Abstract
Quantum technologies use entanglement to outperform classical technologies, and often employ strong cooling and isolation to protect entangled entities from decoherence by random interactions. Here we show that the opposite strategy—promoting random interactions—can help generate and preserve entanglement. We use optical quantum nondemolition measurement to produce entanglement in a hot alkali vapor, in a regime dominated by random spinexchange collisions. We use Bayesian statistics and spinsqueezing inequalities to show that at least 1.52(4) × 10^{13} of the 5.32(12) × 10^{13} participating atoms enter into singlettype entangled states, which persist for tens of spinthermalization times and span thousands of times the nearestneighbor distance. The results show that high temperatures and strong random interactions need not destroy manybody quantum coherence, that collective measurement can produce very complex entangled states, and that the hot, stronglyinteracting media now in use for extreme atomic sensing are well suited for sensing beyond the standard quantum limit.
Introduction
Entanglement is an essential resource in quantum computation, simulation, and sensing^{1}, and is also believed to underlie important manybody phenomena such as highT_{c} superconductivity^{2}. In many quantum technology implementations, strong cooling and precise controls are required to prevent entropy—whether from the environment or from noise in classical parameters—from destroying quantum coherence. Quantum sensing^{3} is often pursued using lowentropy methods, for example, with cold atoms in optical lattices^{4}. There are, nonetheless, important sensing technologies that operate in a highentropy environment, and indeed that employ thermalization to boost coherence and thus sensor performance. Notably, vaporphase spinexchangerelaxationfree (SERF) techniques^{5} are used for magnetometry^{6,7}, rotation sensing^{8}, and searches for physics beyond the standard model^{9}, and give unprecedented sensitivity^{10}. In the SERF regime, strong, frequent, and randomlytimed spinexchange (SE) collisions dominate the spin dynamics, to produce local spin thermalization. In doing so, these same processes also decouple the spin degrees of freedom from the bath of centreofmass degrees of freedom, which increases the spin coherence time^{5}. Whether entanglement can be generated, survive, and be observed in such a high entropy environment is a challenging open question^{11}.
Here, we study the nature of spin entanglement in this hot, stronglyinteracting atomic medium, using techniques of direct relevance to extreme sensing. We apply optical quantum nondemolition (QND) measurement^{12,13}—a proven technique for both generation and detection of nonclassical states in atomic media—to a SERFregime vapor. We start with a thermalized spin state to guarantee the zero mean of the total spin variable and use a [1, 1, 1] direction magnetic field (see Fig. 1a) to achieve QND measurements on three components of the total spin variable. We track the evolution of the net spin using the Bayesian method of Kalman filtering^{14}, and use spin squeezing inequalities^{15,16} to quantify entanglement from the observed statistics. We observe that the QND measurement generates a macroscopic singlet state^{17}—a squeezed state containing a macroscopic number of singlettype entanglement bonds. This shows that QND methods can generate entanglement in hot atomic systems even when the atomic spin dynamics include strong local interactions. The spin squeezing and thus the entanglement persist far longer than the spinthermalization time of the vapor; any given entanglement bond is passed many times from atom to atom before decohering. We also observe a sensitivity to gradient fields that indicates the typical entanglement bond length is thousands of times the nearestneighbor distance. This is experimental evidence of longrange singlettype entanglement bonds. These experimental observations complement recent predictions of coherent interspecies quantum state transfer by spin collision physics^{18,19}.
Results
Material system
We work with a vapor of ^{87}Rb contained in a glass cell with buffer gas to slow diffusion, and housed in magnetic shielding and field coils to control the magnetic environment, see Fig. 1a. The density is maintained at n_{Rb} = 3.6 × 10^{14} atoms/cm^{3}, and the magnetic field, applied along the [1, 1, 1] direction, is used to control the Larmor precession frequency ω_{L}/2π. At this density, the spinexchange collision rate is 325 × 10^{3} s^{−1}. For ω_{L} below about 2π × 5 kHz, the vapor enters the SERF regime, characterized by a large increase in spin coherence time.
Spin thermalization
The spin dynamics of such dense alkali vapors^{5} is characterized by a competition of several local spin interactions, diffusion, and interaction with external fields, buffer gases, and wall surfaces. While the full complexity of this scenario has not yet been incorporated in a quantum statistical model, in the SERF regime an important simplification allows us to describe the state dynamics in sufficient detail for entanglement detection, as we now show.
