Vortices and vortex lattices in quantum ferrofluids

arXiv:1606.07107v1 [cond-mat.quant-gas] 22 Jun 2016

Topical Review

Vortices and vortex lattices in quantum ferrofluids A. M. Martin1 , N. G. Marchant1 , D. H. J. O’Dell2 and N. G. Parker1,3 1

School of Physics, University of Melbourne, Victoria 3010, Australia

2

Department of Physics and Astronomy, McMaster University, Hamilton, Ontario, L8S 4M1, Canada 3

Joint Quantum Centre Durham–Newcastle, School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne, NE1 7RU, United Kingdom. E-mail: [email protected]

Abstract. The achievement of quantum-degenerate Bose gases composed of atoms with sizeable magnetic dipole moments has realized quantum ferrofluids, a form of fluid which combines the extraordinary properties of superfluidity and ferrofluidity. A hallmark of superfluids is that they are constrained to circulate through vortices with quantized circulation. These excitations underpin a variety of rich phenomena, including vortex lattices, quantum turbulence, the Berenzinksii-Kosterlitz-Thouless transition and Kibble-Zurek defect formation. Here we provide a comprehensive review of the theory of vortices and vortex lattices in quantum ferrofluids created from dipolar Bose-Einstein condensates, exploring the interplay of magnetism with vorticity and contrasting this with the established behaviour in non-dipolar condensates. Our discussion is based on the mean-field theory provided by the dipolar Gross-Pitaevskii equation, from analytic treatments based on the Thomas-Fermi and variational approaches to full numerical simulations. We cover single vortex solutions, including their structure, energy and stability, and the interactions and dynamics of vortex pairs. Routes to generate vortices in dipolar condensates are discussed, with particular attention to rotating condensates, where surface instabilities drive the nucleation of vortices, and lead to the emergence of rich and varied vortex lattice structures. Finally we present an outlook, including potential extensions to degenerate Fermi gases, quantum Hall physics, toroidal systems and the Berenzinkskii-Kosterlitz-Thouless transition.

CONTENTS

2

Contents 1 Introduction

2

2 Quantum Ferrofluids: Theory and Basic Properties 3 2.1 The dipolar interaction . . . . . . . . . . 4 2.2 The dipolar Gross-Pitaevskii equation . 4 2.3 Dipolar hydrodynamic equations . . . . 5 3 Vortex-Free Solutions and Stability 3.1 Homogeneous condensate . . . . . . . . 3.1.1 Three-dimensional case . . . . . 3.2 Trapped dipolar condensates . . . . . . 3.2.1 Thomas-Fermi solutions . . . . . 3.2.2 Outside the Thomas-Fermi regime: rotons and density oscillations . 3.3 Quasi-two dimensional system . . . . . .

6 6 6 6 6 8 8

4 Single Vortices 10 4.1 Energetics of vortex formation . . . . . 10 4.2 General features of a vortex in a quantum ferrofluid . . . . . . . . . . . . 11 4.3 Vortex in a trapped dipolar BEC in the hydrodynamic regime . . . . . . . . . . 12 4.4 Vortex in the quasi-two-dimensional system . . . . . . . . . . . . . . . . . . . 13 4.4.1 Dipolar mean-field potential due to a vortex: giant anti-dipoles . 14 5 Vortex Pairs: Interactions and Dynamics 15 5.1 Interaction between vortices . . . . . . . 15 5.2 Dynamics of vortex pairs . . . . . . . . . 16 6 Generation of Vortices 17 6.1 Summary of vortex generation methods 17 6.2 Stationary solutions of rotating dipolar condensates in elliptical traps . . . . . . 18 6.2.1 Circular trapping in the x − y plane: = 0 . . . . . . . . . . . 19 6.2.2 Elliptical trapping in the x − y plane: > 0 . . . . . . . . . . . . 20 6.3 Dynamical stability of stationary solutions . . . . . . . . . . . . . . . . . . . 21 6.4 Routes to instability and vortex lattice formation . . . . . . . . . . . . . . . . . 22 6.4.1 Does the final state of the system contain vortices? . . . . . . . . . 23

7 Vortex Lattices 24 7.1 Vortex lattice in a non-dipolar BEC . . 25 7.2 Vortex lattice in a quantum ferrofluid: dipoles perpendicular to the plane . . . 28 7.3 Vortex lattice in a quantum ferrofluid: dipoles not perpendicular to the plane . 31 7.4 Vortex lattices in two-component dipolar BECs . . . . . . . . . . . . . . . . . 32 8 Summary and Outlook 33 8.1 Summary . . . . . . . . . . . . . . . . . 33 8.2 Outlook . . . . . . . . . . . . . . . . . . 33 8.2.1 Dipolar BECs in toroidal traps . 33 8.2.2 Fractional quantum Hall physics in dipolar BECs . . . . . . . . . 34 8.2.3 Dipolar fermions . . . . . . . . . 34 8.2.4 Berezinskii-Kosterlitz-Thouless transition . . . . . . . . . . . . . . . 35 8.2.5 Vortex lattices in the supersolid phase . . . . . . . . . . . . . . . 35 8.2.6 The Onsager vortex phase transition . . . . . . . . . . . . . . . 36 9 Acknowledgements

36

10 References

36

1

Introduction

Ferrohydrodynamics describes the motion of fluids comprised of particles with significant magnetic (or electric) dipole moments [1, 2]. In these fluids the inter-particle interaction includes a dipole-dipole (from henceforth: dipolar) contribution which is long-range and anisotropic, giving rise to unique behaviour such as magnetostriction (or electrostriction), geometric pattern formation and surface ripple instabilities [3]. The distinctive properties of ferrofluids are exploited in applications from tribology to information display and medicine [1]. The first quantum ferrofluid was realized in 2005 with the creation of a Bose-Einstein condensate (BEC) in a vapour of 52 Cr atoms by the Stuttgart group [4]. These atoms have a magnetic dipole moment of 6µB , six times larger than that found in the alkalis which are used in the majority of BEC experiments. 52 Cr atoms therefore have dipolar interactions which are 36 times larger than in standard BECs. Other groups

CONTENTS have also studied 52 Cr BECs [5, 6], as well as BECs made of atoms with even larger magnetic dipoles such as 164 Dy [7, 8], and 168 Er [9]. Many of the signatures of ferrohydrodynamic behaviour have now been observed in these gases, including magnetostriction [10] (this paper coined the term quantum ferrofluid), collapse due to dipolar interactions [11, 12], and recently the quantum analog of the Rosensweig instability, characterised by the formation of droplet states [8, 13]. Additionally, the production of ultracold fermionic 40 K87 Rb [14] polar molecules and the cooling of fermonic 161 Dy [15] and 167 Er [16], all with significant dipole moments, paves the way for a new generation of quantum degenerate Fermi gas experiments, where dipolar interactions dominate. In Fermi gas systems the partially attractive nature of the dipolar interaction opens up the possibility of BCS pairing at sufficiently low temperatures [17, 18, 19, 20, 21, 22, 23, 24, 25]. Excellent reviews of the field of ultracold dipolar gases can be found in Refs. [26, 27, 28, 29, 30]. Vortical structures have been generated experimentally in non-dipolar condensates in the form of single vortices [31, 32], vortex-antivortex pairs [33, 34], vortex rings [35] and vortex lattices [36, 37], as well as disordered vortex distributions characteristic of quantum turbulence [38, 39, 40]. In geometries approaching the one-dimensional limit, so-called solitonic vortices have been formed [41, 42] which share properties between vortices and their one-dimensional analogs: dark solitons. Several reviews exist which summarise the significant experimental and theoretical aspects of vortices and vortex lattices in non-dipolar BECs [43, 44, 45, 46, 47]. Vortices have yet to be observed in quantum ferrofluids, although numerical simulations suggest the formation of vortex rings in the dipolar collapse experiment of Ref. [11] and the formation of vortex-antivortex pairs [48] in the crystallization dynamics of experiment of Ref. [8]. Here we establish the properties of vortices and vortex lattices in quantum ferrofluids, reviewing the theoretical progress that has been made over the last decade. Whilst it is possible to also consider the properties of vortices and vortex lattices in dipolar Fermi gases, this review is confined to the bosonic case and only a brief discussion of fermionic systems will be given in the Summary and Outlook (Section 8). The structure of this review is built upon the philosophy of taking the reader on a journey. This journey starts in Sections 2 and 3 which provide a brief introduction to the properties of dipolar BECs in the absence of vortices. In Section 2 we examine the mathematical form of the dipolar

3 interaction in quantum ferrofluids, and present the most widely used model of quantum ferrofluids - the dipolar Gross-Pitaevskii equation (GPE) - along with its hydrodynamical interpretation. Section 3 builds on this theory to consider the stability of dipolar BECs. Specifically, we look at stability in the ThomasFermi limit, where interactions dominate, and more generally dipolar GPE solutions, in three dimensional and quasi-two dimensional systems. Section 4 focuses on the properties of single vortex lines in three dimensional condensates and single vortices in quasitwo dimensional systems. Section 5 focuses on vortex-vortex and vortex-antivortex dynamics in quasitwo dimensional dipolar BECs. Section 6 addresses the routes to vortex and vortex lattice formation. This focuses primarily on stationary solutions (in the rotating frame) and their dynamical stability, enabling us to ascertain under what conditions it might be expected that vortices will nucleate into the dipolar BEC. Section 7 focuses on how dipolar interactions can induce changes to vortex lattice structures. This revisits previous work and presents a variational approach to elucidate the properties of vortex lattice structures in dipolar condensates. In Section 8 we provide a brief summary and provide an outlook to several topical aspects for future development which have not been covered in the main body of the review. 2

Quantum Ferrofluids: Theory and Basic Properties

The successful Bose-Einstein condensation of gases of 52 Cr atoms [4, 5], 164 Dy [7, 8] and 168 Er [9] have realized BECs with significant dipolar interactions. These long-range and anisotropic interactions introduce rich physical effects, as well as new opportunities to control the gas. A basic example is how the net dipolar interactions depend on the shape of the BEC, as illustrated in Figure 1. In a prolate (elongated) dipolar gas [Figure 1(a)] with the dipoles polarised along the long axis, the net dipolar interaction is attractive, whereas for an oblate (flattened) configuration [Figure 1(b)] with the dipoles aligned along the short axis, the net dipolar interaction is repulsive. As a result, in comparison to s-wave BECs (which we define as systems in which atomatom scattering is dominated by the s-wave channel), a dipolar BEC undergoes magnetostriction by elongating along the direction of an applied polarizing field [49, 50, 51].

CONTENTS

4 Udd (r)e−ik·r dr of the dipolar interaction is [52], ˜dd (k) = Cdd cos2 α − 1 , U 3

R

(2)

where α is the angle between k and the polarization direction.

Figure 1. (a) For a prolate trapped condensate the net dipolar interaction is attractive. (b) For an oblate system the net dipolar interaction is repulsive.

2.1

The dipolar interaction

Consider a dilute BEC composed of atoms of mass m with magnetic dipole moment d and in the limit of zero temperature. The atomic dipoles are polarized in a common direction by an external magnetic field. In the Gross-Pitaevskii theory the low energy atomic scattering between two atoms at r and r0 can be represented by the effective interatomic potential [28, 50], U (r − r0 ) = UvdW (r − r0 ) + Udd (r − r0 ) Cdd 1 − 3 cos2 θ = gδ(r − r0 ) + . (1) 4π |r − r0 |3 The first term is a Fermi pseudo-potential which models the van der Waals interactions through a contact potential with coupling strength g = 4π~2 as /m, where as is the s-wave scattering length. The second is the bare dipolar interaction, where θ is the angle between the polarization direction and the inter-atom vector r − r0 . The parameter Cdd characterises the strength of the dipoles, and for magnetic dipoles is Cdd = µ0 d2 [52], where µ0 is the permeability of free space. Equation (1) also holds for electric dipoles induced by a static electric field E = ˆ for which the coupling constant is Cdd = E 2 α2 /0 kE, [53, 54], where α is the static polarizability and 0 is the permittivity of free space; however, condensates of electric dipoles have yet to be realized. The dipolar interaction Udd , illustrated in Figure 2, is negative for θ = 0, representing the attraction of head-to-tail dipoles, and positive for θ = π/2, representing the repulsion of side-by-side dipoles. At the “magic angle”, √ θm = arccos(1/ 3) ≈ 54.7◦ , the dipolar interaction is zero. It is often convenient to work in momentum ˜dd (k) = F[Udd ] = space. The Fourier transform U

Figure 2. Illustration of the dipole-dipole interaction.

The strength of the dipolar interactions is conveniently parameterized via the ratio [28], εdd = Cdd /3g,

(3)

where g can be tuned between −∞ and +∞ via a Feshbach resonance [55, 56]. In effect εdd parametrizes the relative importance of the anistropic, long-range dipole-dipole interactions to the isotropic, short-range van der Waals interactions. It is defined with a factor of 3 in the denominator such that the homogeneous dipolar condensate is unstable for εdd > 1 - see Section 3.1.1. For 52 Cr, 168 Er and 164 Dy, the natural value of εdd is 0.16 [57], 0.4 [9] and 1.45 [7, 58], respectively. While Cdd is conventionally positive and set to the natural value of the given atom, it is predicted to be possible to reduce Cdd below its natural value, including to negative values, by tilting the polarization direction off-axis and rotating it rapidly [59]. Hence it is feasible to consider −∞ < εdd < ∞, with both negative and positive Cdd . Note that for Cdd < 0 the dipole-dipole interaction becomes repulsive for head-to-tail dipoles and attractive for side-by-side dipoles. 2.2

The dipolar Gross-Pitaevskii equation

Working in the limit of zero temperature, the condensate can be described by the mean-field wavefunction Ψ(r, t), which specifies the atomic density n(r, t) = |Ψ(r, t)|2 . The wavefunction is

CONTENTS normalized to the number of atoms, Z N = |Ψ|2 dr.

5

(4)

The wavefunction obeys the dipolar GPE [49, 52, 53], ∂Ψ ~2 2 i~ = − ∇ + V + g|Ψ|2 + Φ Ψ, (5) ∂t 2m where V ≡ V (r) is the external potential acting on the condensate (which in principle may also be timedependent, but here we consider it static). The van der Waals interactions lead to the purely local meanfield term g|Ψ|2 , while the dipolar interactions give rise to the non-local mean-field potential Φ given by [50], Z Φ(r, t) = Udd (r − r 0 )n(r 0 , t) dr 0 . (6) If we take the dipoles to be polarized along the zdirection, then using identities from potential theory the dipolar potential can be expressed as [60, 61], 2 1 ∂ (7) φ(r) + n(r) , Φ(r) = −3gεdd ∂z 2 3 where φ is a fictitious ‘electrostatic’ potential defined as Z 1 n(r0 ) φ(r) = dr0 . (8) 4π |r − r0 | This effectively reduces the problem of calculating the dipolar potential Φ to one of calculating an electrostatic potential of the form (8) which is easier to compute because the Green’s function 1/ |r − r0 | has no angular dependence. Furthermore, hundreds of years of literature exist providing efficient methods for solving electrostatic and gravitational problems. Alternatively, Φ can be evaluated in momentum space by exploiting the convolution theorem, h i ˜dd (k)˜ Φ(r, t) = F −1 U n(k, t) , (9)

where γ = ωz /ω⊥ is the so-called trap ratio. When γ 1 the BEC shape will typically be oblate (flattened) while for γ 1 it will typically be prolate (elongated). Time-independent solutions of the GPE satisfy, Ψ(r, t) = ψ(r)e−iµt/~ ,

(12)

where µ is the chemical potential‡. Inserting this into Eq. (5), the time-independent dipolar GPE for the time-independent wavefunction ψ(r) is ~2 2 ∇ ψ + V ψ + g|ψ|2 ψ + Φψ. (13) 2m Solutions of the time-independent GPE are stationary solutions of the system, and the lowest energy solution is the ground state. µψ = −

The energy Z 2of the condensate is given by, g 4 Φ 2 ~ 2 2 |∇Ψ| + V |Ψ| + |Ψ| + |Ψ| dr E= 2m 2 2 = Ekin + Epot + EvdW + Edd . (14) The terms represent (from left to right) kinetic energy Ekin , potential energy Epot , the van der Waals interaction energy EvdW and the dipolar interaction energy Edd . Providing that the potential V is independent of time, then the total energy E is conserved during the time evolution of the GPE. Comparing the relative size of the kinetic term and the net interaction term in the dipolar GPE defines a length-scale termed the healing length, ξ=√

~ . mµ

(15)

It may be interpreted as the minimum length-scale over which the wavefunction changes appreciably. Efficient methods for solving the dipolar GPE are available [62, 63] and progress has been made on extending this treatment to finite temperatures [64, 65, 66].

where n ˜ (k, t) = F[n(r, t)].

2.3

Dipolar hydrodynamic equations

In condensate experiments, the external potential V is typically harmonic and static, with the general form, 1 V (r) = m ωx2 x2 + ωy2 y 2 + ωz2 z 2 , (10) 2 where ωj (j = x, y, z) are the trap angular frequencies in each direction. The characteristic trap length scale is p provided by the harmonic oscillator lengths, `j = ~/mωj . Cylindrically-symmetric traps are common, defined as 1 1 2 2 2 V (r) = m ω⊥ ρ + ωz2 z 2 = mω⊥ ρ2 + γ 2 z 2 , (11) 2 2

There is a deep link between the GPE and fluid dynamics. Indeed, the condensate can be thought of as a fluid, characterised by its density and velocity distributions. This is revealed by writing the condensate wavefunction through the Madelung p transform Ψ(r, t) = n(r, t)eiS(r,t) , where the phase distribution, S(r, t), defines the fluid velocity field v(r, t), v(r, t) =

~ ∇S(r, t). m

(16)

‡ Throughout this review Ψ (ψ) denotes the time (in)dependent condensate wavefunction.

CONTENTS

6

Inserting the Madelung transform into the GPE, and separating real and imaginary terms yields two equations which together are exactly equivalent to the GPE. The first is the continuity equation, ∂n + ∇ · (nv) = 0. (17) ∂t This embodies the conservation of the number of atoms. The second equation is √ 1 ∂v ~2 ∇2 n 2 √ = −∇ mv + V + gn + Φ − m . ∂t 2 2m n (18) √ √ The term ∝ ∇2 n/ n is the quantum pressure, arising from the zero-point kinetic energy of the atoms. It can be dropped when the interactions and external potential dominate the zero-point motion, leading to the Thomas-Fermi approximation. In the absence of the quantum pressure term (and also dipolar interactions; Φ = 0), Eqs. (17) and (18) are commonly known as the superfluid hydrodynamic equations [67, 68, 69, 70, 71] since they resemble the equation of continuity and the Euler equation of motion from inviscid fluid dynamics. Here they have been extended to include dipolar interactions. 3

energy EB and momentum p of a perturbation is given by, s 2 2 p 2 2 , (21) EB = c (θ) p + 2m where c(θ) is the speed of sound, gn c(θ) = 1 + εdd 3 cos2 θ − 1 . (22) m The angle θ is that between the excitation momentum and the polarization direction. For low momenta the spectrum is linear EB ≈ c(θ) p and characteristic of phonons with a phase velocity c(θ) that depends on direction. For higher momenta the relation becomes quadratic in p which is characteristic of free-particle excitations. A mode of energy EB evolves in time as exp(−iEB t/~), and so when EB becomes imaginary, the modes grow exponentially, signifying a dynamical instability. This is known as the phonon instability, familiar from nondipolar attractive (g < 0) condensates [69]. Examining the parameter space over which Eq. (21) is real-valued indicates that the three dimensional homogeneous system is stable to the phonon instability in the range −0.5 ≤ εdd ≤ 1 for g > 0, and εdd ≤ −0.5, εdd > 1 for g < 0.

Vortex-Free Solutions and Stability 3.2

Before discussing vortices, we next describe the solutions and stability of the dipolar condensates themselves, in homogeneous and trapped systems, and introduce some key analytical tools and physical concepts. 3.1 3.1.1

Homogeneous condensate Three-dimensional case

For V (r) = 0 (uniform condensate of infinite extent), the stationary solution is, √ µ = n g (1 − εdd ) , (19) ψ = n, i.e. a state of uniform density n. The two contributions to the chemical potential µ are the uniform meanfield potentials generated by the van der Waals and the dipolar interactions, respectively. In the absence of dipoles, the corresponding solution has chemical potential µ = ng. By comparison, the homogeneous dipolar system is akin to a non-dipolar system but with an effective coupling geff = g (1 − εdd ) .

