Wavefront Curvature and Transverse Atomic Motion in Time-Resolved Atom Interferometry: Impact and Mitigation

TL;DR

Time-resolved atom interferometry faces phase noise from wavefront curvature that couples to transverse atomic motion. The authors develop a semi-classical framework to quantify the per-trajectory perturbation $\delta\varphi$ and its ensemble average $\overline{\delta\varphi}$ in Gaussian beams, validating with Monte Carlo simulations. They show that the beam focus position $f$ can suppress curvature noise (notably near $|f|\approx z_R$) but at a cost to LMT-pulse efficiency, giving HEHN and LELN design regimes. To mitigate residual bias, they propose a position-resolved phase-shift readout that learns wavefront-induced biases from measurable quantities like the phase-shift gradient $\kappa$ and final position $\mu_x$, enabling restoration of high sensitivity for next-generation baselines up to $\sim$1 km.

Abstract

Time-resolved atom interferometry, as employed in applications such as gravitational wave detection and searches for ultra-light dark matter, requires precise control over systematic effects. In this work, we investigate phase noise arising from shot-to-shot fluctuations in the atoms' transverse motion in the presence of the wavefront curvature of the interferometer beam, and analyse its dependence on the laser-beam geometry in long-baseline, large-momentum-transfer atom interferometers. We use a semi-classical framework to derive analytical expressions for the effective phase perturbation in position-averaged measurements and validate them using Monte Carlo simulations. Applied to 100-m and 1-km atom gradiometers representative of next-generation experiments, the model shows that configurations maximizing pulse efficiency also amplify curvature-induced phase noise, requiring micron-level control of the atom cloud's centre-of-mass position and sub-micron-per-second control of its centre-of-mass velocity to achieve sub-$10^{-5}$ rad phase stability. Alternative beam geometries can suppress this noise by up to two orders of magnitude, but at the cost of reduced pulse efficiency. To address this limitation, we propose a mitigation strategy based on position-resolved phase-shift readout, which empirically learns and corrects the wavefront-induced bias from measurable quantities such as the phase-shift gradient and final cloud position. This approach restores high-sensitivity operation in the maximum-pulse-efficiency configuration without detailed beam characterisation, providing a practical route towards next-generation, time-resolved atom interferometers operating at the $10^{-5}$ rad noise level.

Figures (11)

• Figure 1: Schematic of a symmetric large-momentum-transfer (LMT, $n=3$) Mach--Zehnder atom interferometer. Blue (solid) and red (dotted) lines denote atoms in the ground ($\ket{g}$) and excited ($\ket{e}$) states. The vertical axis shows position $z$ and the horizontal axis time $t$, in a uniform gravitational field. Vertical arrows indicate laser pulses, with duration in units of $\Omega_0^{-1}$ indicated at the top; upward and downward arrows correspond to beams propagating along $\pm z$ with spatially varying phases $\phi^{\pm}(\bm{x})$. Gray arrows mark the LMT pulse blocks that generate momentum separation and recombination. After the final beam-splitter pulse at $2T$, the two arms recombine in the output ports $\ket{g}$ and $\ket{e}$, detected at time $t_{\mathrm{det}}$.
• Figure 2: Schematic of the long-baseline atom gradiometer configuration considered in this work. Two vertically separated Mach-Zehnder atom interferometers (top and bottom) are operated by a common Gaussian laser beam focused at position $z = f$. The beam originates from the top of the chamber (red arrow) and is retro-reflected by a mirror at $z = 0$, producing counter-propagating waves with spatially varying phases $\phi^{\pm}(x)$. The atomic ensembles (represented by blue disks), launched with initial centre-of-mass coordinates $(\mu_{x_0}, \mu_{z_0})$ and velocities $(\mu_{v_{x0}}, \mu_{v_{z0}})$, follow ballistic trajectories under uniform gravity. The gradiometer baseline $L$ is defined by the initial vertical separation between the two clouds.
• Figure 3: Phase shift for a single atom in a Gaussian beam at different focus positions $f$: comparison between numerical simulations and the analytical model of Eq. \ref{['eq:one-atom-phase']}. Each panel shows the phase shift $\Delta\varphi(x_0,v_{x_0})-\Delta\varphi_0$ as a function of the initial transverse position $x_0$ for several initial transverse velocities $v_{x_0}$, with $y_0 = v_{y_0} = 0$. Dashed lines correspond to the analytical model using the nominal interrogation time $T$, and dotted lines to the model with the effective interrogation time $T_{\mathrm{eff}}=\sqrt{T(T-n\tau)}$. Simulations (circles) were performed for a symmetric LMT Mach--Zehnder sequence on the $698$ nm clock transition of $^{87}$Sr with $n=1001$, Rabi frequency $\Omega_0=2\pi\times1$ kHz ($\tau=0.5$ ms), beam waist $w_0=3$ cm, launch velocity $v_{z_0}=19.62$ m/s, and interrogation time $T=2.225$ s ($T_{\mathrm{eff}}=1.99$ s). The simulations account for the finite duration of the LMT pulses, while the analytical model assumes instantaneous pulses. The global phase offset $\Delta\varphi_0$ is extracted from the simulation. Note that throughout the paper, the range of initial positions and velocities in simulations is chosen to be much larger than the typical fluctuations expected in a real experiment, to emphasize the effect of wavefront curvature.
• Figure 4: normalised ground-state population as a function of the initial transverse COM position and velocity in a $1001\hbar k$ MZ sequence: comparison between Monte Carlo simulations and our analytical model. The normalised population is computed for $10^6$ initially Gaussian-distributed atoms with varying initial transverse COM position $\mu_{x_0}$ and velocity $\mu_{v_{x0}}$. Simulation parameters are the same as in Fig. \ref{['fig:teff-plots']}. Dashed lines represent the analytical prediction $\frac{1}{2}(1+\cos(\Delta\varphi_0+\overline{\delta\varphi}))$, where $\Delta\varphi_0$ is a global phase offset obtained using the simulation results, and $\overline{\delta\varphi}$ is the analytical model for the effective phase perturbation in Eq. \ref{['eq:pa-bias-gaussian']}, using the effective interrogation time $T_{\text{eff}} = \sqrt{T(T-n\tau)}$ (Eq. \ref{['eq:teff']}), and assuming $f=0$. Confidence intervals are obtained using bootstrapping.
• Figure 5: Curvature-induced differential phase noise in position-averaged readout versus focus position $f$, and assuming fixed COM shot-to-shot fluctuations: $\Delta\mu_{x_0}=10$$\mu$m, $\Delta\mu_{v_{x0}}=10$$\mu$ms$^{-1}$. Interrogation time and initial launch velocity as in Fig. \ref{['fig:teff-plots']}.