Phase transition from straight into twisted vortex lines in dipolar Bose–Einstein condensates

The non-local nonlinearity introduced by the dipole–dipole interaction plays a crucial role in the physics of dipolar Bose–Einstein condensates. In particular, it may distort significantly the stability of straight vortex lines due to the rotonization of the Kelvin-wave spectrum. In the present paper, we analyze this instability, showing that it leads to a second-order-like phase transition from a straight vortex line into novel helical or snake-like configurations, depending on the dipole orientation.


Introduction
The physics of ultracold atomic and molecular gases is crucially determined by the interparticle interactions.Typical experiments in quantum gases have studied up to now particles which interact dominantly via short-range isotropic potentials.At the very low energies involved in these experiments, such interactions are characterized by a single parameter, namely the s-wave scattering length.However, a new generation of recent experiments is opening a fascinating new research area, namely that of dipolar gases, for which the dipole-dipole interaction (DDI) plays a significant or even dominant role.These experiments include on one side those dealing with the DDI effects due to magnetic dipoles in degenerate atomic gases, as it is the case of recent exciting experiments in Chromium Bose-Einstein condensates (BECs) [1] and Rubidium spinor BECs [2].On the other side, recent experiments on the creation of heteronuclear molecules in the lowest vibrational states [3], although not yet brought to quantum degeneracy, open fascinating perspectives for the generation of polar molecules with very large electric dipole moments.Last but not least, the DDI in Rydberg atoms is extremely large [4] and may allow for e.g. the construction of fast quantum gates [5].
Contrary to the isotropic van-der-Waals-like short-range interactions, the DDI is long-range and anisotropic (partially attractive), and leads to fundamentally new physics in ultra cold gases, modifying e.g. the stability and excitations of BECs [6,7], the properties of Fermi gases [8], and the physics of strongly-correlated gases [9].Timeof-flight experiments in Chromium condensates allowed for the first observation of DDI effects in quantum degenerate gases [10], which have been remarkably enhanced recently by means of Feshbach resonances [11].In addition, the DDI has been recently shown to play an important role in the physics of spinor Rubidium BEC [2], and to lead to an observable damping of Bloch oscillations of Potassium atoms in optical lattices [12].
The long-range character of the DDI leads to nonlocal nonlinearity in dipolar BECs, similar as that encountered in e.g.plasmas [13] and nematic liquid crystals [14].This nonlocality leads to novel nonlinear physics in dipolar BECs, including the possibility of obtaining stable 2D bright solitons [15] and stable 3D dark solitons [16].In addition the partially attractive character of the nonlocal nonlinearity due to the anisotropy of the DDI has remarkable consequences for the stability of dipolar BECs, which may become unstable against collapse in 3D traps, as recently shown in experiments with Chromium BECs [17].On the contrary 2D traps may allow for an instability without collapse, characterized by the formation of a gas of inelastic 2D solitons [18].
Vortices and vortex-lines constitute a prominent topic among the fascinating properties of interacting quantum gases.When rotated at sufficiently large angular frequency, a superfluid develops vortex lines of zero density [19,20], around which the circulation is quantized due to the single-valued character of the corresponding wavefunction [21].Quantized vortices constitute indeed one of the most important consequences of superfluidity, playing a fundamental role in various physical systems, as superconductors [22] and superfluid Helium [23].Vortices and even vortex lattices have been created in alkali BECs in a series of milestone experiments [24,25,26].
Similar to strings, vortex lines are 3D structures which may present transverse helical excitations, which are called Kelvin modes or waves [27].These excitations were studied for quantized vortices in superfluids by Pitaevskii [28].Interestingly, the dispersion law for Kelvin modes at small wave vector q follows a characteristic dependence ǫ(q) ∼ −q 2 ln qξ, where ξ is the healing length.Kelvin modes play an important role in the physics of superfluid Helium [23,29], and even of neutron stars [30].Recently, Kelvin modes were experimentally observed in BEC [31].
The DDI interaction may distort significantly the physics of vortices in dipolar BECs, in particular the critical angular frequency for vortex creation [32].In addition, dipolar gases under fast rotation develop vortex lattices, which due to the DDI may be severely distorted [33], and even may change its configuration from the usual triangular Abrikosov lattice into other arrangements [34,35].
In a recent Letter [36] we analyzed the case of a dipolar BEC in a one-dimensional optical lattice, and in particular the physics of straight vortex lines perpendicular to the two-dimensional planes defined by the lattice sites.We showed that due to the longrange character of the DDI, different parts of the vortex line interact with each other, and hence the 3D character of the vortices plays a much more important role in dipolar gases than in usual short-range interacting ones.Specifically, we discussed that, interestingly, the DDI may severely modify the Kelvin-wave dispersion, which may even acquire a roton-like minimum.This minimum may touch zero energy for sufficiently large DDI and strong lattices, leading to the instability of the straight vortex line even for those situations in which the BEC as a whole is stable.However, the certainly very relevant question concerning the nature of this instability was not addressed in Ref. [36].In the present paper we discuss in detail this instability, showing that, interstingly, it has a thermodynamical character, and it is linked to a second-order-like phase transition from a straight vortex line into an helical or snake-like vortex line depending on the dipole orientation.
The structure of the paper is as follows.In Sec. 2 we discuss the effective model that describes a dipolar BEC in a sufficiently strong optical lattice.In Sec. 3 we briefly summarize the main results of [36].In Sec. 4 we discuss the phase transition from straight into helical vortex-lines for the case of dipole oriented parallel to the vortex line, whereas in Sec. 5 we discuss the transition into a snake-like vortex for dipoles perpendicular to the vortex line.Finally, in Sec.6 we summarize our conclusions.

