Optimal control of complex atomic quantum systems

Abstract

Quantum technologies will ultimately require manipulating many-body quantum systems with high precision. Cold atom experiments represent a stepping stone in that direction: a high degree of control has been achieved on systems of increasing complexity. However, this control is still sub-optimal. In many scenarios, achieving a fast transformation is crucial to fight against decoherence and imperfection effects. Optimal control theory is believed to be the ideal candidate to bridge the gap between early stage proof-of-principle demonstrations and experimental protocols suitable for practical applications. Indeed, it can engineer protocols at the quantum speed limit - the fastest achievable timescale of the transformation. Here, we demonstrate such potential by computing theoretically and verifying experimentally the optimal transformations in two very different interacting systems: the coherent manipulation of motional states of an atomic Bose-Einstein condensate and the crossing of a quantum phase transition in small systems of cold atoms in optical lattices. We also show that such processes are robust with respect to perturbations, including temperature and atom number fluctuations.

Figure 1

(a) The optimal control algorithm (CRAB, see Sec. 1.1) applies a first control field V_guess(t) to a numerically simulated experiment. Taking into account experimental constraints, it optimizes the control field V(t) relying on the figure of merit F after time evolution. The final control field obtained after optimization, V_OPT(t), optimally steers the system in the minimal possible time T_OPT compatible with the theoretical and experimental limitations. (b) Vienna atomchip experimental setup: illustration of the experimental setup with the atomchip (top) used to trap and manipulate the atomic cloud (middle) and the light sheet as part of the imaging system (bottom). The trapping potential, centered on a DC wire, is made slightly anharmonic by alternating currents in the two radiofrequency (rf) wires. It is then displaced (black arrow) along the optimal control trajectory V_OPT(t), using an additional parallel wire located far away from the DC wire and carrying a current proportional to V_OPT(t). By this mechanical displacement of the wavefunction, a transition is realized from the ground to the first excited state of the trap. The atomic cloud is imaged after a 46 ms time-of-flight. (c) Garching lattice experiment setup: an optical lattice is applied along an array of tubes and drives the superfluid to Mott-insulator transition with one atom per site (top) following the optimized control field V_OPT(t) applied to the lattice depth (black arrows). The distribution of atoms in the Mott regime is probed by fluorescence imaging through a high-resolution microscope objective with single-site resolution and single-atom sensitivity (bottom right).

Figure 2

(a) Theoretical prediction of the optimal figure of merit F₁ (infidelity with respect to the goal state, see Eq. (2)) achieved by optimal driving of the ground-to-first-excited state transfer of a condensate for N = 700 atoms, as a function of the transformation time T (blue squares). The blue solid lines are fits of the numerical results according to with and (left and right curve respectively), determining the QSL. The green region represents the smallest measurable infidelity in the experiment. Its intercept with the blue line defines the optimal time T_OPT = 1.09 ms (gray vertical line), which is the fastest time compatible with both the QSL and the experimental limitations. The yellow cross marks the experimental result performed at T_OPT. (b) Theoretical prediction of the optimal figure of merit F₂ (averaged atom number fluctuations in each site at the center of the trap, see Eq. (4)) as a function of the control field duration T (blue squares) for the crossing of the SF-MI crossover. The blue solid line is a fit determining the QSL, the crossing point between the numerical result. The green region (estimated experimental limitations) defines the optimal time T_OPT = 12.0(2) (gray vertical line).

Figure 3. Optimal control fields for transformation times T = 1.09 ms for N = 1,700 and 7000 atoms obtained via full CRAB optimization (respectively red, blue, and yellow line).

Inset: Fourier spectrum of the optimal control field for T = 1.09 ms and N = 700 (blue solid line). The vertical lines correspond to single particle transitions from the ground state (red).

Figure 4

(a) Theoretical predictions for the final infidelity F₁ as a function of the atom number when using the control fields optimized for 700 atoms. The shown numerical results are obtained for total transformation times T = 1.09, 5.01 ms (blue and red lines) and are compared to experimental results (circles) obtained with the optimal control field for T_OPT = 1.09 ms. (b) Transverse distribution after time-of-flight during the optimal process (t < T_OPT) and after (T_OPT < t < 2 ms) for N = 700: the experiment (center) and the corresponding GPE simulation (left), plus the residual difference between experiment and simulation (right, only plot with negative values). The gray horizontal line highlights T_OPT.

Figure 5

(a) Theoretical prediction for the final atom number fluctuation in the center of the trap for different atom numbers N for the linear (red) and optimal (blue) control fields. The optimized control field works best for 16 atoms, but a small deviation only slightly decreases the figure of merit (blue). The linear ramp results in a higher figure of merit for all atom numbers (red). (b) Lattice ramps used in the experiment. Top: the lattice is first slowly ramped to 3 E_r (grey) before either the fast linear control field (red) or fast optimized control field (blue) is applied. The typical adiabatic control field (yellow) is much longer. Bottom: magnification of the comparison between the linear (red) and the optimal control field (blue).

Figure 6

(a) Experimental mean parity profiles P_i resulting from the adiabatic (yellow points), optimized (blue points) and linear (red points) lattice ramps compared to the rescaled numerical results (blue and red lines). The short linear lattice ramp has a dip in the parity profile at the center due to non-adiabatic effects. Inset: magnification of the central part of the main panel. Standard deviations of the measured data are smaller than data points in the main plots and therefore only shown in the inset. (b) The red and blue shaded areas display the numerically computed atom number fluctuations for the linear and the optimized ramps lasting T_OPT = 11.75 ms.