(20)

For a three dimensional homogeneous dipolar condensate, the Bogoliubov dispersion relation between the

Trapped dipolar condensates

A full theoretical treatment of a trapped BEC involves solving the dipolar GPE, given in Eq. (5) [69, 70]. The non-local nature of the mean-field potential describing dipolar interactions means that this task is more challenging than for purely s-wave BECs. Moreover, the stability of the condensate becomes non-trivial, becoming dependent on the geometry of the trap and the number of atoms (as well as the dipole strength). Additionally, a dipolar condensate can suffer from a density-wave instability associated with a novel type of excitation called a roton in analogy with a similar type of excitation in superfluid helium [72, 73, 74]. To characterise the stability of a dipolar condensate we first derive and examine the Thomas-Fermi ground state solutions for a dipolar BEC. 3.2.1

Thomas-Fermi solutions

The problem of finding the ground state solution (as well as low-energy dynamics) is greatly simplified by making use of the Thomas-Fermi approximation, whereby density gradients in the GPE (or, equivalently, the hydrodynamic equations) are ignored, allowing analytic solutions [67]. For a non-dipolar condensate, with repulsive van der Waals interactions, this is valid for N as /`¯ 1, where `¯ = (`x `y `z )1/3 is the geometric

CONTENTS

7

mean of the harmonic oscillator lengths [69, 70]. This regime is relevant to many experiments. In the dipolar case, the Thomas-Fermi approximation is valid when the net interactions are repulsive and the number of atoms is large; rigorous criteria have been established for certain geometries in Ref. [75]. Although the governing equations for a dipolar BEC contain the non-local potential Φ(r), exact solutions known from the pure s-wave case hold, in modified form, in the dipolar case too [60, 61], and we make extensive use of them throughout this review. Consider a dipolar condensate with dipoles polarized in the z-direction, repulsive van der Waals interactions (as > 0), and confined by an cylindrically-symmetric trap of the form of Eq. (11). We limit the analysis to the regime of −0.5 < εdd < 1, where the ThomasFermi approach predicts that stationary solutions are robustly stable [60]. Outside of this regime the condensate becomes prone to collapse [76, 77]. Under the Thomas-Fermi approximation the timeindependent GPE (13) then reduces to, 1 2 mω⊥ ρ2 + γ 2 z 2 + Φ(r) + gn(r) = µ. (23) 2 Making use of the formulation of the dipolar potential Φ(r) in terms of the electrostatic potential given in Eqs. (7) and (8), exact solutions of Eq. (23) can be obtained for any general parabolic trap, as proven in Appendix A of Ref. [61]. In particular, the solutions of n(r) take the form ρ2 z2 n(r) = n0 1 − 2 − 2 for n(r) ≥ 0, (24) R⊥ Rz 2 where n0 = 15N/(8πR⊥ Rz ) is the central density, and Rz and R⊥ are the Thomas-Fermi radii of the condensate in the axial and transverse directions. Remarkably, Eq. (24) is the general inverted parabolic density profile familiar from the Thomas-Fermi limit of non-dipolar BECs [69, 70]. An important distinction, however, is that for the dipolar BEC the aspect ratio of the parabolic solution, κ = R⊥ /Rz , differs from the trap aspect ratio γ. The condensate aspect ratio can be determined from the transcendental equation [60, 61], 3κ2 εdd

γ2 +1 2

f (κ) − 1 + (εdd − 1)(κ2 − γ 2 ) = 0, 1 − κ2

where, √ 1 + 2κ2 3κ2 arctanh 1 − κ2 f (κ) = − , 1 − κ2 (1 − κ2 )3/2

(25)

which takes the value f = 1 at κ = 0, and monotonically decreases towards f = −2 as κ → ∞, passing through zero at κ = 1. This is a robust feature: the same transcendental equation is recovered using a variational approach based on a gaussian ansatz for

Figure 3. Top: Aspect ratio κ of the trapped cylindricallysymmetric dipolar condensate in the Thomas-Fermi regime, for specific trap ratios γ (equally spaced on a logarithmic scale in the range γ = [0.1, 10]). Global, metastable and unstable solutions are indicated by white, light grey and dark grey shading. Reprinted figure with permission from [78]. Copyright 2010 by the American Physical Society. Bottom: Stability diagram of the purely dipolar harmonically-trapped condensate (ground state), as a function of the trap aspect ratio λ ≡ γ = ωz /ω⊥ and the dipolar interaction parameter D = (N − 1)md2 /~2 l⊥ . The shaded region denotes stability against collapse. The dark shaded regions indicate biconcave condensates. Reprinted figure with permission from [80]. Copyright 2007 by the American Physical Society.

the condensate wave function [50, 51]. For a nondipolar (εdd = 0) condensate one finds the expected result that κ = γ, i.e. the condensate has the same aspect ratio as the trap. The dipolar interactions, however, lead to magnetostriction of the condensate, such that κ < γ for εdd > 0 and κ > γ for εdd < 0. This behaviour is shown in Figure 3 (top) [78]. Note that, within the range −0.5 ≤ εdd ≤ 1 these are global solutions; elsewhere the solutions are either metastable (light grey shading) or unstable (dark grey shading). For conventional dipoles (Cdd > 0, εdd > 0), the condensate is least stable in prolate (γ < 1) traps; here the dipoles lie predominantly in the attractive head-to-tail configuration and undergo collapse when εdd becomes too large. By contrast, in oblate (γ > 1) traps stability is enhanced since the dipoles lie predominantly in the repulsive sideby-side configuration. Meanwhile the opposite is true for anti-dipoles (Cdd < 0, εdd < 0). Away from the instabilities, these solutions agree well with numerical

CONTENTS solutions of the full dipolar GPE in the ThomasFermi regime [76]. Close to the instabilities, zeropoint kinetic energy (neglected within the ThomasFermi approach) can enhance the stability of the solutions. Once the BEC aspect ratio κ is found from the transcendental equation, the Thomas-Fermi radii are determined by the expressions, 1/5 2 3 κ f (κ) 15N gκ , (26) −1 1 + εdd R⊥ = 2 4πmω⊥ 2 1 − κ2 R⊥ Rz = (27) κ and the total energy by, is given γ2 N 2 2 mωx Rx 2 + 2 ETF = 14 κ 2 15 N g + [1 − εdd f (κ)] . (28) 28π Rx2 Rz The first term corresponds to the trapping energy and second to the s-wave and dipolar interaction energies. Finally, the dipolar potential inside the condensate can be explicitly obtained as [61],2 2z n0 Cdd ρ2 − 2 ΦTF (ρ, z) = 3 Rx2 Rz 2 3 ρ − 2z 2 . (29) − f (κ) 1 − 2 Rx2 − Rz2 This is generally a saddle-shaped function that reflects the anisotropic nature of the dipolar interactions and drives the elongation of the BEC along the polarization direction. A more general version of ΦTF (r) for the case of a dipolar BEC without cylindrical symmetry is given later in Eq. (70). 3.2.2

Outside the Thomas-Fermi regime: rotons and density oscillations

According to the Thomas-Fermi approach, a trap which is sufficiently prolate (γ > ∼ 5.2) is stable to collapse even in the limit εdd → ∞. However, numerical solutions reveal a different fate, whereby the condensate undergoes instability even for γ → ∞ [73]. This is associated with the development of a roton minimum in the dispersion relation [72, 73], reminiscent of rotons in superfluid Helium [79]. For certain parameters, this minimum can approach zero energy, triggering an instability at finite k known as the roton instability. The Thomas-Fermi approach, which is limited to the class of inverted parabolic solutions, is unable to account for this phenomenon. The roton is a strict consequence of the non-local interactions, and does not arise for conventional condensates. The effect of this in trapped and purely dipolar condensates was revealed by Ronen et al., with the stability diagram shown in Figure 3 (bottom) [80].

8 When the dipolar interaction parameter D = (N − 1)md2 /~2 l⊥ exceeds a critical value, for any trap ratio, the system is unstable to collapse. The condensate becomes unstable to modes with increasingly large number of radial and angular modes as the trap aspect ratio increases, signifying that collapse proceeds on a local, rather than global, scale [81]. Of particular interest is the appearance, close to the instability boundary and under oblate traps, of ground state solutions with a biconcave, red blood cell-like, shape [see Figure 3 (bottom)] [80, 82]. Subsequent works confirmed these density oscillations as being due to the roton, which, for certain parameters, mixes with the ground state of the system [83]. More generally, when van der Waals interactions are included [84, 85], both biconcave and dumbbell shapes can arise [86]. Under box-like potentials, which have been realized in recent years [87, 88], density oscillations associated with the roton can arise at the condensate edge [50]. An intuitive interpretation of the roton in an oblate trap was put forward by Bohn et al. [89]. As the dipole strength is increased, it is energetically favourable for the dipoles to locally move out of the plane and align head-to-tail perpendicular to the plane, thereby taking advantage of this attractive configuration. This leads to a periodic density in the plane, with a wavenumber corresponding to that of the roton minimum. 3.3

Quasi-two dimensional system

For a condensate strongly confined in one dimension it is possible to reduce the effective dimensionality of the system to form a quasi-two dimensional condensate. This offers a simplified platform to study vortices and vortex lattices in dipolar condensates, while still retaining the key physics. Consider the dipoles to be polarized at an angle α to the z-axis, lying in the xz plane, and strong harmonic confinement V (z) = 21 mωz2 z 2 in the z direction which satisfies ~ωz µ, i.e. the trapping energy dominates over the condensate energy scale. This set-up is illustrated in Figure 4 [90]. In this regime, one can approximate the wavefunction by the ansatz, Ψ(ρ, z, t) = Ψ⊥ (ρ, t)ψz (z).

(30)

Axially, the condensate is taken to be frozen into the axial ground harmonic oscillator state ψz (z) = 2 2 (π`2z )−1/4 e−z /2`z . The dynamics then become planar, parametrised by the two-dimensional time-dependent wavefunction, Ψ⊥ (ρ, t). Note that R Ψ⊥ is normalized to the number of atoms, i.e. N = |Ψ⊥ |2 dρ. Inserting this ansatz into the dipolar GPE, Eq. (5), and integrating out the axial direction then leads to the

CONTENTS

Figure 4. Schematic of the quasi-two dimensional dipolar condensate, with strong harmonic trapping along z. The dipoles are taken to be polarized at angle α to the z-axis in the x − z plane. The condensate is assumed to follow the static ground harmonic oscillator state along z. Figure reproduced from Ref. [90] under a CC BY licence.

effective two dipolar GPE [91], dimensional ~2 2 g ∂Ψ⊥ = − ∇⊥ + V (ρ) + √ |Ψ⊥ |2 + Φ⊥ Ψ⊥ . i~ ∂t 2m 2πlz (31) √ The g/ 2πlz coefficient characterises the effective van der Waals interactions in the plane, and Φ⊥ is the effective planar dipolar potential, Z ⊥ Φ⊥ (ρ, t) = Udd (ρ − ρ0 ) n⊥ (ρ0 , t) dρ0 , (32) where n⊥ = |Ψ⊥ |2 is the two dimensional density. The real-space form of the effective two dimensional ⊥ dipolar interaction potential Udd is given elsewhere [92], while in this review its Fourier transform is used [91, 93], 4πCdd ˜ ⊥ (˜ U Fk (˜ q) sin2 α + F⊥ (˜ q) cos2 α , (33) dd q) = √ 9 2πlz √ q˜2 2 where Fk (˜ q) = −1 + 3 π q˜x eq˜ erfc(˜ q ), F⊥ (˜ q) = 2 − √ √ q˜2 3 π q˜e erfc(˜ q ) and q ˜ = qlz / 2 with q being the projection of k onto the x − y plane, i.e. the reciprocal space analogue of ρ§. From this momentum space representation Φ⊥ can then be evaluated using the convolution theorem as in Eq. (9). An important parameter is the ratio σ = lz /ξ, where ξ is the healing length. The two dimensional approximation requires σ < 1. Under a cylindrically-symmetric harmonic trap and for dipoles polarized along z, the above “two dimensional mean-field regime” is formally entered when N as (1 + 4 2εdd )lz3 /l⊥ 1. In the opposing regime, when 4 N as (1 + 2εdd )lz3 /l⊥ 1, the system enters the three dimensional Thomas-Fermi regime [75]. A more general analysis of flattened condensates in Ref. [94], has established the validity of the two dimensional mean-field regime for arbitrary § Throughout this review k (q) are the projection of k onto the x − y − z (x − y) plane, i.e. the reciprocal space analogue of r (ρ).

9

Figure 5. Stability diagram in εdd − α space for the homogeneous quasi-two dimensional dipolar BEC (σ = 0.5) with (a) g > 0 and (b) g < 0. Shown are the regions of stability (white), phonon instability (pink) and roton instability (blue). The vertical dashed line indicates the magic angle αm . For α > π/2 the results are the mirror image of the presented region. Figure reproduced from Ref. [90] under a CC BY licence.

polarization direction. In the absence of any planar trapping potential [V (ρ) = 0], the stationary solution of the quasi-two dimensional dipolar condensate is the homogeneous state [91], √ n⊥ Cdd (3 cos2 α − 1) , (34) g+ ψ⊥ = n⊥ , µ = √ 3 2πlz where n⊥ is the two dimensional density. The quasi-two dimensional system undergoes the phonon instability when the net local interactions become attractive in the plane, i.e. when g + Cdd [3 cos2 α − 1]/3 < 0. The phonon unstable regions in the εdd − α plane are shown in Figure 5. These can be understood by considering the trade-off between the van der Waals and the dipolar interactions [90, 95]. Note the divergent behaviour at the magic angle αm , across which the planar dipolar interactions switch between repulsive and attractive. The roton instability also arises in this quasi-two dimensional system [73], as indicated in Figure 5(a) and (b) (blue shaded regions). For g > 0 the roton instability is induced by the attractive part of the dipolar interaction and is only possible for α 6= 0 (for α = 0 the condensate cannot probe the attractive part ⊥ of Udd [93]). For g < 0 the roton exists for all α [96] (excluding the magic angle). For small α it is induced by the attractive van der Waals interactions, while for larger α it is driven by the attractive axial component of the dipolar interactions. The regimes of roton stability/instability are affected by σ. For a narrower (wider) condensate, the regimes of roton instability in the phase diagram shrink (expand). This is because the atoms experience less (more) of the ⊥ out-of-plane component of Udd [90].

CONTENTS 4

10

Single Vortices

Quantized vortices are a consequence of the condensate’s phase coherence. To preserve the single valuedness of the condensate wavefunction, the total change in phase around some closed path C must be 2πqv , with qv = 0, ±1, ±2, .... If qv 6= 0 then there exists one or more phase singularities within C. These singularities are the quantized vortices, and qv is the vortex charge. Consider an isolated vortex at the origin in a uniform system, which is straight along z. The condensate phase about the vortex follows the azimuthal angle, S(ρ, θ, z) = θ = arctan(y/x). From Eq. (16) this gives rise to a circulating azimuthal flow with speed, v=

qv ~ . mρ

(35)

This corresponds to irrotational flow with zero vorticity, i.e. ∇ × v = 0. This can also be seen directly from the definition of the fluid velocity, which is curlfree ∇ × v = 0 and thus very different to the classical solid-body rotation, for which v ∝ ρ. At ρ = 0, the atomic density vanishes, ensuring that infinite flow velocities are not present. The axi-symmetric flow associated with the velocity field given in Eq. (35) carries a total angular momentum Lz = N ~qv . 4.1

Energetics of vortex formation

Even single vortices are giant excitations involving a considerable fraction of the entire BEC. The energy associated with the formation of a vortex Ev ≡ E −E0 , where E0 is the energy of the non-rotating (vortexfree) state, is generically much larger than the energy of elementary excitations described by the Bogoliubov spectrum given in Eq. (21). In a frame rotating at angular frequency Ω, the total energy of the system is shifted to E − Ω · L, where L is the angular momentum in the laboratory frame, and hence it only becomes energetically favourable to form a vortex if the angular momentum is such that |Ω·L| > Ev leading to a critical rotation frequency, Ωv =

Ev . N ~qv

(36)

Ev can be computed analytically in certain situations as we discuss below. Before we do so, it is important to point out that Eq. (36) considers the energetics but not the kinetics of vortex formation. Both theory and experiment reveal that the true value of Ωv is often considerably higher than predicted by Eq. (36). This is because the vortex-free state can remain a local energy minimum separated by an energy barrier

from the global minimum corresponding to the vortex state. The kinetics of vortex formation are examined in Section 6. In the simplest case of an infinite system with a vortex the condensate wavefunction can be written, ψ(ρ, θ, z) = f (ρ)eiqv θ

(37)

and substituting into the dipolar GPE (5) an equation for the amplitude f about the vortex is obtained, ∂f ~2 qv2 ~2 1 ∂ f + gf 3 + Φf, (38) ρ + µf = − 2m ρ ∂ρ ∂ρ 2mρ2 where the Laplacian has been expressed in its cylindrically symmetric form, 1 ∂ ∂2 ∂ 1 ∂2 ∇2 = (39) ρ + 2 2 + 2. ρ ∂ρ ∂ρ ρ ∂θ ∂z The second term on the right-hand side of Eq. (38) is the only difference from the non-vortex case, and is associated with the kinetic energy of the circulating flow, giving rise to a centrifugal barrier. Note that |qv | > 1 vortices are energetically unstable compared to multiple singly-charged vortices, and rarely arise unless engineered; for this reason we will consider only |qv | = 1. In any system uniform along the polarization direction the dipolar potential reduces to a local potential Φ(r) → −gεdd n(r). This can be seen from Eq. (7) which introduces the fictitious electrostatic potential φ(r), and noting that ∂ 2 φ/∂z 2 must equal zero. Thus, the last two terms in Eq. (38) can be combined into a single contact term g(1 − εdd )f 3 . Results that hold for the usual s-wave case in this context therefore also hold for the dipolar case providing one replaces g by g(1 − εdd ). For example, analysis of Eq. (38) for the s-wave case reveals that the centrifugal barrier term dictates that the density √ relaxes as 1/ρ2 to the asymptotic background value n as ρ → ∞, and for ρ → 0 the density tends to zero as ρ|qv | [69]. Furthermore, although Eq. (38) can not be solved in terms of known functions, it can be solved numerically and with appropriate scaling the result for f (ρ) is universal. This solution for f (ρ) can then be used in the energy functional given in Eq. (14) and the extra energy per unit length due to the introduction of a vortex evaluated. In the pure s-wave case one obtains [69] b ~2 . (40) v = πn log 1.464 m ξ This result was originally obtained for superfluid 4 He by Ginzburg and Pitaevskii in 1958 [97]. In this expression the healing length ξ, which gives the size of the vortex core, forms a lower cutoff and the length

CONTENTS

11

b, which could be the system radius, is the upper cutoff. Although b ξ, their finite values avoid a logarithmic singularity that originates from the centrifugal barrier term. The only place that interactions enter this expression for v is through the healing length ξ = √ ~/ mµ which can immediately be adapted to the dipolar case using µ = ng(1 − εdd ). The critical rotation frequency for this case can now be evaluated by replacing Ev by v and N by N/L (number of atoms per unit length) in Eq. (36). 4.2

General features of a vortex in a quantum ferrofluid

In order to obtain analytic results it can sometimes be useful to use the following approximate solution for f (ρ) which incorporates the correct behaviour for ρ → 0 and ρ → ∞, n(ρ, z) = n

ρ02 , 1 + ρ02

(41)

where ρ0 = ρ/ξ. It will be convenient later to write this in the form of a homogeneous background density n and a negative vortex density nv , n(ρ, z) = n + nv ,

(42)

where nv = −n/(1 + ρ02 ). A schematic of a straight singly-charged vortex line through a non-dipolar harmonically-trapped condensate is shown in Figure 6. The vortex has a well-defined core, with zero density and a phase singularity at its centre, relaxing to the background condensate density over length-scale of the order of √ the conventional healing length ξ = ~/ mµ. This is typically of the order of 0.1 − 1µm but can be tuned by means of a Feshbach resonance.

Figure 6. Three-dimensional density (isosurface plot) of a trapped non-dipolar condensate featuring a vortex line along the z-axis. The corresponding two-dimensional phase profile and central one-dimensional density profile are also depicted.

