Entanglement-enhanced phase estimation without prior phase information

TL;DR

This paper demonstrates how planar quantum squeezed states generated via quantum non-demolition measurements can enhance phase estimation accuracy without prior phase knowledge, outperforming classical and traditional squeezed states.

Contribution

It introduces a method to generate and calibrate planar quantum squeezed states that provide a metrological advantage in phase estimation without prior phase information.

Findings

PQS states achieve at least 3.1 dB advantage over classical states.

PQS states outperform traditional squeezed states for most phase values.

Spin-spin entanglement underpins the metrological advantage.

Abstract

We study the generation of planar quantum squeezed (PQS) states by quantum non-demolition (QND) measurement of a cold ensemble of $^{87}$Rb atoms. 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, revealing the dual squeezing characteristic of PQS. 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 gives a metrological advantage of at least 3.1 dB relative to classical states. The PQS also beats traditional squeezed states generated with the same QND resources, except for a narrow range of phase values. Using spin squeezing inequalities, we show that spin-spin entanglement is responsible for the metrological advantage.

Figures

Equations

$$\Delta F_y\Delta F_z\ge \frac{1}{2}|⟨[F_y,F_z]⟩|=\frac{1}{2}|⟨F_x⟩|.$$

$${\xi }_{||}^2\equiv \frac{{\Delta }^2{F}_{||}}{F_{||}}.$$

$${\Delta }^2\varphi$$

$$H_{\mathrm{eff}}=g{S}_z{F}_z$$

$$\varphi(t)=g\Big{(}F_{z}(t_{\rm e})\cos\phi-F_{y}(t_{\rm e})\sin\phi\Big{)}e^{-t_{\rm r}/T_{2}}+\varphi_{0}$$

$$\Gamma_{\mathbf{F}_{2}\mid\mathbf{F}_{1}}=\left[\left(\begin{array}[]{cc}2.32&0.64\\ 0.64&3.00\\ \end{array}\right)\pm\left(\begin{array}[]{cc}0.21&0.16\\ 0.16&0.28\\ \end{array}\right)\right]\times 10^{5}.$$

$$\Gamma_{0}=\left[\left(\begin{array}[]{cc}1.02&0.14\\ 0.14&1.03\\ \end{array}\right)\pm\left(\begin{array}[]{cc}0.07&0.05\\ 0.05&0.07\\ \end{array}\right)\right]\times 10^{5}.$$

Full text

spin squeezing, phase estimation, quantum metrology, entanglement, optical magnetometry, interferometry

Estimation of interferometric phases is at the heart of precision sensing, and is ultimately limited by quantum statistical effects Wiseman and Milburn (2010). Entangled states can improve sensitivity beyond the “classical limits” that restrict sensing with independent particles, and a diversity of entangled states have been demonstrated for this task, including photonic squeezed states Slusher et al. (1985); Wu et al. (1986) and spin-squeezed states Meyer et al. (2001). These give improved sensitivity for a narrow range of phases, but worsened sensitivity for most phases. Optical “NOON” states Mitchell et al. (2004) give improved sensitivity over the whole phase range, but introduce additional phase ambiguity that increases with the size, and thus sensitivity advantage, of the NOON state. Recent proposals Tóth et al. (2009); He et al. (2011, 2012) suggest using planar quantum squeezed (PQS) states to obtain an entanglement-derived advantage for all phase angles, with no additional phase ambiguity. A natural application is in high-bandwidth atomic sensing Shah et al. (2010); Vasilakis et al. (2011); Sewell et al. (2012), in which the precession angle may not be predictable in advance. PQS states may also be valuable for ab initio phase estimation using feedback Xiang et al. (2011); Yonezawa et al. (2012); Berni et al. (2015).

