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- Two-dimensional multisolitons and azimuthons in Bose-Einstein condensates with attraction
- Generating vortex rings in Bose-Einstein condensates in the line-source approximation
- Three-dimensional vortex dynamics in Bose-Einstein condensates
- Three-dimensional solitons and vortices in dipolar Bose-Einstein condensates
- Controlled vortex-sound interactions in atomic Bose-Einstein condensates
- Vortex stability of interacting Bose-Einstein condensates confined in anisotropic harmonic
- Spontaneous soliton symmetry breaking in two-dimensional coupled Bose-Einstein condensates
- Vortex precession in Bose-Einstein condensates observations with filled and empty cores
- Three-dimensional dynamics of vortex lattice formation in Bose-Einstein condensates
- Two-component gap solitons in two- and one-dimensional Bose-Einstein condensates

Vortex stability in nearly two-dimensional Bose-Einstein condensates with attraction

arXiv:cond-mat/0603793v1 [cond-mat.other] 29 Mar 2006

Dumitru Mihalache1,2 , Dumitru Mazilu1,2 , Boris A. Malomed3 , and Falk Lederer1 ,

1

Institute of Solid State Theory and Theoretical Optics,

Friedrich-Schiller Universit¨t Jena, Max-Wien-Platz 1, D-077743 Jena, Germany a

2

National Institute of Physics and Nuclear Engineering,

Institute of Atomic Physics, Department of Theoretical Physics, P.O. Box MG-6, Bucharest, Romania and

3

Department of Interdisciplinary Studies, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel

Abstract

We perform accurate investigation of stability of localized vortices in an e?ectively twodimensional (“pancake-shaped”) trapped BEC with negative scattering length. The analysis combines computation of the stability eigenvalues and direct simulations. The states with vorticity S = 1 are stable in a third of their existence region, 0 < N < (1/3)Nmax , where N is the number of atoms, and Nmax

(S=1) (S=1)

is the corresponding collapse threshold. Stable vortices easily self-trap from ar(S=1)

bitrary initial con?gurations with embedded vorticity. In an adjacent interval, (1/3)Nmax

(S=1) 0.43Nmax , (S=1)

<N <

the unstable vortex periodically splits in two fragments and recombines. At N >

0.43Nmax , the fragments do not recombine, as each one collapses by itself. The results are compared with those in the full 3D Gross-Pitaevskii equation. In a moderately anisotropic 3D con√ ?guration, with the aspect ratio 10, the stability interval of the S = 1 vortices occupies ≈ 40% of their existence region, hence the 2D limit provides for a reasonable approximation in this case. For the isotropic 3D con?guration, the stability interval expands to 65% of the existence domain. Overall, the vorticity heightens the actual collapse threshold by a factor of up to 2. All vortices with S ≥ 2 are unstable.

PACS numbers: 03.75.Lm,03.65.Ge,05.45.Yv

1

I.

INTRODUCTION

Vortices are fundamental dynamical objects in Bose-Einstein condensates (BECs) [1]. They can be easily created in large numbers, forming vortex lattices [2]. A review of basic results for the BEC vortices can be found in Ref. [3]. In most works, vortices were studied in self-repulsive BECs, with a positive scattering length characterizing collisions between atoms. In the experiment, vortices are created stirring the condensate by a properly designed laser beam [1], or by imprinting an appropriate phase pattern onto a condensate trapped in a Jo?e-Pritchard magnetic trap [4]. Recently, more complex vortical structures were predicted in repulsive condensates, such as vortex dipoles [5] and quadrupoles [6], and patterns in the form of globally linked topological defects [7]. Vortices in repulsive BECs may be regarded as dark solitons on a ?nite background, similar to optical vortices in a self-defocusing dielectric medium [8]. A challenging issue is the study of vortex solitons (completely localized objects with embedded vorticity) in attractive BECs, with a negative scattering length [9]. In particular, attractive interaction occurs in the 7 Li condensate [10], where quasi-one-dimensional solitons were created [11]. The most natural setting for vortex solitons is a “pancake” condensate, strongly con?ned in one direction (z) and weakly trapped in the radial direction (r) in the transverse plane. In the experiment, the “pancake” is created by a superposition of a tight optical trap, with a con?nement thickness az , and a loose radial magnetic trap with a trapping frequency ?r in the transverse plane [12]. The theoretical description of the “pancake” con?guration assumes that the underlying three-dimensional (3D) Gross-Pitaevskii equation (GPE) may be reduced to its 2D counterpart. The condition necessary for the reduction is that the squared harmonic-oscillator length determined by the radial trapping frequency ?r , a(ho) r

2

= h/ (m?r ) ?

