Entanglement-Enhanced Phase Estimation without Prior Phase Information

We study the generation of planar quantum squeezed (PQS) states by quantum nondemolition (QND) measurement of an ensemble of ${}^{87}\mathrm{Rb}$ atoms with a Poisson distributed atom number. Precise calibration of the QND measurement allows us to infer the conditional covariance matrix describing the $F_y$ and $F_z$ components of the PQS states, revealing the dual squeezing characteristic of PQS states. PQS states have been proposed for single-shot phase estimation without prior knowledge of the likely values of the phase. We show that for an arbitrary phase, the generated PQS states can give a metrological advantage of at least 3.1 dB relative to classical states. The PQS state also beats, for most phase angles, single-component-squeezed states generated by QND measurement with the same resources and atom number statistics. Using spin squeezing inequalities, we show that spin-spin entanglement is responsible for the metrological advantage.

Figure 1

(a) Experimental setup. A cloud of laser-cooled ${}^{87}\mathrm{Rb}$ atoms is held in a singe-beam optical dipole trap. The atoms precess in the $y\text{−}z$ plane due to an external magnetic field $B_x$. Optical probe pulses experience Faraday rotation by an angle $\theta \propto F_z(t)$, detected via measurement of the output Stokes parameter $S_y^{\prime }$ using a balanced polarimeter that consists in a wave plate (WP), a polarizing beam splitter (PBS), and photodiodes ${\mathrm{PD}}_2$ and ${\mathrm{PD}}_3$. The input $S_x$ polarization is recorded with a reference photodetector (${\mathrm{PD}}_1$). (b) Illustration of a PCS state (green) with ${\mathrm{\Delta }}^2{F}_z=⟨N_A⟩/2$ and ${\mathrm{\Delta }}^2{F}_y=⟨N_A⟩$; a PSS state (red) with reduced ${\mathrm{\Delta }}^2{F}_z$ and increased ${\mathrm{\Delta }}^2{F}_y$; and a PQS state (blue) with both ${\mathrm{\Delta }}^2{F}_y$ and ${\mathrm{\Delta }}^2{F}_z$ reduced. Additional spin noise due to measurement back-action is directed into the $F_x$ spin component, i.e., out of the plane, and does not enter into the measurement record. (c) Recorded measurements of the Faraday rotation angle from a precessing PCS state. We use the measurement record to predict the $F_z$ and $F_y$ components at a time $t=t_e$ using two sequential measurements $M_1$ and $M_2$ of duration $\mathrm{\Delta }t$.

Figure 2

(a) Spin state ${\mathbf{F}}_1$ (red dots) estimated at time $t_e$ for an input state with $⟨N_A⟩=1.75\times 10^6$ atoms from the 453 repetitions of the experiment. For comparison, we illustrate the corresponding measurement made without atoms in the trap, used to quantify the read-out noise (yellow dot). (b) Error in the best linear predictor, $\mathit{F}$, of ${\mathbf{F}}_2$ given ${\mathbf{F}}_1$ (blue dots). The blue ellipse shows the measured $2\sigma$ radii of the distribution. The yellow ellipse shows the standard quantum limit ${\mathrm{\Delta }}^2{F}_y={\mathrm{\Delta }}^2{F}_z=F_{||}/2$ with $2\sigma$ radii, where ${\sigma }^2=(F_{||}/2{)}^2+{\mathrm{\Delta }}^2{\theta }_0$ and ${\mathrm{\Delta }}^2{\theta }_0$ is the measured read-out noise. (c) Linear predictor $\mathit{F}$ estimated from repeating the experiment without atoms in the trap, allowing quantification of the measurement read-out noise. The dashed ellipse shows the measured $2\sigma$ radii of the distribution.

Figure 3

(a) Semi-log plot of the planar squeezing parameter, ${\xi }_{||}^2$, as a function of the in-plane coherence $F_{||}$ of the atomic ensemble. We vary $F_{||}$ by changing the number of atoms loaded in the optical dipole trap. A PQS state is detected for ${\xi }_{||}^2<1$ (shaded region). Entanglement is detected for ${\xi }_e^2=(F_{||}/⟨\widetilde{N_A}⟩){\xi }_{||}^2<7/16$ (dashed line). Error bars represent $\pm 1\sigma$ statistical errors. (b) Calculated phase sensitivity of the PQS state as a function of the measurement phase $\varphi$ (red solid line). The standard quantum limit $2F_{||}{\mathrm{\Delta }}^2\varphi =1$ is indicated by the solid yellow line. We also show the phase sensitivity of the input PCS (blue dotted line), a PSS state produced with the same resources (green dashed line), and an ideal spin-squeezed state (SSS) not subject to number fluctuations (dark yellow dot-dashed line). (c) Metrologically significant enhancement in phase sensitivity relative to that of the PCS, ${\mathrm{\Delta }}^2\varphi /{\mathrm{\Delta }}^2{\varphi }_{\mathrm{PCS}}$, for the PQS (red solid line), PSS (green dashed line) and SSS (dark yellow dot-dashed line) states. Shaded bands indicate $\pm 1\sigma$ confidence intervals.