## Introduction

Quantum gases of ultracold atoms offer an unprecedented platform to study complex, multicomponent quantum fluids in- and out-of-equilibrium1,2,3. In quantum gases with several Zeeman components simultaneously confined in an optical dipole trap, Van der Waals4,5,6,7,8,9,10 or dipole–dipole11,12 interactions drive internal conversion between the Zeeman components. For bosonic atoms, this leads at very low temperatures to Bose–Einstein condensation in a superposition of the internal states where long-range phase coherence, superfluidity and magnetic ordering can all take place. For instance, Josephson-like spin oscillations due to spin-changing interactions have been observed experimentally7,8,9, and spin superfluidity demonstrated in recent experiments with sodium atoms13,14.

A major question that arises for multicomponent fluids—quantum or classical—is the stability of spatially homogeneous phases toward phase separation1. In cold atom experiments, phase separation has been observed in numerous multicomponent systems, either in dual species Bose–Bose or Bose–Fermi mixtures15,16,17,18,19,20,21,22,23 or for single-species quantum gases with several hyperfine components, for example two-component imbalanced Fermi gases with strong interactions24 or bosonic mixtures of hyperfine states25,26,27,28. Reaching equilibrium in dual species mixtures can be difficult if inelastic losses are strong, for instance near a Feshbach resonance. In that context, metastable phase-separated configurations were reported in ref. 20. Furthermore, in many cases the different components experience different trapping potentials due to different magnetic moments or masses. A species- or spin-dependent trapping potential can strongly influence phase separation in a trapped system, to the point where it becomes the main factor deciding its occurrence instead of interatomic interactions1,29.

In this work, we study the formation of spin domains in a quasi-one-dimensional (1d) spinor Bose–Einstein condensate (BEC) in an external, spatially uniform magnetic field without any external magnetic force. The condensate is made from sodium atoms carrying a hyperfine spin F = 1. The spin-dependent interactions have an antiferromagnetic character that leads to phase separation30,31,32,33,34. Early experiments observed spin domains in a F = 1 sodium BEC immersed in a magnetic field gradient around 10 mG/cm30,32,33. The magnetic force produced by the gradient makes the mF = ±1 Zeeman components migrate to opposite sides of the trap, with the mF = 0 component in-between. Without applied gradient, only the miscible mF = ±1 phase was observed in ref. 30. References 35,36,37 pointed out theoretically that mF = 0 spin domains should also form without any applied gradient. For a gas in a box, the domain should preferentially move to one side of the box to have only one interface, with mF = 0 on one side of the box and mF = ±1 on the other. For a trapped gas, the gain in interaction energy when the mF = 0 domain is located in the center of the trap outweights the energetic cost of a second interface30,31,37.

Another important question besides the nature of the equilibrium state is whether this state can be reached on a timescale compatible with the lifetime of the atomic sample. References 32,33 studied relaxation in a strong applied magnetic field gradient, observing that metastable configurations can persist for seconds. Several experiments, mostly using F = 1 87Rb atoms with ferromagnetic interactions27,28,34,38,39,40,41,42, studied the dynamical formation of nonequilibrium spin domains after a quench. Reaching an equilibrium state appears difficult for 87Rb atoms due to the weakness of spin interactions43. Other experiments with F = 1 sodium atoms observed the formation of short-lived domains across a quantum phase transition and studied their equilibration dynamics44,45. However, heating due to the experimental arrangement prevented to study the long-time regime and the approach to the expected equilibrium state. The relation between the formation of spin domains after a quench and the Kibble–Zurek mechanism has also been discussed46.

The spin-1 quantum gas in our experiments is confined in a spin-independent and highly elongated trap, realizing an effectively 1d spinor gas where phase separation occurs only along the weak axis. We take special care to compensate for magnetic field gradients along that axis (canceling them below the mG/cm level) to ensure the domains form in a negligible magnetic force. We measure the equilibrium spatial distributions, which reflect (up to interface effects that we quantify) the phase boundaries for systems with homogeneous particle density. We find qualitative agreement but quantitative differences between the measured equilibrium distributions and T = 0 mean-field theory. We attribute these differences to thermal fluctuations, which play an important role due to the low-energy scales associated with spin ordering, and the low dimensionality. In the second part of the article, we adress the issue of relaxation to equilibrium in a gradient-free situation. We prepare a spin configuration far from equilibrium and monitor how it relaxes to equilibrium. We observe a slow relaxation on a time scale of several seconds, and a spin dynamics that points to spin-mixing collisions as the underlying relaxation mechanism.

## Results

### Experimental system

Our experiments are performed with a gas of 23Na atoms trapped in a spin-independent optical trap with frequencies ωxωy = ωz (Fig. 1a). With a total atom number around N ≈ 104, the chemical potential of a single-component gas at low temperatures is μ ~ 0.5ħωy. This implies a quasi-1d regime where the transverse motion is almost frozen to the ground state of the harmonic potential. The total atom number decays with a measured 1/e lifetime around 50 s, presumably limited by residual evaporation and three-body recombination.

