Measurement-induced nonlocal entanglement in a hot, strongly-interacting atomic system

Jia Kong,1 Ricardo Jimenez-Martınez,1 Charikleia Troullinou,1 Vito Giovanni Lucivero,2 and Morgan W. Mitchell1, 3

1ICFO–Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain∗

2Department of Physics, Princeton University, Princeton, New Jersey 08544, USA3ICREA–Institucio Catalana de Recerca i Estudis Avancats, 08010 Barcelona, Spain

(Dated: April 24, 2018)

Quantum entanglement underlies quantumtechnologies1, and is notoriously sensitive to de-cohering processes. For this reason, many imple-mentations take elaborate measures to remove en-tropy from their environments, e.g. cryogenics2

or optical cooling3–5 to reach mK to nK temper-atures. Here we demonstrate that the oppositestrategy, actively promoting strong interactions andthermalization, can excel in generating and pre-serving entanglement. We work with a vapourof 87Rb heated to 450 K and a number density of3.6× 1014 atoms/cm3. Under these conditions randomspin-exchange collisions dominate the spin physics,giving a single-atom coherence time of ∼ 25 µs andrapid local spin thermalization6. This same physicsparadoxically generates a maximum-entropy many-body state with much longer coherence time, viathe so-called spin-exchange-relaxation-free (SERF)effect7,8. To generate and quantify entanglementin this system, we perform optical quantum non-demolition9,10 (QND) measurement and use Kalmanfiltering techniques to reconstruct the spin dynamicsin real-time. We observe extremely efficient entan-glement generation, including projection of at least2.3× 1013 atoms into non-local singlet states that lastfor ∼ 1.8 ms, i.e., tens of thermalization times. Inaddition to breaking all records for entanglementgeneration11,12, and synthesising for the first timea many-body state resembling a spin liquid13, thisshows that the unique properties of SERF-regimeensembles are extremely attractive for QND-basedquantum technologies, with potential applicationsin quantum memories14, quantum sensing15,16 andquantum simulation13,17–20.

Different quantum technologies require different fea-tures of entanglement: Device-independent communica-tion protocols21 require long-range entangled pairs, moti-vating satellite entanglement distribution22 and quantumrepeater23 strategies. Universal quantum computing1 re-quires entanglement of all the qubits in a register, motivat-ing quantum error correction methods24,25. Quantum simu-lation has still other entanglement requirements. For exam-ple, analogue simulation17 of Anderson’s spin-liquid modelfor high-temperature superconductivity13 requires an entan-gled state in which all spins are entangled with all others,independently of their relative positions, to form a com-plex spin singlet. This requires both complex entanglement,as in computing, and distributed entanglement of subsys-tems that do not directly interact, as in quantum repeaters.While strong cooling into entangled ground states of an en-gineered Hamiltonian is a direct way to produce such states,the challenges to such cooling are formidable3–5. This mo-

tivates searches for new ways to generate large-scale atomicentanglement.

One such method is optical quantum non-demolition(QND) measurement9,10: a beam of light is coherently cou-pled to all of the target atoms and then measured. Bythe coupling, an atomic collective observable, e.g. the totalspin of the ensemble, becomes entangled with an observableproperty of the light such as phase or polarization angle.When the light is subsequently detected, the atoms experi-ence entanglement swapping that leaves them in a state ap-proximating an eigenstate of the collective observable, e.g.a Dicke state. QND measurements can be remarkably effi-cient, for example causing 5.5× 105 atoms to form singletsin a population of 1.1× 106 atoms11. In the quantum sim-ulation context, QND methods were first proposed18,19 asa means to detect and distinguish unpolarized phases con-taining singlet-type entanglement bonds, and later shown toalso be capable of producing such phases20. QND methodshave been used to generate entanglement in low-density coldgases11,15,16,26–30, a scenario in which inter-atomic interac-tions are negligible.

Here we report three main advances in the use of QNDmeasurements to generate atomic entanglement: We showthat the method can work with hot atomic systems, thatthe method can operate also when the atomic dynamics in-clude strong local interactions, and that the entanglementis generated nonlocally, in our case over mm length scaleswhereas the nearest-neighbour distance is µm. Combiningthese, we use QND measurement to generate an entangledstate with several features of the spin-liquid state: a largenumber of atoms, at least 2.3× 1013 as detected by spinsqueezing inequalities, participating in long-range singlets,in a system that is both hot and dominated by short-rangeinter-atomic spin dynamics.

