Non-Gaussian distribution of collective operators in quantum spinchains

Moreno-Cardoner, M., Sherson, J. F., & De Chiara, G. (2016). Non-Gaussian distribution of collective operatorsin quantum spin chains. New Journal of Physics, 18. https://doi.org/10.1088/1367-2630/18/10/103015

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New J. Phys. 18 (2016) 103015 doi:10.1088/1367-2630/18/10/103015

PAPER

Non-Gaussian distribution of collective operators in quantum spinchains

MMoreno-Cardoner1,3, J F Sherson2 andGDeChiara1

1 Centre for Theoretical Atomic,Molecular andOptical PhysicsQueen’s University, Belfast BT7 1NN,UK2 Department of Physics andAstronomy,NyMunkegade 120, AarhusUniversity, 8000AarhusC,Denmark3 Author towhomany correspondence should be addressed.

E-mail:mm.cardoner@gmail.com

Keywords: ultracold atoms, Faraday spectroscopy, spin systems, probability distribution functions, Binder cumulant

AbstractWenumerically analyse the behavior of the full distribution of collective observables in quantum spinchains.Whilemost of previous studies of quantum critical phenomena are limited to the firstmoments, herewe demonstrate howquantum fluctuations at criticality lead to highly non-Gaussiandistributions. Interestingly, we show that the distributions for different system sizes collapse on thesame curve after scaling for awide range of transitions: first and second order quantum transitions andtransitions of the Berezinskii–Kosterlitz–Thouless type.We propose and analyse the feasibility of anexperimental reconstruction of the distribution using light–matter interfaces for atoms in opticallattices or in optical resonators.

1. Introduction

The understanding of phase transitions and critical phenomena lies at the very heart of condensed-matterphysics [1]. Standard quantumphase transitions, i.e. those following Landau’s theory, are signalled by the onsetof a local order parameter when local symmetries are broken [2]. This theory sets themean value of the orderparameter at the center of stage.However, going beyond thefirst ordermoment reveals interesting informationabout themany-body state without performing a full state tomography. For instance: in experiments withultracold atoms, noise correlations reveal antiferromagnetic ordering [3, 4]; variances of collective operators inthe formof structure factors allow to distinguish between quantumphases such as the superfluid andMottinsulator (MI) [5], and to detect antiferromagnetic or crystal ordering [6] and collective entanglement [7, 8]; thekurtosis, related to higher ordermoments, provides information about quasiparticle dynamics and theirinteractions [9]; the full probability distribution of contrast in interference experiments can reveal stronglycorrelated atomic states [10, 11]; and high-order correlation functions are needed to describe properly thephysical outcome in the single-shots of several experiments of quantummany-body dynamics [12].Moreover,in recent experiments with trapped ions, the full-counting statistics of spinfluctuations allows for the detectionof entangled, over-squeezed states [13].

In this paperwe go beyond the firstmoments of the order parameter and analyse the full probabilitydistribution function (PDF) of collective operators in spin chains. The PDFof the order parameter plays a centralrole in statisticalmechanics.When the correlation length isfinite, deep in an ordered phase, the system can beregarded as the sumof independent subblocks offinite length and the central limit theorem leads to aGaussianPDF, in agreement with Landau’s paradigm. Instead, at criticality, the divergence of the correlation length leadsto a non-trivial highly non-Gaussian function, which characterizes the transition. If hyperscaling holds, the PDFshowsfinite size scalingwith the universal critical exponents of themodel [14, 15].Moreover, all statisticalmoments, related tomany-body correlation functions, can be extracted from this function. The PDF thuscontains non-local information of the system, and it can be connected to non-local order parameters in certainquantumphases.

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The PDFofmagnetization has been exhaustively studied in classical spin systems undergoing phasetransitions [16–25], and for thosemodels, non-Gaussian distributions exhibiting finite size scaling at criticalityhave been found.However,much less attention has been paid to its quantum counterpart, and only fewmodelssuch as the transverse Isingmodel have been previoulsy investigated [26, 27]. Non-Gaussian distributions oflightfluctuations have been observed for cold atomic ensembles [28], but nevertheless, a systematic treatment inthe case of strongly correlated atoms is currentlymissing. In this context, quantum coherent fluctuations of theindividual constituents can lead to non-Gaussian PDFs, which are a necessary resource for continuous variablesquantum information processing [29].

