Spin nematic order in antiferromagnetic spinor condensates

T. Zibold,1 V. Corre,1 C. Frapolli,1 A. Invernizzi,1 J. Dalibard,1 and F. Gerbier1

1Laboratoire Kastler Brossel, College de France, CNRS, ENS-PSL Research University,

UPMC-Sorbonne Universites, 11 place Marcelin Berthelot, 75005 Paris

(Dated: October 13, 2015)

Large spin systems can exhibit unconventional types of magnetic ordering di↵erent from the fer-romagnetic or Neel-like antiferromagnetic order commonly found in spin 1/2 systems. Spin-nematicphases, for instance, do not break time-reversal invariance and their magnetic order parameter ischaracterized by a second rank tensor with the symmetry of an ellipsoid. Here we show direct ex-perimental evidence for spin-nematic ordering in a spin-1 Bose-Einstein condensate of sodium atomswith antiferromagnetic interactions. In a mean field description this order is enforced by lockingthe relative phase between spin components. We reveal this mechanism by studying the spin noiseafter a spin rotation, which is shown to contain information hidden when looking only at averages.The method should be applicable to high spin systems in order to reveal complex magnetic phases.

Magnetic order in spin 1/2 systems is commonly asso-ciated with either a ferromagnetic phase or a Neel antifer-romagnet, depending on the sign of the exchange interac-tions. The situation is richer for spins greater than 1/2,and other types of magnetic order can arise at low tem-peratures. Spin 1 systems, for instance, can support spinnematic phases with vanishing average spin hsi [1]. Themagnetic order is then characterized by a non-zero spinquadrupole tensor, Q

ij

⌘ 1

2

hsi

sj

+ sj

si

i which deviatesfrom isotropy even without applied field, i.e. it describesan object with the symmetries of an ellipsoid. In the sim-plest case, with axial symmetry, the spin quadrupole ten-sor has the same mathematical form as the orientationalorder parameter of nematic liquid crystals [2]. There is apreferred axis in space (the director) without a preferreddirection along that axis.

Spin nematic phases have been identified and exten-sively studied theoretically in lattice spin 1 models (see,e.g, [3–7]) or in dilute atomic Bose-Einstein condensates(BECs) composed of spin 1 atoms [8] with antiferromag-netic spin-exchange interactions [9–13]. Experimentally,a direct evidence is still lacking both in the solid state[5–7, 14–16] and in atomic gases [17–21]. In solid statesystems, most magnetic probes couple only to the mag-netization and are therefore unsuitable to reveal spin ne-matic order. In spin 1 condensates, equilibrium proper-ties have been characterized by measuring only the pop-ulations of each Zeeman state; This is not su�cient to es-tablish the nature of the magnetic order. For instance, inthe so-called broken axisymmetry phase where all threeZeeman sublevels are populated, ferromagnetic or spinnematic behavior cannot be distinguished from the aver-age populations alone.

In this Letter, we propose and experimentally demon-strate a method to reveal spin-nematic ordering (or pos-sibly other types of unconventional magnetic order). Weshow that the spin noise following a spin rotation con-tains information about the initial state, which can beretrieved with a suitable statistical analysis. In spinorcondensates, magnetic order follows from the emergence

a b

c d e

100µm

FIG. 1: (Color online): (a): Sketch of the experimental setup.7500 Bose-condensed 23Na atoms are confined in a crossedoptical dipole trap with a homogeneous static magnetic fieldalong z. A resonant oscillating magnetic field along y drives aspin rotation of the initial equilibrium state. (b): Absorptionimage of the atomic cloud after Stern-Gerlach expansion in amagnetic field gradient. (c)-(e): Classical picture explain-ing the principle of our measurement. (c): The average spinhsi of the condensate created in a single realization can bedecomposed into a longitudinal component m

z

= hsz

i and atransverse component hs?i = hs

x

iex

+hsy

iey

, the direction ofwhich is given by the angle ↵. (d): From realization to real-ization, the angle ↵ varies randomly while m

z

and |hs?i| stayconstant. The mean spin vector hsi thus samples a horizontalcircle of radius |hs?i|. (e): A spin rotation of the initial staterotates this circle by an angle ⌦t along the y axis. The fluc-tuations �s0

z

after rotation are proportional to the squaredradius of the circle through a simple geometrical relation.

of a well-defined phase relation between the componentsof the spin wavefunction in the equilibrium state. Weshow evidence for such a phase-locking mechanism whichis not caused by any external field, but emerges from theinteractions between the atomic spins.

Our experiment is performed with ⇠ 7500 23Na atomsforming a spin-1 Bose-Einstein condensate with largecondensed fraction (> 80%). We prepare spinor conden-sates in a well-controlled homogeneous static magnetic

arX

iv:1

506.

0617

6v2

[con

d-m

at.q

uant

-gas

] 10

Oct

201

5

2

field B oriented along the z axis [see Fig. 1a]. We probethe sample using absorption imaging after free expan-sion in a magnetic field gradient, as shown in Fig. 1b, andmeasure the normalized populations n

mF of each Zeemancomponent m

F

= 0,±1 [8]. Importantly, the magnetiza-tion m

z

= n+1

� n�1

is conserved by binary collisionsdriving the system to its equilibrium state [8, 21]. Weprepare a spin mixture well before the BEC forms in ourevaporation sequence, allowing us to adjust the longitu-dinal magnetization m

z

between 0 and 1 (see the SM formore details).

In the single mode approximation where atoms in dif-ferent spin states share the same spatial mode [22], weparametrize the spin state of the condensate as

|⇣i =

0

BB@

q1�n0+mz

2

ei(⇥+↵)/2

pn0q

1�n0�mz2

ei(⇥�↵)/2

1

CCA , (1)

where ⇥ and ↵ are relative phases [33]. Alternatively,the quantum state |⇣i can be characterized completely bythe average spin vector hsi = m

z

ez

+ hs?i and the spinquadrupole tensor Q

ij

[7, 12, 23]. The transverse spinhs?i = hs

x

iex

+ hsy

iey

points in a direction determinedby ↵ and its length is determined by ⇥,

hs?i2 = 2n0

⇣1� n

0

+p(1� n

0

)2 �m2

z

cos⇥⌘. (2)

The spin quadrupole tensor Q is characterized by itsreal eigenvalues, (1 � A)/2, (1 + A)/2, 1, and eigenvec-tors u,v,w, where w is parallel to hsi (see SM for moredetails). Fully magnetized states with maximum aver-age spin |hsi| = 1, |⇣i = |m

F

= +1iw, correspond toA = 0. Fully spin nematic states with vanishing averagespin hsi = 0, |⇣i = |m

F

= 0iu, correspond to A = 1. In

the latter case, the eigenvalues of Q become 0, 1, 1 withfirst eigenvector u called the director. More generally,the alignment A can be used to characterize the prox-imity of a generic, partially magnetized state |⇣i from aspin nematic state. In addition, the relation

hsi2 +A2 = 1 (3)

shows that measuring the length of the mean spin vectorhsi2 is fully equivalent to measuring the alignment A.

