Abstract
Manybody systems at low temperatures generally organize themselves into ordered phases, whose nature and symmetries are captured by an order parameter. This order parameter is spatially uniform in the simplest cases, for example the macroscopic magnetization of a ferromagnetic material. Nonuniform situations also exist in nature, for instance in antiferromagnetic materials, where the magnetization alternates in space, or in the socalled stripe phases emerging for itinerant electrons in strongly correlated materials. Understanding such inhomogeneously ordered states is of central importance in manybody physics. Here we study experimentally the magnetic ordering of itinerant spin1 bosons in inhomegeneous spin domains at nanoKelvin temperatures. We demonstrate that spin domains form spontaneously, that is purely because of the antiferromagnetic interactions between the atoms and in the absence of external magnetic forces, after a phase separation transition. Furthermore, we explore how the equilibrium domain configuration emerges from an initial state prepared far from equilibrium.
Introduction
Quantum gases of ultracold atoms offer an unprecedented platform to study complex, multicomponent quantum fluids in and outofequilibrium^{1,2,3}. In quantum gases with several Zeeman components simultaneously confined in an optical dipole trap, Van der Waals^{4,5,6,7,8,9,10} or dipole–dipole^{11,12} 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 longrange phase coherence, superfluidity and magnetic ordering can all take place. For instance, Josephsonlike spin oscillations due to spinchanging interactions have been observed experimentally^{7,8,9}, and spin superfluidity demonstrated in recent experiments with sodium atoms^{13,14}.
A major question that arises for multicomponent fluids—quantum or classical—is the stability of spatially homogeneous phases toward phase separation^{1}. In cold atom experiments, phase separation has been observed in numerous multicomponent systems, either in dual species Bose–Bose or Bose–Fermi mixtures^{15,16,17,18,19,20,21,22,23} or for singlespecies quantum gases with several hyperfine components, for example twocomponent imbalanced Fermi gases with strong interactions^{24} or bosonic mixtures of hyperfine states^{25,26,27,28}. Reaching equilibrium in dual species mixtures can be difficult if inelastic losses are strong, for instance near a Feshbach resonance. In that context, metastable phaseseparated 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 spindependent 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 interactions^{1,29}.
In this work, we study the formation of spin domains in a quasionedimensional (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 spindependent interactions have an antiferromagnetic character that leads to phase separation^{30,31,32,33,34}. Early experiments observed spin domains in a F = 1 sodium BEC immersed in a magnetic field gradient around 10 mG/cm^{30,32,33}. The magnetic force produced by the gradient makes the m_{F} = ±1 Zeeman components migrate to opposite sides of the trap, with the m_{F} = 0 component inbetween. Without applied gradient, only the miscible m_{F} = ±1 phase was observed in ref. ^{30}. References ^{35,36,37} pointed out theoretically that m_{F} = 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 m_{F} = 0 on one side of the box and m_{F} = ±1 on the other. For a trapped gas, the gain in interaction energy when the m_{F} = 0 domain is located in the center of the trap outweights the energetic cost of a second interface^{30,31,37}.
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 ^{87}Rb atoms with ferromagnetic interactions^{27,28,34,38,39,40,41,42}, studied the dynamical formation of nonequilibrium spin domains after a quench. Reaching an equilibrium state appears difficult for ^{87}Rb atoms due to the weakness of spin interactions^{43}. Other experiments with F = 1 sodium atoms observed the formation of shortlived domains across a quantum phase transition and studied their equilibration dynamics^{44,45}. However, heating due to the experimental arrangement prevented to study the longtime 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 discussed^{46}.
The spin1 quantum gas in our experiments is confined in a spinindependent 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 meanfield theory. We attribute these differences to thermal fluctuations, which play an important role due to the lowenergy 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 gradientfree 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 spinmixing collisions as the underlying relaxation mechanism.