(1)

(m is the atomic mass), must be much larger than a2 , i.e., z ?r ? π 2 h/ ma2 . ? z (2)

For Li atoms trapped in the di?raction-limited gap between two parallel strongly repulsive (blue-detuned) light sheets, with az ? 2 ?m, this means ?r ? 10 KHz, which is readily met in the experiment, where con?ning frequencies are measured in tens of Hz. Condition 2

(2) also rule out bending instability of vortex cores in the pancake geometry (in the full 3D case, the instability results in decay of straight vortices into vortex rings, in repulsive condensates [13]), as az is much smaller than the vertical length necessary for the bending. The condition (2) for the applicability of the 2D approximation can be cast in another form, which sets a limit on the number of atoms N in the pancake-shaped condensate. As shown below, this form is h ? , (3) m?r |a|az where a is the (negative) scattering length. Taking the same value az = 2 ?m as above, N

< ?

together with experimentally relevant ?r = 10 Hz [in this case, Eq. (1) yields ar (3) yields N

< ?

(ho)

for Li atoms], and a = ?0.1 nm, which is typical for soliton experiments in 7 Li [10, 11], Eq. 104 .

? 10 ?m

Generally, vortex solitons are unstable against collapse (catastrophic self-compression of the wave function), and also against azimuthal perturbations which break the axial symmetry of the solitons. In fact, condition (3) provides for the stability of the quasi-2D solitons against the 3D collapse. Within the framework of the 2D description, both the ordinary (zero-vorticity) solitons and their vortical counterparts can be stabilized by a square-shaped optical lattice (OL) [14, 15]. The same OL can also support weakly localized vortex solitons of the gap type in a repulsive BEC [16, 17]; a quasi-1D lattice stabilizes ordinary solitons, but not vortices, in the 2D model with self-attraction [16, 18]. In attractive condensates, ordinary 3D solitons can be stabilized by low-dimensional OLs [viz., a quasi-2D lattice [16, 18, 19], or an axially symmetric one [20]], as well as by the full 3D lattice [21] (for a recent review of multidimensional solitons in optics and BEC, see Ref. [22]). In addition, 2D zero-vorticity solitons can be stabilized by means of the Feshbach-resonance management [23], which amounts to time-periodic modulation of the scattering length between positive and negative values [24]. However, the latter mechanism cannot stabilize vortex solitons [25], although it may inhibit the development of their instability [26]. While vortex solitons of the 2D GPE with a negative scattering length are strongly unstable in free space, they can be stabilized by the radial trapping potential, (m/2)?2 r 2 . r This issue was considered in an early work [27] and more recently [28, 29]. It is well known that the radial trap stabilizes zero-vorticity (S = 0) states against collapse, provided that the norm of the solution (which is proportional to the number of atoms in the condensate) is limited from above. In the 3D case with the isotropic radial trap, the maximum norm 3

depends on the corresponding trapping frequency ?r . This maximum norm was found by means of the variational approximation and from numerical computations [27, 28]. In contrast to that, the maximum 2D norm, Nmax , does not depend on ?r , being exactly equal to the norm of the corresponding Townes soliton in the free 2D space [30]. Similarly, the maximum value of the norm for the trapped fundamental vortex (one with S = 1), which does not lead to collapse, depends on ?r in the 3D case (the shape of the 3D trap which maximizes the collapse threshold under additional conditions was studied in Ref. [31]), but not in the 2D situation, where it coincides with the known collapse threshold, Nmax , of the fundamental vortex soliton in free space [32]. Formally, the vorticity gives rise to a dramatic increase of the 2D collapse threshold: Nmax

(S=1) (S=1) (S=0)