In three dimensions the vortices may bend, e.g. into tangles and rings, carry linear or helical Kelvin wave excitations and undergo reconnections. However, under strong axial confinement of the condensate, the dynamics become effectively two-dimensional. Being topological defects, vortices can only disappear via annihilation with an oppositely-charged vortex (the two-dimensional analog of a reconnection) or by exiting the fluid at a boundary. In trapped condensates, an offcentre vortex precesses about the trap centre; this can be interpreted in terms of a Magnus force [32]. Thermal dissipation causes a precessing vortex to spiral out of a trapped condensate [98, 99, 100, 101]. Acceleration of a vortex (or an element of a three dimensional vortex line) leads to emission of phonons, analogous to the Larmor radiation from an accelerating charge, although under suitable confinement these phonons can be reabsorbed to prevent net decay of the vortex [102]. Optical absorption imaging of the vortices is typically preceded by expansion of the cloud to enlarge the cores [31, 103]. This method has been extended to provide real-time imaging of vortex dynamics [32]. While this imaging approach detects density only, the vortex circulation is detectable via gyroscopic techniques [104]. Yi and Pu [105] performed the first study of vortices in a dipolar BEC, obtaining numerical solutions for a quasi-two dimensional trapped dipolar condensate featuring a vortex. For dipoles polarized perpendicular to the plane they found the striking result that density ripples form about the vortex core for trap ratios γ ∼ 100 and attractive van der Waals interactions. These ripples are not contained in the simple ansatz given above in Eq. (41) and seem to be a rather special feature associated with non-local interactions. Indeed, they had previously been seen in numerical simulations of vortices in superfluid 4 He where nonlocal potentials are employed [106, 107, 108, 109]. For purely dipolar (g = 0), oblate condensates, Wilson et al. [81, 83] numerically found vortex ripples for moderate trap ratios γ ∼ 17, see Figure 7(top) [81], and established the link to roton mixing into the vortex solution, similar to the biconcave structure that they found was induced in vortex-free dipolar condensates (energetic favourability of dipoles aligning head-totail). Vortex ripples have since been studied in other works [84, 90, 95], and similar ripples arise in the presence of other localized density depletions, such as due to localized repulsive potentials [83, 91] and dark solitons [110, 111]. The presence of the vortex slightly reduces the stable parameter space for the condensate relative to the vortex-free condensate [81, 105]. For dipoles tilted perpendicular to axis of the vortex, the

CONTENTS

12 For a general vortex line in three dimensions, the vortex elements interact with each other at long-range via the dipolar interactions [114], as well as the usual hydrodynamic interaction [115, 116]. This modifies the Kelvin-wave (transverse) modes of the vortex line, and can support a roton minimum in their dispersion relation. For large dipolar interactions, the Kelvin waves can undergo a roton instability, leading to novel helical or snake-like configurations [117]. With Cdd < 0 and tight axial trapping, stable two-dimensional bright solitons have been predicted [118, 119], i.e. wavepackets which are self-trapped by interactions in two dimensions. This idea was extended by Tikhonenkov et al. [120] to predict stable twodimensional vortex solitons, which may be considered as a two-dimensional bright soliton carrying a central vortex. 4.3

Figure 7. Top: Stability diagram of the trapped purely dipolar condensate, with dipoles polarized along the z-axis and featuring an axial vortex, as a function of the trap ratio (λ ≡ γ = ωz /ω⊥ ). Below the solid line the condensate is dynamically stable. Ripples about the vortex arise in the pink region. The inset shows an isosurface of the density for such a solution. Reprinted figures with permission from [81]. Copyright 2009 by the American Physical Society. Bottom: Density (a-b) and phase (c) profiles of a trapped quasi-two dimensional condensate with dipoles polarized along x. Reprinted figures with permission from [105]. Copyright 2006 by the American Physical Society.

Vortex in a trapped dipolar BEC in the hydrodynamic regime

In the hydrodynamic (Thomas-Fermi) regime appropriate for large condensates, the problem of a dipolar BEC with single vortex in a three dimensional trap can be tackled semi-analytically [121]. In this case the energy associated with the curvature of the density due both to the trapping and the vortex core can be ignored in comparison to each of the rotational, interaction, and trapping energies. These remaining energies can be evaluated analytically by assuming a density profile much like the one given in Eq. (41), i.e. an unperturbed background piece plus a negative vortex “density”. The difference is that we now take the unperturbed background density to be the inverted parabola nTF (ρ, z) given in Eq. (24) which is an exact solution of the vortex-free Thomas-Fermi problem. Thus, the density profile reads, n(ρ, z) = nTF (ρ, z) + nv (ρ, z)

(43)

where, vortex core becomes elliptical, see Figure 7(bottom) [105] due to the anisotropic dipolar potential in the plane. The properties of an off-axis straight vortex line in a trapped dipolar condensate were considered in [112] in the Thomas-Fermi regime, showing that the dipolar interaction lowers (raises) the precession speed in an oblate (prolate) trap. In the presence of thermal dissipation, making the dipolar interactions partially attractive by changing the polarization direction leads to a reduction in the condensate size and a faster decay rate of the precessing vortex [113].

nv (ρ, z) = −nTF (ρ, z)

β2 . β 2 + ρ2

(44)

The length scale β parameterizes the size of the vortex core and is one of three variational parameters {β, R⊥ , κ} with respect to which the total energy functional must be minimized in order to find their stationary values. Notice that this ansatz does not include ripples which are beyond the Thomas-Fermi approximation. The energy functionals for the rotational, s-wave interaction, and trapping energies can all be evaluated analytically, albeit laboriously, using the above density profile. The results are given in Ref. [121] and will not

CONTENTS

13

Figure 8. Vortex solutions in the infinite quasi-two dimensional dipolar condensate, as a function of εdd (with σ = 0.5). The lefthand side of the main plots shows the normalised density profile along x (with y = 0) and the right-hand side shows the normalised density profile along y (with x = 0). (a) Dipoles polarized along z (α = 0). (b) Dipoles polarized off-axis at α = π/4. (c) Dipoles polarized off-axis at α = π/2. Grey bands indicate the unstable regimes of εdd . Insets: normalised density profile n⊥ (x, y) for example values of εdd over an area (40ξ)2 . Figure reproduced from Ref. [90] under a CC BY licence.

be repeated here. Obtaining an analytic R result for the dipolar interaction energy Edd = (1/2) Φ(r)n(r)dr is more difficult. However, in the hydrodynamic regime we have β R⊥ , implying that the contribution to Φ(r) from nv is negligible in comparison to that from nTF . Thus, to a very good approximation we can write, Z 1 Edd ≈ ΦTF (r) [nTF (r) + nv (r)] dr (45) 2 i.e. replace the true Φ(r) by that purely due to the unperturbed background ΦTF (r) which is known analytically and is given in Eq. (29). Since ΦTF (r) is a quadratic function of the coordinates this integral can be done exactly [121]. Finally, to find the energy Ev associated with exciting a vortex it is necessary to subtract from E the energy E0 of the vortex-free state, but this latter energy is also known analytically and is given as ETF in Eq. (28). In this way Ev and hence Ωv can be computed for the trapped dipolar BEC. It is found that dipolar interactions increase Ωv in prolate traps and lower it in oblate traps when compared to the pure s-wave case [121, 122]. Intuitively, this makes sense because in the prolate case dipolar interactions tend to reduce R⊥ but in the oblate case they increase it. The rotational energy density (1/2)n(ρ, z)v 2 (ρ) is lower at larger radii because v ∝ 1/ρ and hence Ev is lowered if atoms are moved to larger radii, like in the oblate case, and vice-versa in the prolate case. This interpretation is backed up by the following expression for the critical rotation frequency derived in the pure s-wave case in

the Thomas-Fermi limit [123] Ωv =

5 ~ 0.67R⊥ . 2 ln 2 mR⊥ ξ

(46)

The numerical factors 5/2 and 0.67 arise from the inverted parabola of the Thomas-Fermi density profile: since the parabolic profile is maintained in the dipolar case we expect a similar expression to hold there. If R⊥ in Eq. (46) is replaced by its dipolar version as given in Eq. (26), the resulting prediction for Ωv is in very close agreement with the variational calculation described above [121]. We should in principle also modify the healing length to its dipolar expression but this only leads to a logarithmic correction. 4.4

Vortex in the quasi-two-dimensional system

We now review the vortex solutions in the simpler context of the homogeneous quasi-two dimensional dipolar condensate, following the work of Refs. [90, 95]. Figure 8 [90] plots these solutions, found by numerical solution of the quasi-two dimensional dipolar GPE, as a function of εdd . The vortex profile has a non-trivial dependence on the polarization angle and εdd . For α = 0, the vortex density is axisymmetric. For εdd = 0 it has the standard form of non-dipolar condensates [see left inset of Figure 8(a)], consisting of a circularly-symmetric √ core of vanishing density of width ξ = ~/ mµ [69]. For most of the εdd parameter space the vortex solution is essentially identical to that for εdd = 0 (due the choice of the dipolar healing length as the length scale):

CONTENTS the system behaves like a non-dipolar system with modified contact interactions. However, as the phonon instability boundary at εdd = −0.5 is approached from the stable side, the vortex core takes on a funnellike profile [middle inset of Figure 8(a)]. This is associated with the cancellation of explicit s-wave van der Waals interactions in the system, i.e. the van der Waals interactions cancel the contact contribution from the dipolar interactions. Additionally, as the roton instability is approached, density ripples appear around the vortex [right inset of Figure 8(a)]. The ripples decay with distance from the core and have an amplitude of up to ∼ 20%n⊥ . The ripple wavelength, which reflects the roton wavelength, is approximately 4ξ. For α 6= 0, see Figures 8(b,c), the vortex profile becomes anisotropic. In particular, as the roton instability is approached, density ripples again form, but now aligned in the direction of the attractive dipolar interactions (along the polarization direction for Cdd > 0 and perpendicular for Cdd < 0). These anisotropic ripples are related to the anisotropic mixing of the roton into the ground state [91]. 4.4.1

Dipolar mean-field potential due to a vortex: giant anti-dipoles

Considered as a density defect in a homogenous background, the vortex gives rise to its own dipolar mean-field potential. Furthermore, because the creation of a vortex core displaces a large number of atomic dipoles, the vortex can be treated as a single giant anti-dipole [114, 121]. Take the case of a vortex within an otherwise uniform background of density n, with the density expressed using the decomposition as in Eq. (42). Then, by noting that the dipolar potential Φ is a linear functional of density, the total dipolar potential can be written as, Φ[n](r) = Φ[n + nv ](r) = Φ[n](r) + Φ[nv ](r) = Φ0 + Φ[nv ](r), (47) i.e. a contribution Φ0 , which is just a constant, from the uniform background and a spatially dependent contribution Φ[nv ] from the hole created by the vortex core, i.e. the giant anti-dipole. This decomposition, illustrated in Figure 9, assists in understanding the interaction between vortices in dipolar systems. Consider a vortex in the quasi-two dimensional dipolar condensate. For α = 0 and away from the roton and phonon instabilities, the vortex profile is wellapproximated by the non-dipolar ansatz given by Eqs. (41) and (42), where the healing length is taken to be the dipolar healing length ξ = √ ~/ mµ. The dipolar potential generated by the density defect can be determined via the convolution

14

Figure 9. Schematic of the decomposition of a density featuring a vortex into a uniform density n and a negative vortex density nv .

h i ˜ ⊥ (q)˜ result Φ⊥ (ρ, t) = F −1 U n⊥ (q, t) . Due to dd the cylindrical symmetry of the α = 0 case, one can perform the Fourier transforms through Hankel transforms. The Hankel transform of the two dimensional equivalent of Eq. (42) ! is, √ δ(q) K0 (qξ⊥ / 2) + , (48) n ˜ ⊥ (q) = n⊥ q 2 where K0 (·) is a modified Bessel function of the second kind. Expanding the dipolar interaction potential ˜ ⊥ (q) given in Eq. (33) with α = 0 to first order U dd in the condensate widthr parameter, ! σ, gives 9π ˜ ⊥ (q) = 4πC √ dd 2 − qlz σ + O σ 2 . (49) U dd 2 9 2πlz Then, to first order in σ = lz /ξ and third order in 1/ρ0 = ξ/ρ, the dipolar potential due to a vortex is [95, 90], A ln ρ0 + B 1 Φ⊥ (ρ) = Φ0 1 − 02 + σ , (50) ρ ρ03 p with constants A = − 9π/8 ≈ −1.88, B = (ln 2 − 1)A ≈ 0.577, and Φ0 the dipolar potential at infinity. This result indicates that the vortex causes a reduction in the mean-field dipolar potential which is consistent with the reduced density of dipoles in the vicinity of the vortex. Alternatively, it supports the interpretation of a vortex as a collection of anti-dipoles. One also sees from Eq. (50) that the dipolar potential generated by the vortex relaxes predominantly as 1/ρ02 to the background value Φ0 . This scaling is confirmed by numerical solutions, as shown in Figure 10(c) [90]. This scaling is because the vortex density itself relaxes as 1/ρ02 , and the leading contribution to Φv is from the local density. Indeed, the mean-field potential due to van der Waals interactions from the vortex also scale in proportion to the local density with a 1/ρ02 dependence [124]. The terms linear in σ represent the non-local contribution to Φv , arising from the long-range contributions from all anti-dipoles in the vortex. In the true two dimensional limit (σ = 0) this contribution vanishes since the volume of anti-dipoles in the vortex core vanishes. This illustrates the point

CONTENTS

Figure 10. Vortex in the quasi-two dimensional dipolar condensate, with the dipoles polarized at α = π/4 along x (εdd = 5 and σ = 0.5). (a) Density profile n⊥ (x, y). (b) Dipolar potential Φ⊥ (x, y), rescaled by the homogeneous value Φ0 . (c) The decay of Φ⊥ along x = 0 (black line) and y = 0 (red line) recovers the 1/ρ02 scaling at large distance (grey line). Figure reproduced from Ref. [90] under a CC BY licence.

that the dipolar potential generated by a vortex is not a topological quantity like the potential associated with the hydrodynamic flow around the vortex, but instead depends on the number of dipoles excluded from the region by the presence of the vortex. Also, the fact that the non-local vortex potential scales dominantly as ln ρ0 /ρ03 , and not 1/ρ03 , informs us that the vortex does not strictly behave as a point-like collection of dipoles at long-range. This is due to the slow powerlaw recovery of the density away from the vortex core. For dipoles tilted away from the vertical (α 6= 0), the vortex core and its dipolar potential become anisotropic and an analytic treatment is challenging. Figure 10(a,b) shows an example numerical solution of the vortex density and dipolar potential for α 6= 0. The dipolar potential is indeed anisotropic about the vortex. Remarkably, the modification to the dipolar potential induced by the vortex mimics the dipoledipole interaction itself, with an angular dependence of ∼ 1 − 3 cos2 θ, being reduced (attractive) along y and increased (repulsive) along x. Thus, at least in its angular dependence, the vortex shares qualitatively the characteristics of a mesoscopic dipole. Like the above α = 0 case, the dipolar potential is found to decay at long-range as 1/ρ02 to the background value. At short range, ρ < ∼ 10ξ, the dipolar potential is dominated by the core structure. 5

5.1

Vortex Pairs: Interactions and Dynamics

Interaction between vortices

Two vortices (or indeed two elements of the same three dimensional vortex line [114, 115, 116]) have a well-known hydrodynamic interaction due to the kinetic energy associated with the mutual cancellation/reinforcement of their velocity fields. Consider a cylindrical condensate of radius R and height L, featur-

15 ing two straight vortices at planar positions ρ1 and ρ2 , a distance d apart. The vortices have charge q1 and q2 , and individual velocity fields v1 and v2 , respectively. The net velocity field of the two vortices is v1 +v2 . The energy of the vortices can be estimated by integrating the total Z kinetic energy across the system, 1 mn(ρ)|v1 (ρ) + v2 (ρ)|2 dρ. (51) Ekin = L 2 For simplicity, one can ignore the vortex core density and set n(ρ) = n. Assuming ξ d R then the (kinetic) energy of the pair is, Lπn~2 2 R R R Ekin = q1 ln + q22 ln + 2q1 q2 ln . (52) m ξ ξ d Note that, to avoid singularities in the velocity field the integration region excludes a disc of radius of one healing length about each vortex centre. The first two terms are the energies of the individual vortices if they were isolated, and the third term is the pair interaction energy. For a vortex-antivortex pair (q1 = −q2 ) the interaction energy is negative, whilst for a co-rotating pair (q1 = q2 ) it is positive. This may be explained physically by the fact that for a vortex-antivortex pair the flow fields tend cancel out in the bulk, reducing the net kinetic energy in the bulk, whilst for a co-rotating pair the flow fields tend to reinforce, increasing the total kinetic energy. In the presence of dipolar interactions the vortices will feature an additional long-range interaction. This interaction can be pictured as the interaction between two lumps of anti-dipoles in empty space, as illustrated in Figure 11. Before we review how two vortices interact in the presence of dipolar interactions, we first make more precise the definition of the vortex energy introduced in Section 4, and hence allow the identification of vortexvortex interaction energy. In non-dipolar condensates, the vortex energy is conventionally defined as the energy difference between a system with and without vortex, where both systems have the same number of particles [69]. Imagine first a system (quasi-two dimensional) with a vortex and N particles covering an area A. If the asymptotic density is n⊥ , then the number of particles in this system can be expressed as, Z N = An⊥ − n⊥ − |ψ⊥ |2 dρ. (53) A

Figure 11. The dipolar interaction between two vortices may be interpreted as the interaction between two collections of antidipoles.

CONTENTS

16

Figure 12. Vortex interaction energy E12 versus separation d for (a) vortex-antivortex and (b) vortex-vortex pairs in the quasitwo dimensional dipolar condensate, with dipoles polarized along z (α = 0), shown for various values of εdd . Insets show deviation from the non-dipolar value. Reprinted figure with permission from [95]. Copyright 2013 by the American Physical Society.

Now consider the system without a vortex, but with the same number of particles. It has constant density n⊥ = N/A and its energy is N2 geff , E0 = √ 2 2πlz A

(54)

where geff = g + (Cdd /3)[3 cos2 α − 1]. Inserting Eq. (53) into Eq. (54) gives,   Z geff  2 E0 ≈ √ An⊥ − 2n⊥ n⊥ − |ψ⊥ |2 dρ , (55) 2 2πlz A

to the polarization direction, η. This is illustrated for VA and VV pairs by the examples in Figure 13 [95]. For small separations, the angular dependence is dominated by local effects arising from the density profile of the pairs, particularly by any density ripples. 0 However, for d ξ, ∆E12 = E12 (η) − E12 (η = 0) approaches a sinusoidal dependence on η, analogous to the dipolar interaction itself. 5.2

where the last term which has been omitted is negligible in the limit A ξ 2 . Defining the energy of a single vortex as E1 = E − E0 , the interaction energy between two vortices must be, E12 (ρ1 − ρ2 ) = E2 (ρ1 , ρ2 ) − E1 (ρ1 ) − E1 (ρ2 ),

Figure 13. Angular dependence of the vortex interaction energy 0 (η) = E (η) − E (η = 0), for (a) VA and (b) VV pairs, ∆E12 12 12 for various separations d. Parameters: α = π/4, σ = 0.5 and dd = 5. Example (c) VA and (d) VV pair density profiles for d = 5ξ over an area (20ξ)2 . Reprinted figure with permission from [95]. Copyright 2013 by the American Physical Society.