The atomic system is an atomic vapour of 87Rb in thespin-exchange-relaxation-free (SERF) regime7 (see Spin dy-namics part in Methods ). SERF-regime vapours offer theunusual combination of high-optical depth and long coher-ence times, both of which are important for QND measure-ment. In ordinary atomic vapours, high number density,and thus high optical depth (OD), implies also high deco-herence rates, principally through spin-exchange (SE) col-lisions. In SERF, however, the SE collisions are made todominate the spin dynamics through even higher densitiesand low magnetic fields, leading to a kind of motional aver-aging that prevents spin decoherence. This effect has pre-viously been translated into orders-of-magnitude improve-ments in the sensitivity of optical magnetometers31,32. Herewe demonstrate the utility of the SERF regime for entan-glement generation.

The spin dynamics of dense alkali vapours33,34 is charac-terized by a competition of several local spin interactions,

arX

iv:1

804.

0781

8v1

[qu

ant-

ph]

20

Apr

201

8

2

FIG. 1. Long-range entanglement of hot atoms. a) Physical principles: (upper panel) atomic spin ensemble (red arrows) is ahigh-density alkali vapour in which hyperfine interactions and spin-exchange collisions cause rapid thermalization of electron andnuclear spin degrees of freedom, while conserving local total spin. (middle panel) Faraday rotation probing (red beam) performs aquantum non-demolition measurement of the collective spin of the ensemble, reducing the total spin uncertainty below the standardquantum limit, and creating spin singlets, represented as matched-color spheres, with brackets indicating short, medium, and long-range singlet pairs. (lower panel) a magnetic field gradient, if present, causes differential Larmor precession that converts low-noisesinglets into high-noise triplets, providing evidence of long-range entanglement. b) Experimental setup. A linearly polarized probebeam red detuned by 44 GHz from the 87Rb D1 line, passes through a glass cell containing a hot 87Rb ensemble and 100 torr ofN2 buffer gas housed in a low-noise magnetic enclosure. The transmitted light is detected with a shot-noise-limited polarimeter (aWollaston prism (WP) plus differential detector), which indicates Jz, the projection of the collective spin J on the probe direction,plus optical shot noise. A static magnetic field along the [1, 1, 1] direction causes the spin components to precess as Jz → Jx → Jyevery one-third of a Larmor cycle. In this way the polarimeter record contains information about all three components11. c)Representative sample of polarimeter signal (blue) including both shot noise and atomic spin signal, and recovered atomic spinsignal (red) found by Kalman filtering. d) Symbols: Total variation |∆J|2 of the Kalman filter estimate as a function of decay andapplied gradient for d|B|/dz = 0 nT mm−1 (violet, bottom) to 57.2 nT mm−1 (red, top). Curves show exponential fits. The fasterloss of singlets caused by larger gradient reveals the non-local character of the generated entanglement bonds.

diffusion, and interaction with external fields, buffer gases,and wall surfaces. Perhaps due to this complexity, to datethere has been little study of the quantum statistical prop-erties of such dense vapours35,36. If j(k) and i(k) are the kthatom’s electron and nuclear spins, respectively, the spin dy-namics, including sudden collisions, can be described by thetime-dependent Hamiltonian

H = hAhf

∑k

j(k) · i(k) + hθn∑kln

δ(t− t(k,l)n )j(k) · j(l)

+hψm

∑km

δ(t− t(k)m )j(k) · d(k)

m + hγe∑k

j(k) ·B (1)

where the terms describe the hyperfine interaction, spin-exchange (SE) collisions, spin-destruction (SD) collisionsand Zeeman interaction, respectively. Ahf is the hyperfine

(HF) splitting and t(k,l)n is the (random) time of the n-th SE

collision between atoms k and l, which causes mutual preces-sion of j(k) and j(l) by the (random) angle θn. We indicatewith RSE the rate at which such collisions decohere a typi-cal atom. Similarly, the last term describes rotations aboutthe random direction dm by random angle ψm, and causesspin depolarization at a rate RSD. γe = 2π× 28 GHz T−1 isthe electron spin gyromagnetic ratio. We neglect the muchsmaller i ·B coupling.

In the SERF regime we have the hierarchy Ahf RSE γe|B|, RSD, with the following consequences: on short times,the combined action of the HF and SE terms rapidly ther-malizes the state, i.e., generates the maximum entropy con-sistent with the ensemble total angular momentum F, which

is conserved by these interactions. We indicate this F-

parametrized max-entropy state by ρ(th)F . We note that en-

tanglement survives the thermalization process; for exam-

ple ρ(th)F=0 is a singlet and thus necessarily describes entan-

gled atoms. On longer time-scales, F experiences precessionabout B due to the Zeeman term and diffusive relaxationdue to the depolarization term.

