Spin-domain formation in spinor Bose-Einstein condensation

Abstract

We observe multi-step condensation of sodium atoms with spin F = 1, where the different Zeeman components mF = 0, ±1 condense sequentially as the temperature decreases. The precise sequence changes drastically depending on the magnetization mz 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 mF = ±1, a phenomenon that cannot occur for an ideal gas with q > 0. 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 3 He [1] or some unconventional superconduc-tors with spin-triplet Cooper pairing [2]. 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 [3-6]. The macroscopic population of the condensate enhances the role of small energy scales that are negligible for normal gases. This mechanism (some-times termed Bose-enhanced magnetism [6]) 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 [7] pointed out for two-component BECs [7] 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 [8]. This conclusion was later generalized to spin-F bosons without [9] or with spin-independent [10] 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 [6]. Spin-exchange collisions can redistribute populations among the Zeeman states [11-13], 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 [14]), and where magnetic dipole-dipole interactions are negligible (in contrast with dipolar atoms [15]), 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...

Structure of the perturbation series of the spin-1 Bose gas at low temperatures

“…Due to the rotational symmetry in the coordinate space the Green's functions (10) and all the correlation functions (14) and (16) depend only on the modulus of the momentum.…”

“…The expression (10) is the generalization of the well known normal and anomalous Green's functions in case of a complex scalar field [40,30]. The propagators (10) are periodic in τ with period βℏ so their Fourier series can be defined as…”

“…The theoretical investigation of spinor systems is growing rapidly these days concerning the ground state structure and symmetry braking [6][7][8][9], the various phases in traps [10,11], the different vortex states [12,13], the exploration of collective excitations [14][15][16][17] and a number of other problems [18][19][20][21][22][23][24][25][26][27]. Compared to the BEC realized in magnetic traps [28,29], where the spin degree of freedom of the particle field is frozen, such systems, since the spinor nature of the particles is preserved, have a wider variety of excitations including spin density waves, transverse spin waves or quadrupolar spin waves.…”

Spinor Condensates and Light Scattering from Bose-Einstein Condensates

“…no barrier) Goldstone excitation mode. Recent calculations by Isoshima, Machida and Ohmi [108] confirm the disappearance of the energy barrier when the populations in the |m F = ±1 states become equal. These changes of effective interaction parameters in the presence of the |m F = −1 state provide a mechanism for the variation of the tunneling rates with magnetic fields.…”

“…Thus, our time-offlight imaging technique cannot properly characterize the boundary between neighboring spin domains. In future work, it would be interesting to examine such boundaries with an in situ imaging technique, perhaps to observe the spatial structures recently predicted by Isoshima, Machida and Ohmi [108]. It is interesting to note that the radial expansion of the cigar-shaped condensate occurs at a velocity near the speed of Bogoliubov sound, which describes the propagation of density waves, while the axial expansion of a spin domain boundary occurs at a "spin sound velocity" which would describe the propagation of spin waves.…”