Dipolar BEC in an optical lattice. Effective model
In the following, we consider a dipolar BEC of particles with mass m and electric dipole d (the results are equally valid for magnetic dipoles) oriented in the z-direction by a sufficiently large external field, and that hence interact via a dipole-dipole potential: 3 , where θ is the angle formed by the vector joining the interacting particles and the dipole direction.The coefficient α can be tuned within the range −1/2 ≤ α ≤ 1 by rotating the external field that orients the dipoles much faster than any other relevant time scale in the system [37].We consider the dipolar BEC as placed in a 1D optical lattice (see Fig. 1).At sufficiently low temperatures (and away from shape resonances [38]) the physics of the dipolar BEC is provided by a non-local non-linear Schrödinger equation (NLSE) of the form: where g = 4π 2 a/m, with a > 0 the s-wave scattering length, and is the recoil energy and Q l is the laser wave vector.
In the tight-binding regime (sufficiently strong lattice) we can write Φ( r, t) = j w(z − bj)ψ j ( ρ, t), where b = π/Q l , ρ = {x, y} and w(z) is the Wannier function associated to the lowest energy band.Substituting this Ansatz in Eq. ( 1) we obtain a discrete NLSE.We may then return to a continuous equation, where the presence of the lattice amounts for an effective mass m * along the lattice direction and for a renormalized coupling constant g [39]: where Figure 1.(a) Vortex-line in an optical lattice with lattice spacing b.The wave-number of the Kelvin mode is q, whereas the corresponding wavelength satisfies 2π/q ≫ b.(b) The vortex-line has a finite core with healing length ξ and a velocity field v perpendicular to the line. with where w is the Fourier-transform of w(z).The validity of Eq. ( 2) is limited to z-momenta k z ≪ 2π/b, in which one can ignore the discreteness of lattice.In addition, note that the single-band model breaks down if the gap between the first and second band becomes comparable to other energy scales in the problem (m/m * → 1).
As mentioned above, the partially attractive character of the DDI may lead to different types of instability in a dipolar BEC.We consider first an homogeneous 3D solution Ψ 0 ( r, t) = √ n exp[−iµt/ ], where n denotes the condensate density, and µ = (g + Ṽd (0))n the chemical potential, with Ṽd From the corresponding Bogoliubov-de Gennes (BdG) equations one gets that the energy ǫ( k) corresponding to an excitation of wave number k fulfills: phonons with k lying on the xy plane are unstable if β > 2, while for g d < 0 phonons with k along z are unstable if β < −1.Hence absolute phonon stability demands −1 < β < 2.