During such precession, the collective electron spin J re-mains proportional to F, with its evolution described by theLangevin equation (see Methods - Spin dynamics)

dJ = γJ×Bdt− ΓJdt+√

2ΓQdW (2)

where γ = γe/q is the SERF-regime gyromagnetic ratio,q = 6 is the nuclear slowing-down factor34, Γ is the net re-laxation rate including diffusion, spin-destruction collisionsand probe-induced decoherence, Q is the equilibrium vari-ance (see below) and dWh, h ∈ x, y, z are independentWiener increments.

Faraday rotation probing detects the collective electronspin J ≡

∑k j(k) by coupling to the probe polarization. On

passing through the ensemble the optical polarization ex-periences rotation by an angle gJz(t) 1, where z is thepropagation axis of the probe and g is light-atom couplingconstant. This rotation is detected by a balanced polarime-ter (BP), which gives a signal proportional to the Stokesparameter

S(out)y (t) = S(in)

y (t) + gJz(t)Sx (3)

3

where Sx is the Stokes component along which the input

beam is polarized37. S(in)y (t) is a zero-mean Gaussian pro-

cess, whose variance is dictated by photon shot-noise andis characterized by a power-spectral analysis of the BPsignal38.

Based on Eqs. (2) and (3), we employ the Bayesian esti-mation technique of Kalman filtering39 (KF) to recover J(t)as shown in Fig. 1 c). The KF (see Methods - Kalman filter)gives both a best-estimate and a covariance matrix ΓJ(t)for the components of J(t), which gives an upper boundon the variances of the post-measurement state. In partic-ular, the total variation |∆J|2 ≡ Tr[ΓJ] can be comparedagainst spin squeezing inequalities40–42 to detect and quan-tify entanglement: Defining the spin-squeezing parameterξ2 ≡ |∆J|2/SQL, where SQL ≡ NA/2 is the standard quan-tum limit, ξ2 < 1 detects entanglement, indicating a macro-scopic singlet state11. The minimum number of entangledatoms43 is (1− ξ2)NA.

The cell temperature was stabilized at 450 K to give analkali number density of nRb = 3.6× 1014 atoms/cm3, cal-ibrated as described in Methods - Density calibration, andthus NA = 5.3× 1013 atoms within the 3 cm × 0.049 cm2

effective volume of the beam. At this density, the SE col-lision rate is RSE ≈ 41× 103 s−1. By varying B we canobserve the transition to the SERF regime, and the conse-quent development of squeezing. Fig. 2 a) shows spin-noise

spectra (SNS)36,38, i.e., the power spectra of S(out)y , for dif-

ferent values of B, from which we determine the resonancefrequency ωL = γB, Γ and the vapor alkali density. Us-ing these as parameters in the KF (see Methods - Kalmanfilter), we obtain |∆J|2 as shown in Fig. 2 b), including atransition to squeezed/entangled states as the system entersthe SERF regime. At a Larmor frequency of 1 kHz and op-tical power 1 mW, we observe ξ2 = 0.57 or 2.4 dB of spinsqueezing, which implies at least 2.3× 1013 of the 5.3× 1013

participating atoms have entered singlets as a result of themeasurement. This number exceeds the current records formeasurement-induced entanglement11 by about eight ordersof magnitude, and entanglement generation by any means12

by about two orders of magnitude. We use this power andfield condition for the experiments described below.

We now study the spatial distribution of the induced en-tanglement. As concerns the observable J =

∑i j(i), the

relevant dynamical processes, including precession, decoher-ence, and probing, are permutationally-invariant: Eqs. (2)and (3) are unchanged by any permutation of the atomicstates. This suggests that any two atoms should be equallylikely to become entangled, and entanglement bonds shouldbe generated for atoms separated by ∆z ∈ [0, L], whereL = 3 cm is the length of the cell. Indeed, such permuta-tional invariance is central to proposals18–20 that use QNDmeasurement to interrogate and manipulate many-body sys-tems. There are other possibilities, however, such as opticalpumping into entangled sub-radiant states44,45, that couldproduce localized singlets.