For a given magnetization mz

the equilibrium state|⇣i minimizes the spin mean field energy E

MF

, the sum ofthe spin-exchange interaction energy and of the quadraticZeeman energy (QZE) energy in an applied magnetic fieldB [8],

EMF

N=

Us

2hs?i2 � qn

0

, (4)

up to terms that depend only on mz

. In the experi-ments reported in this Letter, the independently mea-sured interaction strength is U

s

/h ⇡ 38Hz (see SM)

and q/h ⇡ 4Hz to 34Hz. Antiferromagnetic interactions(U

s

> 0, the case of sodium atoms) favor minimizingthe transverse spin length. According to Eq. (2), this isachieved by locking the relative phase ⇥ to ⇡ indepen-dently of the value taken by n

0

,mz

,↵ (ferromagnetic in-teractions would lock ⇥ to 0 instead). This is equivalentto maximizing the alignment A introduced above.For a partially magnetized system, the competition be-

tween the two terms in Eq. (4) drives a phase transitionat a critical magnetic field B

c

(mz

) [18, 19, 21, 24]. Atzero temperature, n

0

is zero below Bc

and assumes a fi-nite value above. Although the average transverse spinis not zero above B

c

[see Eq. (2)], its value remains smallbecause ⇥ stays locked to ⇡. As a result, the alignment,which would reach 1 in the absence of other constraints(thus realizing pure spin nematic states), stays very closeto the maximum value given the conservation of m

z

,A

max

=p

1�m2

z

(see SM).In contrast to ⇥, the phase ↵ is expected to take ran-

dom values from one realization to the next. When deal-ing with many realizations of the same experiment, theinitial many-body state is thus characterized by a statis-tical mixture

⇢ =

Z4⇡

0

d↵

4⇡|⇣N ih⇣N | (5)

rather than a pure state |⇣N i with N bosons in the spinstate |⇣i. Only three parameters (e.g., n

0

,mz

, hs?i2)are needed to characterize the ensemble, down from fourto specify completely each member |⇣i. In spite of therandomness of the spin orientation, these three param-eters can still be measured using spin rotation providedone goes beyond single-particle observables and measuresspin noise (recent experiments used similar techniques toreveal squeezing [25–28]).Figure 1c-e illustrates the method geometrically in

terms of the mean spin vector hsi. The ensemble of pos-sible initial states with a uniform distribution for ↵ lieon a circle of radius |hs?i| around the z axis. In order tomeasure this radius, we rotate the state by a known angle⌦t around the y axis and measure the magnetization m0

z

after rotation (panel e). The mean magnetization afterrotation,

hm0z

i↵

= cos(⌦t)mz

, (6)

is independent of |hs?i|. The h · i↵

symbol indicates adouble average (see Eq. 5), first over the quantum state|⇣N i, and then over ↵. However, the variance of m0

z

,�m02

z

= hm02z

i↵

� hm0z

i2↵

is given by

�m02z

=1

2sin2(⌦t) hs?i2 + · · · . (7)

The right hand side of Eq. (7) give the dominant contri-bution, proportional to the initial transverse spin length.

3

0 0.2 0.40

0.2

0.4

0.6

0.8

1

t [ms]0 0.2 0.4

−0.5

0

0.5

t [ms]

a b

FIG. 2: (Color online) (a): Magnetization m0z

and (b): rel-ative population n0

0

in the mF

= 0 state versus duration ofthe Rabi pulse (or equivalently, rotation angle). The smallblue dots correspond to single-shot measurements, while thelarger red circles correspond to the average (m0

z

, n00

) over allmeasurements for each pulse duration. The data correspondto an initial state prepared with m

z

= 0.33 at q/h = 6.0 Hz(B = 147mG).

The dots indicate smaller noise terms (see SM). We there-fore expect the variance to oscillate with the rotation an-gle ⌦t and to be maximum for a rotation angle ⌦t = ⇡/2.

We implement this method experimentally using aradio-frequency (RF) magnetic field along y inducing res-onant Rabi oscillations with Rabi frequency ⌦ for an evo-lution time t [34], and measure the final populations n0

mF

after spin rotation. Fig. 2 shows typical raw data for therelative magnetization m0

z

(a) and the relative popula-tion n0

0

(b) for di↵erent rotation times t. As a result ofthe random nature of ↵, large fluctuations from shot toshot are observed. Fig. 3a shows a cosine fit [see Eq. (6)]to the mean population m0

z

from which we extract theRabi frequency ⌦.

Fig. 3b shows the variance of m0z

, displaying the pre-dicted oscillations. We compare the experimental resultsto the prediction of Eq. (2,7) (blue solid line). The trans-verse spin length hs?i2 is computed with ⇥ = ⇡, withthe measured m

z

and with the population n⇤0

found byminimizing E

MF

[35]. For comparison, we also show thetransverse spin length for the samem

z

, n⇤0

but ⇥ = 0 (reddotted line) and for random ⇥ with uniform distribution(green dash-dotted line), that would correspond to a fer-romagnetic system and to a non-interacting system (nophase locking), respectively. Our measurements are bestdescribed by ⇥ = ⇡, as expected for antiferromagneticsystems in equilibrium. This shows that the system at-tempts to minimize its transverse spin, or equivalentlymaximize its alignment, thereby revealing spin nematicordering.

We attribute the slight di↵erence between the mea-sured amplitude of the variance oscillations and the pre-diction of Eq. (7) for ⇥ = ⇡, to the non-zero temperatureof our sample. We addressed this point using a variant ofthe Hartree-Fock treatment of [29], which is detailed in

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

t [ms]

−0.5

0

0.5

t [ms]

t [ms]

a

b

0 0.50

0.01

0.02

0.03

0.04c

FIG. 3: (Color online) (a): Average magnetization hm0z

i↵

after rotation. The solid line shows a cosine fit to the data,allowing us to extract the Rabi frequency ⌦ [see Eq. (6)] (b):Variance of m0

z

(blue circles) versus duration of the Rabi pulseoscillating at twice the Rabi frequency. The blue solid linecorresponds to the theoretical prediction at zero temperaturefor an initial phase ⇥ = ⇡. The red dashed line (⇥ = 0) andgreen dash-dotted line (random ⇥) are shown for illustrativepurposes. (c): Close-up view of the data (b). The shadedarea corresponds to the prediction of our Hartree-Fock modelat finite temperature assuming a condensed fraction of f

c

�80%. The dashed line gives the best agreement correspondingto a temperature T ⇡ 80 nK. The data set in (a)-(c) is thesame as in Fig. 2.

the SM. Generally, increasing the temperature lowers thetransverse spin per atom. Experimentally, the condensedfraction can only be estimated as f

c

& 0.8 from the stan-dard analysis techniques where a two-component profileis fitted to the experimental images. We show in Fig. 3c ashaded area where the lower limit corresponds to f

c

= 0.8and the upper one to f

c

= 1, indicating that even a smallnon-condensed fraction leads to a measurable decrease ofthe oscillation amplitude. In fact, the oscillation vari-ance can be seen as a low-temperature thermometer. Atemperature T ⇡ 80 nK (condensed fraction f

c

⇡ 0.9) isfound to reproduce the observed oscillation level (dashedline in Fig. 3).We now turn to a more general statistical analysis

based on maximum likelihood estimation (MLE), whichallows us to estimate the distribution of the angle ⇥ in amore quantitative way. It takes all available data into ac-count, including the population n0

0

which was not used inthe previous analysis. Given a set of measurements, theMLE method finds the most likely distribution among aset of parameter-dependent model distributions, therebyproviding a statistical estimator for said parameters.We model the initial state by a density matrix

⇢ =

Zd⇣

G(n0

,mz

)P (⇥)

4⇡|⇣N ih⇣N |, (8)

4

0.8 0.9 1 1.1 1.20

0.05

0.1

0.15

0.2

−500

−400

−300

−200

−100

0

67%95%

99%

FIG. 4: (Color online) Color-coded log likelihood (logL) ofthe mean ⇥ and width �

⇥

of the distribution of the relativephase ⇥. The data are the same as plotted in Fig. 2 and 3.The most likely values, indicated by a cross, are found at⇥ = 1.01⇡ and �

⇥

= 0.08 ⇡. The contour lines indicate the67%, 95% and 99% confidence bounds.

with an integration measure d⇣ = dn0

dmz

d⇥d↵. Thejoint probability density G(n

0

,mz

) is peaked around theaverage value (n⇤

0

,mz

) with n⇤0

the population minimiz-ing E

MF

, with a finite width mostly due to experimentalimperfections in the preparation sequence (see SM). Weassume for simplicity that the probability density func-tions G(n

0

,mz

) and P (⇥) are Gaussians. The covariancematrix characterizing G(n

0

,mz

) is extracted from the ex-perimental data (see SM). The mean value ⇥ and stan-dard deviation �

⇥

of P (⇥) are the unknown parametersto be estimated. Due to the periodic nature of ⇥, ourchoice is sensible only when P (⇥) is peaked around themean, i.e. �

⇥

⌧ 2⇡.