≈ 4Nmax . However, we will demonstrate that

(S=0)

the actual increase of the 2D stability threshold, Nthr , in the fundamental vortex is a more modest e?ect, amounting to Nthr ? 2Nmax ; in the remaining part of the existence interval for the vortex, Nthr < N < Nmax , the azimuthal instability splits the vortex into a pair of fragments, which then collapse intrinsically. It is relevant to mention that, in various models of nonlinear optics (see Refs. [33] and brief reviews [34]), vortex solitons, while being stable against radial perturbations, are easily split into fragments by an azimuthal instability. Nevertheless, azimuthally stable vortical solitons have been predicted too, in models with competing [35] or nonlocal nonlinearities [36], and in defocusing media with an imprinted cylindrical lattice [37]. In this work, we aim to investigate the stability of vortices in the 2D GPE equation with self-attraction and weak radial trapping potential, through computation of a full set of the corresponding stability eigenvalues. The results will be veri?ed by direct simulations. We conclude that all vortices with S ≥ 2 are unstable (as conjectured in Ref. [28]), while the fundamental ones (S = 1) are completely stable for 0 < N < Ncr ≈ (1/3)Nmax

(S=1) (S=1) (S=1) (S=0)

[in

Ref. [28], it was conjectured that Ncr ? (1/4)Nmax ]. At N = Ncr , the S = 1 vortex is destabilized by symmetry-breaking oscillatory perturbations. Further, in an adjacent interval, (1/3)Nmax

(S=1)

< N < 0.43Nmax

(S=1)

, the evolution of the unstable vortex does not

lead to collapse; instead, it features nearly periodic splittings in two fragments followed by their recombinations back into the vortex. This regular dynamical regime was not known before. In the remaining part of the existence interval, 0.43Nmax

(S=1) (S=1)

< N < Nmax , fragments

produced by the splitting of the vortex do not recombine, but rather blow up intrinsically, as their individual norm exceeds the collapse threshold for the Townes soliton. 4

We will also verify that the results obtained within the framework of the 2D model are valid indeed in the full 3D GPE with the corresponding anisotropic trap. In particular, in a moderately anisotropic 3D model, with the radial-con?nement length ar [see Eq. (1)] √ (ho) larger by a factor of 10 than its counterpart, az , in the orthogonal (tightly con?ned) direction, the relative size of the stability area inside the existence region for 3D solitons with the embedded vorticity S = 1 is ≈ 0.4, which is to be compared to the above-mentioned relative size of the stability region, 1/3, in the 2D limit. Approaching the isotropic limit, az

(ho) (ho)

/ar

(ho)

→ 1, the stability area expands, attaining the relative size ≈ 0.65.

The paper is organized as follows. In Section 2, we formulate the 2D model, brie?y recapitulating its derivation from the 3D GPE. In Section 3, we present basic results for the 2D model: shapes of the vortex states (actually, we ?nd them by continuation of exact states with a de?nite value of the angular momentum in the linear 2D Schr¨dinger equation), and o eigenvalues that determine their stability. In Section 4, we verify the predicted stability conditions by direct simulations. In Section 5, we consider the full 3D model with the anisotropic trap, and Section 5 concludes the paper.

II.

THE MODEL

We assume a self-attractive condensate which is loaded in a nonrotating loose trap in the (x, y) plane, and tightly con?ned in the z direction. The corresponding 3D GPE for the single-atom wave function Ψ is [38]: i? h ?Ψ = ?t ? 4π? 2 a 2 h h2 2 m 2 2 ? ?r x + y 2 + ?2 z 2 + ? + |Ψ| Ψ, z 2m 2 m (4)

where ?2 is the 3D Laplacian and a is the negative s-scattering length accounting for the attraction between atoms. In the pancake con?guration, the con?nement frequency ?z is assumed to be much larger than ?r . If the tight con?nement in the z-direction is provided by parallel light sheets repelling the atoms, the corresponding potential term, instead of m?2 z 2 /2, is a deep rectangular potential well in the region of 0 < z < az . z The 3D equation can be reduced to an e?ective GPE in two dimensions, provided that the transverse quantum pressure is much stronger than self-attraction. In other words, the energy of the atom in the ground state of the vertical trap, if the latter is taken as the deep

5

potential well of width az , E0 = (π? )2 / 2ma2 , h z (5)

must be much larger than the contribution of the attraction to the atomic chemical potential, ?? ? ?2π? 2 (a/m) n, where n ≡ |Ψ|2 is the 3D atom density [if the tight vertical h con?nement is provided by large ?z in Eq. (4), then the respective harmonic-oscillator length, az az =

(ho)

=

h/ (m?z ), taken as per Eq. (1), de?nes the e?ective con?nement size as ?

(ho) πaz ].