We test for long-range singlets by applying a weak gradi-ent B′ ≡ d|B|/dz during the cw probing process. As shownin Fig. 1 a), bottom, singlets with separation ∆z will convertinto triplets and back at angular frequency46 Ω = γB′∆z.The range δ∆z of induced separations then induces a rangeδΩ = γB′δ∆z which describes a relaxation rate. In Fig. 1

0 5 10 151.5

2

2.5

3

3.5

4

Frequency ( Hz)

Var

ianc

e (

1013

spin

2 )

k

TSSSQL

|∆ J |2 (0.5 mW)

|∆ J |2 (1 mW)

0 5 10 15

10−2

10−1

100

101

Larmor Frequency (KHz)

PS

D (

10−

10

V2/H

z)

a）

b）

FIG. 2. QND detection of collective spin in the strongly-interacting regime. a) Spin noise spectra with atomic spin signaldriven by thermal fluctuations and precessing at the Larmor fre-quency (νL = ωL/2π) rising above shot noise of the Faradayrotation probe. Different spectra correspond to different biasfield strengths. Black lines are single Lorentz fits for the spectra.Comparing the red and blue curves, we see a roughly 100-foldimprovement in signal to noise ratio (SNR) due to suppressionof SE relaxation and consequent line narrowing, which indicateda stronger QND interaction. b) Spin variance versus Larmor fre-quency. Black dashed line shows the noise level of thermal spinstate (TSS = 3NA/4), the black solid-line shows standard quan-tum limit (SQL = NA/2). Round symbols show |∆J|2 measuredwith 0.5 mW of probe light, corresponding to the spectra in a),and diamonds show |∆J|2 measured with 1 mW.

d) we show the KF-estimated |∆J|2 as a function of B′

and of time since the last data point, which clearly showsfaster relaxation toward a thermal state with increasing B′.For ∆z on the order of a wavelength, as would describe sub-radiant states, this gives δΩ ∼ 1 s−1, even at the largest gra-dient B′ = 57.2 nT mm−1. This clearly disagrees with ob-servations and we conclude the singlets cannot all be short-range. In contrast, if any pair of atoms is equally likelyto form a singlet, the r.m.s. separation ∆z = L/

√24 im-

plies δΩ ∼ 1× 104 s−1 for B′ = 57.2 nT mm−1. Consideringthe simplicity of the model used here, this plausibly agreeswith the observed additional relaxation (relative to B′ = 0)of 2.7× 103 s−1, found by an exponential fit. We concludethe typical singlet separation is indeed several millimeters,which is tens of thousands times the nearest-neighbour dis-tance.

We have demonstrated very efficient and large-scale en-tanglement generation by quantum non-demolition mea-surement: at least 2.3 × 1013 atoms entangled every fewmilliseconds. This number exceeds previously-reported en-tanglement records by two to eight orders of magnitude.

4

More important than the number of entanglements is thefact that it was achieved in a strongly thermalized gas, show-ing that thermal interactions need not be deleterious to en-tanglement production or preservation. The result is espe-cially intriguing for quantum simulation, as it strongly mir-rors proposals20,47 that use QND measurements on strongly-interacting systems to detect and engineer long-range en-tanglement of unpolarized phases resulting from frustratedhamiltonians, e.g. spin liquids. Much work remains to bedone to fully understand the spin dynamics underlying theseobservations; in the experiment a typical atom experiencedspin-exchange with its neighbours roughly forty times fasterthan it became entangled with distant atoms, so no few-body picture of the process can adequately describe the en-tanglement dynamics. The work also shows for the firsttime quantum entanglement effects in SERF-regime atomicvapours, and showcases their potential for other QND-basedquantum technologies, including quantum memory14 andquantum-enhanced sensing15,16 applications.

Acknowledgments. We thank Geza Toth and JanKolodynski for helpful discussions. This project has receivedfunding from the European Unions Horizon 2020 researchand innovation programme under the Marie Sklodowska-Curie grant agreements QUTEMAG (no. 654339). Thework was also supported by ICFOnest + Marie Sklodowska-Curie Cofund fellow (FP7-PEOPLE-2013-COFUND); theEuropean Research Council (ERC) projects AQUMET(280169), ERIDIAN (713682); European Union projectsQUIC (Grant Agreement no. 641122) and FET Innova-tion Launchpad UVALITH (800901); the Spanish MINECOprojects MAQRO (Ref. FIS2015-68039-P), XPLICA(FIS2014-62181-EXP), the Severo Ochoa programme (SEV-2015-0522); Agencia de Gestio d’Ajuts Universitaris i deRecerca (AGAUR) project (2014-SGR-1295); Fundacio Pri-vada Cellex and Generalitat de Catalunya (CERCA Pro-gram).

