Vortex lines attached to dark solitons in Bose-Einstein condensates and Boson-Vortex Duality in 3+1 Dimensions
VVortex lines attached to dark solitons in Bose-Einstein condensates and Boson-VortexDuality in 3+1 Dimensions
A. Mu˜noz Mateo , , ∗ Xiaoquan Yu , , † and Jun Nian , ‡ Departament de F´ısica Qu`antica i Astrof´ısica, Universitat de Barcelona,Mart´ı i Franqu`es, 1, E–08028 Barcelona, Spain Department of Physics, Centre for Quantum Science,and Dodd-Walls Centre for Photonic and Quantum Technologies, University of Otago, Dunedin, New Zealand New Zealand Institute for Advanced Study, Centre for Theoretical Chemistry and Physics,Massey University, Auckland 0745, New Zealand Institut des Hautes ´Etudes Scientifiques, Le Bois-Marie,35 route de Chartres, 91440 Bures-sur-Yvette, France and C. N. Yang Institute for Theoretical Physics, Stony Brook University, Stony Brook, NY 11794-3840, United States
We demonstrate the existence of stationary states composed of vortex lines attached to planardark solitons in scalar Bose-Einstein condensates. Dynamically stable states of this type are found atlow values of the chemical potential in channeled condensates, where the long-wavelength instabilityof dark solitons is prevented. In oblate, harmonic traps, U-shaped vortex lines attached by both endsto a single planar soliton are shown to be long-lived states. Our results are reported for parameterstypical of current experiments, and open up a way to explore the interplay of different topologicalstructures. These configurations provide Dirichlet boundary conditions for vortex lines and therebymimic open strings attached to D-branes in string theory. We show that these similarities can beformally established by mapping the Gross-Pitaevskii theory into a dual effective string theory foropen strings via a boson-vortex duality in 3+1 dimensions. Combining a one-form gauge field livingon the soliton plane which couples to the endpoints of vortex lines and a two-form gauge field whichcouples to vortex lines, we obtain a gauge-invariant dual action of open vortex lines with theirendpoints attached to dark solitons.
I. INTRODUCTION
Quantum vortices and planar dark solitons are fre-quently generated in current experiments with ultracoldgases, from Bose-Einstein condensates (BECs) [1, 2] toFermi gases [3 ? , 4]. With phase imprinting techniques,the phase of the superfluid can be arranged to show eithercontinuous 2 π changes around vortex lines, or suddenleaps across the soliton planes. These topological defects,which separate regions with different values of the result-ing order parameter, also appear after the quench cross-ing a second-order phase transition breaking the under-lying continuous symmetry known as the Kibble-Zurekmechanism [6, 7]. The great achievement of techniquesin experiments with ultracold gases has directed re-searchers’ attention towards more complex physical sys-tems presenting nontrivial topologies, which might sim-ulate different types of topological excitations discussedin quantum field theory and string theory. For instance,analogues of cosmic strings in superfluids [7, 8] and ana-logues of Dirac monopoles in the spin-1 BEC [9, 10] havebeen proposed. He A-B interfaces and the boundarysurface of the two-component BEC have been suggestedas analogues of branes in string theory [11–14]. Isolatedmonopoles [15] and Dirac monopoles [16] have been ob-served in recent spin-1 BEC experiments. ∗ [email protected] † [email protected] ‡ [email protected] In this work we show that scalar BECs are also suit-able for studying rich structures of topological defects.In spite of the fact that there have been an intensive re-search about vortices and solitons in scalar BECs, as faras we know, the study of topological structures showingjunctions between them has not been performed. Only aparticular configuration of this type has been addressedin the context of an effective low-dimensional model intrapped condensates [17]. Note that in a trapped BECwithout dark solitons stable vortex lines which do notform loops have to end at the condensate boundary.Here, within a mean field approach, we address the dy-namics of scalar BECs containing vortex lines that ex-tend between soliton stripes. This arrangement makes itpossible to find the ends of a vortex line in the bulk ofthe system, just at the position of the soliton. Config-urations of this type will be referred to as open vortexlines, and they provide Dirichlet boundary conditions forthe vortex ends. It is interesting to note that they mimicopen strings attached to D-branes in string theory, inparticular the so-called BIons. Although the arrange-ments shown in the present work are not generic fromthe string theory point of view, they might still offer aground for simulating particular aspects of it. To estab-lish this connection on more formal terms, we show thatthe Gross-Pitaevskii (GP) theory can be mapped intoan effective string theory via the so-called boson-vortexduality.The rest of the paper is structured as follows. In Sec-tion II, we present the mean field