Entanglement-Enhanced Radio-Frequency Field Detection and Waveform Sensing

We demonstrate a new technique for detecting the amplitude of arbitrarily chosen components of radio-frequency waveforms based on stroboscopic backaction evading measurements. We combine quantum nondemolition measurements and stroboscopic probing to detect waveform components with magnetic sensitivity beyond the standard quantum limit. Using an ensemble of $1.5\times 10^6$ cold rubidium atoms, we demonstrate entanglement-enhanced sensing of sinusoidal and linearly chirped waveforms, with 1.0(2) and 0.8(3) dB metrologically relevant noise reduction, respectively. We achieve volume-adjusted sensitivity of $\delta B\sqrt{V}\approx 3.96\mathrm{fT}\sqrt{{\mathrm{cm}}^3/\mathrm{Hz}}$, comparable to the best rf magnetometers.

Figure 1

(a) Experimental geometry (not to scale), showing dipole-trapped atoms, on-axis Faraday rotation probe pulses, transverse OP, and polarimeter elements: half-wave plate (HWP), polarizing beamsplitter (PBS), and balanced detector (BD). rf magnetic fields in the $\widehat{x}$ direction are produced by a coil, while a dc field of variable strength (not shown) is applied along $\widehat{y}$. (b) Measure-evolve-measure sequence and illustration of the backaction evading measurement of precessing spins experiencing, for (i) fixed ${\omega }_L$ or (ii) chirped ${\omega }_L(t)$. Probe times $t_i$ are illustrated by red dots, timed to give precession angle $\mathrm{\Theta }(t_i)=n\pi$ for integer $n$.

Figure 2

Characterization of the field-atom coupling. From an initially $+y$-polarized state, the ensemble spin is driven with continuous rf excitation $\gamma B_x(t)=\mathrm{\Omega }\cos ({\omega }_{0}t)$ and constant bias field $B_y$. In (a) we plot the generalized Rabi frequency ${\mathrm{\Omega }}^{\prime }=\sqrt{{\mathrm{\Omega }}^2+{\mathrm{\Delta }}^2}$, where $\mathrm{\Delta }\equiv {\omega }_0-{\omega }_L$; and in (b) we plot the normalized minimum spin projection along the $y$ axis ${\min }_t{F}_y(t)/|F|$, an estimate of the amount the spin vector has been tilted away from $F_y$ by the rf field [16]. Insets show representative traces of $F_z$ versus time (blue curves) and fits of the rotating-wave solution of Eq. (2) (red curves) that give best-fit values for $\mathrm{\Omega }$ and ${\omega }_L=\gamma B_y$ for each trace. The average values in these fits $\overline{\mathrm{\Omega }}/2\pi =0.23(3)\mathrm{kHz}$ and ${\overline{\omega }}_L/2\pi =42.28(1)\mathrm{kHz}$ are used to compute the theoretical curves in (a) and (b), which agree well with the observed values. Error bars correspond to $\pm 1\sigma$ standard error of the mean.

Figure 3

Response of the atomic polarization $F_z$ to rf waveforms with varying chirp $\kappa$. A ramped bias field $B_y(t)$ is used to produce a fixed Larmor frequency of ${\omega }_L(t)=2\pi \times 42.2\mathrm{kHz}$ until the end of ${\mathcal{M}}_1$, then rising linearly to ${\omega }_L(t)=2\pi \times 47.5\mathrm{kHz}$ over a period of $800\mu \mathrm{s}$. During this time, a rf signal is applied with constant amplitude and frequency $\omega ={\omega }_0+\kappa t$, and Faraday rotation probing is used to detect the axial polarization $F_z$. Shown is the maximal axial polarization ${\max }_t{F}_z(t)/|F|$ versus chirp of the rf excitation, with representative traces in the insets. The solid line is the result of a Lorentzian fit giving the resonant chirp $\kappa /2\pi =6.68(14)\times 10^6{\mathrm{Hz}}^2$. Error bars show $\pm 1\sigma$ standard error of the mean.

Figure 4

Atomic noise scaling of the stroboscopic QND measurement as a function of $N_A$ for constant bias field. The black dashed line shows calculated atomic projection noise for a coherent spin state with $\mathrm{var}(\mathrm{\Phi })={\tilde{g}}_1^2{N}_A/2$ plus readout noise (optical shot noise). The blue solid line is a quadratic fit to ${\mathrm{\Phi }}_2$ (blue circles) using Eq. (4). The blue shaded region shows atomic technical noise. The red squares indicate the conditional variance $\mathrm{var}({\mathrm{\Phi }}_2|{\mathrm{\Phi }}_1)$ as a function of $N_{\mathrm{A}}$, and the solid red line quadratic fit to it. The gray shaded area shows the region of metrologically relevant spin squeezing. Error bars represent $\pm 1\sigma$ standard error.