We measure the linear integrated densities $$\rho _{1{\mathrm{d}},m_F}(x) = {\int} {\kern 1pt} dydz{\kern 1pt} \rho _{m_F}({\bf{r}})$$ along the weak axis of the trap after a short expansion in a magnetic field gradient that separates all three components mF = 0, ±1 by the Stern–Gerlach effect. Here and in the following, $$\rho _{m_F}$$ denotes the partial density of the Zeeman component with magnetic quantum number mF = 0, ±1, $$\rho = \mathop {\sum}\nolimits_{m_F} {\kern 1pt} \rho _{m_F}$$ the total density, and the subscript “1d” always indicates linear quantities integrated over the transverse coordinates y, z.

Because of its quasi-1d character, the bosonic quantum gases studied in our experiment are quasi-condensates47,48,49,50,51,52: density fluctuations are essentially frozen, as for a true condensate, but phase fluctuations along the weak axis of the trap can be substantial at finite temperatures. These phase fluctuations do not affect the thermodynamic properties of the mixture, but they show up as density stripes in time-of-flight images49,51,52. The density profiles reported in this article are averaged profiles over many (typically several tens) repetitions of the experiment to suppress the signature of phase fluctuations. Because of the very weak expansion along x, the observed average distributions reflect the linear in-trap density distributions to a good approximation. We also take special care to cancel residual magnetic forces that could affect the spatial distributions (see Methods). This is reflected in the nearly symmetric linear distributions of the spin components (Fig. 1b–d).

### Brief review of ultracold spin-1 gases

Before discussing our results, we first review the salient features of F = 1 spinor condensates3. At very low temperatures, Bose–Einstein condensation leads to a macroscopic occupation of a single-particle state Ψ, a superposition of all three Zeeman states behaving as a three-dimensional vector. The equilibrium many-body state is determined by the competition between the interatomic interactions and the Zeeman energy in an applied magnetic field. The total mean-field energy at T = 0 takes the form3

$$E = {\int} {\kern 1pt} d{\bf{r}}{\kern 1pt} \left( {{\bf{\Psi }}^ \ast \hat h{\bf{\Psi }} + \frac{{\overline g }}{2}\rho ^2 + \frac{{g_{\mathrm{s}}}}{2}{\bf{m}}^2} \right).$$
(1)

Here, $$\hat h = - \frac{{\hbar ^2}}{{2m_{{\mathrm{Na}}}}}{\mathrm{\Delta }} \cdot + V + E_{{\mathrm{Zeeman}}}$$ is the single-particle Hamiltonian, mNa is the atomic mass, EZeeman is the Zeeman energy discussed below, and $$V = \frac{1}{2}m_{{\mathrm{Na}}}[\omega _x^2x^2 + \omega _y^2(y^2 + z^2)]$$ the trapping potential. The partial densities are given by $$\rho _{m_F} = \left| {{\mathrm{\Psi }}_{m_F}} \right|^2$$, and the total density by $$\rho = \mathop {\sum}\nolimits_{m = 0, \pm 1} {\kern 1pt} \rho _m$$. The magnetization density m is defined by its Cartesian components $$m_\alpha = \mathop {\sum}\nolimits_{i,j} {\kern 1pt} {\mathrm{\Psi }}_i^ \ast \left( {\hat S_\alpha } \right)_{i,j}{\mathrm{\Psi }}_j$$, with $$\hat S_\alpha$$ (α = x, y, z) the standard spin-1 matrices3. We denote by m|| the component of m along the axis of the applied magnetic field B. For instance, for a spin-polarized gas of atoms in mF = 1, the magnetization density is equal to the total density, m|| = ρ, while m|| = 0 for atoms in mF = 0.

The two coupling constants $$\overline g$$ and gs characterize spin-independent and spin-dependent interactions, respectively. For sodium atoms in the F = 1 hyperfine manifold the spin-dependent interactions are antiferromagnetic (gs > 0), a key feature to observe phase separation30. Furthermore, the spin-dependent term gs, although much weaker than the dominant spin-independent term $$\left( {g_s{\mathrm{/}}\overline g \approx 0.036} \right)$$, is essential to understand spinor gases: This term lifts spin degeneracies left by $$\overline g$$ and determines the magnetic properties at very low temperatures.

Spinor gases are typically immersed in a uniform magnetic field B that shifts the internal energy levels by the Zeeman effect. The interaction Hamiltonian conserves the total magnetization $$M_{||} = {\int} {\kern 1pt} d{\bf{r}}{\kern 1pt} m_{||}$$. As a result, the constant of motion M|| should be viewed as an experimental control parameter and not as a dynamical variable. The conservation of M|| makes the first-order Zeeman shift linear in B irrelevant to the equilibrium properties. The relevant shift comes from the second-order or quadratic Zeeman energy (QZE), $$E_{{\mathrm{Zeeman}}} = - q{\int} {\kern 1pt} d{\bf{r}}{\kern 1pt} \rho _0$$ (up to a constant), with q = αqB2 and αq ≈ h × 277 Hz/G2 for sodium atoms. Note that the physics explored in this work does not depend of the orientation of B with respect to the trap axis, only of its modulus |B| determining the QZE. This conclusion only holds if dipole–dipole interactions are negligible compared to short-ranged van der Waals interactions.

