Vortices and Dynamics in Trapped Bose-Einstein Condensates

I review the basic physics of ultracold dilute trapped atomic gases, with emphasis on Bose-Einstein condensation and quantized vortices. The hydrodynamic form of the Gross-Pitaevskii equation (a nonlinear Schrödinger equation) illuminates the role of the density and the quantum-mechanical phase. One unique feature of these experimental systems is the opportunity to study the dynamics of vortices in real time, in contrast to typical experiments on superfluid ⁴He. I discuss three specific examples (precession of single vortices, motion of vortex dipoles, and Tkachenko oscillations of a vortex array). Other unusual features include the study of quantum turbulence and the behavior for rapid rotation, when the vortices form dense regular arrays. Ultimately, the system is predicted to make a quantum phase transition to various highly correlated many-body states (analogous to bosonic quantum Hall states) that are not superfluid and do not have condensate wave functions. At present, this transition remains elusive. Conceivably, laser-induced synthetic vector potentials can serve to reach this intriguing phase transition.     

3-Dimensional Dynamics of Vortices in a rotating Bose-Einstein Condensate

No Heading We numerically integrate the 3-D time-dependent Gross-Pitaevskii equation suggested by Tsubota et al. [Phys.Rev.A, 65, 023603(2002)] to investigate dynamical transitions of the trapped Bose-Einstein condensate (BEC) from an initial non-vortex state to a steady vortex lattice state after the trigger of the rotation. We use the splitting-fourier method as a numerical scheme to perform stable 3-D numerical simulations with high computational efficiency. We present three stages of characteristic 3-D non-steady dynamics of vortices, i.e., the vortex penetration, the stochastic vortex wandering, the vortex lattice vibration, etc.PACS numbers: 03.75.Lm, 03.75.Kk     

Toward a tomographic picture of a Bose-Einstein condensate

The possibilities of applying tomographic techniques to a Bose-Einstein condensate to reconstruct its ground state are investigated by means of numerical simulations. Two situations for which the density-matrix elements can be retrieved from atom counting probabilities are considered. The methods presented here allow one to distinguish among various possible quantum states.     

Dynamical Properties of Vortices in a Bose-Einstein Condensate in a Rotating Lattice

We present simulation results of the vortex dynamics in a trapped Bose-Einstein condensate in the presence of a rotating optical lattice. Changing the potential amplitude and the relative rotation frequency between the condensate and the optical lattice, we find a rich variety of dynamical phases of vortices. In particular, when the optical lattice rotates faster than the condensate, the competition between the pinning force and the interactions by nucleated interstitial vortices leads to the melting of vortex lattice, yielding a vortex liquid phase.     

Probing first-order spatial coherence of a Bose-Einstein condensate

Abstract

We probe the spatial coherence properties of a magnetically trapped Bose gas. Two matter wave beams are extracted from two spatially separated regions of the trap and overlap outside the trapping region. The visibility of the resulting interference pattern measures the phase coherence between the regions of extraction. By varying the spatial separation between the two regions the first-order spatial correlation function of the trapped Bose gas can be measured. The location of the minima of the interference pattern is reproducible, which experimentally confirms that the trapped Bose-Einstein condensate is not fragmented into individual condensates.     

Collision Dynamics of Non-Abelian Vortices in Spinor Bose-Einstein Condensates

Bose-Einstein condensates with spin degrees of freedom admit various topological phases and excitations. Non-Abelian vortices offer a remarkable example which is expected to be realized in the cyclic phase of the spin-2 spinor Bose-Einstein condensates. We demonstrate that the non-Abelian property of non-Abelian vortices shows the unique effect in their collision dynamics, i.e., unlike Abelian vortices, they do neither reconnect themselves nor pass through each other but create a rung vortex between them.     

Boojums Connecting Singular and Continuous Vortices in Two-Component Bose-Einstein Condensates

A boojum is a topological defect that can form only on the surface of an ordered medium such as superfluid ³He and liquid crystals. By analogy with superfluid ³He, we numerically create boojums between two phases with different vortex structures in two-component BECs where the intercomponent interaction is spatially dependent. The detailed structure of the boojums is revealed by investigating its density distribution, effective superflow vorticity and pseudospin texture.     