model based on theGross-Pitaevskii energy functional, which is used to de- a r X i v : . [ c ond - m a t . qu a n t - g a s ] D ec scribe accurately the dynamics of BECs containing vor-tices and solitons. In Section III, we propose the boson-vortex duality for open vortex lines ending on dark soli-tons. The following sections are dedicated to particularrealistic configurations fitting into both the GP descrip-tion and its dual description. First, in Section IV, weconsider condensates trapped along their transverse di-rections and containing axisymmetric vortex lines sepa-rated by dark solitons that extend over the cross sectionof the system. We demonstrate that a 3D condensateendowed with this structure can be a dynamically stablestationary state. Beyond a threshold value for the inter-action energy such configuration becomes unstable dueto the soliton decay, and the whole system can evolvethrough new emerging vortex lines. Second, in SectionV, we study slightly oblate condensates holding symmet-ric planar solitons, to which U-shaped vortex lines, thatare referred to as half vortex rings, are attached. In thisvariant configuration, each curved vortex line has bothends on a single dark soliton. We show that, althoughsuch stationary state is dynamically unstable, it has along lifetime. The ultimate decay of the soliton showschaotic behaviors of vortex lines, which might be relatedto quantum turbulence processes. A common feature ofboth configurations is the presence of vortex line pairsseparated by the soliton plane. We finish with the con-clusions in Section VI, where we emphasize that the topo-logical states considered in this paper are feasible to berealized under current experimental techniques. II. GROSS-PITAEVSKII MODEL
We follow a mean-field approach for modeling di-lute gases of repulsive interacting bosons making Bose-Einstein condensates at zero temperature. The state ofsuch a system is described by a complex order param-eter Ψ( r , t ), whose dynamics is determined by the non-relativistic GP Lagrangian density: L GP = i (cid:126) (cid:18) Ψ ∂ Ψ ∗ ∂t − Ψ ∗ ∂ Ψ ∂t (cid:19) − H GP , (1)where the energy density is given by H GP = (cid:126) m |∇ Ψ | + g (cid:0) | Ψ | − | Ψ | (cid:1) . (2)Here g = 4 π (cid:126) a/m is the interaction parameter, charac-terized by the s -wave scattering length a and the atomicmass m . Note that the energy is measured relative to theground state Ψ ( r , t ) = ψ ( r ) exp( − iµ t/ (cid:126) ) with chemi-cal potential µ . For homogeneous systems, Ψ is a realconstant, apart from a global phase, and µ fixes the valueof the uniform density | Ψ | = µ/g . In the presenceof an external potential V ( r ), a local chemical poten-tial µ l ( r ) = µ − V ( r ) can be defined, which in the limitof high interaction energy, or Thomas-Fermi regime, al-lows to approximate the non-uniform ground-state den-sity by | Ψ ( r ) | = µ l ( r ) /g . In this way, we will use Eq. (1) as a unified model for both homogeneous andnon-homogeneous systems. In the latter case, this meansthat when the system is closed, the energy is measuredwith respect to the ground state having the same chemi-cal potential, and thus, in general, a different number ofparticles. For non-homogeneous systems, we will considerharmonic trapping potentials with cylindrical symmetry V ( r ) = V z ( z ) + V ⊥ ( r ⊥ ) = mω z z / mω ⊥ r ⊥ /
2, havingan aspect ratio λ = ω z /ω ⊥ , where the transverse coordi-nates are ( θ, r ⊥ ) = (tan − ( y/x ) , (cid:112) x + y ). As a lengthunit, we define the characteristic length of the transversetrap a ⊥ = (cid:112) (cid:126) /mω ⊥ .The equation of motion corresponding to the La-grangian density L GP is the time-dependent GP equa-tion i (cid:126) ∂ Ψ ∂t = (cid:18) − (cid:126) m ∇ + g ( | Ψ | − | Ψ | ) (cid:19) Ψ . (3)Its stationary solutions with chemical potential µ , thatis Ψ( r , t ) = ψ ( z, θ, r ⊥ ), fulfill the time-independent GPequation (cid:18) − (cid:126) m ∇ + g ( | ψ | − | ψ | ) (cid:19) ψ = 0 . (4) III. DUALITYA. Effective string action
The boson-vortex duality is a powerful tool to studythe dynamics of vortex rings [18–22]. It has been demon-strated that the dual description of the GP theory in3+1 dimensions is equivalent to a certain type of effec-tive string theory [19, 20, 23]. In this paper we extend thedual mapping to open vortex lines and argue that darksolitons in the original GP theory could play the role ofD-branes in the effective string theory by introducing apinned boundary condition for vortex lines [24]. In thefollowing, for simplicity we set the trapping potential tobe zero, i.e. V ( r ) = 0.Introducing the Madelung transformation Ψ = √ ρ e iφ ,the equivalent expression of the GP Lagrangian densityEq. (1) in terms of the condensate density ρ and the phase φ reads L GP (cid:39) − (cid:126) ρ ˙ φ − ρ (cid:126) m ( ∇ φ ) − g ρ − ρ ) , (5)where we neglect the quantum pressure term ∼ ( ∇√ ρ ) ,which should be valid at long wavelengths (the hydro-dynamic limit) that we are interested in. We rewritethe quadratic term − ρ (cid:126) ( ∇ φ ) / m via the