The quantum–classical correspondence dictates for the quantumdistribution the same finite size scalingwith the corresponding critical exponents. However, towhat extent the exact shape of the PDF is universal ordependent on themicroscopic details represents an intriguing question, whichwe address in this work.Here,using exact solutions and the densitymatrix renormalization group (DMRG) [30, 31], we obtain the PDF ofcollective spin variables for different types of quantumphase transitions (first and second order, and BKT type),paying special attention to their behavior at criticality, and compare these results with their classicalcounterparts. Furthermore, we give an example on how the PDF can be connected to a non-local orderparameter.

In particular, we investigate realizations using ultra-cold atoms. Although at present relevant energy scalesare still too small compared to the lowest temperatures achieved in current experimental setups, current effortsfocused onfinding efficient protocols to lower the total entropy of the system [32], for instance, by using Ramantransitions [33] or rehaping the trapping potential [34], maywellmake such investigations feasible in the nearfuture.Here, we propose two possible experimental setups based on optical lattices for themeasurement of thePDF, thefirst employing high resolutionmicroscopy and the second using quantumpolarization spectroscopy.

2. Probability density functions andfinite size scaling

The probability to observe an eigenvaluem of an operatorM is given by ( ) ∣ ∣r= å á ñm m mP m m m , where ρ is the

densitymatrix describing the system and {∣ }ñmm is an orthonormal set of eigenstates ofM compatible withm. IfM is the order parameter, this distribution can be related to the free energy functional [ ] m appearing inLandau formalism [2]: ( ) [ ]~ -P m e m . [ ] m can be approximated by a power series ofm and, if the correlationlength does not diverge, the lowest order terms dominate. To lowest order, this leads to a PDFwhich isapproximately Gaussian, a result which can also be understood by the central limit theorem. In contrast, close tothe critical point we expect a highly non-Gaussian PDF.

Close to a continuous (second order) phase transition induced by a parameter g of theHamiltonianwithcritical point gc, the correlation length diverges as x tµ n- , where ∣ ∣t = -g gc and ν is the critical exponent.Close enough to the critical point, thefinite size scaling hypothesis implies that themean value á ñM scales withthe system size L as [35]:

( ) ( )tá ñ = b n n-M L f L , 11

where f is an analytic function. It is oftenmore convenient and accurate for determining the critical point, bothnumerically and experimentally, to compute the Binder cumulant:

( )= -á ñá ñ

UM

M1

3, 2

4

2 2

as its scaling depends only on ν and not onβ. That is, ˜( )t= nU f L1 , with a different scaling function f̃ .Uquantifies theGaussianity of the PDF, being null for aGaussian distribution centered at zero.

Here we go beyond thefirstmoments and consider the full PDF ( )P m g,L . The renormalised PDF isexpected to be a universal function (although still depending on the boundary conditions):

˜( ˜ ) ( ) ( )= b n-P m r L P m g, , 3L

with ˜ = b n-m m L and x=r L . Note that in this expression the parameter g driving the second order phasetransition needs to be changedwhen varying L, in order to keep x Lfixed. This finite-size scaling of the PDFimplies that hyperscaling and finite-size scaling of higher order correlation functions hold [16, 17, 21]. In fact,and aswewill see, when the critical exponents are unknown, or not defined in the standardway as in the BKTtransition, one should instead rescale the PDF and the eigenvaluemwith s = á ñM 2 , directly computed fromthe PDF:

˜( ˜ ) ( ) ( )s=P m g P m g, , 4L

with ˜ s=m m .

2

New J. Phys. 18 (2016) 103015 MMoreno-Cardoner et al

3.Models andmethods

Weconsider a variety of spin-1/2 chainmodels encapsulated by theHamiltonian:

( ) ( )å s s s= +a

a a a a a+H J h , 5

i

i i i

,

1

where the sumon i extends over the L spins in the chain. The Pauli operators of spin i are denoted by sai , while Jα

and hα are coupling constants andmagnetic fields along different directions (a = x y z, , ). Thismodel exhibitsvarious transitions [1, 36, 37], some ofwhichwill be discussed below.Wewill study the behavior of collectiveoperators such as the totalmagnetization s= åa aM Li

i and its staggered counterpart ( ) s= å -a aM L1ii ist .