We use a Monte Carlo method to sample the initialdistribution. A set of initial points is drawn from thedistribution in Eq. (8) and propagated in time using therotation operator. Here we assume that the spin rota-tion is perfectly known, with rotation axis y and a ro-tation angle extracted from the fit to hm0

z

i↵

as before.Spin-mixing dynamics just after the spin rotation slightlychange the relative population n0

0

. We account for thise↵ect in the propagation (see SM). After convolution ofthe final results with our known measurement noise, weget a continuous probability density p

ti(m0z

, n00

|⇥,�⇥

) forthe final state. Given a set of independent observations{m0

z,i

, n00,i

}, we can construct a (log) likelihood functionlogL(⇥,�

⇥

) =P

i

log pti(m

0z,i

, n00,i

|⇥,�⇥

). The distri-bution that accounts best for the observed results is foundby maximizing this function.

For the data shown in Figs. 2,3, the log likelihood isshown versus (⇥,�

⇥

) in Fig. 4. The maximum is foundfor (⇥,�

⇥

) = (1.01⇡, 0.085⇡), in full agreement withthe conclusion drawn from the variance analysis. Fig. 5shows the mean field phase diagram for our experimen-tal situation. The measurements shown so far corre-spond to the point a slightly above the transition line.Points b and c correspond to further measurements ei-

a b

0 0.5 10

10

20

30

40

a c

b

a b c

0.8

0.9

1

1.1

1.2

FIG. 5: (Color online) (a) The phase diagram in the mz

� qplane, where the three sets of experiments we have performedare located, denoted as a,b,c (a: m

z

= 0.33, q/h = 6.00 Hz;b: m

z

= 0.73, q/h = 33.7 Hz; c: mz

= 0.71, q/h = 3.84Hz;). In the gray area above the phase transition line both n

0

and |hs?i| are nonzero, whereas both vanish below the phasetransition in the zero temperature case. (b)Most likely valuesof ⇥ found by the MLE algorithm. Error bars indicate the67% confidence bounds.

ther well above or well below the phase transition line(see SM for details). The results for data sets b and c are(⇥,�

⇥

) = (0.86⇡, 0.347⇡) and (⇥,�⇥

) = (1.05⇡, 0.210⇡),respectively, similar to the results of data set a.In conclusion, we have shown the existence of spin-

nematic ordering in antiferromagnetic spin 1 BECs, orequivalently of a phase locking between the Zeeman com-ponents caused by spin-exchange interactions in the equi-librium state. Our experimental method combines spinrotations with a statistical analysis, either based on thespin moments or on a maximum-likelihood estimation ofthe probability density function characterizing the ini-tial spin state of the condensate. Our method is notrestricted to single-mode condensates or to spin 1 atoms,and could be used to reveal other types of spin ordering.We remark in particular that measuring the spin variance

can provide the “spin singlet amplitude”Da†+1

a†+1

a20

E

[30, 31]. Such a quantity appears in studies of fluctu-ating systems beyond mean field such as spin liquid inone dimension [32], or spin-singlet Mott states in opticallattices [10].We acknowledge support from IFRAF, from DARPA

(OLE program), from the Hamburg Center for UltrafastImaging and from the ERC (Synergy grant UQUAM).

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[33] The full Hilbert space can be parametrized by ↵ 2 [0, 4⇡[,⇥ 2 [0, 2⇡[, n

0

2 [0, 1] and |mz

| 1� n0

.[34] The Rabi frequency is much larger than the frequency

scales Us

/~, q/~. This allows us to neglect the e↵ect ofE

MF

during the spin rotation pulse.

[35] For this comparison we use the complete formula givenin the SM [Eq. (s19)] instead of Eq. [7]. Noise in m

z

was deduced from the measured distribution in the initialstate. Noise in hs?i was deduced from this measurementand Eq. (2) for ⇥ = ⇡.

Supplementary Material for : Spin nematic order in antiferromagnetic spinorcondensates

T. Zibold,1 V. Corre,1 C. Frapolli,1 A. Invernizzi,1 J. Dalibard,1 and F. Gerbier1

1Laboratoire Kastler Brossel, College de France, CNRS, ENS-PSL Research University,

UPMC-Sorbonne Universites, 11 place Marcelin Berthelot, 75005 Paris

(Dated: October 9, 2015)

I. EXPERIMENTAL SEQUENCE AND PREPARATION

A. Evaporative cooling of a spin mixture

a. Preparation of a spin mixture with well-defined magnetization: Our experimental sequence starts with a pre-cooled thermal cloud of 23Na atoms in a crossed optical dipole trap [1]. The atomic cloud is partially magnetized, witha magnetization m

z

⇡ 0.5 resulting from previous cooling steps. We adjust the magnetization by either demagnetizingthe atoms further with resonant RF-magnetic field sweeps, or by magnetizing it by evaporation in a magnetic fieldgradient (“spin distillation”) [2]. We are able to produce final magnetizations ranging from m

z

= 0 to mz

= 1, witha typical error of 2� 3%.b. Evaporative cooling in a crossed dipole trap: After preparing a spin mixture well above the critical temperature

for Bose-Einstein condensation, the depth of the optical trap is lowered in a few seconds to perform evaporative cooling.A hold time of 3 s is added after the end of the ramp to ensure that the cloud reaches equilibrium [2]. At the end ofthe evaporation ramp, the atoms are confined in the crossings of the two beams of the dipole trap, where the trappingpotential is well-approximated by a harmonic trap with average trap frequency !/2⇡ ⇠ 405Hz (the trap frequenciesare in the ratio 1 : 0.85 : 0.5).Experiments reported in the main article are performed with “almost pure” Bose-Einstein condensates (BECs)

containing typically 7500 atoms at a trap depth VT

/kB

⇡ 400 nK. By “almost pure”, we mean that no discerniblethermal component can be observed in absorption images. Figure S1 shows the measured condensed fraction f

c

=N

c

/N for di↵erent trap depths VT

(which sets the temperature obtained by evaporative cooling). The condensedfraction is obtained by fitting a bimodal profile to absorption images of the cloud following the standard procedure[3]. Here N and N

c

denote respectively the total number of atoms and of condensed atoms, irrespective of theirinternal state. For condensed fractions larger than f

c

⇡ 0.8 the contribution of the thermal component becomesdi�cult to detect, and the bimodal fitting procedure unreliable This sets a lower bound f

c

� 0.8 on the condensedfraction for experiments performed without a discernible thermal component.

1 100

0.2

0.4

0.6

0.8

1

FIG. S1: Condensed fraction versus trap depth, extracted from a fit of a bimodal distribution to the absorption images. Thefit function is the sum of a Bose-Einstein thermal distribution for the thermal component and of a Thomas-Fermi profile forthe condensate [3].

c. Stern-Gerlach imaging: The three Zeeman components are imaged after releasing the cloud from the trapin the presence of a magnetic force separating the Zeeman components. Specifically, we apply a quadrupole fieldB

q

= b0(2xex

� yey

� zez

) together with a uniform “separation” field Bx

ex

, with b0 ⇡ 7G/cm and Bx

⇡ 3G. Theresulting adiabatic magnetic potential is given by U

mag

= gF

mF

µB

|Bx

ex

+Bq

| ⇡ gF

mF

µB

|Bx

|+ gF

mF

µB

b0x+ · · · ,with g

F

= �1/2 the Lande factor and with µB

the Bohr magneton. The quadrupole and separation field are ramped

2

up in a few milliseconds, while the bias field Bez

applied during the experiment is simultaneously ramped down. Thetiming of the sequence is shown in Figure S3a.

