Measurement-induced, spatially-extended entanglement in a ...optics.szfki.kfki.hu/~toth/Publications/NatCommun20.pdfa power-spectral analysis of the BP signal21. Spin dynamics and

ARTICLE

Measurement-induced, spatially-extendedentanglement in a hot, strongly-interactingatomic systemJia Kong1,2✉, Ricardo Jiménez-Martínez2, Charikleia Troullinou2, Vito Giovanni Lucivero 2,

Géza Tóth 3,4,5,6 & Morgan W. Mitchell2,7✉

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 inter-

actions—can help generate and preserve entanglement. We use optical quantum non-

demolition measurement to produce entanglement in a hot alkali vapor, in a regime domi-

nated by random spin-exchange collisions. We use Bayesian statistics and spin-squeezing

inequalities to show that at least 1.52(4) × 1013 of the 5.32(12) × 1013 participating atoms

enter into singlet-type entangled states, which persist for tens of spin-thermalization times

and span thousands of times the nearest-neighbor distance. The results show that high

temperatures and strong random interactions need not destroy many-body quantum

coherence, that collective measurement can produce very complex entangled states, and that

the hot, strongly-interacting media now in use for extreme atomic sensing are well suited for

sensing beyond the standard quantum limit.

https://doi.org/10.1038/s41467-020-15899-1 OPEN

1 Department of Physics, Hangzhou Dianzi University, 310018 Hangzhou, China. 2 ICFO–Institut de Ciencies Fotoniques, The Barcelona Institute of Scienceand Technology, 08860 Castelldefels (Barcelona), Spain. 3 Department of Theoretical Physics, University of the Basque Country UPV/EHU, P.O. Box 644, E-48080 Bilbao, Spain. 4 Donostia International Physics Center, E-20018 San Sebastián, Spain. 5 IKERBASQUE, Basque Foundation for Science, E-48011Bilbao, Spain. 6Wigner Research Centre for Physics, Hungarian Academy of Sciences, P.O. Box 49, H-1525 Budapest, Hungary. 7 ICREA–Institució Catalanade Recerca i Estudis Avançats, 08010 Barcelona, Spain. ✉email: [email protected]; [email protected]

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Entanglement is an essential resource in quantum compu-tation, simulation, and sensing1, and is also believed tounderlie important many-body phenomena such as high-Tcsuperconductivity2. In many quantum technology implementa-tions, strong cooling and precise controls are required to prevententropy—whether from the environment or from noise in clas-sical parameters—from destroying quantum coherence. Quantumsensing3 is often pursued using low-entropy methods, forexample, with cold atoms in optical lattices4. There are, none-theless, important sensing technologies that operate in a high-entropy environment, and indeed that employ thermalization toboost coherence and thus sensor performance. Notably, vapor-phase spin-exchange-relaxation-free (SERF) techniques5 are usedfor magnetometry6,7, rotation sensing8, and searches for physicsbeyond the standard model9, and give unprecedented sensitiv-ity10. In the SERF regime, strong, frequent, and randomly-timedspin-exchange (SE) collisions dominate the spin dynamics, toproduce local spin thermalization. In doing so, these same pro-cesses also decouple the spin degrees of freedom from the bath ofcentre-of-mass degrees of freedom, which increases the spincoherence time5. Whether entanglement can be generated, sur-vive, and be observed in such a high entropy environment is achallenging open question11.

Here, we study the nature of spin entanglement in this hot,strongly-interacting atomic medium, using techniques of directrelevance to extreme sensing. We apply optical quantum non-demolition (QND) measurement12,13—a proven technique forboth generation and detection of non-classical states in atomicmedia—to a SERF-regime vapor. We start with a thermalizedspin state to guarantee the zero mean of the total spin variableand use a [1, 1, 1] direction magnetic field (see Fig. 1a) to achieveQND measurements on three components of the total spinvariable. We track the evolution of the net spin using the Bayesianmethod of Kalman filtering14, and use spin squeezinginequalities15,16 to quantify entanglement from the observedstatistics. We observe that the QND measurement generates amacroscopic singlet state17—a squeezed state containing a

macroscopic number of singlet-type entanglement bonds. Thisshows that QND methods can generate entanglement in hotatomic systems even when the atomic spin dynamics includestrong local interactions. The spin squeezing and thus theentanglement persist far longer than the spin-thermalization timeof the vapor; any given entanglement bond is passed many timesfrom atom to atom before decohering. We also observe a sensi-tivity to gradient fields that indicates the typical entanglementbond length is thousands of times the nearest-neighbor distance.This is experimental evidence of long-range singlet-type entan-glement bonds. These experimental observations complementrecent predictions of coherent inter-species quantum statetransfer by spin collision physics18,19.

ResultsMaterial system. We work with a vapor of 87Rb contained in aglass cell with buffer gas to slow diffusion, and housed in magneticshielding and field coils to control the magnetic environment, seeFig. 1a. The density is maintained at nRb= 3.6 × 1014 atoms/cm3,and the magnetic field, applied along the [1, 1, 1] direction, is usedto control the Larmor precession frequency ωL/2π. At this density,the spin-exchange collision rate is 325 × 103 s−1. For ωL belowabout 2π × 5 kHz, the vapor enters the SERF regime, characterizedby a large increase in spin coherence time.