For the pancake-shaped con?guration in the weak radial trap, the density is determined, together with a characteristic radial size R of the pancake itself, by the condition of the balance between the radial quantum pressure, radial trapping force, and nonlinear selfattraction, which yields R? h ? m?r ≡ a(ho) , n ? . r m?r 8πa? h (6)

The substitution of expressions (5) and (6) in the above condition E0 ? |??|, which implies the tight transverse con?nement, leads to Eq. (2) that determines the range of the radial trapping frequencies where the 2D description is applicable. As a consequence of this condition, the aspect ratio of the pancake con?guration is always large, πR/az ? a?1 z h/ (m?r ) ? 1. A limitation on the number of atoms in the pancake state ?

can be derived by noting that it is determined by the product of the density, as given by Eq. (6), and the e?ective volume, ? πR2 az , which leads to Eq. (3). If N exceeds this limit, the condensate will start a nearly-2D collapse, which may later go over into a stronger collapse in three dimensions. The ?nal derivation of the e?ective 2D equation follows a well-known route: the 3D wave function is factorized, Ψ(x, y, z, t) = ψ(x, y, t)φ0(z), (7)

where φ0 = sin (πz/az ) is the ground-state wave function of the transverse state between two hard walls separated by the distance az . After that, averaging of the 3D equation (4) in the z direction leads to a normalized two-dimensional GPE [39], ?ψ 1 1 ?2 ?2 = ? + 2 + ?2 x2 + y 2 + g|ψ|2 ψ, (8) 2 ?t 2 ?x ?y 2 r √ where g ≡ 3πa (or g ≡ 2 2πa for the case of the parabolic trap in the vertical direction), i the transverse-ground-state energy E0 was subtracted from the chemical potential, while ?r 6

and t actually stand for m?r /? and (? /m) t. The variables x, y and |ψ|2 in Eq. (8) are h h measured in the same units as in Eq. (4). We then rede?ne them too, by ψ → |g|/?r ψ, √ (x, y) → ?r (x, y), and additionally rescale time by t → ?r t, to ?nally set ?r = ?g ≡ 1 (unless g = 0). Equation (8) conserves the 2D norm, N = E= 1 2 ?ψ ?x

2

|ψ(x, y, t)|2 dxdy, and energy (9)

∞ 0

?ψ + ?y

2

1 1 ? g|ψ|4 + ?2 (x2 + y 2 )|ψ|2 dxdy. 2 2 r

A straightforward consequence of Eq. (8) is the relationship E = ?N + π|g|

rR4 dr,

which is used to cross-check the accuracy of numerical results, see below. Further, the above rescalings and factorization (7) result in the following relation between the 2D norm and the number of atoms, N (which is de?ned as the norm of the 3D wave function in original units), N = az az N≡ N, 2|g| 6π|a| (10)

where the second equality follows from the above relation g = 3πa, if the vertical con?nement is provided by the repelling light sheets. If, instead, a tight parabolic potential is used to con?ne the condensate in the z-direction, the coe?cient in front of the last term in Eq. (10) √ (ho) (ho) is replaced by az / 2 2|a| , where az is the respective linear-oscillator length as de?ned above. The ?nal normalized form of the 3D GPE, which is also considered below, is ?Ψ 1 i = ? ?t 2 ?2 ?2 ?2 + + ?x2 ?y 2 ?z 2 + 1 2 x + y 2 + ?2 z 2 ? |Ψ|2 Ψ. 2 (11)

Here the nonlinearity coe?cient is again scaled to be ?1, and ?2 ≡ ?2 /?2 measures the z r relative strength of the tight con?nement in the vertical direction, the nearly-2D case corresponding to ?2 ? 1. The norm of the 3D solutions, N = related to the number of atoms as follows: N = where ar

(ho)

|Ψ(x, y, z, t)|2 dxdydz, is

h ? ar N ≡ N, m?r 8π|a| 8π|a|

(ho)

(12)

is the harmonic-oscillator length in the radial direction, de?ned as per Eq. (6).

The conserved energy of the 3D condensate is E= 1 2 ?Ψ ?x

2

?Ψ + ?y

2

?Ψ + ?z

2

1 1 ? |Ψ|4 + (x2 + y 2 + ?2 z 2 )|Ψ|2 dxdydz. 2 2 (13)

7

no free parameters, while its 3D counterpart, Eq. (11), contains one free coe?cient, ?2 .

Note that, after ?r and ?g were normalized to be 1, the two-dimensional GPE (8) features

III.