B. Experimental implementation of Rabi oscillations

0 0.2 0.4 0.6 0.8

0

0.2

0.4

0.6

0.8

1

t [ms]

rela

tive

pop

ula

tion

n+1

n0

n−1

a

0 0.1 0.2 0.3 0.4 0.5

0

0.2

0.4

0.6

0.8

1

t [ms]

rela

tive

pop

ula

tion

n+1

n0

n−1

b

FIG. S2: Rabi oscillation starting either from a state with all atoms in mF

= +1 (a) or mF

= 0 (b). Unlike the data presentedin the main text, the fluctuations are dominated by preparation noise, imperfections in the Rabi rotation parameters and thedetection noise, all with roughly comparable contributions.

To drive Rabi oscillations between the three Zeeman sublevels, we prepare condensates at a chosen bias magneticfield B

z

in the z direction and apply a pulse of radio-frequency (rf) magnetic field at the Larmor frequency in they direction. The bias field is taken small enough to allow us to neglect the quadratic Zeeman shift (q < 100Hz)compared to the Rabi frequency (⌦/2⇡ ⇠ 5 kHz). At the end of the pulse, the separation field B

x

ex

is increased first,folllowed by the magnetic gradient used for SG imaging and by the decrease of the bias field B

z

ez

. The timing ofthe sequence is shown in Figure S3a. Ramping up the separation field B

x

is done with a linear ramp of T = 3 msduration, su�ciently slowly to remain adiabatic with respect to spin flips (!

L

T ⌧ 1). The optical trap is switchedo↵ 10 ms after the end of the RF pulse (see Section IC below).We have tested this sequence in two special cases, where all the atoms are initially in the m

F

= +1 state and orin the m

F

= 0 state. We are able to prepare these two states to a very good approximation thanks to the sequencepresented before (resulting in �m

z

. 1% in these cases). In these two situations there are no fluctuations of themagnetization after rotation due to randomness in the initial state, in contrast to the cases studied in the main paper.The measured oscillations are presented in Figure S2. The contrast is close to 100%, showing that the assumption ofadiabatic following when ramping up the di↵erent magnetic fields is valid, and we do not observe any sizable spreadof the oscillations after several Rabi periods. This shows that the assumption of adiabatic following when rampingup the di↵erent magnetic fields is valid. A key point to achieve good reproducibility of the oscillations has been tosynchronize the start of the RF pulse to the 50Hz oscillations of the mains line frequency.

C. Influence of spin mixing after the spin rotation

The sudden change of the spin state due to the spin rotation should in principle trigger a spin oscillation dynamics[4–8] driven by spin-exchange interactions during the 10ms hold time following the spin rotation. As seen before,the applied magnetic field is also changed after the spin rotation, from B = B

z

ez

to B = Bz

ez

+ Bx

ex

. Thequadratic Zeeman energy q increases during this ramp, according to the curve shown in Fig. S3b. This increaseis fast compared to the time scale set by spin-exchange interactions, h/U

s

⇠ 25ms, and it reduces spin-mixingdynamics due to exchange collisions that would otherwise develop during the 10ms hold time after the RFpulse. Nevertheless, as seen below, a residual dynamics still takes place and modifies slightly the population n

0

measured in SG imaging. Note that the e↵ect of the spin interaction during the RF pulse is negligible (Us

/~⌦ ⇠ 0.008).

3

t 10.1 ms 8.5 ms

3 ms

time

SG

0

Bx

0

Bz0

RF

(a) Release Image

SpinRotation

Hold ToF & SG

0

1000

2000

3000

time after rotation pulse [ms]0 0.5 1 1.5 2 2.5 3 3.5

0.35

0.4

0.45

0.5

FIG. S3: (a): Schematic diagram (not to scale) showing the experimental sequence. “RF” indicates the rf pulse inducing spinrotations, B

z

is the bias field applied before and during the spin rotation, Bx

and “SG” denote respectively the “separationfield” and magnetic field gradient required for SG imaging. (b) The ramp of B

x

after the spin rotation results in a time-dependent Quadratic Zeeman energy (QZE) q increasing within 3ms after the end of the rf pulse (top panel). The evolutionafter the spin rotation of the normalized population n

0

due to spin-mixing interactions calculated from Eq. (s2) is shown in thelower panel. The initial relative angle ⇥ was assumed to be ⇡, and the initial population chosen such that the final populationis given by n

0

⇡ 0.43 as measured in data set a. The spin exchange interaction energy was taken to be Us

/h = 38 Hz (seesection ID).

To quantify the impact of spin-mixing oscillations on the measured n0

, we use the theoretical framework given in[7]. The evolution of an initial state

|⇣i

i =

0

BB@

q1�n0,i+mz

2

ei(⇥i+↵)/2

pn0q

1�n0,i�mz

2

ei(⇥i�↵)/2

1

CCA . (s1)

is described by the two Josephson-like equations [7] ,

~dn0

dt= 2U

s

n0

p(1� n

0

)2 �m2

z

sin(⇥), (s2)

~d⇥dt

= �2q(t) + 2Us

(1� 2n0

) + 2Us

(1� n0

)(1� 2n0

)�m2

zp(1� n

0

)2 �m2

z

cos(⇥), (s3)

with n0

(0) = n0,i

, with ⇥(0) = ⇥i

and with q(t) as shown in Fig. S3b.We solve Eqs. (s2) numerically to compute the evolution of n

0

. An example for ⇥i

= ⇡ is shown in Fig. S3. Themain changes in n

0

occur early in the ramp. Once q has settled at its final value qf

⇠ h ⇥ 2.5 kHz, the dynamicscontinue as a small amplitude oscillation of the population n

0

around an o↵set value. These residual oscillationscorrespond to the quadratic Zeeman regime studied in details in [6]. Their amplitude is small (⇠ U

s

/qf

⇠ 0.015) andcomparable to our detection noise. Changing the magnetic field to higher values would further reduce the oscillationamplitude, without significantly changing the o↵set of n

0

. Taking the long-time o↵set as the measured value of n0

,we find that the e↵ect of the ramp amounts to increase the relative population in n

0

from its initial value by up to0.05 for an initial angle ⇥

i

= ⇡, a small but measurable change.We emphasize that the spin-mixing dynamics does not change the magnetization m

z

of the system, but only theindividual populations n

mF . Therefore, the occurrence of spin mixing does not influence the analysis of the varianceof m

z

after spin rotation. On the other hand, it does a↵ect the maximum likelihood analysis, as detailed further insection IV.

D. Determination of Us

from spin-mixing dynamics

We have measured directly the exchange interaction parameter Us

by deliberately inducing spin-mixing dynamicsand recording the oscillations of the normalized population n

0

after a sudden change (see Fig. S4a). Starting from acondensate with all atoms in the m

F

= 0 state, prepared as explained above at a bias field B ⇡ 282mG [q/h ⇡ 22Hz],

4

we first apply a spin rotation to produce a mixture with roughly balanced populations in all Zeeman states. Thisresults in an initial state as given by Eq. (s1), with n

0,i

⇡ 0.38 and mz

⇡ 0. Spin-changing collisions produce high-contrast oscillations in the Zeeman populations, as observed in previous work for m

z

6= 0 [5, 6, 8, 9]. The oscillationperiod has been predicted analytically in [7], and is a function of n

0,i

,mz

, q, which are known, and of Us

, which isnot. We extract U

s

/h ⇡ 38Hz from the data by comparing the oscillations period Tosc

⇡ 16ms, obtained by a fit tothe data to this prediction (see Fig. S4b).