Spin thermalization. The spin dynamics of such dense alkalivapors5 is characterized by a competition of several local spininteractions, diffusion, and interaction with external fields, buffergases, and wall surfaces. While the full complexity of this scenariohas not yet been incorporated in a quantum statistical model, inthe SERF regime an important simplification allows us to describethe 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

a b40

B

X

ZY

Rot

atio

nsi

gnal

SY (

nW)

20

0

–20

40

60

Rot

atio

nsi

gnal

SY (

nW)

20

0

–200.0 0.1 0.2

Time (ms)

10 2 3 4 5 6Time (ms)

0.3

SQL TSS

0.4 0.5

–40

c

Fig. 1 Experimental principle. a Experimental setup. A linearly polarized probe beam, red detuned by 44 GHz from the 87Rb D1 line, passes through a glasscell containing a hot 87Rb vapor and 100 torr of N2 buffer gas, which is housed in a low-noise magnetic enclosure. The transmitted light is detected with ashot-noise-limited polarimeter (a Wollaston prism plus differential detector), which indicates F z, the projection of the collective spin F on the probedirection, plus optical shot noise. A static magnetic field along the [1, 1, 1] direction causes the spin components to precess as F z ! F x ! F y every one-third of a Larmor cycle. In this way the polarimeter record contains information about all three components17. b Representative sample of the Stokesparameter S outð Þy tð Þ showing raw data (blue dots), optimal estimate for the atomic spin gF z tð ÞSx (red line), and ±4σ confidence interval (pale red region), ascomputed by Kalman filter (KF). Signal clearly shows atomic spin coherence over ms time-scales. c Expanded view of early signal, colors as in (b), barsshow ±4σ confidence regions of gF z tð ÞSx for a thermal spin state (TSS) and the standard quantum limit (SQL) for a spin-polarized state. The KF acquires asub-SQL estimate for F z in 20 μs, far less than the coherence time. Probe power= 2 mW, Larmor frequency= 1.3 kHz, cell temperature= 463 K.

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be described by the time-dependent Hamiltonian

H ¼ �hAhfXl

j lð Þ � i lð Þ þ �hXll0n

θnδ t � t l;l0ð Þ

n

� �j lð Þ � j l0ð Þ

þ �hXlm

ψmδ t � t lð Þm� �

j lð Þ � d lð Þm þ �hγeXl

j lð Þ � Bð1Þ

where the terms describe the hyperfine interaction, SE collisions,spin-destruction (SD) collisions, and Zeeman interaction, respec-

tively. Ahf is the hyperfine (HF) splitting and tl;l0ð Þn is the (random)

time of the n-th SE collision between atoms l and l0, which causesmutual precession of j(l) and j l

0ð Þ by the (random) angle θn. Weindicate with RSE the rate at which such collisions move angularmomentum between atoms. Similarly, the third term describesrotations about the random direction dm by random angle ψm, andcauses spin depolarization at a rate RSD. γe= 2π × 28 GHz T−1 isthe electron spin gyromagnetic ratio. We neglect the much smalleri·B coupling. We note that short-range effects of the magneticdipole–dipole interaction (MDDI) are already included in RSE andRSD, and that long-range MDDI effects are negligible in anunpolarized ensemble, as considered here.

The SERF regime is defined by the hierarchy Ahf≫ RSE≫ γe|B|,RSD. Our experiment is in this regime, as we have Ahf ≈ 109 s−1,RSE ≈ 105 s−1, γe|B| ≈ 104 s−1 and RSD ≈ 102 s−1. The hierarchyimplies the following dynamics: on short times, the combinedaction of the HF and SE terms rapidly thermalizes the spin state,i.e., generates the maximum entropy consistent with the ensembletotal angular momentum F, which is conserved by theseinteractions (see “Methods”, “Spin thermalization” section). We

indicate this F-parametrized max-entropy state by ρ thð ÞF . We notethat entanglement can survive the thermalization process; for

example, ρ thð ÞF¼0 is a singlet and thus necessarily describes entangledatoms. On longer time-scales, F experiences precession about Bdue to the Zeeman term and diffusive relaxation due to thedepolarization term.

Non-destructive measurement. We perform a continuous non-destructive readout of the spin polarization using Faraday rota-tion of off-resonance light. On passing through the cell the opticalpolarization experiences rotation by an angle gF zðtÞ � π, wherez is the propagation axis of the probe, g is a light-atom couplingconstant and F ≡ Fa− Fb, where Fα is the collective spinorientation from atoms in hyperfine state α∈ {1, 2} (see “Meth-ods”, “Observed spin signal” section).

For thermalized spin states 〈F 〉/ Fh i, so that the observedpolarization rotation gives a view into the full spin dynamics. Theoptical rotation is detected by a balanced polarimeter (BP), whichgives a signal proportional to the Stokes parameter

S outð Þy tð Þ ¼ S inð Þy tð Þ þ gF z tð ÞSx; ð2Þwhere Sx is the Stokes component along which the input beam is

polarized20. S inð Þy tð Þ is a zero-mean Gaussian process, whosevariance is dictated by photon shot-noise and is characterized bya power-spectral analysis of the BP signal21.