Kelvin-wave instability of a straight vortex line
In this section we consider a straight vortex line along the z-direction (see Fig. 1).The corresponding wavefunction is of the form Ψ 0 ( r, t) = φ 0 (ρ) exp(iϕ) exp[−iµt/ ], where ϕ is the polar angle on the xy plane.The function φ 0 (ρ) fulfills where ḡ = g − g d /2.Note that, due to the homogeneity of Ψ 0 in the z-direction, the DDI just regularizes the local term.The density of the vortex core is given by |φ 0 (ρ)| 2 , which vanishes at ρ = 0 and becomes equal to the bulk density n at distances larger 0.002 than the healing length ξ = / √ mḡn (see Fig. 1).Note that ξ depends on the DDI.
In the following we consider Kelvin-wave excitations of the straight vortex line of the form (see Fig. 1) Ψ( r, t) = Ψ 0 ( r, t) + χ( r, t) exp[i(ϕ − µt/ )], where [28] ] (as mentioned in Sec. 2, q ≪ 2π/b in order to justify the validity of the effective model).introducing this Ansatz into (2) and linearizing in χ, one obtains the corresponding BdG equations: with , where I l and K l are modified Bessel functions, and For every q we determine the lowest eigen-energy ǫ, that provides the dispersion law discussed below.
The first line at the rhs of Eqs. ( 4) and ( 5) is exactly the same as that expected for a vortex in a short-range interacting BEC [28], but with the regularized value ḡ.Hence, in absence of DDI (or equivalently from Eqs. ( 4) and ( 5) without the last integral term) the dispersion law at low momenta (qξ ≪ 1) is provided by the well-known expression ǫ(q) = −( 2 q 2 /2m * ) ln (m/m * ) 1/2 qξ .On the contrary, the last term at the rhs of both equations is directly linked to the long-range character of the DDI and, as we show below, leads to novel phenomena in the physics of Kelvin modes (l = −1) in dipolar BECs.In the following, and in order to isolate the effect of the DDI on the core size with respect to the effect of the integral terms in Eqs. ( 4) and ( 5), we fix ḡ and change the parameter β which is proportional to the dipole-dipole coupling constant.
The integral terms of Eqs. ( 4) and ( 5) significantly modify the Kelvin-mode spectrum in a different way depending whether β > 0 or β < 0. The different regimes are summarized in Fig. 3.As shown in Ref. [36], for β > 0 the excitation energy increases, i.e. the vortex line becomes stiffer against transverse modulations.The opposite occurs for β < 0, i.e. the Kelvin-modes become softer.However, the softening of the Kelvin-wave spectrum never leads to instability in absence of an additional lattice (m = m * ), since destabilization demands β < −1, for which, as we show in Sec. 2 the whole dipolar BEC is phonon-unstable.Interestingly, an increase of the potential depth of the additional lattice leads to a reduction of the role of the kinetic energy term mq 2 /m * in Eqs. ( 4) and ( 5) that enhances the effect of the dipolar interaction on the dispersion law.As a consequence, as shown in Fig. 2, in addition to the −q 2 ln(qξ) dependence at low q, a roton-like minimum eventually appears at intermediate q.For a sufficiently small (m/m * ) cr (thick curve in Fig. 3) the roton minimum reaches zero energy.For m/m * ≤ (m/m * ) cr the energy at the roton becomes negative, and hence the straight vortex-line becomes thermodynamically unstable against the formation of a new vortex-line configuration, which we discuss in the following sections.
The intriguing dependence of the dispersion of Kelvin-waves on m/m * and β is explained by the competition of the two processes involved in the inter-site physics, namely tunneling and intersite DDI.On one side, the hopping energy is minimized when vortex cores at neighboring sites are placed right on top of each other.Hence tunneling tends to maintain the vortex line straight.On the other side, we may obtain an intuitive picture concerning the intersite DDI by sketching the vortex core as a 1D chain of dipolar holes.Dipolar holes interact in exactly the same way as dipolar particles, and hence for β > 0 attract each other maximally when aligned along the dipole direction, i.e. the z-direction.As a consequence, for β > 0 DDI and hopping add up in keeping the vortex straight, and the vortex line becomes stiffer.On the contrary, for β < 0 the cores maximally repel each other when aligned along the z-direction.Hence DDI and tunneling compete, and the vortex line becomes softer.

