Stochastic Switching in Rydberg Ensembles

We demonstrate stochastic switching in a bistable system. The double-well potential system is implemented in a Rydberg ensemble with cooperative interaction. The state transition is driven by intensity noise carried in the lasers. The Rydberg ensemble system accumulates energy in an equilibrium situation and brings the nonlinear system across the potential barrier or threshold, where switching occurs between the two wells. Probing the Rydberg state populations using electromagnetically induced transparency allows the nonlinear behavior to be studied in an experiment.


Introduction
System resonance is a fundamental phenomenon pervading both nature and society. It reveals the response of a system to store and transfer energy from an external forcing source to an internal mode, where the forcing source includes the driving signal and stochastic noise [1]. Quantum particles have been proposed as powerful constituents that form nonlinear systems. Recent developments in techniques have provided scalable approaches for studying the interplay of pure quantum mechanical systems and their couplings to reservoirs. These techniques apply equally to quantum information processing and quantum sensing [2].
The phenomenon of stochastic switching in a bistable system has been observed in solidstate crystals, ion systems, and double quantum dot systems [3][4][5][6][7][8][9]. For atomic systems, it has been theoretically predicted that intrinsic interactions lead to stochastic resonances as well, a phenomenon referred to as quantum stochastic resonance. For a neutral atom, the properties of the interaction depend on the quantum state, which is important for implementing a controlled quantum nonlinear system. The interaction of the atoms in their ground state is dominated by 1/R 6 van der Waals forces at short range. The excitation of the neutral atom to a high Rydberg state results in a very strong interaction that scales as 1/R 3 . A long-range * E-mail: wwjjmm@sxu.edu.cn 1/9 arXiv:2006.02261v1 [physics.atom-ph] 3 Jun 2020 cooperative interaction makes Rydberg atoms a promising means of implementing a bistable system. Bistability is critical for producing stochastic resonance, where quantum stochastic resonance can be driven by spin noise or quantum fluctuation. Based on nonlinear systems with a cooperative Rydberg interaction, it may be possible to build atom-based sensing. Electric sensitivity of Rydberg atoms is a promising means to achieve weak signal detection. The detection scheme is based on the sensitivity to initial conditions of either stochastic or chaotic systems, in which the states of the system change under very small perturbations. This is different compared with the traditional methods for weak signal detection.
In our study, we have demonstrated stochastic switching in Rydberg ensembles. The double-well potential system was formed under a cooperative interaction of Rydberg atoms. In this bistable system, population transfer occurs between low and high Rydberg states.

