Estimating applied potentials in cold atom lattice simulators

The gist The key result is that time-resolved measurement of diagonals of the correlation matrix Cii(t) provides sufficient information to reconstruct the actual potential landscape

Motivation and Problem

Cold atom quantum simulators are a versatile and highly controllable platform for quantum simulation, capable of realizing a broad family of Hubbard models A key objective in this context is the ability to realize arbitrary site-dependent optical potentials, that is, to independently tune the on-site energy at each lattice site While a lot of progress has been made towards singlesite controllability, it is fundamentally limited by optical diffraction, which means that potentials can only be realized with finite spatial resolution even in state-of-the-art setups This results in a situation where the difference between the expected and actual implemented Hamiltonian will lead to a simulation error that accumulates with time, thus preventing high-precision quantum simulation.

Proposed Protocol Overview

The paper proposes 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.

Learning Methods

The paper presents two complementary approaches for learning the applied potential:

1. Rigorous protocol: This involves fitting a polynomial to measured data, where Equation (4) defines a recursive triangular system that can be solved for the potential differences ∆i via forward substitution To estimate the second derivative of the time-evolved correlation matrix entry Cii(t) at t = 0, robust polynomial interpolation is applied using times distributed according to the Chebyshev measure on [t0, tmax]. However, numerical findings indicate that this approach requires a large number of snapshots R ∼ d 8/ϵ squared > 107 for experimental feasibility.

2. Efficient, heuristic protocol: This method minimizes a cost function K(v; C(0)) defined as the mean squared error between predicted occupation and measured data. The paper finds that a simple charge density wave is sufficient as an initial state, and this approach is adopted in the remainder of the paper.

Numerical Experiments and Scalability

The numerical experiments systematically analyze how maximal evolution time tmax, the number of sampled time points S, and system size N influence reconstruction accuracy The mean reconstruction error εMRE scales as εMRE ∼ 1/√M and remains largely independent of N, confirming the scalability of the protocol. The optimizer converges to the true potential configuration in the absence of imperfections (γ, ∆h) = (0, 0). When state-preparation and hopping errors are present, convergence is still achieved but saturates at a small bias. Joint optimization over both on-site potentials and hopping amplitudes further reduces this bias. The protocol achieves a mean reconstruction error ≤ 5% with about 3 × 10 4 snapshots for systems up to 100 sites, which can be completed in approximately eight hours.

Implications and Outlook

This framework serves as a scalable Hamiltonian potential-learning tool for calibration when only limited experimental data is available The method achieves convergence to the global minimum and remains robust even in the presence of state-preparation errors and uncertainties in the hopping amplitude. It can be implemented using existing cold-atom technology with global quenches, tunable interactions, and site-resolved imaging. One potential application is as a Hamiltonian-driven shadow-calibration method [39]. The protocol also serves as a robust analogcompatible shadow-tomography scheme that numerically learns the true inversion map for Gaussian free fermions.

ACKNOWLEDGEMENTS

We are grateful to Max McGinley and Matt Kiser for collaboration in a related project, and to Daniel Stilck Fran¸ca, Albert H. Werner, Andreas Elben, and Rune Thinggaard Hansen for additional stimulating discussions. We acknowledge financial support by the Novo Nordisk Foundation under grant numbers NNF22OC0071934 and NNF20OC0059939

References

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1. No local control requirement for potential implementation: The protocol is implementable without additional experimental modifications (in particular, no local control), which allows existing cold-atom setups to be used without needing complex hardware additions for site-dependent potentials.

2. Classical efficiency and robustness: The method achieves a cost function minimization approach that avoids any computation of higher-order derivatives, leading to a classically efficient postprocessing scheme robust to errors, as demonstrated by the scaling where the error follows the expected statistical scaling εMRE ∼1/√M, and remains largely independent of N.

3. Scalability for large systems: The protocol confirms that the learning process is scalable because the mean reconstruction error is independent of N, meaning it can be applied to systems up to 100 sites efficiently with a total of about 3 × 104 snapshots.

4. Mitigation of experimental imperfections: Joint optimization over both on-site potentials and hopping amplitudes further reduces this bias, allowing the system to recover the true potential even when state-preparation and hopping errors are present or when the realized potential deviates from the intended one by a factor λ.

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.