Large-Momentum-Transfer Atom Interferometers with $\mathrm{\ensuremath{\mu}}\mathrm{rad}$-Accuracy Using Bragg Diffraction

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Large-Momentum-Transfer Atom Interferometers with $μ rad$ -Accuracy Using Bragg Diffraction

J.-N. Kirsten-Siemß, F. Fitzek, C. Schubert, E. M. Rasel, N. Gaaloul, and K. Hammerer

Phys. Rev. Lett. 131, 033602 – Published 19 July 2023

Abstract

Large-momentum-transfer (LMT) atom interferometers using elastic Bragg scattering on light waves are among the most precise quantum sensors to date. To advance their accuracy from the mrad to the $μ rad$ regime, it is necessary to understand the rich phenomenology of the Bragg interferometer, which differs significantly from that of a standard two-mode interferometer. We develop an analytic model for the interferometer signal and demonstrate its accuracy using comprehensive numerical simulations. Our analytic treatment allows the determination of the atomic projection noise limit of a LMT Bragg interferometer and provides the means to saturate this limit. It affords accurate knowledge of the systematic phase errors as well as their suppression by 2 orders of magnitude down to a few $μ rad$ using appropriate light-pulse parameters.

Received 13 August 2022
Revised 20 February 2023
Accepted 25 May 2023

DOI:https://doi.org/10.1103/PhysRevLett.131.033602

Physics Subject Headings (PhySH)

Atom interferometry Cold atoms & matter waves Quantum optics

Adiabatic approximation Atom diffraction Mach-Zehnder atom interferometry

Atomic, Molecular & Optical

Authors & Affiliations

J.-N. Kirsten-Siemß ^1,2,*, F. Fitzek ^1,2, C. Schubert^2,3, E. M. Rasel², N. Gaaloul ², and K. Hammerer ^1,†

¹Leibniz Universität Hannover, Institut für Theoretische Physik, Appelstraße 2, D-30167 Hannover, Germany
²Leibniz Universität Hannover, Institut für Quantenoptik, Welfengarten 1, D-30167 Hannover, Germany
³Deutsches Zentrum für Luft- und Raumfahrt e.V. (DLR), Institut für Satellitengeodäsie und Inertialsensorik, Callinstraße 30b, D-30167 Hannover, Germany

^*jan-niclas.siemss@itp.uni-hannover.de, he/him/his
^†klemens.hammerer@itp.uni-hannover.de

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Vol. 131, Iss. 3 — 21 July 2023