Results
Experimental system
Our experiments are performed with a gas of ^{23}Na atoms trapped in a spinindependent optical trap with frequencies ω_{x} ≪ ω_{y} = ω_{z} (Fig. 1a). With a total atom number around N ≈ 10^{4}, the chemical potential of a singlecomponent gas at low temperatures is μ ~ 0.5ħω_{y}. This implies a quasi1d 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 threebody 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 m_{F} = 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 m_{F} = 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 quasi1d character, the bosonic quantum gases studied in our experiment are quasicondensates^{47,48,49,50,51,52}: 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 timeofflight images^{49,51,52}. 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 intrap 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 spin1 gases
Before discussing our results, we first review the salient features of F = 1 spinor condensates^{3}. At very low temperatures, Bose–Einstein condensation leads to a macroscopic occupation of a singleparticle state Ψ, a superposition of all three Zeeman states behaving as a threedimensional vector. The equilibrium manybody state is determined by the competition between the interatomic interactions and the Zeeman energy in an applied magnetic field. The total meanfield energy at T = 0 takes the form^{3}
Here, \(\hat h =  \frac{{\hbar ^2}}{{2m_{{\mathrm{Na}}}}}{\mathrm{\Delta }} \cdot + V + E_{{\mathrm{Zeeman}}}\) is the singleparticle Hamiltonian, m_{Na} is the atomic mass, E_{Zeeman} 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 spin1 matrices^{3}. We denote by m_{} the component of m along the axis of the applied magnetic field B. For instance, for a spinpolarized gas of atoms in m_{F} = 1, the magnetization density is equal to the total density, m_{} = ρ, while m_{} = 0 for atoms in m_{F} = 0.
The two coupling constants \(\overline g\) and g_{s} characterize spinindependent and spindependent interactions, respectively. For sodium atoms in the F = 1 hyperfine manifold the spindependent interactions are antiferromagnetic (g_{s} > 0), a key feature to observe phase separation^{30}. Furthermore, the spindependent term ∝g_{s}, although much weaker than the dominant spinindependent 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 firstorder Zeeman shift linear in B irrelevant to the equilibrium properties. The relevant shift comes from the secondorder 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 = α_{q}B^{2} and α_{q} ≈ h × 277 Hz/G^{2} 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 shortranged van der Waals interactions.
Magnetic phase diagram and spontaneous phase separation
We explore in Fig. 2a–c the equilibrium spatial structure of a quasi1d antiferromagnetic spin1 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 m_{F} = ±1 domains coexist in the same region of space in the center of the trap, surrounded by magnetized m_{F} = +1 regions near the edges of the cloud. Above a critical QZE q_{1,exp} ≈ h × 0.72(14) Hz (corresponding to a magnetic field B_{1,exp} ≈ 51(5) mG), the m_{F} = 0 component appears and develops into a domain expelling m_{F} = ±1 from the central region. The quoted experimental value of q_{1,exp} is found by fitting an empirical function—constant below q_{1,exp} and growing as \(1  e^{  q  q_{{\mathrm{1,exp}}}/{\mathrm{\Delta }}q}\) above—to the m_{F} = 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 q ≳ h × 2 Hz (B ≳ 85 mG), the mixed m_{F} = ±1 region essentially disappears and the spin1 gas reduces to a binary mixture of m_{F} = 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 centerofmass of the m_{F} = 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 lowtemperature 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 m_{F} = 0 Zeeman state, with m_{} = 0 and ρ_{0} = ρ,
Phase II or Partially magnetized phase: The components m_{F} = ±1 coexist, with magnetization density 0 < m_{} < ρ and ρ_{0} = 0,
Phase III or Fully magnetized phase: All atoms occupy the m_{F} = +1 Zeeman state, with m_{} = ρ and ρ_{0} = 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 separation^{35}. 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 (phaseseparated) configurations, either I−II or I−III, depending on the value of q^{35}.
A common choice in the literature^{30,35} 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),
with f_{0} 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 f_{0} = 0 and the homogeneous phase II becomes thermodynamically unstable. Above q_{1}, a phase I domain forms. The equilibrium fraction of m_{F} = 0 atoms grows as f_{0}(q ≥ q_{1}) = 1 − (q_{1}/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 (f_{0} = 1 − M_{}/N), one obtains a phaseseparated 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
where η = g_{s}m_{} in phase II. A second, continuous II–III transition occurs at m_{} = ρ^{*}, with the characteristic density
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 quasi1d gas in a harmonic trap where ρ, m_{}, and ρ_{0} 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 \(\rightarrow\) ρ_{1d} 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 q_{1} that corresponds to our measured q_{1,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_{} ≈ ρ_{1d}(0)[1 − (1 − (M_{}/N))^{2/3}]^{2} for q ≤ q_{1}. Using Eq. (3), this gives the LDA prediction for the first critical QZE^{46},
