A fast and accurate method for simulating Bragg atom interferometers

TL;DR

This work tackles systematic effects in Bragg atom interferometers arising from finite momentum width by solving the 1D time-dependent Schrödinger equation for Bragg scattering. It introduces a fast reduction to decoupled ordinary differential equations in momentum space by exploiting the delta-function-like coupling of the sinusoidal Bragg potential, leading to discretized states $\phi_{n,s}=\phi(2\hbar k n+\hbar k s/S)$ and the evolution $i\hbar \dot{\phi}_{n,s} = \hbar \omega_{\mathrm{r}}(2n + s/S)^2 \phi_{n,s} + \hbar \frac{\Omega_{\mathrm{eff}}(t)}{8}[\phi_{n-1,s} e^{-i\delta t} + \phi_{n+1,s} e^{i\delta t} + 2\phi_{n,s}]$. The approach supports adaptive Runge–Kutta integration and can be augmented by a lookup table to accelerate repeated runs, enabling efficient simulation of long-baseline interferometers and realistic atomic clouds with parasitic momentum states. The results show that the method is faster and more accurate than standard position-space solvers for fixed-pulse Bragg dynamics, while noting that split-step remains more versatile for non-periodic potentials.

Abstract

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.

Figures (5)

• Figure 1: The discretization scheme used when setting $\phi_{n,s}=\phi(2\hbar kn+\hbar k s/S)$ for the case where $S=3$. The upper plot shows an initial momentum space wavefunction, and the lower plot shows the same wavefunction after undergoing a 1st order Bragg pulse. Each discretized point is indicated by a vertical line. For each value of $n$ there is a bin between $n2\hbar k-\hbar k$ and $n2\hbar k+\hbar k$ which contains 6 discretized points. The value of $s$ at each of these points is shown on the top axis. The value of $n$ for each bin is also shown on the top axis. The arrows are each $2\hbar k$ long, and indicate which discretized points are coupled to each other in momentum space (coupled discretized points share the same color). Only three arrows are shown here for simplicity, but in reality every discretized point has two arrows pointing out of it in opposite directions as it is coupled to two other momentum states. Note that coupled states share the same value of $s$, but have different values of $n$. The gray bars indicate the population attributed to each particular discretized state.
• Figure 2: The process by which the split-step, Crank-Nicolson, and the method presented here are compared. Each method is used to simulate a full Ramsey-Borde interferometer composed of Bragg beamsplitters. Simulations are performed for a variety of $\omega_\mathrm{m}$ values, where $\omega_\mathrm{m}$ is set to $\omega_\mathrm{m}=8n_\mathrm{Bragg}\omega_\mathrm{r}-\theta/Tn$ for 50 values of $\theta$ spaced from $-\pi$ to $\pi$. This will trace the interferometer phase over a full fringe. The output wavefunction $\psi(x)$ is then Fourier transformed to obtain the momentum space wavefunction $\phi(p)$, which is integrated over $2\hbar k$ to find $p_n=\abs{\phi_n}^2=\int_{2\hbar k(n-1/2)}^{2\hbar k(n+1/2)}\phi(p')^*\phi(p')\mathrm{d}p$. The ratio of population in each output port is calculated via $(p_{n_\mathrm{initial}}-p_{n_\mathrm{final}})/(p_{n_\mathrm{initial}}+p_{n_\mathrm{final}})$, which is fit to a sinusoid to determine the phase $\theta$ of the simulated interferometer. The difference between this phase and the target phase is due to systematic effects such as parasitic momentum states. These plots were generated with $n_\text{Bragg}=4$ using the method presented here.
• Figure 3: A comparison of the Ramsey-Borde interferometer phase computed by the method presented here, the split-step method (ss), and the Crank-Nicolson method (cn). The interferometer parameters are described in the text.
• Figure 4: A simulation of a Ramsey-Borde interferometer composed of 4th order Bragg pulses. The displayed heatmap is $|\psi(x,t)|^2$. The Bragg pulses are Gaussian in time with $\sigma=0.26\omega_\text{r}^{-1}$, the time between the first two and last two Bragg pulses is $T=259.62\omega_\text{r}^{-1}$, and the time between the second and third Bragg pulses is $T'=25.96\omega_\text{r}^{-1}$, and the initial wavepacket is a minimum uncertainty Gaussian with $\sigma_p=0.1\hbar k$. In the case of a Cesium atom these numbers correspond to a Bragg pulse with $\sigma=20\,\mathrm{\mu s}$, $T=20\,\mathrm{ms}$, $T'=2\,\mathrm{ms}$ and a velocity spread of a tenth of a recoil velocity. The insets show the first and last Bragg pulses. The bottom plot contains identical data to the middle plot but the maximum color value has been lowered to make the parasitic interferometers clearer. The top right inset displays an area 8 times larger than the top left inset. In this simulation $S=3041$ and $N=15$. Computing the final wavefunction takes $2500\,\mathrm{s}$ on a single core on an AMD 3990X without a lookup table.
• Figure 5: Left: This graphic illustrates how atoms in a cloud experience a range of potential depths due to their transverse position spread. Atoms are depicted as solid blue circles. The arrows pointing out of the atoms describe the atom velocity, which can have both vertical and transverse components. The XZ and YZ planes show a projection of the atoms (slightly transparent blue circles), along with arrows depicting their velocity. Propagating the wavefunction of these atoms requires solving the Bragg diffraction Hamiltonian for the potential depth experienced at each point in the atom trajectory. Right: These plots show the simulated phase deviation and contrast for a Ramsey-Borde interferometer composed of 4th order Bragg pulses where $\sigma=0.26\omega_\mathrm{r}^{-1}$, $T=64.9\omega_\mathrm{r}^{-1}$, $T'=64.9\omega_\mathrm{r}^{-1}$, $S=1102$, $N=9$, $\Omega_{\mathrm{eff},0}=20\omega_\mathrm{r}^{-1}$, $\sigma_\mathrm{cloud}=1.5\,\mathrm{mm}$, and $\sigma_{v_T,\mathrm{cloud}}=3.5\,\mathrm{mm/s}$. In the case of a Cesium atom these numbers correspond to a Bragg pulse with $\sigma=20\,\mathrm{\mu s}$, $T=5\,\mathrm{ms}$, and $T'=5\,\mathrm{ms}$. 1000 atoms are sampled for the cloud simulations. The phase deviation and contrast curves are generated using the same process described in Fig.(\ref{['fig:phase_extraction']}). Each simulated interferometer is prepared to zero the accumulated Ramsey-Borde interferometer phase. The meaning of "masked" is described in the text. Using a single core on an AMD 3990X it takes $0.6\,\mathrm{s}$ to simulate a full interferometer for each atom in the cloud using the lookup table method. Generating the above plot took 35 hours with multiple cores utilized and required simulating 800,000 single interferometers.