Quantized vortices in atomic Bose-Einstein condensates
aa r X i v : . [ c ond - m a t . o t h e r] A ug QUANTIZED VORTICES IN ATOMICBOSE-EINSTEN CONDENSATES
KENICHI KASAMATSU
Department of General Education, Ishikawa National College of Technology,Tsubata, Ishikawa 929-0392, Japan
MAKOTO TSUBOTA
Department of Physics, Osaka City University, Sugimoto 3-3-138, Osaka, Japan
Abstract
In this review, we give an overview of the experimental and theoretical advancesin the physics of quantized vortices in dilute atomic-gas Bose–Einstein condensatesin a trapping potential, especially focusing on experimental research activities andtheir theoretical interpretations. Making good use of the atom optical technique,the experiments have revealed many novel structural and dynamic properties ofquantized vortices by directly visualizing vortex cores from an image of the densityprofiles. These results lead to a deep understanding of superfluid hydrodynamics ofsuch systems. Typically, vortices are stabilized by a rotating potential created by alaser beam, magnetic field, and thermal gas. Finite size effects and inhomogeneityof the system, originating from the confinement by the trapping potential, yieldunique vortex dynamics coupled with the collective excitations of the condensate.Measuring the frequencies of the collective modes is an accurate tool for clarifyingthe character of the vortex state. The topics included in this review are the mech-anism of vortex formation, equilibrium properties, and dynamics of a single vortexand those of a vortex lattice in a rapidly rotating condensate.
Preprint submitted to Tsubota et al. (Originally based on Elsevier)29 October 2018 ontents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32. Introduction to ultra-cold atomic-gas BECs . . . . . . . . . . . . . . . . . . . . . . 42.1. General information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2. Atomic Species . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3. Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4. Manipulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.5. Basic theory of trapped BECs . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.5.1. The Gross–Pitaevskii equation . . . . . . . . . . . . . . . . . . . . . 82.5.2. The Bogoliubov–de Gennes equation . . . . . . . . . . . . . . . . 103. Vortex formation in atomic BECs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.1. Theoretical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2. Vortex formation in a stirred condensate . . . . . . . . . . . . . . . . . . . 133.2.1. Experimental scheme to rotate condensates . . . . . . . . . . . 133.2.2. Theory of vortex nucleation and lattice formation . . . . . . 153.3. Phase engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3.1. Phase-imprinting method . . . . . . . . . . . . . . . . . . . . . . . . . 193.3.2. Topological vortex formation . . . . . . . . . . . . . . . . . . . . . . 203.3.3. Stimulated Raman process . . . . . . . . . . . . . . . . . . . . . . . . 214. A single vortex in an atomic BEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.1. Equilibrium properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.2. Dynamical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.2.1. Precession and decay of an off-centered vortex . . . . . . . . 244.2.2. Vortex dynamics coupled with collective modes . . . . . . . 244.2.3. Splitting of a multiply quantized vortex . . . . . . . . . . . . . 265. A lattice of quantized vortices in an atomic BEC . . . . . . . . . . . . . . . . . . 295.1. Equilibrium properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.1.1. Lattice inhomogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.1.2. Attainment of the lowest Landau level regime . . . . . . . . . 305.2. Collective dynamics of a vortex lattice . . . . . . . . . . . . . . . . . . . . . 335.2.1. Vortex lattice dynamics coupled with collective modes . . 335.2.2. Transverse oscillation of a vortex lattice: Tkachenko mode 335.3. Vortices in an anharmonic potential . . . . . . . . . . . . . . . . . . . . . . . 355.4. Vortex pinning in an optical lattice . . . . . . . . . . . . . . . . . . . . . . . 386. Other topics and future studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386.1. A vortex in an attractively interacting BEC . . . . . . . . . . . . . . . . . 396.2. Vortices in dipolar condensates . . . . . . . . . . . . . . . . . . . . . . . . . . . 396.3. Melting state of vortex lattices: beyond the LLL regime . . . . . . . 406.4. Spontaneous vortex generation associated with phase transitions 406.5. Skyrmions in multi-component BECs . . . . . . . . . . . . . . . . . . . . . . 416.6. Vortices in Fermion condensates . . . . . . . . . . . . . . . . . . . . . . . . . . 427. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
1. Introduction
The achievement of Bose–Einstein condensation in trapped atomic gases atultra-low temperatures has stimulated intense experimental and theoreticalactivity in modern physics, as seen by the award of the Nobel Prize in Physicsin 2001 (Cornell 2002, Ketterle 2002). The Bose–Einstein condensate (BEC),a state of matter predicted by Einstein in 1925, is created by the condensationof a macroscopically large number of bosons into one of the eigenstates of thesingle-particle density matrix below the Bose–Einstein transition temperature.A remarkable consequence of the condensation is an extension of microscopicquantum phenomena into the macroscopic scale. This is an essential origin ofsuperfluidity and superconductivity, in which macroscopically extended phasecoherence allows a dissipationless current to flow.Superfluidity is closely related to the existence of quantized vortices. Forweakly interacting BECs the superfluid velocity v is given by the gradientof the phase θ of a “condensate wave function” v = ( ~ /m ) ∇ θ with the Planckconstant ~ = h/ π and particle mass m . Since the wave function remainssingle-valued, the change in the phase around a closed contour must be aninteger multiple of 2 π . Thus, the circulation Γ around a closed contour isgiven by Γ = H v · d l = ( h/m ) q ( q = 0 , , , · · · ), which shows that circulationof a vortex is “quantized” in units of h/m . Realization of weakly-interactingatomic-gas BECs has provided an ideal testing ground to study the physicsof quantized vortices; up to now, several experimental groups have reportedmany interesting results. This experimental work has been followed by con-siderable theoretical activity, leading to proposals and new problems to betackled. (For a review of the early research stages of quantized vortices, see(Fetter 2001a).)In this article, we review the physics of quantized vortices in atomic-gas BECs,especially focusing on the progress of the experimental research. We restrictourselves to arguments on trapped condensates with inhomogeneous densityprofiles. The aim of this review is to stimulate further developments of thisfield. By reflecting on the history of the current research and on unresolvedproblems, we hope to encourage researchers in low temperature physics toinvestigate quantized vortices in this system. While quantized vortices havebeen extensively studied in the field of superfluid helium (Donnelly 1991),there has been a resurgence of interest in vortices in atomic BECs becauseof the following reasons. First, the diluteness of a gas yields a relatively largehealing length that characterizes the vortex core size, thus enabling the visu-alization of vortex cores by imaging techniques characteristic of this system(see Sec.2.3). Because of this observational capability, the ability to manip-ulate a condensate wave function, and the tunability of the rotation over awide range, these systems provide a unique approach to studying quantizedvortices and their dynamics. Second, the finite size effect due to the trappingpotential causes novel properties of vortices. Finally, multicomponent BECsprovide new possibilities for studying unconventional vortex states that have § He, anisotropic su-perconductors, and theories in high-energy physics and cosmology (Volovik2003).Since this is the first time that the topic of atomic BECs has appeared inthe Progress of Low Temperature Physics, we start with a basic introductionto ultra-cold atomic systems, including how a condensate is formed and howthey are manipulated. Although it is desirable to refer to the experimentaland theoretical studies of such systems in detail, we will only mention thebasic ideas necessary to understand some important issues in experiment andtheory because of space restrictions. These issues are described in Sec. 2. Moredetailed accounts can be found in the comprehensive text books by Pethickand Smith (Pethick 2002), Pitaevskii and S. Stringari (Pitaevskii 2003), andin the review paper by Leggett (Leggett 2001). In Sec. 3, we review the basictheory and experiments on quantized vortices in atomic BECs, addressing howvortices are created in this system and how they are detected. We also showthe intrinsic mechanism of vortex nuclation and lattice formation in a trappedBEC. There are two interesting regimes classified by the rotation rate of thesystem: one at a slow rotation rate close to a critical rotation frequency wherethere is a single vortex, and another at high rotation rates for which a latticeof a large number of vortices is formed. The details of these two regimes arediscussed in Sec. 4 and Sec. 5, respectively. Further interesting topics thatcannot be explained sufficiently in this review and remaining future problemsare presented in Sec. 6. We devote Sec. 7 to conclusions and outlook.
2. Introduction to ultra-cold atomic-gas BECs
While progress toward the achievement of Bose–Einstein condensation in a di-lute atomic gas had proceeded for the past few decades, researchers succeededin creating a condensate in 1995 (Anderson 1995). This realization broughtgreat sensation in modern physics and opened a new research field combin-ing condensed matter physics and atomic, molecular, optical (AMO) physics.Here, we briefly summarize the basic introduction about a system of ultra-cold atomic-gas BECs, giving background information for understanding theexperiments and theories of quantized vortices in the following sections.
A typical system considered here is a collection of neutral atoms with particlenumber N ∼ to 10 , trapped by a potential created by a magnetic fieldor an optical laser field. The density of the atomic gas is of the order of n ≃ cm − , which is lower than that of air on the earth ( ∼ cm − ).The transition temperature to Bose-Einstein condensation can be estimatedfrom a dimensional analysis of the relevant physical quantities ( m , n , ~ ) as kT c ≃ ~ n / /m , which is in a range from 100 nK to a few µ K. At such low µ K, with 10 atoms. In thecase of a trap created by a magnetic field, the atoms are trapped by the Zee-man interaction of the electron spin with an inhomogeneous magnetic field.Thus, atoms with electron spins parallel to the magnetic field are attracted tothe minimum of the magnetic field (weak-field seeking state), while ones withelectron spin antiparallel are repelled (strong-field seeking state). Laser cool-ing alone cannot produce sufficiently high densities and low temperatures forcondensation. The second step, evaporative cooling (a process in some sensesimilar to blowing on coffee to cool it), allows the removal of more energeticatoms, thus further cooling the cloud. The evaporation is effected by apply-ing a radio-frequency magnetic field which flips the electron spin of the mostenergetic atoms. At the end of the process, the final temperature is about 100nK and about 10 -10 atoms remain.Experimentally, the atomic-gas systems are attractive, since they can be ma-nipulated by the use of lasers and magnetic fields. The cold gas is confinedin a trap without microscopic roughness, as it is an extremely clean system.In addition, interactions between atoms may be affected either by using dif-ferent atomic species or by changing the strength of an applied magnetic orelectric field for species that exhibit Feshbach resonance. A further advantageis that, because of low density, the microscopic length scales are so large thatthe structure of the condensate wave function may be investigated by opticalmethods. Finally, the mean collision time τ coll ∼ ( σnv ) − ∼ − sec betweenatoms ( σ is the cross section and v the speed of atom) is comparable to thecharacteristic time of the collective mode, which prohibits a local equilibriumof the system. Thus, this system is ideal for studying nonequilibrium relaxationdynamics. The characteristics of BECs are mainly determined by atom–atom interac-tions, which depend crucially on the species of the condensed atoms. Most BECexperiments have been performed using alkali atoms because their groundstate electronic structure is simple; all electrons except one occupy closed shellsand the remaining electron is in an s orbital in a higher shell. This structure iswell suited to laser-based manipulation because its optical transitions can beexcited by available lasers and the internal energy-level structure is favorable § Rb (Anderson 1995), Na (Davis 1995), Li (Bradley 1995),H (Fried 1998), Rb (Cornish 2000), K (Modugno 2001),
Cs (Weber2003), and K (Roati 2007), and the non-alkali atoms metastable He (Robert2001, Dos Santos 2001),
Yb (Takasu 2003), and Cr (Griesmaier 2005). Rb and Na atoms are stable and have long lifetimes against inelastic col-lisional decay and are thus popular atomic species for BEC experiments.