Spin dynamics and spin tracking. The evolution of F tð Þ isdescribed by the Langevin equation (see “Methods”, “Spin dynamics”section)

dF ¼ γF ´Bdt � ΓFdt þffiffiffiffiffiffiffiffiffi2ΓQ

pdW ð3Þ

where γ= γe/q is the SERF-regime gyromagnetic ratio, i.e., that of abare electron reduced by the nuclear slowing-down factor5, whichtakes the value q= 6 in the SERF regime22. Γ is the net relaxation rateincluding diffusion, spin-destruction collisions, and probe-induced

decoherence, Q is the equilibrium variance (see below) and dWh, h∈{x, y, z} are independent temporal Wiener increments.

Based on Eqs. (3) and (2), we employ the Bayesian estimationtechnique of Kalman filtering (KF)14 to recover F tð Þ, which isshown as gF z tð ÞSx to facilitate comparison against the measuredS outð Þy tð Þ in Fig. 1b). The KF (see “Methods”, “Kalman filter”section) gives both a best estimate and a covariance matrix ΓF tð Þfor the components of F tð Þ, which gives an upper bound on thevariances of the post-measurement state. Figure 1c shows that theF z component of ΓF tð Þ is suppressed rapidly, to reach a steadystate value which is below the SQL. The other components aresimilarly reduced in variance by the measurement, and the totalvariance ΔFj j2� Tr ΓF½ � can be compared against spin squeezinginequalities15,16 to detect and quantify entanglement: Definingthe spin-squeezing parameter ξ2 � ΔFj j2=SQL, where SQL≡NA13/8 is the standard quantum limit, ξ2 < 1 detects entangle-ment, indicating a macroscopic singlet state17. The minimumnumber of entangled atoms15 is NA(1− ξ2)13/16 (see “Methods”,“Entanglement witness” section).

Experimental results. The cell temperature was stabilized at463 K to give an alkali number density of nRb= 3.55(6) × 1014atoms cm−3, calibrated as described in “Methods”, “Densitycalibration” section, and thus NA= 5.32(12) × 1013 atoms withinthe 3 cm × 0.0503(8) cm2 effective volume of the beam. At thisdensity, the SE collision rate is RSE ≈ 325 × 103 s−1. By varying Bwe can observe the transition to the SERF regime, and the con-sequent development of squeezing. Figure 2a shows spin-noisespectra (SNS)21, i.e., the power spectra of detected signal from BP,for different values of B, from which we determine the resonancefrequency ωL= γB, relaxation rate Γ and the number density.Using these as parameters in the KF (see “Methods”, “Kalmanfilter” section), we obtain ΔFj j2 as shown in Fig. 2b, including atransition to squeezed/entangled states as the system enters theSERF regime.

At a Larmor frequency of 1.3 kHz, we observe ξ2= 0.650(2) or1.88(1) dB of spin squeezing at optimal probe power 2 mW (see“Methods”, “Kalman filter” section), which implies that at least1.52(4) × 1013 of the 5.32(12) × 1013 participating atoms havebecome entangled as a result of the measurement. This greatlyexceeds the previous entanglement records: 5 × 105 cold atoms insinglet states using a similar QND strategy17 and a Dicke stateinvolving 2 × 1011 impurities in a solid, made by storing a singlephoton in a multi-component atomic ensemble23. This is also thelargest number of atoms yet involved in a squeezed state; see Baoet al. for a recent record for polarized spin-squeezed states24. Weuse this power and field condition for the experiments describedbelow, and note that the spin-relaxation time greatly exceeds thespin-thermalization time. In this condition, the entanglementbonds are rapidly distributed amongst the atoms by SE collisionswithout being lost.

We now study the spatial distribution of the inducedentanglement. As concerns the observable F , the relevantdynamical processes, including precession, decoherence, andprobing, are permutationally-invariant: Eqs. (3) and (2) areunchanged by any permutation of the atomic states. This suggeststhat any two atoms should be equally likely to become entangled,and entanglement bonds should be generated for atoms separatedby Δz∈ [0, L], where L= 3 is the length of the cell. Indeed, suchpermutational invariance is central to proposals25,26 that useQND measurement to interrogate and manipulate many-bodysystems. There are other possibilities, however, such as opticalpumping into entangled sub-radiant states27, that could producelocalized singlets.

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We test for long-range singlet-type entanglement by applyinga weak gradient B0 � d Bj j=dz during the cw probing process. Amagnetic field gradient, if present, causes differential Larmorprecession that converts low-noise singlets into high-noisetriplets, providing evidence of long-range entanglement. Forexample, singlets with separation Δz will convert into tripletsand back at angular frequency28 Ω ¼ γB0Δz. The range δΔz ofseparations then induces a range δΩ ¼ γB0δΔz of conversionfrequencies, which describes a relaxation rate. In Fig. 3 we showthe KF-estimated ΔF zj j2 as a function of B0 and of time sincethe last data point, which clearly shows faster relaxation towarda thermal spin state with increasing B0. The observed additionalrelaxation for B0 = 57.2 nT mm−1 (relative to B0 = 0) is δΩ=1.54 × 103 s−1, found by an exponential fit. For Δz on the orderof a wavelength, as would describe sub-radiant states, we wouldexpect δΩ ~ 1 s−1 at this gradient, which clearly disagrees withobservations. The observed r.m.s. separation δΔz is about onemillimeter, which is thousands of times the typical nearest-

neighbor distance n�1=3Rb � 0:14 μm.