### Magnetic phase diagram and spontaneous phase separation

We explore in Fig. 2a–c the equilibrium spatial structure of a quasi-1d antiferromagnetic spin-1 Bose gas in a spatially uniform applied field B. We set the total magnetization to M|| ≈ 0.5N and vary the QZE q. We find that the spatial structure of the spinor gas undergoes a marked change as q increases. For low q, we observe a mixed phase where mF = ±1 domains coexist in the same region of space in the center of the trap, surrounded by magnetized mF = +1 regions near the edges of the cloud. Above a critical QZE q1,exp ≈ h × 0.72(14) Hz (corresponding to a magnetic field B1,exp ≈ 51(5) mG), the mF = 0 component appears and develops into a domain expelling mF = ±1 from the central region. The quoted experimental value of q1,exp is found by fitting an empirical function—constant below q1,exp and growing as $$1 - e^{ - |q - q_{{\mathrm{1,exp}}}|/{\mathrm{\Delta }}q}$$ above—to the mF = 0 density in the trap center (Fig. 2b—see also the Supplementary Fig. 5 that shows the behavior of the reduced populations). Error bars denote the 90% uncertainty level of the fit obtained by standard error analysis assuming Gaussian noise. Furthermore, for qh × 2 Hz (B 85 mG), the mixed mF = ±1 region essentially disappears and the spin-1 gas reduces to a binary mixture of mF = 0, +1. Our data are summarized in Fig. 2a, where we plot the linear partial densities $$\rho _{1{\rm d},m_F}$$ versus q. We observe similar behavior for other values of the magnetization M||.

Besides the stripes due to phase fluctuations discussed earlier, we also observe substantial position fluctuations of the spin domains. For instance, in the examples shown in Fig. 1d, we find that the center-of-mass of the mF = 0 component fluctuates by ~16 μm for B = 45 mG and by ~6 μm for B = 150 mG. We believe that this behavior is due to thermal fluctuations of the domain, and not to a technical artifact such as a magnetic gradient fluctuating around the compensated value. The fluctuations of the position of the spin domains and their possible use for low-temperature thermometry will be explored in more detail in a future publication.

### Phase coexistence in homogeneous systems

The observed characteristics of the phase diagram can be qualitatively understood by considering first a uniform system in the thermodynamic limit enclosed in a box of volume $${\cal V}$$. Three homogeneous phases can be realized depending on the magnetization $$M_{||} = m_{||}{\cal V}$$30,31,35,37,

Phase I or Unmagnetized phase: All atoms occupy the mF = 0 Zeeman state, with m|| = 0 and ρ0 = ρ,

Phase II or Partially magnetized phase: The components mF = ±1 coexist, with magnetization density 0 < m|| < ρ and ρ0 = 0,

Phase III or Fully magnetized phase: All atoms occupy the mF = +1 Zeeman state, with m|| = ρ and ρ0 = 0. Note that phase II evolves continuously into phase III when the magnetization increases.

The properties of the various phases are summarized in Table 1. A completely homogeneous phase where the three Zeeman components coexist is always unstable toward phase separation35. For a partially magnetized system with 0 < M|| < N, phase II is the only possible homogeneous phase compatible with the conservation of the total magnetization M||. However, it competes with inhomogeneous (phase-separated) configurations, either I−II or I−III, depending on the value of q35.

A common choice in the literature30,35 is to describe the evolution of the system for fixed ρ, m||, and with varying QZE q. For low QZE, phase II minimizes the interaction energy and is the stable equilibrium phase. As the QZE increases, a mixed configuration where part of the system is in phase II and part in phase I becomes energetically competitive. The preferred equilibrium configuration can be determined by comparing the energies of the competing possibilities (neglecting the energy cost of the I−II interface),

$$\delta E = E_{{\mathrm{I}} - {\mathrm{II}}} - E_{{\mathrm{II}}} = {\cal V}f_0 \times \left[ {\frac{{g_sm_{||}^2}}{{2\left( {1 - f_0} \right)}} - q\rho } \right],$$
(2)

with f0 the fraction of the available volume occupied by phase I in the mixed configuration. When $$q \ge q_1 = g_{\mathrm{s}}m_{||}^2{\mathrm{/}}(2\rho )$$, δE becomes negative for f0 = 0 and the homogeneous phase II becomes thermodynamically unstable. Above q1, a phase I domain forms. The equilibrium fraction of mF = 0 atoms grows as f0(q ≥ q1) = 1 − (q1/q)1/2. The conservation of the total magnetization M|| then requires that the magnetization density in subregion II decreases as $$m_{||} = M_{||}{\mathrm{/}}[{\cal V}(1 - f_0)]$$. When m|| = ρ (f0 = 1 − M||/N), one obtains a phase-separated I−III mixture which remains the same when ρ increases further. The sequence of transitions is illustrated in Fig. 3a.

