We study the dynamics of vortex lattice formation of a rotating trapped Bose-Einstein condensate by numerically solving the two-dimensional Gross-Pitaevskii equation, and find that the condensate undergoes elliptic deformation, followed by unstable surface-mode excitations before forming a quantized vortex lattice. The origin of the peculiar surface-mode excitations is identified to be phase fluctuations at the low-density surface regime. The obtained dependence of a distortion parameter on time and that on the driving frequency agree with the recent experiments by Madison et al. [Phys. Rev. Lett. 86, 4443 (2001)].PACS numbers: 03.75. Fi, 67.40.Db Quantized vortices have long been studied in superfluid 4 He as the topological defects characteristic of superfluidity [1,2]. However, the relatively high density and strong repulsive interaction complicate the theoretical treatments of the Bose-Einstein condensed liquid, and the healing length of the atomic scale makes the experimental visualization of the quantized vortices difficult. The recent achievement of Bose-Einstein condensation in trapped alkali-metal atomic gases at ultra low temperatures has stimulated intense experimental and theoretical activity. The atomic Bose-Einstein condensates(BECs) have the weak interaction because they are dilute gases, thus being free of the above difficulties that superfluid 4 He is subject to. Quantized vortices in the atomic BECs have recently been created experimentally by Matthews et al. By rotating an asymmetric trapping potential, Madison et al. at ENS succeeded in forming a quantized vortex in 87 Rb BEC for a stirring frequency that exceeds a critical value [4]. Vortex lattices were obtained for higher frequencies. The ENS group subsequently observed that vortex nucleation occurs via a dynamical instability of the condensate [6]. For a given modulation amplitude and stirring frequency, the steady state of the condensate was distorted to an elliptic cloud, stationary in the rotating frame, as predicted by Recati et al. [7]. An intrinsic dynamical instability [8] of the steady state transformed the elliptic state into a more axisymmetric state with vortices. However, the origin of that instability and how it leads to the formation of vortex lattices remains to be investigatedThe ENS group found that the minimum rotation frequency Ω nuc at which one vortex appears is 0.65ω ⊥ , where ω ⊥ is the transverse oscillation frequency of the cigar-shaped trapping potential, independent of the number of atoms or the longitudinal frequency ω z [9]. There is a discrepancy between the observations and the theoretical considerations based on the stationary solution of the Gross-Pitaevskii equation(GPE) [10,11]; Ω nuc is significantly larger than its equilibrium estimates.The present paper addresses these issues by numerically solving the GPE that governs the time evolution of the order parameter ψ(r, t):Here g = 4πh 2 a/m is the coupling constant, proportional to the 87 Rb scattering length a ≈5.77 nm. The high anisotropy of the cig...

We review the topic of quantized vortices in multicomponent Bose–Einstein condensates of dilute atomic gases, with an emphasis on the two-component condensates. First, we review the fundamental structure, stability and dynamics of a single vortex state in a slowly rotating two-component condensates. To understand recent experimental results, we use the coupled Gross–Pitaevskii equations and the generalized nonlinear sigma model. An axisymmetric vortex state, which was observed by the JILA group, can be regarded as a topologically trivial skyrmion in the pseudospin representation. The internal, coherent coupling between the two components breaks the axisymmetry of the vortex state, resulting in a stable vortex molecule (a meron pair). We also mention unconventional vortex states and monopole excitations in a spin-1 Bose–Einstein condensate. Next, we discuss a rich variety of vortex states realized in rapidly rotating two-component Bose–Einstein condensates. We introduce a phase diagram with axes of rotation frequency and the intercomponent coupling strength. This phase diagram reveals unconventional vortex states such as a square lattice, a double-core lattice, vortex stripes and vortex sheets, all of which are in an experimentally accessible parameter regime. The coherent coupling leads to an effective attractive interaction between two components, providing not only a promising candidate to tune the intercomponent interaction to study the rich vortex phases but also a new regime to explore vortex states consisting of vortex molecules characterized by anisotropic vorticity. A recent experiment by the JILA group vindicated the formation of a square vortex lattice in this system.

We investigate the structure of vortex states in rotating two-component Bose-Einstein condensates with equal intracomponent but varying intercomponent-coupling constants. A phase diagram in the intercomponent-coupling versus rotation-frequency plane reveals rich equilibrium structures of vortex states. As the ratio of intercomponent to intracomponent couplings increases, the interlocked vortex lattices undergo phase transitions from triangular to square, to double-core lattices, and eventually develop interwoven "serpentine" vortex sheets with each component made up of chains of singly quantized vortices.