DiscussionOur observation of complex, long-lived, spatially-extended entan-glement in SERF-regime vapors has a number of implications.

First, it is a concrete and experimentally tractable example of asystem in which entanglement is not only compatible with, but,in fact, stabilized by entropy-generating mechanisms—in thiscase strong, randomly-timed spin-exchange collisions. It is par-ticularly intriguing that the observed macroscopic singlet stateshares several traits with a spin liquid state2, which is conjecturedto underlie high-temperature superconductivity, a prime exam-ple of quantum coherence surviving in an entropic environment.Second, the results show that optical quantum non-demolitionmeasurement can efficiently produce complex entangled stateswith long-range entanglement. This confirms a critical assump-tion of QND-based proposals25,26 for QND-assisted quantumsimulation of exotic antiferromagnetic phases. Third, the resultsshow that SERF media are compatible with both spin squeezingand QND techniques, opening the way to quantum enhancementof what is currently the most sensitive approach to low-frequency magnetometry and other extreme sensing tasks.

MethodsDensity 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

πΔνSE ¼ ω2L2I �3þ I 1þ 4I I þ 2ð Þð Þ½ �

3 3þ 4I I þ 1ð Þ½ �RSE; ð4Þ

where for 87Rb atomic samples the nuclear spin I= 3/2, and ωL= γe|B|/q. In Eq. (4)the spin-exchange collision rate RSE ¼ σSEnRbV is proportional to the alkali densitynRb with proportionality dictated by the SE collision cross-section σSE and therelative thermal velocity between two colliding 87Rb atoms V . Using the reportedvalue31 of σSE= 1.9 × 10−14 cm2 and V ¼ 4:75 ´ 104 cm s�1, which is computed for87Rb atoms at a temperature of 463 K, we then calibrate the alkali density by fittingthe measured linewidth Δν as a function of ωL. The model uses Δν= Δν0+ ΔνSE,where ΔνSE is given by Eq. (4), and Δν0 describes density-independent broadeningdue to power broadening and transit effects. nRb and Δν0 are free parameters foundby fitting, with results shown in Fig. 4.

Observed spin signal. For a collection of atoms, we define the collective totalatomic spin F � Pl f lð Þ, where f(l) is the total spin of the lth atom. We identify thecontributions of the two hyperfine ground states Fa= 1 and Fb= 2, defined asFα �

Pl f

lð Þα , where f

lð Þα describes the contribution of atoms in state Fα, such that

f lð Þ ¼ f lð Þa þ f lð Þb .The Faraday rotation signal arises from an off-resonance coupling of the probe

light to the collective atomic spin. To lowest order in F, as appropriate to theregime of the experiment, the polarization signal Sy is related to the collective spin

10−2

10−1

100

101P

SD

(*1

0−10

V2 /

Hz)

0 5 10 15

6

8

10

12

Frequency (kHz)

Var

ianc

e (1

013

spin

2 )

SQL

a

b

⏐Δ ⏐2 (0.5 mW)

⏐Δ ⏐2 (1 mW)

⏐Δ ⏐2 (2 mW)

Fig. 2 Quantum non-demolition detection of collective spin in thestrongly-interacting regime. a Spin noise spectra with atomic spin signaldriven by thermal fluctuations and precessing at the Larmor frequency (νL=ωL/2π) rising above shot noise of the Faraday rotation probe. Differentspectra correspond to different bias field strengths. Black lines are singleLorentz fits for the spectra. Comparing the red and purple curves, we see aroughly 100-fold improvement in signal to noise ratio (SNR) due tosuppression of SE relaxation and consequent line narrowing, which indicateda stronger quantum non-demolition (QND) interaction. b Spin varianceversus Larmor frequency. Black solid-line shows the standard quantum limitof total spin (SQL=NA13/8). Dashed horizontal lines indicate standardquantum limit (SQL) ±1σ statistical uncertainty. Round symbols show ΔFj j2measured with 0.5mW probe light, corresponding to the spectra in (a).Diamonds and squares show ΔFj j2 measured with 1 mW and 2mW probelight respectively. All error bars show ±1σ uncertainty due to uncertainty inatomic number, including uncertainties in atomic density and effectivevolume (see “Methods”, “Density calibration” section).

0.0 0.2 0.4 0.6 0.8 1.0

20

40

60

80

Elapsed time (ms)

B' (nT/mm) = 57

38

19

4.8

0

TSS

SQL

〈Δz2

〉 (1

012

spin

s2)

Fig. 3 Evidence for long-range entanglement. Points show the KF-obtainedvariances ΔF 2z

� �of the rotating-frame spin component F z as a function of

delay sinceF z was last aligned along the laboratory z axis, and thus subjectto measurement by Faraday rotation. Error bars show ±4σ uncertainty,originating in the uncertainty of atom number. As seen in these data,increased gradient B0 causes a faster relaxation toward the thermal spinstate (TSS) value, as is expected for a gas of singlets with a range ofseparations (along the z axis) in the mm range. The blue and gray dashedlines are the standard quantum limit (SQL) (2.88(7) × 1013 spins2) and TSS(7.99(18) × 1013 spins2) noise levels, respectively. Probe power= 2mW.