Anticipating the discussion of the trapped case within the framework of the local density approximation (LDA), we now adopt a slightly different point of view and consider the properties of the system for fixed q and varying ρ, m|| (Fig. 3b). It is convenient to chose a thermodynamic ensemble characterized by two thermodynamic potentials enforcing the conservation of N and of M||, respectively: the chemical potential μ and another potential η (that could be called the thermomagnetic potential). The equation of state of the various phases are given in terms of μ and η in Fig. 3a. Phase II (respectively phase I) is the stable equilibrium phase for densities below (resp. above) a critical value defined by

$$\left( {\rho q} \right)_1 = \frac{{\eta ^2}}{{2g_{\mathrm{s}}}} = \frac{{g_{\mathrm{s}}m_{||}^2}}{2}{,}$$
(3)

where η = gsm|| in phase II. A second, continuous II–III transition occurs at m|| = ρ*, with the characteristic density

$$\rho ^ \ast = \frac{{2q}}{{g_{\mathrm{s}}}},$$
(4)

with the fully magnetized phase III realized for densities lower than ρ*.

### Spatial structure of a trapped system

The preceding discussion is directly relevant to determine the spatial structure of a quasi-1d gas in a harmonic trap where ρ, m||, and ρ0 vary with position. We consider in this Section the purely 1d limit $$\mu \ll \hbar \omega _y$$ where the transverse motion is frozen in the transverse ground state of the trap. We first perform our analysis within the LDA, and discuss effects beyond the LDA at the end of this section. The equalities established in the previous section remain valid substituting linear densities ρ$$\rightarrow$$ρ1d and $$\left( {\overline g ,g_{\mathrm{s}}} \right) \to \left( {\overline g ^{{\mathrm{1d}}},g_{\mathrm{s}}^{{\mathrm{1d}}}} \right)$$, with effective 1d coupling constants $$\left( {\overline g ^{{\mathrm{1d}}},g_{\mathrm{s}}^{{\mathrm{1d}}}} \right) = \left( {\overline g ,g_{\mathrm{s}}} \right) \times 1/\left( {2\pi a_y^2} \right)$$. Here, $$a_y = \sqrt {\hbar {\mathrm{/}}\left( {m_{{\mathrm{Na}}}\omega _y} \right)}$$ is the transverse harmonic oscillator size. We keep the same notation m|| for the integrated magnetization density with a slight abuse of notation. The pure 1d limit is not strictly realized in our experiment, as noted earlier. However, we have evaluated corrections to this limit and found that they only change marginally the conclusions (see Supplementary Fig. 4 and Methods). As a result we stick to the 1d description in the core of the article to keep the discussion as simple as possible.

In general, one can find three regions corresponding to the three homogeneous phases. Moving from the edges to the center, phase III first appears for low densities in a region where $$0 \le \rho _{1{\rm d}}(x) \le \rho _{1{\rm d}}^ \ast$$, followed by phase II where $$\rho _{1{\rm d}}^ \ast \le \rho _{1{\rm d}}(x) \le \rho _{c2}$$, and eventually by phase I near the trap center where $$\rho _{1{\rm d}}(x) \ge \frac{{m_{||}^2}}{{\rho _{1{\rm d}}^ \ast }}$$. The magnetization density m|| is zero in region I, equal to $$\eta {\mathrm{/}}g_{\mathrm{s}}^{{\mathrm{1d}}}$$ and spatially uniform in region II, and equal to the total density (and therefore nonuniform) in region III. A corresponding trajectory in the $$\left( {\rho ,m_{||}} \right)$$ plane is shown in Fig. 3c.

For given N, M|| the condition for the appearance of phase I in the center of the trap given in Eq. (3) can only be fulfilled for sufficiently high QZE q. Similarly to the homogeneous case, this leads to a first critical value q1 that corresponds to our measured q1,exp. The magnetization density m|| in region II is uniform, but not directly proportional to M|| as it was in the homogeneous case. For a purely 1d system, we find following ref. 46 that m|| ≈ ρ1d(0)[1 − (1 − (M||/N))2/3]2 for q ≤ q1. Using Eq. (3), this gives the LDA prediction for the first critical QZE46,

$$q_{{\mathrm{1,LDA}}} = \frac{{g_{\mathrm{s}}\mu }}{{2\overline g }}\left[ {1 - \left( {1 - \frac{{M_{||}}}{N}} \right)^{2/3}} \right]^2.$$
(5)

Using our experimental parameters (μ/h ≈ 120 Hz), we obtain q1,LDA ≈ h × 0.3 Hz, substantially below the observed q1,exp ≈ h × 0.72 Hz. The same conclusion holds when taking the deviations from the purely 1d case into account (see Supplementary Methods).

The quantitative difference between the observations and the LDA prediction can be expected, as the latter completely neglects the energy cost of the interface between two immiscible phases. This cost comes from the balance between the kinetic energy, increasing when the width of the interface decreases, and the interaction energy, increasing when the interface spreads out due to the increased overlap between the two components. The interface energy is proportional to its width (typically several times the spin healing length) $$\left( {\zeta _s = \sqrt {\hbar ^2\overline g {\mathrm{/}}\left( {2m_{{\mathrm{Na}}}g_{\mathrm{s}}\mu } \right)} \sim 7\,\mu {\mathrm{m}}} \right)$$, and therefore not extensive and negligible for infinitely large systems. However it can be significant in a gas of finite extent as in our experiment where a typical cloud half-length is $$L = \sqrt {2\mu {\mathrm{/}}\left( {m\omega _x^2} \right)} \sim 100\,\mu {\mathrm{m}}$$.