so-calledHubbard-Stratanovich transformation [25], and obtainthe dual representation of the original theory Eq. (5) : L ∗ GP (cid:39) − (cid:126) ρ ˙ φ − (cid:126) ∇ φ · f + m ρ f · f − g ρ − ρ ) = − (cid:126) f µ ∂ µ φ + m ρ f · f − g ρ − ρ ) , (6)where ∂ µ ≡ ( ∂ t , ∇ ), and f is an auxiliary three-vector,which is also the space-like components of the four-vector f µ ≡ ( ρ, f ). The physical meaning of the field f becomesclear by evaluating its equation of motion f = ρ (cid:126) m ∇ φ ,which matches the superfluid current J in the GP theorydescribed by Eq. (1) [18].Let us now decompose the phase φ into a non-singularpart and a few singular parts related to the topologicaldefects: φ = φ g + φ cv + φ ov + φ s , (7)where φ g is the smooth phase, which is the Goldstonemode associated with the U (1) symmetry breaking, φ cv is the phase of vortex rings, φ ov is the phase of openvortex lines, and φ s represents the phase of dark solitons.Note that Eq. (7) just means that φ cv , φ ov and φ s havesingularities, and they may still contain smooth parts.The action of the non-singular part φ g reads S g = (cid:90) d x ( − (cid:126) f µ ∂ µ φ g ) (8)= (cid:126) (cid:90) d x [ ∂ µ f µ φ g − ∂ µ ( f µ φ g )] . By integrating out the smooth phase φ g in the bulk, weobtain ∂ t ρ + ∇ · f = ∂ µ f µ = 0 , (9)which can be solved by f µ = 12 (cid:15) µνλσ ∂ ν B λσ , (10)where (cid:15) µνλσ is the totally-antisymmetric tensor, and B λσ is an antisymmetric rank-2 gauge field. Note that f µ isinvariant under the gauge transformation B µν → B µν + ∂ µ Λ ν − ∂ ν Λ µ , where Λ ν is an arbitrary four vector.In the dual description, the action of a vortex ring in3+1 dimensions has already been proposed [18–20, 23].In this paper we focus on open vortex lines. For an openvortex line the action reads S ov = (cid:90) d x ( − (cid:126) f µ ∂ µ φ ov ) (11)= − (cid:126) (cid:90) d x(cid:15) µνλσ ∂ ν B λσ ∂ µ φ ov = (cid:126) (cid:90) d x (cid:2) ∂ µ ( (cid:15) µνλσ B λσ ∂ ν φ ov ) − B µν (cid:15) µνλσ ∂ λ ∂ σ φ ov (cid:3) . We elaborate on these terms by considering a vortex linewhich is parallel to the z-axis, whose topological naturecan be seen from the term (cid:90) d x (cid:15) ztλσ ∂ λ ∂ σ φ ov = 2 π. (12)The topological defects produced by vortex lines can beintroduced explicitly through (cid:15) µνλσ ∂ λ ∂ σ φ ov (13)= − π (cid:88) i (cid:90) Σ i dτ dσ(cid:15) αβ ∂ α X µ ∂ β X ν δ ( x µ − X µ ) , where Σ i is the worldsheet spanned by the i -th vortexline, and α, β ∈ { , } label the worldsheet coordinates. (cid:15) αβ is a rank-2 antisymmetric tensor, and we set (cid:15) = 1.The space σ ∈ [ σ , σ ], and the time τ is chosen to be t . X µ = X µ ( τ, σ ) stands for the coordinates of a vortexline, namely X = t , X = ( x, y, z ). Hence, the bulkaction of the open vortex line reads S vbulk = − (cid:126) (cid:90) d xB µν (cid:15) µνλσ ∂ λ ∂ σ φ ov (14)= η (cid:88) i (cid:90) Σ i dτ dσB µν (cid:15) αβ ∂ α X µ ∂ β X ν , where η ≡ π (cid:126) and B µν = B µν ( X µ ).Suppose that an ( x - y )-planar dark soliton, to which avortex line can be attached, is located at z = z . Wemake use of the soliton property of producing a localized π phase jump by crossing the soliton plane, i.e. ∂ z φ s ( t, x, y, z ) = π (cid:90) d σ δ ( x a − Y a ( σ i )) δ ( z − z ) ,∂ a φ s ( t, x, y, z ) = (cid:96) (cid:90) d σ ∂ a ˆ φ s ( σ i ) δ ( x a − Y a ( σ i )) δ ( z − z ) , (15)where σ i ( i = 1, 2, 3) are the coordinates on the darksoliton plane, Y a ( σ i ) ∈ { t, x, y } is the map from σ i tothe spacetime coordinates, (cid:96) is a length scale inserted fordimensional reasons, and ˆ φ s ( σ i ) is the transverse phasedefined only along the dark soliton plane, which in turncontains a smooth part and a singular part. In addition,the superfluid current along the z -direction vanishes atthe soliton plane J z ( t, x, y, z ) = 0 , (16)which implies f z ( t, x, y, z ) = 0 and thus imposes a con-straint on the B -field, according to Eq. (10). Conse-quently, the boundary term in the action Eq. (8) vanisheson the dark soliton surface − (cid:126) (cid:82) d xf z ( z , x, y, t ) φ g = 0.As a result, the action for the dark soliton is S s = (cid:90) d x ( − (cid:126) f µ ∂ µ φ s ) = − (cid:126) (cid:90) d x (cid:16) (cid:96)f a ∂ a ˆ φ s (cid:17) . (17)Combining the boundary term in the action of the openvortex Eq. (11): (cid:126) (cid:90) d x (cid:15) zabc B bc ∂ a φ ov , (18)with a,b,c ∈ { t, x, y } , and the dark soliton actionEq. (17), we obtain the boundary action S vboundary = − (cid:126) (cid:90) d x (cid:18) (cid:96)f a − (cid:15) abc B bc (cid:19) ∂ a ˆ φ s . (19)Here we made the identification φ ov ( z → z ± , x, y, t ) =ˆ φ s ( x, y, t ), which implies that the endpoints of the openvortex lines are the vortex excitations living on the soli-ton surface ( z → z ± ). As it has previously been done for φ g , integrating out the smooth part of ˆ φ s leads to ∂ a ( (cid:96)f a − (cid:15) abc B bc ) = 0 , (20)whose solution is (cid:96)f a − (cid:15) abc B bc = 12 (cid:15) abc F bc , (21)where F ab ≡ ∂ a A b − ∂ b A a is the field strength of A a , and A a is a one-form