If j^{(l)} and i^{(l)} are the lth atom’s electron and nuclear spins, respectively, the spin dynamics, including sudden collisions, can be described by the timedependent Hamiltonian
where the terms describe the hyperfine interaction, SE collisions, spindestruction (SD) collisions, and Zeeman interaction, respectively. A_{hf} is the hyperfine (HF) splitting and \(t_n^{\left( {l,l^{\prime}} \right)}\) is the (random) time of the nth SE collision between atoms l and \(l^{\prime}\), which causes mutual precession of j^{(l)} and \({\mathbf{j}}^{\left( {l^{\prime}} \right)}\) by the (random) angle θ_{n}. We indicate with R_{SE} the rate at which such collisions move angular momentum between atoms. Similarly, the third term describes rotations about the random direction d_{m} by random angle \({\it{\uppsi }}_m\), and causes spin depolarization at a rate R_{SD}. γ_{e} = 2π × 28 GHz T^{−1} is the electron spin gyromagnetic ratio. We neglect the much smaller i·B coupling. We note that shortrange effects of the magnetic dipole–dipole interaction (MDDI) are already included in R_{SE} and R_{SD}, and that longrange MDDI effects are negligible in an unpolarized ensemble, as considered here.
The SERF regime is defined by the hierarchy A_{hf} ≫ R_{SE} ≫ γ_{e}B, R_{SD}. Our experiment is in this regime, as we have A_{hf} ≈ 10^{9} s^{−1}, R_{SE} ≈ 10^{5} s^{−1}, γ_{e}B ≈ 10^{4} s^{−1} and R_{SD} ≈ 10^{2} s^{−1}. The hierarchy implies the following dynamics: on short times, the combined action of the HF and SE terms rapidly thermalizes the spin state, i.e., generates the maximum entropy consistent with the ensemble total angular momentum F, which is conserved by these interactions (see “Methods”, “Spin thermalization” section). We indicate this Fparametrized maxentropy state by \(\rho _{\mathbf{F}}^{\left( {{\mathrm{th}}} \right)}\). We note that entanglement can survive the thermalization process; for example, \(\rho _{{\mathbf{F}} = 0}^{\left( {{\mathrm{th}}} \right)}\) is a singlet and thus necessarily describes entangled atoms. On longer timescales, F experiences precession about B due to the Zeeman term and diffusive relaxation due to the depolarization term.
Nondestructive measurement
We perform a continuous nondestructive readout of the spin polarization using Faraday rotation of offresonance light. On passing through the cell the optical polarization experiences rotation by an angle \({g{\mathbf{\cal{F}}}_z(t) \ll \pi}\), where z is the propagation axis of the probe, g is a lightatom coupling constant and \({{{\mathbf{\cal{F}}}}}\) ≡ F_{a} − F_{b}, where F_{α} is the collective spin orientation from atoms in hyperfine state α ∈ {1, 2} (see “Methods”, “Observed spin signal” section).
For thermalized spin states 〈\({{{\mathbf{\cal{F}}}}}\) 〉\(\propto \left\langle {\mathbf{F}} \right\rangle\), so that the observed polarization rotation gives a view into the full spin dynamics. The optical rotation is detected by a balanced polarimeter (BP), which gives a signal proportional to the Stokes parameter
where S_{x} is the Stokes component along which the input beam is polarized^{20}. \(S_y^{\left( {{\mathrm{in}}} \right)}\left( t \right)\) is a zeromean Gaussian process, whose variance is dictated by photon shotnoise and is characterized by a powerspectral analysis of the BP signal^{21}.
Spin dynamics and spin tracking
The evolution of \({\boldsymbol{{\cal{F}}}}\left( t \right)\) is described by the Langevin equation (see “Methods”, “Spin dynamics” section)
where γ = γ_{e}/q is the SERFregime gyromagnetic ratio, i.e., that of a bare electron reduced by the nuclear slowingdown factor^{5}, which takes the value q = 6 in the SERF regime^{22}. Γ is the net relaxation rate including diffusion, spindestruction collisions, and probeinduced decoherence, Q is the equilibrium variance (see below) and dW_{h}, h ∈ {x, y, z} are independent temporal Wiener increments.
Based on Eqs. (3) and (2), we employ the Bayesian estimation technique of Kalman filtering (KF)^{14} to recover \({\mathbf{\cal{F}}}\left( t \right)\), which is shown as \(g{\mathbf{\cal{F}}}_z\left( t \right)S_x\) to facilitate comparison against the measured \(S_y^{\left( {{\mathrm{out}}} \right)}\left( t \right)\) in Fig. 1b). The KF (see “Methods”, “Kalman filter” section) gives both a best estimate and a covariance matrix \({{\Gamma }}_{\boldsymbol{{\cal{F}}}}\left( t \right)\) for the components of \({\boldsymbol{{\cal{F}}}}\left( t \right)\), which gives an upper bound on the variances of the postmeasurement state. Figure 1c shows that the \({\mathbf{\cal{F}}}_z\) component of \({{\Gamma }}_{\boldsymbol{{\cal{F}}}}\left( t \right)\) is suppressed rapidly, to reach a steady state value which is below the SQL. The other components are similarly reduced in variance by the measurement, and the total variance \(\left {{{\Delta }}{\boldsymbol{{\cal{F}}}}} \right^2 \equiv {\mathrm{Tr}}\left[ {{{\Gamma }}_{\boldsymbol{{\cal{F}}}}} \right]\) can be compared against spin squeezing inequalities^{15,16} to detect and quantify entanglement: Defining the spinsqueezing parameter \(\xi ^2 \equiv \left {{{\Delta }}{\boldsymbol{{\cal{F}}}}} \right^2/{\mathrm{SQL}}\), where SQL ≡ N_{A}13/8 is the standard quantum limit, ξ^{2} < 1 detects entanglement, indicating a macroscopic singlet state^{17}. The minimum number of entangled atoms^{15} is N_{A}(1 − ξ^{2})13/16 (see “Methods”, “Entanglement witness” section).