Once a BEC has been created in a harmonic trap, it is probed for its prop-erties. This can be achieved either in situ , i.e., with the condensate insidethe trap, or using a time-of-flight (TOF) technique. Although in situ diag-nostics, such as nondestructive phase-contrast imaging (Andrews 1996), arevaluable tools for some applications, the TOF technique is more often used invortex experiments. The TOF technique involves switching off the trappingfield (magnetic or optical) at time t = 0 and taking an image of the BECa few (typically 5 to 25) milliseconds later. Switching off the trap allows thesample to expand before applying the laser beam probe, because the probeis difficult to apply at high densities. Images of the sample are most oftentaken by absorption, i.e., shining a resonant laser beam into the atomic cloudand using a CCD camera to observe the shadow cast by the absorption ofphotons, from which can be determined the integrated atomic density. Thismethod is inherently destructive since real absorption processes are involvedby spontaneous radiation and the accompanying heating. (i) Laser created potential
Atoms in a laser field experience a force, due mainly to the interaction of thelaser field with the electric dipole moment induced in the atoms. The force onatoms in a laser field is used in a variety of ways in BEC experiments.The character of the force is determined by the detuning given by ∆ ≡ ~ ω las − ( E e − E g ), where ω las is the laser frequency and E g ( E e ) the ground(excited) state energy of an atom. The force also depends on the laser-beamintensity I , given by I = ǫ c Γ /d , where ǫ is the dielectric constant, c the speed of light, d an appropriately defined dipole matrix element for thetransition in question, and Γ ≡ ~ /τ e with the lifetime of the excited state τ e .In the limit Γ ≪ ∆, the change in energy of the atom in the laser field is∆ E laser ( r ) = ( I ( r ) /I )Γ / ∆. A region of high laser intensity thus provides anattractive potential for ∆ < > Hyperfine state
Atomic BECs can have internal degrees of freedom, attributed to the hyperfinespin of atoms. A hyperfine-Zeeman sublevel of an atom with total electronicangular momentum J and nuclear spin I may be labeled by the projection m F of total atomic spin F = I + J on the axis of the field B and by the valueof total F , which can take a value from | I − J | to | I + J | . This is becausethe hyperfine coupling, which is proportional to I · J , is much larger thanthe typical temperature of an ultra-cold atomic system. The hyperfine stateis denoted by | F, m F i with m F = − F, − F + 1 , · · · , F − , F . The simulta-neous trapping of atoms with different hyperfine sublevels makes it possibleto create multicomponent (often called “spinor”) BECs with internal degreesof freedom (Ho 1998, Ohmi 1998), characterized by multiple order parame-ters (Hall 1998, Stenger 1998, Barrett 2001, Schmaljohann 2004, Chang 2004,Kuwamoto 2004).An external field can couple the internal sublevels of the atom and cause co-herent transition of the population. This coherent transition can be used tocontrol the spatial variation of the condensate wave functions, resulting in an“imprinting” of a phase pattern onto the condensate. In most schemes, thespatial configuration of the field, the intensity and detuning of the laser fields,and the phase relationship between the different fields need to be carefullycontrolled to create the right phase pattern, that takes full advantage of thecomplex internal dynamics.(iii) Feshbach resonance
A salient feature of cold atom systems is that field-induced Feshbach resonancecan tune the scattering length between atoms (Inouye 2001), which determinesthe atom–atom interaction. A Feshbach resonance occurs when a quasi-boundmolecular state in a closed channel has energy equal to that of two collid-ing atoms in an open channel. Such resonances can greatly effect elastic andinelastic collisions such as dipolar relaxation and three-body recombination.Scattering near the resonance can be quantified by perturbation theory. Tofirst order in the coupling between open and closed channels, the scatteringis unaltered, because there are no continuum states in the closed channels.However, two particles in an open channel can scatter to an intermediatestate in a closed channel, which subsequently decays to give two particles inan open channel. Considering such second-order processes, we can obtain thecontribution to the scattering length as ∼ ( E op − E cl ) − , where E op is theenergy of the particles in the open channel and E cl is the energy of a state inthe closed channels. Consequently, there are large effects if the energy E op ofthe two particles in the entrance channel is close to the energy E cl of a boundstate in the closed channels. Therefore, coupling between the channels yieldsa repulsive interaction if the energy of the scattering particles is greater than § From a theoretical point of view, a major advantage of weakly-interactingatomic-gas BECs is that almost all the atoms in the system occupy the samequantum state and the condensate may be described very well in terms ofmean-field theory. This is in contrast to liquid He, for which a mean-fieldapproach is inapplicable due to the strong correlations induced by the inter-actions between the atoms. Bogoliubov’s treatment of a uniform Bose gas atzero temperature provides a useful mean-field description of a condensate. Sub-sequently, Gross and Pitaevskii independently considered an inhomogeneousdilute Bose gas, generalizing Bogoliubov’s approach to include nonuniformstates, which includes quantized vortices. Such nonuniform states of a diluteBose gas can be understood by considering the second-quantized many-bodyHamiltonianˆ H = Z d r ˆΨ † ( r ) " − ~ ∇ m + V ex ( r ) + 12 Z d r ′ ˆΨ † ( r ′ ) V int ( r − r ′ ) ˆΨ( r ′ ) ˆΨ( r ) , (1)expressed in terms of Boson field operators ˆΨ( r ) and ˆΨ † ( r ) that obey Bose–Einstein commutation relations [ ˆΨ( r ) , ˆΨ † ( r ′ )] = δ ( r − r ′ ), [ ˆΨ( r ) , ˆΨ( r ′ )] =[ ˆΨ † ( r ) , ˆΨ † ( r ′ )] = 0. Here, V ex ( r ) is a trapping potential. The interparticle po-tential V int is approximated by a short-range interaction V int ≃ gδ ( r − r ′ ),where g = 4 π ~ a/m is a coupling constant, characterized by the s-wave scat-tering length a , because only binary collisions at low energy are relevant ina dilute cold gas and these collisions are independent of the details of thetwo-body potential.In three dimentions, a remarkable feature of a dilute Bose gas at zero temper-ature is the existence of a macroscopic wave function Ψ (an “order parame-ter”). The macroscopic occupation of condensed particles makes it natural towrite the field operator as a sum ˆΨ( r , t ) = Ψ( r , t ) + ˆ φ ( r , t ) of a classical fieldΨ( r , t ) that characterizes the condensate and a quantum field ˆ φ ( r , t ) repre-senting the remaining noncondensed particles. In order to derive the equationof motion for the order parameter, we write the time evolution of the operatorˆΨ( r , t ) = exp( i ˆ Ht/ ~ ) ˆΨ( r ) exp( − i ˆ H t/ ~ ) using the Heisenberg equation withthe many-body Hamiltonian i ~ ∂ ˆΨ( r , t ) ∂t = [ ˆΨ( r , t ) , ˆ H ] = " − ~ ∇ m + V ex + g ˆΨ † ( r , t ) ˆΨ( r , t ) ˆΨ( r , t ) . (2)To leading order, the Bogoliubov approximation neglects the noncondensed i ~ ∂ Ψ( r , t ) ∂t = " − ~ ∇ m + V ex + g | Ψ( r , t ) | Ψ( r , t ) (3)for the condensate wave function Ψ( r , t ). The GP equation (3) can be usedto explore the dynamic behavior of the system, characterized by variationsof the order parameter over distances larger than the mean distance betweenatoms. This equation is valid when the s-wave scattering length is much smallerthan the average distance between atoms, and the number of atoms in thecondensate is much larger than unity.The ground state of a trapped BEC can be expressed within the formalismof the GP theory. We can write the condensate wave function as Ψ( r , t ) =Φ( r ) e − iµt/ ~ , where Φ( r ) obeys the time-independent GP equation " − ~ ∇ m + V ex + g | Φ( r ) | Φ( r ) = µ Φ( r ); (4)Φ is normalized to the number of condensed particles R d r | Φ( r ) | = N , whichdetermines the chemical potenteial µ . Typically, studies of trapped atomicgases involve the dilute limit (the gas parameter ¯ n | a | is typically less than10 − , where ¯ n is the average density of the gas), so that depletion of thecondensate is small with N ′ = N − N ∝ q ¯ n | a | N ≪ N . Hence, most of theparticles remain in the condensate such that N ≃ N . The time-independentGP equation (4) is also derived by minimizing the GP energy functional E [Ψ , Ψ ∗ ] = Z d r Ψ ∗ − ~ ∇ m + V ex + g | Ψ | ! Ψ ≡ E kin + E tr + E int , (5)subject to the constraint of a fixed particle number N . This constraint is takeninto account by the Lagrange multiplier method; we write the minimizationprocedure as δ ( E − µN ) /δ Ψ ∗ = 0, where the chemical potential µ is theLagrange multiplier that ensures a fixed N .Equation (4) provides a starting point for studying the structure of a conden-sate in a harmonic confining potential V ex = m ( ω x x + ω y y + ω z z ) /
2. Thisintroduces the length scale a ho = q ~ /mω with ω = ( ω x ω y ω z ) / . Although theexact ground state can be obtained only by solving Eq. (4) numerically, anapproximate analytic solution can be gained when the interaction energy E int is much larger than E kin (Baym 1995). To see this argument, let us neglectthe anisotropy of the harmonic potential and assume that the cloud occupiesa region of radius ∼ R , so that n ∼ N/R . Then, the scale of the harmonicoscillator energy per particle is ∼ mω R / ∼ gN/R . By comparing theseenergies, the radius is found to be R ∼ a ho (8 πN a/a ho ) / . The kinetic energyis of order ~ / mR , so that the ratio of the kinetic to interaction (or trap)0 § ∼ ( N a/a ho ) − / . In the limit N a/a ho ≫
1, which is relevant tocurrent experiments on trapped BECs, the repulsive interactions significantlyexpand the condensate, so that the kinetic energy associated with the densityvariation becomes negligible compared to the trap and interaction energies.As a result, the kinetic-energy operator can be omitted in Eq. (4), giving theThomas–Fermi (TF) parabolic profile for the ground-state density n ( r ) ≃ | Ψ TF ( r ) | = µ − V ex ( r ) g Θ [ µ − V ex ( r )] , (6)where Θ( x ) denotes the step function. The resulting ellipsoidal density inthree-dimensional (3D) space is characterized by two types of parameters:the central density n = µ/g and the three condensate radii R j = 2 µ/mω j ( j = x, y, z ). The chemical potential µ is determined by the normalization R d r n ( r ) = N as µ = ( ~ ω/ N a/a ho ) / . The spectrum of elementary excitations of a condensate is an essential ingre-dient in calculations of the thermodynamic properties. To study the low-lyingcollective-excitation spectrum of trapped BECs, the Bogoliubov–de Gennes(BdG) equation coupled with the GP equation is a useful formalism.Let us consider the equation of motion for a small perturbation around thestationary state Φ, which is a solution of Eq. (4). The wave function takes theform Ψ( r , t ) = [Φ( r ) + u j ( r ) e − iω j t − v ∗ j ( r ) e iω j t ] e − iµt . By inserting this ansatzinto Eq. (3) and retaining terms up to first order in u and v , we obtain theBdG equation: L ( r ) − g Φ( r ) g Φ ∗ ( r ) −L ( r ) u j ( r ) v j ( r ) = ~ ω j u j ( r ) v j ( r ) , (7)where L ( r ) = − ~ ∇ / m + V ex − µ + 2 g | Φ( r ) | , and ω j are the eigenfrequenciesrelated to the quasiparticle normal-mode functions u j ( r ) and v j ( r ). The modefunctions are subject to the orthogonality and symmetry relations R d r [ u i u ∗ j − v i v ∗ j ] = δ ij and R d r [ u i v ∗ j − v i u ∗ j ] = 0.Since the energy ~ ω j of this quasiparticle is defined with respect to the con-densate energy, in Eq. (5) with the stationary solution Φ, the presence ofquasiparticles with negative frequencies implies an energetic (thermodynamic)instability for the solution Φ. If there is energy dissipation in the system, theexcitation of negative-energy modes lowers the total energy of the system andΦ relaxes to a more stable solution. We note that, since the matrix element ofEq. (7) is non-hermitian, the eigenfrequencies can be complex-valued. Whenthere are complex-valued frequencies, small-amplitude fluctuations of the cor-responding eigenmodes grow exponentially during the energy-conserving timedevelopment. This is known as dynamical instability and is a main origin of the
3. Vortex formation in atomic BECs
In this section, we describe some basic properties of quantized vortices intrapped BECs and experimental procedures to create them. We also describethe nucleation mechanisms of quantized vortices in trapped BECs, includingcontrolled schemes of phase engineering. Typically, experiments have used asmooth rotating potential, which is created by a laser or magnetic field, witha small transverse anisotropy to rotate the condensate. This potential inducesa low-energy collective oscillation or shape deformation of the condensate.Such global motions of the condensate are responsible for the instability ofthe vortex nucleation, producing interesting nonequilibrium dynamics in thesystem. This is contrary to superfluid helium systems, where vortex nucleationoccurs locally through roughness or impurities in the rotating container.