(56)

where E2 (ρ1 , ρ2 ) is the energy of the 2-vortex system. This is plotted in Figure 12 for α = 0 and for (a) vortex-antivortex (VA) pairs and (b) vortex-vortex (VV) pairs, as a function of their separation d, based on numerical dipolar GPE two-vortex solutions [95]. In the absence of dipoles, E12 increases with d for the VA pair and decreases for the VV pair. For d ξ, E12 closely follows the logarithmic scaling of the hydrodynamic prediction, while for d < ∼ ξ the overlap of the cores causes a breakdown of the logarithmic behaviour. With dipoles, E12 is significantly modified up to moderate length-scales d< ∼ 5ξ, but at larger scales the effects of the dipoles are washed out by the dominant hydrodynamic effects. The modification due to the dipoles arises from a nontrivial combination of the modified density profile and non-local interactions. When the dipoles are tilted in the plane, E12 becomes dependent on the in-plane angle of the pair relative

Dynamics of vortex pairs

In a two dimensional system, a vortex co-moves with the local fluid velocity. Thus, for a vortex pair, each vortex is carried along by the flow field of the other vortex. For a VA pair this means that the vortices move in the same direction, perpendicular to the intervortex axis. This solitary wave has speed v = ~/md for well-separated vortices. When the VA separation is too small, d ∼ ξ, the vortex and anti-vortex are unstable to annihilation, resulting in a burst of density waves. Numerical simulations [95] show that dipoles modify this separation threshold [Figure 14(a)]. Moreover, since for α 6= 0 the speed of sound varies with angle [91], one can expect the pair speed to be anisotropic in space. For a VV pair, the flow which carries each vortex now acts in opposite directions (again, perpendicular to the line separating the vortices and with the above speed), resulting in the co-rotation of the vortices about their mid-point. Viewed another way, the vortices follow a path of constant energy; since the interaction energy in the absence of dipoles depends only on d, this path is circular. The same is true for axisymmetric dipoles, α = 0. However, for α 6= 0, the vortices co-rotate on an anisotropic path, as shown in Figure 14(b) [95], due to the anisotropic

CONTENTS

Figure 14. (a) For a given initial separation, a non-dipolar VA pair annihilates while dipolar interactions (εdd = −0.4, α = 0) support stable pair propagation (red lines). (b) A non-dipolar VV pair co-rotates in a circular path (black line). Off-axis (α = π/4) dipolar interactions lead to anisotropic paths (blue line: εdd = −1.5; magenta line: εdd = 5) and suppression of corotation altogether (green line: εdd = 10). Reprinted figure with permission from [95]. Copyright 2013 by the American Physical Society.

interaction energy of the pair [refer to Figure 13]. Moreover, for extreme cases where the vortex is highly elliptical with significant ripples, corotation can be prevented altogether, with the vortices being localized [Figure 14(b), green curves]. In this limit, the vortices act as extended, highly-elongated objects, with effective geometrical restrictions on their motion past each other, reminiscent of the smectic phase of liquid crystals. Gautam [113] numerically considered the dynamics of a corotating VV pair in a dipolar BEC in the presence of dissipation. For symmetric configurations in the trap, the vortices decay with equal decay times, while for asymmetric initial configurations the decay is modified with one vortex decaying slower at the expense of the other. 6

Generation of Vortices

6.1

Summary of vortex generation methods

In conventional condensates, vortices have been generated through several mechanisms. Below we list the main mechanisms, as well as relevant considerations in the presence of dipoles. • The most common and intuitive approach to generate vortices is via mechanical rotation of the system [31, 36, 37], analogous to the rotating bucket in Helium II [79]. Both the thermodynamic threshold for vortices to be favoured, as well as the process by which vortices nucleate [125, 126, 127] into the condensate, are sensitive to dipolar interactions; this will be analysed in detail below. • Motion of a localized obstacle potential (as

17 generated by a tightly-focussed blue-detuned laser beam) through a condensate (or, equivalently, motion of the condensate relative to a static obstacle potential) leads to the nucleation of vortices above a critical relative speed [33, 34, 103], forming a quantum wake downstream of the obstacle. The critical speed is related to the Landau criterion which predicts the formation of elementary excitations in the fluid for relative speeds exceeding vc = min[ω(k)/k] [69, 70]. Ticknor et al [91] examined this process in a quasitwo dimensional dipolar condensate. For dipoles titled into the plane (α > 0), the critical speed becomes anisotropic, a consequence of Landau’s criterion and the anisotropic dispersion relation in the plane. The critical velocity for vortex nucleation can also be derived by considering the energetics of a vortex-antivortex pair [68], implying that the aniostropic critical velocity is directly related to the anisotropic vortex interaction energy. • The phase of the condensate can be directly engineered via optical imprinting to produce vortex phase singularities, as employed to generate both singly- and multiply-charged vortices [128]. This mechanism is independent of the dipoles themselves. • Following a rapid quench through the transition temperature for the onset of Bose-Einstein condensation, the growth of local phase-coherent domains leads to the entrapment of phase singularities and hence vortices (i.e. the KibbleZurek mechanism [129, 130]) [32, 42, 131]. A relevant consideration is the effect of the dipoles on the critical temperature. This shift is sensitive to the shape of the trap, relative to the polarization direction, but is only up to a few percent for 52 Cr [132]. However, this shift may be more significant in 168 Er and 164 Dy condensates. • Dark solitons are dimensionally unstable to decay into vortex pairs or vortex rings [35, 133, 134] via the so-called snake instability, previously established in nonlinear optics [135]. As shown by Nath et al. [136] the nonlocal character of dipolar interactions can stabilize the dark soliton against this instability. • Instead of rotating the system it is possible to introduce synthetic magnetic fields to nucleate vortices [137]. As in the case of mechanical rotation, vortex nucleation is dependent on the properties of the stationary solutions [138]. Numerical investigations of the dipolar GPE, carried out by Zhao and Gu [139], find that

CONTENTS

18

the nucleation of vortices depends on the dipole strength, the strength of the synthetic magnetic field, the potential geometry, and the orientation of the dipoles, with anisotropic interactions significantly altering vortex nucleation. 6.2

Stationary solutions of rotating dipolar condensates in elliptical traps

The most common approach to generate vortices and vortex lattices in trapped condensates is via rotation. Since a cylindrically-symmetric trap set into rotation applies no torque to the condensate, the trap is made anisotropic in the plane of rotation. In the simplest case, this leads to a trap which is weakly elliptical in the plane of rotation [31, 140], with a potential of the form, 1 2 (1 − )x2 + (1 + )y 2 + γ 2 z 2 , (57) V (r) = mω⊥ 2 where rotation is performed about the z-axis. For typical parameters in the absence of dipolar interactions, vortices become energetically favourable in harmonically-trapped condensates for rotation frequencies Ω ∼ 0.3ω⊥ . Surprisingly, in non-dipolar BEC experiments the observed nucleation of vortices occurs at considerably larger rotation frequencies Ω ∼ 0.7ω⊥ . Theoretical analysis based on the hydrodynamic equations reveals the important role of collective modes. At low rotation frequencies the elliptical deformation excites stable low-lying collective modes (quadrupole etc.). However, at a certain rotation frequency, these modes become unstable [141, 142]. Comparison with experiments [140, 143] and full numerical simulations of the GPE [144, 145, 146] have shown that the instabilities are the first step in the entry of vortices into the condensate and the formation of a vortex lattice. The hydrodynamic description of condensates in rotating elliptical traps can be extended to include dipolar interactions. For rotation about the z-axis, described by the rotation vector Ω where Ω = |Ω| is the rotation frequency, the Hamiltonian in the rotating frame is given by, ˆ Heff = H0 − Ω · L,

(58)

where H0 is the Hamiltonian in absence of the rotation ˆ = −i~(r × ∇) is the quantum mechanical and L angular momentum operator. Applying this result for Ω = (0, 0, Ω), the dipolar GPE in the rotating frame is [147, 148], ∂Ψ ~2 2 = − ∇ + V + Φ + g|Ψ|2 i~ ∂t 2m

~ ∂ ∂ x −y Ψ. (59) i ∂y ∂x Note that all spatial coordinates r are those of the rotating frame and the time independent trapping potential V , given by Eq. (57), is stationary in this frame. Momentum coordinates, however, are expressed in the laboratory frame [147, 148, 149]. −Ω

Using the Madelung transform, as per Section 2.3, leads to the following hydrodynamical equations in the rotating-frame ∂n = −∇ · [n (v − Ω × r)] , (60) ∂t ∂v 1 m = −∇ mv 2 + V + gn + Φ − mv · [Ω × r] , ∂t 2 (61) where the quantum pressure is assumed to be small and is neglected, i.e. the Thomas-Fermi limit. Stationary solutions of Eqs. (60) and (61) satisfy the equilibrium conditions, ∂v ∂n = 0 and = 0. ∂t ∂t Assuming the velocity field ansatz [141], v = α∇(xy),

(62)

(63)

permits us to examine the rotating solutions in terms of the velocity field amplitude α. Note that v is the velocity field in the laboratory frame expressed in the coordinates of the rotating frame. It also satisfies the irrotationality condition ∇ × v = 0. Actually, the velocity field amplitude α can be given even more physical meaning by noting that, according to the continuity equation (60), it can be written as [70], α = −DΩ,

(64)

where D is the deformation of the BEC in the x − y plane, D=

κ2y − κ2x hy 2 − x2 i = . hy 2 + x2 i κ2y + κ2x

(65)

Here h. . .i denotes the expectation value in the stationary state. The parameters κx = Rx /Rz and κy = Ry /Rz represent the aspect ratios pf the BEC along x and y with respect to z. Combining Eqs. (61) and (63) gives, m 2 2 ω ˜xx + ω ˜ y2 y 2 + ωz2 z 2 + gn(r) + Φ(r), (66) µ= 2 where the effective trap frequencies ω ˜ x and ω ˜ y are given by, 2 ω ˜ x2 = ω⊥ (1 − ) + α2 − 2αΩ, (67) 2 ω ˜ y2 = ω⊥ (1 + ) + α2 + 2αΩ. (68) The breaking of cylindrical symmetry means that the BEC has an ellipsoidal shape and in the Thomas-Fermi

CONTENTS approximation it adopts an inverted parabolic density profile of the form, y2 z2 x2 for n(r) ≥ 0, (69) n(r) = n0 1 − 2 − 2 − 2 Rx Ry Rz where n0 = 15N/(8πRx Ry Rz ). This is an exact solution of the stationary hydrodynamic equations given in Eqns. (60) and (61) with velocity field (63), and it only remains to find the radii {Rx , Ry , Rz } and velocity amplitude α. The dipolar potential due to a parabolic density distribution of general ellipsoidal symmetry, with dipoles aligned in the z-direction, i.e. the generalized version of that given in Eq. (29), is derived in the appendices of Refs. [61] and [78] : Φ = 3gεdd × (70) 2 2 2 n x β101 + y β011 + 3z β002 n0 κx κy − β001 − , 2 Rz2 3 where the Z ∞coefficients βijk are given by, 1 ds, βijk = j+ 21 1 i+ k+ 1 2 2 2 0 κy + s (κx + s) (1 + s) 2 (71) for integer-valued i, j and k. Thus, Eq. (66) can be rearranged to obtain an expression for the density profile [125, 126, 127], µ− m ˜ y2 y 2 + ωz2 z 2 ˜ x2 x2 + ω 2 ω n= g (1 − εdd ) n0 κx κy 2 3gεdd 2R2 x β101 + y 2 β011 + 3z 2 β002 − Rz2 β001 z + . g (1 − εdd ) (72) Matching the coefficients of the x2 , y 2 and z 2 terms in Eq. (69) and Eq. (72) three self-consistency relations are found, which define the size and shape of the condensate: 2 1 + εdd 32 κ3x κy β101 − 1 ωz 2 κx = , (73) ω ˜x ζ 2 1 + εdd 32 κ3y κx β011 − 1 ωz , (74) κ2y = ω ˜y ζ 2gn0 Rz2 = ζ, (75) mωz2 h i 9κ κ where ζ = 1 − εdd 1 − x2 y β002 . Furthermore, by inserting Eq. (72) into Eq. (60) the stationary solutions satisfy the condition [125, 126, 127], 3 ω 2 κx κy γ 2 0 = (α + Ω) ω ˜ x2 − εdd ⊥ β101 2 ζ 2 ω⊥ κx κy γ 2 3 2 β011 . (76) + (α − Ω) ω ˜ y − εdd 2 ζ Equation (76) gives the velocity field amplitude α for a given εdd , Ω and trap geometry. For εdd = 0, α is independent of the s-wave interaction strength g and the trap ratio γ. However, in the presence of dipolar interactions the velocity field amplitude becomes dependent on both εdd and γ. For fixed εdd and trap geometry, Eq. (76) leads to branches of α as a

19 function of rotation frequency Ω. These branches are significantly different between traps that are circular ( = 0) or elliptical ( > 0) in the x − y plane, and so below each case is considered in turn. Note that the analysis is restricted to the range Ω < ω⊥ : for Ω ∼ ω⊥ the static solutions can disappear, with the condensate becoming unstable to a centre-of-mass instability [141]. 6.2.1

Circular trapping in the x − y plane: = 0

In Figure 15(a) [127] the solutions of Eq. (76) are plotted as a function of rotation frequency Ω for a spherically-symmetric trap γ = 1 and for various values of εdd . Before dicussing the specific cases, let us first point out that for each εdd the solutions have the same qualitative structure. Up to some critical rotation frequency only one solution exists corresponding to α = 0. At this critical rotation frequency the solution bifurcates, giving two additional solutions for α > 0 and α < 0 on top of the original α = 0 solution. We term this critical frequency the bifurcation frequency

Figure 15. (a) Irrotational velocity field amplitude α of the static condensate solutions as a function of the trap rotation frequency Ω in a spherically-symmetric trap (γ = 1 and = 0). Various values of εdd are presented: εdd = −0.49, 0, 0.5 and 0.99. Insets illustrate the geometry of the condensate in the x−y plane. (b) The bifurcation frequency Ωb (the point at which the solutions of α in (a) bifurcate) according to Eq. (77) versus trap ratio γ. Plotted are the results for εdd = −0.49, −0.4, −0.2, 0, 0.2, 0.4, 0.6, 0.8, 0.9 and 0.99. In (a) and (b) εdd increases in the direction of the arrow. Reprinted figure with permission from [127]. Copyright 2009 by the American Physical Society.

CONTENTS Ωb . For εdd = 0 the results of Refs. [141, 142] are √ reproduced with a bifurcation point at Ωb = ω⊥ / 2 and, p for Ω > Ωb , non-zero solutions given by α = 2 [141]. The physical significance of the ± 2Ω2 − ω⊥ bifurcation frequency has been established for the nondipolar case and is related to the fact that the system becomes energetically unstable to the spontaneous √ excitation of quadrupole modes for Ω ≥ ω⊥ / 2. In the Thomas-Fermi limit, a general surface excitation with angular momentum ~l = ~ql R, where R is the Thomas-Fermi radius and ql is the quantized wave number, obeys the classical dispersion relation ωl2 = (ql /m)∇R V involving the local harmonic potential 2 2 V = mω⊥ R /2 evaluated at R, see page 183 of [70]. Consequently, √ for the non-rotating and nondipolar BEC ωl = lω⊥ . Meanwhile, inclusion of the rotational term in the Hamiltonian (58) shifts the mode frequency by −lΩ. Then, in the rotating frame, the frequency of the l √ = 2 quadrupole surface excitation becomes ω2 (Ω) = 2ω⊥ − 2Ω [70]. The bifurcation frequency thus coincides with the vanishing of the energy of the quadrupolar mode in the rotating frame, and the two additional solutions arise√from excitation of the quadrupole mode for Ω ≥ ω⊥ / 2. Remarkably, for the non-dipolar BEC Ωb does not depend on the interactions. This feature arises because the surface excitation frequencies ωl are themselves independent of g. However, in the case of longrange dipolar interactions the potential Φ of Eq. (7) gives non-local contributions, breaking the simple dependence of the force −∇V on R [60]. Thus it is expected that the resonant condition for exciting the quadrupolar mode, i.e. Ωb = ωl /l (with l = 2), varies with εdd . Figure 15(a) shows that this is the case: as dipole interactions are introduced, the solutions change and the bifurcation point, Ωb , moves to lower (higher) frequencies for εdd > 0 (εdd < 0). Note that the parabolic solution still satisfies the hydrodynamic equations providing −0.5 < εdd < 1. Outside of this range the parabolic solution may still exist but it is no longer guaranteed to be stable against perturbations. Figure 15(a) shows that as the dipolar interactions are increased the bifurcation point Ωb moves to lower frequencies. The bifurcation point can be calculated analytically as follows. First, note that for α = 0 the condensate is cylindrically symmetric and κx = κy = κ. In this case the aspect ratio κ is determined by the transcendental Eq. (25) [60, 61] For small α → 0+ , the first order corrections to κx and κy with respect to κ from Eqs. (73) and (74) can be calculated. Inserting these values in Eq. (76) and solving for Ω (noting that

20 Ω → Ωs b as α → 0+ ) gives, Ωb 1 3 2 κ2 β201 − β101 . = + κ εdd γ 2 (77) ω⊥ 2 4 1 − εdd 1 − 92 κ2 β002 Figure 15(b) shows Ωb [Eq. (77)] as a function of γ for various values of εdd . For εdd = √ 0 the bifurcation point remains unaltered at Ωb = ωx / 2 as γ = ωz /ωx is changed [141, 142]. As εdd is increased the value of γ for which Ωb is a minimum changes from a trap shape which is oblate (γ > 1) to prolate (γ < 1). Note that for εdd = 0.99 the minimum bifurcation frequency occurs at Ωb ≈ 0.55, which is over a 20% deviation from the non-dipolar value. For more extreme values of εdd it is expected that Ωb will deviate even further, although the validity of the inverted parabola ThomasFermi solution does not necessary hold. For a fixed γ as εdd increases the bifurcation frequency decreases monotonically. 6.2.2

Elliptical trapping in the x − y plane: > 0

Rotating elliptical traps have been created experimentally with lasers and magnetic fields [31, 140]. Following the experiment by Madison et al. [31] a weak trap ellipticity of = 0.025 is considered. Figure 16(a) [127] shows the solutions to Eq. (76) for various values of εdd in a γ = 1 trap. As has been predicted for the nondipolar case [141, 142], the solutions become heavily modified for > 0. There is an upper branch of α > 0 solutions which exists over the whole range of Ω, and a lower branch of α < 0 solutions which back-bends and is double-valued. The frequency at which the lower branch back-bends is denoted as the back-bending frequency Ωb . The bifurcation frequency in non-elliptical traps can be regarded as the limiting case of the backbending frequency, with the differing nonclamenture employed to emphasise the different structure of the solutions at this point. However, for convenience we shall use the same parameter, Ωb , to refer to both. Note that for non-zero Ω, there are no α = 0 solutions, unlike the circular case exists (for any non-zero Ω). In the absence of dipolar interactions the effect of increasing the trap ellipticity is to increase the back-bending frequency Ωb . Turning on the dipolar interactions, as in the case of = 0, reduces Ωb for εdd > 0, and increases Ωb for εdd < 0. This is more clearly seen in Figure 16(b) where Ωb is plotted versus εdd for various values of the trap ratio γ. Importantly, the back-bending of the lower branch can introduce an instability. Consider the BEC to be on the lower branch at some fixed rotation frequency Ω. Now consider decreasing εdd . The back-bending frequency Ωb increases and at some point can exceed Ω. In other words, the static solution of the BEC suddenly

CONTENTS

21 δn and S = S(0) + δS. Then, by linearizing the hydrodynamic equations Eqs. (60, 61), the dynamics of such perturbations can be described as [125, 126, 127], ∂ δS vc · ∇ g (1 + εdd K) /m δS =− ∇ · n(0) ∇ [(∇ · v) + vc · ∇] δn ∂t δn

Figure 16. (a) Irrotational velocity field amplitude, α, as a function of the trap rotation frequency, Ω, for a trap ratio γ = 1 and ellipticity = 0.025. Various values of εdd are presented, εdd = −0.49, 0, 0.5 and 0.99, with εdd increasing in the direction of the arrow. Insets illustrate the geometry of the condensate in the x − y plane. (b) Backbending point Ωb versus εdd for = 0.025 and γ = 0.5 (solid curve), 1.0 (long dashed curve) and 2.0 (short dashed curve). Reprinted figure with permission from [127]. Copyright 2009 by the American Physical Society.

disappears and the BEC finds itself in an unstable state. We will see in Section 6.4 that this type of instability can also be induced by variations in γ and . As in the = 0 case, increasing εdd decreases both κx and κy , i.e. the BEC becomes more prolate. This distortion is expected because of the anisotropy of dipolar interactions. However, because the dipolar interactions are isotropic in the x−y plane it is perhaps surprising to find that they increase the deformation of the BEC in that plane. This can be clearly seen in Figure 16(a) where it is seen that the magnitude of α increases as εdd is increased (for any fixed value of Ω). See Eq. (64) for the relationship between α and the deformation of the BEC in the x − y plane. 6.3

Dynamical stability of stationary solutions

Although the solutions derived above are static solutions in the rotating frame, they are not necessarily stable, and so in this section their dynamical stability is analyzed. Consider small perturbations in the BEC density and phase of the form n = n(0) +

(78) where vc = v − Ω × r and the integral operator K is defined as Z δn(r0 ) ∂2 dr0 − δn(r). (79) (Kδn)(r) = −3 2 ∂z 4π |r − r0 | To investigate the stability of the BEC the eigenfunctions and eigenvalues of the operator in Eq. (78) can be examined. Dynamical instability arises when one or more eigenvalues λ possess a positive real part. The size of the real eigenvalues dictates the rate at which the instability grows. Note that the imaginary eigenvalues of Eq. (78) relate to stable collective modes of the system [150], e.g. sloshing and breathing [127]. All operators in Eq. (78), when acting on polynomials of degree N , result in polynomials of (at most) the same degree, including the operator K [151, 152, 153]. Hence, the perturbation evolution operator can be rewritten as a scalar matrix operator, acting on vectors of polynomial coefficients, for which finding eigenvectors and eigenvalues is a trivial computational task [125, 126, 127, 142]. Recall the general form of the branch diagram, i.e. Figure 16(a). In the α < 0 half-plane, the static solutions nearest the α = 0 axis never become dynamically unstable, except for a small region Ω ' ω⊥ due to a centre-of-mass instability of the condensate [154]. The other lower branch solutions are always dynamically unstable and therefore expected to be irrelevant to experiment. Thus, only the dynamical instability for the upper branch solutions is considered, i.e. the branch in the upper half plane (where α > 0). Figure 17 [127] plots the maximum positive real eigenvalues of the upper branch solutions as a function of Ω for a fixed ellipticity = 0.025. The maximum polynomial perturbation, δn = xp y q z r was set at p + q + r ≤ 3 = N , since for this ellipticity higher order perturbations do not alter the region of instability. For a given εdd and γ there exists a dynamically unstable region in the − Ω plane. An illustrative example is shown in Figure 17(inset) for εdd = 0 and γ = 1. The instability region (shaded) consists of a series of crescents [142]. Each crescent corresponds to a single value of the polynomial degree N , where higher values of N add extra crescents from above. At the high frequency end these crescents merge to form a main region of instability, characterised by large eigenvalues. At the low frequency end the crescents

CONTENTS

Figure 17. The maximum positive real eigenvalues of Eq. (78) (solid curves) for the upper-branch solutions of α as a function of Ω. We assume = 0.025, γ = 1 and N = 3, and present various dipolar strengths εdd = −0.49, 0, 0.5 and 0.99, with εdd increasing in the direction of the arrow. The inset shows the full region of dynamical instability in the − Ω plane for εdd = 0. The narrow regions have negligible effect and so we only consider the main instability region (bounded by the dashed line). Reprinted figure with permission from [127]. Copyright 2009 by the American Physical Society.