Experimental results
The cell temperature was stabilized at 463 K to give an alkali number density of n_{Rb} = 3.55(6) × 10^{14} atoms cm^{−3}, calibrated as described in “Methods”, “Density calibration” section, and thus N_{A} = 5.32(12) × 10^{13} atoms within the 3 cm × 0.0503(8) cm^{2} effective volume of the beam. At this density, the SE collision rate is R_{SE} ≈ 325 × 10^{3} s^{−1}. By varying B we can observe the transition to the SERF regime, and the consequent development of squeezing. Figure 2a shows spinnoise spectra (SNS)^{21}, i.e., the power spectra of detected signal from BP, for different values of B, from which we determine the resonance frequency ω_{L} = γB, relaxation rate Γ and the number density. Using these as parameters in the KF (see “Methods”, “Kalman filter” section), we obtain \(\left {{{\Delta }}{\boldsymbol{{\cal{F}}}}} \right^2\) as shown in Fig. 2b, including a transition to squeezed/entangled states as the system enters the SERF regime.
At a Larmor frequency of 1.3 kHz, we observe ξ^{2} = 0.650(2) or 1.88(1) dB of spin squeezing at optimal probe power 2 mW (see “Methods”, “Kalman filter” section), which implies that at least 1.52(4) × 10^{13} of the 5.32(12) × 10^{13} participating atoms have become entangled as a result of the measurement. This greatly exceeds the previous entanglement records: 5 × 10^{5} cold atoms in singlet states using a similar QND strategy^{17} and a Dicke state involving 2 × 10^{11} impurities in a solid, made by storing a single photon in a multicomponent atomic ensemble^{23}. This is also the largest number of atoms yet involved in a squeezed state; see Bao et al. for a recent record for polarized spinsqueezed states^{24}. We use this power and field condition for the experiments described below, and note that the spinrelaxation time greatly exceeds the spinthermalization time. In this condition, the entanglement bonds are rapidly distributed amongst the atoms by SE collisions without being lost.
We now study the spatial distribution of the induced entanglement. As concerns the observable \({\boldsymbol{{\cal{F}}}}\), the relevant dynamical processes, including precession, decoherence, and probing, are permutationallyinvariant: Eqs. (3) and (2) are unchanged by any permutation of the atomic states. This suggests that any two atoms should be equally likely to become entangled, and entanglement bonds should be generated for atoms separated by Δz ∈ [0, L], where L = 3 is the length of the cell. Indeed, such permutational invariance is central to proposals^{25,26} that use QND measurement to interrogate and manipulate manybody systems. There are other possibilities, however, such as optical pumping into entangled subradiant states^{27}, that could produce localized singlets.
We test for longrange singlettype entanglement by applying a weak gradient \(B^{\prime} \equiv d\left B \right/dz\) during the cw probing process. A magnetic field gradient, if present, causes differential Larmor precession that converts lownoise singlets into highnoise triplets, providing evidence of longrange entanglement. For example, singlets with separation Δz will convert into triplets and back at angular frequency^{28} \({\mathrm{\Omega }} = \gamma B^{\prime}{{\Delta }}z\). The range δΔz of separations then induces a range \(\delta {\mathrm{\Omega }} = \gamma B^{\prime}\delta {{\Delta }}z\) of conversion frequencies, which describes a relaxation rate. In Fig. 3 we show the KFestimated \(\left {{{\Delta }}{\mathbf{\cal{F}}}_z} \right^2\) as a function of \(B^{\prime}\) and of time since the last data point, which clearly shows faster relaxation toward a thermal spin state with increasing \(B^{\prime}\). The observed additional relaxation for \(B^{\prime}\) = 57.2 nT mm^{−1} (relative to \(B^{\prime}\) = 0) is δΩ = 1.54 × 10^{3} s^{−1}, found by an exponential fit. For Δz on the order of a wavelength, as would describe subradiant states, we would expect δΩ ~ 1 s^{−1} at this gradient, which clearly disagrees with observations. The observed r.m.s. separation δΔz is about one millimeter, which is thousands of times the typical nearestneighbor distance \(n_{{\mathrm{Rb}}}^{  1/3} \approx 0.14\) μm.