As a simple example, let us consider the structure of a single vortex in a con-densate trapped by an axisymmetric harmonic potential V ex ( r, z ) = mω ⊥ ( r + λ z ) / λ = ω z /ω ⊥ . The condensate wave function with aquantized vortex line located along the z -axis takes the form Φ( r ) = φ ( r, z ) e iqθ with winding number q and cylindrical coordinate ( r, θ, z ). φ is a real functionrelated to the condensate density as n ( r, z ) = φ . The velocity field aroundthe vortex line is v s = ( q ~ /mr )ˆ θ . Equation (4) becomes " − ~ m ∂ ∂r + 1 r ∂∂r + ∂ ∂z ! + q ~ mr + V ex ( r, z ) + gn φ = µφ. (8)Here, the centrifugal term q ~ / mr arises from the azimuthal motion of thecondensate. Equation (8), solved numerically, gives the structure of the vortex.In the TF limit N a/a ho ≫
1, we can omit the terms involving derivatives withrespect to r and z in Eq. (8), and the density can be obtained approximatelyas n ( r, z ) = n − r R ⊥ − z R z − q ξ r ! Θ − r R ⊥ − z R z − q ξ r ! , (9)where n = µ/g is the density at the center of the vortex-free TF profile. Wedefine the TF radius R ⊥ = 2 µ/mω ⊥ and R z = 2 µ/mω z and the healing length ξ = ( ~ / mgn ) / = (8 πan ) − / . Equation (9) shows that the condensatedensity vanishes at the center, out to a distance of order ξ , due to the cen-trifugal term ξ /r (numerical solution shows that the density grows as r away from the center), whereas the density in the outer region has the form2 § ξ characterizes thevortex core size; for typical BEC parameters, ξ ∼ . µ m. In the TF limit,the core size is very small because ξ/R ⊥ = ~ ω ⊥ / µ = (15 N a/a ho ) − / ≪ q widens the core radius due to the centrifugaleffects.The energy associated with a single vortex line is an important quantity todetermine the stability of the vortex state. The dominant contribution to thisenergy is the kinetic energy of the superfluid flow by a vortex. The energy isestimated as E = Z mnv d r ≃ m ¯ n R z R ⊥ Z ξ v πrdr = q R z π ~ ¯ nm ln R ⊥ ξ ! , (10)where we assume the spatially uniform density ¯ n and the size along the z -axis R z . Since E ∝ q , vortices with q > q = 2 vortex is higher than that tocreate two q = 1 vortices. Therefore, a stable quantized vortex usually has q = 1, except for that in non-simply connected geometry, and we will mainlyconcentrate on the q = 1 vortex in the following discussions. In atomic BECs,however, such a multiply quantized vortex can be created experimentally byusing topological phase imprinting (Shin 2004) and exhibits interesting disin-tegration dynamics, as discussed in Sec. 4.2.3.It is necessary to ensure the stability of a vortex in a trapped BECs againsta non-vortex state to investigate its behavior. Imposing rotation on the sys-tem is a direct way to achieve stabilization. If the system is under rotation,it is convenient to consider the corresponding rotating frame; for a rotationfrequency Ω = Ωˆ z , the integrand of the GP energy functional (5) acquires anadditional term, E ′ = Z d r Ψ ∗ − ~ ∇ m + V ex + g | Ψ | − Ω L z ! Ψ , (11)where L z = − i ~ ( x∂ y − y∂ x ). The corresponding GP equation becomes i ~ ∂ Ψ( r , t ) ∂t = " − ~ ∇ m + V ex + g | Ψ( r , t ) | − Ω L z Ψ( r , t ) . (12)If there is a quantized vortex along the trap axis, h L z i = N ~ , so that thecorresponding energy of the system in the rotating frame is E ′ = E − N ~ Ω.The difference between E ′ and the vortex-free energy E ′ gives the favorablecondition for a vortex to enter the condensate. Since E ′ is equal to the energy E in the laboratory frame, the difference is given by ∆ E ′ = E ′ − E ′ = E − E − N ~ Ω. Thus, the critical rotation frequency Ω c for the existenceof an energetically stable vortex line is given by Ω c = ( E − E ) /N ~ . Abovethe critical rotation frequency Ω c the single vortex state is ensured to be thermodynamically stable. E more quantitatively, it is necessary to take into account theinhomogeneous effect of the condensate density (Lundh 1997). In the TF limit,for a condensate in a cylindrical trap ω z = 0 (an effective 2D condensate), thecritical frequency is given by Ω c = (2 ~ /mR ⊥ ) ln(0 . R ⊥ /ξ ). For an axisym-metric trap V ex ( r, z ), the critical frequency is Ω c = (5 ~ / mR ⊥ ) ln(0 . R ⊥ /ξ ).Ω c for a nonaxisymmetric trap is slightly modified by a small numerical fac-tor, and has been discussed analytically (Svidzinsky 2000a) and numerically(Feder 1999a).When the rotation frequency is significantly higher than Ω c , further vorticeswill appear in the form of a triangular lattice, in analogy to the Abrikosovlattice of magnetic fluxes in type-II superconductors. The nature of the equi-librium state changes, first to a state with two vortices rotating around eachother, then to three vortices in a triangle form, and subsequently to arraysof more vortices (Butts 1999, Feder 2001b). The detail of this state will bedescribed in Sec. 5. Rotation effects of atomic BECs were first studied based on the knowledgeof deformed atomic-nuclei systems. A slight rotation of a deformed trap ex-cites so-called “scissors modes”, which are closely related to the irrotational-ity ∇ × v s = 0 of the superfluid hydrodynamics (Gu`ery-Odelin 1999, Marago2000). Although these experiments demonstrated the superfluidity of atomicBECs, more intuitive evidence can be gained by observing quantized vor-tices. However, since the first experimental realization of BECs, there havebeen technical hurdles to rotating a system. For atomic gases trapped by animpurity-free external potential, it was supposed that rotation of the potentialcould not transfer a sufficient angular momentum into the condensate. Thus,the first experimental detection of a vortex in an atomic BEC, by Matthews etal. (Matthews 1999a), was made by using a complicated phase imprinting tech-nique proposed by Williams and Holland (Williams 1999). Subsequently, vor-tices have been created by stirring a condensate mechanically with an “opticalspoon”; the first success of this “rotating bucket” experiment was reported byMadison et al . (Madison 2000). In the following, we detail the rotating bucketexperiments. Madison et al. at Ecole Normale Sup´erieure (ENS) succeeded in observing avortex and a vortex lattice in an atomic BEC (Madison 2000), using a methodsimilar to the observation of superfluid helium in a rotating bucket (Yarmchuk1979). A schematic illustration of their experimental setup is shown in Fig. 1.Since the magnetic trap is axially symmetric, its rotation cannot impart anangular momentum to the condensate. In this scheme, a laser beam is propa-gated along the z -axis of the condensate. This beam rapidly oscillates aroundthe z -axis with a frequency much larger than the typical trapping frequency,4 § stirring laser cigar shaped condensate~ 100 m m ~ 7 m m W y x z Fig. 1. Schematic illustration of the experimental setup used by the ENS group.A cigar-shaped condensate is rotated by an effective dipole potential made by thelaser beam. The laser beam oscillates rapidly with a very large frequency aroundthe z -axis, with an amplitude ∼ µ m. The beam width is about 20 µ m. The dipolepotential rotates with an angular frequency Ω. which is effectively regarded to be as if two laser beams are located at anequilibrium position. Thus, the two laser beams break the axisymmetry ofthe condensate, allowing an angular-momentum transfer into the condensate.An optical spoon is then realized by rotating the two beams around the z -axis. Since the beam width is larger than the radial size of the condensate,the condensate is trapped in a trap combining an axisymmetric harmonicpotential and a nonaxisymmetric harmonic potential created by a stirringlaser beam. In a rotating frame, the combined potential can be written as(1 / mω ⊥ [(1 + ǫ ) X + (1 − ǫ ) Y ] + (1 / mω z z , where X and Y are the co-ordinates in the rotating frame, ǫ = ( ω X − ω Y ) / ( ω X + ω Y ) is the anisotropicparameter, and ω ⊥ = q ( ω X + ω Y ) /
2. By rotating this potential at a frequencyΩ, a vortex is formed above a certain critical value of Ω after the equilibration.When Ω is increased further, multiple vortices appear, forming a triangularlattice. The quantized vortices can be directly visualized as “dips” in thetransverse density profile of the TOF image.
Fig. 2. Typical density profiles of a rotating condensate taken by TOF measurement.The condensates contain approximately 16, 32, 80, and 130 vortices from left toright. From J.R. Abo-Shaeer et al., SCIENCE 292, 476 (2001). Reprinted withpermission from AAAS.
Following the experiments of the ENS group, other groups have also observedquantized vortices using slightly different methods under the concept of therotating bucket. Abo-Shaeer et al. at Massachusetts Institute of Technology(MIT) observed a vortex lattice consisting of up to 100 vortices in a Na con- Na condensates can be much larger than Rbcondensates (Abo-Shaeer 2001). Hodby et al. created a vortex lattice by rotat-ing the anisotropic magnetic trap directly without using a laser beam, which issimilar to the rotating bucket experiment (Hodby 2002). This method has theadvantage that a wider range of the anisotropic parameter ǫ can be selectedthan for the optical spoon. Rotating an optical spoon made by multiple-spotlaser beams or narrow focusing beams has also used for nucleating vortices(Raman 2001).In the above methods, the condensate was rotated by an external potential.Haljan et al. at Joint Institute for Laboratory Astrophysics (JILA), in con-trast, created a vortex state by cooling an initially rotating thermal gas ina static confining potential (Haljan 2001). Thermal gas above the transitiontemperature was rotated by a slightly anisotropic trap. After recovering theanisotropy of the potential, the rotating thermal gas was evaporatively cooleduntil most of the atoms were condensed. Although the atom number decreasedthrough the evaporative cooling, the condensate continued to rotate becausethe angular momentum per atom did not change and thus the vortex latticewas created. This method allows the investigation of the intrinsic mechanismof vortex nucleation, which is independent of the character of the stirring po-tential. In addition, since atoms can be selectively removed during the evapo-ration, spinning up of the condensate can be efficiently achieved by removingatoms extending in the axial direction, as opposed to the radial direction, andhence a BEC with a high rotation rate can be obtained. (i) Surface mode instability
The critical rotation frequency Ω c indicates the energetic stability of the cen-tral vortex state and provides a lower bound for the critical frequency. Vortexnucleation of a non-rotating condensate occurs when the trap is rotated ata higher frequency than Ω c , to overcome the energy barrier that stops thetransition from the nonvortex state to the vortex state (Isoshima 1999). Thethreshold of the rotation frequency for instability, leading to vortex nucle-ation, is related to the excitation of surface modes of the trapped condensate(Feder 1999b, Dalfovo 2000, Anglin 2001, Williams 2002, Simula 2002b). Ithas been shown that, according to the Landau criterion for rotationally in-variant systems, the critical frequency is given by Ω v = min( ω ℓ /ℓ ), where ω ℓ is the frequency of a surface mode with multipolarity ℓ . Above Ω v , negative-energy surface modes appear with high multipolarities in the spectrum of anon-rotating condensate (Isoshima 1999, Dalfovo 2000), which may lead tovortex generation. The negative-energy modes can grow only in the presenceof dissipation, caused by e.g., thermal atoms (Williams 2002). This mechanismoccurred in the experiment of the JILA group (Haljan 2001). In this experi-ment, vortices were formed by cooling a rotating thermal cloud to below T c ,where the surface modes were excited by the “wind” of the rotating thermal6 § Fig. 3. Dependence of the vortex number (angular momentum per atom) on therotation frequency. (a) Result of the ENS group (Chevy 2000, 2001). The parametervalues are N = 2 . × , ω ⊥ = 2 π ×
172 Hz, and an anisotropic parameter ǫ (blackdots: ǫ = 0 .
01, white square: ǫ = 0 . ≃ ω ⊥ / √ N ∼ and ω ⊥ = 2 π ×
86 Hz. Thearrows below the graph show the positions of the surface mode resonance ω ⊥ / √ ℓ .The inset shows 2-, 3-, and 4-point potentials produced by a laser beam. (Takenfrom (Chevy 2001) and (Raman 2001). Reprinted with permission from AmericanPhysical Society (APS).) The nucleation frequency Ω v is insufficient to explain the results of the groupsusing external stirring potentials (Madison 2000, Abo-Shaeer 2001, Hodby2002). For example, in the case of the ENS group, the number of nucleatedvortices has a peak near Ω = 0 . ω ⊥ , as shown in Fig. 3(a). These experimentsconfirm that instability occurs when a particular surface mode is resonantlyexcited by a deformed rotating potential. The optical spoon of the ENS groupmainly excites the surface mode with ℓ = 2 (quadrupole mode). In a rotatingframe with frequency Ω, the frequency of the surface mode is increased by − ℓ Ω.This resonance thereby occurs close to the rotation frequency Ω = ω ℓ /ℓ . In theTF limit, the dispersion relation for the surface mode is given by ω ℓ = √ ℓω ⊥ (Stringari 1996). Hence, it is expected that the quadrupole mode with ℓ = 2 isresonantly excited at Ω = ω ⊥ / √ ≃ . ω ⊥ . A theoretical study has revealedthat, when the quadrupole mode is resonantly excited, an imaginary compo-nent appears in frequencies of fluctuations with high multipolarities (Sinha2001). This indicates that dynamic instability can trigger vortex nucleationeven at zero temperature. This scheme is supported further by the experimentof the MIT group (Raman 2001), where surface modes with higher multipo-larities ( ℓ = 3 ,
4) were resonantly excited using multiple laser-beam spots,where the largest number of vortices were generated at frequencies close tothe expected values Ω = ω ⊥ / √ ℓ , as seen in Fig. 3(b). > ω ⊥ / √ et al. followed these stationary states experimentallyby adiabatically introducing trap ellipticity and rotation. They observed vor-tex nucleation in the expected dynamically unstable region (Madison 2001).An overview on the problem of vortex nucleation in a trapped BEC can beseen in the review paper (Ghosh 2004).(ii) Formation dynamics of a vortex lattice
Fig. 4. (Left) Measurement of the time dependence of α (see text) when the stirringanisotropy is turned on rapidly from ǫ = 0 to ǫ = 0 .