Stochastic switching in a bistable system
Stochastic switching in a bistable system is similar to that in a threshold-crossing excitable system. The system accumulates energy in the resting state. Perturbations above a certain threshold may induce large excursions triggering transitions. When noise is injected into a bistable system, the collective interaction assists the intrinsic oscillator in eliciting an efficient response by overcoming the potential barrier. Then, the resting condition transforms into a firing condition. The nonlinear system can be described by the universal scaling theory of the FitzHugh-Nagumo model [10], which is a simplified version of the Hodgkin-Huxley model [11,12]. Similarly, the interface between a nonlinear system and an atomic system could take its starting points in the cooperative long-range interaction of a Rydberg atom.
Under incoherence, the Rydberg atoms are considered as independent superatoms. Under conditions of low atomic density, the Rydberg excitations are independent, and the number of excitations increases linearly. The mutual interactions do not affect the system dynamics. As the atomic density increases, nearest neighbors interact strongly preventing the generation of new Rydberg atoms, and hence the atoms in the ground state located in the available space have a low probability of being excited. The low and high Rydberg populations form the two stable states of system, corresponding to the double-well potential in the bistable system.
When the driving energy is small, there is no cross-well motion. When the cumulative energy of the system is above the potential barrier, switching between the two wells occurs.
The experimental apparatus ( Fig.1) consists of an external cavity diode laser (ECDL) with a wavelength of 852 nm and used as a probe laser with a typical linewidth in the megahertz scale. The power of the 1018 nm laser was amplified to 5 W by the fiber amplifier, and 2/9   Fig. 2(b)]. Owing to the Doppler mismatch, the hyperfine splitting of the 6P 3/2 states scale as ∆ c = ∆ p · ω c /ω p . For atoms with velocity moving in the same direction as the probe field, the detuning of the probe laser is ∆ p → −ω p · ϑ/c and that of the coupling laser is ∆ c = ∆ p · ω c /ω p . For a ladder-type EIT, the matrix element for the population may be derived 4/9 by solving the steady-state optical Bloch equations using a perturbation approximation under the weak probe field regime. The line shape of the transmission signal amounts to a convolution of the Holtsmark probability distribution with the EIT line shape: [13,14].
The cooperative interaction in a Rydberg ensemble is as a function of the power and detuning parameter of driving lasers. As shown in Fig. 2(c), we exhibit the nonlinear transition by modeling the Rydberg population as a function of the coupling laser frequency detuning, along with semiclassical Maxwell-Bloch equations [15]. The Rabi frequency of the coupling laser and the probe laser is 11.5 MHz. The interaction energy shift is ±49 MHz. When there is no interaction in the Rydberg ensemble, the transmission signal is described by a Lorentzian line shape [gray dotted line in Fig. 1(c)]. When there are strong interactions, the system response is governed by quantum state-dependent nonlinear dynamics. At critical frequency, there is a sharp switching in the transmission signal[green dash lines in Fig. 2(c)].  Fig. 3(a). The Rabi frequency of the 5/9 probe laser beams are change from 3.4 MHz to 12.5 MHz, as shown in Fig. 3(b). When the power of probe and coupling laser are weak, the line shapes of transmission signal are similar to a Lorentzian function or a Voigt function. Under conditions of strong laser power, at critical frequency, there is a sharp switching in the transmission signal. The switching of nonlinear effect is caused by cooperative interactions in the Rydberg ensembles. These interactions include the dipole-dipole interaction between Rydberg atoms [15,16], the charge-induced interaction between the Rydberg atoms, and the charge produced by the spontaneous ionization of the Rydberg atoms [17][18][19][20]. This may be the reason that there is no nonlinear effect for 6S 1/2 (F = 4) → 6P 3/2 (F = 4).
The optimized cell temperature of the atomic gas is 35 • C, which corresponds to an atomic density of 1.3×10 11 cm −3 . As shown in Fig 3(b), the transmission windows of the EIT signal are shifted, which may arise from Stark shift due to the spontaneous ionization of the Rydberg atoms. The ionization rate depends on the atomic density of the ground state and the Rydberg state. The effect of optical pumping increases with the power of the probe laser. The strength of the cooperative interaction is renormalized by factor Ω 1 / (Ω 2 1 + Ω 2 2 ) [20,21]. In addition, strong driving from the probe laser inhibits the ensemble occupation of the Rydberg state, which effectively suppresses avalanche ionization, resulting in a shift in the EIT window with probe laser power.
The nonlinear interaction of the Rydberg atoms enables the creation of a bistable system with a double-well potential. The barrier heights of the double-well can be controlled by tuning the laser parameters; here, the system stores and transfers energy received from an external forcing source into an internal mode. We assume laser beams with white Gaussian noise provide the perturbations or driving force. When noise acts on the system, the cumulative energy of the system in the equilibrium state is above the barrier height, and a transition occurs between the two wells. Fig. 4(a) shows the experimental implementation of the stochastic resonance for Rydberg ensembles in the state, as detailed in Fig. 4(b). The spectroscopy of a time series plot are obtained by tuning the coupling laser frequency with all other parameters fixed. The Rabi frequencies of the probe laser and the coupling laser are ∼11.46 MHz and 6/9

Summary and outlook
We explored the stochastic switching of a bistable system in the Rydberg ensemble. The cooperative interaction of the Rydberg atoms produces a bistable double-well system. The collective state of the Rydberg atoms accumulates energy in an equilibrium situation. External noise occasionally gives the system a kick that is large enough to cross the barrier of the double well. An indicator of stochastic resonance is that the flow of information through a system is maximized when the input noise intensity matches the system response, which is one of the fundamental laws in physics, engineering, and biology. A quantum nonlinear system will be useful for implementing resonance sensing and precision measurements. Note that cooperative interactions of the Rydberg ensemble are used to establish a bistable system, but the driving force of the stochastic switching is not quantum noise. Nevertheless, stochastic resonance driven by quantum noise has not been realized. The proposal of Rydberg ensemble bistability does realize quantum stochastic resonance using spin noise.