Discussion
Our observation of complex, longlived, spatiallyextended entanglement in SERFregime vapors has a number of implications. First, it is a concrete and experimentally tractable example of a system in which entanglement is not only compatible with, but, in fact, stabilized by entropygenerating mechanisms—in this case strong, randomlytimed spinexchange collisions. It is particularly intriguing that the observed macroscopic singlet state shares several traits with a spin liquid state^{2}, which is conjectured to underlie hightemperature superconductivity, a prime example of quantum coherence surviving in an entropic environment. Second, the results show that optical quantum nondemolition measurement can efficiently produce complex entangled states with longrange entanglement. This confirms a critical assumption of QNDbased proposals^{25,26} for QNDassisted quantum simulation of exotic antiferromagnetic phases. Third, the results show that SERF media are compatible with both spin squeezing and QND techniques, opening the way to quantum enhancement of what is currently the most sensitive approach to lowfrequency magnetometry and other extreme sensing tasks.
Methods
Density calibration
In the SERF regime, and in the low spin polarization limit, decoherence introduced by SE collisions between alkali atoms is quantified by ^{5,29,30}
where for ^{87}Rb atomic samples the nuclear spin I = 3/2, and ω_{L} = γ_{e}B/q. In Eq. (4) the spinexchange collision rate \(R_{{\mathrm{SE}}} = \sigma _{{\mathrm{SE}}}n_{{\mathrm{Rb}}}\overline V\) is proportional to the alkali density n_{Rb} with proportionality dictated by the SE collision crosssection σ_{SE} and the relative thermal velocity between two colliding ^{87}Rb atoms \(\overline V\). Using the reported value^{31} of σ_{SE} = 1.9 × 10^{−14} cm^{2} and \(\overline V = 4.75 \times 10^4\,{\mathrm{cm}}\;{\mathrm{s}}^{  1}\), which is computed for ^{87}Rb atoms at a temperature of 463 K, we then calibrate the alkali density by fitting the measured linewidth Δν as a function of ω_{L}. The model uses Δν = Δν_{0} + Δν_{SE}, where Δν_{SE} is given by Eq. (4), and Δν_{0} describes densityindependent broadening due to power broadening and transit effects. n_{Rb} and Δν_{0} are free parameters found by fitting, with results shown in Fig. 4.
Observed spin signal
For a collection of atoms, we define the collective total atomic spin \({\mathbf{F}} \equiv \mathop {\sum}\nolimits_l {{\mathbf{f}}^{\left( l \right)}}\), where f^{(l)} is the total spin of the lth atom. We identify the contributions of the two hyperfine ground states F_{a} = 1 and F_{b} = 2, defined as \({\mathbf{F}}_\alpha \equiv \mathop {\sum}\nolimits_l {{\mathbf{f}}_\alpha ^{\left( l \right)}}\), where \({\mathbf{f}}_\alpha ^{\left( l \right)}\) describes the contribution of atoms in state F_{α}, such that \({\mathbf{f}}^{\left( l \right)} = {\mathbf{f}}_a^{\left( l \right)} + {\mathbf{f}}_b^{\left( l \right)}\).
The Faraday rotation signal arises from an offresonance coupling of the probe light to the collective atomic spin. To lowest order in F, as appropriate to the regime of the experiment, the polarization signal S_{y} is related to the collective spin variables F_{a,z}, F_{b,z} through the inputoutput relation^{20,32,33,34}
where \(S_\alpha \equiv ( {E_ + ^{( )},E_  ^{(  )}} )\sigma _\alpha ( {E_ + ^{( + )},E_  ^{( + )}} )^T/2\) are Stokes operators, σ_{α}, α ∈ {x, y, z} are the Pauli matrices and \(E_\beta ^{\left( \pm \right)}\) is the positivefrequency (negativefrequency) part of the quantized electromagnetic field with polarization β = ± for sigmaplus (sigmaminus) polarized light. The factor (g_{a}F_{a,z} − g_{b}F_{b,z}) ≡ Θ_{FR} plays the role of a Faraday rotation angle, which in this smallangle regime can be seen to cause a displacement of S_{y}(t) from its input value. It should be noted that Θ_{FR} is operatorvalued, enabling entanglement of the spin and optical polarizations, and that the hyperfine ground states F_{a}, F_{b} contribute differentially to it.
The coupling constants are^{14,33,35}
where r_{e} = 2.82 × 10^{−13} cm is the classical electron radius, f_{osc} = 0.34 is the oscillator strength of the D_{1} transition in Rb, c is the speed of light, and ν − ν_{α} is the optical detuning of the probelight. \(\Upsilon = 2.4\) GHz is the pressurebroadened fullwidth at halfmaximum (FWHM) linewidth of the D_{1} optical transition for our experimental conditions of 100 Torr of N_{2} buffer gas. For a fardetuned probe beam, such that ν − (ν_{a} + ν_{b})/2 ≫ ν_{a} − ν_{b} (as in this experiment) one can approximate g ≡ g_{a} ≈ g_{b}, such that
Spin thermalization
In a local region containing a mean number of atoms N_{A}, the SE and HF mechanisms will rapidly produce a thermal state ρ. We note that this process conserves F, and thus also conserves the statistical distribution of F, including possible correlations with other regions. ρ is then the maximumentropy state consistent with a given distribution of F. Partitioning arguments then show that, for weakly polarized states such as those used in this experiment, the mean hyperfine populations are \(\left\langle {N_a} \right\rangle /N_{\mathrm{A}} = 3/8\) and \(\left\langle {N_b} \right\rangle /N_{\mathrm{A}} = 5/8\), and the polarisations are \(\left\langle {{\mathbf{F}}_a} \right\rangle = \left\langle {\mathbf{F}} \right\rangle /6\), \(\left\langle {\mathbf{F}} \right\rangle _b = \left\langle {\mathbf{F}} \right\rangle 5/6\), from which the FR signal is \(\left\langle {{\mathrm{\Theta }}_{{\mathrm{FR}}}} \right\rangle = g\left\langle {{\mathbf{\cal{F}}}_z} \right\rangle =  g\left\langle F \right\rangle _z2/3\). The same relations must hold for spin observables that sum F over larger regions, including the region of the beam, which determines which atoms contribute to the observed signal.