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Figure 1
Space-time diagrams of MZ interferometers. Top row: $n$ th-order Bragg beam splitters (BS) populating the main trajectories (solid black lines), open ports, and parasitic paths (dashed lines; the dominant ones for $n = 5$ are thick blue lines), all affecting the MZ signal recorded in ports $a$ , $b$ . Trajectories are shown in the optical lattice frame. Bottom row: Gaussian pulses $Ω (t)$ with pulse widths $τ_{BS, M}$ spaced by time $T$ . While in panel (a) we choose identical peak Rabi frequencies $Ω$ (which we refer to as MZ type A), panel (b) illustrates deflection of dominant spurious paths with a tailored choice for $Ω_{M}, τ_{M}$ (MZ type B).
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Figure 2
MZ signal phase scan in port $a$ . (a),(b) Comparison of numerical simulations $P_{a}^{meas} (ϕ)$ (symbols) with analytical predictions of Eqs. (1) (orange dashed line) and (2) (green dotted line) and with the sinusoidal signal, $P_{a}^{ideal} (ϕ) = (1 + \cos n ϕ) / 2$ (gray solid line). The parameters $Ω, τ_{BS}, τ_{M} = 28.5 ω_{r}, 0.309 ω_{r}^{- 1}, 0.681 ω_{r}^{- 1}$ cause approximately 1.4% beam-splitter losses and amplify signal distortions. Here, $ω_{r} = ℏ k^{2} / 2 m$ is the recoil frequency of an atom with mass $m$ , and we scan the relative phase $ϕ_{L} (t) = ϕ_{L}$ of the final beam splitter. Zooms in panels (b) and (c) reveal a mrad shift relative to $P_{a}^{ideal} (ϕ)$ . Using adapted mirror parameters $Ω_{M}, τ_{M} = 31.8 ω_{r}, 0.463 ω_{r}^{- 1}$ as in Fig. 1, panel (c) confirms agreement between the numerics and both analytical models at the mrad level, which we quantify below.
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Figure 3
Phase estimation error. Diffraction phases $δ ϕ$ in Eq. (3) evaluated for numerically simulated MZ interferometers with different separation times $T$ . Operation at the so-called “magic” Bragg beam-splitter duration for $n = 5$ (see main text) reduces oscillation amplitudes for both MZ configurations to less than or equal to $200 μ rad$ due to small diffraction losses of approximately 0.18% (cf. Refs. [48, 49]). Open circles (MZ type A) assume parameters $Ω, τ_{BS}, τ_{M} = 30.75 ω_{r}, 0.218 ω_{r}^{- 1}, 0.519 ω_{r}^{- 1}$ , while the adapted Bragg mirror $Ω_{M}, τ_{M} = 31.8 ω_{r}, 0.463 ω_{r}^{- 1}$ (closed circles, MZ type B) further suppresses the peak-to-peak value by a factor of 5 to less than $40 μ rad$ . Fits to the data (solid lines) show identical offsets $B_{0} \approx - 27 μ rad$ (dotted line) for both sets.
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Figure 4
Diffraction phase suppression. (a,b) Data points characterizing oscillations in $δ ϕ$ for MZ configurations A (open symbols) and B (closed symbols) as in Fig. 3, but for a range of pulse parameters; see SM [36]. Lines between symbols serve as a guide for the eye. (a) Offsets $| B_{0} | \leq 30 μ rad$ (symbols), which are largely identical for both MZ types. Panel (b) shows that in case B, peak-to-peak (PP) values are mostly below 1 mrad and less than $10 μ rad$ for sufficiently long pulse durations. (c) Numerically determined diffraction loss of a fifth-order Bragg beam splitter (coupling $| \pm 5 ℏ k ⟩$ ). Parameters used in Fig. 2 (Fig. 3) correspond to the visible local maximum (minimum) denoted by the solid (dotted) vertical line.
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Figure 5
Sensitivity bounds for uncorrelated particles. Cramér-Rao bound (dashed lines) and quantum Cramér-Rao bound (solid lines) for $n = 5$ scanning $Ω_{BS}$ , $τ_{BS}$ with $T = 10 ms$ . Results are scaled to the projection noise of an ideal two-mode MZ, $n \sqrt{N_{atoms}}$ . Symbols represent the statistical uncertainty $Δ ϕ^{est} (ϕ)$ of phase estimates based on Eq. (2) applied to numerical MZ signals. Both CRB and $Δ ϕ^{est}$ are obtained at midfringe $(ϕ = \frac{3 π}{2} \frac{1}{5})$ . The adapted Bragg mirror $Ω_{M}, τ_{M} = 31.8 ω_{r}, 0.463 ω_{r}^{- 1}$ (case B) suppresses parasitic interference and, compared to case A in the inset, is less susceptible to the finite velocity spread of the wave packet $σ_{p} = 0.01 ℏ k$ ( $0.05 ℏ k$ ) in blue (purple) at longer pulse durations. The dotted (solid) vertical line again corresponds to the local minimum (maximum) in the losses in Fig. 4.
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Physical Review Letters

Large-Momentum-Transfer Atom Interferometers with μrad-Accuracy Using Bragg Diffraction

J.-N. Kirsten-Siemß, F. Fitzek, C. Schubert, E. M. Rasel, N. Gaaloul, and K. Hammerer

Phys. Rev. Lett. 131, 033602 – Published 19 July 2023

Abstract

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Large-Momentum-Transfer Atom Interferometers with $μ rad$ -Accuracy Using Bragg Diffraction