variables Fa,z, Fb,z through the input-output relation20,32–34

S outð Þy tð Þ � S inð Þy tð Þ þ gaFa;z � gbFb;z� �

S inð Þx tð Þ; ð5Þ

where Sα � ðEð�Þþ ; Eð�Þ� ÞσαðEðþÞþ ; EðþÞ� ÞT=2 are Stokes operators, σα, α∈ {x, y, z} arethe Pauli matrices and E ±ð Þβ is the positive-frequency (negative-frequency) part ofthe quantized electromagnetic field with polarization β= ± for sigma-plus (sigma-minus) polarized light. The factor (gaFa,z− gbFb,z)≡ΘFR plays the role of a Faradayrotation angle, which in this small-angle regime can be seen to cause adisplacement of Sy(t) from its input value. It should be noted that ΘFR is operator-valued, enabling entanglement of the spin and optical polarizations, and that thehyperfine ground states Fa, Fb contribute differentially to it.

The coupling constants are14,33,35

gα ¼1

2I þ 1crefoscAeff

ν � ναν � ναð Þ2þ ϒ=2ð Þ2

; ð6Þ

where re= 2.82 × 10−13 cm is the classical electron radius, fosc= 0.34 is theoscillator strength of the D1 transition in Rb, c is the speed of light, and ν− να is theoptical detuning of the probe-light. ϒ ¼ 2:4 GHz is the pressure-broadened full-width at half-maximum (FWHM) linewidth of the D1 optical transition for ourexperimental conditions of 100 Torr of N2 buffer gas. For a far-detuned probebeam, such that |ν− (νa+ νb)/2|≫ |νa− νb| (as in this experiment) one canapproximate g≡ ga ≈ gb, such that

ΘFR � g Fa � Fbð Þz� gF z : ð7Þ

Spin thermalization. In a local region containing a mean number of atoms NA, theSE and HF mechanisms will rapidly produce a thermal state ρ. We note that thisprocess conserves F, and thus also conserves the statistical distribution of F,including possible correlations with other regions. ρ is then the maximum-entropystate consistent with a given distribution of F. Partitioning arguments then showthat, for weakly polarized states such as those used in this experiment, the meanhyperfine populations are Nah i=NA ¼ 3=8 and Nbh i=NA ¼ 5=8, and the polarisa-tions are Fah i ¼ Fh i=6, Fh ib¼ Fh i5=6, from which the FR signal isΘFRh i ¼ g F zh i ¼ �g Fh iz2=3. The same relations must hold for spin observablesthat sum F over larger regions, including the region of the beam, which determineswhich atoms contribute to the observed signal.

Entanglement witness. We can construct a witness for singlet-type entangle-ment16 as follows: we define the total variance

ΔFj j2� var F xð Þ þ var�F y

�þ var F zð Þ: ð8ÞSeparable states of NA atoms will obey a limit ΔFj j2 ≥NAC, where C is a

constant, meaning that ΔFj j2N1

NpF1; if Np ≤ N1

( ð15ÞThe 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, sincethe bound in Eq. (13) is convex in (N1, Np). Then, in our formulas N1, must bereplaced by N1h i. We also have to define the number of entangled particles Nefor the case of a mixed state. A mixed state has Ne entangled particles, if it cannotbe constructed as a mixture of pure states, which all have fewer than Ne entangledparticles15.

We know that in our experiments F1= 1, F2= 2, and N1h i ¼ 3=8NA. Fromthese, we obtain the minimum number of entangled atoms as

Ne ≥1316 1� ξ2� �

NA; ξ2 ≥ 313 ;

1� 138 ξ2� �

NA; ξ2< 313 :

( )ð16Þ

The bound in Eq. (16) is plotted in Fig. 5b. Here, again ξ2 ¼ ΔFj j2=SQL andthe standard quantum limit SQL in this case is NA × 13/8. For our experiment, theξ2 ≥ 3/13 case is relevant.

The macroscopic singlet state gives a metrological advantage in estimatinggradient fields28,36 and in detecting displacement of the spin state, e.g. by opticalpumping14,37.

Balanced polarimeter signal. The photocurrent I(t) of the balanced polarimetershown in Fig. 1a is

I tð Þ ¼ <ZAdxdy S outð Þy x; y; tð Þ; ð17Þ

where the detector’s responsivity is < ¼ qeη=Eph in terms of the detector quantumefficiency η, charge of the electron qe, and photon energy Eph. To account for itsspatial structure in Eq. (17) the integral is carried over the area of the probe. FromEq. (2) and Eq. (17) one obtains the differential photocurrent increment

I tð Þdt ¼ ηg 0 _NF z tð Þdt þ dwsn tð Þ; ð18Þwhere g 0 ¼ gqe, the stochastic increment dwsn(t), due to photon shot-noise, is givenby dwsn tð Þ ¼

ffiffiffiffiffiffiffiffiffiffiffiηq2e _N

qdW with _N being the photon-flux and dW N 0; dtð Þ

0 1 2 3 4�L /2� (kHz)

