The Bogoliubov theory is extended to a Bose-Einstein condensation with internal degrees of freedom, realized recently in 23 Na gases where several hyperfine states are simultaneously cooled optically. Starting with a Hamiltonian constructed from general gauge and spin rotation symmetry principles, fundamental equations for condensate are derived. The ground state where time reversal symmetry is broken in some cases and low-lying collective modes, e.g. spin and density wave modes, are discussed. Novel vortex as a topological defect can be created experimentally. 4 He. One of the most notable differences in these two systems lies in their mutual interaction, either very weak in the former or very strong in the latter system. This enables us to construct a microscopic theory for BEC in the present system from first principles. In fact a mean-field theory or HartreeFock-Bogoliubov framework developed by Bogoliubov, 6)Gross, 7) Pitaevskii 8) and others has been quite successful in explaining fundamental physics of BEC realized in magnetically trapped atomic gases such as condensate fraction, transition temperature, or low-lying collective modes. Only the s-wave scattering length a is required to be fed into these microscopic theories as a material parameter, which itself is known rather accurately. 4, 5)Most experiments for BEC so far were performed by the magnetic traps which necessarily select one of several possible atomic ground states, or the so-called weak field seeking state, such as F z = −1 among F z = ±1, 0 of F = 1 in the 23 Na case, where the spin degrees of freedom are "frozen". Recently Stamper-Kurn et al.9) have succeeded in cooling 23 Na atoms purely optically in an optical dipole trap and achieved BEC, where the three substates F z = ±1, 0 are simultaneously "Bose-condensed". They demonstrated this by a "Stern-Gerlach" experiment: The F = 1 magnetic three sublevels are all visible as a split image by passing the condensate through a field gradient. This is remarkable. This finding opens an interesting possibility to explore BEC with internal degrees of freedom where not only the gauge symmetry U (1) but also "spin" symmetry SO(3) for the F = 1 case are involved, a situation similar to the superfluid 3 He problem where the orbital (l = 1) and spin (s = 1) degrees of freedom give rise to a 3 × 3 × 2 dimensional manifold of complex order parameter space. 10)The purposes of this paper are (1) to establish fundamental equations for describing such a situation, leading to an extended Gross-Pitaevskii equation, (2) to examine low-lying collective modes in order to extract basic physical properties of the ground state and finally (3) to point out several interesting possible topological excitations or textural spatial structures under special circumstances, e.g. when releasing BEC from an optically plugged quadrupole magnetic trap, a novel vortex may be created experimentally. Till date, no one has succeeded in observing a vortex even for one-component BEC although there have been several theoretical discuss...

It is argued that the pairing symmetry realized in a ferromagnetic superconductor UGe(2) must be a nonunitary triplet pairing. This particular state is free from the Pauli limitation and can survive under a huge internal molecular field. To check our identification we examine its basic properties and several experiments are proposed. In particular, the external field is used to raise T(c) by controlling the internal spontaneous dipole field.

It is shown theoretically that a persistent current can be continuously created in a Bose-Einstein condensate (BEC) of alkali atoms confined in a multiply connected region by making use of a spindegree of freedom of the order parameter of a BEC. We demonstrate that this persistent current is easily transformed into a vortex. Relaxation processes of these BEC after the confining field is turned off are also studied so that our analyses are compared with time of flight experiments. The results are shown to clearly reflect the existence of a persistent current.

We consider the splitting mechanism of a multiply charged vortex into singly charged vortices in a Bose-Einstein condensate confined in a harmonic potential at zero temperature. The Bogoliubov equations support unstable modes with complex eigenfrequencies (CE modes), which cause the splitting instability without the influence of thermal atoms. The investigation of the excitation spectra shows that the negative-energy (NE) mode plays an important role in the appearance of the CE modes. The configuration of vortices in splitting is determined by the angular momentum of the associated NE mode. This structure has also been confirmed by the numerical simulation of the time-dependent Gross-Pitaevskii equation.ized order parameter = a HO / ͱ N, where a HO is the har-*Electronic address: yuki@scphys.kyoto-u.ac.jp PHYSICAL REVIEW A 70, 043610 (2004)

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...

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The orbital symmetry of the superconducting order parameter in UPt3 is identified by evaluating the directionally dependent thermalconductivity and ultrasound attenuation in the clean limit and compared with the existing data for both basal plane and the c-axis of a hexagonal crystal. The resulting two component orbital part expressed by (λx(k), λy(k)) is combined with the previously determined triplet spin part, leading to the order parameter of either the non-unitary bipolar state of the form: d(k) = ˆ bλx(k) + i ˆ jλy(k) or the unitary planar state of the form: d(k) = ˆ bλx(k) + ˆ jλy(k) wherê b ⊥ ˆ j = ˆ c, orâorˆorâ with the hexagonal unit vectorsâvectorsˆvectorsâ, ˆ b andˆcandˆ andˆc. The d vector is rotatable in the plane spanned byâbyˆbyâ andˆcandˆ andˆc perpendicular tô b under weak applied c-axis field because of the weak spin orbit coupling. Experiments are proposed to distinguish between the equally possible these states.

Quantum vortices in the multi-component Bose-Einstein condensation (BEC) are investigated theoretically. It is found that three kinds of the vortex configurations are possible and their physical properties are discussed in details, including the density distribution and the spin texture. By using the Bogoliubov theory extended to the three component BEC, the collective modes for these vortices are evaluated. The local vortex stability for these vortices are examined in light of the existence of the negative eigenvalue, yielding a narrow magnetization window for the local intrinsic stable region where the multi-components work together to stabilize a vortex in a self-organized way.