Discussion of such states under the name “intelligent spin states” Aragone et al. (1974) predates modern squeezing terminology, and analogous states have been studied with optical polarization Korolkova et al. (2002); Schnabel et al. (2003); Predojević et al. (2008). Generation of PQS states in material systems has been proposed using two-well Bose-Einstein condensates with tunable and attractive interactions He et al. (2011, 2012), and using quantum non-demolition (QND) measurements Puentes et al. (2013). Here we take the latter approach, using Faraday rotation QND measurements Mitchell et al. (2012); Sewell et al. (2013) applied to an ensemble of cold atomic spins with $f=1$. As the ensemble spin precesses about the $x$ axis in an external magnetic field Behbood et al. (2013a, b, 2014), we measure the $y$ and $z$ spin components to generate measurement-induced squeezing in these two components, creating a PQS state. The resulting state has enhanced sensitivity to precession angle, i.e., to Zeeman-shift induced phase. The demonstrated PQS state beats the best possible classical state at any precession angle, and beats traditional spin-squeezed states when averaged over the possible angles. Spin-squeezing inequalities He et al. (2011, 2012); Vitagliano et al. (2011) detect spin entanglement in the PQS state, showing the sensing advantage is due to spin entanglement Beduini et al. (2015).

A spin $\mathbf{F}$ obeys the Robertson uncertainty relation

[TABLE]

Unlike the canonical Heisenberg uncertainly relation, the rhs of Eq. (1) may vanish, e.g. for $\langle F_{x}\rangle=0$, with the consequence that two spin components, e.g. $F_{y}$ and $F_{z}$, may be simultaneously squeezed, with the uncertainty absorbed by the third component, $F_{x}$. We refer to a state fulfilling this condition as a PQS state.

Following the approach of He et al. He et al. (2011, 2012), we adopt an operational definition planar squeezing. We take $\Delta^{2}F_{y}=\Delta^{2}F_{z}=F_{||}/2$ as the standard quantum limit, where $F_{||}\equiv\sqrt{F_{y}^{2}+F_{z}^{2}}$, so that $F_{||}$ is the magnitude of the in-plane spin components. We define the planar variance $\Delta^{2}F_{||}\equiv\Delta^{2}F_{y}+\Delta^{2}F_{z}$, with standard quantum limit $\Delta^{2}F_{||}=F_{||}$, and the planar squeezing parameter

[TABLE]

A PQS state has $\xi_{||}^{2}<1$, and has individual component variances below the standard quantum limit, i.e., $\xi_{y}^{2}<1$, and $\xi_{z}^{2}<1$ , where $\xi_{i}^{2}\equiv{2\Delta^{2}F_{i}}/{F_{||}}$, so that $\xi_{||}^{2}=(\xi_{y}^{2}+\xi_{z}^{2})/2$.

Entanglement is detected using the witness $\xi_{\rm e}^{2}\equiv\Delta^{2}F_{||}/\langle\tilde{N_{\rm A}}\rangle$, derived in Ref. He et al. (2011); for $f=1$ atoms, entanglement is detected if $\xi_{\rm e}^{2}<7/16$. Here $\tilde{N}_{\rm A}\equiv(\eta_{\rm sc}+p(1-\eta_{\rm sc}))N_{\rm A}$ is the number of atoms remaining in the $f=1$ state after probing, $\eta_{\rm sc}$ accounts for off-resonant scattering of atoms, and $p$ is the fraction of scattered atoms that return to $f=1$ Colangelo et al. (2013). We also define a metrological squeezing parameter $\xi_{\rm m}^{2}\equiv F\Delta^{2}F_{||}/F_{||}^{2}$, where $F\equiv\langle N_{\rm A}\rangle$ is the input spin coherence, similar to the Wineland criterion Wineland et al. (1992, 1994), in that it compares noise to the magnitude of the coherence $F_{||}$. A PQS with $\xi_{\rm m}^{2}<1$ gives enhanced metrological sensitivity to arbitrary phase shifts.