These effects beyond the LDA can be explored at zero temperature using the mean-field theory of spin-1 gases, which takes the form of three coupled Gross–Pitaevskii equations. We have solved these equations numerically to find the lowest energy solution (see Methods). Examples of the density profiles that we obtain numerically are shown in Fig. 1e–g. Using the same fitting procedure as for the experimental data in Fig. 2d, we find that the first critical QZE predicted by the GP approach is $$q_{\mathrm{1}}^{{\mathrm{GP}}} \approx h \times 0.36\,{\mathrm{Hz}}$$. Therefore the discrepancy between the measured and predicted first critical QZE is not resolved by upgrading the theory from LDA to GP. A second critical QZE $$q_{\mathrm{2}}^{{\mathrm{GP}}} \approx h \times 2\,{\mathrm{Hz}}$$ where mF = −1 disappears can also be identified in the GP calculation. This is consistent with the experimental results, although we find experimentally that the population of the mF = −1 component decreases smoothly with q and does not completely vanish at high q. This prevents us to clearly identify a critical value q2,exp analogous to $$q_{\mathrm{2}}^{{\mathrm{GP}}}$$.

To summarize, we used the LDA to describe the spatial structure of a trapped gas in terms of the phase diagram of the uniform system at T = 0. We were able to account qualitatively for the observations, but found quantitative differences. In particular, the measured critical field where spin domains appear is higher than predicted. In the next section, we show that the discrepancy between the measured q1 and the T = 0 prediction, as well as the difficulty in identifying q2 in experiments, can be understood qualitatively by considering the role of a finite temperature of the sample.

### Role of the thermal components

To compare the experimental results with the prediction of the spin-1 GP theory in more detail and discuss the role of a thermal component, we define an effective size for each Zeeman component as the root-mean-square (rms) radius restricted to the condensate region [−L, L],

$$R_{m_F} = \frac{1}{{{\cal N}_{m_F}}} \int_{ - L}^L {\kern 1pt} x^2\rho _{1{\mathrm{d}},m_F}(x)dx,$$
(6)

where the half-length L of the condensate is found by a parabolic fit to ρ1d(x), and with a normalization factor $${\cal N}_{m_F} = {\int}_{ - L}^L {\kern 1pt} \rho _{m_F,1{\mathrm{d}}}(x)dx$$. We show in Fig. 4a–c the size $$R_{m_F}$$ computed from the measured profiles and from the calculated ones. The size of mF = +1 increases only slightly with q, and stays close to the T = 0 GP prediction for all values of q. In contrast, both $$R_{m_F = 0}$$ and $$R_{m_F = - 1}$$ differ substantially from the T = 0 predictions. Focusing on mF = 0, the rms radius starts from a large value at low q, then decreases above q1,exp before settling to an asymptotic value above qh × 2 Hz. The agreeement between experiments and T = 0 theory improves with increasing q.

The differences between experiment and theory can be explained qualitatively by thermal excitations. Low-energy excitations of homogeneous spin-1 BECs have been studied using the Bogoliubov approach2,3,53. In general, one expects for q ≠ 0 that the Bogoliubov spectrum consists of three modes. For low values of $$q \ll q_1$$, where the (quasi-)condensate occupies the mF = ±1 states, one spin mode essentially reduces to excitations of atoms in the mF = 0 state with a gap Eg ≥ q2. In a Hartree–Fock picture appropriate for $$k_BT \gg g_{\mathrm{s}}\rho$$, the effective potential seen by the uncondensed mF = 0 atoms is almost flat (up to small terms gs): The mean-field from the condensate in mF = ±1 cancels almost exactly the trapping potential54,55. Uncondensed excitations in mF = ±1 experience a different mean-field potential that expels them from the trap center. As a result one expects that below q1 the thermal component occupies mostly the mF = 0 Zeeman state. In Fig. 4d, we show a magnified view of the linear density profiles for q < q1,exp. A subtantial population is present in mF = 0 (in contrast to the T = 0 prediction, see Fig. 4e) and shows a flat-top profile within the volume where the mF = ±1 condensate is present, in agreement with the Hartree–Fock description. For a flat density confined within the condensate region [−L, L], the rms radius is $$\approx \sqrt {1/3} L \approx 62\,\mu {\mathrm{m}}$$, in good agreement with the measured $$R_{m_F = 0} \approx 58\,\mu {\mathrm{m}}$$ for low q.

