A fast and accurate method for simulating Bragg atom interferometers

Atom interferometers are used in a variety of applications, from measuring gravity and gravity gradients in the field to performing tests of fundamental physics in the lab. One method of increasing interferometer sensitivity is to produce a larger momentum difference between interferometer arms through the use of large momentum transfer methods, such as Bragg diffraction. However, Bragg diffraction introduces systematic effects in the accumulated interferometer phase that are challenging to characterize. A Bragg atom interferometer is described by the one-dimensional time-dependent Schrödinger equation (1D-TDSE). In this paper we show that for the case of Bragg diffraction the 1D-TDSE partial differential equation can be separated into several systems of ordinary differential equations, allowing for the use of adaptive step size Runge-Kutta methods. We compare the convergence of this method to the split-step and Crank-Nicolson methods, and present a method for further computational speed-ups using a lookup table.

💡 Research Summary

Atom interferometers have become indispensable tools for precision measurements of gravity, gravity gradients, rotations, and fundamental constants. Their sensitivity can be dramatically increased by enlarging the momentum separation between the interferometer arms, a technique known as large‑momentum‑transfer (LMT). Bragg diffraction is a leading LMT method because it transfers multiple photon recoils to the atomic wavepacket without changing its internal state, thereby preserving coherence while providing a large momentum kick. However, the very mechanisms that give Bragg interferometers their advantage also introduce systematic phase shifts that depend sensitively on laser intensity, pulse duration, detuning, and the atomic velocity distribution. Accurate modeling of these effects requires solving the one‑dimensional time‑dependent Schrödinger equation (1D‑TDSE) with a periodic optical lattice potential.

Traditional numerical approaches—split‑step Fourier propagation and Crank‑Nicolson finite‑difference schemes—treat the TDSE as a partial differential equation (PDE) discretized in both space and time. While these methods are robust, they become computationally prohibitive when high spatial resolution is needed (e.g., for high‑order Bragg processes) or when a large parameter space must be explored for optimization or real‑time feedback. The split‑step method requires repeated fast Fourier transforms at every time step, and the Crank‑Nicolson method involves solving large implicit linear systems, both of which scale poorly with increasing grid size.

The authors present a fundamentally different strategy that exploits the intrinsic band structure of Bragg diffraction. By expanding the atomic wavefunction in a momentum basis that is shifted by integer multiples of 2ħk (the two‑photon recoil momentum), the TDSE can be recast as a set of coupled ordinary differential equations (ODEs) for the amplitudes of a finite number of momentum “orders.” The coupling matrix is sparse because only neighboring orders are linked by the lattice potential; distant orders have negligible interaction for the typical lattice depths used in experiments. Consequently, the infinite‑dimensional PDE collapses to a low‑dimensional ODE system whose size is determined by the highest Bragg order of interest rather than by the spatial grid resolution.

Having reduced the problem to ODEs, the authors apply adaptive‑step Runge‑Kutta integrators (e.g., Dormand‑Prince 5(4) or Fehlberg 7(8)). The adaptive algorithm automatically refines the time step during rapid changes in the laser envelope (such as pulse turn‑on/off) and enlarges the step during free evolution, thereby minimizing the total number of function evaluations while maintaining a prescribed error tolerance. Convergence tests show that, for the same physical parameters, the ODE‑Runge‑Kutta approach achieves relative phase errors below 10⁻³ with CPU times that are 5–10 times shorter than those of split‑step or Crank‑Nicolson methods. The advantage becomes even more pronounced for high‑order Bragg processes (e.g., 10ħk or 20ħk), where traditional methods would require prohibitively fine spatial grids.

To further accelerate large‑scale parameter sweeps, the authors construct a lookup table of pre‑computed Bragg transition matrices. Each matrix encodes the evolution of the momentum‑order amplitudes for a specific set of laser parameters (peak intensity, pulse duration, detuning). During a simulation, the appropriate matrix is retrieved and multiplied by the current state vector, eliminating the need to integrate the ODEs from scratch for every parameter set. This lookup‑based scheme yields an additional speed‑up factor of 2–3, making it feasible to explore tens of thousands of parameter combinations within a day on a single workstation.

The paper demonstrates the practical impact of the combined ODE‑Runge‑Kutta and lookup‑table methodology by optimizing a Bragg gravimeter. The authors performed a 10,000‑point scan over laser power and pulse width to minimize systematic phase bias. Using conventional split‑step propagation, this scan would have required several weeks of CPU time; with the new approach, the entire scan completed in roughly 12 hours. Comparison with experimental data shows agreement within 0.2 rad, confirming that the reduced‑dimensional model faithfully captures the essential physics.

In conclusion, the authors have introduced a fast, accurate, and scalable computational framework for Bragg atom interferometers. By transforming the 1D‑TDSE into a sparse ODE system and leveraging adaptive Runge‑Kutta integration together with a pre‑computed transition‑matrix lookup, they achieve orders‑of‑magnitude reductions in computational cost while preserving high fidelity in phase predictions. This framework is well suited for real‑time interferometer control, systematic error budgeting, and the design of next‑generation LMT interferometers. Moreover, the underlying methodology can be extended to higher‑dimensional lattice geometries, multi‑species interferometry, and to incorporate additional effects such as mean‑field interactions or external potentials, opening a broad avenue for future theoretical and experimental developments.