Entanglement witness
We can construct a witness for singlettype entanglement^{16} as follows: we define the total variance
Separable states of N_{A} atoms will obey a limit \(\left {{{\Delta }}{\boldsymbol{{\cal{F}}}}} \right^2 \ge N_{\mathrm{A}}{\cal{C}}\), where \({\cal{C}}\) is a constant, meaning that \(\left {{{\Delta }}{\boldsymbol{{\cal{F}}}}} \right^2 < N_{\mathrm{A}}{\cal{C}}\) witnesses entanglement. To find \({\cal{C}}\), we note that a product state of N_{a} atoms in state F_{a} and N_{b} atoms in state F_{b} has \(\left {{{\Delta }}{\boldsymbol{{\cal{F}}}}} \right^2 \ge \mathop {\sum}\nolimits_\alpha {N_\alpha F_\alpha }\). Separable states are mixtures of product states. For such states, due to the concavity of the variance, \(\left {{{\Delta }}{\mathbf{\cal{F}}}} \right^2 \ge \mathop {\sum}\nolimits_\alpha {\left\langle {N_\alpha } \right\rangle F_\alpha }\) holds^{16}. In light of the 3:5 ratio resulting from spin thermalization, this gives
or \({\cal{C}} = 13/8\). Therefore, the standard quantum limit (SQL) is N_{A}13/8. We define the degree of squeezing \(\xi ^2 \equiv \left {{{\Delta }}{\boldsymbol{{\cal{F}}}}} \right^2/\left( {\mathop {\sum}\nolimits_\alpha {\left\langle {N_\alpha } \right\rangle F_\alpha } } \right)\). Meanwhile the “thermal spin state (TSS),” i.e. the fullymixed state, has \(\left {{{\Delta }}{\boldsymbol{{\cal{F}}}}} \right^2 = N_{\mathrm{A}}9/2\).
Our condition provides also a quantitative measure of the number of entangled atoms. We consider a pure entangled quantum state of the form
where \( {{\it{\uppsi }}^{\left( l \right)}} \rangle\) are single particle states. Here, N_{p} particles are in a product state, while N_{e} = N_{A} − N_{p} particles are in an entangled state denoted by \( {\Phi _{\mathrm{e}}} \rangle\). For the collective variances of \( \kappa \rangle\) we can write that
for h = x, y, z. Let us try to find a lower bound on (11).
Let us assume that all atoms are in state F_{α}. Then, we know that \(\left( {{{\Delta }}{\mathbf{\cal{F}}}_h} \right)_{{\it{\uppsi }}^{\left( l \right)}}^2 \ge F_\alpha\) while \(\left( {{{\Delta }}{\mathbf{\cal{F}}}_h} \right)_{\Phi _{\mathrm{e}}}^2\) can even be zero, if the entangled state \( {\Phi _{\mathrm{e}}} \rangle\) is a perfect singlet. Hence,
Based on these, the number of entangled atoms in this case is bounded from below as N_{e} ≥ (1 − ξ^{2})N_{A}, where \(\xi ^2 = \left {{{\Delta }}{\boldsymbol{{\cal{F}}}}} \right^2/{\mathrm{SQL}}\) and the standard quantum limit SQL in this case is F_{α}N_{A}.
Let us now consider the case when some atoms have F_{1} others have F_{2} In particular, let us consider a state of the type (10) such that N_{α} particles have spin F_{α} with α = 1, 2 such that F_{1} ≤ F_{2}. Then, for such a pure state,
holds, where
Note that the bound in Eq. (13) is sharp, since it can be saturated by a quantum state of the type (10). In order to minimize the lefthand side of Eq. (13), the particles corresponding to the product part must have as many spins in F_{1} as possible, since this way we can obtain a small total variance. In particular, if N_{p} ≥ N_{1} then all atoms in the product part must have an F_{1} spin, otherwise at least N_{1} atoms of the N_{p} atoms.
It is instructive to rewrite Eq. (10) with a piecewise linear bound as
The bound in Eq. (15) is plotted in Fig. 5a.