A PQS state may be used to measure arbitrary phase angles with quantum-enhanced precision. For example, we consider an ensemble of atomic spins precessing in the $y$–$z$ plane in an external magnetic field $B_{x}$. The spin projection onto the $z$-axis is given by $F_{z}(t)=F_{z}\cos\phi-F_{y}\sin\phi$, where $F_{y}$ and $F_{z}$ are evaluated at $t=0$ and the phase $\phi=\omega_{\rm L}t$ is proportional to the magnetic field. The uncertainty in estimating $\phi$ of the atomic precession is

[TABLE]

where $\Delta^{2}F_{z}(\phi)\equiv\Delta^{2}F_{y}\sin^{2}\phi+\Delta^{2}F_{z}\cos^{2}\phi+\mathrm{cov}(F_{y},F_{z})\sin 2\phi$, and $\mathrm{cov}(A,B)\equiv\frac{1}{2}\langle AB+BA\rangle-\langle A\rangle\langle B\rangle$ is the covariance. The standard quantum limit is $\Delta^{2}\phi_{\rm SQL}=1/2F_{||}$. We note that PQS states reduce the planar variance for arbitrary angles on a finite interval, except where the denominator in Eq. (3) is equal to zero. In contrast, squeezing a single spin component is only beneficial to refine the estimate of a phase over a limited range of angles, and requires prior knowledge of the phase, or adaptive procedures to determine the phase during the measurement He et al. (2012).

We work with an ensemble of up to $1.75\times 10^{6}$ laser-cooled {}^{87}\mathrm{Rb}\atoms held in a single beam optical dipole trap Kubasik et al. (2009); Koschorreck et al. (2010a, b), as illustrated in Fig. 1. The atoms are initially polarized via high efficiency ($\sim 98\%$) stroboscopic optical pumping, in the presence of a small magnetic field applied along the $x$-axis, such that $\langle F_{y}\rangle\simeq\langle N_{\rm A}\rangle$. $N_{\rm A}$ is subject to Poissonian fluctuations because accumulation of independent atoms into the ensemble is a stochastic process limited by Poisson statistics $\Delta^{2}N_{\rm A}=\langle N_{\rm A}\rangle$. We refer to this kind of state as a Poissonian coherent spin state (PCSS), with variances $\Delta^{2}F_{x}=\Delta^{2}F_{z}=\langle N_{\rm A}\rangle/2$ and $\Delta^{2}F_{y}=\langle N_{\rm A}\rangle$. Generating sub-Poissonian atom number statistics, either via strong interaction among the atoms during accumulation Schlosser et al. (2001); Sortais et al. (2012); Chuu et al. (2005); Itah et al. (2010); Sanner et al. (2010); Whitlock et al. (2010); Hofmann et al. (2013), or precise non-destructive measurement Stockton (2007); Takano et al. (2009); Appel et al. (2009); Schleier-Smith et al. (2010); Hume et al. (2013); Béguin et al. (2014); Bohnet et al. (2014); Gajdacz et al. (2016); Hosten et al. (2016); Zhang et al. (2012); Stroescu et al. (2015), remains a significant experimental challenge.

We probe the atomic spins via off-resonant paramagnetic Faraday-rotation. The effective atom-light interaction is given by the hamiltonian

[TABLE]

Here, the atoms are described by the collective spin operators $\mathbf{F}\equiv\sum_{i}\mathbf{f}^{(i)}$, with $\mathbf{f}^{(i)}$ the spin orientation of individual atoms. The optical polarization of the probe pulses is described by the Stokes operators $S_{k}=\frac{1}{2}(a_{L}^{\dagger},a_{R}^{\dagger})\sigma_{k}(a_{L},a_{R})^{T}$, with Pauli matrices $\sigma_{k}$. The coupling constant $g$ depends on the detuning from the resonance of the probe beam, the atomic structure, the geometry of the atomic ensemble and probe beam Kubasik et al. (2009); Koschorreck et al. (2010a); Deutsch and Jessen (2010); Kuzmich et al. (2000); Appel et al. (2009).

Equation (4) describes a quantum non-demolition measurement of the collective atomic spin $F_{z}$: an input $S_{x}$-polarized optical pulse interacting with the atoms experiences a rotation by an angle $\varphi=gF_{z}$. The transformation produced by the interaction is $S_{y}^{\prime}=S_{y}\cos\varphi+S_{x}\sin\varphi$. In our experiment we measure $S_{x}$ at the input by picking off a fraction of the optical pulse and sending it to a reference detector, and $S_{y}^{\prime}$ using a fast home-built balanced polarimeter Ciurana et al. (2016). Both signals are recorded on a digital oscilloscope, from which we calculate $\hat{\varphi}=\arcsin\left(S_{y}^{\prime}/S_{x}\right)$, the estimator for $\varphi$. We correct for slow drifts in the polarimeter signal by subtracting a baseline from each pulse, estimated by repeating the measurement without atoms in the trap.