TWO-DIMENSIONAL VORTEX SOLUTIONS AND THEIR LINEAR STA-

BILITY

Stationary vortex solutions to Eq. (8) are looked for as ψ = R(r) exp (iSθ ? i?t) , (14)

with an integer vorticity S, real chemical potential ?, and function R(r) obeying the equation R′′ + r ?1 R′ + 2? ? S 2 r ?2 ? r 2 R ? 2gR3 = 0,

(15)

where the prime stands for d/dr. In the linear case, g = 0, Eq. (15) is tantamount to the usual stationary quantum-mechanical Schr¨dinger equation for the 2D harmonic oscillator, o which gives rise to an in?nite set of discrete eigenvalues, ? = j + k + 1, with j, k = 0, 1, 2, ... . In the Cartesian coordinates, the corresponding eigenfunctions are (recall we have set ? ≡ 1) ψjk (x, y, t) = e?i(j+k+1)t Φj (x)Φk (y),

(16)

where Φj (x) and Φk (x) are stationary wave functions of the 1D harmonic oscillator associated with the energy eigenvalues j + 1/2 and k + 1/2, respectively. The set of states (16) is degenerate, as all the states corresponding to a ?xed value of j + k appertain to the same eigenvalue. Making use of this degeneracy, the eigenfunctions may be rearranged into combinations with a de?nite value of the vorticity, which is S = j + k ≡ ? ? 1. In particular, the combination ψ0

(S=1)

(17)

≡ ψ10 (x, y, t) + iψ01 (x, y, t) ≡ r exp ?2it + iθ ? r 2 /2

(18)

is the wave function with the lowest value of the chemical potential for S = 1, ? ≡ j +k +1 = 2, and ψ0

(S=2)

≡ ψ20 (x, y, t) ? ψ02 (x, y, t) + 2iψ11 (x, y, t) ≡ r 2 exp ?3it + 2iθ ? r 2 /2 8

(19)

is the wave function with the lowest chemical potential, ? ≡ j + k + 1 = 3, for S = 2. Nonlinear solutions for vortices can be obtained by a continuation method, starting at g = 0 with expressions (18) and (19). Note that, pursuant to Eq. (15), the shift of ? due to the self-attractive nonlinearity (with g < 0) is negative, hence from the comparison with the linear result (17) it follows that ?(N = 0) < ?(N = 0) ≡ ?max = S + 1. (20)

To specify solutions to Eq. (15) in the nonlinear case (recall we set g = ?1), Eq. (15) was solved numerically. The dependences ?(N) obtained from the numerical solutions for S = 0, 1 and 2 are plotted in Fig. 1(a) (in a part, these dependences are tantamount to those reported earlier in Ref. [28]). Naturally, ? approaches the linear value ?max for N → 0, see Eq. (20), while for large negative ? the norm of the S = 0 state asymptotically approaches the value Nmax

(S=0)

≈ 5.85, which is the above-mentioned collapse threshold corresponding

(S)

to the Townes soliton [30]. Similarly, for the states with S ≥ 1 the asymptotic values N(? = ?∞) are the above-mentioned ones, Nmax , which determine the (formal) collapse threshold for the vortices. To study stability of the stationary solutions, we search for a perturbed solution of Eq. (8) as ψ(x, y, t) = [R(r) + u(r) exp(λt + iLθ) + v ? (r) exp(λ? t ? iLθ)] exp (iSθ ? i?t) , (21)

where (u, v) and λ are eigenmodes and the instability growth rate corresponding to an integer azimuthal index L of the perturbation. The linearization around the stationary solution leads to equations 1 1 ′′ u + r ?1 u′ ? (S + L)2 r ?2 u + ?u + R2 (v + 2u) ? r 2 u = 0, 2 2 1 1 ′′ ?iλv + v + r ?1 v ′ ? (S ? L)2 r ?2 v + ?v + R2 (u + 2v) ? r 2 v = 0, 2 2 iλu +

(22)

which were solved numerically, with boundary conditions demanding that u(r) and v(r) decay exponentially at r → ∞, and decay as r |S±L| at r → 0. The most important result following from the numerical computation of eigenvalues λ is that the vortex soliton with S = 1 is stable, i.e., real parts of all the eigenvalues remain equal to zero, only in a ?nite interval of values of the chemical potential adjacent to the linear limit [see Eq. (20)], ?max (S = 1) ≡ 2 > ? > ?cr ≈ 1.276, and outside this interval, 9

i.e., for ?∞ < ? < ?cr , the states with S = 1 are unstable. At the critical point (? = ?cr ), the corresponding critical norm is Ncr the region