So far, we have been discussing a bound for a pure state of the form (10). The results can be extended to a mixture of such states straightforwardly, since the bound in Eq. (13) is convex in (N_{1}, N_{p}). Then, in our formulas N_{1}, must be replaced by \(\left\langle {N_1} \right\rangle\). We also have to define the number of entangled particles N_{e} for the case of a mixed state. A mixed state has N_{e} entangled particles, if it cannot be constructed as a mixture of pure states, which all have fewer than N_{e} entangled particles^{15}.
We know that in our experiments F_{1} = 1, F_{2} = 2, and \(\left\langle {N_1} \right\rangle = 3/8N_{\mathrm{A}}\). From these, we obtain the minimum number of entangled atoms as
The bound in Eq. (16) is plotted in Fig. 5b. Here, again \(\xi ^2 = \left {{{\Delta }}{\boldsymbol{{\cal{F}}}}} \right^2/{\mathrm{SQL}}\) and the standard quantum limit SQL in this case is N_{A} × 13/8. For our experiment, the ξ^{2} ≥ 3/13 case is relevant.
The macroscopic singlet state gives a metrological advantage in estimating gradient fields^{28,36} and in detecting displacement of the spin state, e.g. by optical pumping^{14,37}.
Balanced polarimeter signal
The photocurrent I(t) of the balanced polarimeter shown in Fig. 1a is
where the detector’s responsivity is \(\Re = q_{\mathrm{e}}\eta /E_{{\mathrm{ph}}}\) in terms of the detector quantum efficiency η, charge of the electron q_{e}, and photon energy E_{ph}. To account for its spatial structure in Eq. (17) the integral is carried over the area of the probe. From Eq. (2) and Eq. (17) one obtains the differential photocurrent increment
where \(g' = gq_{\mathrm{e}}\), the stochastic increment dw_{sn}(t), due to photon shotnoise, is given by \(dw_{{\mathrm{sn}}}\left( t \right) = \sqrt {\eta q_{\mathrm{e}}^2\dot N} dW\) with \(\dot N\) being the photonflux and \(dW\sim {\cal{N}}\left( {0,dt} \right)\) representing a differential Wiener increment. In our experiments the photocurrent I(t) is sampled at a rate Δ^{−1} = 200 k Samples/s. To formulate the discretetime version of Eq. (17) we consider the sampling process as a shortterm average of the continuoustime measurement. The photocurrent I(t_{k}) recorded at t_{k} = kΔ, with k being an integer, can then be expressed as
where the Langevin noise ξ_{D}(t_{k}) obeys \(E\left[ {\xi _D\left( t \right)\xi _D\left( {t^{\prime}} \right)} \right] = \delta \left( {t  t^{\prime}} \right)\eta q_{\mathrm{e}}^2\dot N/{{\Delta }}\), with Δ^{−1} quantifying the effective noisebandwidth of each observation.
Spin dynamics
We model the dynamics of the average bulk spin of our hot atomic vapor in the SERF regime,^{5,29,38} and in the presence of a magnetic field B in the [1,1,1] direction, i.e. \({\mathbf{B}} = {\mathrm{B}}\left( {{\hat{\mathrm{x}}} + {\hat{\mathrm{y}}} + {\hat{\mathrm{z}}}} \right)/\sqrt 3\), as
where the matrix A includes dynamics due to Larmor precession and spin relaxation. It can be expressed as A_{ij} = −γB_{h}ε_{hij} + Γ_{ij}, where h, i, j = x, y, z. The relaxation matrix Γ has eigenvalues \(T_1^{  1}\) and \(T_2^{  1} = T_1^{  1} + T_{{\mathrm{SE}}}^{  1}\) for spin components parallel and transverse to B, respectively. We note that in the SERF regime the decoherence introduced by SE collisions between alkali atoms is quantified by Eq. (4)^{5,29}.
To account for fluctuations due to spin noise in Eq. (20) we add a stochastic term \(\sqrt {\boldsymbol{\sigma} } d{\mathbf{W}}\) where dW_{h}, h ∈ {x, y, z) are independent Wiener increments. Thus the statistical model for spin dynamics reads
where the strength of the noise source σ, the matrix A, and the covariance matrix in statistical equilibrium \(Q = {\mathrm{E}}\left[ {{\mathbf{\cal{F}}}\left( t \right){\mathbf{\cal{F}}}\left( t \right)^{\rm{T}}} \right]\) are related by the fluctuationdissipation theorem
from which we obtain σ = 2ΓQ.
Kalman filter
Kalman filtering is a signal recovery method that provides continuouslyupdated estimates of all physical variables of a stochastic model, along with uncertainties for those estimates. For linear dynamical systems with gaussian noise inputs, e.g. the spin dynamics of Eq. 3 with the readout of Eq. 5, the Kalman filter estimates are optimal in a leastsquares sense. The KF estimates, e.g. those shown in Fig. 1b and c, indicate our evolving uncertainty about the values of the physical quantities, e.g. \({\mathbf{\cal{F}}}_z\). As such, they provide an upper bound on the intrinsic uncertainty of these same quantities due to, e.g. quantum noise. As information accumulates, the uncertainty bounds on \({\mathbf{\cal{F}}}_x\), \({\mathbf{\cal{F}}}_y\) and \({\mathbf{\cal{F}}}_z\) contract toward zero, implying the production of squeezing and entanglement. This is measurementinduced, rather than dynamicallygenerated entanglement. The measured signal, i.e. the optical polarization rotation, indicates a joint atomic observable: the sum of the spin projections of many atoms. For an unpolarized state such as we use here the physical backaction—which consists of small random rotations about the \({\mathbf{\cal{F}}}_z\) axis induced by quantum fluctuations in the ellipticity of the probe—has a negligible effect.