Analytical results for the PDF can be obtained formodels that can bewrittenwith a free fermionicrepresentation after the Jordan–Wigner transformation, i.e. = =J h 0z x y, in equation (5) assuming periodicboundary conditions (PBC), and operators that can bewritten as the sumof single-site operators in this basis, asfor exampleMz. The order parameterMx however, is not separable and contains the string order operator. Inprinciple all its powers could be computed bymeans ofWick’s theorem, but the evaluation becomes involved asthe order increases. At criticality and for the quantum Ising spin chain, the PDF can be exactly obtained byexploiting the relationwithKondo physics [27], but in general, reconstructing the PDF analytically is a verychallenging task.Here instead, to obtain the PDF,we combine twodifferent numericalmethods.We use exactdiagonalisation for sizes up to 20 sites, and also the time-dependentDMRG (t-DMRG) [30]with open boundaryconditions (OBC). The t-DMRGmethod provides an efficient way to evaluate the characteristic function,defined as the Fourier transformof the PDF. For a pure quantum state ∣Y ñ0 this can bewritten as:

( ) ( ) ∣ ∣ ∣ ∣ ∣ ( )å åc y y= = á Y ñ = á ñu P m me e e 6m

um

m

um uMi0

2 i0

i0

beingM the operator of interest and { }m the corresponding set of eigenvalues. The expression for ( )c u isequivalent to the overlap between the initial state ∣y ñ0 and its evolution under afictitiousHamiltonian equal toM at different times u, and thus, it can be computedwith t-DMRG. Since there is a one-to-one correspondencebetween the characteristic function and the PDF, this can be recovered by inverse Fourier transforming ( )c u .Moreover, it allows to directly evaluate all centralmomenta and cumulants.

4. Results

4.1. Second order quantumphase transitionWe set = =J h 0y z x y, , , = <J J 0x and =h Jgz in equation (5), corresponding to the ferromagnetic transverseIsingmodel, which exhibits a second order phase transition at = g 1 separating a ferromagnetic ordered phase(FM) at lowfields and a paramagnetic disordered phase (PM) at high fields. Infigure 1(a)we show the numericalresults for the PDF of the spontaneousmagnetizationMx, the order parameter in FM. Remarkably, we haveobtained data collapse already for relatively small system sizes, assuming the ansatz of equation (4)with thepredicted values b = 1 8 and n = 1 (see e.g. [14, 15]). This result reinforces the scaling hypothesis of all thestatisticalmoments of the order parameter. Away from the critical point and in the thermodynamic limit( ¥r ) the PDF isGaussian, in agreement with Landau theory and the central limit theorem. In the disordered

Figure 1.Rescaled PDF of the order parameter in the transverse Isingmodel. Data collapse is observed for different system sizes( =L 20, 30, 40, 60, 80, 100 correspond to+, star,×, circle, square and diamond symbol, respectively). (a) In absence oflongitudinal field and for different phases atfixed r: FM (blue) at r=4 (equivalent to g=0.75 for L = 16), critical (green) at r=0(g=1.0) and PM (red) (at r=4, equivalent to g=1.25 for L=16). (b) In presence of the longitudinal field in the FMphase atr=4.8 (equivalent to g=0.7 for L = 16) and q=0 (h=0, blue), q=0.5 (green, equivalent to = ´ -h 6 10 5 for L=16) andq=2 (red, equivalent to = ´ -h 2 10 4 for L=16). The solid lines correspond to aGaussian fit as discussed in the text. The PDF isnon-Gaussian close to the critical point.

3

New J. Phys. 18 (2016) 103015 MMoreno-Cardoner et al

phase, it is centered aroundm=0, whereas in the ordered phase the distribution is bimodal with the two peakscorresponding approximately to the broken-symmetry values of the order parameter. In contrast, at criticality( r 0) the distribution becomes non-Gaussian. The central limit theoremdoes not necessarily hold anymoredue to quasi long-range fluctuations in this regime.Wenote that, despite the PDF is universal at criticality, it stilldepends on the boundary conditions that are chosen.