METHODS

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

I(t) = <∫Adxdy S(out)

y (x, y, t), (4)

where the detector’s responsivity < = qη/Eph is propor-tional to its quantum efficiency η, charge of the electron q,and photon energy Eph. To account for its spatial struc-ture in Eq.(4) the integral is carried over the area of theprobe. From Eq.(3) and Eq.(4) one obtains the differentialphotocurrent increment

I(t)dt = dwsn(t) + ηgNJz(t)dt, (5)

where the sthochastic increment dwsn(t), due to photon

shot-noise, is given by dwsn(t) =

√ηNdW with N being

the photon-flux and dW ∼ N (0, dt) representing a differen-tial Wiener increment. In our experiments the photocurrentI(t) is sampled at a rate ∆−1 = 200 kSamples/s. To formu-late the discrete-time version of Eq. (4) we consider the sam-pling process as a short-term average of the continuous-timemeasurement. The photocurrent I(tk) recorded at tk =k∆,

0 5 10 15 20 250

1

2

3

4

0

1

2

ωL

(2π• kHz)

∆ν

(kH

z)

DataFit

FIG. 3. The measured full width half maximum linewidth ∆ν ofSNS as a function of ωL. The blue crosses are data and red lineis the fitting with Eq. (7).

with k being an integer, can then be expressed as

I(tk) =1

∆

∫ tk

tk−∆

I(t′)dt′ = ξD(tk) + ηgNJz(tk), (6)

where the Langevin noise ξD(tk) obeys E[ξD(t)ξD(t′)] =

δ(t − t′)ηN/∆, with ∆−1 quantifying the effective noise-bandwidth of each observation.

Density calibration. In the SERF regime, and in thelow spin polarization limit, decoherence introduced by SEcollisions between alkali atoms is quantified by7,31,34

π∆νSE = ω2L

2I[−3 + I(1 + 4I(I + 2))]

3[3 + 4I(I + 1)]RSE, (7)

where for 87Rb atomic samples the nuclear spin I = 3/2,and ωL = γe|B|/q. In Eq. (7) the spin-exchange colli-sion rate RSE = σSEnRbV is proportional to the alkali den-sity nRb with proportionality dictated by the SE collisioncross-section σSE and the relative thermal velocity betweentwo colliding 87Rb atoms V . Using the reported value6 ofσSE = 1.9× 10−14 cm2 and V = 4.75× 104 cm/s, which iscomputed for 87Rb atoms at a temperature of 450 K, wethen calibrate the alkali density by fitting the measuredlinewidth ∆ν as a function of ωL. The model used consistsof ∆ν = ∆ν0 +∆νSE , where ∆νSE is given by Eq. (7), withnRb and ∆ν0 as the free parameters. In Fig. (3) we show afit to the data taken at 450 K with a probe optical power of1 mW from which we obtain nRb = 3.6× 1013 cm−3.

Spin dynamics. We model the dynamics of the av-erage bulk spin of our hot atomic vapour in the SERFregime31,34,48 and in the presence of a magnetic field B inthe [1,1,1] direction, i.e. B = B(x + y + z)/

√3, as

dJ = −AJdt, (8)

where the matrix Aij , with i, j = x, y, z, includes dy-namics due to Larmor precession and spin relaxation, itcan be expressed as Aij = −γBxεxij + Γij with the ele-

ments of the relaxation matrix Γ proportional to T−11 and

T−12 = T−1

1 + T−1SE , the relaxation rates appropriate for

spin components parallel and transverse to B, respectively.

5

0 1 21

2

3

4

t (ms)

Var

ianc

e (1

013sp

in2 )

SQL

|∆ J|2

TSS

FIG. 4. Spin variance versus KF tracking time. The dashedblack line and solid-black line are TSS noise level and SQL for Jsystem, and the red curve is |∆J|2 from Kalman filter estimation.

We note that in the SERF regime the decoherence intro-duced by SE collisions between alkali atoms is quantified byEq.(7)31,34.

To account for fluctuations due to spin noise in Eq. (8) weadd a stochastic term

√σdW where dWh, h ∈ x, y, z are

independent Wiener increments. Thus the statistical modelfor spin dynamics reads

dJ = −AJdt+√σdW (9)

where the strength of the noise source σ, the matrix A,and the covariance matrix in statistical equilibrium Q =E[Ji(t)Jj(t

′)] are related by the fluctuation-dissipation the-orem

AQ+QAT = σ, (10)

from which we obtain σ = 2ΓQ. Defining the state transi-tion matrix Φ(t, t0) as the solution to Eq. (8) and assumingJk−1 as the state of the spin at time t = tk−1, then thestochastic, unconditional evolution during the time intervaltk − tk−1 = ∆ is described by the multi-normal distributionN (Φk,k−1Jk−1, Q

∆) with mean Φk,k−1Jk−1 and covarianceQ∆.

Kalman filter. We construct the estimator Jt of themacroscopic spin vector using the continuous-discrete ver-sion of Kalman filtering39. This framework relies on a two-step procedure to construct the estimate xt, and its errorcovariance matrix Σ = E

[(xt − xt)(xt − xt)

T], of the state

xt of a continuous-time linear-Gaussian process, in our caseJ(t), that is observed at discrete-time intervals.