(2D) 0 ≤ N < Ncr = 7.79, (2D)

= 7.79. Thus, the vortices with S = 1 are stable in

(23)

and in the remaining two-thirds of their existence interval, i.e.,

(S=1) 7.79 < N < Nmax ≈ 24.1,

(24)

they are unstable. The conclusion that “less nonlinear” solutions, i.e., ones with smaller N, are more stable is quite natural, as they continue the stable linear solutions which correspond to N = 0 [40]. The energy [see Eq. (9)] at ? = ?cr is Ecr = 12.913. The energy as a function of N is displayed, for the S = 1 soliton family, in Fig. 1(b). It is noteworthy that, while the ?(N) dependence in Fig. 1(a) is monotonic, its E(N) counterpart is not. To further characterize the (in)stability of the vortices with S = 1 and 2, in Fig. 2(a) we show the largest real part of the eigenvalues λL as a function of ? for di?erent values of the perturbation azimuthal index L, see Eq. (21). Extensive numerical calculations demonstrate that the perturbation mode that actually determines the stability border always has L = 2. In particular, Re λL=2

(S=2) (S)

is positive in the entire existence domain of the S = 2 vortices,

up to the linear limit, ?max (S = 2) = 3 [see Eq. (20)], thus making the vortices with S = 2 completely unstable and con?rming a conjecture put forward in Ref. [28]. All vortices with S > 2 are unstable too. For the fundamental vortex solitons with S = 1, the numerical calculation of the eigenvalues λL

(S=1)

was performed for values of the azimuthal index up to L = 6, in a broad range

(S=1)

of values of the soliton’s chemical potential, ?20 < ? < 2. It was found that the maximum value of Re λL is nonzero for L = 1, 2, 3, and this value is zero (up to the numerical accuracy) for higher values of the azimuthal index, L = 4, 5, and 6. To further check the accuracy of the numerical results, we have compared the eigenvalues produced by the numerical code running on 400 and 800 discretization points, respectively, and concluded that four signi?cant digits are identical in both cases. The bifurcation responsible for the destabilization of the S = 1 vortex at ? = ?cr is identi?ed by results displayed in Fig. 2(b). As seen from the ?gure, the bifurcation is of the Hamiltonian-Hopf type [41], i.e., it is accounted for by a quartet of complex eigenvalues, 10

(λ, λ? , ?λ, ?λ? ), cf. earlier reported results for solitons in scalar [42, 43] and vectorial [44, 45] models. The unstable quartet is generated by a collision of two pairs of imaginary (stable) eigenvalues, which is typical to the Hamiltonian-Hopf bifurcation.

IV.

DIRECT SIMULATIONS OF TWO-DIMENSIONAL STABLE AND UNSTA-

BLE VORTICES

To verify the above predictions concerning the stability of the vortex solitons in the 2D approximation, we have performed direct simulations of the perturbed evolution of vortex solitons in Eq. (8). The simulations were carried out by means of the Crank-Nicholson discretization scheme. The system of the corresponding nonlinear ?nite-di?erence equations was ?rst solved by means of the Picard iteration method [46], and the resulting linear system was then handled using the Gauss-Seidel iterative scheme. For good convergence we typically needed ?ve Picard iterations and eight Gauss-Seidel iterations. We employed a grid with 600 × 600 points (typical stepsizes were ?x = ?y = 0.025 and ?t = 0.0008). To test the stability of the S = 1 vortex, random noise was added to it at the initial moment – typically, with a 5% relative amplitude. Figure 3 shows how a vortex with S = 1, that was predicted above to be stable against small perturbations, completely recovers after the addition of the initial perturbations. To further explore the robustness of the vortex which is predicted to be stable, in Fig. 4 we display its self-trapping from an arbitrarily chosen Gaussian input with a nested vortex, u0 = Ar exp (?αr 2 + iθ), that has A = 1 and α = 1. The shape of the input is far from the exact stable vortex soliton. In this case, we used a grid with 391 × 391 points and stepsizes ?x = ?y = 0.028, ?t = 0.001. Note that, at the values of N corresponding to the de?nitely stable solitons displayed in Figs. 3 and 4, a rather inaccurate stability condition proposed in Ref. [28] for the S = 1 vortices would imply instability. For the S = 1 vortices which are predicted above to be unstable, we have found that, in the interval of 7.79 < N < 10.30, (25)

