Scalable Spin Squeezing for Quantum-Enhanced Magnetometry with Bose-Einstein Condensates

A major challenge in quantum metrology is the generation of entangled states with a macroscopic atom number. Here, we demonstrate experimentally that atomic squeezing generated via nonlinear dynamics in Bose-Einstein condensates, combined with suitable trap geometries, allows scaling to large ensemble sizes. We achieve a suppression of fluctuations by 5.3(5) dB for 12 300 particles, from which we infer that similar squeezing can be obtained for more than ${10}^7\mathrm{atoms}$. With this resource, we demonstrate quantum-enhanced magnetometry by swapping the squeezed state to magnetically sensitive hyperfine levels that have negligible nonlinearity. We find a quantum-enhanced single-shot sensitivity of 310(47) pT for static magnetic fields in a probe volume as small as $90\mu {\mathrm{m}}^3$.

Figure 1

Scaling squeezing to large ensemble sizes. (a) Independent squeezing (displayed on generalized Bloch sphere) of 25 binary BECs in a 1D lattice gives access to large atom numbers by summing over adjacent lattice sites. (b) Scaling of the squeezed state after 20 ms of nonlinear evolution. A relative analysis using adjacent parts of the lattice (inset) yields ${\xi }_{\text{rel}}^2=-5.3(5)\mathrm{dB}$ for the full sample of 12 300 atoms (blue circles), compared to ${\xi }_{\mathrm{N}}^2=-1.3(6)\mathrm{dB}$ using direct analysis (open squares). The corresponding measurement for an initial coherent spin state (black diamonds) is at the classical shot noise limit. (c) Analysis for different tomography rotation angles $\alpha$ and 12 300 atoms shows the sinusoidal behavior from the redistributed uncertainties for direct (open squares) and relative analysis (filled circles). Error bars are the statistical 1 s.d. error.

Figure 2

Swapping squeezed states for a quantum-enhanced Ramsey scheme. (a) After 20 ms of nonlinear evolution on all lattice sites (left panel), the squeezed states are rotated to the phase sensitive axis (middle panel). By swapping the population from state $|b⟩$ to $|c⟩$, the nonlinearity is effectively switched off and the magnetic field sensitivity is significantly enhanced (right panel). After magnetic field-dependent phase evolution, the population is swapped back before readout. (b) Squeezing tomography after a hold time of $1\mu \mathrm{s}$ (negligible magnetic field phase) confirms relative squeezing of ${\xi }_{\text{rel}}^2=-{5.1}_{-0.7}^{+0.6}\mathrm{dB}$ [$-3.8(5)\mathrm{dB}$ with detection noise] after swapping and readout. (c) Ramsey fringe for a final $\pi /2$ pulse with variable readout phase $\mathrm{\Phi }$ yields a visibility of $\mathcal{V}=0.950(5)$. This implies metrologically relevant spin squeezing of $-3.4(5)\mathrm{dB}$ and direct applicability of the system for quantum-enhanced gradiometry. Error bars are the statistical 1 s.d. error.

Figure 3

Ramsey magnetometry beyond the standard quantum limit. (a) A magnetic field gradient translates into a phase shift of the corresponding Ramsey fringes for left and right parts of the full sample (upper panel for $t_{\text{int}}=342\mu \mathrm{s}$). The difference in population imbalance $\delta z=z_{\text{left}}-z_{\text{right}}$ is maximal at the zero crossings of the individual fringes (lower panel). (b) Magnetic field sensitivity of the Ramsey magnetometer versus interrogation time. For the full sample, we find quantum-enhanced performance up to interrogation times of $342\mu \mathrm{s}$ with a single shot sensitivity of 310(47) pT for static magnetic fields. For longer times, we are limited by fluctuations of the magnetic field. The shaded area depicts the region which is accessible for nonentangled states, the dashed line indicates the standard quantum limit including detection noise. Error bars show the statistical 1 s.d. confidence intervals obtained from a resampling method.

Figure 4

Magnetic field gradiometry with an array of condensates. (a) A magnetic field gradient is deduced from the fringe amplitude of $\delta z$ (insets), which grows linearly with interrogation time (blue circles and linear fit). We find $\partial B_z/\partial x=19.6(6)\mathrm{pT}/\mu \mathrm{m}$. (b) Gradiometric sensitivity for $t_{\text{int}}=342\mu \mathrm{s}$. For increasing baseline length $d$ between single wells, we find a gain of gradient sensitivity (triangles) below the classical standard quantum limit (SQL, upper dashed line). The deviation from the expected linear behavior is due to the decrease of atom number toward the edges of the array. The sensitivity can be further increased (squares) by summing the populations of lattice sites as indicated in the inset, and beats the corresponding standard quantum limit for the summed ensemble (lower dashed line). Error bars are the 1 s.d. confidence from a resampling method.