025 in 20 msec and Ω = 0 . γ = 0 .
03) and the experimentally relevant parameters.The density profile in the bottom panels is integrated along the z -axis. Madison et al . directly observed nonlinear processes such as vortex nucle-ation and lattice formation in a rotating condensate (Madison 2001). Theleft panel of Fig. 4 depicts the time development of the condensate elliptic-ity α = Ω( R X − R Y ) / ( R X + R Y ), where R X,Y is the TF radius measuredfrom the image. By suddenly turning on the rotation of the potential, theinitially axisymmetric condensate undergoes a collective quadrupole oscilla-tion in which the condensate deforms elliptically. This oscillation continues8 § i − γ ) ∂/∂t ,which caused the lattice configuration to settle. Other works have simulatedvortex lattice formation using dissipation derived from the microscopic ap-proach, such as quantum kinetic theory (Penckwitt 2002) or the classical fieldformalism (Lobo 2004). Long numerical propagation of the energy-conservingGP equation can cause crystallization of a lattice through the vortex–phononinteraction (Parker 2005)(iii) Vortex nucleation by a moving object
Vortices can also be nucleated in BECs by a moving localized potential. Nu-merical simulations of the GP equation for a 2D uniform condensate flowaround a circular hard-walled potential show that vortex–antivortex pairs nu-cleate when the flow velocity exceed a critical value (Frisch 1992). In trappedBECs, a similar situation can be realized experimentally using a narrow blue-detuned laser potential, being studied theoretically (Jackson 1998, Crescimanno2000). In experiments by the MIT group, a repulsive laser beam was oscillatedin an elongated condensate to study the dissipationless flow of a Bose gas(Raman 1999, Onofrio 2000). Although vortices were not observed directly,the measurement of condensate heating and drag above a critical velocity wasconsistent with the nucleation of vortices (Jackson 2000). Focused laser beamsmoving in a circular path around the trap center can also stir the condensateby nucleating vortices (Caradoc-Davies 1999, 2000, Lundh 2003). This schemewas used in the experiment detailed in Ref. (Raman 2001), where vortices weregenerated at lower stirring frequencies than the critical value given by surfacemode instability.
Atom optics techniques allow the controlled creation of vortices by imprintinga spatial phase pattern of the condensate wave function. Several ideas for thecreation of vortices by this technique have been proposed theoretically, basedon the coherent control of the time evolution of the wave function, instead of
The first observation of a quantized vortex in an atomic-gas BEC was achievedin a two-component BEC consisting of Rb atoms with hyperfine spin states | F = 1 , m F = − i ≡ | i and | F = 2 , m F = 1 i ≡ | i (Matthews 1999a), whichwere confined simultaneously in almost identical magnetic potentials. Sincethe scattering lengths between atoms of | i and | i , | i and | i , and | i and | i are all different, the two states are not equivalent, and the two-componentcondensate is characterized by two-component order parameters.In this experiment, condensed atoms are initially trapped in one state, say,the | i state. Then, a two-photon microwave field is applied, inducing coherentRabi transitions of atomic populations between the | i state and the | i state.For a homogenous system in which both components have uniform phases,interconversion takes place at the same rate everywhere. However, the timevariation of the spatially inhomogeneous potential changes the nature of theinterconversion. This is a key point of the phase-imprinting method for vortexcreation.The underlying physics can be understood by considering a co-rotating framewith an off-centered perturbation potential at the rotation frequency Ω ′ . Inthis frame, the energy of a vortex with one unit of angular momentum isshifted by ~ Ω ′ relative to its value in the laboratory frame. When this energyshift is compensated for by the sum of the detuning energy of an applied mi-crowave field and the small chemical potential difference between the vortexand non-vortex states, a resonant transfer of population can occur. Experi-mentally, the rotating perturbation is created by a laser beam with a spatiallyinhomogeneous profile, rotating around the initial nonrotating component, say | i . By adjusting the detuning and Ω ′ , the | i component is resonantly trans-ferred to a state with unit angular momentum by precisely controlling thetime when the coupling drive is turned off (Williams 1999). This procedureresults in a “composite” vortex, where the | i component has a vortex at thecenter, whereas the nonrotating | i component occupies the center and worksas a pinning potential that stabilizes the vortex core (Kasamatsu 2005b).As shown in Sec. 2.4, a far-off-resonant laser beam can create an externalpotential V las ( r ) in the condensate. By directly applying a laser pulse withan inhomogeneous intensity to the condensate, the condensate phase can bemodulated. This can be easily understood by observing the evolution of thephase by inserting the form Ψ( r , t ) = | Ψ( r , t ) | e iθ ( r ,t ) into Eq. (3). When thelaser intensity is much stronger than the other terms and the duration of thepulse τ is sufficiently short, the evolution of the phase is governed by θ ( r , t ) = − ~ − R τ dtV las ( r , t ). Since the potential amplitude of V las is proportional tothe laser intensity, a suitable spatial variation of the intensity can imprint the0 § dark soliton (Burger 1999, Denschlag 2000), whichis a topological excitation with a complete density dip across which the phasechanges by π . It is known that a dark soliton in a dimensional space largerthan 2D experiences dynamical instability, called “snake instability”. Thisinstability causes the decay of the dark solitons into a form of a vortex ring (Anderson 2001, Dutton 2001). Leanhardt et al. (Leanhardt 2002, 2003) used a method called “topologicalphase imprinting” (Nakahara 2000, Isoshima 2000, Ogawa 2002) to create avortex in a trapped BEC. In this experiment, Na condensates were preparedin either a lower, | F, m F i = | , − i , or upper, | , +2 i , hyperfine state andconfined in a Ioffe–Pritchard magnetic trap, described by B = B ′ ( x ˆ x − y ˆ y ) + B z ˆ z . A vortex was created by adiabatically inverting the axial bias field B z along the trap axis.To interpret the mechanism, consider an alkali atom with a hyperfine-spin | F | = 1. The order parameter has three components Ψ ± and Ψ correspond-ing to F z = ± ,
0, respectively. The basis vectors in this representation are {|±i , | i} . We introduce another set of basis vectors | x i , | y i and | z i , whichare defined by F x | x i = F y | y i = F z | z i = 0. These vectors are related to theprevious vectors as | ± i = ∓ (1 / √
2) ( | x i ± i | y i ) and | i = | z i . When the z -axis is taken parallel to the uniform magnetic field, the order parameter ofthe weak field seeking state takes the form Ψ − = ψ and Ψ = Ψ = 0, orin vectorial form as Ψ = ( ψ/ √
2) (ˆ x − i ˆ y ). When the magnetic field points inthe direction ˆ B = (sin β cos α, sin β sin α, cos β ), a rotational transformationwith respect to the Euler angle ( α, β, γ ) gives Ψ = ( ψ/ √ e iγ ( ˆ m + i ˆ n ), whereˆ m = (cos β cos α, cos β sin α, − sin β ) and ˆ n = (sin α, − cos α, l = ˆ m × ˆ n = (cos α sin β, − sin α sin β, − cos β ) is the direction of the spinpolarization. The three real vectors { ˆ l , ˆ m , ˆ n } form a triad, analogous to theorder parameter of the orbital part of superfluid He. The same amplitudes inthe basis {| i , |±i} are Ψ = ( ψ/ − cos β ) e − iα + iγ , Ψ = − ( ψ/ √
2) sin βe iγ ,and Ψ − = ( ψ/ β ) e iα + iγ .When the field B z is strong compared to the quadrupole field, the trappedcondensate has an order parameter Ψ = ( ψ/ √ x − i ˆ y ) without vorticity.This configuration corresponds to β = 0 and γ = − α = φ , where φ is theazimuthal angle. Then, B z is adiabatically decreased, where the adiabaticcondition is required for atoms to remain in the weak field seeking state so thatˆ l is always antiparallel to B . In the final step, the external field B z is graduallyincreased in the opposite ( − z ) direction. Then, the ˆ l -vector points so that β = π . Substituting these angles into Ψ ± and Ψ , we obtain Ψ − = Ψ = 0 andΨ = ψe iφ , which corresponds to a vortex with winding number q = 2. Thisresult can be reinterpreted in terms of Berry’s phase (Ogawa 2002, Leanhardt2002) (which is why it is called “topological phase imprinting”). When the F in general, we obtain a vortex with a winding number 2 F since Ψ − F and Ψ F have phases F ( α + γ ) and F ( − α + γ ), respectively (Shin2004, Kumakura 2006).When the bias field vanishes during the inversion ( B z = 0), a spin textureknown as cross disgyration appears in the Ioffe–Pritchard trap. Here, the angle β increases from 0 to π/ γ and − α are identified with φ , where theˆ l -vector aligns with a hyperbolic distribution around the singularity at thecenter. This texture has a nonvanishing vorticity n when γ = nφ . This spintexture has been observed as a coreless vortex composed of three-componentorder parameters Ψ ± , of a spinor BEC (Leanhardt 2003). Some papers proposed generating vortices in a BEC using stimulated Ramanprocesses with configurations of optical fields that have orbital angular mo-mentum (OAM) (Marzlin 1997, 1998, Dum 1998, Nandi 2004, Kapale 2005). Alight beam with a phase singularity, such as a Laguerre-Gaussian (LG) beam,has a well-defined OAM along its propagation axis. The set of LG modesLG lp ( r, φ ) = s pπ ( | l | + p ) 1 w √ rw ! | l | L | l | r r w ! e − r /w + ilφ (13)defines a possible basis set to describe paraxial laser beams, where w is thebeam width, l the winding number, and p the number of radial nodes for radius r >
0. Each photon in the LG lp mode carries OAM l ~ along its direction ofpropagation.A group at NIST (Anderson 2006, Ryu 2007) used a 2-photon stimulated Ra-man process with a Gaussian laser beam propagating along + x and a LG beam, carrying ~ of OAM, propagating along − x . Interference of counter-propagating Gaussian beams generates a moving sinusoidal optical dipole po-tential, which can give a directed linear momentum (LM) to Bose-condensedatoms via Bragg diffraction. The potential generated by interference of thecounter propagating LG and Gaussian beams is not sinusoidal but corkscrew-like, due to the radial intensity profile and the helical phase of the LG beam.Diffraction off this corkscrew potential produces a vortex state with a center-of-mass motion, where atoms that absorb a photon from one beam and simu-latedly emit a photon into the other beam acquire both LM and OAM differ-ence of the beams, which in this case was 2 ~ k ( k the photon wavevector) and ~ , respectively.They generated vortices of higher charge by transferring to each atom the an-gular momentum from several LG photons (Anderson 2006, Ryu 2007). Thisexperiment directly demonstrated that the OAM of a photon is transferredcoherently to an atom in quantized units of ~ . In some situations it might bedesirable to generate rotational states with no net LM. This could be accom-plished by using an initial Bragg diffraction pulse to put atoms in a non-zeroLM state from which they could subsequently be transferred to a rotational2 §
4. A single vortex in an atomic BEC
In this section, we concentrate on the problem of a single vortex state in atrapped BEC. As described in Sec. 3.1, vortex stability is ensured by the ro-tation of the system. Studying the motion of a vortex line is the first steptowards understanding superfluid hydrodynamics in such a system. TrappedBECs are mesoscopic systems in the sense that the healing length is not signif-icantly smaller than the sample size. Thus, vortex dynamics have noticeableeffects on the collective excitation of the condensate, in contrast to the casefor traditional superfluid helium systems.