For comparison, it is worth noting that for the transversemagnetizationMz, which is not the orderparameter, the PDF is alwaysGaussian. For this observable, themoment of nth order can be directly evaluatedfrom the characteristic function and it always scales as -L n1 . This leads to aGaussian distribution in thethermodynamic limit.

As previously discussed, it is not clear a priori that the PDF is a universal function (only depending on theuniversality class), and thus, that it can be retrieved from the classicalmodel analogue. The 1D transverse Isingmodelmaps onto the classical 2D Isingmodel with spatially anisotropic couplings. The anisotropy depends onthe temperature of the quantum system, and therefore, on the system size of the equivalent classicalmodel.Performingfinite size scaling in the 2Dmodel simultaneously in all dimensions (as it is usually done for a z=1transition) changes the anisotropy at each step, and onemight doubt whether this process leads to a universalfunction. To elucidate this, we compare the previous results with those obtained for the classicalmodel byMonte Carlo in [20, 38]. A direct comparison shows that the quantum results for the PDF at the critical pointhavemore fluctuations aroundm=0, leading to a flatter distributionwith non-zero value, in contrast to theclassical case.

Gaussianity of the PDF—we can characterize the gaussianity of the distribution by fitting the data with thesumof twoGaussians as

( ) [ ] ( )( ) ( ) ( ) ( )

ps= +s s- - - +P m

1

2 2e e 7m m m m

fit2

2 202 2

02 2

with s2 andm0 as free parameters (see solid lines infigure 1(a)). As shown in thefigure, wefind very goodagreement away from the critical point, whereas close to the critical point the fitting yields poor results and oneshould use instead the exponential of a higher-order polynomial. This can be quantified by the nonlinear leastsquarefitting correlation coefficient defined as:

[ ] ( )d= -c y y1 Var , 82

where dy2 is the squared normof the residuals of the data y and [ ]yVar the corresponding variance. Thisquantity is close to onewhen the fittingworks well, whereas it drops to smaller values for a poorfitting.Figure 2(a) shows c as a function of ˜ ( ) ( )= - = -r r g L gsgn 1 1 . Clearly, there is a sudden decrease close tothe critical point (˜ =r 0), whereas it tends to one away from the critical point (˜ ¥r ).

An alternative way to quantify theGaussianity of the PDF is by evaluating the Binder cumulantU,equation (2).U vanishes for aGaussian distribution centered at zero, whereas it tends to U 2 3 for adistribution close to two symmetric delta functions. EvaluatingU is convenient for locating the critical point,since its finite size scaling only depends on the critical exponent ν and not onβ, that is, ˜( )=U f r , and in this case

˜(∣ ∣ · )= -U f g g Lc . Thus, when plotted as a function of the transverse field g, the crossing point for datacorresponding to different sizes tends to the actual critical point gc.When instead, plotted as a function of r̃ , theycollapse to the same curve, which depends, however, on the boundary conditions.

Figure 2. (a)Correlation coefficient c of a nonlinear least square fitting of the PDFP(m) to theGaussian function ( )P mfit in thetransverse Isingmodel. The coefficient c suddenly decreases close to the critical point, indicating the PDF is non-Gaussian. (b)–(c)Binder cumulantU in the transverse isingmodel, for different system sizes denoted by different symbols (from L=0 to L= 100). (b)As a function of ˜ · ( ) ( )x= - = -r L g g L g gsgn c c , data collapse is observed. (c)As a function of g forOBC, the lines cross at thecritical point. Black and brown symbols are forOBC and PBC, respectively.

4

New J. Phys. 18 (2016) 103015 MMoreno-Cardoner et al

In the samefigure we also plot the result ofU for the transverse Isingmodel withOBC as a function of r̃ (b)and g (c) for different system sizes, ranging from L=20 to L=100. The result for PBC is also shown forcomparison.

4.2. First order phase transitionWeconsider the sameHamiltonian as before butwith an additional longitudinal field = ¹h Jh 0x . Atfixedvalue of the transverse field = g 1and approaching the Ising critical point by varying only h, the quantumphase transition is characterized by a different set of exponents (b = 1 15 and n = 8 15 [14]). Our results (notshown) support strong evidence of scaling of the PDF.