A vortex molecule is predicted in rotating two-component Bose-Einstein condensates whose internal hyperfine states are coupled coherently by an external field. A vortex in one component and that in the other are connected by a domain wall of the relative phase, constituting a "vortex molecule", which features a nonaxisymmetric (pseudo)spin texture with a pair of merons. The binding mechanism of the vortex molecule is discussed based on a generalized nonlinear sigma model and a variational ansatz. The anisotropy of vortex molecules is caused by the difference in the scattering lengths, yielding a distorted vortex-molecule lattice in fast rotating condensates.PACS numbers: 03.75. Lm, 03.75.Mn, 05.30.Jp Topological defects appear in cross-disciplinary subfields of physics as long-lived excitations, constrained by the topology of the order parameter [1]. A prime example is quantized vortices which play a key role in understanding of superfluidity [2]. When a system has a multicomponent order parameter, it is possible to excite various exotic topological defects which have no analogue in systems with a single-component order parameter.An atomic-gas Bose-Einstein condensate (BEC) offers an ideal testing ground to investigate such topological defects, because almost all parameters of the system can be controlled to the extent that state engineering is possible. Because alkali atoms have a spin degree of freedom, multicomponent BECs can be realized if more than one hyperfine spin state is simultaneously populated [3,4]. A quantized vortex and a vortex lattice in BECs have been created experimentally by several techniques [5,6,7,8]. The methods reported in Ref. [5,6] utilized internal degrees of freedom of BECs, creating unconventional vortices described by multicomponent order parameters. The structure of single vortex states in systems with multicomponent order parameters was investigated in Refs. [9,10,11]. In addition, it has been predicted theoretically that fast rotating two-component BECs exhibit a rich variety of unconventional vortex structures [11,12,13].In this Letter, we study the vortex structure of rotating two-component BECs whose internal states are coupled coherently by an external driving field. This coupling can be achieved experimentally as reported in Refs. [14,15], where Rabi oscillations between the two components were observed. If the strength of the coupling drive is increased gradually from zero and its frequency is gradually ramped to resonance, one can obtain a stationary state with a nearly equal-weight superposition of the two states [15]. Here, we study stationary states of two-component BECs with an external rotation as well as internal coupling. Combination of these two effects enables us to explore a new regime of rich vortex structures beyond the conventional binary system [9,12,13,16]; the two components interact not only through their meanfield interactions but also through the relative phase of the order parameters. We find that such two-component BECs exhibit unique vortex struct...

The energy spectrum of superfluid turbulence is studied numerically by solving the GrossPitaevskii equation. We introduce the dissipation term which works only in the scale smaller than the healing length, to remove short wavelength excitations which may hinder the cascade process of quantized vortices in the inertial range. The obtained energy spectrum is consistent with the Kolmogorov law.PACS numbers: 67.40. Vs, 47.37.+q, 67.40.Hf The physics of quantized vortices in liquid 4 He is one of the most important topics in low temperature physics [1]. Liquid 4 He enters the superfluid state at 2.17 K. Below this temperature, the hydrodynamics is usually described using the two-fluid model in which the system consists of inviscid superfluid and viscous normal fluid. Early experimental works on the subject focused on thermal counterflow in which the normal fluid flowed in the opposite direction to the superfluid flow. This flow is driven by the injected heat current, and it was found that the superflow becomes dissipative when the relative velocity between two fluids exceeds a critical value [6] in superfluid 4 He at temperatures above 1 K, and their results were consistent with the Kolmogorov law. The Kolmogorov law is one of the most important statistical laws [7] of fully developed CT, so these experiments show a similarity between ST and CT. This can be understood using the idea that the superfluid and the normal fluid are likely to be coupled together by the mutual friction between them and thus to behave like a conventional fluid [8]. Since the normal fluid is negligible at very low temperatures, an important question arises: even without the normal fluid, is ST still similar to CT or not?To address this question, we consider the statistical law of CT [7]. The steady state for fully developed turbulence of an incompressible classical fluid follows the Kolmogorov law for the energy spectrum. The energy is injected into the fluid at some large scales in the energycontaining range. This energy is transferred in the inertial range from large to small scales without been dissipated. The inertial range is believed to be sustained by the self-similar Richardson cascade in which large eddies are broken up to smaller ones, having the Kolmogorov lawHere the energy spectrum E(k) is defined as E = dk E(k), where E is the kinetic energy per unit mass and k is the wave number from the Fourier transformation of the velocity field. The energy transferred to smaller scales in the energy-dissipative range is dissipated by the viscosity with the dissipation rate, which is identical with the energy flux ǫ of Eq. (1) in the inertial range. The Kolmogorov constant C is a dimensionless parameter of order unity.In CT, the Richardson cascade is not completely understood, because it is impossible to definitely identify each eddy. In contrast, quantized vortices in superfluid are definite and stable topological defects. A BoseEinstein condensed system yields a macroscopic wave function Φ(x, t) = ρ(x, t)e iφ(x,t) , whose dynamics is g...