which is adjacent to stability region (23) and occupies ≈ 1/8 of the entire existence region of the vortices [cf. Eq. (24)], initial perturbations initiate regular quasi-periodic evolution, which features recurrent splittings of the soliton into two segments and their recombina11

tions. The regular character of the evolution in this case is also manifest in nearly periodic oscillations of the soliton’s amplitude, see typical examples in Figs. 5 and 6. This dynamical regime (which, to the best of our knowledge, has never been reported before) can be easily understood, as, unlike the known scenario of the instability development for vortex solitons in nonlinear-optical models in free space, where the splinters emerging after the breakup of the vortex soliton separate, moving away in tangential directions, in the present setting they cannot do it, being con?ned by the parabolic trap. We stress that Eq. (25 ) shows that the introduction of the vorticity makes it possible to heighten the actual collapse threshold for solitons in the trapped self-attractive condensate by a factor of two: from Nmax the spinless solitons to N ≈ 11 for their vortical counterparts. For larger values of the norm, 10.3 < N < Nmax

(S=1) (S=0)

= 5.85 for

≈ 24.1, the evolution of the unstable

vortex again starts with its splitting into two fragments; however, in this case they do not recombine, but rather quickly blow up (not shown here). The fact that the corresponding collapse threshold for the S = 1 vortex solitons, found at N = 10.3, is much smaller than the above-mentioned formal threshold, Nmax

(S=1)

≈ 24.1, can be readily explained. Indeed, it has

been shown above that the azimuthal instability sets in earlier than the radial instability that would directly lead to collapse. If the norm of each of the zero-vorticity fragments, into which the vortex is broken by the azimuthal instability, is close enough to the collapse threshold for the S = 0 solitons, i.e., Nmax

(S=0)

= 5.85, the fragments start collapsing and

therefore fail to recombine into the vortex, unlike the situation with the weak instability in interval (25). This explanation is consistent with the fact that the actual collapse threshold for the vortex, N = 10.3 , is fairly close to 2Nmax . As concerns the S = 2 solitons which were predicted above to be unstable at all ?nite values of N, the simulations (not shown here) demonstrate their irregular (chaotic) evolution for 0 < N < 10.4. For N > 10.4, they exhibit collapse, again through the splitting into a set of two S = 0 solitons which then blow up intrinsically. Lastly, we have also checked that the zero-vorticity solitons in Eq. (8) exhibit collapse exactly when N > Nmax was assumed above. At a late stage of the collapse, when the condensate will shrink to a size ? az , the above derivation of the 2D equation (8) from its 3D counterpart (4) will break down, and the initially quasi-2D collapse will switch into a faster 3D blow-up mode. Direct simulations of the full 3D equation (11) corroborate this expectation, as will be reported in detail elsewhere. 12

(S=0) (S=0)

= 5.85, as

V.

VORTEX SOLITONS IN THE THREE-DIMENSIONAL GROSS-PITAEVSKII

EQUATION

In this section, we aim to compare the above results, obtained in the framework of the 2D approximation, with what can be found from the full 3D equation (11), where the quasi-2D case corresponds to large ?. Systematic presentation of the results for the 3D model will be a subject of a separate work, while here we report, in a brief form, ?ndings which are most relevant for the comparison with the 2D approximation. Stationary solutions for the 3D solitons with embedded vorticity S and chemical potential ? are looked for in the form [cf. Eq. (14)] Ψ = R(r, z) exp (iSθ ? i?t), where r, z, θ are cylindrical coordinates, and the real function R obeys the equation ? 2 R 1 ?R ? 2 R S2 + + + 2? ? 2 ? r 2 ? ?2 z 2 R ? 2R3 = 0, ?r 2 r ?r ?z 2 r (26)

supplemented by obvious boundary conditions: R(r, z) → 0 when r → ∞ or |z| → ∞, and R ? r S for r → 0 at ?xed z (we again de?ne S to be ≥ 0). After ?nding a family of the vortices from numerical solution of Eq. (26), their stability was investigated via the computation of eigenvalues for small perturbations. First of all, the linear version of Eq. (26) can be considered similar to how it was done above for the 2D case: the solutions are built as linear combinations of products of the wave functions of the harmonic oscillator, cf. Eq. (16): √ Ψjkl (x, y, z, t) = e?i[j+k+1+(l+1/2)?]t Φj (x)Φk (y)Φl ( ?z), (27)