If instead, the longitudinal field h is varied across zero, but atfixed value of the transverse field ∣ ∣ <g 1, thesystemundergoes a first order transition between two ferromagnetic states with oppositemagnetizationMx. Atthis transition, neither the correlation length diverges nor the gap closes. Nevertheless, inspired by [39], wepropose afinite size scaling of the PDF.

In afinite size chain, two different energy gaps are present in thismodel. One is the real energy gap in thethermodynamic limit, which at h=0 is given by ∣∣ ∣ ∣D = - nE g 1 z with z=1, and it closes at the Ising criticalpoint ( = g 1). The other gap δ separates the two lowest eigenstates and it isminimumat h=0, with value

∣ ∣( )d = -J g g2 1 L0

2 , decreasing exponentially with L (∣ ∣ <g 1). In [39], following dimensional arguments, theauthors assume that δ andMx only depend on the longitudinal field h and L through the ratio q between the twoenergy scales: the first associatedwith the longitudinal field hLMx,0, being ( )= -M g1x,0

2 1 8 themagnetization at h=0, and the second d0:

( )( )

d= =

-q

hLM hL

g g

2

1. 9x

L

,0

02 7 8

Thus, ( ) ( )d d~ g f q0 1 and ( ) ( )~M m g f qx 0 2 , where ( )f q1,2 are analytic functions at q=0.Here, we conjecture a similar dependence for the correlation length: ( ) ( )x x~ g f q0 , where f is analytic at

q=0, and ( ) ( )x x= =g q g0,0 diverges close to the Ising critical point with the usual power law:( ) ∣∣ ∣ ∣x ~ - n-g g 10 .Moreover, any observable ( )M g h,L depends only on the ratios x=r L 0 and qwhen

rescaledwith the proper critical exponents, e.g. for themagnetization ˜ ( ) ( )= b n-M r q L M g h, ,x x L, . Indeed, atfixed values of r and q (and for large enough chains, or equivalently weak enough external fields), we observeagain data collapse for the PDF for different lengths, as shown infigure 1(b).Wefind the distribution to bealways bimodal with the relative height of the two peaks ruled by h.Wefitted the data with the sumof twoGaussians and found reasonable agreement except at the critical point, due to nonlinear effects in the Landaupotential.

4.3. BKT transitionFor completeness, wefinish this analysis with a different type of transition, the Berezinski–Kosterlitz–Thouless(BKT) transition in the spin-1/2XXZmodel.We set =h 0x y z, , , = >J J 0x y, and = DJ Jz inHamiltonian (5).The phase diagram iswell known: the ground state is ferromagnetic forD < -1, critical for ∣ ∣D < 1, andwithNéel order forD > 1. The BKT transition is atD = 1. This transition is of the same universality class as theclassical 2DXYmodel, for which previous calculations on the PDF [23] and on the Binder cumulant [24, 25],show that the order parameter distribution is non-Gaussian at criticality. Herewe compute the PDF for theground state of the XXZmodel, for both staggeredmagnetizationsMx

st andMzst, corresponding to the order

parameters in the critical andNéel phases respectively, and for the full range of values ofΔ. In contrast to theprevious sections, we rescale the quantitiesm andP(m)withσ as defined in equation (4). ForMst

z we alsofix

x=r L , being x = p D-e 1 [37], and observe data collapse for different system sizes, as shown infigure 3(a).As expected, the distribution tends in the thermodynamic limit to a double- and single-peakedGaussian forD > 1 (ordered phase with respect toMz

st) andD < 1, respectively.As shown infigure 3(b), the situation ismuchmore interesting for the PDF ofMst

x , for which the spin–spincorrelations do not decay exponentially in the critical phase (∣ ∣D < 1). Outside this interval the distribution isGaussian and centered at zero, because the operatorMst

x is disordered, while in the critical phase it is highly non-Gaussian.We observe again data collapse for different chain lengths, but nowwhen fixing the value ofΔ. ThescalingwithfixedΔ is expected to occur in a critical phase, where the energy gap has already closed.Whencrossing thefirst order transition atD = -1, the PDF shows also a discontinuity, with a sudden jump from aGaussian to a function that has a singular behavior. In contrast to the usual behavior of the PDF, this functiondoes not tend to zero as s ¥m , but it increases its value and its derivative when increasing the system size(see blue symbols infigure 3(b)). Across the critical phase, the function changes continuously from a double-peak structure forD < 0 to a single peak distribution forD > 0. At the BKTpoint the PDF tends again to aGaussian in the thermodynamic limit. Note that the results for the ferromagneticmodel <J 0 are equivalentwhen analyzingMx instead, and changingD « -D.

