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...

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Focusing on a quantum-limit behavior, we study a single vortex in a clean s-wave type-II superconductor by selfconsistently solving the Bogoliubov-de Gennes equation. The discrete energy levels of the vortex bound states in the quantum limit is discussed. The vortex core radius shrinks monotonically up to an atomic-scale length on lowering the temperature T , and the shrinkage stops to saturate at a lower T . The pair potential, supercurrent, and local density of states around the vortex exhibit Friedel-like oscillations. The local density of states has particle-hole asymmetry induced by the vortex. These are potentially observed directly by STM. PACS number(s): 74.60. Ec, 61.16.Ch, Growing interest has been focused on vortices both in conventional and unconventional superconductors from fundamental and applied physics points of view. This is particularly true for high-T c cuprates, since it is essential that one understands fundamental physical properties of the vortices in the compounds to better control various superconducting characteristics of some technological importance. Owing to the experimental developments, it is not difficult to reach low temperatures of interest where distinctive quantum effects associated with the discretized energy levels of the vortex bound states are expected to emerge. The quantum limit is realized at the temperature where the thermal smearing is narrower than the discrete bound state levels [1]: T /T c ≤ 1/(k F ξ 0 ) with ξ 0 =v F /∆ 0 the coherence length (∆ 0 the gap at T = 0) and k F (v F ) the Fermi wave number (velocity). For example, in a typical layered type-II superconductor NbSe 2 with T c = 7.2 K and k F ξ 0 ∼ 70, the quantum limit is reached below T < 50 mK. As for the high-T c cuprates, the corresponding temperature is rather high: T < 10 K for YBa 2 Cu 3 O 7−δ (YBCO).Important microscopic works to theoretically investigate the quasiparticle spectral structure around a vortex in a clean limit are put forth by Caroli et al. The purposes of the present paper are to reveal the quantum-limit aspects of the single vortex in s-wave superconductors and to discuss a possibility for the observation of them. The present study is motivated by the following recent experimental and theoretical situations: (1) The socalled Kramer and Pesch (KP) effect [1,3,7,8]; a shrinkage of the core radius upon lowering T (to be exact, an anomalous increase in the slope of the pair potential at the vortex center at low T ) is now supported by some experiments [9]. The T dependence of the core size is studied by µSR on NbSe 2 and YBCO [10], which is discussed later. The KP effect, if confirmed, forces us radically alter the traditional picture [11] for the vortex line such as a rigid normal cylindrical rod with the radius ξ 0 . (2) The scanning tunneling microscopy (STM) experiment on YBCO by Maggio-Aprile et al. [12], which enables us to directly see the spatial structure of the low-lying quasiparticle excitations around the vortex, arouses much interest. They claim that surprisingly enough, ...

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We study the vortex structure and its field dependence within the framework of the quasi-classical Eilenberger theory to find the difference between the d x 2 −y 2 -and s-wave pairings. We clarify the effect of the d x 2 −y 2 -wave nature and the vortex lattice effect on the vortex structure of the pair potential, the internal field and the local density of states. The d x 2 −y 2 -wave pairing introduces a fourfold-symmetric structure around each vortex core. With increasing field, their contribution becomes significant to the whole structure of the vortex lattice state, depending on the vortex lattice's configuration. It is reflected in the form factor of the internal field, which may be detected by small angle neutron scattering, or the resonance line shape of µSR and NMR experiments. We also study the induced s-and dxy-wave components around the vortex in d x 2 −y 2 -wave superconductors.