0

1

2

3

Δ

(kH

z)Fit

Data

Fig. 4 Density calibration by spin noise spectroscopy. Graph shows thefull-width at half maximum linewidth Δν as a function of ωL, across thetransition into the spin-exchange relaxation free (SERF) regime. Blue dotsand error bars show the mean ±1σ standard error of the mean, obtainedfrom the fitting of 20 spectra. Red line shows Eq. (4) fit to the data withdensity nRb as a free parameter. Probe power= 1 mW, T= 463 K.




representing a differential Wiener increment. In our experiments the photocurrentI(t) is sampled at a rate Δ−1= 200 k Samples/s. To formulate the discrete-timeversion of Eq. (17) we consider the sampling process as a short-term average of thecontinuous-time measurement. The photocurrent I(tk) recorded at tk= kΔ, with kbeing an integer, can then be expressed as

I tkð Þ ¼1Δ

Z tktk�Δ

I t0ð Þdt0 ¼ ηg 0 _NF z tkð Þ þ ξD tkð Þ; ð19Þ

where the Langevin noise ξD(tk) obeys E ξD tð ÞξD t0ð Þ½ � ¼ δ t � t0ð Þηq2e _N=Δ, with Δ−1quantifying the effective noise-bandwidth of each observation.

Spin dynamics. We model the dynamics of the average bulk spin of our hot atomicvapor in the SERF regime,5,29,38 and in the presence of a magnetic field B in the[1,1,1] direction, i.e. B ¼ B x̂þ ŷ þ ẑð Þ= ffiffiffi3p , as

dF ¼ �AFdt; ð20Þwhere the matrix A includes dynamics due to Larmor precession and spinrelaxation. It can be expressed as Aij=−γBhεhij+ Γij, where h, i, j= x, y, z. Therelaxation matrix Γ has eigenvalues T�11 and T

�12 ¼ T�11 þ T�1SE for spin compo-

nents parallel and transverse to B, respectively. We note that in the SERF regimethe decoherence introduced by SE collisions between alkali atoms is quantified byEq. (4)5,29.

To account for fluctuations due to spin noise in Eq. (20) we add a stochasticterm

ffiffiffiσ

pdW where dWh, h ∈ {x, y, z) are independent Wiener increments. Thus

the statistical model for spin dynamics reads

dF ¼ �AFdt þ ffiffiffiσp dW ð21Þwhere the strength of the noise source σ, the matrix A, and the covariance matrix in

statistical equilibrium Q ¼ E F tð ÞF tð ÞTh i

are related by the fluctuation-

dissipation theorem

AQþ QAT ¼ σ; ð22Þfrom which we obtain σ= 2ΓQ.

Kalman filter. Kalman filtering is a signal recovery method that providescontinuously-updated estimates of all physical variables of a stochastic model,

along with uncertainties for those estimates. For linear dynamical systems withgaussian noise inputs, e.g. the spin dynamics of Eq. 3 with the readout of Eq. 5, theKalman filter estimates are optimal in a least-squares sense. The KF estimates, e.g.those shown in Fig. 1b and c, indicate our evolving uncertainty about the values ofthe physical quantities, e.g. F z . As such, they provide an upper bound on theintrinsic uncertainty of these same quantities due to, e.g. quantum noise. Asinformation accumulates, the uncertainty bounds on F x , F y and F z contracttoward zero, implying the production of squeezing and entanglement. This ismeasurement-induced, rather than dynamically-generated entanglement. Themeasured signal, i.e. the optical polarization rotation, indicates a joint atomicobservable: the sum of the spin projections of many atoms. For an unpolarizedstate such as we use here the physical back-action—which consists of small randomrotations about the F z axis induced by quantum fluctuations in the ellipticity ofthe probe—has a negligible effect.

We construct the estimator eF t of the macroscopic spin vector using thecontinuous-discrete version of Kalman filtering14. This framework relies on a two-step procedure to construct the estimate ~xt , and its error covariance matrix

Σ ¼ E xt � ~xtð Þ xt � ~xtð ÞTh i

, of the state xt of a continuous-time linear-Gaussian

process, in our case F tð Þ, that is observed at discrete-time intervals Δ= tk − tk−1.Measurement outcomes are described by the observations vector zk, in our case thescalar Ik, which is assumed to be linearly related to xt via the coupling matrix Hkand to experience independent stochastic Gaussian noise as described previously14.

In the first step of the Kalman filtering framework, also called the predictionstep, the values at t= tk, eF kjk�1 and Σkjk�1, are predicted conditioned on theprocess dynamics and the previous instance, eF k�1jk�1 and Σk�1jk�1, as follows:eF kjk�1 ¼ Φk;k�1 eF k�1jk�1; ð23Þ

Σkjk�1 ¼ Φk;k�1Σk�1jk�1ΦTk;k�1 þ QΔk ; ð24Þwhere

Φk;k�1 ¼ 13 e�tT1 þ 13 e�

tT2

2cos ωLΔð Þ �cos ωLΔð Þ �ffiffiffi3

psin ωLΔð Þ �cos ωLΔð Þ þ

ffiffiffi3

psin ωLΔð Þ

�cos ωLΔð Þ þffiffiffi3

psin ωLΔð Þ 2cos ωLΔð Þ �cos ωLΔð Þ �

ffiffiffi3

psin ωLΔð Þ

�cos ωLΔð Þ �ffiffiffi3

psin ωLΔð Þ �cos ωLΔð Þ þ

ffiffiffi3

psin ωLΔð Þ 2cos ωLΔð Þ

0B@1CA

ð25Þis the state transition matrix describing the evolution of the dynamical model Eq.(20) within the time interval Δ, and