5

New J. Phys. 18 (2016) 103015 MMoreno-Cardoner et al

Weemphasize that while the collapse is expectedwith the correct universal exponents as in the classical case,the specific collapse function can be dependent on themodel or on the boundary conditions. This is clear byvisual inspectionwhen comparing the previous results at different values ofΔ to those in the 2D classical XYmodel reported in [23] (figure 2).

The non-Gaussianity in the critical phase is also evident from the analysis of the Binder cumulant. Infigure 4we show the result ofU for the spin-1/2XXZmodel for the two observablesMz

st (left) andMxst (right), as a

function ofΔ. The Binder cumulant forMzst tends to the value U 2 3 in theNéel phase in the

thermodynamical limit, whereas the one forMxst remainsfinite in the critical phase.

5.Non-local order parameters

The BKT transition is particularly relevant in the context of 1Doptical lattice gases because it rules the superfluid(SF) toMI transition in the Bose–Hubbardmodel. Close to theMI, where density fluctuations are small, and forlarge enough integer fillings, themodel can be approximated to an effective spin-1model [40, 41]. The SF andMI aremapped respectively to the critical ferromagnetic phase, and to a state with localmagnetization =S 0z

i

perturbedwith tightly bound particle–holefluctuations. The nature of density fluctuations is however differentin the two phases. In theMI the non-local correlations can be characterized by the parity operator defined as

( ) = á ñpd

¥ +

lim e , 10l k j k l

nP2 i j

where dnj is the local excess density from the average filling [42–44], and corresponds to the localmagnetizationin themagneticmodel.P

2 is finite in theMI, while it vanishes in the SF, and it has been experimentallyreconstructed in [44] using single sitemicroscopy.

In this context, an additionalmotivation for the study of PDFs is that it captures these non-local correlations.Indeed, it is easy to see that the characteristic function ( ) ( )= åX u P mem

umi for the suitable collective variable(in this case, the averagemagnetization of a subblock of length l) at frequency p=u l is equal to the parity

Figure 3.Rescaled PDF in the spin-1/2XXZmodel. Data collapse is observed for different system sizes ( =L 10, 20, 30, 40, 60correspond to dot, cross, star, circle and square symbol, respectively). (a) ForMz

st atfixed r: (critical phase) r=0.9 (blue) and r=0(cyan); (Néel ordered phase) r=0.35 (orange) and r=1.3 (purple). (b) ForMx

st atfixedΔ: (critical phase) D = -0.99 (blue),D = -0.5 (green),D = 0 (red) andD = 1 (cyan).

Figure 4.Binder CumulantU in the spin-1/2XXZmodel, for different system sizes denoted by different symbols( =L 10, 20, 30, 40, 60 correspond to dot, cross, star, circle and square symbol, respectively). (a) ForMz

st it tends to =U 2 3 in theNéel ordered phase in the thermodynamic limit. (b) ForMx

st isfinite in the critical phase in the thermodynamic limit.

6

New J. Phys. 18 (2016) 103015 MMoreno-Cardoner et al

operator ( ) p= =¥X u lliml P2 . Thus, ameasurement of a collective variable and the reconstruction of its

PDF provides an alternative route to evaluate some particular non-local order parameters.

