Dissipative Double-Well Potential for Cold Atoms: Kramers Rate and Stochastic Resonance

We experimentally study particle exchange in a dissipative double-well potential using laser-cooled atoms in a hybrid trap. We measure the particle hopping rate as a function of barrier height, temperature, and atom number. Single-particle resolution allows us to measure rates over more than 4 orders of magnitude and distinguish the effects of loss and hopping. Deviations from the Arrhenius-law scaling at high barrier heights occur due to cold collisions between atoms within a well. By driving the system periodically, we characterize the phenomenon of stochastic resonance in the system response.

Figure 1

Hybrid optical trap and hopping rate measurements. (a) The hopping rate $r$ between the two sites of the dissipative double-well potential depends on both the potential barrier height $\mathrm{\Delta }V$ and the temperature $T$. Fluorescence imaging with single-atom resolution yields the individual atom number $N_1$ and $N_2$. (b) An initial atom number imbalance equilibrates over time. For large atom numbers, the initial slope is a measure for the rate of light-assisted collisions. (c) Single-atom resolution allows the precise identification of hopping events as the anticorrelated atom number changes. Increasing the barrier height for a given temperature reduces the hopping rate.

Figure 2

The hopping rate dependence on small barrier heights is given by the Kramers rate (dashed line). At large enough barrier heights, the thermal activation is suppressed, and collisionally activated hopping starts to dominate for large mean atom numbers, such as 2500 (blue circles) and 320 (red diamonds). Including the high-energy scale of light-assisted collisions in the velocity distribution describes the scaling with atom number (solid lines). For small atom numbers, however, the measured hopping rate (black dots) is significantly higher than expected from light-assisted collisions.

Figure 3

Temperature control and Arrhenius’s law. (a) The temperature of the laser-cooled atomic ensemble can be varied via the duty cycle of blue detuning, which can reach a maximum of 27% (dashed line). (b) An Arrhenius plot of $\log (r)$ versus the inverse temperature shows the expected linear dependence over 4 orders of magnitude.

Figure 4

Stochastic resonance. (a) The hopping rate $r$ can be tuned via the temperature to match the time scale $T_{\mathrm{\Omega }}=2\pi /\mathrm{\Omega }$ of the external modulation of the system, resulting in a stochastic resonance. (b) The linear response of the system extracted from the atom number imbalance shows the resonance feature in the amplitude as well as in the signal-to-noise ratio. The solid lines are fitted varying the driving amplitude $A_0$. The phase of the linear response exhibits the expected shift for temperatures below the resonance and approaches $\pi /2$ for vanishing thermal noise. The insets show the linear response (black dots) relative to the drive (green line). (c) Increasing the barrier height effectively probes lower temperatures, where the effects of intrawell motion are observed (solid lines), causing a deviation from the simple stochastic resonance theory (dashed lines). Here, the linear response is obtained from the motion of the atomic centroid. Intrawell motion is induced by a shift $\delta x_m=x_m-x_m^{\prime }$ in the potential minima.