Stepwise Bose-Einstein condensation in a spinor gas
C. Frapolli T. Zibold A. Invernizzi K. Jiménez-García J. Dalibard F. Gerbier








		
		

		Current address: Department of Physics, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland
		


		
		

		


		
		

		
		





We observe multi-step condensation of sodium atoms with spin F=1, where the different Zeeman components m_F=0,± 1 condense sequentially as the temperature decreases. The precise sequence changes drastically depending on the magnetization m_z and on the quadratic Zeeman energy q (QZE) in an applied magnetic field. For large QZE, the overall structure of the phase diagram is the same as for an ideal spin 1 gas, although the precise locations of the phase boundaries are significantly shifted by interactions. For small QZE, antiferromagnetic interactions qualitatively change the phase diagram with respect to the ideal case, leading for instance to condensation in m_F=± 1, a phenomenon that cannot occur for an ideal gas with q>0.

Stepwise Bose-Einstein condensation in a spinor gas
    F. Gerbier
    December 30, 2023
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Multi-component quantum fluids described by a vector or tensor order parameter are often richer than their scalar counterparts. Examples in condensed matter are superfluid ^3He <cit.> or some unconventional superconductors with spin-triplet Cooper pairing <cit.>. In atomic physics, spinor Bose-Einstein condensates (BEC) with several Zeeman components m_F inside a given hyperfine spin F manifold can display non-trivial spin order at low temperatures <cit.>. The macroscopic population of the condensate enhances the role of small energy scales that are negligible for normal gases. This mechanism (sometimes termed Bose-enhanced magnetism <cit.>) highlights the deep connection between Bose-Einstein condensation and magnetism in bosonic gases, and raises the question of the stability of spin order against temperature.

In simple cases, magnetic order appears as soon as a BEC forms. Siggia and Ruckenstein <cit.> pointed out for two-component BECs <cit.> that a well-defined relative phase between the two components implies a macroscopic transverse spin. BEC and ferromagnetism then occur simultaneously, provided the relative populations can adjust freely. A recent experiment confirmed this scenario for bosons with spin-orbit coupling <cit.>. This conclusion was later generalized to spin-F bosons without <cit.> or with spin-independent <cit.> interactions. These results indicate that without additional constraints, bosonic statistics favors ferromagnetism.

In atomic quantum gases with F>1/2, this type of ferromagnetism competes with spin-exchange interactions, which may favor other spin orders such as spin-nematics <cit.>. 
Spin-exchange collisions can redistribute populations among the Zeeman states <cit.>, but are also invariant under spin rotations. The allowed redistribution processes are therefore those preserving the total spin, such as 2× (m_F=0) ↔ (m_F=+1)+(m_F=-1).


For an isolated system driven to equilibrium only by binary collisions (in contrast with solid-state magnetic materials <cit.>), and where magnetic dipole-dipole interactions are negligible (in contrast with dipolar atoms <cit.>), the longitudinal magnetization m_z is then a conserved quantity. This conservation law has deep consequences on the thermodynamic phase diagram.

The thermodynamics of spinor gases with conserved magnetization has been extensively studied theoretically using various assumptions and methods  <cit.>. A generic conclusion is that Bose-Einstein condensation occurs in steps, where BEC occurs first in one specific component and magnetic order appears at lower temperatures when two or more components condense. 
Natural questions are the number of steps that can be expected, and the nature of the magnetic phases realized at different temperatures. 


In this Letter, we report on the observation of multi-step condensation in an antiferromagnetic F=1 condensate of sodium atoms. Fig. <ref> illustrates four situations that occur when lowering the temperature starting from a normal Bose gas. Without loss of generality, we focus in this work on the case of positive magnetization, given that the case of m_z < 0 can be deduced by symmetry. In all cases with m_z≠ 0, we find a sequence of transitions where different Zeeman components condense at different temperatures. Depending on the applied magnetic field B and on the magnetization, we find either two or three condensation temperatures. The purpose of this paper is to explore this rich landscape of transitions in a bosonic spinor system and to elucidate the role of atomic interactions.