This discussion, although qualitative, explains the increase of the observed critical field from the T = 0 value. For qq1, the small domain expected at T = 0 does not actually form but rather dissolve inside the existing mF = 0 thermal component. The suppression of phase separation at finite temperatures has been noted in a theoretical study of a two-component gas56, and is also consistent with our previous experimental work on three-dimensional spin-1 gases57. We are not aware of theoretical studies of antiferromagnetic spin-1 gases in 1d that can explain our observations quantitatively. Our experiments could be modeled using classical field methods (see ref. 58 for a review) and perhaps used to benchmark such methods. To ease such comparison, we have measured the temperature of the thermal component by fitting the equation of state obtained from the wings of the linear profiles59 to a Hartree–Fock model of our quasi-1d gas60. Here, the wings correspond to the nondegenerate region of the cloud where the one-dimensional phase space density ρ1dλT ≤ 1, with $$\lambda _T = \sqrt {2\pi \hbar ^2{\mathrm{/}}(m_{{\mathrm{Na}}}k_BT)}$$ the thermal De Broglie wavelength and kB the Boltzmann constant. We find T ≈ 30–40 nK without any obvious dependence on q. Note that the measured temperature is substantially higher than the spin-dependent energies η, q explored in this work. Hence, even if quantum and thermal fluctuations probably lead to qualitatively similar effects on the domain formation, we expect the latter to be dominant in our experimental conditions.

### Long-time relaxation of out-of-equilibrium spin textures

Having characterized the equilibrium properties of a spin-1 antiferromagnetic gas, we now turn to nonequilibrium behavior. We investigate how an initial, highly nonequilibribrium configuration relaxes to a final equilibrium configuration. The experiment is performed at a uniform bias field B = 600 mG (q/h ≈ 100 Hz), well above $$q_{\mathrm{2}}^{{\mathrm{GP}}}$$. We prepare the system at a magnetization M|| ≈ 0.66(2)N using the same procedure as before, except for an applied magnetic potential Vmag = gFmFμBbx along x controlled by an applied magnetic gradient b′ (μB is the Bohr magneton and gF = −1/2 the Landé g-factor). Using b′ = 24 mG/cm, the net effect of the combined action of the magnetic force and of spin-dependent interactions is to pull the mF = +1 Zeeman component to the right side of the cloud while pushing the mF = 0 component to the left one. Atoms in mF = −1 are purely thermal and barely discernible in this regime.

We remove the applied magnetic force at t = 0, leaving the spin-1 gas in a purely optical potential independent of the Zeeman state but also in a highly nonequilibrium configuration. The first consequence is an excitation of the center-of-mass (c.o.m.) motion of the cloud that persists up to 20 s, the longest time we explored (see Fig. 5a). This motion is common-mode to the mF = 0 and +1 components and occurs at the expected dipole mode frequency ωx. In contrast, the relative positions of the two Zeeman components do not display any detectable oscillation and evolve on a much longer time scale than the axial period, as pictured in Fig. 5b. To quantify the relaxation we introduce the c.o.m. displacements

$${\mathrm{\Delta }}\bar x_{m_F} = \frac{1}{{N_{m_F}}}{\int} \left( {x - \bar x} \right)\rho _{1{\mathrm{d}},m_F}(x)dx,$$
(7)

of the mF = +1 and mF = 0 components from the center of mass $$\bar x = (1{\mathrm{/}}N){\int} {\kern 1pt} x\rho _{1{\mathrm{d}}}(x)dx$$ of the whole cloud. Here, $$N_{m_F} = {\int} {\kern 1pt} \rho _{1{\mathrm{d}},m_F}(x)dx$$ is the total population of the mF component. We report in Fig. 5c the relative displacement of mF = 0, which remains mostly constant for several periods of the c.o.m. oscillations before decaying to zero within a timescale of ~4 s.

The profiles shown in Fig. 5b indicate that this relaxation occurs progressively, with the mF = 0 component penetrating slowly into the mF = +1 majority component. This behavior could be surprising for a truly immiscible binary mixture, where the repulsion between the species acts as an effective barrier preventing relaxation. Figure 5d shows that the mF = −1 component, altough weakly populated, plays an important role in the relaxation process. The relative populations of the Zeeman states evolve on the same time scale as the relaxation, with a decrease in the population of mF = 0 and a roughly equal increase in the populations of mF = ±1. This indicates that spin-changing collisions of the form 2 × (mF = 0) $$\rightarrow$$ (mF = +1) + (mF = −1) are involved in the mechanism enabling mF = 0 atoms to cross the effective energy barrier due to spin-dependent interactions.

The process is most likely dominated by excitations residing initially in the inferface between the mF = 0 and mF = +1 regions, and seeding the long-time dynamics33. The effective mean-field potential experienced by atoms in mF = −1 inside the volume of the mF = +1 domain is not exactly flat but slightly attractive, $$V_{{\mathrm{eff}}}(x)$$ = $$\frac{1}{2}M_{{\mathrm{Na}}}x^2 + \overline g \rho (x)$$ − $$g_{\mathrm{s}}\rho _{ + 1}(x) \approx \mu - g_{\mathrm{s}}\rho _{ + 1}(x)$$. Atoms in mF = −1 created in the interface by the process 2 × (mF = 0) → (mF = +1) + (mF = −1) can thus freely migrate toward the trap center, where they can recombine with mF = +1 atoms to generate mF = 0 atoms by the inverse process. This mechanism effectively enables mF = 0 atoms to crawl through the otherwise impenetrable mF = +1 component. This mechanism could be coherent (preserving the initial relative phase), but need not be. Given that $$T \gg g_{\mathrm{s}}\rho$$, we believe it is likely that the relaxation process is seeded by the thermal component initially present near the interface.