gauge field living on the dark solitonsurface ( z → z ± ). Therefore, we are left with the singularpart ˆ φ ss of ˆ φ s S vboundary = − (cid:126) (cid:90) d x (cid:15) abc F bc ∂ a ˆ φ ss = − (cid:126) (cid:90) d x(cid:15) abc ( ∂ b A c − ∂ c A b ) ∂ a ˆ φ ss = − (cid:126) (cid:90) d xA a ( (cid:15) abc ∂ b ∂ c ˆ φ ss ) , (22)and for the endpoints of vortex lines, similar to Eqs. (12)–(13), we get (cid:15) abc ∂ b ∂ c ˆ φ ss = − π (cid:88) i (cid:90) ∂ Σ i dτ ∂ τ X a δ ( x a − X a ( τ )) , (23)where ∂ Σ i is the boundary of the worldsheet Σ i , and X a ( τ ) are the coordinates of an endpoint of a vortexline, namely X = t , X = ( x, y ). Then we obtain S vboundary = − (cid:126) (cid:90) d xA a ( (cid:15) abc ∂ b ∂ c ˆ φ ss )= η (cid:88) i (cid:90) ∂ Σ i dτ A a ∂ τ X a , (24)where A a = A a ( X a ).In terms of the field strengths, the last two terms ofthe action Eq. (6) can be written as S gauge = m ρ f · f − g ρ − ρ ) = − g (cid:96) (cid:90) d x ( (cid:101) F + (cid:101) B ) − g (cid:90) d x h . (25)Near ρ (cid:39) ρ , H µνλ ≡ ∂ µ B νλ + ∂ ν B λµ + ∂ λ B µν (cid:39) H µνλ + h µνλ , where H µνλ is the background field with H = ρ , and the fluctuations are given by h = h µνλ h µνλ / η µν = diag {− c s , , , } determinedby the speed of sound c s = (cid:112) gρ /m . (cid:101) F and (cid:101) B are thefluctuations of F and B on the soliton plane respectively. Collecting all the contributions, we finally obtain thefollowing dual description of open vortex lines: S ∗ = S vbulk + S vboundary + S gauge = η (cid:88) i (cid:90) Σ i dσdτ B µν (cid:15) αβ ∂ α X µ ∂ β X ν + η (cid:88) j (cid:90) ∂ Σ j dτ A a ∂ τ X a − g (cid:96) (cid:90) d x ( (cid:101) F + (cid:101) B ) − g (cid:90) d x h . (26)The summation over Σ i includes all the vortex lines andthe summation over ∂ Σ j includes all the endpoints ofvortex lines attached to dark solitons. It is important tonote that the resulting action Eq. (26) is invariant underthe gauge transformations [26]: B µν → B µν + ∂ µ Λ ν − ∂ ν Λ µ , (27) A a → A a − Λ a . (28)The action given in Eq. (26) has two aspects. Firstof all, it provides a hydrodynamic description of openvortex lines in scalar BECs, which is useful to study thedynamics of open vortex lines with Dirichlet boundaryconditions and vortex-sound interactions. On the otherhand, it can be viewed as an effective open string actionwithout the tension term, or equivalently an action in thelarge B -field limit, in the presence of D-branes. Hence,the dual theory Eq. (26) might provide a possibility totest some aspects of string theory in BEC experiments. B. Equation of motion
In the following we will discuss a simple example ofthe action Eq. (26). From now on we will only con-sider a single vortex line whose endpoints are attachedto two parallel dark solitons. Here, for simplicity, we ig-nore the dynamics of the dark solitons and treat themas hard walls. We will also assume that the fluctuationsof B µν and A a in space-time are small, and hence thefield strength parts in S gauge can be neglected since theyprovide higher order terms.In order to have non-trivial dynamics of a single vor-tex line, we need to introduce in the action Eq. (26) aphenomenological tension term, which plays the role ofkinetic energy of vortices. This term has been neglectedup to now because of the assumption of infinitely thintopological defects. Such assumption implies the lack ofvortex cores and also the absence of a “physical” mass ofthe vortex lines, which is responsible, among other dy-namical effects, for the buoyancy-like forces in inhomo-geneous backgrounds [27]. As a consequence, the tensionterm can be seen as a remnant of the core structure ofthe vortices in the hydrodynamic limit.We introduce the tension term in Eq. (26) by adding aPolyakov string action [26] proportional to the the stringtension T : S single = T c s (cid:90) Σ dσdτ √− h h αβ η µν ∂ α X µ ∂ β X ν + η (cid:90) Σ dσdτ B µν (cid:15) αβ ∂ α X µ ∂ β X ν + η (cid:90) ∂ Σ dτ A a ∂ τ X a , (29)where h αβ = diag {− / c , } is the worldsheet metric,and √− h = (cid:112) − det( h αβ ) = c s . T has the standarddimension [ E ] / [ L ] for the string tension and in terms ofBEC units T ∼ π (cid:126) ρ / m .Considering a small variation δX µ on top of the back-ground vortex configuration X µ [28], the variation of theaction to the leading order in δX µ reads δS single = T (cid:90) Σ dσ dτ ∂ α ( δX µ ) ∂ α X µ + η (cid:90) Σ dσ dτ B µν (cid:15) αβ ( ∂ α δX µ ) ∂ β X ν + η (cid:90) Σ dσ dτ B µν (cid:15) αβ ( ∂ α X µ )( ∂ β δX ν )+ η (cid:90) Σ dσ dτ ( ∂ ρ B µν )( δX ρ ) (cid:15) αβ ∂ α X µ ∂ β X ν + η (cid:90) ∂ Σ dτ ( ∂ b A a )( δX b ) ∂ τ X a + η (cid:90) ∂ Σ dτ A a ∂ τ ( δX a )= − T (cid:90) Σ dσdτ ( δX µ ) ∂ α ∂ α X µ + η (cid:90) Σ dσdτ H µλν ( δX µ ) (cid:15) αβ ∂ α X λ ∂ β X ν + T (cid:90) ∂ Σ dτ ( δX a ) ∂ σ X a + η (cid:90) ∂ Σ dτ ( B ab + F ab )( δX a ) ∂ τ X b . (30)Hence the equation of motion of the vortex line in thebulk is given by − T ∂ α ∂ α X µ + η (cid:15) αβ H µλν ∂ α X λ ∂ β X ν = 0 , (31)and the dynamics of the endpoints reads T ∂ σ X a (cid:12)(cid:12) σ + η ( B ab + F ab ) ∂ τ X b (cid:12)(cid:12) σ = 0 (32)and − T ∂ σ X a (cid:12)(cid:12) σ − η ( B ab + F ab ) ∂ τ X b (cid:12)(cid:12) σ = 0 . (33)We now consider a special situation. In the bulk H ij = 0, and H ijk = ρ (cid:15) ijk . On the boundaries A = 0, ∂ τ A a ( σ ) = ∂ τ A a ( σ ) = 0 and B nm ( σ ) = B nm ( σ ) = ρ (cid:15) nm . Here m, n = 1 ,