become vanishingly thin and are characterised by very small eigenvalues which are at least one order of magnitude smaller than in the main instability region [146]. As such these regions will only induce instability in the condensate if they are traversed very slowly. This was confirmed by numerical simulations for nondipolar BECs [146] where it was shown that the narrow instability regions have negligible effect when ramping 2 Ω at rates greater than dΩ/dt = 2 × 10−4 ω⊥ . For this reason the narrow regions of instability can be ignored and the instability region will be defined to be the main region, as bounded by the dashed line in Figure 17(inset). For the experimentally relevant trap ellipticities 6 0.1 the unstable region is defined solely by the N = 3 perturbations. The lower bound of the instability region is defined to be Ωi (this corresponds to the dashed line in the inset in Figure 17). This is the key parameter to characterise the dynamical instability. As εdd is increased, Ωi decreases and the unstable range of Ω widens accordingly. Note that the upper bound of the instability region is defined by the endpoint of the upper branch at Ω ' ω⊥ . What if higher order perturbations N > 3 are considered? As the size of the scalar matrix operator (78) is increased to N = 4, 5, . . . the higher lying modes that are thereby described also develop real eigenvalues as Ω is increased. However, these higher lying modes fall within the region of instability already shown in Figure 17 for N = 3 and so do not alter the region of instability, as mentioned above. Significantly, these higher lying modes, i.e. the perturbations contained in N = 4 that are not present in N = 3 become

22 unstable at higher values of Ω than those in N = 3. The same is true for the higher lying modes in N = 5 in comparison to those in N = 4 etc. This can be taken as circumstantial evidence that there is no “roton” minimum in the energy spectrum for the parameters considered. As discussed in Section 3.2.2, the roton minimum refers to a minimum in the energy spectrum at a finite value of the eigenvalue (e.g. momentum p uniform case) labelling the excitation. A roton minimum means that, counter-intuitively, some higher lying modes can have lower energy than lower lying modes and this causes important effects in flowing systems. In particular, Pitaevskii [155] discussed the case of superfluid 4 He flowing through a pipe at velocity v. In the laboratory frame the energy spectrum is Galilean-shifted such that E → E − pv and this leads, for large enough v, to the roton mode pr being brought down to zero energy first. Crudely speaking, this is expected to trigger an instability to the formation of a density wave with wavelength ∝ p−1 r . In the present rotational flow the Galileanshifted energy E → E − LΩ can presumably result in angular roton modes [80] becoming unstable as Ω is increased. However, the empirical observation that the modes become unstable in order as Ω is increased seems to rule out the presence of an angular roton minimum at finite angular momentum Lr for the parameters considered. This is not surprising because roton minima in dipolar BECs have so far only been predicted to occur outside of the range −0.5 < εdd < 1. A proper treatment outside this stable range requires going beyond the Thomas-Fermi approximation since the zero point energy must be included. 6.4

Routes to instability and vortex lattice formation

For a non-dipolar BEC the static solutions and their stability in the rotating frame depend only on the rotation frequency Ω and trap ellipticity . Adiabatic changes in and Ω can be employed to evolve the condensate through the static solutions and reach a point of instability. Indeed, this has been realized both experimentally [140, 143] and numerically [144, 145], with excellent agreement to the hydrodynamic predictions. For the case of a dipolar BEC the static solutions and their instability depend additionally on the trap ratio γ and the interaction parameter εdd , see Sections 6.2 and 6.3. Since all of these parameters can be experimentally tuned in time, one can realistically consider each parameter as a distinct route to traverse the parameter space of solutions and induce instability in the system. Examples of these routes are presented in Figure 18

CONTENTS

23 6.4.1

Figure 18. Stationary states in the rotating trap characterised by the velocity field amplitude α, determined from Eq. (76). Dynamically unstable solutions are marked with red circles. In each of the figures the trap rotation frequency Ω (a), trap ellipticity (b), dipolar interaction strength εdd (c) and axial trapping strength γ (d) are varied adiabatically, whilst the remaining parameters remain fixed at Ω = 0.7ω⊥ , = 0.025, εdd = 0.99, and γ = 1. The adiabatic pathways to instability (onset marked by red asterisk) are schematically shown by the dashed and solid arrows. Dashed arrows indicate a route towards dynamical instability, whereas solid arrows indicate an instability due to disappearance of the stationary state. Reprinted with permission from [127].

[127]. Specifically, Figure 18 shows the static solutions α of Eq. (76) as a function of Ω [Figure 18(a)], [Figure 18(b)], εdd [Figure 18(c)] and γ [Figure 18(d)]. In each case the remaining three parameters are fixed at = 0.025, γ = 1, Ω = 0.7ω⊥ , and εdd = 0.99. Dynamically unstable solutions are indicated with red circles. Grey arrows mark routes towards instability (the point of onset of instability being marked by an asterisk), where the free parameter Ω, , εdd , or γ is varied adiabatically until either a dynamical instability is reached, or the solution branch backbends and so ceases to exist. For solutions with α > 0, the instability is always due to the system becoming dynamically unstable (dashed arrows), whereas for α < 0 the instability is always due to the solution branch backbending on itself (solid arrows) and so ceasing to exist. Numerical studies [145] indicate that these two types of instability, in the absence of dipolar interactions, involve different dynamics and possibly have distinct experimental signatures.

Does the final state of the system contain vortices?

Having revealed the regions of parameter space where a rotating dipolar condensate becomes unstable the question of whether this instability leads to a vortex lattice is now considered. First, let us review the situation for a non-dipolar BEC. The presence of vortices in the system becomes energetically favourable when the rotation frequency exceeds the critical frequency Ωv defined in Eq. (36). Working in the Thomas-Fermi limit where the background density takes a parabolic form, Ωv is given by Eq. (46), which assumes cylindrical symmetry. For typical condensate parameters Ωv ∼ 0.3ω⊥ . It is observed experimentally, however, that vortex lattice formation occurs at considerably higher frequencies, typically Ω ∼ 0.7ω⊥ . This difference arises because the vortex-free solutions remain remarkably stable above Ωv . It is only once a hydrodynamic instability occurs (which occurs in the locality of Ω ≈ 0.7ω⊥ ) that the condensate has a mechanism to deviate from the vortex-free solution and relax into a vortex lattice. Another way of visualising this is as follows. Above Ωv the vortex-free condensate resides in some local energy minimum, while the global minimum represents a vortex or vortex lattice state. Since the vortex is a topological defect, there typically exists a considerable energy barrier for a vortex to enter the system. However, the hydrodynamic instabilities offer a route to navigate the BEC out of the vortexfree local energy minimum towards the vortex lattice state. Note that vortex lattice formation occurs via nontrivial dynamics. The initial hydrodynamic instability in the vortex-free state that was discussed in Section 6.4 is only the first step [145]. For example, if the condensate is on the upper branch of hydrodynamic solutions (e.g. under adiabatic introduction of Ω) and undergoes a dynamical instability, this leads to the exponential growth of surface ripples in the condensate [31, 143, 145]. Alternatively, if the condensate is on the lower branch and the static solutions disappear (e.g. following the introduction of ) the condensate undergoes large and dramatic shape oscillations. In both cases the destabilisation of the vortex-free condensate leads to the nucleation of vortices into the system. A transient turbulent state of vortices and density perturbations then forms, which subsequently relaxes into a vortex lattice configuration [145, 156]. In the presence of dipolar interactions, however, the critical frequency for a vortex depends crucially on the trap geometry γ and the strength of the dipolar interactions εdd as discussed in Section 4.3 based upon the results in Ref. [121]. There it was assumed that

CONTENTS

24 [121]. This dependence is very weak for γ = 10, and throughout the range of εdd presented, it maintains the approximate value Ωv ≈ 0.1ω⊥ . Importantly these results show that when the condensate becomes unstable a vortex/vortex lattice state is energetically favoured. As such, it is expected that in an oblate dipolar BEC a vortex lattice will ultimately form when these instabilities are reached.

Figure 19. The relation between the instability frequencies, Ωb (long dashed red curve) and Ωi (short dashed curve), and the critical rotation frequency for vorticity Ωv (solid curve) for (a) an oblate trap γ = 10 and (b) a prolate trap γ = 0.1. The instability frequencies are based on a trap with ellipticity = 0.025 while Ωv is obtained from Eq. (46) under the assumption of a 52 Cr BEC with 150, 000 atoms and scattering length as = 5.1nm in a circularly symmetric trap with ω⊥ = 2π × 200Hz. Reprinted figure with permission from [127]. Copyright 2009 by the American Physical Society.

the system was cylindrically symmetric, however, if we assume a very weak ellipticity = 0.025, it is expected that the correction to the critical frequency will be correspondingly small. As an example, take the parameter space of rotation frequency Ω and dipolar interactions εdd and consider the behaviour in a rather oblate trap with γ = 10. Figure 19(a) plots the instability frequencies Ωi and Ωb for this system as a function of the dipolar interactions εdd . Depending on the specifics of how this parameter space is traversed, either by adiabatic changes in Ω (vertical path) or εdd (horizontal path), the condensate will become unstable when it reaches one of the instability lines (short and long dashed lines). These points of instability decrease weakly with dipolar interactions and have the approximate value Ωi ≈ Ωb ≈ 0.75ω⊥ . On the same plot the critical rotation frequency Ωv according to Eq. (46) is plotted. In order to calculate this, a BEC of 150,000 52 Cr atoms confined within a trap with ω⊥ = 2π × 200Hz has been assumed. In this oblate system the dipolar interactions lead to a decrease in Ωv , as noted in

Figure 19(b) makes a similar plot but for a prolate trap with γ = 0.1. The instability frequencies show a somewhat similar behaviour to the oblate case. However, Ωv is drastically different, increasing significantly with εdd . This qualitative behaviour occurs consistently in prolate systems [121]. This introduces two regimes depending on the dipolar interactions. For εdd 6 0.8, Ωi,b > Ωv , and so it is expected that a vortex/vortex lattice state forms following the instability. However, for εdd > 0.8 a new regime emerges in which Ωi,b < Ωv . In other words, while the instability in the vortexfree parabolic density profile still occurs, a vortex state is not energetically favourable. The final state of the system is therefore not clear. Given that a prolate dipolar BEC is dominated by attractive interactions (since the dipoles lie predominantly in an attractive end-to-end configuration), one might expect similar behavior to the case of non-dipolar BECs with attractive interactions (g < 0) where the formation of a vortex lattice can also be energetically unfavourable. Suggestions for final state of the condensate in this case include centre-of-mass motion and collective oscillations, such as quadrupole modes or higher angular momentum-carrying shape excitations [?, 158, 159]. However, the true nature of the final state warrants further investigation. Numerical results of the dipolar GPE [160, 161] show that, for an adiabatic introduction of the rotation frequency, the strength of the dipolar interaction influences the rotation frequency at which vortices are admitted into the condensate, in the quasi-two dimensional regime. In agreement with analysis presented above it is also found that as the dipolar interaction is increased the rotation frequency required to nucleate vortices is reduced. 7

Vortex Lattices

To date there have been several examinations of the properties of vortex lattices in quasi-two dimensional rotating dipolar BECs [105, 162, 163, 164, 165, 166, 167]. Work by Cooper et al. [162, 164, 165] found, by numerical minimization of the interaction energy with the wavefunction constrained to states in the

CONTENTS Lowest Landau Level (LLL) regime, that the dipolar interaction could modify the symmetry of the vortex lattice. Specifically they found as opposed to the conventional triangular lattice structure of non-dipolar BECs, new phases could emerge with square, stripe (rectangular) and bubble phases. These new phases emerge in the regime as . −0.13Cdd m/~2 . This result coincided with the work of Zhang and Zhai [163] who also found that the triangular phase is not favoured when the contact interactions are attractive (as < 0), with stronger dipolar interactions leading to a square then rectangular lattice. Additionally, recent work by Kishor Kumar et al. [167], using numerical solutions of the three dimensional purely (as = 0) dipolar GPE, found both triangular and square vortex lattice configurations, with both the strength of the dipolar interactions and the rotation frequency determining the symmetry of the final state. Each of these investigations assumed that the axis of rotation of the BEC was the same as the alignment direction of the dipoles. Numerical simulations of the dipolar GPE, carried out by Yi and Pu [105] did not find any evidence of this change of lattice structure for such a configuration. However, for dipoles aligned off axis, they did find evidence of a change in the symmetry of the vortex lattice. This work also found evidence for ripply vortex lattices, where density modulations arise in the vicinity of a vortex. This can be understood in the context of the analysis presented in Section 4 where we deduced that as the BEC approaches the roton instability density ripples appear around a single vortex. This is consistent with numerical work presented by Jona-Lasinio et al. [85]. Below we present new results for the structure of a vortex lattice in a dipolar BEC in the quasi-two dimensional regime. We consider the case where the dipole alignment is perpendicular to the condensate plane (in reference to Figure 4 α = 0) and generalise this to include configurations where the alignment has a component into the plane (in reference to Figure 4 α > 0). The treatment considered draws on the works of Cooper et al. [162, 165], Zhang and Zhai [163] and [168, 169]. For clarity, we shall begin our analysis by considering vortex lattices for the case of a nondipolar BEC. Although this case has been covered rather extensively in the literature, it will be useful for the reader to provide a comprehensive, self-contained review here. We shall then build on the methodology presented to consider dipolar BECs.

25 7.1

Vortex lattice in a non-dipolar BEC

Assuming that a condensate is confined in a cylindrically-symmetric harmonic trap, the energy functional Z in the rotating frame is given by, E[ψ] = ψ(r)H 0 ψ(r) dr Z 1 n(r1 )U (r1 − r2 )n(r2 ) dr1 dr2 , (80) + 2 where, 1 2 H0 = (i~∇ + Ωmˆ z × ρ) 2m m 2 m + ω⊥ − Ω2 ρ2 + ωz2 z 2 . (81) 2 2 There are two distinct contributions to the total energy: a single-particle energy contribution, E0 (the first term in Eq. (80)) and an interaction energy contribution, which in the absence of dipolar interactions is EvdW (the second term in Eq. (80)). To study the properties of vortex lattices it is appropriate to consider the quasi-two-dimensional regime. Physically, this corresponds to a situation where the condensate is rapidly rotating so that centrifugal spreading is significant, and where the longitudinal trapping is strong (~ωz gn(0)). In such circumstances it is appropriate to assume that the longitudinal motion is described by the ground state of the z-confinement, so that the condensate wavefunction can be approximated via Eq. (30). The resulting form for the energy functional, in the absence of dipolar interactions is, Z N ? 0 ? ψ⊥ (ρ) dρ E = ~ωz + ψ⊥ (ρ)H⊥ 2 Z g + √ n2⊥ (ρ) dρ, (82) 2 2πlz where, 1 2 0 (i~∇⊥ + Ωmˆ z × ρ) H⊥ = − 2m m 2 + ω⊥ − Ω2 ρ2 . (83) 2 Consider the fast-rotating limit where Ω → ω⊥ . This is the point at which the centrifugal spreading due to rotation almost overwhelms the confinement due to the radial trap. In this limit, the singleparticle Hamiltonian in quasi-two dimensions tends to −(i~∇⊥ + Ωmˆ z × ρ)2 /2m, up to a constant. The eigenfunctions of this Hamiltonian are well known: they are the Landau level orbitals um,n (x, y) with corresponding eigenenergies n = ~Ω(n + 1/2). The n quantum number labels the Landau level, and may take on any non-negative integer value. Each Landau level is infinitely degenerate since n does not depend on m (which also takes on non-negative integer values). Although the Landau level orbitals do not necessarily

CONTENTS

26

form a complete basis for the full quasi-two dimensional Hamiltonian, in the limit of weak interactions then they are a good approximate basis choice [157]. In searching for the ground state of the condensate it is assumed that it is adequately described by a superposition of n = 0 Landau level orbitals, i.e. the lowest Landau level (LLL) approximation. Using an unrestricted minimization this assumption implies that [47], √ 2πlz Ω . . (84) 1− ω⊥ 2N as In this limit the two-dimensional ground state wavefunction of the condensate can be written as, ∞ X ψ⊥ (ρ) = cm um,0 (ρ) =

m=0 ∞ X m=0

√

cm 2πm!

x + iy l⊥

m

2 x + y2 , exp − 2 2l⊥

(85) where the right-hand side follows from the p explicit form of um,0 (x, y). The length scale l⊥ = ~/mΩ characterises the radial extent of the condensate and is effectively equal to the transverse trap length since Ω → ω⊥ . At this point it turns out to be convenient to convert to a complex number representation, rather than working with the components of a two-dimensional vector. To this end ρ = xˆ x + yˆ y is mapped onto the complex number w = x + iy. Then the ground state wavefunction of Eq. (85) 2may be written as, |w| ψ⊥ (w) = h(w) exp − 2 , (86) 2l⊥ where h is an analytic function of w. With this definition, the coefficients of the superposition cm have been absorbed into h. It is possible to fully specify h in terms of its roots since it is an analytic function, where each root specifies the location of a vortex core at the corresponding x and y coordinates in the condensate [169]. A vortex lattice ground state corresponds to a situation where the roots of h lie on a lattice. The next step is to construct an analytic function with roots that satisfy this property. Fortunately, there is a wellstudied function in complex analysis which will be useful here: the Jacobi theta function θ1 (z, ζ). The roots of θ1 (z, ζ) are able to describe any regular lattice up to a rotation by making an appropriate choice for the parameter ζ. This means that h may be expressed in terms of θ1 to obtain a vortex lattice ground state wavefunction. However, in order to make this connection we must introduce a way to describe a regular lattice mathematically.

Figure 20. Illustration of the lattice basis vectors and the lattice parameters.

A two-dimensional lattice may be fully specified by a pair of basis vectors: b1 and b2 . The points of the lattice are obtained from these basis vectors by constructing all possible linear combinations of the form m1 b1 + m2 b2 where m1 and m2 are integers. In general, four real parameters are required to fully specify a lattice: two real components for each of the two basis vectors. However, one of these parameters may be fixed since there is no need to distinguish between lattices which are equivalent up to a rotation. This is justified because the Hamiltonian which describes the condensate is cylindrically symmetric. In order to fix one of the parameters the first basis vector b1 is chosen to be orientated along the x-axis. The three remaining parameters which describe the lattice are depicted in Figure 20, along with the lattice basis vectors. The first basis vector b1 is specified solely in terms of its magnitude, which is denoted by the parameter b1 , since its direction is fixed along x ˆ. The second basis vector b2 is defined with reference to the first basis vector through a rotation and rescaling. This requires two additional parameters: a rotation angle η and a scaling factor τ . Writing out the basis vectors explicitly in terms of the parameters b1 , τ and η, results in b1 = b1 x ˆ and b2 = τ b1 (cos ηˆ x+ sin ηˆ y). The other parameter that appears in Figure 20 is the area of the unit cell, given by vc = b21 τ sin η. This parameter is redundant in specifying the lattice because it depends on the other parameters: b1 , τ and η. However, it is useful to mention it because it appears in a number of places in the subsequent analysis. Another useful concept regarding our mathematical description of the vortex lattice is the reciprocal lattice. It is needed to represent a function which is defined on a lattice as a Fourier series. The reciprocal lattice is obtained from the original lattice by a transformation of the lattice basis vectors. Denoting the basis vectors of the reciprocal lattice as q1 and q2 , we have q1 = 2πb2 × ˆ z/vc = 2πb1 τ (sin ηˆ x − cos ηˆ y)/vc and q2 =

CONTENTS

27

2πˆ z ×b1 /vc = 2πb1 y ˆ/vc . As for the case of the original lattice, the reciprocal lattice is constructed from its basis vectors by considering all linear combinations of the form m1 q1 + m2 q2 where m1 and m2 are integers. Each one of these linear combinations is a reciprocal lattice vector, which is denoted in general by q. From this, it is possible to describe the two-dimensional density as, N − χρ22 e g(ρ), (87) n⊥ (ρ) = πχ2 P where g(ρ) = q g˜q exp(iq · ρ) and, 2

(−1)m1 +m2 e−vc q /8π √ , (88) τ sin η with q = m1 q1 + m2 q2 . In other words, the two-dimensional condensate density is the product of P a Gaussian envelope and a function g(ρ) = ˜q exp(iq · ρ) which is periodic on the vortex qg lattice. The Fourier coefficients of g(ρ) are rescaled compared to those in Eq. (88) so that g˜q = P gq / v g˜v exp(−iv 2 χ2 /4). The radial extent of the condensate cloud is quantified by the length-scale χ = −2 (l⊥ − πvc−1 )−1/2 which is related to the number of vortices in the system.

gq =

From this ansatz for the vortex lattice ground state, it is possible to calculate the energy of the condensate as a function of the lattice parameters. Using the fact that ψ⊥ (ρ) is in the LLL, the single-particle energy contribution, in the limit of large vortex number, is independent of the vortex lattice parameters. The interaction energy contribution may be rewritten in the following form, N 2g EvdW = I[n]. (89) (2π)3/2 lz χ2 R Here I[n] = π 2 χ2 /N 2 n2⊥ (ρ)dρ is a dimensionless analogue of the interaction energy contribution. Substituting the ansatz for the vortex lattice ground state from Eq. (87) gives, χ2 |q+v|2 1X I(τ, η) = g˜q g˜v e− 8 . (90) 2 q,v This quantity depends on τ and η implicitly through its dependence on the reciprocal lattice vectors. Considering the limit of large vortex number where χ2 q2 1, it is possible to simplify this expression significantly. In fact, one may assume that the χ2 |q+v|2

is so sharply peaked at q = −v exponential e− 8 that it may be approximated by a Kronecker delta: δq,−v . This collapses the double sum to a single sum over q, which is much simpler to evaluate, 1X 2 I(τ, η) ≈ (˜ gq ) 2 q ∞ X π 1 = e 2 m ,m =−∞ 1

2

2m1 m2 cot η−m21 τ csc η−

m2 2 csc η τ

.