We probe the atoms using a train of $\tau=$0.6\text{,}\mathrm{\SIUnitSymbolMicro s}$$ duration pulses of linearly polarized light, with a detuning of $700\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ to the red of the {}^{87}\mathrm{Rb}\D2 line, sent through the atomic cloud at $3\text{\,}\mathrm{\SIUnitSymbolMicro s}$ intervals. The probe pulses are $V$-polarized, with on average $n_{\rm l}=2.74\times 10^{6}$ photons. Between the probe pulses, we send $H$-polarized compensation pulses with on average $n_{\rm l}^{(H)}=1.49\times 10^{6}$ photons through the atomic cloud to compensate for tensor light shifts Koschorreck et al. (2010b); Sewell et al. (2013); Colangelo et al. (2013). During the measurement, an external magnetic field $B_{x}$ coherently rotates the atoms in the $y$–$z$ plane at the larmor frequency $\omega_{\rm L}$. The time taken to complete a single-pulse measurement is small compared to the Larmor precession period, i.e. $\tau\ll T_{L}$. Off-resonant scattering of probe photons during the measurement leads to decay of the atomic coherence at a rate $\eta=3\times 10^{-10}$ per photon.

The measurable signal is described by the free induction decay model Behbood et al. (2013b)

[TABLE]

where $t_{\rm r}\equiv t-t_{\rm e}$ and the phase $\phi=\omega_{\rm L}t_{\rm r}$ is proportional to the magnetic field. We record a set of measurements $\varphi(t_{k})$, and detect the PQS state at time $t_{\rm e}$. A typical free induction decay signal is illustrated in Fig. 2. An independent measurement is used to calibrate $g$, while $\omega_{\rm L}$, $T_{2}$, and $\varphi_{0}$ are found by fitting the measured $\varphi(t_{k})$ over all the data points.

The model described in Eq. (5) allows a simultaneous estimation of $\mathbf{F}_{1}=(F_{y}^{(1)},F_{z}^{(1)})$ at a time $t=t_{\rm e}$ by fitting the the data using the measurements from an interval $\Delta t$ prior to $t_{\rm e}$ (labeled $M_{1}$ in Fig. 2), producing a conditional PQS at time $t_{\rm e}$. We detect the PQS by comparing the first measurement outcome to a second estimate $\mathbf{F}_{2}=(F_{y}^{(2)},F_{z}^{(2)})$ using the measurements from an interval $\Delta t$ after to $t_{\rm e}$ (labeled $M_{2}$ in Fig. 2). The classical parameters $g$, $\omega_{\rm L}$, $T_{2}$ and $\varphi_{0}$ are fixed beforehand. As a result, these are two linear, least-squares estimates of the vector $\mathbf{F}$ obtained from disjoint data sets 111Further details of the fitting procedure are given in Ref. Colangelo et al. (2017). Statistics are gathered over 450 repetitions of the experiment, taking into account the inhomogeneous atom-light coupling Appel et al. (2009); Schleier-Smith et al. (2010); Colangelo et al. (2017).

The estimate of the state from the two independent measurements is subject to technical noise due to amplitude and phase fluctuations of the input state, and shot-to-shot variations of the magnetic field. In Fig. 3 a) we plot the estimate of $\mathbf{F}_{1}$ at time $t_{\rm e}$ for an input state with $\langle N_{\rm A}\rangle=1.75\times 10^{6}$ atoms. In contrast, the conditional uncertainty of $\mathbf{F}_{2}$ given $\mathbf{F}_{1}$ is limited mainly by the measurement read-out noise, as shown in Figs. 3 b) and c).