QΔ ¼ NA2

1� e�2ΔT1� �

þ NA2

1� e�2ΔT2� � 2 �1 �1

�1 2 �1�1 �1 2

0B@1CA ð26Þ

is then the effective covariance matrix of the system noise14.In the second step, or update step, the information gathered through the fresh

photocurrent observation Ik is incorporated into the estimate:eF kjk ¼ eF kjk�1 þ Kk Ik �Hk eF kjk�1� � ð27ÞΣkjk ¼ 1� KkHkð ÞΣkjk�1; ð28Þ

where Hk ¼ ηg _N; 0; 0

�

and the Kalman gain Kk is defined as

Kk ¼ Σkjk�1HTk RΔ þHkΣkjk�1HTk� ��1 ð29Þ

with sensor covariance RΔ= R/Δ dictated by the power-spectral-density, R, of thephotocurrent noise, i.e. due to photon shot-noise, and the sampling period, Δ. Asdicussed in previous work14 the KF is initialised according to a distribution thatrepresents our prior knowledge about the system at time t= t0 and fixeseF 0j0 N μ0;Σ0� �, where μ0, and Σ0 are the mean value and total variance of theobserved data. After initialization KF estimates for the covariance matrix Σk|kundergo a transient and once this transient has decayed they converge to a steadystate value Σss.

In Fig. 6 we observe this behavior for the total variance ΔFj j2 � Tr ΓFð Þ as afunction of time t= tk, where ΓF ¼ Σkjk . After about 0.8 ms, the total variancereaches steady state value which is used to compare with SQL and indicatessqueezing degree. Figure 7 shows squeezing degree at different probe power, andpresents the optimal probe power we observed is 2 mW.

Validation. To validate the sensor model we employ three validation techniquessensitive to both the statistics of the optical readout and spin noise. First, weanalyse the statistics of the sensor output innovation, i.e., the difference betweenobservations Ik (data) and Kalman estimates (~yk ¼ Ik �Hk eF kjk�1). In Fig. 8, weshow the ~yk histogram with the sensor output estimation error, which is describedby zero-mean Gaussian process with variance equal to RΔ þHkΣkjk�1HTk . We find94% of ~yk data lie within a two-sided 95% confidence region of the expected

0

0.5

1

1.5Lo

wer

bou

nd/N

AN

e/N

A

N1/NANp/NA

2

0 0.5 1

0 0.5313 �2

10

0.2

0.4

0.6

0.8

1

a

b

Fig. 5 Entanglement witness. a Lower bound on the sum of the threevariances given in Eq. (13) as a function of Np for a quantum state of thetype given in Eq. (10). We set N1/NA= 3/8, F1= 1, F2= 2, corresponding tothe experiment. If we had only particles with the same spin, it would just bea straight line. b The lower bound on the number of entangled spins, givenin Eq. (16), as a function of the spin-squeezing parameter ξ2.




Gaussian distribution, thus indicating a very close agreement of the model andobserved statistics. We note that while being a standard technique in the validationof Kalman filtering39, this technique for our experimental conditions is moresensitive to photon shot noise than to spin noise. Therefore, to further validate ourestimates we also include two other validation techniques, designed to be sensitiveto the atomic statistics on a range of time-scales.

Particularly, we perform Monte Carlo simulations based on the model describedby Eqs. (2), (3) and (17) and fed with the operating conditions of our experimentsand 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 thestatistics of the spin dynamics model.

Finally, we employ the Kalman filter to identify the evolution of the atomic statevariables based on the Simulation. We can then compare the distribution of Kalmanspin-estimates from the Data versus that from Simulation. The results are shown inFig. 9b and c, respectively. The similarity in the statistics of these two spin estimatesvalidates the spin dynamics model. Together with the above validations, it providesa 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 thethree components of ΔFj j2: ΔF zj j2� ΓF 1; 1ð Þ, ΔF xj j2� ΓF 2; 2ð Þ, andΔF y

2� ΓF 3; 3ð Þ, as a function of gradient field. Here ΓF i; ið Þ ¼ Σss i; ið Þ. Weobserve that the variance of each component increases towards the TSS noise levelwith gradient field. We note that due to the bias field along the [1,1,1] direction, thecurrent (t= tk) sensor reading indicates F z at that time, while F x and F ydescribe components that were measured 1/3 and 2/3 Larmor cycles earlier,respectively. The combined variance is used to compute ΔF zj j2, as in Fig. 3. Wenote that the Stern–Gerlach (SG) effect, in which a gradient causes wave-functionscomponents to separate in accordance with their magnetic quantum numbers, alsocontributes to the loss of coherence. The SG contribution is negligible, however,due to the weak gradients used here and the rapid randomization of momentumcaused by the buffer gas.

0 0.5 1 1.5 2 2.5

t (ms)

10

20

30

40

50V

aria

nce(

1013

spi

n2) TSS

SQL

⏐Δ ⏐2

Fig. 6 Spin variance versus Kalman filter (KF) tracking time. The dashedblack line and solid-black line are thermal spin state (TSS) noise leveland standard quantum limit (SQL) for total spin. Probe power= 2mW,νL= 1.3 kHz.