2. For this special case we obtainthe following equations0 = T (cid:18) ∂ X i ∂σ − c s ∂ X i ∂τ (cid:19) − η (cid:15) αβ ρ (cid:15) ijk ∂ α X j ∂ β X k , T ∂ σ X n + η ρ (cid:15) nm ∂ τ X m . (34)Here we only look for a static solution. For a staticvortex line, ∂ X i ∂σ = 0 , ∂ σ X n = 0 . (35)The general solution would be x = a σ + a , y = b σ + b , z = c σ + c , (36)where a , a , b , b , c and c are constants. By choos-ing the proper coordinates and taking into account theboundary conditions, the static solution reads x = a , y = b , z ( σ ) = c σ + c , (37)which describes a static open vortex line between twodark solitons. Note that this vortex line satisfies Neu-mann boundary conditions along x and y directions,while fulfills Dirichlet boundary condition along z di-rection. If we replace the current boundary conditionswith periodic boundary conditions, Eq. (35) would givea trivial solution, which means that vortex rings mustpropagate as it is well-known.In the next two sections we show that such staticvortex-soliton composite topological excitations can in-deed be found in the GP theory. For the sake of compari-son with realistic parameters used in current experimentsof ultracold gases, in what follows we report numericalresults for systems composed of Rb atoms, with scatter-ing length a = 5 .
29 nm, confined by harmonic potentialswith angular frequencies in the range 2 π × [10 − IV. OPEN VORTEX LINESA. Configurations
In this section, we consider elongated condensatesalong the axial z -direction (assuming ω z = 0) andisotropic trapping in the transverse plane, providing thesystem with a channeled structure. In particular, wewill consider stationary states containing vortex lines at-tached to solitons. The simplest configuration of thistype, containing a single dark soliton, is shown in Fig. 1.In order to realize this simple configuration we have de-vised a channeled, cylindrical geometry with an axisimet-ric vortex line: ψ ( z, θ, r ⊥ ) = ψ ( z, r ⊥ ) e iθ , (38)where ψ ( z, r ⊥ ) is a real function. After substituting inEq. (4), the vortex solution gives − (cid:126) m (cid:18) ∂ z + ∇ ⊥ − r ⊥ (cid:19) ψ + V ⊥ ψ + g | ψ | ψ = µψ , (39)where ∇ ⊥ = ∂ r ⊥ + ∂ r ⊥ /r ⊥ and V ⊥ accounts for the trans-verse confinement. This is the stationary equation fora vortex state that generates a tangential velocity field v = (cid:126) ∇ arg( ψ ) /m around the z -axis: v ( r ⊥ ) = (cid:126) u θ /mr ⊥ ,where u θ is the unit tangent vector.Next, we search for solutions to the nonlinear Eq. (39)including a dark soliton along the axial direction. Ananalytical ansatz for this configuration is ψ ( z, r ⊥ ) = r ⊥ χ ( r ⊥ ) tanh (cid:18) zξ ( r ⊥ ) (cid:19) , (40)where ξ ( r ⊥ ) = (cid:126) / (cid:112) mg | χ ( r ⊥ ) | defines a radius-varyinghealing length through the Thomas-Fermi density profile | χ ( r ⊥ ) | = µ l ( r ⊥ ) /g of a system without the vortex. Thisexpression follows the ansatz introduced in Ref. [29] for3D dark solitons in channeled condensates, and gives agood estimate in the strongly interacting regime, wherethe Thomas-Fermi approximation applies. Although the r ⊥ factor of Eq. (40) accounts for the vortex core in aquite simple manner, which is characteristic of the cor-responding noninteracting system, it will turn out to beefficient in getting numerical convergence to the real sta-tionary state.By using the ansatz Eq. (40) for open vortex lines, wehave followed a Newton method to find the exact nu-merical solution to the full GP Eq. (4). Fig. 1 showsthe features of this configuration (without axial confine-ment) around a vortex-soliton junction, which leads toDirichlet boundary conditions for the vortex end points.The panel Fig. 1(a), presenting a semi-transparent den-sity isocontour of the condensate at 5 % of maximum den-sity, shows how the presence of the dark soliton breaksthe system into two phase-separated subsets containingcorresponding axisymmetric vortices. These vortices aredifferent entities, as can be seen in the detailed view ofpanel Fig. 1(b). The dark soliton twists their relativephase in π radians along the z -axis for every value of theazimuthal coordinate θ , and their end points lay alignedon opposite sides of the soliton membrane. Characteris-tic features of the system are depicted in Fig. 1(b)-(c):the axial phase jump for a given azimuthal angle, and theaxial density of the condensate after integration over thetransverse coordinates n ( z ) = (cid:82) | ψ ( z, r ⊥ , θ ) | r ⊥ dr ⊥ dθ times the scattering length. The stability of this andmore complex configurations within the GP theory willbe analyzed in Sections IV-V in realistic condensates.It is important to note that for scalar BECs, a hypo-thetical configuration with a