Using our experimental parameters (μ/h ≈ 120 Hz), we obtain q_{1,LDA} ≈ h × 0.3 Hz, substantially below the observed q_{1,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 halflength 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 meanfield theory of spin1 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 m_{F} = −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 m_{F} = −1 component decreases smoothly with q and does not completely vanish at high q. This prevents us to clearly identify a critical value q_{2,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 q_{1} and the T = 0 prediction, as well as the difficulty in identifying q_{2} 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 spin1 GP theory in more detail and discuss the role of a thermal component, we define an effective size for each Zeeman component as the rootmeansquare (rms) radius restricted to the condensate region [−L, L],
where the halflength L of the condensate is found by a parabolic fit to ρ_{1d}(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 m_{F} = +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 m_{F} = 0, the rms radius starts from a large value at low q, then decreases above q_{1,exp} before settling to an asymptotic value above q ≳ h × 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. Lowenergy excitations of homogeneous spin1 BECs have been studied using the Bogoliubov approach^{2,3,53}. 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 m_{F} = ±1 states, one spin mode essentially reduces to excitations of atoms in the m_{F} = 0 state with a gap E_{g} ≥ q^{2}. In a Hartree–Fock picture appropriate for \(k_BT \gg g_{\mathrm{s}}\rho\), the effective potential seen by the uncondensed m_{F} = 0 atoms is almost flat (up to small terms ∝g_{s}): The meanfield from the condensate in m_{F} = ±1 cancels almost exactly the trapping potential^{54,55}. Uncondensed excitations in m_{F} = ±1 experience a different meanfield potential that expels them from the trap center. As a result one expects that below q_{1} the thermal component occupies mostly the m_{F} = 0 Zeeman state. In Fig. 4d, we show a magnified view of the linear density profiles for q < q_{1,exp}. A subtantial population is present in m_{F} = 0 (in contrast to the T = 0 prediction, see Fig. 4e) and shows a flattop profile within the volume where the m_{F} = ±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 q ≳ q_{1}, the small domain expected at T = 0 does not actually form but rather dissolve inside the existing m_{F} = 0 thermal component. The suppression of phase separation at finite temperatures has been noted in a theoretical study of a twocomponent gas^{56}, and is also consistent with our previous experimental work on threedimensional spin1 gases^{57}. We are not aware of theoretical studies of antiferromagnetic spin1 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 profiles^{59} to a Hartree–Fock model of our quasi1d gas^{60}. Here, the wings correspond to the nondegenerate region of the cloud where the onedimensional phase space density ρ_{1d}λ_{T} ≤ 1, with \(\lambda _T = \sqrt {2\pi \hbar ^2{\mathrm{/}}(m_{{\mathrm{Na}}}k_BT)}\) the thermal De Broglie wavelength and k_{B} 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 spindependent 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.
Longtime relaxation of outofequilibrium spin textures
Having characterized the equilibrium properties of a spin1 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 V_{mag} = g_{F}m_{F}μ_{B}b′x along x controlled by an applied magnetic gradient b′ (μ_{B} is the Bohr magneton and g_{F} = −1/2 the Landé gfactor). Using b′ = 24 mG/cm, the net effect of the combined action of the magnetic force and of spindependent interactions is to pull the m_{F} = +1 Zeeman component to the right side of the cloud while pushing the m_{F} = 0 component to the left one. Atoms in m_{F} = −1 are purely thermal and barely discernible in this regime.
We remove the applied magnetic force at t = 0, leaving the spin1 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 centerofmass (c.o.m.) motion of the cloud that persists up to 20 s, the longest time we explored (see Fig. 5a). This motion is commonmode to the m_{F} = 0 and +1 components and occurs at the expected dipole mode frequency ω_{x}. 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
of the m_{F} = +1 and m_{F} = 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 m_{F} component. We report in Fig. 5c the relative displacement of m_{F} = 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 m_{F} = 0 component penetrating slowly into the m_{F} = +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 m_{F} = −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 m_{F} = 0 and a roughly equal increase in the populations of m_{F} = ±1. This indicates that spinchanging collisions of the form 2 × (m_{F} = 0) \(\rightarrow\) (m_{F} = +1) + (m_{F} = −1) are involved in the mechanism enabling m_{F} = 0 atoms to cross the effective energy barrier due to spindependent interactions.