0 1 2 3 4Power (mW)

–2

–1.8

–1.6

–1.4

–1.2

–1

Squ

eezi

ng (

dB)

Fig. 7 Degree of squeezing versus probe power. We observe a squeezingoptimum as 2mW. νL= 1.3 kHz. Error bars show ±1σ uncertainty in thevariance, originating in the uncertainty of the atomic number.

0.02Sensor output

0.01

0–0.004 –0.002 0 0.002 0.004

Fig. 8 Kalman model validation based on sensor output. Histogram (cyan)of observed sensor output innovation and Kalman estimates collected overa period of 0.35 s, compared to the zero-mean Gaussian distribution withcomputed sensor output estimation error (red lines). Probe power= 2mW,νL= 1.3 kHz.

−1 0 1

Spin estimation 108−1 0 1

0

0.02

0.04

0.06

Spin estimation 108

100 10110−12

10−10

10−8

Frequency (kHz)

PS

D (

V2 /

Hz)

Data

Fit − Data

Simulation

Fit − Simulation

SNSa

b cData Simulation

Fig. 9 Kalman model validation based on real data (Data) and simulateddata (Simulation). a Spin noise spectroscopy (SNS) of Data (blue dots)and Simulation (green dots). The Lorentz fittings of Data (black line) andSimulation (red line) are totally overlapped. The spin distributions (seetext) from Data (b) and Simulation (c) are shown as histograms. Error barsindicate plus/minus one standard deviation of histograms of 20 traces.Probe power= 2mW, νL= 1.3 kHz.

0 10 20 30 40 50 60

Gradient field (nT/mm)

0

20

40

60

80V

aria

nce

(101

2 sp

ins2

)

⏐Δ y⏐2

⏐Δ x⏐2

⏐Δ z⏐2

TSSz

SQLz

Fig. 10 Reduction of squeezing with increasing magnetic field. Spinvariance components increase with gradient field strength. Red dots, yellowsquares, blue diamonds represent ΔF 2z

� �, ΔF 2x� �

, and ΔF 2y

D E,

respectively. Error bars show ±4σ uncertainty in the variance, originating inthe uncertainty of the atomic number. The dashed line and solid line showthermal spin state (TSS) (7.99(18)×1013spins2) and standard quantum limit(SQL) (2.88(7) × 1013spins2) noise levels, respectively for one spincomponent F h; h 2 x; y; zf g.




Data availabilityThe data that support the findings of this study are available from the correspondingauthor upon reasonable request. Open-access datasets from this work available at https://doi.org/10.5281/zenodo.3694692.

Received: 18 October 2019; Accepted: 16 March 2020;

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AcknowledgementsWe thank Jan Kolodynski for helpful discussions. This project has received fundingfrom the European Union’s Horizon2020 research and innovation programme underthe Marie Skłodowska-Curie grant agreements QUTEMAG (no. 654339). The work wasalso supported by ICFOnest + Marie Skłodowska-Curie Cofund (FP7-PEOPLE-2013-COFUND), the National Natural Science Foundation of China (NSFC) (grant no.11935012), and ITN ZULF-NMR (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); 17FUN03-USOQS, which has received funding from the EMPIR programmeco-financed by the Participating States and from the European Union’s Horizon 2020research and innovation programme; the Spanish MINECO projects MAQRO (Ref.FIS2015-68039-P), XPLICA (FIS2014-62181-EXP), OCARINA (Grant Ref. PGC2018-097056-B-I00) and Q-CLOCKS (PCI2018-092973), MCPA (FIS2015-67161-P), theSevero Ochoa programme (SEV-2015-0522); Agència de Gestió d’Ajuts Universitaris ide Recerca (AGAUR) project (2017-SGR-1354); Fundació Privada Cellex and Gen-eralitat de Catalunya (CERCA program, QuantumCAT), the EU COST ActionCA15220 and QuantERA CEBBEC, the Basque Government (Project No. IT986-16),and the National Research, Development and Innovation Office NKFIH (Contract No.K124351, KH129601). We thank the Humboldt Foundation for a BesselResearch Award.

Author contributionsJ.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 Kalmanfilter model. G.T. and M.W.M. built the entanglement witness. M.W.M. supervised theproject.

Competing interestsThe authors declare no competing interests.

Additional informationSupplementary information is available for this paper at https://doi.org/10.1038/s41467-020-15899-1.

Correspondence and requests for materials should be addressed to J.K. or M.W.M.

Peer review information Nature Communications thanks Denis Sukachev, IannisKominis and the other, anonymous, reviewer for their contribution to the peer review ofthis work. Peer reviewer reports are available.

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Measurement-induced, spatially-extended entanglement in a hot, strongly-interacting atomicsystemResultsMaterial systemSpin thermalizationNon-destructive measurementSpin dynamics and spin trackingExperimental results

DiscussionMethodsDensity calibrationObserved spin signalSpin thermalizationEntanglement witnessBalanced polarimeter signalSpin dynamicsKalman filterValidationGradient field tests

Data availabilityReferencesAcknowledgementsAuthor contributionsCompeting interestsAdditional information

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