0.00 0.02 0.04 0.06 0.08 0.10 0.12

t[s]

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

n0

a

5 10 15 20 25 30 35 40 45 50

Us/h[Hz]

0.014

0.016

0.018

0.020

0.022

0.024

0.026

Tosc

[ms]

b

FIG. S4: Spin-mixing oscillations (a) and calculated oscillation period (b). A fit to a damped sinusoid is shown in (a) as solidline, and yields an oscillation period T

osc

⇡ 16ms indicated by dashed lines in (b).

E. Description of the data sets

In total we have taken three data sets for di↵erent initial magnetizations and magnetic fields which we label a, b, c(see Fig. 5 in the main text). The experimental parameters were: a: m

z

= 0.33, q/h = 6.00 Hz; b: mz

= 0.73,q/h = 33.7 Hz; c: m

z

= 0.71, q/h = 3.84 Hz. We show the results for these data sets in section III C and IVE.

II. EQUILIBRIUM STATE AT FINITE TEMPERATURES

A. Finite temperature description of a spinor condensate

In the main text, the initial spin state in equilibrium is described in terms of a zero temperature mean field state|⇣N i, with N atoms occupying the single-particle spin 1 state

|⇣i =

0

BB@

q1�n0+mz

2

ei(⇥+↵)/2

pn0q

1�n0�mz2

ei(⇥�↵)/2

1

CCA . (s4)

In a recent paper [10], we have generalized this description to a finite temperature spin ensemble, which was shownto be described to leading order in the small parameter 1/N by the density operator

⇢ ⇡ 1

Z

Ze��K |⇣N ih⇣N |d⇣, (s5)

with d⇣ = dn0

dmz

d⇥d↵, Z the partition function and � = 1/kB

T the inverse temperature. The generalized freeenergy K, accounting for the conservation of magnetization, can be written as

�K =�0z

2(m

z

�m⇤z

)2 �N�qn0

+N�Us

n0

⇣1� n

0

+p(1� n

0

)2 �m2

z

cos(⇥)⌘. (s6)

We have introduced two Lagrange multipliers �0z

,m⇤z

determined by the two constraints mz

= 1

ZRm

z

e��Kd⇣ and

�m2

z

= 1

ZR(m

z

�mz

)2 e��Kd⇣.

5

0

0.2

0.4

0.6

0.8

1

0 2 4 6 80

0.02

0.04

0.06

0.08

a

b

FIG. S5: (a) Condensed fraction normalized to the total number of atoms for each Zeeman component mF

= +1 (dashed red),m

F

= 0 (dash doted green) and mF

= �1 (dotted blue line) and total condensed fraction (black solid). (b) Total transversespin length hs?i2 versus temperature. The contribution of the thermal cloud is found to be negligible in this scenario. Thecalculation was done for a spherical trap of frequency !/(2⇡) = 405Hz, N = 7500 atoms, m

z

= 0.33 and q/h = 6Hz. Inphysical units, k

B

T = 8~! corresponds to T ⇡ 150 nK.

Close to the phase transition at qc

⇡ Us

⇣1�

p1�m2

z

⌘, the population n⇤

0

> 0 which minimizes the free energy is

small. An expansion around (n0

,mz

) = (n⇤0

,mz

) leads to

�K ⇡ �0z

2(m

z

�mz

)2 + �0 qc

Us

p1�m2

z

(n0

� n⇤0

)2 +�0

2n⇤0

q1�m2

z

(⇥� ⇡)2 + · · · (s7)

Equations (s5,s7) show that the initial distribution can be approximated by a Gaussian distribution in n0

,mz

,⇥,provided the fluctuations and n⇤

0

are small [10].

B. Hartree-Fock description of the uncondensed component

The single-mode approximation only describes the lowest energy “spatial mode” into which the atoms condense.Higher energy modes can be thermally populated, leading to a condensed fraction f

c

lower than one. The thermalcomponent interacts with the condensate and in principle could also develop a magnetic behavior. To describe thethermal component of the non-condensed cloud, we have adapted the Hartree-Fock (HF) description proposed in [11]in the uniform case to our experimental situation.The model of [11] treats the non-condensed cloud as a gas of non-interacting free particles evolving in a self-

consistent mean field potential accounting for spin-exchange interactions [11]. Importantly, this mean field potentialis not diagonal in the Zeeman basis due to spin-mixing interactions. The thermal component can in principle developnon-zero coherences due to interactions with the condensate and therefore a non-zero average spin. The quantity ofinterest is the single-particle density matrix

⇢(1)m,n

(r) = �⇤m

(r)�n

(r) + ⇢0(1)m,n

(r), (s8)

with � the condensate wavefunction, with ⇢0(1)m,n

the contribution of the thermal component, and where m,n = 0,±1.

The density in each Zeeman component m is determined by the diagonal terms ⇢(1)m,m

and the transverse spin by the

o↵-diagonal coherences ⇢(1)0,±1

.With respect to the full HF model laid out in [11], we make two additional simplifying assumptions. First, we

assume that the single-mode approximation holds [? ]. This amounts to setting �m

(r) =pN

c

�(r)⇣m

, as done in the

6

main text. The single mode wavefunction � determining the condensate spatial distribution is computed numericallyby solving the GP equation

µ� = � ~22M

Na

��+ V (r)�+ gNc

|�|2�. (s9)

with MNa

the mass of Sodium atoms. The spinor part ⇣m

is found from the single-mode theory using Us

=N

c

gs

Rd(3)r|�|4. The coupling constants g, g

s

are proportional to the scattering lengths a ⇡ 2.79 nm and as

⇡ 0.1 nm[12] with a proportionality factor 4⇡~2/M

Na

. Second, we neglect the interaction energy of the thermal cloud incomparison to the one of the condensate (“semi-ideal” model [13]). Far from T

c

, this is expected to be an accurateapproximation [14]. Finally, we perform the calculations for a spherical trap. Although the trapping potential used inthe experiment is not exactly isotropic, we do not expect that this a↵ects strongly the results (in the Thomas-Fermiregime, for instance, only the average trap frequency matters to compute thermodynamic quantities [14]).The excitations modes u(⌫) and energies E

⌫

are solutions of the eigenproblem

E⌫

u(⌫) =

✓� ~22M

Na

� ·+V (r) +A(r)

◆u(⌫) (s10)

where the matrix A, explicitely given in [11], depends on the condensate wavefunction �(r) and on g, gs

. Diagonalizingthis equation, we obtain the single-particle density matrix ⇢0(1) of the thermal component as

⇢0(1)m,n

(r) =X

⌫

⇣u(⌫)

m

(r)⌘⇤

u(⌫)

n

(r)NBE

(E⌫

) (s11)

with NBE

(E) = 1/(eE/kBT � 1) the occupation number for each mode ⌫.The results of this calculation are shown in figure S5. The parameters were chosen to correspond to the experimental

situation of data set a in the main article. The condensed fraction in mF

= 0 decreases first. Above kB

T � 7.8~!,the m

F

= 0 component is purely normal and the condensate is formed by mF

= ±1 only. The transverse spin isreduced with increasing temperature. As found in [11], the contribution of the thermal component to the average spinvector is oriented opposite to the average spin of the condensate. The transverse spin is thus naturally reduced withincreasing temperature. In the regime we have investigated, the temperatures fulfill k

B

T � q, Us

. As a result, thenon-condensate spin vector is always much smaller in magnitude than its condensed counterpart, and we find thatthe main e↵ect that reduces the length of the transverse spin vector is the reduction of the condensed fraction. Fora total condensed fraction f

c

= 0.8 the transverse component hs?i2 of the spin is reduced to about 57% of its zerotemperature value, corresponding to the lower limit of the shaded area in figure 3 of the main text. The data arecloser to a value f

c

⇡ 0.9.