The solution of the GP equation shows that a vortex has a core with a sizeof the order of the healing length ξ , in which the condensate density is zero.When Ω = Ωˆ z , vortices are identified by a density dip in the transversedensity distribution (in the xy plane). TOF observations by the ENS group(Madison 2000), however, show that the density is not completely zero in thedip. This result implies that the vortex line is not necessarily straight, becausethe condensate density along the z -axis (rotation axis) is integrated for thetransverse image. Surprisingly, such vortex bending remains stationary in theground state of a cigar-shaped condensate (Garc´ıa-Ripoll 2001, Aftalion 2002,Modugno 2003, Aftalion 2003).Evidence of vortex bending in the ground state was observed by the ENSgroup (Rosenbusch 2002). They prepared a single vortex state slightly aboveΩ c and equilibrated it for a sufficient long time. In the TOF measurements,two imaging beams were aligned along the y and z directions and probed theatom distribution simultaneously [Fig. 5 (a)]. The transverse image in Fig. 5(b) shows the vortex line, corresponding to the lower atom density; it is notstraight and has the shape of a wide “U”. Figure 5 (c) shows the decay of a Uvortex for which the angular momentum has decreased significantly comparedto Fig. 5 (b). In the longitudinal view, we can see a vortex off-center. In thetransverse view, a narrow U can be seen.This result is supported by theoretical analysis based on the 3D GP equa-tion with experimentally appropriate parameters (Garc´ıa-Ripoll 2001, Aftalion2002, Modugno 2003, Aftalion 2003), where the ground state with a rotation Ω = Ωˆ z was calculated by minimizing the energy functional in Eq. (11).The central vortex is generally bent if the trap aspect ratio λ = ω z /ω ⊥ ismuch less than unity. A simple physical picture of the bending can be gainedby viewing a cigar-shaped condensate as a series of 2D sheets at various z (Modugno 2003). For each sheet, there is a corresponding 2D vortex stabilityproblem with a critical frequency Ω c ( z ) (see Sec. 3.1) above which a cen- Fig. 5. (a) Schematic of the imaging method of a vortex line. Two beams image theatom cloud simultaneously along the longitudinal ( z ) and transverse ( y ) directionsof the initial cigar. In (b)–(d), the left column shows the “longitudinal” view along z , representing the atom distribution in the xy plane. The right column depicts the“transverse” view along the y direction, representing the atom distribution in the xz plane. The images were taken after equilibration times of (b) 4 s, (c) 7.5 s, (d)and 5 s. In the atom distribution of the transverse image, the ellipticity is invertedwith respect to the initial cigar form, caused by transverse expansion during theTOF. (Taken from (Rosenbusch 2002). Reprinted with permission from APS.) tered vortex is the stable solution. Since the effective 2D chemical potentialis µ ( z ) = µ − mω z z /
2, the radius of the 2D condensate at z becomes R ⊥ ( z ) = 2 µ D ( z ) /mω ⊥ = R ⊥ (0) − λ z . Thus, Ω c ( z ) ∝ R ⊥ is a decreasingfunction from z = 0 to | z | = R z . For a given rotation frequency Ω, the vortexline minimizing the total energy is well centered for | z | < z c and is stronglybent for | z | > z c , where Ω c ( z c ) = Ω. This bending is a symmetry-breakingeffect, which does not depend on the presence of rotating anisotropy andwhich occurs even in a completely axisymmetric system (Garc´ıa-Ripoll 2001).A precursor of this bending effect can be found in the excitation spectrum ofa condensate with a centered straight vortex (Svidzinsky 2000b, Feder 2001a),in which negative-energy modes localized at the core (so-called “anomalousmodes”) appear with increasing λ . As these modes grow, the central vortex ispushed outward. This indicates that the bending instability needs a dissipationmechanism and that it takes a long time at low temperatures.Figure 5 (d) shows a vortex line in the shape of an “S”, which can be regardedas a U vortex with a half part rotated by 180 ◦ . An S single vortex can also befound by numerical simulation as the stationary state of an elongated conden-sate for a given rotation frequency (Aftalion 2003), having an energy higherthan that of the U vortex (Komineas 2005).4 § Precession of a vortex core off-center in a condensate is a simple example ofvortex motion. Core precession can be described in terms of a Magnus forceeffect. A net force on a quantized vortex core creates a pressure imbalance,resulting in core motion perpendicular to both the force and the vortex quan-tization axis. In the case of trapped BECs, these net forces can be causedby either condensate density gradients (Svidzinsky 2000a,b, Jackson 1999,McGee 2001) or drag due to thermal atoms (Fedichev 1999). The former maybe thought of as a sort of effective buoyancy. Typically, the total buoyancyforce is towards the condensate surface and the net effect is a precession ofthe core around the condensate axis via the Magnus effect. The latter causesradial drag and spiraling of the core towards the condensate surface due toenergy dissipation and damping processes.Core precession has been investigated in detail by the JILA group (Anderson2000). Starting with a composite vortex created by the phase-engineeringmethod of Sec. 3.3, they selectively removed components filling the vortexcore with resonant light pressure. In the limit of complete removal, a single-component vortex state with a bare core can be obtained. The vortex wasoff-center because of the instability of the formation process. The precessionfrequency was determined from the vortex position taken directly from thedensity profile. The vortex core precessed in the same direction as the vortexfluid flow around the core. The obtained result of 1.8 Hz agrees well with an-alytical results based on the Magnus force picture (Svidzinsky 2000a,b) andmore precise numerical simulations (Jackson 1999, McGee 2001, Feder 2001a).In some results, the vortex core disappeared from the observed images dur-ing the time evolution. However, these were not associated with the decayof vortices because there was no evidence of radial spiral motion due to en-ergy dissipation, which could be caused by the thermal drag (Fedichev 1999)or the sound radiation from a moving vortex (Lundh 2000, Parker 2004).Subsequently, an experiment using the surface-wave spectroscopic techniquerevealed that vortices were actually present for a long time in the condensate(Haljan 2001). The main cause of the disappearance was the tilting motion ofa vortex, the lowest odd-order normal mode of a single-vortex state, which issensitive to small trap anisotropy (Svidzinsky 2000b).
Vortex dynamics are greatly affected by the overall collective motion of thecondensate because of the mesoscopic nature of the system. Here, we detailseveral interesting results of coupled dynamics. The determination of the fre-quency of the collective modes allows precise measurement of the angularmomentum carried by the quantized vortices.
Transverse quadrupole mode
The collective modes of a trapped condensate can be classified by expressingthe density fluctuations in terms of polynomials of degree in the Cartesian co-ordinates ( x , x , x ) = ( x, y, z ). The quadrupole modes are characterized bya density fluctuation with a polynomial of second order, e.g., δn = P p ij x i x j ,which gives six normal modes. In an axisymmetric harmonic potential, linearcombinations of the diagonal components p xx , p yy , and p zz describe three nor-mal modes: one transverse mode with m z = 2 and two radial-breathing modeswith m z = 0, where m z is the projected angular momentum on the symmetryaxis. The remaining three normal modes are associated with the off-diagonalcomponents p xy , p yz , and p zx , which are scissors modes (Gu`ery-Odelin 1999,Marago 2000).The first study was done for the excitation of two transverse quadrupole modeswith m z = ± m z = ± ω + − ω − =2 h L z i /mN h x + y i because of broken rotational symmetry (Zambelli 1998),where h i stands for the average within the condensate. The increase causesprecession of the eigenaxes of the quadrupole mode at an angular frequency˙ θ = ( ω + − ω − ) / | m z | . By measuring the angular velocity of this precession,we can determine the mean angular momentum h L z i of the condensate. Thisspectroscopic method has also been used to characterize the tilting motion ofa vortex (Haljan 2001) and the winding number of a single vortex (Leanhardt2002).Excitation of the transverse quadrupole mode yields further interesting vor-tex dynamics. The ENS group observed that when the superposition of the m z = ± m z = − m z = +2 mode (Bretin2003). A possible physical origin of this phenomenon is that the m z = − ω K ≃ ( ~ k / m ) ln(1 /kξ ) ( kξ ≪ − ~ . Because of the negative angular momentum with respect to thevortex winding number, this mechanism is effective only for the m z = − ω − = 2 ω K and angular momentum conservation, anexcitation of the quadrupole mode m z = − k and − k , while angular momentum conservationforbids the decay of the m z = +2 mode.(ii) Gyroscope motion
What happens when the other quadrupole modes are excited in a condensatewith a vortex? The Oxford group studied the response of a condensate with6 §
4a vortex when the xz or yz scissors modes are excited (Hodby 2003). Similarto the case of transverse quadruople modes, in the presence of the vortex, theplane of oscillation of a scissors mode precesses slowly around the z axis. Inpolar coordinates, the scissors oscillation is in the θ direction and the pre-cession is in the φ direction, as shown schematically in Fig. 6. This can beregarded as a kind of gyroscope motion of the vortex line (Stringari 2001). Fig. 6. Left: schematic picture of gyroscope motion of the experiment of Ref. (Hodby2003). The scissors mode involves a fast oscillation of the small angle θ between thecondensate normal axis and the z axis. When a vortex is present, the plane of thisoscillation (initially the xz plane with φ = 0) slowly precesses through angle φ aboutthe z axis. Right: data of the evolution of the tilt angle θ projected onto the xz plane, when the scissors mode is initially excited (a) in the xz plane and (b) in the yz plane. (Taken from (Hodby 2003). Reprinted with permission from APS.) The relationship between the precession rate and h L z i can be derived by con-sidering the scissors mode as an equal superposition of two counter-rotating m z = ± z axis at the frequency of thescissors oscillation, ω ± = ω sc . The symmetry and degeneracy of these modesare also broken by the axial angular momentum h L z i . By applying a similarargument as that for transverse quadrupole modes (Stringari 2001), the pre-cession rate is related to the frequency splitting, ω + − ω − = h L z i /mN h x + z i ,allowing the angular momentum h L z i to be determined.The precession associated with gyroscope motion can be seen in the resultsof Fig. 6(a) and (b), corresponding to a slowly varying oscillation component.The pattern of increase and decrease of the amplitude is exchanged between (a)and (b), with different directions of the excitation. This is clear evidence of aslow precession along the φ -direction. The results also show that the motion ofthe vortex core exactly follows the axis of the condensate. These observationscan be well reproduced by direct numerical simulations of the 3D GP equation(Nilsen 2003). From the precession rate, the measured angular momentum perparticle associated with a vortex line was found to be 1 . ~ ± . ~ . The energy cost to create a q > q single-quantized vortices, as seen in Eq. (10). This raises an interesting questionas to what happens when such an unstable vortex is created. As it happens, q , angular momentum conservationleads to constraints on the normal mode functions u j ( r ) = u j ( r ) e i ( κ j + q ) θ and v j ( r ) = v j ( r ) e i ( κ j − q ) θ , where κ j denotes the angular momentum quantum num-ber of the mode. For q = 2 there are alternating stable and unstable regionswith respect to the interaction parameter an z = a R | Ψ | dxdy ; the first andsecond regions appears for 0 < an z < . < an z <
16. Numerical sim-ulations demonstrate that when a system is in an unstable region, a doublyquantized vortex decays into two singly quantized vortices (M¨ott¨onen 2003).An experiment by the MIT group studied the splitting process of a doublyquantized vortex and its characteristic time scale as a function of an z =0 (Shin2004). The results show that a doubly quantized vortex decays, but that thelifetime increases monotonically with an z =0 , showing no periodic behavior.This contrary to the above theoretical prediction. Recent numerical studiesof 3D GP equations reveal this mysterious observation, emphasizing that thedetailed dynamical behavior of a vortex along the entire z -axis is relevant forcharacterizing the splitting process in an elongated condensate (Huhtam¨aki2006a, Mu˜noz Mateo 2006).The trigger for splitting instability is likely to be gravitational sag during theformation process with a reversing axial bias field B z (Huhtam¨aki 2006a). Inexperiments, absorption images were restricted to a 30- µ m thick central slice ofthe condensate to increase the visibility of the vortex cores. The experimentalresults in Fig. 7(a) shows that the fastest decay occurs at an z ≃ .