In the first step, also called the prediction step, the valuesat t = tk, Jk|k−1 and Σk|k−1, are predicted conditioned on

the process dynamics and the previous instance, Jk−1|k−1

and Σk−1|k−1, as follows:

Jk|k−1 = Φk,k−1Jk−1|k−1, (11)

Σk|k−1 = Φk,k−1Σk−1|k−1ΦTk,k−1 +Q∆

k , (12)

where Φk,k−1 and Q∆k are defined above.

0 10 20 30 40 50 602

6

10

14

Gradient Field (nT/mm)

Var

ianc

e (1

012sp

in2 )

|∆Jz|2

|∆Jx|2

|∆Jy|2

TSS

SQL

FIG. 5. Spin variance versus gradient field strength. The dashedblack line and solid-black line are TSS noise level and SQL forJj component. The red dots, yellow square and blue diamondare |∆Jz|2, |∆Jx|2 and |∆Jy|2 respectively.

In the second step, or update step, the information gath-ered through the fresh photocurrent observation Ik is incor-porated into the estimate:

Jk|k = Jk|k−1 + Kk

(Ik −HkJk|k−1

)(13)

Σk|k = (11−KkHk) Σk|k−1, (14)

where Hk = [0, 0, ηgN ] and the Kalman gain Kk is definedas

Kk = Σk|k−1HTk

(R∆ + HkΣk|k−1H

Tk .)−1

. (15)

As dicussed in previous work39 the KF is initialised accord-ing to a distribution that represents our prior knowledgeabout the system at time t = t0 and fixes J0|0 ∼ N (µ0,Σ0).After initialization KF estimates for the covariance matrixΣk|k undergo a transient and once this transient has decayedthey converge to a steady state value Σss. In Fig. (4) weobserve this behaviour for the total variation |∆J|2 ≡ Tr[ΓJ]as a function of time t = tk, where ΓJ = Σk|k.

Gradient field tests. A weak gradient magnetic field isapplied along the probe (z) direction by coils implementedinside the magnetic shields. In Fig. (5) we plot all thethree components of |∆J|2 : |∆Jz|2 ≡ ΓJ(1, 1), |∆Jx|2 ≡ΓJ(2, 2), and |∆Jy|2 ≡ ΓJ(3, 3), as a function of gradientfield. Here ΓJ(i, i) = Σss(i, i). We observe that the varianceof each component increases towards the TSS noise levelwith gradient field. According to our experimental geometryand KF model, at time t = tk, Jz is probed, while thedetected components Jx and Jy correspond to their statesat 1/3 and 2/3 Larmor cycle intervals earlier, therefore theycan be used to indicate the evolution of Jz in a Larmor cycleunder gradient field but without probing as shown in Fig. 1d).

1 M. Nielsen and I. Chuang,Quantum Computation and Quantum Information, Cam-

bridge Series on Information and the Natural Sciences

6

(Cambridge University Press, 2000).2 A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang,

J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf,“Strong coupling of a single photon to a superconductingqubit using circuit quantum electrodynamics,” Nature 431,162 EP (2004).

3 A. E. Leanhardt, T. A. Pasquini, M. Saba, A. Schirotzek,Y. Shin, D. Kielpinski, D. E. Pritchard, and W. Ketterle,“Cooling bose-einstein condensates below 500 picokelvin,”Science 301, 1513 (2003).

4 D. C. McKay and B. DeMarco, “Cooling in strongly corre-lated optical lattices: prospects and challenges,” Reports onProgress in Physics 74, 054401 (2011).

5 R. Olf, F. Fang, G. E. Marti, A. MacRae, and D. M. Stamper-Kurn, “Thermometry and cooling of a bose gas to 0.02 timesthe condensation temperature,” Nature Physics 11, 720 EP(2015).

6 W. Happer, Y. Y. Jau, and T. Walker,Optically Pumped Atoms (Wiley, 2010).

7 W. Happer and H. Tang, “Spin-exchange shift and narrow-ing of magnetic resonance lines in optically pumped alkalivapors,” Phys. Rev. Lett. 31, 273 (1973).

8 I. Kominis, T. Kornack, J. Allred, and M. Romalis, “A sub-femtotesla multichannel atomic magnetometer,” Nature 422,596 (2003).

9 P. Grangier, J. A. Levenson, and J.-P. Poizat, “Quantum non-demolition measurements in optics,” Nature 396, 537 (1998).