We construct the estimator \(\widetilde {\mathbf{\cal{F}}}_t\) of the macroscopic spin vector using the continuousdiscrete version of Kalman filtering^{14}. This framework relies on a twostep procedure to construct the estimate \({\tilde{\mathbf{x}}}_t\), and its error covariance matrix \({\mathbf{\Sigma }} = {\mathrm{E}}\left[ {\left( {{\mathbf{x}}_t  {\tilde{\mathbf{x}}}_t} \right)\left( {{\mathbf{x}}_t  {\tilde{\mathbf{x}}}_t} \right)^T} \right]\), of the state x_{t} of a continuoustime linearGaussian process, in our case \({\boldsymbol{{\cal{F}}}}\left( t \right)\), that is observed at discretetime intervals Δ = t_{k} − t_{k−1}. Measurement outcomes are described by the observations vector z_{k}, in our case the scalar I_{k}, which is assumed to be linearly related to x_{t} via the coupling matrix H_{k} and to experience independent stochastic Gaussian noise as described previously^{14}.
In the first step of the Kalman filtering framework, also called the prediction step, the values at t = t_{k}, \(\widetilde {\boldsymbol{{\cal{F}}}}_{kk  1}\) and \({\mathbf{\Sigma }}_{kk  1}\), are predicted conditioned on the process dynamics and the previous instance, \(\widetilde {\boldsymbol{{\cal{F}}}}_{k  1k  1}\) and \({\mathbf{\Sigma }}_{k  1k  1}\), as follows:
where
is the state transition matrix describing the evolution of the dynamical model Eq. (20) within the time interval Δ, and
is then the effective covariance matrix of the system noise^{14}.
In the second step, or update step, the information gathered through the fresh photocurrent observation I_{k} is incorporated into the estimate:
where \({\mathbf{H}}_k = \left[ {\eta g\dot N,0,0} \right]\) and the Kalman gain K_{k} is defined as
with sensor covariance R^{Δ} = R/Δ dictated by the powerspectraldensity, R, of the photocurrent noise, i.e. due to photon shotnoise, and the sampling period, Δ. As dicussed in previous work^{14} the KF is initialised according to a distribution that represents our prior knowledge about the system at time t = t_{0} and fixes \(\widetilde {\boldsymbol{{\cal{F}}}}_{00}\sim {\cal{N}}\left( {{\boldsymbol{\mu }}_0,{\mathbf{\Sigma }}_0} \right)\), where μ_{0}, and Σ_{0} are the mean value and total variance of the observed data. After initialization KF estimates for the covariance matrix Σ_{kk} undergo a transient and once this transient has decayed they converge to a steady state value Σ_{ss}.
In Fig. 6 we observe this behavior for the total variance \(\left {{{\Delta }}{\boldsymbol{{\cal{F}}}}} \right^2\, \equiv\, {\mathrm{Tr}}\left( {{{\Gamma }}_{\boldsymbol{{\cal{F}}}}} \right)\) as a function of time t = t_{k}, where \({{\Gamma }}_{\boldsymbol{{\cal{F}}}} = {\mathbf{\Sigma }}_{kk}\). After about 0.8 ms, the total variance reaches steady state value which is used to compare with SQL and indicates squeezing degree. Figure 7 shows squeezing degree at different probe power, and presents the optimal probe power we observed is 2 mW.
Validation
To validate the sensor model we employ three validation techniques sensitive to both the statistics of the optical readout and spin noise. First, we analyse the statistics of the sensor output innovation, i.e., the difference between observations I_{k} (data) and Kalman estimates (\(\tilde y_k = I_k  {\mathbf{H}}_k\widetilde {\boldsymbol{{\cal{F}}}}_{kk  1}\)). In Fig. 8, we show the \(\tilde y_k\) histogram with the sensor output estimation error, which is described by zeromean Gaussian process with variance equal to \({\mathbf{R}}^{{\Delta }} + {\mathbf{H}}_k{\mathbf{\Sigma }}_{kk  1}{\mathbf{H}}_k^T\). We find 94% of \(\tilde y_k\) data lie within a twosided 95% confidence region of the expected Gaussian distribution, thus indicating a very close agreement of the model and observed statistics. We note that while being a standard technique in the validation of Kalman filtering^{39}, this technique for our experimental conditions is more sensitive to photon shot noise than to spin noise. Therefore, to further validate our estimates we also include two other validation techniques, designed to be sensitive to the atomic statistics on a range of timescales.