5, consistentwith the BdG and numerical analysis. As the particle number increases, thefirst instability region (0 < an z <
3) moves progressively away from the centralslice toward the edges of the condensate because of the trapping potential. Asa consequence, the splitting instability of the vortex core has to propagatefrom those regions to the center. This process is responsible for the monotonicincrease in the lifetime for an z >
3. According to the theory, a second minimumis expected about an z =0 ≃ .
75. Even though no such minimum occurs,a change in the slope of the predicted curve at an z =0 ≃ .
75 is seen as an z =0 enters the second instability region (Huhtam¨aki 2006a, Mu˜noz Mateo2006). Figures 7 (b)–(e) show the time evolution of the splitting process for an z =0 = 13 .
75. The first and second instability regions correspond to theshaded zones in Fig. 7 (b). This clearly shows that at t = 25 ms, the splittingprocess has already begun in both the edges and the center of the condensate.The different precession frequency along different z slices causes inter-windingof two single-quantized vortices. However, the two vortex cores near the centralslice still overlap (Fig. 7 (c)) and thus cannot be experimentally resolved untilmuch longer times. At t ≈
70 ms (Fig. 7 (d)), the vortex cores begin to8 § Fig. 7. Splitting process of a doubly quantized vortex. (a) Experimental data com-bined with theoretically predicted splitting times as a function of an z =0 for a 4%quadrupolar perturbation acting during 0.3 ms. The splitting is identified by thenumber of visible vortex cores from the density profile of the axial absorption imagestaken from a 30- µ m thick central slice of the condensate. (b)–(e) Time evolutionof the splitting process for an z =0 = 13 .
75, obtained by a 3D simulation of the GPequation. The shaded zones in (b) indicate the instability regions. The correspond-ing axial absorption images of the central slice are also shown at the bottom ofthe figure. The lengths are in units of 6.05 µ m. (Taken from (Mu˜noz Mateo 2006).Reprinted with permission from APS.) disentangle so that they can be unambiguously resolved at t = 75 ms (Fig.7(e)). Thus, there is no contradiction with the theoretical prediction.The physical origin of the periodic appearance of the unstable region is anoma-lous modes with negative eigenvalues. When the eigenvalue of positive-energymodes coincides with the absolute value of the eigenvalue of negative-energymodes, a complex eigenvalue mode are produced through mutual annihilationof these two excitations (Skryabin 2000, Kawaguchi 2004, Jackson 2005, Lundh2006); the total angular momentum of these excitations is also vanished. Sincethe an z dependence of the negative-energy eigenvalues is very different fromthat of positive-energy ones, the above matching condition can be satisfied insequence with increasing an z , which results in the periodic appearance of thecomplex-eigenvalue modes. Thus, splitting instability of a multiply quantizedvortex can be suppressed for a particular trap asymmetry and interactionstrength because the collective excitations depend strongly on the characterof the trapping potential (Huhtam¨aki 2006b). It has also been predicted thatmultiply quantized vortices can be stabilized by introducing a suitable local-ized pinning potential (Simula 2002a) or non-simply connected geometry suchas quartic confinement (Lundh 2002). Very recently, splitting dynamics of aquadruply quantized ( q = 4) vortex was reported (Isoshima 2007).
5. A lattice of quantized vortices in an atomic BEC
We now address the issue of a rapidly rotating BEC where many vortices havebeen nucleated and arranged into a regular triangular lattice (Abo-Shaeer2001, Coddington 2004). We first present a discussion of the equilibrium prop-erties of a rapidly rotating condensate, and then present the basic theoreticalbackground for its description. The equilibrium properties of vortex lattices ina trapped BEC have been extensively studied by the JILA group (Schweikhard2004a, Coddington 2004). We next discuss the collective dynamics of an as-sembly of vortices in a trapped BEC. Finally, we discuss an unconventionalvortex phase which occurs in the presence of an externally applied potentialcreated by laser beams.
For very large Ω, the rotation of the superfluid mimics a rigid body rotationwith ∇ × v s = 2 Ω by forming a vortex lattice. Using the fact that the vorticityis given by the form ∇ × v s = κδ (2) ( r ⊥ )ˆ z , we find that the average vorticityper unit area is given by ∇× v s = κn v ˆ z , where n v is the number of vortices perunit area. Hence, the density of the vortices is related to the rotation frequencyΩ as n v = 2Ω /κ (Feynman 1955). This relation can be used to estimate themaximum possible number of vortices in a given area as a function of Ω. Asshown below, the properties of a vortex lattice can be characterized by thenearest-neighbor lattice spacing ∼ b = ( ~ /m Ω) / , defined by the area pervortex n − v = πb , and by the radius of each vortex core ∼ ξ .Note that the GP energy functional of Eq. (11) in a rotating frame can berewritten as E ′ = Z d r ~ m (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) − i ∇ − m ~ Ω × r (cid:19) Ψ (cid:12)(cid:12)(cid:12)(cid:12) + V eff | Ψ | + g | Ψ | ! , (14)where V eff = m ( ω ⊥ − Ω ) r / mω z z / ω ⊥ . Because the first term in Eq. (14)reads ~ ( ∇| Ψ | ) / m + m ( v s − Ω × r ) | Ψ | /
2, it can be neglected in the TF limitand the rigid-body rotation limit v s = Ω × r . Then, the TF radius is given by R ⊥ (Ω) = R ⊥ / [1 − (Ω /ω ⊥ ) ] / with R ⊥ for nonrotating condensate and anaspect ratio of λ rb = R ⊥ (Ω) /R z = λ/ [1 − (Ω /ω ⊥ ) ] / . Thus, measuring λ rb will give the rotation rate of the condensate (Raman 2001, Haljan 2001). Also,in the high rotation limit Ω → ω ⊥ , the condensate flattens out and reaches aninteresting quasi-2D regime; current experiments have reached Ω /ω ⊥ ≈ . § Experimental observations (Abo-Shaeer 2001, Engels 2002) and numerical sim-ulation of the 3D GP equation (Feder 2001b) have revealed that for a finite-sizetrapped BEC, the vortex density in a lattice is lower than the rigid-body esti-mate and the lattice is remarkably regular. Sheehy and Radzihovsky explainedthese points analytically in the TF limit (Sheehy 2004a,b); they derived asmall, radial-position-dependent, inhomogeneity-induced correction term tothe vortex density as n v ( r ) = 2Ω κ − π R ⊥ (Ω) ( R ⊥ (Ω) − r ) ln e − κ π Ω ξ . (15)This result indicates that the vortex density is always lower than the rigid-body estimate of the first term in Eq. (15). Also, the vortex density is higher inregions where the condensate density is most uniform, that is, the central partof a harmonically trapped gas. However, the position-dependent correction issmall ( n v changes less than a few % over a region in which the atom densityvaries by 35%), which seemingly causes regularity of the lattice. These resultshave been confirmed by a detailed experimental study (Coddington 2004). Itshould be also noted that inhomogeneity in the area density of vortices canalso be derived in the limit of the lowest Landau level (Watanabe 2004, Cooper2004, Aftalion 2005, Baym 2007), as explained below. Note that the first term in Eq. (14) can be identified as the Hamiltonian H L = ( − i ~ ∇ − e A /c ) / m of a charge − e particle moving in the xy planeunder a magnetic field B ˆ z with a vector potential A = ( mc/e ) Ω × r . If theinteraction is neglected ( g = 0), the eigenvalues of the Hamiltonian of Eq. (14)forms Landau levels as ǫ n,m / ~ = ω ⊥ + n ( ω ⊥ + Ω) + m ( ω ⊥ − Ω), where n isthe Landau level index and m labels the degenerate states within a Landaulevel. The lowest energy states of two adjacent Landau levels are separatedby ~ ( ω ⊥ + Ω), whereas the distance between two adjacent states in a givenLandau level is ~ ( ω ⊥ − Ω); when Ω = ω ⊥ , all states in a given Landau levelare degenerate. Physically, this corresponds to the case where the centrifugalforce exactly balances the trapping force in the x - y plane, and only the Coriolisforce remains. The system is then invariant under translation and hence hasmacroscopic degeneracy. This formal analogy has led to the prediction thatquantum Hall-like properties would emerge in rapidly rotating BECs (Ho 2001,Viefers 2000, Cooper 2001, Sinova 2002, Regnault 2003, 2004, Cazalilla 2005,Chang 2005, Rezayi 2005, Morris 2006).Interaction effects mix different ( m , n ) states. Because the density ¯ n of thesystem drops as Ω → ω ⊥ , the interaction energy ∼ g ¯ n can become small com- ~ ω ⊥ . In this limit, particles shouldcondense into the lowest Landau level (LLL) with n = 0. Then, the systementers the “mean field” quantum Hall regime, where the wave function can bedescribed by only the LLL orbitals with the form Ψ LLL = A Q j ( z − z j ) e − r / a ,where z = x + iy , z j is the positions of vortices (zeros), and A is a normaliza-tion constant. The minimization of Eq. (14) using the ansatz Ψ LLL is a use-ful theoretical prescription to tackle the properties of rapidly rotating BECs(Watanabe 2004, Cooper 2004, Aftalion 2005, Sonin 2005b, Aftalion 2006,Cozzini 2006).Schwaikhard et al . created rapidly rotating BECs by spinning condensatesto Ω /ω ⊥ > .