10 R. J. Sewell, M. Napolitano, N. Behbood, G. Colangelo, andM. W. Mitchell, “Certified quantum non-demolition measure-ment of a macroscopic material system,” Nat Photon 7, 517(2013).

11 N. Behbood, F. Martin Ciurana, G. Colangelo, M. Napoli-tano, G. Toth, R. J. Sewell, and M. W. Mitchell, “Genera-tion of macroscopic singlet states in a cold atomic ensemble,”Phys. Rev. Lett. 113, 093601 (2014).

12 P. Zarkeshian, C. Deshmukh, N. Sinclair, S. K. Goyal, G. H.Aguilar, P. Lefebvre, M. G. Puigibert, V. B. Verma, F. Mar-sili, M. D. Shaw, S. W. Nam, K. Heshami, D. Oblak, W. Tit-tel, and C. Simon, “Entanglement between more than twohundred macroscopic atomic ensembles in a solid,” NatureCommunications 8, 906 (2017).

13 P. W. Anderson, “The resonating valence bond state inLa2CuO4 and superconductivity,” Science 235, 1196 (1987).

14 J. Appel, E. Figueroa, D. Korystov, M. Lobino, and A. I.Lvovsky, “Quantum memory for squeezed light,” Phys. Rev.Lett. 100, 093602 (2008).

15 R. J. Sewell, M. Koschorreck, M. Napolitano, B. Dubost,N. Behbood, and M. W. Mitchell, “Magnetic sensitivity be-yond the projection noise limit by spin squeezing,” Phys. Rev.Lett. 109, 253605 (2012).

16 O. Hosten, N. J. Engelsen, R. Krishnakumar, and M. A.Kasevich, “Measurement noise 100 times lower than thequantum-projection limit using entangled atoms,” Nature529, 505 (2016).

17 J. I. Cirac and P. Zoller, “Goals and opportunities in quantumsimulation,” Nature Physics 8, 264 EP (2012).

18 K. Eckert, L. Zawitkowski, A. Sanpera, M. Lewenstein, andE. S. Polzik, “Quantum polarization spectroscopy of ultracoldspinor gases,” Phys. Rev. Lett. 98, 100404 (2007).

19 K. Eckert, O. Romero-Isart, M. Rodriguez, M. Lewenstein,E. S. Polzik, and A. Sanpera, “Quantum non-demolition de-tection of strongly correlated systems,” Nature Physics 4, 50EP (2007).

20 P. Hauke, R. J. Sewell, M. W. Mitchell, and M. Lewen-stein, “Quantum control of spin correlations in ultracold lat-tice gases,” Phys. Rev. A 87, 021601 (2013).

21 R. Colbeck, “Quantum and relativistic protocols for secure

multi-party computation,” Ph.D. Thesis, Cambridge Univer-sity (2007), https://arxiv.org/abs/0911.3814.

22 S.-K. Liao, W.-Q. Cai, W.-Y. Liu, L. Zhang, Y. Li, J.-G. Ren,J. Yin, Q. Shen, Y. Cao, Z.-P. Li, F.-Z. Li, X.-W. Chen, L.-H. Sun, J.-J. Jia, J.-C. Wu, X.-J. Jiang, J.-F. Wang, Y.-M.Huang, Q. Wang, Y.-L. Zhou, L. Deng, T. Xi, L. Ma, T. Hu,Q. Zhang, Y.-A. Chen, N.-L. Liu, X.-B. Wang, Z.-C. Zhu,C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan,“Satellite-to-ground quantum key distribution,” Nature 549,43 EP (2017).

23 H. J. Kimble, “The quantum internet,” Nature 453, 1023(2008).

24 J. Chiaverini, D. Leibfried, T. Schaetz, M. D. Barrett, R. B.Blakestad, J. Britton, W. M. Itano, J. D. Jost, E. Knill,C. Langer, R. Ozeri, and D. J. Wineland, “Realization ofquantum error correction,” Nature 432, 602 EP (2004).

25 P. Schindler, J. T. Barreiro, T. Monz, V. Nebendahl, D. Nigg,M. Chwalla, M. Hennrich, and R. Blatt, “Experimental repet-itive quantum error correction,” Science 332, 1059 (2011).

26 T. Takano, M. Fuyama, R. Namiki, and Y. Takahashi, “Spinsqueezing of a cold atomic ensemble with the nuclear spin ofone-half,” Physical Review Letters 102, 033601 (2009).

27 J. Appel, P. J. Windpassinger, D. Oblak, U. B. Hoff, N. Kjær-gaard, and E. S. Polzik, “Mesoscopic atomic entanglement forprecision measurements beyond the standard quantum limit,”Proc. Nat. Acad. Sci. 106, 10960 (2009).