Figure 21. Contour plot of I(τ, η), a dimensionless analogue of the interaction energy. The white areas are regions where I(τ, η) is greater than the cut-off value of 0.6. The purple areas correspond to regions where I(τ, η) approaches its minimum value. Multiple local minima are visible, although they become difficult to see in the lower left-hand region of the plot. The three minima labelled by red numbers are mentioned in the text.

(91) In order to minimise I(τ, η) as defined in Eq. (91) a numerical treatment is required, which results in a cut off the sums over m1 and m2 at some upper and lower bound. Defining the upper and lower bounds to be at M and −M respectively, with M set to 15 ensures that I(τ, η) is accurate to double precision for values of τ and η greater than about 0.05. To explore regions in which τ or η is less than 0.05, M needs to be increased above 15 to include higher frequency terms in the Fourier series. Fortunately, it is reasonable to exclude these regions, in the absence of dipolar interactions, because they correspond to an unphysical situation where the vortex lattice begins to collapse onto a line. It is illustrative to perform the minimisation of I(τ, η) graphically, by generating a contour plot of I(τ, η) as a function of τ and η. The result is shown in Figure 21. In this plot, the dark purple shading corresponds to regions where I(τ, η) approaches its minimum value. The white areas correspond to regions where I(τ, η) is greater than the cut-off value, which is in this case set to 0.6. In Figure 21 each local minimum has the same value of I equal to 0.5797. However, a careful consideration shows that each of these solutions in fact corresponds to the same type of lattice; just at a different scale and orientation. The solution labelled ‘1’ in Figure 21 is the standard parametrisation of the triangular lattice. It is illustrated in Figure 22, along with the standard

CONTENTS (a)

28 7.2

(b)

Triangular (τ = 1, η = π/3) (c)

Square (τ = 1, η = π/2) (d)

Vortex lattice in a quantum ferrofluid: dipoles perpendicular to the plane

For the case of dipolar BECs the dipolar interaction potential, Udd (r), needs to be included. This is the only modification to the theory that is required to account for the effect of the dipoles. Assuming that the dipoles are aligned perpendicular to the plane of rotation, i.e. along the z-axis, implies that the cylindrical symmetry of the Hamiltonian is maintained. In this limit the criterion for being in the LLL becomes √ 2πlz Ω q i . 1− h ω⊥ 9π dd 2N as 1 + 16πε 1 − σ 3 8 +

Rectangular (τ < 1, η = π/2)

Parallelogrammic

Figure 22. A lattice may be classified as one of four types according to the geometry of the unit cell. The triangular, square and rectangular lattices are all special cases of the parallelogrammic lattice. For the triangular, square and rectangular lattice, we have given the corresponding (τ ,η) values in the so-called standard parametrisation.

parametrisations of the three other types of lattice: square, rectangular and parallelogrammic. In general, the standard parametrisation is defined to be the one for which τ is closest to 1. Alternative parametrisations of the same lattice have smaller values of τ compared to the standard one, and correspond to trivial rotations and rescalings of the lattice. For example, the second solution, denoted as ‘2’ in Figure 21 is an equally valid parametrisation of the triangular lattice. The above calculation verifies that a triangular vortex lattice geometry is always favoured in non-dipolar BECs. Of course, this was to be expected based on the results of numerous experiments and previous theoretical studies. It is interesting to note that the triangular lattice geometry is not significantly favoured over other possible lattice geometries. In particular, the energy corresponding to the square lattice geometry at τ = 1 and η = π (the saddle point in Figure 21) is only 1.8% larger than that of the triangular lattice geometry. It is therefore conceivable that the energy minimum may shift to a non-triangular lattice geometry if the functional form of the interaction energy contribution is altered. With this motivation the above calculation is generalized below to include dipolar interactions.

π 2 lz εdd σ q i2 , h 16πεdd 1 − σ 9π as N 1 + 3 8

(92)

where σ = lz /R⊥ . The above has been obtained by ˜ ⊥ (q) in terms of considering the expansion of the U dd the width of the BEC to first order, i.e. Eq. (49), and then calculating the dipolar potential to second order in ρ/R⊥ . In the LLL regime an additional contribution to the energy functional arises, Z 1 Edd = n(r1 )Udd (r1 − r2 )n(r2 ) dr1 dr2 . (93) 2 Adding this to the single-particle and contact interaction energy contributions gives the total energy in the rotating frame for a dipolar BEC, E = E0 + EvdW + Edd . (94) The results obtained in Section 7.1 for the singleparticle (E0 ) and contact interaction (EvdW ) energy contributions are unchanged for the dipolar case. All that remains then, is to perform the calculation for Edd . Rewriting the dipolar interaction energy contribution in reciprocal space by applying the Fourier convolution theorem leads to, Z 1 1 ˜dd (k) dk. Edd = n ˜ (k)˜ n(−k)U (95) 2 (2π)3 To evaluate this in quasi-two dimensions, we substitute the Fourier transform of the quasi-two dimensional 2 2 condensate density, n ˜ (k) = e−kz lz /4 n ˜ ⊥ (q), such that Z 1 1 ⊥ ˜dd n ˜ ⊥ (q)˜ n⊥ (−q)U (q) dq, (96) Edd = 2 (2π)2 ˜ ⊥ (q) is given by Eq. (33), with α = 0. where U dd Now that an expression for the dipolar interaction energy in quasi-two dimensions has been derived, it is possible to evaluate the energy assuming that the condensate is in the vortex lattice ground state. Performing the Fourier transform on the vortex lattice condensate density specified in Eq. (87) results

CONTENTS

29

in, X 2 0 2 n ˜ (q) = N g˜q0 e−χ |q −q| /4 .

(97)

resulting in, X χ2 q 2 W(τ, η) ≈ (˜ gq )2 e− 2 Aa2 (2q) + Ab2 (2q) ,

q0

This enters into the expression for the dipolar interaction energy, leading to, 2 2 2 N 2 Cdd X Edd (τ, η) = g˜q g˜v e−χ (q +v )/4 A(v − q), 3 2 3 3 (2π) lz q,v

where, e−

χ2 q 2 2

Aa2 (2q) =

(98) 2

√ 2 − 3 πueu erfc(u) du 2 2 2 ∞ χ qu − χ l2u du e z uI0 √ 2lz2 0 2 Z ∞ χ2 u 2 √ χ qu − −3 π e l2z u2 eu erfc(u)I0 √ du,(99) 2lz2 0 and I0 (·) is the modified Bessel function of the first kind. At this point, the integral may be separated into two terms A1 (q) + A2 (q). The first term has a simple solution, 2 2 2 2 Z ∞ 2 2 χ q χ qu lz − χ l2u e 8 . A1 (q) = 2 ue z I0 √ du = 2 χ 2lz 0 1 2π Z =2

Z

e

− χl2

z

u2 + lz√q·u 2

(100) Returning to the expression for the dipolar interaction energy given in Eq. (98), and substituting the simplified result for A1 (q) gives 2N 2 Cdd N 2 Cdd Edd (τ, η) = I(τ, η) + W(τ, η), 3 3 3 (2π) 2 lz χ2 3 (2π) 2 lz3 (101) where I(τ,X η) is as defined in Eq. (90) and, 2 2 2 W(τ, η) = g˜q g˜v e−χ (q +v) /4 A2 (v − q). (102) q,v

This expression shows that the dipolar energy has been separated into two distinct contributions: a local contribution which is proportional to I(τ, η) and a non-local contribution which is proportional to W(τ, η). In principle, it is now possible to calculate the dipolar interaction energy as a function of the lattice parameters, however the function for the non-local contribution, W(τ, η) turns out to present a number of difficulties. Specifically, since this integral is not tractable, it must be evaluated numerically. In order to resolve this the difficult integral it is broken up into a series of simpler integrals, each of which is analytically solvable. This can be done by expressing the complementary error function, which appears in the integrand, as a power series, ∞ 1 X (−1)n u2n+1 (103) erfc(u) = 1 − √ π n=0 n!(2n + 1) and bringing the sum outside of the integral. Additionally, as in the case of the contact interactions it is possible to reduce the double sum, in Eq. (102) to a single sum in the limit of large vortex number,

3π 5 2

e

−

(β 2 −2)q 2 χ2 4(β 2 −1)

8(β 2 − 1) 42 2 β lz q × β 4 lz2 q 2 I1 4 − 4β 2 4 2 2 β lz q − 2β 2 + β 4 lz2 q 2 − 2 I0 4 − 4β 2 (105)

where, A(q) =

(104)

q

and, χ2 q 2

−

q 2 χ2

e− 2 Ab2 (2q) = 3e 2(β2 −1) × 42 2 ∞ X (−1)n (n + 1) β lz q , L 2 − 1)2+n (2n + 1) n+1 (β 2 − 2β 2 n=0

(106)

where β = χ/lz and Ln (·) is the nth Laguerre polynomial. With the results of Section 7.1 it is now feasible, from a computational perspective, to numerically minimise the condensate energy, E(τ, η) = E0 + Eint (τ, η), with respect to τ and η to determine the optimal vortex lattice geometry. In minimising the condensate energy, as in Section 7.1, only the interaction energy contribution needs to be considered, N 2 Cdd Eint (τ, η) = × 3 3 (2π) 2 lz3 # " 2 lz 1 I(τ, η) + W(τ, η) , (107) 2+ εdd χ since the single-particle contribution does not depend on τ and η. Assuming Cdd > 0 so that the factor outside the square brackets is positive the vortex lattice configuration is determined by the the minimization of 2 1 lz 2+ I(τ, η) + W(τ, η), (108) εdd χ with respect to τ and η. The results of this minimization are shown in Figure 23 for a particular choice of length scale parameters whoch are given in the figure caption. In both figures, the optimal values of τ and η are plotted against ε−1 dd . The optimal value of τ is represented by a solid line with reference to the scale on the left vertical axis, while the optimal value of η is represented by a dotted line with reference to the scale on the right vertical axis. Taken together, the optimal values of τ and η describe the optimal vortex lattice geometry. By comparing the optimal values of τ and η to the standard lattice parametrisations given in Figure 22, it is possible to classify the geometry of the vortex lattice. For example, at the point ε−1 dd = −1.82 in Figure 23(a), a square lattice is favourable since the optimal values

CONTENTS

30 Phase Collapse Rectangular Square Triangular

Figure 23(a) ε−1 dd < −1.895 −1.895 < ε−1 dd < −1.825 −1.825 < ε−1 dd < −1.815 ε−1 > −1.815 dd

Figure 23(b) ε−1 dd < −1.946 −1.946 < ε−1 dd < −1.911 −1.911 < ε−1 dd < −1.906 ε−1 > −1.906 dd

Table 1. Definition of the four regions in phase space shown in Figure 23.

(a)

(b)

between the two cases: the regions in Figure 23(b) are contracted and shifted to the left compared to those in Figure 23(a). A more accurate description of the phase regions is given in Table 1 in terms of inequalities. In addition to the three colour-shaded regions, there is also a grey-shaded region for which the condensate is in the so-called collapse phase. In the collapse phase, the optimal value of τ tends to zero and the vortex lattice analysis begins to break down. Physically, this phase corresponds to a situation where the vortices are arranged in densely-packed lines, with the spacing between the lines being larger than the extent of the condensate in the x − y plane. Since the unit cell of the lattice extends beyond the boundaries of the condensate in this situation, the vortex lattice can be consider as collapsed. When the system enters the collapse phase, there are also signs that the analysis becomes invalid. Since τ approaches zero in this phase, the expression for interaction energy becomes inaccurate because only enough terms in the Fourier decomposition to consider values of τ greater than about 0.05 are included (as in Section 7.1 M = 15).

Figure 23. Plots showing the optimal values of τ and η in dipolar BECs with on-axis polarisation. Plot (a) assumes the 2 ) = 1.0191, l /l = 40 and χ/l = 292.33 length scales vc /(πl⊥ ρ z z 2 ) = 1.0191, l /l = 80 (b) assumes the length scales vc /(πl⊥ ⊥ z and χ/lz = 584.66. In each plot, the black solid (dashed) line represents the optimal value of τ (η) with reference to the scale on the left (right) vertical axis. By classifying the (τ ,η) parameters, four distinct regions in phase space are identified: a collapse phase, rectangular lattice (stripe) phase, square lattice phase, and triangular lattice phase. These regions are indicated by coloured shading. The red line with arrows on the left side specifies the region for which the interaction energy is less than zero.

Apart from the stability of the vortex lattice, we may also assess the stability of the condensate itself by looking at the sign of the interaction energy. If the interaction energy is negative, then it can approximately be regarded that the condensate as being prone to collapse. The region of phase space for which interaction energy is negative is the area to the left of the red line in Figures 23 (a) and (b). Interestingly, the interaction energy becomes negative at roughly the value of ε−1 dd where the optimal value of τ approaches zero. This suggests that there may be a link between the collapse of the condensate and the collapse of the vortex lattice.

of τ and η at that point are 1 and π respectively. Continuing in this way, three types of lattice emerge depending on the value of ε−1 dd : triangular, square or rectangular. Each of these lattice geometries occurs in a distinct region of the phase space, indicated by coloured shading in the Figure 23. Comparing Figures 23(a) and (b), the relative size of each region is the same for both sets of length scales. There is however an overall translation and scaling difference

In this Section the vortex lattice geometry in dipolar BECs for the special case of on-axis polarisation has been analysed. The results show that three lattice geometries are possible, depending on the value of ε−1 dd and the values of the length scales l⊥ , lz and χ. In general, a triangular lattice geometry is favoured in regions where the local interaction contribution dominates, as was seen in the non-

CONTENTS dipolar case. However, in regions where the nonlocal interaction contribution becomes significant, the favoured lattice geometry changes from triangular to square or rectangular. Below a certain value of ε−1 dd (corresponding to reasonably strong, attractive contact interactions) the vortex lattice and the condensate appear to collapse concurrently. The results that we have obtained qualitatively agree with previous results of Zhang and Zhai [163] and Cooper et al. [162, 164]. Zhang and Zhai also find that the lattice geometry undergoes a transition from triangular → square → rectangular → collapse as the value of ε−1 dd decreases. Cooper et al. also find the same transitions between lattice geometries. However, they do not find that the lattice collapses after passing through the rectangular lattice phase. Instead, they observe a bubble phase - a different kind of periodic vortex structure in which the vortices are arranged around bubbles of high particle density. Such states do not occur in the above analysis since they fall outside the scope of the analytic treatment used. Yi and Pu [105] also conducted a similar study of vortex lattice geometry based on numerical simulations of the GPE, however their results are not in agreement with those obtained above, nor with those of Zhang and Zhai [163] and Cooper et al. [162, 164]. They only observe triangular lattice geometries in their simulations, and conclude that the square and rectangular lattice geometries do not exist. A possible explanation for this discrepancy, is that the particular parameter values they chose for their simulations did not fall in the square and rectangular lattice regions. 7.3

Vortex lattice in a quantum ferrofluid: dipoles not perpendicular to the plane

Although it is no longer appropriate to assume cylindrical symmetry for the case of off-axis polarisation, there is still another useful symmetry that can be exploited: reflection symmetry. This reflection symmetry occurs about the x−z plane – the plane which contains both the polarisation vector and the axis of rotation. In order to derive a new ansatz for the vortex lattice ground state which assumes reflection symmetry, only minor modifications need to be made to the derivation given in Section 7.2. Specifically, it is necessary to introduce two new variational parameters: λ and ζ. The parameter λ is required to describe the deviation of the condensate cloud from cylindrical symmetry. It is the ratio of the width of the condensate cloud along the y-direction divided by the width along the x-direction.

31 If the density profile of the cloud is expressed in the form exp[−x2 /lx2 − y 2 /ly2 ], then the aspect ratio would be written as λ = ly /lx . For a cylindrically symmetric BEC, the width of the cloud along the x- and ydirections must be the same, which implies that λ = 1. For α > 0 [see Figure 4], since dipolar BECs elongate along the direction of polarisation, it is expected that lx > ly and hence 0 < λ < 1. The other new parameter, ζ, is required to allow the vortex lattice to adopt any orientation with respect to the polarisation direction. It is defined to be the angle between the first lattice basis vector, b1 , and the projection of the dipole polarisation onto the plane of rotation. For a cylindrically symmetric BEC, the energy will be independent of ζ. However, for noncylindrically symmetric dipolar BEC (α > 0), it is conceivable that the energy may depend on ζ. By modifying the derivation of the ansatz for the vortex lattice ground state to incorporate the new parameters, it is possible to show that the condensate density must be of the following form, 2 2 2 λ x +y 2 2N λ − 1+λ 2 χ2 e n⊥ (ρ) = 2 2 πχ (1 + λ ) X ˆ × g q eiq·Rζ ρ , (109) q

ˆ represents the standard two-dimensional rowhere R tation operator, χ = [(lx ly )−1 − πvc−1 ]−1/2 and, gq h i. gq = P χ2 (1+λ2 ) ˆζ x ˆζ y v·R ˆ/λ2 + v · R ˆ v gv exp(− 8 (110) From this starting point is possible to generalize the approach presented in Section 7.2 to the case where α > 0. The total energy can be broken into two components, E(τ, η, λ, ζ) = E0 (λ) + Eint (τ, η, λ, ζ). (111) The non-interacting component of the energy is given by, 1 E0 (λ) = N ~Ω + ~ωz 2 2 2 2 N ~χ ω⊥ − Ω2 1 + λ2 + , (112) 2λ 2Ω lx2 + ly2 which in the limit Ω → ω⊥ tends to a constant. Following the same procedure as in Section 7.2, using Eq. (87), the interaction energy by, " isgiven 2 N 2 Cdd χ Eint (τ, η, λ, ζ) = 2 W(τ, η, λ, ζ) lz 6(2π)3/2 lz χ2 2 + 1 + 3 cos(2α) + I(τ, η, λ) , εdd (113)

CONTENTS

32

where, λ2 X 2 (˜ gq ) I(τ, η, λ) ≈ 1 + λ2 q and, W(τ, η, λ, ζ) ≈

X

2

−χ2

(˜ gq ) e

(114) (1+λ2 )

"

2

2 (−ζ)

qx λ

Interestingly, the optimal value of λ does not depend on the lattice geometry. ε−1 dd 0.9 0.9 0.95 0.95

# +qy(−2ζ)

q

!

(−ζ)

× A2 (−ζ)

2qx λ

, 2qy(−ζ)

,

(115)

(−ζ)

where qx (qy ) represents the x-component (yˆ −ζ q and, component) of R Z ∞ Z 2π 2 3 du duφ u2 eu erfc(u) A2 (qx , qy ) = √ 2 π 0 0 × sin2 α cos2 uφ − cos2 α " # χ2 1 + λ2 u2 cos2 uφ × exp − 2lz2 2 # " χ2 1 + λ2 2 2 u sin uφ × exp − 2lz2 " # χ2 1 + λ2 lz qx u cos uφ √ × exp − 2lz2 2λ " # 2 2 χ 1+λ lz qy u sin uφ √ × exp − . 2lz2 2 (116) Although there are severe computational limitations surrounding the minimisation of the above expression for the condensate energy, it is still possible to make some meaningful calculations if only triangular and square lattices are considered. For example, it is possible to address the question of whether there a transition between triangular and square lattices. By considering the minimization at two points in −1 parameter space: (ε−1 dd = 0.9, α = π/2) and (εdd = 0.95, α = π/2) we see there is evidence for a transition. In the limit Ω → ω⊥ (E0 (λ) → N [~Ω + ~ωz /2]) Table 2 shows Eint (τ, η, λ, ζ) for the two vortex lattices at the two points in parameter space considered. Looking at the results, there is a phase transition in the vortex lattice geometry from triangular to square as a function of εdd . It is also found that the variational parameter ζ is essentially irrelevant, since the minimum value of Eint (τ, η, λ, ζ) is found to be the same for any choice of ζ. This suggests that the orientation of the vortex lattice is unaffected by the broken cylindrical symmetry due to the off-axis polarisation. However, the optimal value of λ does deviate from the cylindrically symmetric value of 1. At both points considered, the optimal value is less than 1, which indicates, as expected, that the dipolar BEC is elongated along the direction of polarisation.

Phase Square Triangular Square Triangular

Optimal λ 0.93 0.93 0.78 0.78

˜int E −4979.49 −4788.31 −1377.22 −1426.81

˜int = Table 2. Results of the minimisation of E 3 (3(2π) 2 lz3 )/(N 2 Cdd )Eint with respect to λ and ζ for triangular and square lattices at two values of εdd with p α = π/2. The specific parameters used are vc /(πlx ly ) = 1.0191, lx ly /lz = 40 and χ/lz = 292.33. The optimal value of ζ is not included, be˜int was found to be independent of ζ. The results show cause E that the optimal value of λ is the same for both triangular and square lattice geometries.