Particularly, we perform Monte Carlo simulations based on the model described by Eqs. (2), (3) and (17) and fed with the operating conditions of our experiments and compare the power spectral density (PSD) of the simulated sensor output (Simulation) to the observed PSD of the measurements (Data), as shown in Fig. 9a. The observed agreement between Data and Simulation suggests the validity of the statistics of the spin dynamics model.
Finally, we employ the Kalman filter to identify the evolution of the atomic state variables based on the Simulation. We can then compare the distribution of Kalman spinestimates from the Data versus that from Simulation. The results are shown in Fig. 9b and c, respectively. The similarity in the statistics of these two spin estimates validates the spin dynamics model. Together with the above validations, it provides a full validation of both the optical and spin parts of the model.
Gradient field tests
A weak gradient magnetic field is applied along the probe (z) direction by coils implemented inside the magnetic shields. In Fig. 10 we plot the three components of \(\left {{{\Delta }}{\boldsymbol{{\cal{F}}}}} \right^2\): \(\left {{{\Delta }}{\mathbf{\cal{F}}}_z} \right^2 \equiv {{\Gamma }}_{\boldsymbol{{\cal{F}}}}\left( {1,1} \right)\), \(\left {{{\Delta }}{\mathbf{\cal{F}}}_x} \right^2 \equiv {{\Gamma }}_{\boldsymbol{{\cal{F}}}}\left( {2,2} \right)\), and \(\left {{{\Delta }}{\mathbf{\cal{F}}}_y} \right^2 \equiv {{\Gamma }}_{\boldsymbol{{\cal{F}}}}\left( {3,3} \right)\), as a function of gradient field. Here \({{\Gamma }}_{\boldsymbol{{\cal{F}}}}\left( {i,i} \right) = {\mathbf{\Sigma }}_{ss}\left( {i,i} \right)\). We observe that the variance of each component increases towards the TSS noise level with gradient field. We note that due to the bias field along the [1,1,1] direction, the current (t = t_{k}) sensor reading indicates \({\mathbf{\cal{F}}}_z\) at that time, while \({\mathbf{\cal{F}}}_x\) and \({\mathbf{\cal{F}}}_y\) describe components that were measured 1/3 and 2/3 Larmor cycles earlier, respectively. The combined variance is used to compute \(\left {{{\Delta }}{\mathbf{\cal{F}}}_z} \right^2\), as in Fig. 3. We note that the Stern–Gerlach (SG) effect, in which a gradient causes wavefunctions components to separate in accordance with their magnetic quantum numbers, also contributes to the loss of coherence. The SG contribution is negligible, however, due to the weak gradients used here and the rapid randomization of momentum caused by the buffer gas.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request. Openaccess datasets from this work available at https://doi.org/10.5281/zenodo.3694692.
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Acknowledgements
We thank Jan Kolodynski for helpful discussions. This project has received funding from the European Union’s Horizon2020 research and innovation programme under the Marie SkłodowskaCurie grant agreements QUTEMAG (no. 654339). The work was also supported by ICFOnest + Marie SkłodowskaCurie Cofund (FP7PEOPLE2013COFUND), the National Natural Science Foundation of China (NSFC) (grant no. 11935012), and ITN ZULFNMR (766402); the European Research Council (ERC) projects AQUMET (280169), ERIDIAN (713682); European Union projects QUIC (Grant Agreement no. 641122) and FET Innovation Launchpad UVALITH (800901); Quantum Technology Flagship projects MACQSIMAL (820393) and QRANGE (820405); 17FUN03USOQS, which has received funding from the EMPIR programme cofinanced by the Participating States and from the European Union’s Horizon 2020 research and innovation programme; the Spanish MINECO projects MAQRO (Ref. FIS201568039P), XPLICA (FIS201462181EXP), OCARINA (Grant Ref. PGC2018097056BI00) and QCLOCKS (PCI2018092973), MCPA (FIS201567161P), the Severo Ochoa programme (SEV20150522); Agència de Gestió d’Ajuts Universitaris i de Recerca (AGAUR) project (2017SGR1354); Fundació Privada Cellex and Generalitat de Catalunya (CERCA program, QuantumCAT), the EU COST Action CA15220 and QuantERA CEBBEC, the Basque Government (Project No. IT98616), and the National Research, Development and Innovation Office NKFIH (Contract No. K124351, KH129601). We thank the Humboldt Foundation for a Bessel Research Award.
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J.K. and M.W.M. designed the experiment, analyzed the data and wrote the paper. J.K., C.T., and V.G.L. performed the experiment. J.K., R.J.M., and M.W.M built the Kalman filter model. G.T. and M.W.M. built the entanglement witness. M.W.M. supervised the project.
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Kong, J., JiménezMartínez, R., Troullinou, C. et al. Measurementinduced, spatiallyextended entanglement in a hot, stronglyinteracting atomic system. Nat Commun 11, 2415 (2020). https://doi.org/10.1038/s41467020158991
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