiently strong interactions via a global Rydberg density mean field cause frequency and phase entrainment. The emergent oscillations in the vapor’s bulk quantities are detected in the transmission of the probe laser for a two-photon excitation scheme.

Figure 1

Single velocity class model. The basic model with the relevant parameters is shown in (a). An example steady-state solution of the resulting nonlinear OBEs is shown in (b) where the dark-red steady-state branch is repulsive and green indicates the limit cycle region. For a fixed detuning ${\mathrm{\Delta }}_c/{\mathrm{\Gamma }}_{ge}=-1$, indicated by the dashed line, the time evolution from an initial state $|\mathrm{\Psi }{⟩}_{t=0}=(1-x)|g⟩+x|r⟩$ with $x\in [0,1]$ toward a limit cycle is shown in (c). For the same time traces, a phase space projection of the limit cycle in the ${\rho }_{ge}$-plane is shown in (d). The other model parameters were set to ${\mathrm{\Delta }}_p=0$, ${\mathrm{\Omega }}_p/{\mathrm{\Gamma }}_{ge}=3.8$, ${\mathrm{\Omega }}_c/{\mathrm{\Gamma }}_{ge}=2$, $V/{\mathrm{\Gamma }}_{ge}=-12$, ${\mathrm{\Gamma }}_{er}/{\mathrm{\Gamma }}_{ge}={10}^{-5}$, ${\mathrm{\Gamma }}_{gr}/{\mathrm{\Gamma }}_{ge}={10}^{-2}$, and $\beta =3$.

Figure 2

Thermal vapor simulation showing emergence of synchronization. A thermal vapor simulation for uncoupled (a) and coupled (b) velocity classes shows the emergence of synchronization via the Rydberg density induced mean field. The time evolution and corresponding steady-state spectrum are shown on the left and right, respectively. Simulation parameters were ${\mathrm{\Omega }}_p/{\mathrm{\Gamma }}_{ge}=6$, ${\mathrm{\Omega }}_c/{\mathrm{\Gamma }}_{ge}=4$, ${\mathrm{\Delta }}_p=0$, ${\mathrm{\Delta }}_c/{\mathrm{\Gamma }}_{ge}=-11$, ${\mathrm{\Gamma }}_{er}/{\mathrm{\Gamma }}_{ge}={10}^{-5}$, ${\mathrm{\Gamma }}_{gr}/{\mathrm{\Gamma }}_{ge}={10}^{-2}$, $V/{\mathrm{\Gamma }}_{ge}=-800$, $\beta =2$ and $N_{\mathrm{vel}}=101$ velocity classes with equal populations. The atomic velocity distribution corresponds to that of a rubidium vapor on the ${\mathrm{D}}_2$ line at $48°\mathrm{C}$.

Figure 3

Setup and example onset of oscillations. (a) The counterpropagating probe and coupling lasers are polarization cleaned with a polarizing beamsplitter (PBS) after exiting the fibers. The subsequent acousto-optic modulator (AOM) and aperture are used to remote control the laser powers incident on the heated, 4 cm long rubidium cell. The probe light is detected by a photodetector (PD). (b) The relevant level scheme for two-photon spectroscopy in rubidium. The coupling laser addresses either the $|n^{\prime }{S}_{1/2}⟩$ or the $|n{D}_{m_J}⟩$ state(s). In (c), an example set of traces obtained for fixed probe Rabi frequency ${\mathrm{\Omega }}_p/2\pi =191\mathrm{MHz}$ and increasing coupling Rabi frequencies is shown. The Rydberg laser is coupled to the $|43D_{5/2}⟩$ state, and the number density is ${\rho }_{87\text{Rb}}=(4.7\pm 0.2)\times {10}^{10}{\mathrm{cm}}^{-3}$. Here, the oscillatory regime is preceded by an onset of bistability.

Figure 4

Change in oscillation shape and frequency along coupling laser scan. (a) The oscillation region for a scan across resonance with $|43D_{5/2}⟩$ at $T=(52.0\pm 0.5)°\mathrm{C}$ with ${\mathrm{\Omega }}_p/2\pi =191\mathrm{MHz}$, ${\mathrm{\Omega }}_c/2\pi =37\mathrm{MHz}$, and a scan rate of $2\pi \times 10\mathrm{MHz}/\mathrm{ms}$. The colored insets show an enlargement of the trace in the color-shaded regions, each of width $2\pi \times 4.8\mathrm{MHz}$. Different shapes of the oscillations can be distinguished. (b) Pointwise integrated spectrum with errorbars denoting the amplitude of the oscillations. The time evolution toward a limit cycle is shown in (c) with the inset showing only the limit cycles approached after an integration time of $t=5000{\mathrm{\Gamma }}_{ge}^{-1}$. In (d), the oscillations in Rydberg population ${\rho }_{rr}^r$ (solid) and in the imaginary part of the $\mathrm{c}{\rho }_{ge}^i$ (dashed) are shown. The case $\mathrm{\Delta }=-3{\mathrm{\Gamma }}_{ge}$ did not approach a limit cycle within the maximum integration time but behaves similar to a system near a strange attractor. The simulation assumes a thermal vapor with $N_{\mathrm{vel}}=101$ velocity classes with equal populations, ${\mathrm{\Omega }}_p=1.5$, ${\mathrm{\Omega }}_c=1$, ${\mathrm{\Delta }}_p=0$, ${\mathrm{\Gamma }}_{er}={10}^{-6}$, ${\mathrm{\Gamma }}_{gr}={10}^{-3}$, and $V=-300$, in units of ${\mathrm{\Gamma }}_{ge}$, and $\beta =2$.