Time-optimal force sensing with ultracold atoms

TL;DR

This work tackles time-efficient force sensing with a Bose-Einstein condensate in a shaken optical lattice. It introduces a fingerprinting (Fp)–based, time-optimal quantum sensing protocol that steers the system into interferometer-like momentum superpositions to maximize sensitivity, connecting Fp to the Quantum Fisher Information (QFI) and Classical Fisher Information (CFI). The authors show that the final-state manifold for a small parameter interval forms a geodesic in Hilbert space, yielding a constant QFI along the path and a minimum time scaling of $t^* \,\propto\, (\delta f)^{-1/3}$, with a robust, two-fold interferometric structure arising when momentum dispersion is finite. They demonstrate the approach experimentally for inertial and magnetic forces on $^{87}$Rb atoms in a one-dimensional lattice, achieving high sensitivity and robustness and outlining a general route to time-optimal quantum sensing across platforms.

Abstract

We develop a time-optimal approach to force sensing using a Bose-Einstein condensate in a shaken optical lattice. Optimal control protocols are derived from a Fisher information framework and yield optimal dynamics that spontaneously organize in intereferometer-like structures, where multiple interferences combine to maximize sensitivity. We analyse how measurement precision scales with control time and how the finite momentum dispersion of the condensate changes the optimal dynamics, observing an abrupt change of conformation from single- to double-folded interference structures for robust controls. The protocols are implemented experimentally for cold atoms subjected to inertial and magnetic forces, demonstrating high sensitivity and robustness. Our approach establishes a general route to time-optimal quantum sensing beyond standard interferometric architectures, applcable across all quantum platforms.

Figures (5)

• Figure 1: (Left) Schematic illustration of the convergence of the fingerprinting approach to a geodesic curve $\ket{\psi_\lambda(t^*)}$ between the target states $|\psi_1\rangle$ and $|\psi_2\rangle$. For a given parameter interval $\delta\lambda$, the time $t^*$ is the minimum control time required to reach the two states for two systems characterized by the values $\lambda_1$ and $\lambda_2$ chosen for fingerprinting. The times $t^{(1)}_{\rm f}$ and $t^{(2)}_{\rm f}$ verify $t^{(2)}_{\rm f}>t^{(1)}_{\rm f}>t^*$. (Right) Representation of an optimal control curve $\{\ket{\psi_\lambda(t^*)}\}$ on the Bloch sphere of the subspace generated by the states $|\psi_1\rangle$ and $|\psi_2\rangle$. The color of the curve denotes the fraction of the final state contained in the subspace. The control protocol producing this curve was derived for force sensing with a BEC in a lattice (see main text), and approaches a geodesic curve, which corresponds to a great circle between the poles of the Bloch sphere.
• Figure 2: (a). Experimental setup: the BEC is produced in a glass cell and trapped in an optical lattice potential produced by counter-propagating beams (red arrows) and subjected to the combined magnetic field of quadrupole coils along the $z-$axis, and horizontal coils along the lattice $x-$axis that allow to displace the zero of the field. (b) The lattice potential is characterized by its spacing $d$ and depth $s$. The control parameter $\varphi(t)$ translates the lattice along $x$, and an external constant force $\lambda$ is applied. (c-e) Typical numerical optimization results: (c) control $\varphi(t)$, (d) evolution of the quantum (red) and classical (blue) Fisher informations, and (e) final probability distribution over the states $\ket{\ell}$ for $\lambda=\lambda_1$ (orange) and $\lambda_2$ (green). The corresponding final distributions obtained in the absence of control are superimposed in faded colors. Parameters: $\lambda_1 = 0, \lambda_2 = 7.5\cdot 10^{-4}$.
• Figure 3: (a) Emerging interferometer pattern in the optimal time evolution of the momentum population in an infinite lattice with $q=0$, for $\lambda_1=0$ and $\lambda_2 = 7\cdot10^{-4}$. The dynamics are shown for $\lambda=\lambda_1$. (b) Optimal trajectory for a direct optimization of the QFI (see text). (c) Evolution of the minimum time $t^*$ as a function of $\delta \lambda$ for Fp (blue diamonds) and a direct maximization of the QFI (red diamonds). Numerical uncertainties are typically smaller than the symbol size. Linear regressions are shown in blue and red lines, yielding slopes of $-0.34 \pm 0.01$ and $-0.37 \pm 0.01$ respectively. The scaling $t^*\propto(\delta\lambda^*)^{-1/3}$ is shown by the dashed black line.
• Figure 4: (a) Evolution of the minimum time $t^*$ as a function of $\delta \lambda$ for the Fp approach (blue diamonds) at $q=0$, with $\lambda_1=0$ and a quasi-momentum robustness constraint with $\Delta q=0.002$ (yellow diamonds). With the robustness constraint, a sharp change of behaviour occurs (grey stripe). Insets show typical momentum distribution evolutions in the two regimes of the robust case. Linear regressions are shown in blue and yellow lines, yielding slopes of $-0.349(4)$ (blue) and $-0.35(3)$ and $-0.42(1)$ (yellow, for low and high values of $\delta\lambda^*$, respectively). (b-d) Experimental demonstration of a robust Fp control with $\lambda_1=10^{-3}$, $\lambda_2=7.15\cdot10^{-3}$, $t^*=400$, $\ket{\psi_1}=\ket{-1}$, $\ket{\psi_2}=\ket{1}$ and $\Delta q=0.06$. (b) Measured populations $p_{\pm1}$ in states $\ket{\pm1}$ as the inertial force $\lambda$ is varied. Measured standard errors on probability are smaller than marker size. Full lines show the numerical prediction, including a correction for the impact of scattering halos during imaging (see Appendix \ref{['app:exp']}). (c) Numerically computed Quantum (upper red curve) and Classical (lower blue curve) Fisher information. (d-e) Experimentally measured (blue, panel 1) and numerical (red, panel 2) momentum distribution evolutions for $\lambda=\lambda_1$, $\lambda_0$ and $\lambda_2$ respectively.
• Figure 5: (a) Measured populations $p_{\pm 1}$ in states $\ket{\pm 1}$ after the Fp protocol, as the initial position of the condensate in the quadrupole magnetic field is varied. Measured standard errors on probability and position are smaller than marker size. (b) Deduced local estimate of the dimensionless force $\lambda$ (blue disks). The curve shows the expected behavior from an offset quadrupole field, with the light blue area denoting the uncertainties in fitting parameters.