99 (Schweikhard 2004a). When the condensate enters the LLLregime, characteristic equilibrium properties appear, as described below.(i)
Global structure
In the LLL limit, the radial condensate density profile with uniform vortex dis-tribution has been predicted to change from a parabolic TF profile to a Gaus-sian profile as Ω is increased (Ho 2001). However, no signs of such crossoverhave been found in experiments; even when the dynamics were restricted tothe LLL, the density profile remained a parabolic TF profile as Ω → ω ⊥ (Schweikhard 2004a, Coddington 2004). This result can be seen qualitativelyfrom the energy minimization under the LLL limit with nonuniform vortexdensity (Watanabe 2004, Cooper 2004, Aftalion 2005). As long as the totalnumber of vortices is much larger than unity ( N v ≫ E ′ = Ω N + Z d r " ( ω ⊥ − Ω) r a n ( r ) + bg n ( r ) , (16)plus terms involving the trapping potential in the z -direction. Here, n ( r ) = h| Ψ LLL | i is the coarse-grain averaged density profile in order to smooth therapid variations at the vortex cores. Then, the interaction parameter g isrenormalized to bg , where b = h| Ψ LLL | i / h| Ψ LLL | i is the Abrikosov param-eter. The energy (16) is then minimized by the TF profile, n ( r ) = [ µ − Ω − ( ω ⊥ − Ω) r /a ] /bg .Since the energy (16) depends only on the smoothed density, the vorticesadjust their locations so that the smoothed density becomes an invertedparabola. In the LLL regime, the relation between the condensate density n ( r )and the mean vortex density, n v ( r ) = P j δ ( r − r j ) is given by 4 − ∇ ln n ( r ) = − a − + πn v ( r ) (Ho 2001, Aftalion 2005). If the density profile is Gaussian, thevortex density is constant. However, for a TF profile, n v ( r ) = 2Ω κ − πR ⊥ − r /R ⊥ ) . (17)This result is similar to Eq. (15) in the low rotation regime, where the coef-ficient of the second term is different. Since the second term is smaller than2 § ∼ a /R ⊥ ≃ N − v , the density of the vortex lattice is basicallyuniform, consistent with the argument in Sec. 5.1.1. Turning the argumentaround, very small distortions of the vortex lattice from perfect triangular canresult in large changes in the global density distribution such that the TFform is energetically favored rather than the Gaussian.(ii) Vortex core structure and fractional area f r a c t i ona l c o r e a r ea W / m = ( G LLL ) -1 (cid:1) Increasing WW =0.976 W =0.37 (cid:1) ~~~ Fig. 8. Fraction of the condensate surface area occupied by vortex cores A (see text)versus the inverse of the LLL parameter Γ − , measured after condensate expansion.The data clearly show a saturation of A as Ω /ω ⊥ →
1. The dashed line representsthe prediction for the pre-expansion value in the case of a low rotation rate. Thedotted line shows the results for a saturated value of A in the LLL limit. (Takenfrom (Schweikhard 2004a). Reprinted with permission from APS.) Another interesting characteristic of the LLL is that the vortex core is of thesame size as the distance b = ( ~ /m Ω) / between adjacent vortices. The ra-dius of a single vortex core is of order ξ = 1 / √ πna , so that a vortex corewould begin to overlap the next at ξ ∼ b . This gives an upper-critical rotationfrequency Ω c ∼ πna ~ /m ∼ − rad/sec, which is an experimentallyaccessible rate. However, there is no phase transition associated with vor-tex cores overlapping in a rotating condensate. Rather, vortex cores begin toshrink as the intervortex spacing becomes comparable to the healing length ξ , and eventually the core radius scales with the intervortex spacing (Fischer2003, Baym 2004b, Watanabe 2006).Figure 8 shows the measured fractional area, defined as A = r v /b = n v πr v , asa function of the inverse of the LLL parameter Γ LLL = µ/ ~ Ω (Schweikhard2004a, Coddington 2004). The linear rise of A at small Ω occurs becausethe core size remains constant, while n v increases linearly with Ω. Explicitly,the core radius was estimated numerically as r v = 1 . ξ (Schweikhard 2004a),and n v = m Ω /π ~ by neglecting the effect of inhomogeneity. These values yield A = 1 . − , shown by the dashed line in Fig. 8, where n = 0 . n peak and µ = gn peak were used for the estimation. The flattening of A with increasingΩ is a consequence of the vortex radius scaling with the intervortex spacing. A = 0 . p -state structure | Ψ core ( r ) | ∼ [( r/b ) exp( − r / b )] as the profileof a vortex core in the LLL limit, where b is regarded as the radius of the(cylindrical) Wigner–Seitz cell around a given vortex. The data in Fig. 8 showthe expected initial linear rise, with the predicted scaling of the core radiuswith intervortex spacing. More detailed theoretical studies which treat the corestructure explicitly obtain excellent agreement with the experimental results(Cozzini 2006, Watanabe 2006). As shown in Sec. 4.2.2, vortex states undergo interesting responses to excita-tion of the transverse quadrupole modes with m z = ±
2. Similar studies havebeen made for rapidly rotating BECs. In this case, the dispersion relationof the quadrupole modes is given by ω ± = q ω ⊥ − Ω ± Ω (Cozzini 2003),which has been measured experimentally (Haljan 2001). When Ω → ω ⊥ , wehave ω +2 → ω ⊥ and ω − →
0, reflecting the tendency of the system to becomeunstable against quadrupole deformation. Excitation of the quadrupole modefor Ω → ω ⊥ induces large deformations of the condensate and nonequilibruimdynamics of vortex lattices (Engels 2002). Interestingly, when the m z = − m z = +2 modedissolved the regular lattice, where the vortex lines were randomly arrangedin the x - y plane but were still strictly parallel along the z -axis.As stated in Sec. 5.1, a centrifugal force distorts the cloud into an extremelyoblate shape, and thus the rotating cloud approaches the quasi-2D regime.Excitation of an axial breathing mode ( m z = 0) has been used to confirmthe 2D signature of a rapidly rotating BEC (Schweikhard 2004a). For a BECin the axial TF regime, an axial breathing frequency ω B = √ ω z has beenpredicted in the limit Ω /ω ⊥ → ω B = 2 ω z is expectedfor a noninteracting gas, expected for µ < ~ ω z . Schweikhard et al . observeda crossover of ω B from √ ω z to 2 ω z with increasing Ω ( µ ∼ ~ ω z ). Also,excitation of the scissors mode in a condensate with a vortex lattice induces acollective tilting mode of the vortex array (the lowest-energy Kelvin wave ofthe lattice) (Smith 2004), referred to as an anomalous scissors mode (Chevy2003). The dynamics of vortex lattices itself raises many interesting problems. Itshould be possible to propagate collective waves in a transverse direction tothe vortex lattice in the superfluid, called Tkachenko (TK) modes. For an in-4 § ω TK ( k ) = q ~ Ω / mk .The TK modes of a vortex lattice in a trapped BEC have been analyzed theo-retically (Baym 2003, Mizushima 2004, Baksmaty 2004, Baym 2004a, Cozzini2004, Gifford 2004, Sonin 2005a,b) and observed experimentally (Coddington2003).Experimentally, TK modes have been excited by the selective removal of atomsat the center of a condensate with a resonant focused laser beam, or by theinsertion of a red-detuned optical potential at the center to draw atoms intothe middle of the condensate. The former method has also been used to createlong-lived vortex aggregates (Engels 2003). In the experiment, the TK modeswere identified by the sinusoidal displacement of the vortex cores with the ori-gin at the center of the condensate; see Fig. 9(A). TK modes can be classifiedby the quantum number ( n, m ), associated with radial and angular nodes, ina presumed quasi-2D geometry.To explain the observed frequency of the TK mode ω ( n,m ) , the effects of com-pressibility should be taken into account. According to the elastohydrody-namic approach developed by Baym (Baym 2003), the TK frequency is de-scribed by the compressional modulus C and shear modulus C of the vortexlattice, included in the elastic energy E el = Z d r C ( ∇ · ǫ ) + C ∂ǫ x ∂x − ∂ǫ y ∂y ! + ∂ǫ x ∂y + ∂ǫ y ∂x ! , (18)where ǫ ( r , t ) is the continuum displacement field of the vortices from theirhome positions. In the incompressible TF regime, C = − C = n ~ Ω /
8. Then,the upper branch of the energy spectrum follows the dispersion law ω =4Ω + c k with sound velocity c = q gn/m , being the standard inertial modeof a rotating fluid and having a gap at k = 0. Conversely, the low frequencybranch corresponds to the TK mode and has ω − = ~ Ω4 m c k + c k . (19)For large k , this reproduces the original TK frequency ω TK , while for small k it exhibits the quadratic behavior ω − ≃ q ~ / m Ω ck . The transition between k and k dependence occurs at k ∼ Ω /c > R − ⊥ . This suggests that the effectsof compressibility, characterizing the k dependence, play a crucial role in theTK mode. Thus, this regime is distinguished from the usual incompressible TFregime as the “soft” TF regime. When the finite compressibility is included,the observed values of ω (1 , are well explained (Baym 2003, Cozzini 2004,Sonin 2005a). First-principle simulations based on the GP formalism also agreeexcellently with the experimental data (Mizushima 2004, Baksmaty 2004).In the LLL limit, corrections to the elastic shear modulus C of the vortexlattice are important; with increasing Ω, its value eventually reaches C LLL2 ≃ (81 / π ) mc n (Baym 2003). Using this value, Schweikhard et al. compared Fig. 9. (A) TK mode ( n, m ) = (1 ,
0) at N = 1 . × and Ω = 0 . ω ⊥ , fittedby sine fits. (B) Comparison of measured TK mode frequency ω (1 , (solid symbols)versus the theoretical value (Baym 2003), using the vortex lattice shear modulus C TF2 in the TF limit (circles) and C LLL2 in the LLL regime (stars). Note that both N and Γ LLL decrease as ˜Ω = Ω /ω ⊥ increases. For Γ LLL ≃ N = 7 . × and ˜Ω ≃ . the measured ω (1 , with the theoretical prediction, finding a crossover of ω (1 , from the TF results to the LLL results, as shown in Fig. 9(B). However, moredetailed theoretical analysis in the LLL limit revealed that the value of theshear modulus is estimated as 0 . mnc , which is a factor of 10 larger than(81 / π ) mnc (Sinova 2002, Sonin 2005b, Cozzini 2006). This indicates thateven though the equilibrium properties in the experiment are consistent withthe LLL picture, the data of the TK frequency are still far from the LLLlimit. One possible explanation of this discrepancy is an underestimate of therotation rate from the aspect ratio of the cloud due to the defocus of theimaging camera or the breakdown of the TF theory (Watanabe 2007). For a rotating condensate with a frequency Ω in a harmonic potential (1 / mω ⊥ r ,the centrifugal potential cancels the confinement, thus preventing a BEC fromrotating at Ω beyond ω ⊥ . This restriction can be avoided by introducing anadditional quartic potential, so that the combined trapping potential in the xy plane becomes V ex ( r ) = (1 / mω ⊥ ( r + λr /a ), where the dimensionlessparameter λ characterizes the relative strength of the quartic potential. Theproperties of a rotating condensate in an anharmonic potential have recentlyattracted a lot of theoretical attention (Fetter 2001b, Lundh 2002, Fischer2003, Kasamatsu 2002a, Kavoulakis 2003, Aftalion 2004, Jackson 2004a,b,Fetter 2005, Danaila 2005, Kim 2005, Bargi 2006, Fu 2006).The vortex phases in an anharmonic trap are quite different from those in aharmonic trap, since it is possible to rotate the system arbitrarily fast. Thepredicted phase diagram of the vortex states as a function of interparticle in-teraction strength versus rotation rate is shown in Fig. 10. For small Ω, theequilibrium state is the usual vortex lattice state. As Ω increases, the vortices6 § Fig. 10. Phase diagram of the vortex state with an additional quartic potentialfor λ = 1 / g = 4 πn z a ) versusrotation rate (Ω c = Ω). The dashed curve denotes the onset of a central densityhole (VLH) in the uniform vortex lattice state (VL), obtained by TF analysis asΩ h = 1 + 2 √ λ (3 √ λg/ π ) / (Fetter 2005). The dashed-dotted curve and the solidpoints ( • ) joined by solid lines show the phase boundary Ω c between the annularcondensate with a circular array of vortices (AA) and the giant vortex (GV) state,which are determined by two different analytical methods (Fu 2006). The opentriangles ( △ ) are the values of Ω c determined by the GP solution (Fetter 2005) andthe open squares ( (cid:3) ) are the results using an improved variational approach (Kim2005). The filled square ( (cid:4) ) with error bars gives the approximate bounds on Ω c determined numerically for g = 125 (Kasamatsu 2002a). For a weakly interactinglimit g ∼
10, a much richer structure was revealed (Jackson 2004a,b). (Taken from(Fu 2006). Reprinted with permission from APS.) begin to merge in the central region and the centrifugal force pushes the parti-cles towards the edge of the trap. This results in a new vortex state consistingof a uniform lattice (multiple circular arrays of vortices) with a central densityhole. The central hole becomes larger with increasing Ω, and the condensateforms an annular structure with a single circular array of vortices. A furtherincrease of Ω stabilizes a giant vortex , where all vortices are concentrated inthe single hole (Fischer 2003).A combined harmonic-plus-quartic potential was formulated by the ENS groupby superimposing a blue detuned laser with the Gaussian profile (Bretin 2003).Since the waist w of the beam propagating along the z -axis is larger than thecondensate radius, the potential created by the laser U exp( − r /w ) canbe written as U ( r ) ≃ U (1 − r /w + 2 r /w ). The second term leads to areduction of the transverse trapping frequency ω ⊥ and the third term providesthe desired quartic confinement, giving ω ⊥ / π = 65 Hz and λ ≃ − for V ex ( r ). Figure 11 shows experimental images of the condensate density as therotation frequency Ω is increased (Bretin 2003). For Ω < ω ⊥ , the vortex latticeis clearly visible. When Ω > ω ⊥ , however, the vortices becomes graduallydifficult to observe and the images become less clear for Ω = 1 . ω ⊥ (= 2 π × (a)(b) Fig. 11. (a) Density profiles of a rapidly rotating condensate in a quadratic plusquartic potential for various stirring frequencies Ω / π . For these data ω ⊥ / π = 65Hz. (b) Ground state structure obtained by numerical simulations with a parametercorresponding to the experiment of Ref. (Bretin 2003). The rotation frequency isΩ / π = 60, 64, 66, 70.6, 73 Hz (respectively, Ω /ω ⊥ = 0.92, 0.98, 1.01, 1.08, 1.11)from left to right, where Ω / π = 70 . h . The first two rows show3D views of the vortex lattice as isosurfaces of low atomic density. In the bottomrow, the density distribution is integrated along the z -axis. (Taken from (Bretin2003) and (Danaila 2005). Reprinted with permission from APS.) Despite the visibility of the cores, the angular momentum of the condensatemonotonically increased, confirmed by the measurement of R TF and surfacewave spectroscopy. Hence, the most plausible explanation for this mysteriousobservation is that the vortex lines are still present, but strongly bent whenΩ > ω ⊥ . This bending may occur due to the finite temperature effect onthe fragile vortex lattice at a high rotation rate; numerical simulations of the3D GP equation show that, when looking for the ground state of the systemusing imaginary time evolution of the GP equation, much longer imaginarytimes were required to reach a well ordered vortex lattice for Ω > ω ⊥ than forΩ < ω ⊥ . Compared with the numerical results shown in the bottom in Fig.11, the condensate should still have an ordered visible lattice even for Ω ≥ ω ⊥ .To observe the density hole at the center, it is necessary to rotate at a slightlyfaster rate than the upper frequency used in this experiment.8 § Rotating BECs combined with an optical lattice are an interesting system,which has two competing length scales, vortex separation and the periodicityof the optical lattice. The structure of the vortex lattice is strongly depen-dent on the externally applied optical lattice. Various vortex phases appeardepending on the number of vortices per pinning center, i.e., the filling factor.The JILA group formulated a rotating optical lattice using a rotating mask(Tung 2006). Such a rotating optical lattice provides a periodic pinning po-tential which is static in the corresponding rotating frame. The authors haveobserved a structural crossover from a triangular to a square lattice by in-creasing the potential amplitude of the optical lattice, as shown in Fig. 12.These observations are consistent with theoretical studies (Reijnders 2004, Pu2005). The rotating optical lattice provides new phenomena in vortex physicsfor rotating bosons; such as the realization of a driven vortex system in a peri-odic array (Kasamatsu 2006) or strongly correlated phases in rotating bosons(Bhat 2006).
