Estimating applied potentials in cold atom lattice simulators

TL;DR

The paper tackles the challenge of calibrating arbitrary site-dependent potentials in cold-atom lattice experiments, where diffraction limits precise implementation. It introduces a protocol that leverages interaction-off dynamics via a Feshbach resonance and time-resolved site occupations $D_i(t)=C_{ii}(t)$ measured with a quantum gas microscope to infer the actual potential landscape from a known initial state. Two learning strategies are developed: a rigorous polynomial-derivative reconstruction that certifies informational completeness and a practical gradient-based data-fitting approach that uses a simple initial state to achieve robust, scalable calibration. Numerical experiments demonstrate favorable sample complexity and robustness to state-preparation errors and hopping miscalibration, with mean reconstruction error scaling as $\varepsilon_{\mathrm{MRE}} \sim 1/\sqrt{M}$ and minimal dependence on system size $N$ for moderate $M$, enabling high-fidelity quantum simulation with potentially up to $\sim$100 sites and extensive measurement campaigns. The framework also suggests extensions to interacting or time-dependent potentials and connects to shadow-tomography concepts for analog quantum simulators.

Abstract

Cold atoms in optical lattices are a versatile and highly controllable platform for quantum simulation, capable of realizing a broad family of Hubbard models, and allowing site-resolved readout via quantum gas microscopes. In principle, arbitrary site-dependent potentials can also be implemented; however, since lattice spacings are typically below the diffraction limit, precisely applying and calibrating these potentials remains challenging. Here, we propose a simple and efficient experimental protocol that can be used to measure any potential with high precision. The key ingredient in our protocol is the ability in some atomic species to turn off interactions using a Feshbach resonance, which makes the evolution easy to compute. Given this, we demonstrate that collecting snapshots from the time evolution of a known, easily prepared initial state is sufficient to accurately estimate the potential. Our protocol is robust to state preparation errors and uncertainty in the hopping rate. This paves the way toward precision quantum simulation with arbitrary potentials.

Figures (3)

• Figure 1: Schematic of an optical lattice formed by counter-propagating lasers, generating periodic wells, with on-site energies $v_i$. The atoms can hop from one site to the next. At the end of the experiment, a quantum gas microscope (QGM) is used to read out the location of all the atoms in the lattice, which yields a snapshot. By running the evolution for different times, we can infer the applied potential from the snapshots.
• Figure 2: (a) Blue: Mean reconstruction error $\varepsilon_{\mathrm{MRE}}$ [\ref{['MRE']}] of the estimated potential differences averaged across all sites and many instances of the noise as a function of the maximal evolution time $t_{\max}$ for fixed $S = 10$ and $M = 10^{5}$, and for increasing levels of state-preparation and hopping errors $(\gamma, \Delta h)$. Errors accumulate with time for the dark blue curve due to miscalibration of the hopping strength. Red: Additionally fitting the hopping strength removes this error. (b) Mean reconstruction error $\varepsilon_{\mathrm{MRE}}$ plotted against the number of sampled time points $S$ for increasing system sizes $N$, with fixed $M = 10^{5}$, and $t_{\max} = 5$. The error decreases rapidly and saturates for $S\geq 5$ at the statistical noise floor determined by $M$ and $t_{\max}$, independent of system size. In both plots, the initial guess for gradient descent is the true potential profile.
• Figure 3: (a) Scaling of the mean reconstruction error ${\varepsilon_{\text{MRE}}}$ with the total number of samples $M$ and system size $N$. The error exhibits the expected statistical scaling $\varepsilon_{\text{MRE}}\!\sim\!1/\sqrt{M}$ and remains independent of $N$, confirming the scalability of the learning protocol. In the absence of experimental imperfections (blue curves), the optimizer reliably converges to the true potential profile. When state-preparation and hopping errors $(\gamma,\Delta h)\neq(0,0)$ are introduced (green curves), the optimization still converges but saturates at a biased minimum of the loss landscape. The red dashed curve shows the performance when both on-site potentials and hopping amplitudes are jointly optimized The true potential is generated as $v_i+\lambda\,\Omega_i$, with the optimizer initialized at $v_i$ and $\lambda=0.5$. (b) Dependence of ${\varepsilon_{\text{MRE}}}$ on the initialization uncertainty ${\lambda}$. Convergence to the true potential is maintained for $\lambda\!\lesssim\!1$, demonstrating that the optimization remains robust against substantial initialization mismatch. At larger $\lambda$, the errors grow as the initial guess departs too far from the true potentials.