A recent experiment by Eto et al.61 appears similar at first sight, but reports drastically different results: bouncing of the spin components against each other in the strongly immiscible regime, followed by relaxation in a few hundreds of ms to a highly excited state with large kinetic energy. We believe the mechanism at play in our experiments is different from the ones studied in ref. 61, where the experimental parameters differ by orders of magnitude. Eto et al. apply a bias field of 11G which supresses completely spin-changing collisions (we use 600 mG), and a gradient of 900 mG/cm (we use 10 mG/cm). It is then not really surprising that we do not observe the bouncing motion, nor the formation of stripes reported by Eto et al. in the strongly immiscible regime.

Figure 5e displays histograms of the c.o.m. of the mF = 0, +1 components as a function of the relaxation time. We observe a gradual change over time from a distribution peaked near the cloud edges to a distribution peaked near the cloud center. The distribution of $$\overline x _0$$ appears smooth and single-peaked at all times. These observations rule out a scenario where relaxation is explained by a macroscopic quantum tunneling of the mF = 0 component. In that case, we expect at intermediate times that the mF = 0 component is in a superposition of two domains, one localized on the left side of the mF = +1 cloud and one localized near its center. This would lead to a bimodal spatial distribution for which we find no evidence.

## Discussion

Magnetically ordered many-body systems characterized by non-uniform order parameters are plentiful. The simplest example corresponds to antiferromagnetism of localized spins (here the adjective localized means that the mobility of the spin-carrying particles is irrelevant, and that the physics reduces to a pure spin problem). A second, richer case occurs in itinerant magnetism when the spin-carrying particles are mobile. Stripe phases of electrons in strongly correlated materials are prominent examples62. These materials are hole-doped antiferromagnets, which organize for certain doping levels in antiferromagnetic domains separated by nanometer-size conducting domain walls. This structure is believed to arise from the competition between the kinetic energy of the holes and the exchange interactions between the spins. This example illustrates that one can expect (and often finds) nonuniform phenomena in the magnetism of itinerant quantum particles. In this article, we have investigated the formation and relaxation of inhomogeneous spin domains in a quasi-1d spin F = 1 Bose gas with antiferromagnetic interactions. The low-field configuration is a mixed phase of the mF = ±1 components stabilized by antiferromagnetic spin-exchange interactions. An applied uniform bias magnetic field favors the appearance of mF = 0 atoms through the QZE shift. The two influences compete agains each other, and this competition leads to a critical value of the QZE above which mF = 0 atoms appear and spontaneously organize in a spin domain near the trap center. We experimentally measured the critical value q1,exp where a mF = 0 domain appears and characterized the phase diagram in detail.

We found that the T = 0 mean-field theory of spin-1 Bose gases describes qualitatively well our observations. However, there exist discrepancies between the predicted and measured values of the critical fields. The finite temperature of our samples, although very low, could explain these discrepancies. Indeed, energy scales in spinor gases are naturally low in comparison to the natural scale set by the chemical potential of the BEC. Therefore, we expect that thermal fluctuations are able to suppress the formation of spin domains near the transition where different spin configurations are close in energy. The quasi-1D nature of our experimental system may further enhance thermal effects.

Finally, we studied the nonequilibrium dynamics and relaxation of spin domains in the phase-separated, high-q regime. In contrast to the miscible regime14,63, we observe no spin–dipole oscillations in the phase-separated regime. Instead we find that spin dynamics is frozen on short time scales on the order of the trap period, and undergoes slow relaxation toward an equilibrium configuration on long times scales of several tens of axial trap periods (about 10 s). We found evidence that relaxation takes place through spin-changing collisions, enabling atoms from immiscible Zeeman components to pass through the effective barrier created by mean-field interactions with the other component. Our results could be explained by a thermally-assisted process where a scarcely populated, but not empty thermal component in mF = −1 seeds the relaxation dynamics. We found no clear evidence of macroscopic quantum tunneling.

## Methods

### Optical dipole trap

Our experiments start with a spinor gas of ultracold 23Na atoms with a fixed total magnetization M|| and immersed in a uniform magnetic field B. The spinor gas is held in a crossed dipole trap (CDT) created at the intersection of two Gaussian beams propagating along the x- and z-axes. We prepare a normal gas with a well-defined magnetization far above the critical temperature using spin-distillation in an applied magnetic field gradient at the beginning of the evaporation64,65. We then cool the sample to degeneracy using standard evaporative cooling, and obtain a three-dimensional condensate in the CDT with >90% condensate fraction. We transfer the condensate to the 1d trap by adiabatically turning off one of the dipole beams in 5 s (see Supplementary Fig. 1 and Methods for more details).

### Stern–Gerlach imaging

We measure the density profiles of each Zeeman component by removing suddenly the trapping potential and letting the cloud expand for a time-of-flight (t.o.f.) of t = 8 ms in a magnetic field gradient (applied only during the t.o.f.). We extract linear density profiles from the two-dimensional absorption images (Supplementary Methods and Supplementary Fig. 2). Owing to the large trap anisotropy, the expansion is essentially in the radial direction (At T = 0, the condensate expands along its weak axis by a factor ≈10−4 66). The finite width ζ of the spin domains entails a finite quantum confinement energy. When the gas is released from the trap, the domains are therefore expected to expand at a speed ~ħ/(mNaζ33. However, in our experiments we have ħt/(mNaζ) ≈ 1–2 μm ζ, so that we can safely neglect this expansion.