The present work is to the best of our knowledge the first comprehensive measurement of thermodynamic properties of spinor condensates with conserved magnetization. 
Previous experimental works exploring finite temperatures in spinor gases mostly studied spin dynamics in thermal gases <cit.>, or demonstrated cooling of a majority Zeeman component by selective evaporation of the minority components <cit.>. The realization of dipolar spinor gases with free magnetization <cit.> was limited to the study of spin-polarized condensed phases in equilibrium due to dipolar relaxation. More recently, a gas of spin excitations in a spin-polarized (m_z ≈ 1) ferromagnetic Bose-Einstein condensate was observed to equilibrate and even condense at sufficiently low temperatures <cit.>.





Our experiments are performed with ultracold ^23Na atoms confined in a crossed optical dipole trap (ODT). The longitudinal magnetization m_z=(N_+1-N_-1)/N acts as an external control parameter independent of the externally applied magnetic field B. Here, N_m_F is the reduced population in Zeeman state m_F and N the total atom number. We vary m_z between unmagnetized (m_z≈ 0) and fully magnetized samples (m_z≈ 1) using a preparation sequence performed far above T_c <cit.>. An applied magnetic field B shifts the single-atom energy by Δ E_m_F=p m_F+q(m_F^2-1). The conservation of magnetization makes the linear Zeeman effect ∝ p irrelevant in the equilibrium state. The quadratic Zeeman energy (QZE), which lowers the energy of m_F=0 with respect to m_F=±1, is the relevant term, and is given by  q= α_q B^2 with α_q/h ≈ 277Hz/G^2 for sodium atoms. 

The depth V_0 of the ODT determines the temperature T and total atom number N for a given V_0. We find that the magnetization m_z also varies with V_0 (by up to 15%), a byproduct of evaporative cooling. Once a condensate forms in one of the Zeeman components, evaporation tends to eliminate preferentially atoms in the other Zeeman states. The evaporative cooling dynamics is very slow compared to the microscopic thermalization time on which the gas returns to thermal equilibrium. As a result, the kinetic equilibrium state for the quantum gases studied in this work is still determined by a magnetization-conserving Hamiltonian. Furthermore, the ODT is tight enough such that a condensate forms in the so-called single-mode regime <cit.>, where the spatial shape of the condensate wavefunction is independent of the Zeeman state. In the following, we characterize our data for a given value of q by an evaporation “trajectory” (N,T,m_z)_V_0, taking four experimental realizations for each point in the trajectory. 







Absorption images as shown in Fig. <ref> are recorded after 3ms of expansion in an applied magnetic field gradient <cit.>. We perform a fit to a bimodal distribution for each component to extract the temperature, the populations N_m_F, and the condensed fraction f_c,m_F per component <cit.>. We found that low condensed fractions < 5 % are difficult to detect with the fit algorithm due to a combination of low signal-to-noise ratio and the complexity of fitting the three Zeeman components simultaneously. The signature of BEC, the appearance of a dense, narrow peak near the center of the atomic distribution, can instead be tracked by monitoring the peak optical density (OD) taken as a proxy for the condensed fraction <cit.>. This procedure avoids relying on bimodal fits or other indirect analyses with uncontrolled systematic biases. 

Fig. <ref> shows such a measurement for a particular evaporation trajectory. The peak OD increases sharply when Bose-Einstein condensation is reached, demonstrating in this particular example a two-step condensation where m_F=+1 condenses first, followed by m_F=0. For a given evaporation trajectory, we identify the critical trap depth V_ 0,c where condensation is reached by a piece-wise linear fit to the data, taking the intercept point as the experimentally determined V_ 0,c (see Fig. <ref>). We interpolate numerically the atom number, magnetization and temperature to obtain the critical values N_ c, T_ c, m_z, c from V_ 0,c.

