Time series learning in a many-body Rydberg system with emergent collective nonlinearity

Abstract

Interacting Rydberg atoms constitute a versatile platform for the realization of non-equilibrium states of matter. Close to phase transitions, they respond collectively to external perturbations, which can be harnessed for technological applications in the domain of quantum metrology and sensing. Owing to the controllable complexity and straightforward interpretability of Rydberg atoms, we can observe and tune the emergent collective nonlinearity. Here, we investigate the application of an interacting Rydberg vapour for the purpose of time series prediction. The vapour is driven by a laser field whose Rabi frequency is modulated in order to input the time series. We find that close to a non-equilibrium phase transition, where collective effects are amplified, the capability of the system to learn the input becomes enhanced. This is reflected in an increase of the accuracy with which future values of the time series can be predicted. Using the Lorenz time series and temperature data as examples, our work demonstrates how emergent phenomena enhance the capability of noisy many-body quantum systems for data processing and forecasting.

Figures (4)

• Figure 1: Time series prediction with a Rydberg vapour. The time-dependent input signal $x(t)$ is sampled at discrete time steps $t_n$, yielding a step-wise constant function with values $x_n=x(t_n)$. The signal is converted into the Rabi frequency $\Omega_c(t_n)$ of the coupling laser, which couples the intermediate state $6 P_{3/2}$ to the Rydberg state $48 D_{5/2}$ detuned by $\Delta_c$. This Rabi frequency ranges within the interval $[\Omega_\mathrm{min}, \Omega_\mathrm{max}]$. The caesium (Cs) atoms are first excited by a probe laser with a constant Rabi frequency $\Omega_p$, which drives the transition from the ground state $6 S_{1/2}$ to the intermediate $6 P_{3/2}$. Subsequently, the atoms are excited by the modulated coupling laser. The optical response of the Cs vapour is measured by monitoring the transmission $T(t)$ of this probe laser. The measured transmission $T(t)$ is post-processed using a Savitzky-Golay filter and down-sampled by a factor of $20$, yielding the transmission signals $T^{(i)}=T(t_{i+20n})$. Here, $i\in[1,20]$ labels the index of the downsampled time series and $s_n=t_{i+20n}$ denotes the corresponding time. The transmission signal within the time interval $[s_{n-M}, s_n]$ is used to predict the signal $x(s_{n+1})$ (denoted as $\hat{y}(s_{n+1})$). This prediction is generated by a linear regression layer, which was previously trained (see text for details).
• Figure 2: Time series prediction.(a) Amplitude modulated signal $\Omega_c/2\pi$ in MHz which is input into the Rydberg system. (b-d) Output transmission signal $T(t)$ in arbitrary units recorded for different detunings $\Delta_c$. The detuning is chosen such that the system is out of (panel b) and within the bistable region (panels c-d). (e, f) Time series prediction for the Lorenz series $x_\mathrm{Lor}$. A Rydberg vapour operating in the bistable region (e) results in a better prediction (red curve), compared to outside (f). The ground truth is shown in blue. (g-h) Prediction results on daily temperature of Beijing for detuning $\Delta_c$, near (g) and out (h) of the bistable region. (i) The lower yellow and lower green curves correspond to the minimal Rabi frequency $\Omega_{\mathrm{min}}$ of the coupling laser but with different scan directions of the detuning $\Delta_c$ (as indicated by arrows and colours). The upper yellow and upper green curves correspond to $\Omega_{\mathrm{max}}$. A closed hysteresis loop formed by the two scan directions characterizes the bistable phenomenon. The detuning interval where the bistability occurs is marked by a gray shaded box. (j) Mean square error (MSE) for varying detuning $\Delta_c$ on Lorenz and temperature time series prediction tasks. Inside the bistable region (gray shaded box), the MSE is lower than outside. The plotted MSE values correspond to the average over the 20 sampled time series, while the error bar corresponds to their standard deviation (see text for details).
• Figure 3: Mean-field dynamical response and learning capacity.(a) Phase diagram of the mean-field equation (\ref{['eq:n-dot-eliminated']}). The blue region marks the bistable domain where two stable stationary solutions exist. The yellow (green) horizontal lines indicate the minimum (maximum) Rabi frequency of the input signal of the learning protocol, see panel (d). The gray shadowed area indicates the region in which collective effects are most prominent for the considered Rabi frequencies. Its left (right) boundary corresponds to the detuning at which the relaxation time for the minimum (maximum) Rabi frequency displays a maximum, see panel (c). (b) Stationary Rydberg density $n_\mathrm{ss}$ for varying detuning along the horizontal cuts $\Omega/\gamma=1.1$ and $\Omega/\gamma=1.21$. For the case $\Omega/\gamma=1.21$ hysteresis is observed. We denote the solution with the lower density with a dashed line. (c) Relaxation time $\tau_\mathrm{relax}$ along the same cuts as in (b) for the stable stationary solutions. Within the bistable region, two relaxation times are possible, one for each stable solution. The solid (dashed) line corresponds to the solution with higher (lower) density. The longest relaxation times occur near the boundary of the bistable region. (d) Mean square error (MSE) for the prediction of the Lorenz input signal fixing the parameters to $\Omega/\gamma=1.1$, modulation amplitude $\delta \Omega/\Omega=0.1$, modulation period $\gamma T=20$, noise strength $D/\gamma=0.0001$ and for different detunings (points). In all panels $V/\gamma=100$ and $\gamma_\mathrm{d}/\gamma=10$.
• Figure 4: Experimental analysis of the relaxation times. Measurement of the transmission of the probe laser when the Rabi frequency $\Omega_c$ of the coupling light is amplitude modulated by a square pulse with frequency $1kHz$, duty cycle 50%, and modification depth 15%. In panel (a), the detuning of the coupling light $\Delta_c$ is swept from $\SIrange{6}{22}{\mega\hertz}$ linearly in time. Instead, in panels (b) and (c) the detuning is fixed to $6MHz$, and $12MHz$, respectively. Panel (b) corresponds to a point outside the bistable regime, while panel (c) to a point within the bistable regime. In panel (c) an exponential relaxation is fitted to the experimental data (see text) at three different periods obtaining the relaxation times $\tau_1=0.139\,ms$, $\tau_2=0.148\,ms$ and $\tau_3=0.125\,ms$.