Diffraction phase-free Bragg atom interferometry

Abstract

Bragg Diffraction of matter waves is an established technique used in the most accurate quantum sensors. It is also the method of choice to operate large-momentum-transfer, high-sensitivity atom interferometers. It suffers, however, from an intrinsic multi-path character. Optimal control theory (OCT) has recently led to an improved robustness of atom interferometers to a range of challenging environmental effects such as vibrations or platform accelerations. In this theoretical work, we apply OCT protocols to control the Bragg diffraction phase shifts thereby enhancing the metrological accuracy of the interferometer. We show a minimization of the diffraction phase for realistic conditions of finite temperature of the incoming wavepacket in a multi-path, high-order Bragg interferometer in a Mach-Zehnder configuration. We study input states with different momentum widths and find that our approach mitigates diffraction phases below the microradian level in the case of $1\%$ of the photon recoil, thereby eliminating one of the leading systematic effects in atom interferometry.

Figures (9)

• Figure 1: a) Multi-path nature of Bragg atom interferometer for a MZ interferometer, showing an $n$th-order Bragg beam splitter populating the main trajectories (solid black lines), open ports, and parasitic paths (dashed lines; the dominant ones for $n=5$ are thicker), all affecting the Mach-Zehnder (MZ) signal recorded in ports a and b. Fig. adapted from JanniPRL. b) MZ signal phase scan of the population in port a ($P_a$) by scanning over the relative phase $\Phi_\mathrm{L}$ of the final beam splitter. The red lines represent the extraction of the mid-fringes by solving $P_a(\Phi)= 0.5$, where $P_a(\Phi_\mathrm{L})$ is the fitted two-mode estimator for the population in port a, see Eq. \ref{['eq:signalmodel']}.
• Figure 2: Study of an optimized beam splitter for the case of an initial state with momentum distribution of $\sigma_p = 0.1 \hbar k$ and Bragg order $n=5$. a) Top: Populations after the interaction with an optimal Gaussian pulse with a peak Rabi frequency of $\Omega_0 =34 \omega_r$ and a pulse width of $\tau=0.145 \omega_r^{-1}$, where we have the relative populations in $\ket{5\hbar k}$ and $\ket{-5\hbar k}$ of $0.5458$ and $0.2848$, respectively. Bottom: Populations after the interaction with an optimal OCT pulse where we have the relative populations in $\ket{5\hbar k}$ and $\ket{-5\hbar k}$ of $0.4997$ and $0.4980$, respectively b) OCT pulse parameters, where a cutoff frequency for the OCT of $\omega_c = 95$ kHz has been used for the optimization.
• Figure 3: Diffraction phase for Bragg diffraction order $n=3$. The first and second rows show the diffraction phase at the first and second mid-fringe, respectively. The columns, from left to right, correspond to $\sigma_p \in \{0.01,0.1,0.3\}\,\hbar k$. In each plot, the solid blue horizontal line indicates the systematic phase shift, the empty circles represent the simulated data, and the orange curve corresponds to Eq. \ref{['eq:OscFitModel']} fitted to the simulations. Only very small residual diffraction-phase oscillations are visible, demonstrating the performance of OCT pulses. For $\sigma_p \in \{0.01,0.1,0.3\}\,\hbar k$, the populations and contrasts $(P_{\text{out}},C)$ are $(0.99998,0.99997)$, $(0.99309,0.99567)$, and $(0.87722,0.70157)$, respectively.
• Figure 4: Diffraction phase for Bragg diffraction order $n=5$. The first and second rows show the diffraction phase at the first and second mid-fringe, respectively. The columns, from left to right, correspond to $\sigma_p \in \{0.01,0.1,0.3\}\,\hbar k$. In each plot, the solid blue line indicates the model shift, the empty circles represent the simulated data, and the orange curve corresponds to Eq. \ref{['eq:OscFitModel']} fitted to the simulations. Only very small residual diffraction-phase oscillations are visible, demonstrating the performance of OCT pulses. For $\sigma_p \in \{0.01,0.1,0.3\}\,\hbar k$, the populations and contrasts $(P_{\text{out}},C)$ are $(0.99994,0.99997)$, $(0.99157,0.98874)$, and $(0.78424,0.69266)$, respectively.
• Figure 5: Momentum distribution of an incoming state $\ket{-5 \hbar k}$ with momentum width $\sigma_p = 0.1 \hbar k$ after interacting with the laser that generates a beam splitter pulse. Top: Gaussian pulse optimized for the unitary submatrix of the interaction, yielding $\left(\Omega_0, \tau\right)= \left(34\,\omega_r,0.145\,\omega_r^{-1}\right)$. Bottom: Gaussian pulse optimized for an equal population split between the states $\ket{5 \hbar k}$ and $\ket{-5 \hbar k}$, we obtain $\left(\Omega_0, \tau\right)= \left(30.952\,\omega_r,0.210 \,\omega_r^{-1}\right)$.