Helical vortices
From the intuitive picture discussed in the previous section it becomes clear that when m/m * becomes sufficiently small the tunneling cannot balance the DDI any longer, and as a consequence the straight vortex line becomes thermodynamically unstable, as already mentioned in our discussion of the Bogoliubov analysis in the previous section.In this section we show that this instability is related to a transition from a straight vortex into a twisted vortex-line.This twisting, which we confirm below numerically, may be expected by analyzing the DDI between vortex cores at neighboring sites.This energy is minimized by laterally displacing the vortices with respect to each other a finite distance on the xy-plane.These lateral displacements between nearest neighbors lead to a twisting of the vortex line.Note that due to the symmetry of the problem, we may expect an helical structure, as we confirm in the following.- In order to determine the new lowest-energy configuration of the vortex line, we have performed fully 3D numerical calculations of the GPE (2).We start our calculations from an initially imposed straight vortex line and let the system evolve in imaginary time.We performed our calculations employing a cylindrical numerical box placed several healing lengths apart from the vortex core to minimize boundary effects.The cylindrical box configuration permits a sufficiently flat density in the region where the vortex core is created, allowing us to avoid unwanted density effect, which will obscure the analysis of the vortex stability.As expected, the imaginary time evolution leads to a ground-state straight vortex line configuration for those values of β and m/m * lying inside the stable regime of Fig. 3. On the contrary, for those regions within the instability regime a qualitatively new ground-state is found, where we observe a departure from the straight form into an helical configuration, as in Fig. 4.
Hence at the line (m/m * ) cr there is a phase transition from straight into helical ground-state vortex lines.In order to characterize this phase transition we have performed an exhaustive analysis of the helical configurations, which may be characterized by a pitch with wavenumber Q and a radius r 0 .We confirmed that the wave-number Q is indeed comparable to the minimum q rot in the dispersion law obtained from the Bogoliubov analysis of Sec. 3.For several fixed values of β we performed a large number of 3D numerical simulations for different m/m * , to analyze the behavior of the amplitude r 0 inside the unstable region and at the transition to the stable region.Our results (for β = −0.8)are depicted in Fig. 5.We observe a gradual decrease of r 0 when m/m * approaches the stability border (m/m * ) cr .For values to close to the stability border r 0 ≪ ξ and hence it becomes in practive (both numerically and experimentally) impossible to discern a significant vortex twisting.Our simulations are hence compatible with a second-order-like phase-transition from an helical into a straight lowest-energy vortex-line configuration.
Note that inside the instability region an increase of the lattice strength (i.e. a decrease of m/m * ) results in a larger bending which comes together with a shallower binding energy for the line.If the binding energy becomes lower than other typical energy scales involved in the system (inhomogeneity, boundary effects) then the 3D vortex line breaks into uncorrelated 2D vortices.The latter is, of course, also expected in the absence of DDI if the tunneling becomes sufficiently small.Note that in Fig. 3 we did not consider any other additional energy scale, and hence the 3D vortex line could be considered all the way down to very small m/m * .However, in our numerical simulations we do have boundary effects introduced by our finite cylindrical numerical box.As a consequence we have observed (also for the results of next section) that for very small m/m * inside the instability regime the boundary effects may completely unbound the vortex line into uncorrelated 2D vortices at each layer.However, in the present paper we are more concerned with the phase transition boundary, where the binding is still dominant even when the vortex is bent.

Snake-like vortices
In Secs. 3 and 4, we have discussed the case in which the dipoles are oriented along the z-direction.In that situation, the cylindrical symmetry of the problem largely simplified the Bogoliubov analysis.The helical instability demands β < 0, and it is hence necessary the employ of the tuning mechanism discussed in Sec. 2. However, the tuning is not strictly necessary to observe the destabilization of the straight vortex line.A perhaps experimentally simpler scenario is offered by the case in which the dipoles are oriented perpendicular to the vortex line (and hence also perpendicular to the over-imposed .lattice potential).Then the DDI is again repulsive in z-direction and competes with the kinetic energy in what concerns the stability of the straight vortex-line in a similar way as described in the previous sections.In this section we analyze this particular case (orientation along y).We show that the vortex line may be also destabilized in this experimentally less involved case.As for Sec. 4, we have evolved an initially straight vortex in imaginary time using Eq. ( 2) employing similar numerical conditions.However, the analysis in the new configuration is largely handicapped due to the distortion of the density of the condensate induced by the fact that the dipole in now on the xy plane, and hence breaks the polar symmetry.These density modulations may be reduced (but not fully eliminated) by considering larger numerical boxes.However, the latter may make the 3D numerical simulation prohibitively long.
In spite of these difficulties we observed the departure from the straight vortex line for sufficiently large β (note that in this configuration β > 0 and hence it may take values up to 2 before entering into the regime of phonon instability discussed in Sec. 2).Due to the broken cylindrical symmetry, for sufficiently large β and small m/m * the straight vortex line is destabilized into a snake-like configuration on the yz plane rather than into an helix, as in the previous section, as shown in Fig. 6.We have analyzed the amplitude of the helix when approaching the stability regime.Our results are compatible with a second-order-like behavior as in the previous section, although, as mentioned above, a rigorous analysis is largely handicapped by the appearance of strong density modulations in the simulations.

Conclusions
In this paper we have studied the physics of vortex lines in dipolar gases.In particular, we have analyzed in detail the stability of a straight vortex line with respect to Kelvin waves.We have shown that the presence of an additional optical lattice along the vortex line may allow for the observation of a dipole-induced destabilization of the straight vortex line due to the softening of a roton minimum in the Kelvin-wave spectrum.If this occurs the straight vortex-line configuration ceases to be that of minimal energy, and there is a second-order-like phase transition into an helical or snake vortex-line, depending on the dipole orientation.

Figure 4 .
Figure 4. Vortex-line configurations of the ground state for β = −0.8 and m/m * = 0.15 (straight) and m/m * = 0.04 (helical), respectively.For both cases the dipole is assumed in the z-direction.The unit of length is √ 2ξ

Figure 6 .
Figure 6.Vortex-line configurations of the ground state for β = 1.2 and m/m * = 0.075 (straight) and m/m * = 0.04 (snake-like), respectively.For both cases the dipole is assumed in the y-direction.The unit of length is √ 2ξ