single vortex line attachedto a dark soliton is not stationary. For such a case, thephase difference between the left and the right side of thedark soliton changes continuously along the azimuthalangle from 0 to π . As a consequence, the superfluid den-sity can not be zero all along the soliton plane, and thesuperfluid velocity is non-uniform, which makes this atransient configuration.It is also interesting to consider configurations contain-ing two nearby dark solitons, as the one presented in (b)(c) FIG. 1. Singly-charged open vortex lines in a BEC with µ = 10 (cid:126) ω ⊥ . (a) Semi-transparent density isocontour at 5%of maximum density around the soliton plane. (b) Axial phasefor a fixed value of the transverse polar angle θ = 0, and den-sity isocontour (inset on the left) of the inner part of the sys-tem (capturing the vortex-soliton junction) colored by phase.(c) Dimensionless axial density profile a n ( z ). Fig. 2 for a condensate having µ = 4 (cid:126) ω ⊥ . In this ar-rangement, the interaction between solitons is mediatedin the long range by the vortex lines. Fig. 2(a) showsthe isocontour at 5 % of maximum density colored ac-cording to the complex phase pattern produced by theinterplay of solitons and vortices. As can be deducedfrom Fig. 2(b)-(c), presenting the axial phase and den-sity of the system, the inner region between solitons ischaracterized by a state that differs in the phase fromthat of the outer region. B. Stability
States containing open vortex lines, as exemplified byFigs. 1 and 2, are dynamically stable as long as the darksoliton does not decay. As it is known, the decay of multi-dimensional dark solitons is produced by long-wavelengthmodes excited on the soliton membrane, through the so-called snaking instability [30]. However, such modes canbe prevented to appear by means of a tight transverse π pha s e ( θ = , z ) -10 -5 0 5 10 z / a ⊥ a n ( z ) (b)(c) FIG. 2. Two dark solitons connected by axisymmetric vorticesin a channeled condensate with µ = 4 (cid:126) ω ⊥ . (a) Perspectiveview of the density isocontour at 5 % of maximum density,colored by phase, after removing half condensate for bettervisualization. (b)-(c) Same as in Fig. 1. trap, which confines the system to a reduced cross sec-tion. In terms of the chemical potential, and in the ab-sence of a vortex, a channeled dark soliton is stable upto the value µ = 2 . (cid:126) ω ⊥ [29]. One could expect that,since the zero point energy introduced by an axisymmet-ric vortex in the harmonic trap increases in one energyquantum (cid:126) ω ⊥ relative to the ground state, the stabilitythreshold for dark solitons in the presence of the vortexwould increase by the same amount relative to the casewithout vortex. This argument leads to a threshold lo-calized at µ = 3 . (cid:126) ω ⊥ . As we will see below, it is areasonable estimate, since the unstable frequencies thatcan be found under such value are very small. We haveobserved that these latter instabilities are derived fromthe junction vortex–soliton (i.e. the boundary conditionsimposed by the soliton on the vortex ends), and possessslightly different amplitudes for different axial lengths ofthe computational domain considered. For particular ax-ial lengths, it is possible to find dynamically stable con-figurations with chemical potentials below the mentionedthreshold [as the case shown in Fig. 4(a)], and in the gen-eral case our results show that the system presents longlifetimes in the characteristic units of the trap.A quantitative analysis of the dynamical stability can µ / h / ω ⊥ I m ( ω ) / ω ⊥ q = 2q = 1q = 0q = 1 FIG. 3. Frequencies of unstable modes for open vortex lines asa function of the chemical potential (upper panel). The curvescorrespond to the three lowest azimuthal quantum numbers q of the excitation modes (see text). The q = 1 dashed lineaccounts for excitations derived from the vortex-soliton junc-tion, an example of which is given in the lower panel for acondensate with µ = 3 . (cid:126) ω ⊥ . The density isocontours (lowerpanel) correspond to the condensate (semi-transparent con-tour), and to two different unstable modes (colour contours)having zero (inner blue contour) and 2 (cid:126) (outer red toroidalcontours) angular momentum per particle. be done through the Bogoliubov equations (BE) for thelinear excitations of the condensate. To this aim, weintroduce the linear modes { u ( r ) , v ( r ) } with energy µ ± (cid:126) ω to perturb the equilibrium state, i.e. Ψ( r , t ) = ψ + (cid:80) ω ( u e − iωt + v ∗ e iωt ). After substitution in Eq. (3), andkeeping terms up to first order in the perturbation, weget (cid:0) H L + 2 g | ψ | (cid:1) u + gψ v = (cid:126) ω u , (41a) − g ( ψ ∗ ) u − (cid:0) H L + 2 g | ψ | (cid:1) v = (cid:126) ω v , (41b)where H L is the linear part of the Hamiltonian in Eq. (3),i.e. H L = − (cid:126) ∇ / m + V ( r ) − µ . These equations allowto identify the dynamical instabilities of the stationarystate ψ , which are associated to the existence of ω fre-quencies with non-vanishing imaginary parts.For the vortex state ψ ( z, θ, r ⊥ ) = ψ ( z, r ⊥ ) e iθ the BEEq. (41) read (cid:0) H L + 2 gψ (cid:1) u + gψ e i θ v = (cid:126) ω u , (42a) − gψ e − i θ u − (cid:0) H L + 2 gψ (cid:1) v = (cid:126) ω v . (42b)We search for the modes of the functional form { u ( z, r ⊥ ) e i ( q +1) θ , v ( z, r ⊥ ) e i ( q − θ } with q = 0 , ± , ± , ... . FIG. 4. Real time evolution of open vortex lines after addingrandom perturbations to the corresponding stationary states.