Fig. 12. Images of rotating condensates pinned to a co-rotating optical lattice atΩ = 0 . ω ⊥ with pinning strength U pin /µ = (a) 0.049, (b) 0.084, (c) 0.143, showingthe structural crossover of the vortex lattice. (a)–(c) show absorption images of thevortex lattices after expansion. (d)–(f) are the Fourier transforms of the images in(a)–(c). k is taken by convention to be the strongest peak; k tr1 , k sq , and k tr2 are at60 ◦ , 90 ◦ , and 120 ◦ , respectively, from k . (Taken from (Tung 2006). Reprinted withpermission from APS.)
6. Other topics and future studies
In this section, we discuss other intriguing problems associated with quantizedvortices in atomic BECs. Since many theoretical works have predicted novelproperties of vortices under a variety of situations, it is impossible to refer to
Although we have dealt primarily with vortices in repulsively interactingBECs, vortices in attractively interacting BECs characterized by a negatives-wave scattering length a < a < N lies below a critical value N c ∼ a ho / | a | (Bradley 1995). Thisis because the self-focusing can be balanced by the kinetic energy (quantumpressure), which tends to defocus the wave function.Since a central vortex state reduces the peak density, it may help stabilizationof a trapped condensate with a < < ω ⊥ excitationof a vortex in a harmonically trapped BEC with a < BECs of chromium atoms have recently been created (Griesmaier 2005), ex-hibiting a larger magnetic-dipole moment ( µ d = 6 µ B ; µ B is the Bohr mag-neton) than those of typical alkali atoms ( µ d ≃ µ B ). This opens the doorfor studying the effect of anisotropic long-range interactions in BECs. Theinteraction potential between two magnetic dipoles µ d ˆ e separated by r isgiven by V dd ( r ) = ( µ µ d / π )(1 − θ ) /r , where µ is the vacuum mag-netic permeability and ˆ e · ˆ r = cos θ . Such dipole–dipole interactions con-tribute to the GP equation as a nonlocal mean-field potential as i ~ ∂ψ/∂t =[ − ~ ∇ / m + V ex + g | ψ | + R d r ′ V dd ( r ′ − r ) | ψ ( r ′ ) | ] ψ . Since the scatteringlength can be tuned to zero by a Feshbach resonance technique, we can obtainnovel quantum ferrofluids dominated by the dipole–dipole interaction (Lahaye2007).The principal effect of V dd on the equilibrium properties of a condensate is tocause distortion of its aspect ratio so that it is elongated along the directionof the dipoles. This feature affects the stability of a vortex in a dipolar BEC;the thermodynamic critical rotation frequency Ω c decreases for a condensate0 § ω ⊥ < ω z ), while it increases a cigar-shaped trap( ω ⊥ < ω z ), compared to that of a conventional BEC (Sec. 3.1) (O’Dell 2007).Interestingly, the critical frequency Ω c can become larger than the onset ofthe dynamical instability of a rotating condensate (see Sec. 3.2.2). This isan intriguing regime where a rotating dipolar BEC is dynamically unstablebut vortices will not enter (Bijnen 2007). Numerical simulations show that thestructure of vortices has a craterlike shape for ˆ e k ˆ z and has an elliptical shapefor ˆ e ⊥ ˆ z (Pu 1999). Rapidly rotating dipolar BECs possess a rich variety ofvortex phases characterized by different symmetries of the lattice structure(Cooper 2005, Zhang 2005, Komineas 2007). At sufficiently high rotation rates, a vortex lattice should melt via quan-tum fluctuations (Sinova 2002) and the system should then begin to entera strongly-correlated vortex liquid phase. Exact diagonalization studies forsmall number of bosons have revealed that the ground states exhibit stronganalogy with the physics of electronic fractional quantum Hall states (Viefers2000, Cooper 2001, Paredes 2002, Regnault 2003, 2004, Ghosh 2004a); forspecific filling factors ν = N/N v , i.e., the ratio of the total number and thevortex number, the ground state possesses incompressibility characterized bythe energy gap. For example, at angular momentum L z = N ( N − N v = 2 N , the exact ground state is an N -particle fully symmetric Laughlinwave function adopted for bosons: Ψ( r , r , · · · , r N ) ∼ Q j = k ( z j − z k ) e − Σ l r l / a with z j = x j + iy j .The conditions for the formation of these states have been expressed as ν ≤O (1). It can be seen that an unrealistically high rotation is necessary to satisfythe condition; observed filling factors are always greater than 100 (Schweikhard2004a), which are still deeply within the mean-field GP regime. To overcomethis difficulty, insertion of a 1D optical lattice along the z -direction has beenproposed to enhance the quantum fluctuations of the vortices (Martikainen2003, Snoek 2006). Then, the optical lattice divides the condensate into pan-cake fractions coupled by a tunneling process between near neighbors and N in a single pancake is greatly reduced. Achieving this regime experimentallyremains an important challenge. In 2D systems with continuous symmetry, true long-range order is destroyed bythermal fluctuations at any finite temperature. For 2D Bose systems, a quasi-condensate can be formed with a correlation decaying algebraically in space,where superfluidity is still expected below a certain critical temperature. This2D phase transition is closely connected with the emergence of thermally acti-vated vortex–antivortex pairs, known as the Berezinskii–Kosterlitz–Thouless(BKT) phase transition occurring at T = T BKT (Berezinskii 1972, Kosterlitz
T < T
BKT , isolated free vortices are absent; vortices always existonly in the form of bound pairs, formed by two vortices with opposite circula-tions. The contribution of these vortex pairs to the decay of the correlation isnegligible, and the algebraic decay is dominated by phonons. For
T > T
BKT ,the free vortices form a disordered gas of phase defects and give rise to anexponential decay of the correlation.Recently, the BKT transition was observed experimentally in ultracold atomicgases (Hadzibabic 2006, Schweikhard 2007, Kr¨uger 2007, Hadzibabic 2007).In the ENS experiment, a 1D optical lattice was applied to an elongated con-densate, splitting the 3D condensate into an array of independent quasi-2DBECs (Hadzibabic 2006). The interference technique revealed the temperaturedependence of an exponent of the first-order correlation function of the fluctu-ating 2D bosonic field (Polkovnikov 2006). A universal jump in the superfluiddensity characteristic of the BKT transition was identified by observing thesudden change of the exponent, where the finite size effect causes a finite-width crossover rather than a sharp transition. Surprisingly, the microscopicorigin of this transition, i.e., whether or not it is a BKT type transition, wasdirectly clarified from the image of the interference of the two 2D condensates.If isolated free vortices are present in either of two condensates, the interfer-ence fringes exhibit dislocations. Such a dislocation has been observed in thehigh- T region of the crossover, supported by the theory using classical fieldsimulations (Simula 2006).In contrast, the JILA group inserted a 2D optical lattice into a condensateto create 2D bosonic Josephson junction arrays (Schweikhard 2007). Eachcondensate was localized at a site j . Each had an individual phase θ j and wasseparated by a potential barrier from the nearest neighbors. This system canbe mapped to the XY model, H = − J P h j,j ′ i cos( θ j − θ j ′ ), where J denotesthe tunneling coupling and the sum is restricted within nearest neighbors(Trombettoni 2005). The XY model is expected to exhibit a BKT transitionat T BKT ≃ J from free-energy considerations. Direct imaging of vortex coresand the systematic determination of J revealed evidence for a gradual increasein the number of isolated free vortices at T ≥ J , consistent with the BKTcrossover picture.A related work is vortex formation by merging three uncorrelated BECs thatare initially separated by a triple well potential (Scherer 2007). Dependingon the relative phases between the condensates and merging rate, vorticesformed stochastically without applying rotation. This situation is useful toclarify the mechanism of spontaneous vortex generation through the Kibble-Zurek mechanism (Kibble 1976, Zurek 1985) during rapid phase transition(Leggett 1998, Kasamatsu 2002b). Another important issue in vortex physics is to elucidate the vortex phases inmulticomponent (spinor) BECs. Multicomponent order parameters allow the2 § He and uncon-ventional superconductors, and theories in high-energy physics and cosmology.A few experimental works have investigated the properties of composite vor-tices in spinor BECs (Leanhardt 2003, Schweikhard 2004b, Sadler 2006). Areview of this topic is presented in Ref. (Kasamatsu 2005b) and referencestherein.
Quantum degenerate Fermi gases provide a remarkable opportunity to studystrongly interacting fermions. In contrast to other Fermi systems, such assuperconductors, neutron stars, or the quark–gluon plasma, these gases havelow densities and their interactions can be precisely controlled over a widerange by using a Feshbach resonance technique. For small and negative valuesof the scattering length a the equation of state approaches the limit of anoninteracting Fermion gas, while for small and positive values the systembehaves as bosons of tightly-bound molecules. Therefore, we can study thecrossover from a BEC of molecules to a Bardeen–Cooper–Schrieffer (BCS)superfluid of loosely-bound Cooper pairs when an external magnetic field isvaried across a Feshbach resonance. Recent topics in this rapidly growing fieldare presented in the comprehensive review paper Ref. (Giorgini 2007).Decisive evidence for fermion superfluidity was obtained from observationsof long-lived vortex lattices in a strongly-interacting rotating Fermion gas(Zwierlein 2005). Rotation was applied to an ultracold gas of Li atoms in | F = 1 / , m F = ± / i , in a similar way as for conventional BECs (see Sec.3.2.1), with magnetic fields covering the entire BEC–BCS crossover region. Acrucial problem was detecting the vortex cores in the BCS limit, because asufficient density depletion at the vortex core could not be expected (Nygaard2003). In the experiment, the visibility of the vortex cores was increased bya rapid sweep of the magnetic field from the BCS to the BEC side duringballistic expansion of the TOF measurement. These measurements stronglysupport the existence of vortices before the expansion even on the BCS sideof the resonance.The most striking aspect of this experiment is that it opens up the possibil-ity of studying vortex physics in a strongly-coupled fermion superfluid in asystematically controlled way. In the strong-coupling limit | a | → ∞ at theresonance, called a unitarity limit, the Fermi gas exhibits universal behavior.Along this line, several microscopic calculations of the vortex structure, basedon the BdG formalism, have been carried out (Bulgac 2003, Sensarma 2006, T c superconductors,and eventually those in room-temperature superconductors.
7. Conclusion
Quantized vortices in atomic Bose–Einstein condensates constitute an activeresearch field, which has drawn the continuous attention of researchers in re-lated fields such as superconductors, mesoscopic systems, nonlinear optics,atomic nuclei, and cosmology, as well as superfluid helium. Quantized vorticesin rotating condensates have provided conclusive evidence for superfluiditybecause they are a direct consequence of the existence of a macroscopic wavefunction that describes the superfluid. The direct imaging of the vortex coresand lines helps us understand the fundamentals of superfluid dynamics. Es-pecially, the inhomogeneous effect caused by a confining potential yields newfeatures in both a slowly rotating regime and a rapidly rotating one, not foundin a bulk superfluid system. The observed phenomena are consistent with theprediction of the Gross–Pitaevskii equation without fitting parameters.Finally, this volume addresses mainly the topics of quantum turbulence . Thefeasibility of generating quantum turbulence in a trapped BEC is discussedby one of the authors (Kobayashi 2007).AcknowledgmentsK.K. acknowledges the support of a Grant-in-Aid for Scientific Research fromJSPS (Grant No. 18740213). M.T. acknowledges the support of a Grant-in-Aidfor Scientific Research from JSPS (Grant No. 18340109) and a Grant-in-Aidfor Scientific Research on Priority Areas (Grant No. 17071008) from MEXT.
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