III. SPIN ROTATIONS AND SPIN NOISE

A. Theoretical description of spin rotations

We compute the e↵ect of the spin rotation using the standard angular momentum algebra. In a Rabi-type exper-iment where a resonant RF field polarized along y is applied at t = 0 and in the rotating-wave approximation, theevolution operator is equivalent to a rotation operator along y with a rotation angle ⌦t proportional to the pulselength t and to the Rabi frequency ⌦. The state of a single spin 1 atom transforms under rotation as

| 0i = e�i⌦t

ˆ

Sy | i = Ry

(⌦t)| i. (s12)

Here and in the following, primed variables denote quantities evaluated after the spin rotation is complete. Therelevant rotation matrix for a spin 1 is given by

Ry

(⌦t) = 1 � i sin(⌦t)Sy

+ (cos(⌦t)� 1)S2

y

(s13)

The populations of each Zeeman state after the pulse are given by simple expressions when starting from |mF

= +1i,�� 0+1

��2 = cos4�⌦t

2

�,�� 0

�1

��2 = sin4�⌦t

2

�, and | 0

0

|2 = sin2 (⌦t) /2. Similarly, for an initial state |mF

= 0i, one has�� 0+1

��2 =�� 0

�1

��2 = 1

2

sin2 (⌦t) and | 00

|2 = cos2 (⌦t).

7

B. General formula for the spin variance

We derive in this section a general formula linking the moments of the magnetization after rotation to propertiesof the initial states. We apply the rotation to the spin operator S

z

, and make use of the angular momentum algebrato obtain,

S0z

= R†y

(⌦t)Sz

Ry

(⌦t) = cos(⌦t)Sz

� sin(⌦t)Sx

. (s14)

We now introduce a key assumption: the initial density matrix is invariant under rotation around the z axis, or, usingthe notations in the main article, the phase ↵ is random from one realization of the experiment to the next (with auniform distribution in [0, 4⇡]). The value of an observable measured after averaging over many realizations of theexperiment is

hOi↵

=1

4⇡

Z4⇡

0

d↵ hR†ORi, (s15)

where R = e�i⌦t

ˆ

Sye�i

↵2ˆ

Sz . The h · i↵

symbol is intended to stand for a double average : the first one, denoted byh · i, is the usual average over the quantum state before rotation for each realization, and the second one is done overrandom values of ↵ arising from one experimental realization to the next. Defining an average in this way allows usto obtain formula expressing measurement results without specifying the initial state. We have for instance

S0z

= R†Sz

R = cos(⌦t)Sz

� sin(⌦t)⇣cos(↵/2)S

x

+ sin(↵/2)Sy

⌘. (s16)

Using this result, we find the average and variance of S0z

/N after the pulse,

hm0z

i↵

=1

NhS0

z

i↵

= cos(⌦t)mz

, (s17)

�m02z

= cos2(⌦t)�m2

z

+1

2N2

sin2(⌦t)hS2

x

+ S2

y

i. (s18)

In other words, relying only on the randomness of ↵ we find that the variance of the magnetization �m02z

after thepulse measures the initial transverse spin fluctuations. This result holds for any initial state, and only relies on thefact that the pulse is short enough to neglect any other terms than the oscillating field in the Hamiltonian during theevolution time.Introducing the squared length hs?i2 = (hS

x

i2+hSy

i2)/N2 of the mean transverse spin and �s2? = hS2

x

+ S2

y

i/N2�hs?i2 its variance, we obtain our final result

�m02z

=1

2sin2(⌦t)hs?i2 + cos2(⌦t)�m2

z

+1

2sin2(⌦t)�s2?. (s19)

We thus expect that the variance �m02z

oscillates with the rotation angle ⌦t and reaches its maximum for ⌦t = ⇡/2where the slope of m0

z

versus ⌦t is maximum. The last two noise terms in Eq. (s19) are typically dominated by thepreparation noise on m

z

(which also introduces noise on n0

in the equilibrium state, and thus on hs?i).We note that this measurement gives access to a quantity (the squared transverse spin length) which can be used

to characterize other phases than a fully condensed state. The expression

S2

x

+ S2

y

= N + N0

+ 2N0

⇣N

+1

+ N�1

⌘+⇣a†+1

a†+1

a20

+ h.c.⌘, (s20)

shows that measuring the spin variance gives access to the “spin singlet amplitude”Da†+1

a†+1

a20

E[15, 16], which appears

in studies of fluctuating systems beyond mean field (spin liquid in one dimension [17], or spin-singlet Mott states inoptical lattices, for instance [18]). The results are also easily generalized beyond the single-mode approximation.

C. Complementary data sets for the spin variance

We show here additional data sets not shown in the main article, which have been obtained for the parameterslocated at the points b,c in the mean field phase diagram (see figure 5 in the main text). Point a shown in the main

8

0 0.2 0.40

0.01

0.02

0.03

0.04

0.05a

0 0.2 0.40

0.05

0.1

0.15b

0 0.2 0.40

0.005

0.01

0.015c

FIG. S6: Variance data for measurements a, b and c (see figure 5 in the main text). In panel (a) and (b) the solid blueline is the zero temperature theory for an initial angle of ⇥ = ⇡ (antiferromagnetic interactions). Panel (c) shows data set c

which is below the phase transition line. The zero temperature theory predicts no oscillation in the variance at all since n0

andthereby hs?i2 vanish. The dotted curve is the theoretical expectation from Eq. s19 taking the initial measured n

0

into account(corrected for the e↵ect of the spin-mixing collisions described in Section IC).

article is also reproduced for comparison. The same behavior is observed in all cases, an oscillation of the variancewith fixed amplitude.The case shown in Fig. S6c (data set c), taken below the T = 0 phase transition, deserves a separate discussion.

According to the T = 0 mean field picture, one would expect n0

= 0 and hs?i2 = 0 (see Eq. (2) in the main text).In contrast, we find a small initial population n

0

⇡ 0.04, and an oscillation of the magnetization variance with asmall, but non-zero amplitude. The dotted lines in the figure correspond to the theoretical predictions which take theinitial measured n

0

into account (corrected for the small shift in n0

due to the spin changing collisions discussed inSection IC) and ⇥ = ⇡.A first explanation for this behavior could be the presence of the thermal (uncondensed) component. In the

Bogoliubov framework [19], spin excitations are phase-locked to the condensed components, and a finite transversespin originating from the uncondensed component could contribute to our signal. However, from the Hartree-Fockcalculations described in Section II B, we found that the transverse spin of the uncondensed component remains verysmall for our typical parameters, and cannot explain the measured signal.A second explanation comes from a finite temperature of the initial spin state of the condensate, which is then

described by a statistical ensemble rather than a pure state [10]. This leads to a finite population in mF

= 0 evenbelow the phase transitions [10]. By numerically integrating the thermal distribution described by the free energyin Eq. (s6) for a typical temperature k

B

T = 80nK, we find a finite population n0

= 0.016. This leads to a maximalvariance after rotation of �m02

z

= 0.005. Although this is smaller than the observed fluctuations we want to emphasizethat the amplitude of the variance oscillations found still indicates to the antiferromagnetic order of the state with⇥ = ⇡. Our maximum likelihood estimation algorithm also agrees with these findings as is shown in the next section.

IV. MAXIMUM LIKELIHOOD ESTIMATION OF THE DISTRIBUTION OF ⇥

A. Principle of the method

The maximum likelihood estimation (MLE) provides an estimator for an unknown parameter by comparing theoutcome of a measurement with a model distribution depending on the unknown parameter. The comparison at-tributes a likeliness to each parameter value, by evaluating the model distribution at the result of the measurement.The best estimated parameter given the set of measurements corresponds to the value which maximizes the likelihoodfunction. Since the estimator strongly depends on the chosen probabilistic model, it is important for the model to beclose to the physical reality. In the following we motivate the model chosen for our measurement.In our case the measurement data consists of couples (m0

z,i

, n00,i

) which we find after various rotation angles {⌦ti

}corresponding to evolution times {t

i

} in the RF field. In the framework of MLE, the measurement results are seenas probabilistic variables of an unknown probability distribution. We model this distribution by a set of probabilitydistributions which we derive from the knowledge about our initial state, its transformation during the Rabi rotation

9

and independently measured experimental noise.