6. Experimental reconstruction of the PDF

The PDF for the local number parity operator can be reconstructed using single site resolutionmicroscopy[45, 46]. Recently, novel schemes which allow to circumvent light-assisted pair loss and resolve the internalatomic degree of freedomhave been also demonstrated [47]. An alternative proposal is based on the Faradayeffect, and consists in analysing the polarization fluctuations of a strongly polarized laser beam that interactswith the atomic sample [48–50]. Since this is based on dispersive light–matter interaction, it has the advantage ofbeing less destructive, which could potentially pave theway formultiple probing of the same ensemble. Thus,thismethod could find relevant applications for quantum state control and preparation. In particular, bymonitoring the system in the appropriate way, this could be used in principle to lower its total entropy. Already ithas been experimentally demonstrated thatQPS combinedwith real-time feedback allows for a dramaticreduction of the atomnumber fluctuations [51], increasing the purtity of a coherent spin state [52], and inducingspin squeezing [28].Moreover, QPS can be exploited for quantum state engineering and control [53], inducingnon-trivial quantumdynamics [54], and the conversion of atomic correlations and entanglement into the lightdegree of freedom for quantum information processing [55].

The light–matter interactionwith the collective spin operatorM, ( ˜ )k t~H LP Meff ph , is written in terms ofthemomentum-like light quadrature Pphmeasuring the photonfluctuations in the circular basis with respect tothe strong polarization axis, and τ is the interaction time. Pph is canonically conjugated to its position-likecounterpart: [ ] =X P, iph ph . The coupling constant ( )k ha= 1 2 can be expressed in terms of the single atomexcitation probability or destructivity η, and the resonant optical depth a s= N Across , whereA is the overlaparea between light and atoms, scross is the resonant cross section and for a one-dimensional systemN=L/d,being d the interparticle distance. By adjusting the laser intensity, the destructivity parameter η is typically set tovalues smaller than 0.1 to limit the fraction of excited atoms. The resonant optical depthα should bemaximizedin order to obtain the largest possible coupling between light and atoms. For ultracold atoms in a one-dimensional optical lattice, considering L=100 and m=A 0.5 m2 we obtain a » 8 and k » 1.We definek̃ k= L , whichwill be approximately independent of L and could be in the best case scenario as large ask̃ ~ 0.1.Wewill show that this value ofκ is large enough to reconstruct the PDF of the spin operatorM bylooking at the distribution of the light quadratureXph. Larger values of k̃ could be engineered by coupling atomswith optical cavities [56, 57], nanophotonic crystals [58, 59] or optical nanofibers [60].

It is possible to show [61] that the light distribution is the sumof vacuumGaussian distributions of light eachdisplaced by a quantity proportional to the eigenvaluem of the operatorM and scaled by the probability P(m) toobserve such eigenvalue:

( ) ( ) ( )( ˜ ) ( )åps

= k s- +P X P m1

2e 11

m

X Nmph

ph2

2 ,ph2

ph2

wherewe have considered aGaussian input light beamwith variance sph2 , being 1/2 for the vacuum state.

To evaluate the effectiveness of themethodwe compare the actual atomic spin distributionwith the one ofthe output light. The distance between both distributions decreases exponentially with k̃ sph. Thus, one could inprinciple improve the fidelity by increasing k̃ or using squeezed light [62].We show infigure 5 the result for thetransverse field Isingmodel for an optimal case k̃ = 1and s = 1 2ph

2 and for amore realistic value of k̃ = 0.15,

but squeezed input light with s = 1 4ph2 , recently achieved [62]. For the former, the light distribution faithfully

follows themagnetization, whereas for the later, it only agrees qualitatively, but still captures the peaks.Experimentally, onewould need to repeat themeasurement a number of shots ~N Lshots

2 to estimate the PDF.

7. Summary

In conclusion, we have shown that the distribution of collective variables in spinmodels reveals relevantinformation of quantumphase transitions.We have shown that for a range of quantumphase transitions a non-Gaussian distribution of the order parameter is a clear signature of criticality, and that the scaling hypothesisholds. Finally we have proposed an experimentalmethod for itsmeasurement using light–matter interfaces, anddiscussed its feasibility for realistic values.

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Acknowledgments

The authors want to thankDDagnino, A Ferraro, GMussardo, A Sanpera andR Sewell for very usefulcomments. This work is supported by theUKEPSRC (EP/L005026/1), JohnTempleton Foundation (grant ID43467), the EUCollaborative Project TherMiQ (Grant Agreement 618074).

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Figure 5.Comparison between rescaled light distribution (red solid line) and PDF of the order paramater (blue bar) in the criticalphase in the Transverse Isingmodel for (a) k̃ = 1.0, s = 1 2ph

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