Optimal control in phase space applied to minimal-time transfer of thermal atoms in optical traps

TL;DR

The paper develops a phase-space-based optimal control framework to minimize the time and energy of transporting ultracold atoms between optical tweezers in a 1D array, incorporating deterministic classical dynamics, stochastic noise via Liouville-Fokker-Planck, and quantum effects via the Wigner equation. By solving the corresponding optimality systems for each description, the authors demonstrate near-minimal transfer times (≈7.01 μs theoretical) with very high fidelities (≈99.97% classically and 98.95% quantum) and robustness to bath temperature and perturbations. The approach yields fast, adaptable transport trajectories that can initialize large atom arrays for quantum simulation or computation, while accounting for realistic noise and energy costs. This contributes a versatile, phase-space-based toolkit for robust, high-fidelity, nonadiabatic atom transport in scalable quantum platforms.

Abstract

We present an optimal control procedure for the non-adiabatic transport of ultracold neutral thermal atoms in optical tweezers arranged in a one-dimensional array, with focus on reaching minimal transfer time. The particle dynamics are modeled first using a classical approach through the Liouville equation and second through the quantum Wigner equation to include quantum effects. Both methods account for typical experimental noise described as stochastic effects through Fokker-Planck terms. The optimal control process is initialized with a trajectory computed for a single classical particle and determines the phase-space path that minimizes transport time and ensures high transport fidelity to the target trap. This approach provides the fastest and most efficient method for relocating atoms from an initial configuration to a desired target arrangement, minimizing time and energy costs while ensuring high fidelity. Such an approach may be highly valuable to initialize large atom arrays for quantum simulation or computation experiments.

Figures (8)

• Figure 1: Optimal control of the phase-space trajectories of the atoms driven by the tweezer field. We depict three distinct solutions of the optimality system corresponding to three final times $t_f$. (a) Phase-space trajectories corresponding to different optimal times: blue curve, $t_f=\qty{7.36}{\micro\second}$; red curve, $t_f=\qty{10.45}{\micro\second}$; and green curve, $t_f=\qty{13.17}{\micro\second}$. (b) Potential profile of the static initial and target traps.
• Figure 2: Time evolution of the atom position associated with the optimal time $t_f=\qty{7.36}{\micro\second}$ (blue curve in Fig. \ref{['fig_Opt_traj_02']}(a)). The particle trajectory $x_{\text{opt}}(t)$ is depicted in magenta and the spheres indicate the initial and final positions of the atom. The 3D plot depicts the time evolution of the moving tweezer profile and, as a guide to eye, the yellow dashed curve depicts the evolution of the center of the tweezer's position. For this deterministic simulation the tweezer depth is fixed to $v=\qty{-1.5}{\milli\kelvin}$.
• Figure 3: (a) Optimal time solutions as a function of the weight $\nu_{t_f}$. The dashed horizontal line indicates $t_{\text{lim}}$. The inset shows the estimation of the cost functional $J$ varying the weight $\nu_{t_f}$. (b) Controlled parameter $u$ as a function of time for the three optimal times. The colors refer to $t_f=\qty{7.36}{\micro\second}$ (blue), $t_f=\qty{10.45}{\micro\second}$ (red) and $t_f=\qty{13.17}{\micro\second}$ (magenta). (c) Maximum difference $\lvert \Delta x \rvert$ within $[0,t_f]$ occurring between the atom trajectories as a function of $\nu_u$ with $\log(\gamma_u)=-3$. Our optimal solution at $\log(\nu_u)=-1$ is highlighted by a red box. (d) Maximum difference $\lvert \Delta x \rvert$ within $[0,t_f]$ as a function of $\gamma_u$ with $\log(\nu_u)=-1$. Our optimal solution at $\log(\gamma_u)=-3$ is highlighted by a red box. Within each bar in (c) and (d), the orange area represents the range of variation of the optimal trajectory with respect to the reference case with $\log(\nu_u)=-1$ and $\log(\gamma_u)=-3$.
• Figure 4: Optimal control procedure applied to the transfer of a well localized density of particles to the target region represented by the red rectangle. The plots are made at (a) $\qty{0.0}{\micro\second}$, (b) $\qty{2.4}{\micro\second}$, (c) $\qty{4.8}{\micro\second}$, and (d) $\qty{7.4}{\micro\second}$, considering a bath temperature of $T_{\text{th}}=\qty{0.1}{\milli\kelvin}$. The initial temperature of the atoms distribution is the same as indicated in Table \ref{['tab_par']}.
• Figure 5: (a) Time evolution of the temperature $T$ of the atoms along the trajectory $x_{\text{opt}}(t)$ related to the optimal time $t_f=\qty{7.36}{\micro\second}$ (b) Final temperature $T(t_f)$ of the atoms as a function of the final time $t_f$. We extend our simulations to longer final times $t_f=9,12,20,35,50\,\unit{\micro\second}$, observing that $T(t_f)$ converges towards the initial temperature $T=\qty{0.1}{\milli\kelvin}$, represented by the red dashed line.