(a) Semi-transparent density isocontours at 5 % of maximumdensity around the soliton position for a stable state with µ = 3 . (cid:126) ω ⊥ and ω ⊥ / π = 100 Hz. (b) Same as (a) for anunstable case containing two solitons, with µ = 5 . (cid:126) ω ⊥ and ω ⊥ / π = 71 . q = 1 mode (see text). For every time,along with the density isocontour of the whole system on theright, a narrow slice along the z -axis shows the density in agreyscale, with vortices in black, on the left. After adding and substracting both Eqs. (42), we achieve (cid:18) H + (cid:126) ( q + 1)2 mr ⊥ + g ± ψ (cid:19) f ± = ( (cid:126) ω − (cid:126) qmr ⊥ ) f ∓ , (43)where H = − (cid:126) ( ∂ z + ∂ r ⊥ + ∂ r ⊥ /r ⊥ ) / m + V ⊥ − µ , g ± =(2 ± g , and f ± ( z, r ⊥ ) = u ( z, r ⊥ ) ± v ( z, r ⊥ ).Bifurcations from the dark soliton state occur ifEqs. (43) have non-trivial solutions for ω = 0 [31, 32].In this case, and for q = 0 modes, Eqs. (43) are lin-ear Schr¨odinger equations for f ± , with effective poten-tials given by V ± = V ⊥ + (cid:126) / mr ⊥ + g ± ψ . In particu-lar, the equation for f − is identical to the GP equation,thus admitting the solution f G − = ψ ( z, r ⊥ ), apart froma global phase. This solution is the Goldstone mode as-sociated to the breaking of the continuous symmetry ofthe phase. Since f G − presents an axial node (the one ofthe soliton), there must be another solution to Eq. (43)without axial nodes, and then with lower axial energy.This energy difference, associated to the axial degrees offreedom, can be released for the excitation of transverse modes in the condensate, which can produce the decayof the dark soliton. Following a procedure parallel tothat used in Ref. [29], based in a separable ansatz for f ± within the Thomas-Fermi regime, the bifurcation pointsfor q = 0 can be estimated to appear at chemical po-tential values µ = √ p + 2) (cid:126) ω ⊥ , where p = 1 , , . . . is a radial quantum number. The pair ( p, q ) character-izes the corresponding transverse unstable modes local-ized at the soliton plane, and indicates the number ofradial and azimuthal nodal points, respectively. Specifi-cally, for ( p = 1 , q = 0) the predicted unstable mode willappear at µ = 5 . (cid:126) ω ⊥ , which is close to the value ( ≈ q = 1 , p = 0) appear before thosewith q = 0. The latter excitations introduce radial nodes( p (cid:54) = 0) in the soliton plane, whose energy cost is higherthan the kinetic energy excess ( ∝ (cid:126) q / m ) of the az-imuthal nodes generated by the modes with q = 1 , π a ⊥ , the excita-tion of transverse modes with q = 1 above µ = 3 . (cid:126) ω ⊥ marks the threshold for instability. As previously com-mented, the small bump extended between 3 . (cid:126) ω ⊥ and3 . (cid:126) ω ⊥ on the µ axis is due to small instabilities derivedfrom the vortex–soliton junction, and are represented bythe ( q = 1)–dashed line of Figure 3. To illustrate thispoint, the lower panel of Figure 3 depicts the density iso-contours of an open vortex state with µ = 3 . (cid:126) ω (semi-transparent contour) and its only unstable modes com-ing from the vortex–soliton junction (colored contours).Such unstable modes are not exclusively localized aroundthe junction. This specific instability ceases to act foran intermediate range of chemical potentials, where thesnaking instability makes its appearance through the q = 1 solid curve.We have cross-checked our results obtained from thelinear stability analysis, by evolving in real time the sta-tionary states ψ with different chemical potentials, afteradding a random Gaussian perturbation ψ → ψ + δψ .The upper panel of Fig. 4 presents an example of suchdynamics for a condensate with µ = 3 (cid:126) ω ⊥ , just belowthe instability threshold. The system remains nearly un-altered during the whole evolution, thus stable accordingto the linear analysis of Fig. 3; only smooth oscillationsdue to sound waves can be observed.For states with two solitons, as shown in Fig.2, when-ever the distance between solitons is large in compari-son with the healing length, the stability analysis followsthat of a single soliton. To illustrate their characteris-tic dynamics, we have selected an unstable system with µ = 5 . (cid:126) ω ⊥ , shown in the lower panel of Fig. 4. As canbe seen, at about 20 ms, the snaking instability starts tooperate at the planes of the solitons, while the vorticesbegin to lose their alignment. At later time, two newvortices appear as a remainder of the two initial solitons, (a)(b) (c)(d) FIG. 5. Half vortex rings attached to a dark soliton in anoblate condensate with µ = 6 (cid:126) ω ⊥ confined by a harmonictrap with aspect ratio λ = 1 .