From the measurement record we compute the conditional covariance matrix $\Gamma_{\mathbf{F}_{2}\mid\mathbf{F}_{1}}=\Gamma_{\mathbf{F}_{2}}-\Gamma_{\mathbf{F}_{2}\mathbf{F}_{1}}\Gamma_{\mathbf{F}_{1}}^{-1}\Gamma_{\mathbf{F}_{1}\mathbf{F}_{2}}$ which quantifies the error in the best linear prediction of $\mathbf{F}_{2}$ based on $\mathbf{F}_{1}$ Behbood et al. (2014). $\Gamma_{\bf v}$ indicates the covariance matrix for vector ${\bf v}$, and $\Gamma_{\bf uv}$ indicates the cross-covariance matrix for vectors ${\bf u}$ and ${\bf v}$. The difference between the best linear prediction of $\mathbf{F}$ using $\mathbf{F}_{1}$ and the confirming estimate $\mathbf{F}_{2}$ is visualized using the vector $\bm{\mathcal{F}}=\{\mathcal{F}_{\rm y},\mathcal{F}_{\rm z}\}=\tilde{\mathbf{F}}_{2}-\Gamma_{\mathbf{F}_{2}\mathbf{F}_{1}}\Gamma_{\mathbf{F}_{1}}^{-1}\tilde{\mathbf{F}}_{1}$, where $\tilde{\mathbf{F}_{i}}=\mathbf{F}_{i}-\langle\mathbf{F}_{i}\rangle$. Standard errors in the estimated conditional covariance matrix are calculated from the statistics of $\{\bm{\mathcal{F}}\}$ Kendall and Stuart (1979).

Empirically, we find $\Delta t=$270\text{,}\mathrm{\SIUnitSymbolMicro s}$$ minimizes the total variance ${\rm Tr}(\Gamma_{\mathbf{F}_{2}\mid\mathbf{F}_{1}})$. This reflects a trade-off of photon shot noise versus scattering-induced decoherence and magnetic-field technical noise. At this point $N_{\rm L}=2.47\times 10^{8}$ photons have been used in the measurement and the atomic state coherence has decayed by a factor $\eta_{\rm sc}=0.89$ due to off-resonant scattering, and a factor $\eta_{\rm dec}=0.93$ due to dephasing induced by magnetic field gradients Colangelo et al. (2013). The resulting spin coherence of the PQS is $F_{||}=\eta_{\rm dec}\eta_{\rm sc}N_{\rm A}$ spins.

From $\Gamma_{\mathbf{F}_{2}\mid\mathbf{F}_{1}}$ we estimate the planar squeezing parameter $\xi_{||}^{2}=\mathrm{Tr}(\tilde{\Gamma}_{\mathbf{F}_{2}\mid\mathbf{F}_{1}})/F_{||}$, where $\tilde{\Gamma}_{\mathbf{F}_{2}\mid\mathbf{F}_{1}}=\Gamma_{\mathbf{F}_{2}\mid\mathbf{F}_{1}}-\Gamma_{0}$ and $F_{||}$ is estimated at $t_{\rm e}$. $\Gamma_{0}$ is the read out noise, quantified by repeating the measurement without atoms in the trap. In Fig. 4 we show $\xi_{||}^{2}$ as function of the in-plane coherence $F_{||}$ of the atomic ensemble, which we vary by changing the number of atoms in the optical dipole trap. We detect a PQS for $F_{||}\geq 4\times 10^{5}$ spins. With the maximum coherence $F_{||}=1.45\times 10^{6}$ spins, we observe $\xi_{||}^{2}=0.37\pm 0.03<1$, detecting a PQS with $>20\sigma$ significance, with $\xi_{y}^{2}=0.32\pm 0.03$ and $\xi_{z}^{2}=0.42\pm 0.04$, and $\xi_{\rm e}^{2}=0.32\pm 0.02<7/16$, detecting entanglement among the atomic spins with $>5\sigma$ significance He et al. (2011). The measured conditional covariance (in units of spins2) is

[TABLE]

For comparison, the estimated read-out noise is

[TABLE]

For this state, the observed metrological squeezing parameter is $\xi_{\rm m}^{2}=0.45\pm 0.03$, indicating that entanglement-enhanced phase sensitivity is achievable. To estimate the enhanced phase sensitivity provided by the PQS state, we evaluate Eq. (3) using the conditional covariance $\Gamma_{\mathbf{F}_{2}\mid\mathbf{F}_{1}}$ and the measured coherences. The PQS state achieves a maximum sensitivity $\Delta^{2}\phi=0.38\,\Delta^{2}\phi_{\rm SQL}$ ($\Delta\phi=3.6\times 10^{-4}$ radians) at a phase $\phi=0.68\,\pi$ radians. Note that this phase is determined by the choice of measurement time $t_{\rm e}$.

