Experimental distributed quantum sensing in a noisy environment

Abstract

The precision advantages offered by harnessing the quantum states of sensors can be readily compromised by noise. However, when the noise has a different spatial function than the signal of interest, recent theoretical work shows how the advantage can be maintained and even significantly improved. In this work we experimentally demonstrate the associated sensing protocol, using trapped-ion sensors. An entangled state of multi-dimensional sensors is created that isolates and optimally detects a signal, whilst being insensitive to otherwise overwhelming noise fields with different spatial profiles over the sensor locations. The quantum protocol is found to outperform a perfect implementation of the best comparable strategy without sensor entanglement. While our demonstration is carried out for magnetic and electromagnetic fields over a few microns, the technique is readily applicable over arbitrary distances and for arbitrary fields, thus present a promising application for emerging quantum sensor networks.

Figures (3)

• Figure 1: Experimental demonstration with trapped-ion sensors. (a) Three sensors in a line are exposed to spatially-constant ($B^c$), linear ($B^lx$) and quadratic ($B^qx^2/2$) fields. (b) Encoding of sensor state $\ket{\psi_{(1,-2,1)}^{\mathrm{SWD}}} = (\ket{1,-2,1}+\ket{-1,2,-1})/\sqrt{2}$ into superpositions of the $3^2\textrm{D}_{5/2}$ manifolds of three $^{40}$Ca$^+$ ions. (c) Protocol. $\overline{\mathrm{MS}}$ is equivalent to an entangling Mølmer-Sørensen gate on the optical transition. 'Map' moves electron population to the $3^2\textrm{D}_{5/2}$ manifold. After state preparation, fluctuating noise ($B^c$ and $B^l$) and signal ($B^q$) fields are turned on. $U_{\phi_r}$ is described in the text.
• Figure 2: Experimentally-reconstructed density matrices of three-sensor states. Numbers in cells are matrix elements multiplied by 100. (a) Initial separable state $\rho^{\mathrm{sep}}$. (b)$\rho^{\mathrm{sep}}$ after 80ms of noise from CC coils. Red, yellow and blue boxes show DFSs with respect to spatially-constant noise. (c)$\rho^{\mathrm{sep}}$ after 80ms of noise from both the CC and CG coils. Red box shows DFS with respect to both spatially constant and gradient noise. (d)$\rho^{\mathrm{SWD}}$. (e)$\rho^{\mathrm{SWD}}$ after 80ms noise from both CC and CG coils.
• Figure 3: Sensing results. Data coloring reflects applied signal strength $B^q_{\mathrm{cal}}$. (a) Example parity estimates. Colored shapes show data. Lines show fits of $P(B^q, \phi_r)$ to data. Bold markers are used for estimating $B^q_{\mathrm{est}}$ in (b). (b) Histograms showing that quadratic field strength estimates $B^q_{\mathrm{est}}$ are centred around the calibrated values $B^q_{\mathrm{cal}}$. Grey lines show $B^q_{\mathrm{est}} = B^q_{\mathrm{cal}}$. Coloured curves show Gaussian fits. (c) Markers show $\mathrm{\overline{RMSE}}$ from data in each histogram in (b). Solid lines show corresponding weighted averages. Grey area is achievable by two-level separable estimation protocols. Striped area is beyond the Heisenberg-limit and inaccessible. Errorbars are one standard deviation.