28 M. H. Schleier-Smith, I. D. Leroux, and V. Vuletic, “Statesof an ensemble of two-level atoms with reduced quantum un-certainty,” Phys. Rev. Lett. 104, 073604 (2010).

29 J. G. Bohnet, K. C. Cox, M. A. Norcia, J. M. Weiner,Z. Chen, and J. K. Thompson, “Reduced spin measurementback-action for a phase sensitivity ten times beyond the stan-dard quantum limit,” Nat Photon 8, 731 (2014).

30 F. Martin Ciurana, G. Colangelo, L. Slodicka, R. J.Sewell, and M. W. Mitchell, “Entanglement-enhanced radio-frequency field detection and waveform sensing,” Phys. Rev.Lett. 119, 043603 (2017).

31 J. C. Allred, R. N. Lyman, T. W. Kornack, and M. V.Romalis, “High-sensitivity atomic magnetometer unaffectedby spin-exchange relaxation,” Phys. Rev. Lett. 89, 130801(2002).

32 V. Shah and M. V. Romalis, “Spin-exchange relaxation-freemagnetometry using elliptically polarized light,” Phys. Rev.A 80, 013416 (2009).

33 W. Happer, “Optical pumping,” Rev. Mod. Phys. 44, 169(1972).

34 I. M. Savukov and M. V. Romalis, “Effects of spin-exchangecollisions in a high-density alkali-metal vapor in low magneticfields,” Phys. Rev. A 71, 023405 (2005).

35 I. K. Kominis, “Sub-shot-noise magnetometry with a corre-lated spin-relaxation dominated alkali-metal vapor,” Phys.Rev. Lett. 100, 073002 (2008).

36 V. G. Lucivero, A. Dimic, J. Kong, R. Jimenez-Martınez,and M. W. Mitchell, “Sensitivity, quantum limits, and quan-tum enhancement of noise spectroscopies,” Phys. Rev. A 95,041803 (2017).

37 G. Colangelo, R. J. Sewell, N. Behbood, F. M. Ciurana,G. Triginer, and M. W. Mitchell, “Quantum atom–light in-terfaces in the gaussian description for spin-1 systems,” NewJournal of Physics 15, 103007 (2013).

38 V. G. Lucivero, R. Jimenez-Martınez, J. Kong, and M. W.Mitchell, “Squeezed-light spin noise spectroscopy,” Phys. Rev.A 93, 053802 (2016).

39 R. Jimenez-Martınez, J. Ko lodynski, C. Troullinou, V. G. Lu-civero, J. Kong, and M. W. Mitchell, “Signal tracking beyondthe time resolution of an atomic sensor by kalman filtering,”Phys Rev Lett 120, 040503 (2018).

7

40 O. Guhne and G. Toth, “Entanglement detection,” PhysicsReports 474, 1 (2009).

41 G. Toth and M. W. Mitchell, “Generation of macroscopic sin-glet states in atomic ensembles,” New Journal of Physics 12,053007 (2010).

42 G. Vitagliano, P. Hyllus, I. L. Egusquiza, and G. Toth, “Spinsqueezing inequalities for arbitrary spin,” Phys. Rev. Lett.107, 240502 (2011).

43 P. Hyllus, L. Pezze, and A. Smerzi, “Entanglement and sen-sitivity in precision measurements with states of a fluctuatingnumber of particles,” Phys. Rev. Lett. 105, 120501 (2010).

44 W. Guerin, M. O. Araujo, and R. Kaiser, “Subradiance ina large cloud of cold atoms,” Phys. Rev. Lett. 116, 083601(2016).

45 A. Asenjo-Garcia, M. Moreno-Cardoner, A. Albrecht, H. J.Kimble, and D. E. Chang, “Exponential improvement inphoton storage fidelities using subradiance and “selective ra-diance” in atomic arrays,” Phys. Rev. X 7, 031024 (2017).

46 I. Urizar-Lanz, P. Hyllus, I. L. Egusquiza, M. W. Mitchell,and G. Toth, “Macroscopic singlet states for gradient magne-tometry,” Phys. Rev. A 88, 013626 (2013).

47 K. Eckert, O. Romero-Isart, M. Rodriguez, M. Lewenstein,E. S. Polzik, and A. Sanpera, “Quantum non-demolition de-tection of strongly correlated systems,” Nature Physics 4, 50(2007).

48 M. P. Ledbetter, I. M. Savukov, V. M. Acosta, D. Budker,and M. V. Romalis, “Spin-exchange-relaxation-free magne-tometry with cs vapor,” Phys. Rev. A 77, 033408 (2008)