Dynamics of Vortices and Solitons in a Bose-Einstein Condensate by an Oscillating Potential

We study numerically the dynamics of quantized vortices and solitons induced by an oscillating potential inside a trapped Bose-Einstein condensate. The dynamics of the topological defects is much different from the case for a linear uniform moving object; the metamorphosis between vortices and solitons is characteristic of the dynamics. We discuss how vortices are nucleated by an oscillating potential.     

Dynamics of Quantized Vortices in Bose-Einstein Condensates Trapped in Precessing Potential

We study numerically and analytically dynamics of Bose-Einstein condensates trapped in precessing potential. The precessing potential has two frequencies because the trapping potential rotates around one axis with one frequency, and this axis rotates around the other axis with the other frequency. In this paper, we treat a situation in which the second frequency is much smaller than the first. The vortex lattice formed by the first rotation rotates around the second axis, following the rotating axis. We find numerically an interesting steady state in which the vortex lattice lags by a finite angle behind the rotating axis. We discuss analytically the origin of this steady state.     

Dynamics of Quantized Vortices in Superfluid Helium and Rotating Bose-Einstein Condensates

No Heading We study theoretically and numerically the dynamics of quantized vortices in superfluid helium and rotating Bose-Einstein condensates. After reviewing briefly the recent motivation, we discuss these topics with the emphasis on the research done by our group. One of the modern important problems is how superfluid turbulence relates to classical turbulence. First, we show that superfluid turbulence consisting of a vortex tangle has an energy spectrum consistent with the Kolmogorov law. Second, we discuss the vortex states that appear in a rotating channel with counterflow. In the field of atomic-gas Bose-Einstein condensation, the dramatic observations of quantized vortices were made for rotating condensates. By solving numerically the Gross-Pitaevskii equation, we found a whole story of the vortex lattice formation consistent with the observations.PACS numbers: 67.40.Vs, 47.32.Cc, 47.37.+q.     

Connection of Vortices Between Spatially Different Phases in Two-Component Bose-Einstein Condensates

We study interfacial topological defects called boojums, a vortex ending or a connecting point of two kinds of vortex cores, in rotating two-component Bose-Einstein condensates. First, we show that the boojum exists at a vortex ending that connects to the interface of the strongly phase-separated condensates. Next, we study various boojums appearing between two phases characterized by different vortex structures, where the intercomponent s-wave scattering lengths are spatially varied. Using three-dimensional simulations of the Gross-Pitaevskii equations, we reveal the detailed structure of the boojums by visualizing its density distribution and effective superflow vorticity.     

Dynamical Instability of Coreless Vortices in F=2 Spinor Bose-Einstein Condensates

We theoretically investigate the low-lying excitation spectra of coreless vortex states in Bose-Einstein condensates (BECs) with F=2 hyperfine spin degrees of freedom. Here, we extend the previous work in F=1 spinor BEC. In addition to the dynamical instabilities in F=1 coreless vortex states, we find an another set of dynamical instabilities due to different hyperfine spin interaction. The calculation is carried out in the possible parameter space of the F=2 ⁸⁷Rb atom. This study assists interpretation of experimental data and presents a general characteristics of the dynamical instability of F=2 hyperfine spin system. whether the ground state of the spin interaction is in the cyclic or polar phase. Our study can encourages the experiments to examine our results.     

The enhancement of spontaneous and induced transition rates by a Bose-Einstein condensate

Abstract

In this paper, an exactly solved model for the emission by N atoms is presented, the spontaneous and induced transition rates obtained, are enhanced by a factor which is proportional to the number of atoms n in the volume Λ³(2π²) (A is the transition wavelength of the atom) and dependent on the de-Broglie wavelength Λ_B in a more complicated way.     

Generation of Vortices and Observation of Quantum Turbulence in an Oscillating Bose-Einstein Condensate

We report on the experimental observation of vortex formation and production of tangled vortex distribution in an atomic BEC of ⁸⁷Rb atoms submitted to an external oscillatory perturbation. The oscillatory perturbations start by exciting quadrupolar and scissors modes of the condensate. Then regular vortices are observed finally evolving to a vortex tangle configuration. The vortex tangle is a signature of the presence of a turbulent regime in the cloud. We also show that this turbulent cloud has suppression of the aspect ratio inversion typically observed in quantum degenerate bosonic gases during free expansion.     