Numerical studies [105], based on the dipolar GPE, suggest that for α > 0 the vortex lattice can undergo a phase transition from triangular structure to a nontriangular structure as εdd is increased. This is consistent with analysis presented above, however, to our knowledge there has not, to date, been a thorough study of vortex lattice structures for the regime of α > 0. 7.4

Vortex lattices in two-component dipolar BECs

The theoretical study of non-dipolar two component BECs [169, 170] has shown how interspecies interactions g12 can influence the vortex lattice structure of the two components. The analysis presented in Section 7 can be adapted to analyse the the vortex lattice structure of two component condensates. Doing this Mueller and Ho [169] were able to quantify var√ ious regimes through the parameter β = g12 / g1 g2 , where g12 is the interspecies interaction parameter and g1(2) is the intraspecies interaction parameter for component 1 (2) of the two component BEC. This treatment and subsequent numerical analysis [170] shows that for β < 0 the two components overlap and a single triangular vortex lattice arises. For β > 0 the two components seperate with to form interlaced triangular vortex lattices. As β ∼ 1 the triangular lattice distorts to form square or rectangular arrays. This theoretical analysis is consistent with experiment [171]. It is also possible to consider two component dipolar BEC systems, with both interspecies and an intraspecies contact and dipolar interactions. Work by Shirely et al. [172] showed that such systems (where one of the components has zero dipolar interactions), under rotation, exhibit a rich phase diagram, which includes triangular vortex lattices, square vortex lattices, vortex sheets (where half quantum vortices

CONTENTS of one component align in a winding sheet, which is interwoven with a sheet in the other component [173]), half quantum vortex chains (where vortices, alternating between each component, line up along a chain) and half quantum vortex molecules (where a vortex in a given component pairs up with a vortex in the other component). This analysis is consistent with further studies [174, 175, 176] and has been extended to consider how dipole alignment in the plane of rotation [176] and component-dependent optical lattices [177] influences the phase diagram. 8 8.1

Summary and Outlook Summary

The aim of this review was to take the reader on a journey, starting with the fundamental concepts and methodologies used to understand the properties of dipolar BECs and then how these have been applied to understand the properties of vortices and vortex lattices in these systems. Throughout, the dipolar interactions are seen to enrich the physical properties of the system and vortices therein. The journey started in earnest in Section 2 where we met the dipolar interactions; these introducue a longrange and anisotropic component to the interactions, making a significant departure from conventional swave interactions which manifest as isotropic contact interactions In Section 3 we saw that the dipolar interaction significantly modifies the fundamental stationary solutions of a dipolar condensate, including the introduction of collapse instabilities dependent on the shape of the gas relative to the polarization direction. In Section 4 we found that dipolar interactions can alter the energy and structure of a single vortex. Specifically in quasi-two dimensional systems when the dipole alignment is in the plane of the condensate the vortex core is no longer circularly symmetric. Additionally, density ripples appear in the vicinity of the vortex core, as the roton instability is approached, due to the roton mixing with the ground state of the system. In Section 5 we found that the interaction between vortices can be altered by the absence of dipoles in the vortex core, introducing an additional long-range and anisotropic contribution to the vortex-vortex interaction. This can, for instance, lead to the suppression of the annihilation of vortex-antivortex pairs and induce the co-rotational dynamics of vortex-vortex pairs to become anisotropic. In Section 6 we summarised the methods for generating vortices in condensates, and discussed the role of dipolar interactions in these processes. Concentrating on the properties

33 and instabilities of rotating condensates, dipolar interactions were shown to significantly alter the regimes of stability and the critical rotation frequencies for vortices to be nucleated. This also allowed us to identified routes to vortex formation under rotation. Finally, in Section 7 we found that dipolar interactions lead to newand exotic vortex lattice phases; whereas lattices in non-dipolar condensates are well-known to follow a triangular pattern, dipolar interactions can support rectangular, square and bubbles phases. Of course any journey is a compromise between taking an efficient route and a scenic path, i.e. in this case a compromise between completing the review, in a timely manner, and detailing every contribution to the field. Unfortunately our path has been fairly efficient and as such we have omitted several areas of focus pertaining to vortices and vortex lattices in quantum ferrofluids. Below we, all too briefly, provide a snapshot of some of the scenery we have missed along the way and avenues for further exploration. 8.2 8.2.1

Outlook Dipolar BECs in toroidal traps

The experimental study of persistent superfluid flow in BECs confined to toroidal traps [178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189] has matured significantly over the last decade. As such, this scenario have been the subject of many experimental and theoretical investigations [190, 191, 192, 193, 194, 195] focusing on persistent currents [178, 181, 196], weak links [179, 182], formation of matter-wave patterns by rotating potentials [197], solitary waves [192, 198], and the decay of the persistent current via phase slips [180, 199, 200]. In these studies the persistent flow is created by transferring angular momentum from optical fields [178, 182] or by stirring with a rotating barrier [182, 183]. Within the context of this we review we consider a persistent flow in a toroidal BEC as a giant vortex state. One might suppose that beyond studying the density profile and stability of a dipolar condensate within a toroidal trap [201, 202] that dipolar interactions have a limited influence on the superflow properties in such a geometry. This assumption arises since to a close approximation the wavefunction can be considered to have the form p ψ(r) = n(r) exp[iqv θ], where θ is the azimuthal angle and qv is the vortex charge (charaterising the persistent flow) around the toroid, and the density is independent of qv . As such when considering the energy difference between qv and qv + 1 the dipolar interactions play no role.

CONTENTS Despite this observation work has been carried out on the properties of dipolar BECs in toroidal traps focusing on the generation of persistent flows via the He-McKellar-Wilkens or Aharonov-Casher effect [203] and the properties of two component dipolar BECs in concentrically coupled toroidal traps [204]. In the former case it was shown that for atomic dipolar BECs, where the dipolar interaction is mediated via a magnetic dipole moment, that although it is theoretically possible to induce persistent flow, via the Aharonov-Casher effect [205, 206, 207], the strength of electric field required is prohibitive. In the case of polar molecules, with significant electric dipole moments, the He-McKellar-Wilkens effect [208, 209, 210] could ultimately be used to generate a persistent flow in a dipolar BEC in a toroidal geometry. For the case of a two component dipolar BECs [204] in concentrically coupled toroidal traps it is found that vortex structures such as polygonal vortex clusters and vortex necklaces can be obtained via a proper choice of the dipolar interaction and rotational frequency. 8.2.2

Fractional quantum Hall physics in dipolar BECs

In Section 7 we considered the LLL regime to investigate vortex lattice structures in dipolar BECs. Ultimately in the limit Ω → ω⊥ a rotating BEC is predicted to make a quantum phase transition to a highly correlated, non-superfluid, fractional quantum Hall groundstate. This state emerges when the LLL meanfield vortex lattice melts, i.e. when the number of vortices (Nv ) in the BEC is the same as or greater than the number of atoms (N). This regime occurs approximately when Ω/ω⊥ ∼ 0.999. In the absence of dipolar interactions theoretical evidence for such a transition has come from exact two dimensional groundstate calculations for a small number of bosons with a large angular momentum [157, 211, 212]. For a review of quantum Hall physics in rotating BECs see Ref. [213]. The natural question arises do dipolar interactions influence this phase transition and the properties of the highly correlated state? Work by Rezayi et al. [214] showed that at a filling factor of N/Nv = 3/2, with N = 18, dipolar interactions support an incompressible fluid ground state which possesses nonAbelian statistics for the quasiparticle excitations. Additionally dipolar interactions, in lattice systems, have been shown to increase the gap between the ground state and the first excited state [215], for N/Nv = 1/2. It is expected that this increase in gap will be maintained in the thermodynamic limit, appropriate for experiments. Numerical simulations by Chung and Jolicoeur [216] showed that at N/Nv = 1

34 the ground state is a Moore-Read paired state, as is the case of bosons with purely contact interactions. This state is destabilized when the contact interactions are small enough, i.e. dipolar interactions alone can not support this state. For N/Nv = 1/3 a composite fermion sea emerges, where each boson is bound with three vortices. The robustness of fractional quantum Hall states in artificial gauge fields, in the presence of dipolar interactions, has been investigated by Graß et al. [217]. 8.2.3

Dipolar fermions

There is considerable interest in the properties of dipolar Fermi gas systems. The partially attractive nature of the dipolar interaction in single component dipolar Fermi gases opens up the possibility of BCS superfluid states pairing in three dimensions [18, 19, 20, 24] and two dimensions [21, 22, 25] at sufficiently low temperatures. In these systems the anisotropy of the superfluid order parameter provides a major difference in the properties of the superfluid as compared to the case of a two component BCS superfluid, dominated by van der Waals interactions, where the superfluid order parameter is isotropic (s-wave). It is expected that the anisotropic gap will lead to significant differences in the properties of single and multiple vortex states in these systems. For example, Levinson et al. [25] have proposed a scheme to construct a topological px + ipy superfluid phase in a quasi-two dimensional single component dipolar Fermi gas which can support vortices which carry zero energy Majorana modes on their cores [218, 219, 220]. However, to date, there has been very little research into the properties of vortices and vortex lattices of such states. Assuming that such a state can be experimentally achieved then there is a significant opportunity to revisit much of what has been discussed in this review within the context of vortices and vortex lattices in the BCS state of a dipolar Fermi gas. There have been extensive studies of the properties of rotating single component fermionic quantum ferrofluids [221, 222, 223, 224, 225, 226] away from the BCS superfluid transition. These studies have primarily focused on the emergence of fractional quantum Hall states in the Ω → ω⊥ regime. Specifically, it has been shown [221, 222] that for a 2 filling fraction of ν = 2πl⊥ n⊥ = 1/3 the many-body state is well described by the Laughlin wave function with a significant gap between the ground and the excited states. Further studies [222, 223, 225] have shown that as the filling fraction is reduced further (ν < 1/7) Wigner crystal [227] states may emerge. To our knowledge, these studies have all been carried out in the regime where the plane of rotation is

CONTENTS perpendicular to the orientation of the dipoles. As such an interesting question to ask may be how do such states change when the dipole orientation has a component in the plane of rotation. 8.2.4

Berezinskii-Kosterlitz-Thouless transition

In a strictly homogeneous two-dimensional system at finite temperature phase coherence cannot be established and condensation will not occur. At high enough temperatures free vortices proliferate through the system. However, the BerezinskiiKosterlitz-Thouless transition [228, 229] occurs when the temperature is lowered and there is no longer enough thermal energy to unbind vortex and antivortex pairs, enabling long range order to emerge [230, 231]. The Berezinskii-Kosterlitz-Thouless and BEC transition have been studied in trapped ultracold gases via observations of phase defects [232], vortices [233] and changes in the density profile due to the onset of superfluidity [234]. The BerezinskiiKosterlitz-Thouless transition in the presence of dipolar interactions has been studied using, for a homogeneous system, Monte Carlo methods [235], the meanfield Hartree Fock Bogoliubov Popov model [236]. However, the key physics underpinning our understanding of the Berezinskii-Kosterlitz-Thouless transition comes from asking the question: what is the energy required to seperate a vortex-antivortex pair? For vortex-antivortex pairs in a uniform two dimensional non-dipolar BEC, the energy of a pair, separated by a distance b and calculated from hydrodynamical arguments, is approximately given by V (b) = 2π~2 n⊥ ln(b/ξ)/m. The critical temperature associated with the transition is given by the relation 2π~2 n⊥ /(mkB T ) = 4. This is calculated by determining the average distance between the pairs, R ∞ 3 −βV (b) b e db Γ−2 ξ = ξ2 , (117) hb2 i = R ∞ −βV (b) Γ−4 be db ξ where we have used the short-hand notation Γ = n⊥ 2π~2 /(mkB T ). This result diverges at Γ = 4, signifying the Berenzinksii-Kosterlitz-Thouless transition. The inclusion of dipolar interactions adds three complications. The first complication is that as dipolar interactions are introduced ξ will change. The second complication is, as seen in Section 5, the interaction between a vortex and an antivortex is modified in the presence of dipolar interactions due to the absence of dipoles in the vortex cores. As such the energy scaling of the pair with separation, i.e. V (b), is changed. The third complication arises if the dipole alignment has some component in the

35 two-dimensional plane of the gas. In this case the interaction between the vortex-antivortex pair is no longer just a function of the distance between them it also depends on the in-plane angle of the pair relative to the polarization direction, i.e. V (b) → V (b, η) [see Figure 13(b)]. The above generalisations can be incorporated into Eq. (117) by considering the following analysis [237]. Let the vortices carry a dipole moment ∝ (0, sin α, cos α). If the separation between the vortices is b = b(cos η, sin η, 0) then the interaction between the vortex and the anti-vortex can be written as, 3 b ξ V (b, η) = Γ ln −λ f (α, η). (118) kB T ξ b From this R ∞ 3−Γ R 2π b db 0 exp λf (α, η)/b3 dη ξ 2 (119) hb i = R ∞ R 2π b1−Γ db 0 exp [λf (α, η)/b3 ] dη ξ "P # ∞ −1 n 2 n=0 an λ (Γ + 3n − 4) = ξ P∞ (120) −1 , n n=0 an λ (Γ + 3n − 2) where Z 1 2π n an = f (α, η)dη. (121) n! 0 When λ = 0 the results obtained from Eq. (117) is regained, i.e. hb2 i = ξ 2 (Γ − 2)/(Γ − 4), with the Berezinskii-Kosterlitz-Thouless transition arising from the divergence at Γ = 4. However, the general inclusion of an additional power law interaction between the vortex and antivortex, arising from the absence of dipoles in the region of the cores, does not fundamentally change this result. Specifically, the meansquare separation, given by Eq. (120), diverges first at Γ = 4, i.e. this simple analysis implies that although the dipolar interaction can change the interaction between vortex-antivortex pairs the long-range hydrodynamic interaction always wins, suggesting that the Berezinskii-Kosterlitz-Thouless transition is unaffected by dipolar interactions. However, a more complete treatment, e.g. considering screening of the superfluid flow via the generation of additional vortexantivortex pairs and finite-size effects, is required to firmly establish the role of dipolar interactions on this phase transition. 8.2.5

Vortex lattices in the supersolid phase

Systems in a supersolid phase possess a spontaneously formed crystalline structure along with off-diagonal long range order which characterizes superfluidity. The investigation of supersolid phases in condensed-matter systems has been a focus of research for more than half a century [238, 239, 240, 241, 242]. Until recently, this effort has primarily focused on the possible realization of a supersolid phase in 4 He [243, 244, 245], with the most credible claim for observation [246] now being

CONTENTS withdrawn [247]. The relatively recent realization of dipolar BECs has provided an alternative avenue to investigate supersolid phases in extended BoseHubbard lattice models [248, 249, 250, 251, 252, 253, 254, 255, 256, 257]. Within the context of dipolar BECs, to date, there has been various works focusing on how an artificial gauge field influences the boundaries between Mott-insulator/supersolid and regimes [258, 259] and how staggered fluxes lead to supersolid phases with staggered vortex phases [260]. However, there has only been a minimal investigation of the structural properties of vortices [261] and vortex lattices in the supersolid state. 8.2.6

The Onsager vortex phase transition

By studying a two-dimensional point vortex model Onsager predicted that negative temperature states may be relevant for two-dimensional fluids [262]. While intended as a model of two-dimensional fluids in general, Onsager noted that the model was potentially particularly relevant for two-dimensional superfluids, whose vortices have quantized circulation and uniform size. Simulations of the two-dimensional GPE have shown how it is possible to dynamically evolve to the negative temperature Onsager vortex, starting from a collection of randomly distributed vortices and antivortices [263]. Starting with a random configuration of vortices and antivortices one might expect the vortices and antivortices to evaporate from the BEC via pairwise annihilation and re-thermalization of the emitted sound, simply resulting in a BEC with an increased temperature. However, Simula et al. [263] showed that only some vortices annihilate, and the remaining vortices self-organise into two ordered clusters of like-sign circulation, which represent the Onsager vortices. This outcome was found to be the result of the evaporative heating of quantized vortices where annihilating vortex-antivortex pairs annihilate, leaving the remaining vortices to re-thermalize to a state with higher energy per vortex. This process drives the vortex component of the superfluid to ever higher energies, leading to the Onsager vortex phase transition. The final Onsager vortex state emerges as a clustering of vortices and antivortices. The dynamical process which leads to this final state is non-trivial, but is underpinned by the interaction between the constituent vortices and antivortices in the system. Introducing dipolar interactions, as shown in Section 5, can significantly modify this interaction and as such may result in significant changes to the Onsager vortex phase transition. As such we suspect that dipolar interactions may offer the opportunity to significantly modify the Onsager vortex phase transition.

36 9

Acknowledgements

AMM acknowledges support by the Australian Research Council (Grant No. DP150101704), DOD acknowledges support from the Natural Sciences and Engineering Research Council (Canada), and NGP acknowledges support by the Engineering and Physical Sciences Research Council (Grant No. EP/M005127/1). 10

References

[1] Rosenswieg R E 1997 Ferrohydrodynamics (New York: Dover Publications, New York) [2] Hakim S S and Higham J B 1962 Proc. Phys. Soc. London 80 190 [3] Cowley M D and Rosenswieg R E 1962 J. Fluid Mech. 30 671 [4] Griesmaier A et al. 2005 Phys. Rev. Lett. 94 160401 [5] Beaufils Q et al. 2008 Phys. Rev. A 77 061601(R) [6] Pasquiou B et al. 2011 Phys. Rev. Lett. 106 015301 [7] Lu M, Burdick N Q, Youn S H and Lev B L 2011 Phys. Rev. Lett. 107 190401 [8] Kadau H et al. 2016 Nature 530 194 [9] Aikawa K et al. 2012 Phys. Rev. Lett. 108 210401 [10] Lahaye T et al. 2007 Nature 448 672 [11] Lahaye T et al. 2008 Phys. Rev. Lett. 101 080401 [12] Koch T et al. 2008 Nat. Phys. 4 218 [13] Ferrrier-Barbut I et al. 2016 Nature 530 194 [14] Ni K-K et al. 2008 Science 322 231 [15] Lu M et al. 2012 Phys. Rev. Lett. 108 215301 [16] Aikawa K et al. 2014 Phys. Rev. Lett. 112 010404 [17] You L and Marinescu M 2000 Phys. Rev. A 60 2324 [18] Baranov M A et al. 2002 Phys. Rev. A 66 013606 [19] Baranov M A, Dobrek L and Lewenstein M 2004 New J. Phys. 6 198 [20] Baranov M A, Dobrek L and Lewenstein M 2004 Phys. Rev. Lett. 92 250403 [21] Brunn G M and Taylor E 2008 Phys. Rev. Lett. 101 245301 [22] Cooper N R and Shlyapnikov G V 2009 Phys. Rev. Lett. 103 155302 [23] Fregoso B M, Sun K, Fradkin E and Lev B L 2009 New J, Phys. 11 103003 [24] Zhao C et al. 2010 Phys. Rev. A 81, 063642 [25] Levinson J, Cooper N R and Shlyapnikov G V 2011, Phys. Rev. A 84 013603 [26] Baranov M et al. 2002 Physica Scripta 2002 74 [27] Baranov M A 2008 Physics Reports 464 71 [28] Lahaye T et al. 2009 Rep. Prog. Phys. 72 126401 [29] Carr L D, DeMille D, Krems R V and Ye J 2009 New Journal of Physics 11 055049 [30] Baranov M A, Dalmonte M, Pupillio G and Zoller P, 2012 Chem. Rev. 112 512 [31] Madison K W, Chevy F, Wohlleben W and Dalibard J 2000 Phys. Rev. Lett. 84 806 [32] Freilich D V et al. 2010 Science 329 1182 [33] Neely T W et al. 2010. Phys. Rev. Lett. 104 160401 [34] Kwon W J, Seo, S W and Shin Y 2015 Phys. Rev. A 92 033613 [35] Anderson B P et al. 2001 Phys. Rev. Lett. 86 2926 [36] Hodby E et al. 2001 Phys. Rev. Lett. 88 010405 [37] Abo-Shaeer J R, Raman C, Vogels J M and Ketterle W 2001 Science 292 476 [38] Henn E et al. 2009 Phys. Rev. Lett. 103 045301 [39] Neely T W et al. 2013 Phys. Rev. Lett. 111 235301