Fig. <ref> summarizes the results of this work. We show the peak optical density for each Zeeman component and each value of q in a (T-m_z) plane (Fig. <ref> a-c, e-g and i-k). In this plot, all data taken at a given QZE q are binned with respect to magnetization and temperature. The domains where condensation occurs appear in light colors. For convenience, the temperature is scaled to the critical temperature of a single-component ideal gas
k_B T_c,id=ħω[N/ζ(3)]^1/3, with ω the geometric average of the trap frequencies and ζ the Riemann zeta function <cit.>. The same plot also shows the measured critical temperatures (Fig. <ref> d, h, l)[In one case, m_F=0 when m_z ≈ 0.3 and q/h=2.8Hz, the lowest temperature images do show a condensed component but the critical temperature could not be extracted reliably from the fitting procedure due to sparse sampling. This particular point is not reported in Fig. <ref>l.]. The phenomenon of sequential condensation is always observed for m_z ≠ 0, but the overall behavior changes drastically with q.

We first discuss the cases with largest QZE, q/h≈ 8.9kHz (Fig. <ref> a-d) and q/h≈ 69Hz (Fig. <ref> e-h). 
For q/h≈ 8.9kHz and highly magnetized samples, the majority component m_F=+1 condenses first at a critical temperature T_c,1, followed by the m_F=0 component at a lower temperature T_c,2. For low magnetizations, the condensation sequence is reversed. For q/h≈ 69Hz, we observe only one sequence, a two-step condensation with m_F=+1 first and m_F=0 second.


This behavior can be understood qualitatively from ideal gas theory, taking the QZE and the conservation of magnetization into account <cit.>. For ideal gases, BEC occurs when the chemical potential μ equals the energy of the lowest single-particle state <cit.>. The same criterion holds for a spin 1 gas with μ_0=μ and μ_± 1=μ±λ, where λ is a Lagrange multiplier enforcing the conservation of m_z. For m_z =0 (λ=0) and q >0, the QZE lowers the energy of m_F=0, which is therefore the first component to condense when μ=-q. For m_z > 0, λ is positive and increases with m_z. The energetic advantage of m_F=0 is in balance with the statistical trend favoring the most populated component m_F=+1. Eventually, this trend takes over at a “critical” value m_z^∗ (where λ=q). For m_z>m_z^∗, the m_F=+1 component condenses first.

Coexisting m_F=0 and m_F=± 1 components with a well-defined phase relation correspond to a non-zero transverse spin ⟨Ŝ_x+iŜ_y ⟩≠ 0 (“transverse magnetized” phase – M_⊥). For large q, the condensate is reduced to an effective two-component system m_F=0,+1 with m_F=-1 mostly spectator. The case m_z=m_z^∗ (μ_0=μ_+1) realizes the Siggia-Ruckenstein (S-R) scenario, where condensation and ferromagnetic behavior appear simultaneously. Away from that point, the S-R picture breaks down (μ_0≠μ_+1) and sequential condensation takes place. 

Figure <ref> d-h show the critical temperatures and compare them to ideal gas theory. Although the general trends in the theory are the same as in the experiment, we observe a systematic shift of T_c,1 and T_c,2 towards lower temperatures, and an experimental “critical” m_z^∗∼ 0.3 larger than the ideal gas prediction. The behavior for q/h≈ 69Hz (Fig. <ref> e-h) is qualitatively similar to the largest q case, but with a small m_z^∗ that cannot be resolved experimentally (the ideal gas theory predicts ≈ 0.002).


Repulsive interactions between the atoms can be expected to lower the critical temperatures as in single-component gases <cit.>, with an enhanced shift of T_ c,2 due to the presence of a condensate. 
We use a simplified version of Hartree-Fock (HF) theory to make quantitative predictions <cit.>. Our self-consistent calculations include the trap potential in a semi-classical approximation, and treat the interactions as spin-independent. These approximations are valid only above T_ c,2, where at most one component condenses <cit.>. As a result, the HF model cannot make any prediction for the low-temperature behavior below T_ c,2. The results of the HF calculations, performed for atom numbers and trap frequencies matching the experimental values <cit.>, are shown in Figure <ref>. The HF model qualitatively accounts for the experimental data, explaining in particular the strong downwards shift of T_ c,2 for all q and the shift of m_z^∗ to higher values for q/h≈ 8.9kHz. 
The residual discrepancy around 7-8 % could be partially explained by finite-size and trap anharmonicity effects not included in the Hartree-Fock calculation <cit.>.