B. Model for the initial distribution

0 0.5 10

5

10

−1 −0.5 0 0.5 10

10

20

−1 −0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

0

50

100

150

200

250

300a b

c

FIG. S7: Initial distribution of the populations n0

and mz

for data set a. (a) Measured initial populations are indicated byblack circles. Color-coded is the 2D-histogram of the simulated 106 initial points used in the Monte Carlo method with linearbin width 2.5⇥ 10�3. The initial 2D-Gaussian distribution is estimated from the initial measurements. The dashed lines limitthe allowed area (|m

z

| 1 � n0

). Marginal histograms of measured initial populations for mz

(b) and n0

(c). The red lineindicates the marginal distributions used for the Monte Carlo analysis.

The distribution of initial states is probabilistic due to three di↵erent e↵ects. The first e↵ect is intrinsic to ourtheoretical model which is degenerate in the initial angle ↵. We therefore expect that it takes random values fromrealization to realization with a uniform distribution.The second probabilistic e↵ect is due to experimental imperfections. In an ideal scenario the parameters n

0

and mz

characterizing the initial state ⇣ would be fully determined by minimizing the mean field energy. In our experimenthowever we find fluctuations of these parameters from realization to realization. The fluctuations of m

z

result fromexperimental imperfections in our preparation process and in the subsequent evaporation. Assuming the cloud reachesequilibrium before we probe it with spin rotation, fluctuations in m

z

or in the spin-spin interaction energy Us

(dueto fluctuations of the total atom number or of the confinement strength) result in correlated fluctuations in n

0

dueto the system exploring di↵erent minima of the mean field energy.A third probabilistic mechanism originates from the finite spin temperature as described in Section IIA which allows

the system to explore states situated away from the minimum. The second and third e↵ect are more pronouncedclose to the phase transition [10]. In principle, one could account for both by convolving the initial finite temperaturedensity matrix ⇢ by the probability distribution describing preparation noise on m

z

and the correlated fluctuations ofn0

. For our setup, we estimate that experimental imperfections dominate (in the language of Section IIA, �0z

⌧ �0)and we include only the latter e↵ect.We find empirically that the initial joint distribution of n

0

and mz

is well described by a two-dimensional GaussianG(n

0

,mz

). We thus assume that the initial density operator is of the form

⇢ =

Zdn

0,i

dmz

d⇥d↵G(n

0,i

,mz

)P (⇥)

4⇡|⇣N ih⇣N |. (s21)

We stress that G(n0,i

,mz

) reflects the fluctuations of the preparation, but that the marginal distribution P (⇥) isa priori not a↵ected by them. At T = 0, P (⇥) is given by a Dirac delta, P (⇥) / �(⇥ � ⇡), but acquires a finitewidth at finite T (see Section IVD below). The mean and covariance matrix characterizing G are calculated fromthe measured data without spin rotation. We account for the spin changing collisions discussed in Section IC, whicha↵ect the measured “initial distribution”. Specifically, for each values of ⇥, m

z

and n0

, the mean field equations (s2)are used to find the initial value n

0,i

that leads to the measured one, n0

. The known values of q and the measuredvalue of U

s

are used as fixed inputs for this calculation. The initial distribution G(n0,i

,mz

) deduced in this way isshown in Figure S7.

10

C. Monte Carlo approach

−1 0 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

100

200

300

400

500

600

700

800

900

−1 0 10

50

100

150

200

250

−1 0 10

50

100

150

200

250

300a b c

FIG. S8: Comparison for data set a of the Monte Carlo simulated populations with measured data (black circles) after Rabi-rotation for di↵erent assumed initial angles ⇥. The three panels show the color-coded 2-D histograms of the Monte Carlosimulations for ⇥ = 0 (a), ⇡/2 (b) and ⇡ (c). Black circles indicate the same measured populations in all panels. Bestagreement, i.e. the maximum likelihood is found for ⇥ = ⇡. The example is taken for ⌦t ⇡ ⇡/2 where sensitivity is the highest.

In order to construct a MLE for the parameters characterizing the distribution P (⇥), we need to obtain a distributionof the measurement outcomes from the knowledge of the initial state, including the evolution under spin rotation andmeasurement noise on the final Zeeman state populations. We use a Monte Carlo approach, where the initial densityoperator is sampled by drawing random numbers (n

0

,mz

,↵) according to our assumed probability distributions (seefigure S7) and assuming a certain value for ⇥. This determines an initial mean field state |⇣N i. Using the knownevolution under spin rotations, we propagate this state in time for a given t

i

to arrive at the final mean outcomepopulations (n0

0

,m0z

) as the expectation values of the corresponding operators in the time-evolved mean field state.In our numerical implementation we use a typical number of ⇠ 106 Monte Carlo samples to reconstruct the finalstatistical distribution of the measurement outcomes.As discussed in Section IC, the spin populations change slightly due to spin-mixing collisions in the beginning of

the magnetic field ramp applied just after the spin rotation. This was already included in the determination of theinitial distribution of (m

z

, n0

) and we also take this e↵ect into account when comparing with the final simulateddistributions. In the Monte-Carlo simulation, the spin state found after rotation is used as initial condition to solvethe mean field equations (s2)describing the spin dynamics. We arrive in this way at a distribution of n0

0

corrected forthe e↵ect of spin changing collisions. For initial angles close to ⇥ = ⇡, the population n

0

changes by a few percents.By evaluating the final populations using their expectation values, we neglect the quantum fluctuations on the

final results, which are on the order 1/pN

mF and small for our typical atom numbers of particles (NmF ⇠ a few

thousands) when compared to the noise level of our population measurements. The measurement noise results froma combination of photon shot noise and small spatial intensity fluctuations of the laser pulse used for absorptionimaging, and is typically �n

mF ⇡ 1% for the normalized population in Zeeman state mF

. We include this noise inour model by convolving the simulated measurement outcome by a Gaussian distribution. This leads to a continuousprobability density p

ti(n00

,m0z

|⇥) for the measurement outcome which depends on the initial phase ⇥, and defines thelog-likelihood function as described in the main text.As a side remark, we note that including the spin changing collisions into the MLE analysis changes the symmetry

of the final distribution. Without including them, the final distributions pti(n

00

,m0z

|⇥), and therefore the likelihood,enjoy a reflection symmetry around ⇥ = ⇡ and ⇥ = 0. When spin-mixing collisions are taken into account, thissymmetry is lost. For comparison, we show in figure S10 not only the likelihoods accounting for the e↵ect of the spinchanging collisions (panel a-c) but also the ones which do not include the correction (panel d-f).

11

D. Probabilistic model for ⇥

We model the marginal distribution P (⇥) that is to be estimated by the MLE procedure by a truncated Gaussianwith a mean value ⇥ and a standard deviation �

⇥

. The finite width can be attributed to two e↵ects. The first e↵ectoriginates from the finite temperature of the initial spin ensemble, as discussed in Section IIA. Eq. (s6) shows that themarginal distribution of ⇥ obtained after integrating over n

0

,mz

is approximately Gaussian, with a root-mean-square(rms) width ⇡

pkB

T/NUs

. A numerical integration of the full distribution without approximations for a realistictemperature T/k

B

⇡ 80 nK and the experimental parameters of data set a leads to the distribution shown in Fig. S9,with a width ⇡ 0.1 comparable to the results of the MLE. The second e↵ect originates from an underestimation ofthe noise sources in the system. As seen before, the probability distribution p

ti(n00

,m0z

|⇥) is almost symmetric in⇥ with respect to 0 and ⇡. The presence of fluctuations (induced for example by experimental imperfections) notincluded in our model always bias the estimator away from ⇥ = ⇡. We thus conclude that the finite width of P (⇥)probably partly accounts for underestimated or unconsidered noise in our probabilistic model. This conclusion stillholds qualitatively when spin-mixing collisions are taken into account and the symmetry of L(⇥) around ⇡ is lost.