CONTENTS [40] Kwon W J et al. 2014 Phys. Rev. A 90 063627 [41] C. Becker, K. Sengstock, P. Schmelcher, P. G. Kevrekidis, and R. Carretero-Gonz´ alez, New J. Phys. 15, 113028 (2013). [42] S. Donadello, S. Serafini, M. Tylutki, L. P. Pitaevskii, F. Dalfovo, G. Lamporesi and G. Ferrari, Phys. Rev. Lett. 113, 065302 (2014). [43] Fetter A and Svidzinksy A 2001 J. Phys.: Condens. Matter 13 R135 [44] Kevrekidis P G, Frantzeskakis D J and Carretero-Gonz´ alez R (Eds.) 2008 Emergent Nonlinear Phenomena in Bose-Einstein condensates (Berlin: Springer) [45] Cooper N R 2008 Advances in Physics 57, 539 [46] Kasamatsu K and Tsubota M 2009 Progress in Low Temperature Physics 16 351 [47] Fetter A L 2010 J. Low Temp. Phys. 161 445 [48] Bisset R N and Blakie P B 2015 Phys. Rev. A 92, 061603R [49] Santos L, Shlyapnikov G V, Zoller P and Lewenstein M 2000 Phys. Rev. Lett. 85 1791 [50] Yi S and You L 2001 Phys. Rev. A 63 053607 [51] Yi S and You L 2002 Phys. Rev. A 66 013607 [52] G´ oral K, Rz¸ az˙ ewski K and Pfau T 2000 Phys. Rev. A 61 051601R [53] Yi S and You L 2000 Phys. Rev. A 61 041604R [54] Marinescu M and You L 1998 Phys. Rev. Lett. 81 4596 [55] Feshbach H 1958 Ann. Phys. 5 357; Feshbach H 1962 Ann. Phys. 19 287 [56] Werner J et al. 2005 Phys. Rev. Lett. 94 183201 [57] Griesmaier A et al. 2006 Phys. Rev. Lett. 97 250402 [58] Tang Y et al. 2015 Phys. Rev. A 92 022703 [59] Giovanazzi S, Gorlitz A and Pfau T 2002 Phys. Rev. Lett. 89 130401 [60] O’Dell D H J, Giovanazzi S and Eberlein C 2004 Phys. Rev. Lett. 92 250401 [61] Eberlein C, Giovanazzi S and O’Dell D H J 2005 Phys. Rev. A 71 033618 [62] Bao W, Cai Y and Wang H 2010 J. Comp. Phys. 229 7874 [63] Kishor Kumar R et al. 2015 Comp. Phys. Commun. 195 117 [64] Ronen S and Bohn J L 2007 Phys. Rev. A 76 043607 [65] Blakie P B et al 2009 Phys. Rev. E 80 016703 [66] Bisset R N, Baillie D and Blakie P B 2012 Phys. Rev. A 86 033609 [67] Stringari S 1996 Phys. Rev. Lett. 77 2360 [68] Nozieres P and Pines D, Theory Of Quantum Liquids (Westview, New York, 1999). [69] Pethick C J and Smith H, Bose-Einstein Condensation in Dilute Gases (Cambridge University Press, Cambridge, 2002) [70] Pitaevskii L and Stringari S, Bose-Einstein Condensation (Oxford University Press, Oxford, 2003) [71] C. F. Barenghi and N. G. Parker, A Primer on Quantum Fluids (Springer, Berlin, 2016) [72] O’Dell D H J, Giovanazzi S and Kurizki G 2003 Phys. Rev. Lett. 90 110402 [73] Santos L, Shlyapnikov G V and Lewenstein M 2003 Phys. Rev. Lett. 90 250403 [74] Giovanazzi S and O’Dell D H J 2004 Eur. Phys. J. D 31 439 [75] Parker N G and O’Dell D H J 2008 Phys. Rev. A 78 041601R [76] Parker N G, Ticknor C, Martin A M and O’Dell D H J 2009 Phys. Rev. A 79 013617 [77] Ticknor C et al. 2008 Phys. Rev. A 78 061607R [78] van Bijnen R M W et al. 2010 Phys. Rev. A 82 033612 [79] Donnelly R J 1991 Quantized Vortices in Helium II (Cambridge, Cambridge University Press) [80] Ronen S, Bortolotti D C E and Bohn J L 2007 Phys. Rev. Lett. 98 030406 [81] Wilson R M, Ronen S and Bohn J L 2009 Phys. Rev. A

37 79 013621 [82] Dutta O and Meystre P 2007 Phys. Rev. A 75 053604 [83] Wilson R M, Ronen S, Bohn J L and Pu H 2008 Phys. Rev. Lett. 100 245302 [84] Abad M, Guilleumas M, Mayol R, Pi M and Jezek D M 2009 Phys. Rev. A 79 063622 [85] Jona-Lasinio M, Lakomy K and Santos L 2013 Phys. Rev. A 88 013619 [86] Lu H-Y et al. 2010 Phys. Rev. A 82 023622 [87] Gaunt A L et al. 2013 Phys. Rev. Lett. 110 200406 [88] Chomaz L et al. 2015 Nat. Comm. 6 6162 [89] Bohn J L, Wilson R M and Ronen S 2009 Laser Phys. 19 547 [90] Mulkerin B C, O’Dell D H J, Martin A M and Parker N G 2014 J. Phys.: Conf. Ser. 497 012025 [91] Ticknor C, Wilson R M and Bohn J L 2011 Phys. Rev. Lett. 106 065301 [92] Cai Y, Rosenkranz M, Lei Z and Bao W 2010 Phys. Rev. A 82 043623 [93] Fischer U R 2006 Phys. Rev. A 73 031602 [94] Baillie D and Blakie P B 2015 New J. Phys. 17 033028 [95] Mulkerin B C, van Bijnen R M W, Martin A M and Parker N G 2013 Phys. Rev. Lett. 111 170402 [96] Klawunn M and Santos L 2009 Phys. Rev. A 80 013611 [97] Ginzburg V L and Pitaevskii L P 1958 Zh. Eksp. Teor. Fiz. 34, 1240 [1958 Sov. Phys. JETP 7, 858] [98] Jackson B, Proukakis N P, Barenghi C F and Zaremba E 2009 Phys. Rev. A 79 053615 [99] Rooney S J, Bradley A S and Blakie P B 2010 Phys. Rev. A 81 023630 [100] Allen A J, Zaremba E, Barenghi C F and Proukakis N P 2013 Phys. Rev. A 87 013630 [101] Gautam S, Roy A and Mukerjee S 2014 Phys. Rev. A 89 013612 [102] Parker N G, Proukakis N P, Barenghi C F and Adams C S 2004 Phys. Rev. Lett. 92 160403 [103] Raman C, Abo-Shaeer J R, Vogels J M, Xu K and Ketterle W 2001 Phys. Rev. Lett. 87 210402 [104] Powis A T, Sammut S J and Simula T P 2014 Phys. Rev. Lett. 113 165303 [105] Yi S and Pu H 2006 Phys. Rev. A 73 061602R [106] Dalfovo F 1992 Phys. Rev. B 46 5482 [107] Ortiz G and Ceperley D M 1995 Phys. Rev. Lett. 75 4642 [108] Sadd M, Chester G V, and Reatto L 1997 Phys. Rev. Lett. 79 2490 [109] Berloff N G and Roberts P H 1999 J. Phys. A 32 5611 [110] Bland T et al. 2015 Phys. Rev. A 92 063601 [111] Edmonds M J, Bland T, O’Dell D H J and Parker N G 2016 Phys. Rev. A 93 063617 [112] Yuce C and Oztas Z 2010 J. Phys. B 43 135301 [113] Gautum S 2014 J. Phys. B 47 165301 [114] Klawunn M, Nath R, Pedri P and Santos L 2008 Phys. Rev. Lett. 100 240403 [115] Svidzinsky A A and Fetter A L 2000 Phys. Rev. A 62, 063617 [116] Koens L and Martin A M 2012 Phys. Rev. A 86 013605 [117] Klawunn M and Santos L 2009 New J. Phys. 11 055012 [118] Pedri P and Santos L 2005 Phys. Rev. Lett. 95 200404 [119] Nath R, Pedri P and Santos L 2007 Phys. Rev. A 76 013606 [120] Tikhonenkov I, Malomed B A and Vardi A 2008 Phys. Rev. A 78 043614 [121] O’Dell D H J and Eberlein C 2007 Phys. Rev. A 75 013604 [122] Abad M et al. 2010 Laser Physics 20 1190 [123] Lundh E, Pethick C J and Smith H 1997 Phys. Rev. A 55 2126 [124] Wu H and Sprung D W L 1994 Phys. Rev. A 49 4305 [125] van Bijnen R M W, O’Dell D H J, Parker N G and Martin A M 2007 Phys. Rev. Lett. 98 150401 [126] Martin A M et al. 2008 Las. Phys. 18 322

CONTENTS [127] [128] [129] [130] [131] [132] [133] [134] [135] [136] [137] [138] [139] [140] [141] [142] [143] [144] [145] [146] [147]

[148]

[149]

[150]

[151] [152] [153] [154] [155] [156] [157] [158] [159] [160] [161] [162] [163] [164] [165] [166] [167] [168] [169] [170]

van Bijnen R M W et al. 2009 Phys. Rev. A 80 033617 Leanhardt A E et al. 2002 Phys. Rev. Lett. 89 190403 Kibble T W B 1976 J. Phys. A 9 1387 Zurek W H Nature 317 505 Weiler C N et al. 2008 Nature 455 948 Glaum K, Pelster A, Kleinert H and Pfau T 2007 Phys. Rev. Lett. 98 080407 Scott R G et al. 2003 Phys. Rev. Lett. 90 110404 Scott R G et al. 2004 Phys. Rev. A 69 033605 Proukakis N P, Parker N G, Frantzeskakis D J and Adams C S 2004 J. Opt. B: Quantum Semiclass. Opt. 6 S380 Nath R, Pedri P and Santos L 2008 Phys. Rev. Lett. 101 210402 Lin Y-J et al. 2009 Nature 462 628 Taylor L B et al. 2011 Phys. Rev. A 84 021604R Zhao Q and Gu Q 2015 Chinese Physics B 25 016702 Hodby E et al. 2001 Phys. Rev. Lett. 88 010405 Recati A, Zambelli F and Stringari S 2001 Phys. Rev. Lett. 86 377 Sinha S and Castin Y 2001 Phys. Rev. Lett. 87 190402 Madison K W, Chevy F, Bretin V and Dalibard J 2001 Phys. Rev. Lett 86 4443 Lundh E, Martikainen J P and Suominen K A 2003 Phys. Rev. A 67 063604 Parker N G, van Bijnen R M W and Martin A M 2006 Phys. Rev. A 73 061603R Corro I, Parker N G and Martin A M 2007 J. Phys. B 40 3615 Leggett A J 2006 Quantum Liquids: Bose-Condensation and Cooper Pairing in Condensed Matter Systems (Oxford: Oxford University Press) Leggett A J 2001, in Bose-Einstein Condensation: From Atomic Physics to Quantum Fluids (Proceedings of the Thirteenth Physics Summer School) ed Savage C M and Das M P (Singapore: World Scientific) Landau L D and Lifshitz E M, Course of Theoretical Physics: Mechanics (Butterworth-Heinemann, third edition, 1982) Castin Y 2001, in Coherent Matter Waves (Lecture Notes of Les Houches Summer School) Kaiser R, Westbrook C and David F (Berlin: Springer-Verlag) Ferrers N M 1877 Quart. J. Pure and Appl. Math 14 1 Dyson F W 1891 Quart. J. Pure and Appl. Math 25 259 Levin M L and Muratov R Z 1971 Astrophys. J. 166 441 Rosenbusch P et al. 2002 Phys. Rev. Lett. 88 250403 Pitaevskii L P 1984 Zh. Eksp. Teor. Fiz. 39 423 Parker N G and Adams C S 2005 Phys. Rev. Lett. 95 145301 Wilken N K, Gunn J M F and Smith R A 1998 Phys. Rev. Lett. 80 2265 Mottelson B 1999 Phys. Rev. Lett. 83 2695 Pethick C J and Pitaevskii L P 2000 Phys. Rev. A 62 033609 Kishor Kumar R and Muruganandam P 2012 J. Phys. B 45 215301 Kishor Kumar R and Muruganandam P 2014 Eur. Phys. J. D 68 289 Cooper N R, Rezayi E H and Simon S H 2005 Phys. Rev. Lett. 95 200402 Zhang J and Zhai H 2005 Phys. Rev. Lett. 95 200403 Cooper N R, Rezayi E H and Simon S H 2006 Solid State Comm. 140 61 Komineas S and Cooper N R 2007 Phys. Rev. A 75 023623 Malet F, Kristensen T, Reimann S M and Kavoulakis G M 2011 Phys. Rev. A 83 033628 Kishor Kumar R et al. 2015 arXiv:1506.08184 Butts D A and Rokhsar D S 1999 Nature 397 327 Mueller E J and Ho T-L 2002 Phys. Rev. Lett. 88 180403 Kasamatsu K, Tsubota M and Ueda M 2003 Phys. Rev. Lett. 91 150406

38 [171] Schweikhard V et al. 2004 Phys. Rev. Lett. 92 040404 [172] Shirley W E, Anderson B A, Clark C W and Wilson R M 2014 Phys. Rev. Lett. 113 165301 [173] Kasamatsu K and Tsubota M 2009 Phys. Rev. A 79 023606 [174] Zhao Y, An J and Gong C D 2013 Phys. Rev. A 87 013605 ¨ 2014 Phys. Rev. A 89 [175] Ghazanfari N, Kele¸s, Oktel M O 025601 [176] Zhang X-F et al. 2016 Sci. Rep. 6 19380 [177] Wang L-X et al. 2016 Phys. Lett. A 380 435 [178] Ryu C et al. 2007 Phys. Rev. Lett. 99, 260401 [179] Ramanathan A et al. 2011 Phys. Rev. Lett. 106, 130401 [180] Moulder S et al. 2012 Phys. Rev. A 86, 013629 [181] Beattie S, Moulder S, Fletcher R J and Hadzibabic Z 2013 Phys. Rev. Lett. 110, 025301 [182] Wright K C et al. 2013 Phys. Rev. Lett. 110, 025302 [183] Wright K C et al. 2013 Phys. Rev. A 88, 063633 [184] Ryu C, Blackburn P W, Blinova A A and Boshier M G 2013 Phys. Rev. Lett. 111, 205301 [185] Neely T W et al. 2013 Phys. Rev. Lett. 111, 235301 [186] Jendrzejewski F et al. 2014 Phys. Rev. Lett. 113, 045305 [187] Corman L et al. 2014 Phys. Rev. Lett. 113, 135302 [188] Ryu C, Henderson K C, and Boshier M G 2014 New Journal of Physics 16, 013046 [189] Eckel S et al. 2014 Nature 506, 200 [190] Kasamatsu K, Tsubota M and Ueda M 2002 Phys. Rev. A 66, 053606. [191] Benakli M et al. 1999 Europhysics Letters 46, 275 [192] Brand J and Reinhardt W P 2001 Journal of Physics B: Atomic and Molecular Physics 34, L113 [193] Das A, Sabbatini J and Zurek W H 2012 Scientific Reports 2, 352 [194] Martikainen J-P et al. 2001Phys. Rev. A 64, 063602 [195] Modugno M, Tozzo C and Dalfovo F 2006 Phys. Rev. A 74, 061601 [196] Yakimenko A I, Isaieva K O, Vilchinskii S I and Weyrauch M 2013 Phys. Rev. A 88, 051602 [197] Li Y, Pang W and Malomed B A 2012 Phys. Rev. A 86, 023832 [198] Mason P and Berlo N G 2009 Phys. Rev. A 79, 043620 [199] Piazza F, Collins L A and Smerzi A 2009 Phys. Rev. A 80, 021601 [200] Piazza F, Collins L A and Smerzi A 2013 Journal of Physics B: Atomic and Molecular Physics 46, 095302 [201] Abad M et al. 2010 Phys. Rev. A 81 043619 [202] Adhikari S K 2012 Phys. Rev. A 85 053631 [203] Wood A A, McKellar B H J and Martin A M 2016 Phys. Rev. Lett. 116, 250403 [204] Zhang X-F et al. 2015 Sci. Rep. 5 8684 [205] Aharonov Y and Casher A 1984 Phys. Rev. Lett. 53 319 [206] Anandan J 1982 Phys. Rev. Lett. 48 1660 [207] Cimmino A et al. 1989 Phys. Rev. Lett. 63 380 [208] He X G and McKellar B H J 1993 Phys. Rev. A 47 3424 [209] Wilkens M 1994 Phys. Rev. Lett. 72, 5 [210] Lepoutre S et al. 2012 Phys. Rev. Lett. 109 120404 [211] Wilken N K and Gunn J M F 2000 Phys. Rev. Lett. 84 6 [212] Cooper N R, Wilken N K and Gunn J M F 2001 Phys. Rev. Lett. 87 120405 [213] Viefers S 2008 Journal of Physics: Condensed Matter 20 123202 [214] Rezayi E H, Read N and Cooper N R 2005 Phys. Rev. Lett. 95 160404 [215] Hafezi et al. 2007 Phys. Rev. A 76 023613 [216] Chung B and Jolicoeur Th 2008 Phys. Rev. A 77 043608 [217] Graß T, Baranov M A and Lewenstein 2011 Phys. Rev. A 84 043605 [218] Gurarie V and Radzihovsky L 2007 Annals of Physics 222 2 [219] Read N and Green D 2000 Phys. Rev. B 61 10267 [220] A Stern 2008 Annals of Physics 323 204

CONTENTS [221] Baranov M A, Osterloh K and Lewenstein M 2005 Phys. Rev. Lett. 94 070404 [222] Osterloh K, Barber´ an N and Lewenstein M 2007 Phys. Rev. Lett. 99 160403 [223] Baranov M A, Fehrmann N and Lewenstein M 2008 Phys. Rev. Lett. 100 200402 [224] Eriksson G, Cremon J C, Manninen M and Reimann S M 2012 Phys. Rev. A 86 043607 [225] Jheng S-D, Jiang T F and Cheng S-C 2013 Phys. Rev. A 88 051601R [226] Ancilotto F 2015 Phys. Rev. A 92 061602R [227] Wigner E 1934 Phys. Rev. 46 1002 [228] Berezinskii V L 1972 Sov. Phys. JETP 34 610 [229] Kosterlitz L M and Thouless D J 1973 J. Phys. C 6 1181 [230] Hadzibabic H and Dilibard J 2011 Riv. Nuo. Cime. 34 389 [231] Bagnato V and Kleppner D 1991 Phys. Rev. A 44 7439 [232] Hadzibabic Z et al. 2006 Nature 441 1118 [233] Tung S et al. 2010 Phys. Rev. Lett. 105 230408 [234] Claude P et al. Phys. Rev. Lett. 102 170401 [235] Filinov A et al. 2010 Phys. Rev. Lett. 105 070401 [236] Ticknor C 2012 Phys. Rev. A 85 033629 [237] Dr P. J. Camp, University of Edinburgh, U.K., private communication [238] E. P. Gross E P 1957 Phys. Rev. 106 161 [239] Andreev A F and Lifshitz I M 1969 Zh. Eksp. Theor. Fiz. 56 2057 [Sov. Phys. JETP 29 1107] [240] Thouless D J 1969 Ann. Phys. 52 403 [241] Leggett A J 1970Phys. Rev. Lett. 25, 1543 [242] Chester G V 1970 Phys. Rev. A 2, 256 [243] Balibar S, Fefferman A D, Haziot A, and Rojas X 2012 J. Low Temp. Phys. 168, 221 [244] Nozi` eres P 2006 J. Low Temp. Phys. 142 91 [245] Nozi` eres P 2009 J. Low Temp. Phys. 156 9 [246] Kim E and M. H. W. Chan M H W 2004 Nature 427 225; Science 305 1941 [247] Kim D Y and Chan M H W 2012 Phys. Rev. Lett. 109 155301 [248] G´ oral K, Santos L and Lewenstein M 2002 Phys. Rev. Lett. 88 170406 [249] Kovrizhin D L, Pai G V, and Sinha S 2005 Europhys. Lett. 72 162 [250] Sengupta P et al. Phys. Rev. Lett. 94 207202 [251] Scarola V W and Das Sarma S 2005 Phys. Rev. Lett. 95 033003 [252] Scarola V W, Demler E and Das Sarma S 2006 Phys. Rev. A 73 051601(R) [253] Dalla Torre E G, Berg E and Altman E 2006 Phys. Rev. Lett. 97 260401 [254] Schmidt K P, Dorier J, L¨ auchli A M and Mila F 2008 Phys. Rev. Lett. 100 090401 [255] Sen A, Dutt P, Damle K and Moessner R 2008 Phys. Rev. Lett. 100 147204 [256] Trefzger C, Menotti C and Lewenstein M 2009 Phys. Rev. Lett. 103 035304 [257] Pollet L , Picon J D, B¨ uchler H P and Troyer M 2010 Phys. Rev. Lett. 104 125302 [258] Sachdeva R, Jori S and Ghosh S 2010 Phys. Rev. A 82 063617 [259] Sachdeva R and Ghosh S 2012 Phys. Rev. A 85 013624 [260] Tielman O, Lazarides A and Morias Smith C 2011 Phys. Rev. A 83 013627 [261] Sachdeva R and Ghosh S 2014 Journal of Atomic, Molecular, Condensate and Nano Physics 1 1 [262] Onsager L 1949 Il Nuovo Cimento 6S2 279 [263] Simula T, Davis M J and Helmerson K 2014 Phys. Rev. Lett. 113 165302

39