In Fig. 5 we plot the estimated phase sensitivity $\Delta^{2}\phi$ of the observed PQS state (red solid line). For comparison purposes, we rotate the PQS so that the spin coherence is aligned along the $y$-axis, i.e. $\mathbf{F}\rightarrow R(\theta)\cdot\mathbf{F}$ and $\Gamma_{\mathbf{F}_{2}\mid\mathbf{F}_{1}}\rightarrow R(\theta)\cdot\Gamma_{\mathbf{F}_{2}\mid\mathbf{F}_{1}}\cdot R(\theta)^{T}$, where $\arctan\theta\equiv F_{y}/F_{z}$. We compare this with the sensitivity of a PCSS with input spin coherence $\langle F_{y}\rangle=N_{\rm A}$ (blue dashed line), and an ideal single-variable spin squeezed state (SSS) that would be produced by a single instantaneous quantum non-demolition measurement with the same precision, i.e. with $\Delta^{2}F_{y}=\langle N_{\rm A}\rangle$, $\Delta^{2}F_{z}$ reduced by a factor $1/(1+g^{2}N_{\rm L}N_{\rm A}/2)$, and input coherence $\langle F_{y}\rangle=\eta_{\rm sc}N_{\rm A}$ (green dot-dashed line).

We also plot the calculated enhancement in phase sensitivity $\Delta^{2}\phi$ of both the PQS and SSS states relative to the classical input PCSS. The measured PQS state achieves $\geq$3.1\text{,}\mathrm{d}\mathrm{B}$$ quantum-enhanced, metrologically-significant phase sensitivity with respect to the PCSS for all phases, with a maximum of $4.1\text{\,}\mathrm{d}\mathrm{B}$, enabling quantum-enhanced measurement of an arbitrary phase shift. In contract, the SSS achieves 6.6 dB enhancement relative to the PCSS at $\phi=0$, but performs worse than the PQS state outside the range $-0.09\,\pi<\phi<0.12\,\pi$ radians.

In contrast to the well known spin-squeezed states, planar quantum squeezed states enhance the precision of phase estimation without requiring a priori information about the phase. Here we have shown that QND measurement can efficiently produce such states, demonstrating more than $3\text{\,}\mathrm{d}\mathrm{B}$ of advantage relative to classical states over the full range of phase angles. We also detect spin-spin entanglement underlying the metrological advantage. Such states are attractive for high-bandwidth and high-sensitivity optical magnetometers Kominis et al. (2003); Shah et al. (2010) and other atomic sensing applications employing non-destructive spin detection Lodewyck et al. (2009); Sheng et al. (2013); Hosten et al. (2016).

Acknowledgements

We thank Q. Y. He, M. Reid, P. Drummond, G. Vitagliano, G. Tóth, E. Distante, V.G. Lucivero, L. Bianchet, N. Behbood and M. Napolitano for helpful discussions. Work supported by MINECO/FEDER, MINECO projects MAQRO (Ref. FIS2015-68039-P), XPLICA (FIS2014-62181-EXP) and Severo Ochoa grant SEV-2015-0522, Catalan 2014-SGR-1295, by the European Union Project QUIC (grant agreement 641122), European Research Council project AQUMET (grant agreement 280169) and ERIDIAN (grant agreement 713682), and by Fundació Privada CELLEX. GP gratefully acknowledges funding from the Agencia Nacional de Promoción Científica y Tecnológica (ANPCyT), PICT2014-1543, PICT2015-0710, and UBACYT PDE 2017.