Fast closed-loop optimal control of ultracold atoms in an optical lattice

We present experimental evidence of the successful closed-loop optimization of the dynamics of cold atoms in an optical lattice. We optimize the loading of an ultracold atomic gas minimizing the excitations in an array of one-dimensional (1D) tubes (3D-1D crossover) and we perform an optimal crossing of the quantum phase-transition from a superfluid to a Mott insulator in a 3D lattice. In both cases we enhance the experiment performances with respect to those obtained via adiabatic dynamics, effectively speeding up the process by more than a factor three while improving the quality of the desired transformation.

Figure 1

Scheme of closed-loop optimization experiment. The control-field $s\left(t\right)$ describes the temporal dependence of the lattice depth during the loading of the atomic gas in the lattice. An initial guess $s_0\left(t\right)$ is chosen: For each experimental run a time-of-flight image is recorded. From a double-structure fit of the density distribution we extract the figure of merit $F$. The optimization algorithm provides an updated function $s\left(t\right)$ and the experiment is repeated until reaching an optimal field $s_f\left(t\right)$.

Figure 2

3D-1D crossover. (a) Two-dimensional mapping (blue-colored palette) of the FOM as a function of the parameters $\Delta t$ and $\tau$. Circles (triangles) show the progress during the first (second) run of optimization (gray-colored palette). (b) Measured figure of merit $F$ (ratio between final and initial thermal fraction of the atomic sample) after ramping up and down a 2D optical lattice during the optimization loop. The first (second) experimental run (see text) is represented by blue circles (red triangles). The two best results $F_{\mathrm{opt}}$ measured in the two runs are circled in evidence. For completeness, on the right side of the graph we report also the corresponding values of the ratio between temperature $T_f$ measured after having switched on and off the lattices and temperature $T_i$ measured before loading the lattices (inferred from the thermal fraction assuming thermal equilibrium). The table reports the values of the FOM corresponding respectively to the quasiadiabatic loading, the exponential initial loading ($s_0$) and the best one ($s_{\mathrm{opt}}$) of the two runs, together with the correspondent values of the parameters $\Delta t$ and $\tau$.

Figure 3

Optical lattice depth ramps (in $E_r$ units) $s\left(t\right)$ for the superfluid-Mott insulator transition. Solid lines represent (a) first-run initial guess, (b) first-run optimized ramp, (c) second-run initial guess, and (d) second-run optimized ramp. The dashed black line represents the exponential noncorrected ramp.

Figure 4

Optimization of the superfluid-Mott insulator transition. The FOM $F$ is reported as a function of the iteration number $n$ (second run). The green region shows the FOM $F_{\mathrm{ad}}=(2.16\pm 0.03)$ in the case of the quasiadiabatic loading $\left(\Delta t=140\mathrm{ms},\tau =30\mathrm{ms}\right)$. The blue-circled empty square reports the final optimal loading $F_{\mathrm{opt}}$. In the table are reported the values of the parameters $[a_1,b_1,a_2,b_2]$ of the correction and the FOM $F$ corresponding, respectively, to the quasiadiabatic loading, to the exponential uncorrected loading ($s_{uc}$), and to the optimal loading ($s_{\mathrm{opt}}$) found in the two runs.