Energy Landscape Shaping for Robust Control of Atoms in Optical Lattices

TL;DR

The paper addresses robust quantum control for spin transfer in ultracold atoms by energy landscape shaping, offering robustness advantages over dynamic control. It develops a two-stage hybrid optimization that first selects static site biases $\vec{\Delta}$ to achieve high fidelity, then realizes those biases via DMD-projected potentials using surrogate models. A robustness framework quantifies sensitivity to structured perturbations, including lattice-DMD misalignment and projection power drift, and reveals a critical dependence on the distance of $\Delta$ from the singular point $\Delta = 1$ where $J_{\text{eff}}$ diverges. The results demonstrate feasible, robust spin-transfer controllers in a realistic cold-atom architecture and provide a toolkit for robust energy-landscape quantum control applicable to broader quantum technologies.

Abstract

Robust quantum control is crucial for realizing practical quantum technologies. Energy landscape shaping offers an alternative to conventional dynamic control, providing theoretically enhanced robustness and simplifying implementation for certain applications. This work demonstrates the feasibility of robust energy landscape control in a practical implementation with ultracold atoms. We leverage a digital mirror device (DMD) to shape optical potentials, creating complex energy landscapes. To achieve a desired objective, such as efficient quantum state transfer, we formulate a novel hybrid optimization approach that effectively handles both continuous (laser power) and discrete (DMD pixel activation) control parameters. This approach combines constrained quasi-Newton methods with surrogate models for efficient exploration of the vast parameter space. Furthermore, we introduce a framework for analyzing the robustness of the resulting control schemes against experimental uncertainties. By modeling uncertainties as structured perturbations, we systematically assess controller performance and identify robust solutions. We apply these techniques to maximize spin transfer in a chain of trapped atoms, achieving high-fidelity control while maintaining robustness. Our findings provide insights into the experimental viability of controlled spin transfer in cold atom systems. More broadly, the presented optimization and robustness analysis methods apply to a wide range of quantum control problems, offering a toolkit for designing and evaluating robust controllers in complex experimental settings.

Figures (5)

• Figure 1: A cartoon diagram of the experimental setup The atoms, in a vacuum chamber, are trapped in an optical lattice potential (red) generated by a laser retro-reflecting from a mirror. In practice the trapping is 3D, but we show here a 1D lattice as this work deals only with a single dimension of the lattice potential. A second optical field (blue) generated by a laser is reflected off of a DMD and projected onto the underlying lattice potential via a high-NA microscope objective. This projected potential generates the required biases that generate the desired energy landscape for the trapped atoms, one of these biases $\Delta$ is indicated by the purple dotted lines.
• Figure 2: Example DMD patterns and the resulting potentials. Plots (a) and (c) show the DMD patterns for blue and red light where the height of the superpixel in $y$ was set to $12$ and $1$ DMD pixels, respectively (noting that the apparent difference in superpixel width in (a) is an artifact of the plotting). Plots (b) and (d) show the resulting projected potentials (black, solid line) and the total potential with a $\zeta = 18$ depth lattice on top (blue, dotted). The red dots mark the minima of the potentials used to calculate $\vec{\Delta}$. The potentials here correspond to the filled green diamond and open orange square in Figs. \ref{['fig:stuff_v_sens']} and \ref{['fig:stuff_v_pow']} in (a,b) and (c,d) respectively. In (a,b), the number of superpixels was taken to be $6$, and in (c,d) $4$ superpixels were used. Note that only $4$ superpixels are visible in (a) due to the plot limits, and the effect on the $\vec{\Delta}$ from the two superpixels that are not shown is minimal.
• Figure 3: Fidelity error vs. time curves for three controllers that are representative of the $24$ controller dataset, showing differences in the time evolution behavior of the system. For each controller, the minimum fidelity point is shown with a red dot. The three controllers in (a)-(c) correspond to the filled blue circle, filled green diamond, and open orange square, respectively, in Figs. \ref{['fig:stuff_v_sens']} and \ref{['fig:stuff_v_pow']}, and they use blue-, blue-, and red-detuned potentials, respectively. The time is given in units of $1/J$; by choosing a lattice depth $\zeta$, this can be backed out to find a real, physical time. For $\zeta = 18$, a time unit of $1$ corresponds to a time of $0.186ms$.
• Figure 4: Results for the $24$ DMD-generated potential patterns were found for a $5$-spin chain transfer from site $1$ to site $5$. For each plot, the absolute value of the derivative of the error $\epsilon$ with respect to drifts in the projected potential along the chain direction $x$ (quantifying controller sensitivity) is shown for each controller with respect to (a) the minimum distance of a given bias $\Delta$ to the singularity at $\Delta = 1$, (b) the fidelity error, and (c) the transfer time. Filled (open) points indicate blue- (red-)detuned potentials. The distinct data points for each controller are the same between plots for ease of comparison. The units of the sensitivity are given in $1/a$, where $a$ is the lattice spacing.
• Figure 5: Same as Fig. \ref{['fig:stuff_v_sens']} but showing the absolute value of the derivative of the error $\mathsf{e}$ with respect to changes in the intensity of the projected potential. The distinct data points for each controller are the same between plots and figures for ease of comparison. The units of the sensitivity are given in $1/E_\mathrm{R}$.