At the lowest field we studied, q/h≈ 2.8Hz (Fig. <ref> i-l), we observe a change in the nature of T_ c,2. For high values of m_z, T_ c,2 corresponds to condensation into m_F=-1 while m_F=0 remains uncondensed. 
This phenomenon is incompatible with ideal gas theory <cit.> and with our HF model with spin-independent interactions. It corresponds to a change of the magnetic ordering appearing below T_ c,2. While coexisting m_F=0 and m_F=+1 components form a M_⊥ phase with ⟨Ŝ_x+iŜ_y ⟩≠ 0, coexisting m_F=± 1 components correspond to a phase with ⟨Ŝ_x+iŜ_y ⟩ = 0 but where the spin-rotational symmetry around z is broken by a non-zero spin-quadrupole tensor (“quasi-spin nematic” phase -qSN). 
At T=0 and in the single-mode regime, the M_⊥- qSN transition occurs at a critical magnetization m_z,c=√(1-[1-(q/U_s)]^2), with U_s ≤ q the spin-dependent interaction energy <cit.>. When q>U_s, there is no phase transition and only the M_⊥ phase is present. This explains the qualitative difference between the data for q/h=2.8Hz and the other two values. We estimate U_s/h ≲ 50Hz and m_z,crit≈ 0.3 for a BEC without thermal fraction <cit.>. This agrees well with the lowest temperature measurements reported in Fig. <ref>j-k. 

In the experimental data in Fig. <ref> i-l, the region of the phase diagram occupied by the M_⊥ phase shrinks with increasing temperature. In fact, we find that m_F=-1 condenses at T_c,2 for all parameters we have explored, with m_F=0 condensing at a third, lower critical temperature (except for m_z ≈ 0, where all components appear to condense together within the accuracy of our measurement). 

Finally, the dashed line in Fig. <ref>k shows T_ c,2 predicted by the HF model with spin-independent interactions. Although the model incorrectly predicts that m_F=0 should condense below T_ c,2, the predicted transition closely matches the observed boundary between single-component m_F=+1 BEC and qSN m_F= ± 1 BEC. This indicates that the transition line itself (but not the magnetic order below it) is determined by the thermal component alone.


In conclusion, we have studied the finite-T phase diagram of a spin-1 Bose gas with antiferromagnetic interactions. For condensates in the single-mode regime, we observed a sequence of transitions, two for high QZE and three for low QZE, with the lower two leading to different magnetic orders. We have found that a simplified HF model reproduces the trends observed in the variations of the critical temperatures T_ c,1 and T_ c,2 with magnetization and QZE. A more complete theoretical analysis accounting for all experimental features –in particular the harmonic trap, which is crucial to stabilize an antiferromagnetic condensate in a single spatial mode <cit.>– and elucidating the exact nature of the low-temperature transitions for low QZE remains open. A natural extension of this work would be to study the critical properties of the observed finite-T transitions, in particular near m_z=m_z^∗ and between the M_⊥ and qSN phases at very low q. Two-dimensional systems provide another intriguing direction to explore. Several Berezinskii-Kosterlitz-Thouless transitions mediated either by vortices or spin textures have been predicted <cit.>. We expect that such topological features will further enrich the already complex phase diagram observed in three dimensions.


We acknowledge stimulating discussions with B. Evrard, L. De Sarlo, E. Witkowska, J. Beugnon, L. de Forges de Parny, A. Rançon and T. Roscilde. This work has been supported by ERC (Synergy grant UQUAM). TZ acknowledges funding from the Hamburg Center for Ultrafast Imaging, and KJG from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 701894.

