The process is most likely dominated by excitations residing initially in the inferface between the m_{F} = 0 and m_{F} = +1 regions, and seeding the longtime dynamics^{33}. The effective meanfield potential experienced by atoms in m_{F} = −1 inside the volume of the m_{F} = +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 m_{F} = −1 created in the interface by the process 2 × (m_{F} = 0) → (m_{F} = +1) + (m_{F} = −1) can thus freely migrate toward the trap center, where they can recombine with m_{F} = +1 atoms to generate m_{F} = 0 atoms by the inverse process. This mechanism effectively enables m_{F} = 0 atoms to crawl through the otherwise impenetrable m_{F} = +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 spinchanging 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 m_{F} = 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 singlepeaked at all times. These observations rule out a scenario where relaxation is explained by a macroscopic quantum tunneling of the m_{F} = 0 component. In that case, we expect at intermediate times that the m_{F} = 0 component is in a superposition of two domains, one localized on the left side of the m_{F} = +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 manybody systems characterized by nonuniform order parameters are plentiful. The simplest example corresponds to antiferromagnetism of localized spins (here the adjective localized means that the mobility of the spincarrying particles is irrelevant, and that the physics reduces to a pure spin problem). A second, richer case occurs in itinerant magnetism when the spincarrying particles are mobile. Stripe phases of electrons in strongly correlated materials are prominent examples^{62}. These materials are holedoped antiferromagnets, which organize for certain doping levels in antiferromagnetic domains separated by nanometersize 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 quasi1d spin F = 1 Bose gas with antiferromagnetic interactions. The lowfield configuration is a mixed phase of the m_{F} = ±1 components stabilized by antiferromagnetic spinexchange interactions. An applied uniform bias magnetic field favors the appearance of m_{F} = 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 m_{F} = 0 atoms appear and spontaneously organize in a spin domain near the trap center. We experimentally measured the critical value q_{1,exp} where a m_{F} = 0 domain appears and characterized the phase diagram in detail.
We found that the T = 0 meanfield theory of spin1 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 quasi1D nature of our experimental system may further enhance thermal effects.
Finally, we studied the nonequilibrium dynamics and relaxation of spin domains in the phaseseparated, highq regime. In contrast to the miscible regime^{14,63}, we observe no spin–dipole oscillations in the phaseseparated 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 spinchanging collisions, enabling atoms from immiscible Zeeman components to pass through the effective barrier created by meanfield interactions with the other component. Our results could be explained by a thermallyassisted process where a scarcely populated, but not empty thermal component in m_{F} = −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 ^{23}Na 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 zaxes. We prepare a normal gas with a welldefined magnetization far above the critical temperature using spindistillation in an applied magnetic field gradient at the beginning of the evaporation^{64,65}. We then cool the sample to degeneracy using standard evaporative cooling, and obtain a threedimensional 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 timeofflight (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 twodimensional 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/(m_{Na}ζ) ^{33}. However, in our experiments we have ħt/(m_{Na}ζ) ≈ 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 zdirections. We calibrate the magnetic fields using radiofrequency spectroscopy, with a typical resolution of ~1 mG. We observe magnetic field fluctuations with δB ~ 3 mG rootmeansquare (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 zaxis, orthogonal to the applied bias field B that lies in the x–y 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 zdirections are negligible due to the stronger confinement.
Spin1 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 onedimensional spin1 Gross–Pitaevskii equation can be written as a set of three equations for each Zeeman component, of the form
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 spinindependent 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 splitstep method. The evolution due to the kinetic energy, local spinindependent and local spindependent 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 (firstorder Trotter expansion). We have studied the influence of the time step carefully to make sure the higherorder terms are indeed negligible.
We use harmonic oscillator units where time is rescaled by \(\omega _x^{  1}\), energy by ħω_{x}, 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 30_{ax}, an imaginary time step ω_{x}δt = 2 × 10^{−5} and we compute the imaginary time evolution up to ω_{x}T = 10^{3}. 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 \(g_{\mathrm{s}}^{{\mathrm{1d}}}{\mathrm{/}}\overline g ^{{\mathrm{1d}}} = 0.0357\).
Data availability
The data that support the findings of this study and numerical programs used for data analysis are available from the corresponding author upon reasonable request.
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Acknowledgments
We acknowledge stimulating discussions with Emilia Witkowska, Sandro Stringari and Gabriele Ferrari. This work has been supported by ERC (Synergy Grant UQUAM). K.J.G. acknowledges funding from the European Union’s Horizon 2020 Research and Innovation Program under the Marie SklodowskaCurie Grant Agreement No. 701894. LKB is a member of the SIRTEQ network of Région IledeFrance.
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K.J.G. and A.I. contributed equally to this work. K.J.G. and A.I. configured the existing experimental apparatus to perform 1D experiments. K.J.G., A.I., and B.E. gathered the data with assistance from C.F. K.J.G. and A.I. carried out the analysis of the data. K.J.G., A.I, J.D., and F.G. performed numerical and analytical calculations. All authors contributed to discussions and preparation of the manuscript. This work was supervised by J.D. and F.G.
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JiménezGarcía, K., Invernizzi, A., Evrard, B. et al. Spontaneous formation and relaxation of spin domains in antiferromagnetic spin1 condensates. Nat Commun 10, 1422 (2019). https://doi.org/10.1038/s41467019085056
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