0 0.5 1 1.5 2

FIG. S9: The black solid line shows the marginal probability distribution for the phase ⇥ for a thermal spin ensemble at atemperature of 80 nK. The distribution was obtained by numerical integration of the thermal distribution described by Eqs. (s5)and (s6) for the parameters of data set a. The red dotted line shows the best Gaussian fit.

E. Additional data analysis with the MLE algorithm

We show here the results found by the MLE approach for the three data sets a, b and c in Fig. S10.

V. GEOMETRIC CHARACTERIZATION OF SPIN 1 CONDENSATES

A. Representation of a spin 1 state in the cartesian basis

A given spin 1 state | i can be defined in terms of its components { +1

, 0

, �1

} in the standard basis formedby the eigenvectors of s

z

with eigenvalues +1, 0,�1, respectively. For our purpose, it is convenient to use instead theso-called Cartesian basis {|xi, |yi, |zi} defined as |xi = 1p

2

(|� 1i � |+ 1i), |yi = ip2

(|� 1i+ |+ 1i), and |zi = |0i.From the relation S

a

|bi = i✏abc

|ci (✏abc

is the fully antisymmetric tensor), we deduce that the cartesian state |ai isthe eigenstate of S

a

with eigenvalue 0. The spin 1 state in the Cartesian basis can be written as [20–23]

| i =

0

@1p2

( �1

� +1

)�ip2

( +1

+ �1

)

0

1

A

c

= (u+ iv) · |ri, (s22)

where we introduced two real vectors u, v that are useful to simplify the algebra. From the normalization u2+v2 = 1alone, the vectors u and v are not uniquely defined. Performing a gauge transformation ! 0 = ei� transformsu and v as u0 = cos(�)u� sin(�)v and v0 = cos(�)v + sin(�)u. As a result, we can choose � such that u · v = 0 and|u| � |v|. We make this choice in the following.

12

0.5 1 1.50

0.05

0.1

0.15

0.2

−1500

−1000

−500

0a

0.5 1 1.50

0.1

0.2

0.3

0.4

0.5

−3500

−3000

−2500

−2000

−1500

−1000

−500

0b

0.5 1 1.50

0.1

0.2

0.3

0.4

0.5

−400

−300

−200

−100

0c

0.5 1 1.50

0.05

0.1

0.15

0.2

−1500

−1000

−500

0a’

0.5 1 1.50

0.1

0.2

0.3

0.4

0.5

−3500

−3000

−2500

−2000

−1500

−1000

−500

0b’

0.5 1 1.50

0.1

0.2

0.3

0.4

0.5

−400

−300

−200

−100

0c’

FIG. S10: The first row of panels (a-c)shows the results for the maximum likelihood estimation for the datasets a, b and c(seefigure 5 in the main text). In all cases the maximum, i.e. the estimated phase ⇥, is found close to the theoretical predicted valueof ⇡ (black cross). The contour lines indicate the 67%, 95% and 99% confidence area. The second row of panels (a’-c’) showsthe corresponding results obtained if the small changes of the n

0

population induced by the residual spin-mixing dynamics(Sec. I C) is ignored. These results quantitatively close to the ones in a-c but are fully symmetric with respect to ⇥ = ⇡.

In terms of the unit vectors u, v, the average spin vector is hsi = 2u ⇥ v and the spin quadrupole tensor Qij

⌘1

2

hsi

sj

+ sj

si

i is

Qij

= �ij

� (ui

uj

+ vi

vj

). (s23)

We introduce three unit vectors {u,v,w} defined as u = u|u| , v = v

|v| , w = hˆsi|hˆsi| . Since u · v = 0 and w = u ⇥ v,

{u,v,w} forms an orthornormal, right-handed triad and Qij

is already diagonal. We can rewrite the spin quadrupolar

tensor Q in a more geometric form,

Q =1�A

2u⌦ u+

1 +A2

v ⌦ v +w ⌦w, (s24)

where

A =2|u|2 � 1. (s25)

A generic spin 1 state is uniquely determined by the average spin vector hsi and by the spin quadrupole tensor Qij

given above. There are two limiting cases. The first one is the case of an aligned state (also called spin nematic orpolar state in the context of spinor condensates [24]), where the spin wavefunction takes the form

| i = u · |ri, (s26)

i.e. is the eigenstate of s · u with zero eigenvalue. In such a state, the average spin vanishes, |hsi| = 0, and the spinquadrupole tensor is

Q = 1 � u⌦ u. (s27)

13

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0a b c

FIG. S11: (a) Equilibrium population n⇤0

, (b) transverse spin length |hs?i| and (c) alignment A of an antiferromagneticspin 1 condensate versus longitudinal magnetization m

z

, for a fixed value of q/US

= 0.2 (solid lines). The dotted blue line inpanel c indicates the the maximum alignment A

max

. The critical magnetization separing the broken axisymmetry from theantiferromagnetic phase is m

z,c

= 0.6, marked by the black dashed line. For comparison, the dashed lines show the samequantities but for ferromagnetic interactions.

In the literature, it is common to call u the director field. The tensor Q, or equivalently the director u, plays the roleof the order parameter for spin nematic states. The second limiting case is the one of an oriented or fully magnetizedstate, for which the average spin is maximal, |hsi| = 1. This is achieved when |u| = |v| = 1/

p2, and also corresponds

to a non-zero spin quadrupole tensor

Q =1

2(1 �w ⌦w) . (s28)

For a generic, partially magnetized state, one can quantify the proximity to one or the other limiting cases by thequantity A, which characterizes the amount of alignment present in a given state. For purely aligned states A = 1while for purely oriented states A = 0. Since |hsi|2 = 4|u|2|v|2 = 4|u|2(1 � |u|2), the alignment A and spin length|hsi| are related through the relation

|hsi|2 +A2 = 1. (s29)

B. Application : ground state of spinor condensates

We use the same parametrization of the condensate spin wavefunction as in the main article, but written in thecartesian basis,

|⇣i = ei✓0

0

@ei⇥/2 [�A� cos(↵/2)� iA

+

sin(↵/2)]ei⇥/2 [A� sin(↵/2)� iA

+

cos(↵/2)]B

1

A , (s30)

where A± = 1

2

�p1� n

0

+mz

±p1� n

0

�mz

�, B =

pn0

. Note that A2

� +A2

+

+B2 = 1.The average spin is then given by

hsi = 2A�

0

@�B sin

�↵

2

�

�B cos�↵

2

�

A+

1

A , (s31)

with a spin length |hsi|2 = 4A2

�(1�A2

�). The director u is given by

u =1q

1�A2

�

0

@A

+

sin�↵

2

�

A+

cos�↵

2

�

B

1

A . (s32)

14

Finally, the alignment is

A = 1� 2A2

� = n0

+p(1� n

0

)2 �m2

z

. (s33)

We show in Fig. S11 the equilibrium population n⇤0

in the ground state of an antiferromagnetic condensate, togetherwith the length |hs?i| of the transverse spin and the alignment A. Fig. S11c illustrates that A stays close to themaximum value A

max

=p

1�m2

z

that the alignment can reach, given a magnetization mz

. This justifies using thisquantity to determine the amount of alignment present in the state |⇣i, even when hsi 6= 0.

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equation

20