### Magnetic field generation

We generate uniform magnetic fields using three pairs of bias coils aligned along the x ± y and z-directions. We calibrate the magnetic fields using radio-frequency spectroscopy, with a typical resolution of ~1 mG. We observe magnetic field fluctuations with δB ~ 3 mG root-mean-square (r.m.s.) amplitude and with a typical time scale of several tens of seconds. These fluctuations, coming from a nearby subway line, are along the vertical z-axis, orthogonal to the applied bias field B that lies in the xy plane. The impact of magnetic field fluctuations is minimized by working with applied fields B ≥ 30 mG. The resulting r.m.s. uncertainty on q is then below δq ~ (δB/B)2 ~ 1%.

### Magnetic force cancellation

Our experiments are performed after carefully canceling stray magnetic field gradients (thereby canceling magnetic forces) along the weak axis of the trap. Stray gradients have at least two origins: (i) the residual ambient gradients (arising from inhomogeneously magnetized elements around the experiment, power supplies, etc…) and (ii) the imperfections of the bias coils that produce slightly inhomogeneous fields.

We cancel the residual magnetic force along x by two methods, either by applying a compensation gradient along the weak axis of the trap (more appropriate at low bias fields where effect (i) dominates), or by choosing the direction of the applied field (more appropriate at large bias fields where effect (ii) dominates) [see Supplementary Methods and Supplementary Fig. 3 for more details]. We are able to cancel magnetic gradients to better than a few 100 μG/cm along the weak trapping direction x. Residual magnetic forces along the y- and z-directions are negligible due to the stronger confinement.

### Spin-1 Gross–Pitaevskii equations

In the 1d limit, the complete BEC wavefunction can be written as Ψ = ϕ(y, z)ζ(x) where ϕ(y, z) denotes the transverse harmonic oscillator ground state. The one-dimensional spin-1 Gross–Pitaevskii equation can be written as a set of three equations for each Zeeman component, of the form

$$i\hbar \frac{{\partial \zeta _{ + 1}}}{{\partial t}} = \left[ {{\cal L} + g_s^{1{\mathrm{d}}}\left( {\rho _{{\mathrm{1d,0}}} + m_{||}} \right)} \right]\zeta _{ + 1} + g_s^{1{\mathrm{d}}}\zeta _0^2\zeta _{ - 1}^ \ast ,$$
(8)
$$i\hbar \frac{{\partial \zeta _0}}{{\partial t}} = \left[ {{\cal L} + g_s^{1{\mathrm{d}}}\left( {\rho _{{\mathrm{1d}}} - \rho _{{\mathrm{1d,0}}}} \right)} \right]\zeta _0 + 2g_s^{1{\mathrm{d}}}\zeta _0^ \ast \zeta _{ - 1}\zeta _{ + 1},$$
$$i\hbar \frac{{\partial \zeta _{ - 1}}}{{\partial t}} = \left[ {{\cal L} + g_s^{1{\mathrm{d}}}\left( {\rho _{1{\rm d},0} - m_{||}} \right)} \right]\zeta _{ - 1} + g_s^{1{\mathrm{d}}}\zeta _0^2\zeta _{ + 1}^ \ast ,$$

with $${\cal L} = - \frac{{\hbar ^2}}{{2M_{{\mathrm{Na}}}}}\partial _x^2 + \frac{1}{2}M_{{\mathrm{Na}}}\omega _x^2x^2 + E_{{\mathrm{Zeeman}}} + \overline g ^{{\mathrm{1d}}}\rho _{{\mathrm{1d}}}$$ the spin-independent GP operator and $$m_{||} = \rho _{1{\mathrm{d}}, + 1} - \rho _{1{\mathrm{d}}, - 1}$$ the magnetization density.

We propagate Eq. (8) in imaginary time to obtain the lowest energy state using a split-step method. The evolution due to the kinetic energy, local spin-independent and local spin-dependent terms are calculated separately by exponentiating the corresponding operator. This can be done analytically, either in the momentum or position basis. Then the total evolution at each time step is approximated by multiplying all three evolution operators neglecting their commutation properties (first-order Trotter expansion). We have studied the influence of the time step carefully to make sure the higher-order terms are indeed negligible.

We use harmonic oscillator units where time is rescaled by $$\omega _x^{ - 1}$$, energy by ħωx, and lengths by $$a_x = \sqrt {\hbar /\left( {m_{{\mathrm{Na}}}\omega _x} \right)}$$. For the data shown in this paper, we typically use a grid containing N = 64 points and toal length 30ax, an imaginary time step ωxδt = 2 × 10−5 and we compute the imaginary time evolution up to ωxT = 103. We use dimensionless coupling constants $$N\overline g ^{1{\mathrm{d}}} = N\overline g {\mathrm{/}}(2\pi \hbar \omega _xa_y^2a_x) \approx 378.9$$ and $$g_{\mathrm{s}}^{{\mathrm{1d}}}{\mathrm{/}}\overline g ^{{\mathrm{1d}}} = 0.0357$$.