4. Density isocontours at 5%of maximum density are shown for the semi-transparent topand side views (left panels) and the perspective view coloredby phase (top right). (d) Column density profile along the z -axis. along with manifest oscillations of the initial straight ax-isymmetric vortices. V. HALF VORTEX RING STATES
In this section we consider a variant layout with curvedopen vortex lines having both ends attached to the sameplanar dark soliton. The resulting structures bear someresemblance to the U-shaped vortices found in elongatedsystems [33], where the vortex lines bend near their endpoints. On the contrary, here we focus on half vortexrings. These objects have been studied in the realm ofclassical fluids [34], and can propagate long distances at-tached to the water surface without decaying (see forinstance [35], where they are shown to provide the mech-anism of motion for light insects). This configurationmimics an open string whose both ends are attached tothe same D-brane, which is another basic configurationof an open string.In order to generate half vortex rings in a BEC, weproceed with two steps. We first search for stable vortexrings, which are later split into two halves by imprint-ing a planar dark soliton. To this end, we have cho-sen near-spherical, harmonically trapped condensates,where the conditions for the stability of vortex rings havebeen analytically predicted for aspect ratios in the range
FIG. 6. Real time evolution, after adding a random Gaussianperturbation, of half vortex rings in oblate BECs. Semitrans-parent density isocontours at 5 % of maximum density areshown at different times for condensates with: (a) λ = 1 . µ = 6 (cid:126) ω z , and ω z / π = 11 . λ = 1 . µ = 8 . (cid:126) ω z ,and ω z / π = 57 . ≤ λ ≤ λ = 1 . µ = 6 (cid:126) ω z . The translucentdensity isocontours Fig. 5(a)-(b) show the condensatefrom two perpendicular views, and the colored densityisocontour Fig. 5(c) reproduces the resulting phase pat-tern from the interplay between the soliton (lying acrossthe x -coordinate) and the vortices. By integration of thedensity over the x -coordinate [panel Fig. 5(d)], it is pos-sible to observe the density depletion produced by thevortices at z = 0. As demonstrated in Fig. 6(a), showingthe real time evolution of this state obtained from thenumerical solution of the full GP equation (3), the halfvortex ring is a robust state, which is able to survive un-der perturbations during 100 ms in a harmonic trap with ω z / π = 11 . λ = 1 . µ = 8 . (cid:126) ω z , in atrap with ω z / π = 57 . µ we have observed a chaotic scenario at the later stagesof the half-vortex-ring decay, including the appearance ofnew vortex lines that split or reconnect.0 VI. DISCUSSION AND CONCLUSIONS
In this paper two variants of open vortex lines in scalarBECs have been reported, having either one end or twoends of a vortex attached to a planar soliton. In the lat-ter case, consisting of half vortex rings, we have shownthat these configurations present long lifetimes for typ-ical parameters of current experiments. For channeledcondensates containing straight vortices attached to soli-tons, we have demonstrated the existence of dynamicallystable states and identified the bifurcation points of thefirst unstable modes. In both variants, a necessary condi-tion for the corresponding configurations to be stationaryrequires that the vortex lines appear in pairs, as shownin Figs. 1, 2, 4, 5.The stationary configurations we considered in this pa-per are either stable (vortex lines attached to planar darksolitons) or long-lived (U-shaped vortex lines), which in-dicate their feasibility for BEC experimental realizations.To this aim, the well-established experimental techniquesin BECs developed for the observation of topological de-fects, such as optical phase imprinting [38, 39] or laserstirring [40], can be used. We propose here two possi-ble ways for elongated condensates adequately devised toprevent the snaking instability [41]. By means of phaseimprinting, both a transverse planar soliton and a pinnedstraight vortex along the z -axis could be simultaneouslyseeded. Alternatively, the laser stirring of the atomiccloud could be applied to drive the condensate into ro-tation around the z -axis, so that an energetically stable,singly charged vortex can be generated. After this, aplanar soliton could be imaged onto the condensate byphase imprinting, in order to split the initial vortex intotwo vortex lines attached to the soliton. With regard tothe realization of half vortex rings, the experimental pro-cedures are more elaborate, since they would involve the controlled generation of vortex rings [42], within a regimeof dynamical stability, followed by the phase imprintingof a soliton. These settings can open up a way to advancein the fairly unexplored domain of the interplay betweenvortices and solitons.To summarize, we have demonstrated that stationary,robust states composed of vortex lines attached to planardark solitons can be found in scalar BECs. Among them,we have reported on dynamically stable configurations inelongated systems for small values of the chemical poten-tial. Our results follow from the Gross-Pitaevskii theoryapplied to realistic condensates of ultracold gases, andallow the study of vortex lines with Dirichlet bound-ary conditions. In addition, we have shown that inthe hydrodynamic limit the dynamics of open vortexlines can be well characterized by the dual descriptionof the Gross-Pitaevskii theory, which can be viewed as a(3+1)-dimensional effective string theory. This connec-tion might pave the way to test some analytical predic-tions of string theory with experimental realizations inultracold gases. To advance in this way, the further anal-ysis of solutions to the equation of motion derived fromthe string functional Eq. (26) and the comparison withnumerical simulations from the Gross-Pitaevskii equationare required, which will be reported elsewhere [43]. ACKNOWLEDGMENTS
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