Precursor Phenomena of Nucleations of Quantized Vortices in the Presence of a Uniformly Moving Obstacle in Bose-Einstein Condensates

We investigate excitations and fluctuations of Bose-Einstein condensates in a two-dimensional torus with a uniformly moving Gaussian potential by solving the Gross-Pitaevskii equation and the Bogoliubov equation. The energy gap Δ between the current-flowing metastable state (that reduces to the ground state for sufficiently slowly-moving potential) and the first excited state vanishes when the moving velocity v of the potential approaches a critical velocity $v_{\rm c}$ (>0). We find a scaling law $\Delta\propto(1-|v|/v_{\rm c})^{\frac{1}{4}}$ , which implies that a characteristic time scale diverges toward the critical velocity. Near the critical velocity, we show that low-energy local density fluctuations are enhanced. These behaviors can be regarded as precursor phenomena of the vortex nucleation.     

Velocity of sound in a Bose-Einstein condensate in the presence of an optical lattice and transverse confinement

No Heading We study the effect of the transverse degrees of freedom on the velocity of sound in a Bose-Einstein condensate immersed in a one-dimensional optical lattice and radially confined by a harmonic trap. We compare the results of full three-dimensional calculations with those of an effective 1D model based on the equation of state of the condensate. The perfect agreement between the two approaches is demonstrated for several optical lattice depths and throughout the full crossover from the 1D mean-field to the Thomas Fermi regime in the radial direction.     

Surface tension of liquid4He. Surface energy of the Bose-Einstein condensate

The surface tension of liquid⁴He was measured by means of surface-wave resonance, and the relative variation showed excellent agreement with that previously obtained by a precise capillary-rise method. The absolute value of the surface tension at absolute zero was measured as 354.4±0.5 mdyne/cm, 6% smaller than the previous value. The surface energy associated with the Bose-Einstein condensate wave function was found to dominate the surface tension in the superfluid phase. The condensate fraction other than in the vicinity of the point was estimated asn ₀(0)-n ₀(T)=A(T/T ), withn ₀(0)=0.125±0.025,A=0.177±0.047, and =5.07±0.17.     

Vortices in superconductors

In type II superconductors where the London penetration depthλ is larger than the coherence lengthξ, there is a possibility of flux penetration inside the sample for magnetic field greater than \(H_{0_1 } \left( { = \frac{{\phi _0 }}{{4\pi \lambda ^2 }}ln \lambda /\xi , \phi _0 = \frac{{hc}}{{2e}}} \right).\) The flux penetrates in the form of vortices with core of sizeξ. However these vortices differ from those in superfluid He⁴ in variation of currentj(r) circulating around them. For superconductorsj(r) ～ 1/r only up to a distanceλ and then it falls exponentially whilev(r) ～ 1/r for all distances in superfluids. The reason is that in superconductors vortex carries a magnetic flux which is screened by conduction electrons. This coupling of order parameter field (the pairing wavefunction) with the gauge field has many interesting implications for superconductors and for non-Abelian gauge theories. Some examples are as follows:

1. The energy of the vortices is reduced. The energy of vortex of lengthL (ind = 3 sample) is of orderL lnL for a superfluid, is of orderL in a superconductor, and (in ad = 2 sample) the energy of a vortex point which diverges like lnR (whereR is the size of the sample) in a superfluid becomes finite in a superconductor.
2. The superconducting-normal transition in three dimension is very weak first order, because the fluctuations of the gauge field, when summed over, add to Ginzburg Landau free energy a term proportional to |ψ|³, whereψ is the order parameter.
3. Because of the lnr behaviour of interaction energy of vortices, a two-dimensional superfluid sample can exhibit a Kosterlitz-Thouless type transition whereas a strictlyd = 2 superconductors should not have any. However for dirty superconducting films whereλ is large vortex binding-unbinding transition can be observed with quite a rich phase diagram.

The paper presented at the discussion meeting discusses the above in detail. Here we give only a brief summary of results and some relevant references.     

Pancake Vortices

I describe the magnetic-field and current-density distributions generated by two-dimensional (2D) pancake vortices in infinite, semi-infinite, and finite-thickness stacks of Josephson-decoupled superconducting layers. Arrays of such vortices have been used to model the magnetic structure in highly anisotropic layered cuprate high-temperature superconductors. I show how the electromagnetic forces between pancake vortices can be calculatated, and I briefly discuss the effects of interlayer Josephson coupling.