where l = 0, 1, 2, ... is the quantum number in the z-direction. Being interested here in the “?attest” states (ones closest to the pancake con?guration in the 3D space), we will set l = 0. Then, wave functions with given vorticity S in the (x, y) plane are constructed as combinations of factorized wave functions (27), subject to the same constraint as in the 2D case, i.e., j + k = S. From this restriction and from the form of the chemical potential in the linear-limit solution (27) with l = 0, ? = j + k + 1 + ?/2, it follows that the chemical potential of solutions to the full nonlinear equation (26), obtained as a continuation starting from the linear limit, obeys a limitation ? < ?max = S + 1 + ?/2, cf. Eq. (20). 13 (28)

Numerically found families of the 3D soliton solutions with S = 1 are characterized by ?(N) and E(N) curves displayed in Fig. 7, one for the anisotropic model, with ? = 10, and one for its isotropic counterpart, with ? = 1. In the former case [which, as a matter of fact, is not a strongly anisotropic one, as the corresponding aspect ratio of the con?nement √ lengths in the (x, y) and z directions is 10, pursuant to Eq. (1)], the existence and stability intervals of the fundamental vortices have been found to be, respectively, 0 < N < 14.46, and

(3D) 0 < N < Ncr |?=10 = 5.95

(29)

[cf. Eqs. (23) and (24) in the 2D model], hence the relative size of the stability area is ≈ 0.41. This should be compared to the relative size of the stability area in the 2D limit reported above, which is ≈ 1/3. For the isotropic model (? = 1), the existence and stability intervals are, respectively, 0 < N < 23, and

(3D) 0 < N < Ncr |?=1 ≈ 15,

(30)

their ratio being ≈ 0.65. Thus, we conclude that the 2D limit yields a reasonable approximation for the pancakeshaped con?gurations (the relative di?erence between its prediction and the direct numerical √ result for the moderately anisotropic model with the aspect ratio 10 is ? 20%). Another noteworthy conclusion is that the relative size of the stability area essentially increases as the 3D model approaches the isotropic con?guration; in the isotropic limit, the relative size of the stability region is almost twice as large as in the 2D limit. In terms of the maximum value of the norm, Nmax , up to which the vortex soliton remains stable, the di?erence is still bigger: Nmax for ? = 1 exceeds its counterpart for ? = 10 by a factor of ≈ 2.5. To further illustrate the e?ect of the degree of asymmetry on the stability of vortex solitons, in Fig. 8 we display the spectrum of unstable eigenvalues for the 3D solitons with S = 1, in the same two cases as considered above, i.e., the moderately anisotropic one with ? = 10, and isotropic with ? = 1. Comparison of this ?gure with the respective picture from Fig. 2(a) for the 2D model shows a gradual deformation of the spectrum with the transition from the 2D limit to the isotropic 3D con?guration. An important observation is that the transition from the 2D approximation to the full 3D model does not lead to any new instability eigenmode (at least, up to the isotropic limit, ? = 1); incidentally, the bending instability of the vortex core does not occur either. 14

As well as in the 2D case, vortex solitons with S ≥ 2 [at least, the “?attest” ones, corresponding to l = 0 in the linear limit, see Eq. (27)] are all unstable. As concerns the fundamental solitons with S = 0, we could easily reproduce a known result stating that, as well as in the 2D model, they are stable within their full existence region, which is N < 3.8 for ? = 10, and N < 7.3 for ? = 1. Comparing these values with ones in Eqs. (29) and (30), we conclude that the introduction of the vorticity enhances the stability limit by a factor of 1.6 for ? = 10, and by 2 for ? = 1. Recall that, in the 2D model, the same enhancement factor was 1.3 [if its de?nition does not include the extra region (25) of the quasi-stable dynamical states]; this again demonstrates a general trend to the expansion of stability regions with the transition from the 2D limit to the fully isotropic 3D case.

VI.

CONCLUSION

We have reported results of accurate investigation of the stability of localized vortices in Bose-Einstein condensates with self-attraction, trapped in the nearly two-dimensional con?guration. We have brie?y recapitulated the derivation of the e?ective 2D Gross-Pitaevskii equation from its full 3D counterpart, and demonstrated that, in the 2D limit, the vortices with S = 1 are stable against azimuthal splitting in a third of their existence region, i.e., for 0 < N < (1/3)Nmax , in terms of the solution’s norm (which is proportional to the number of atoms trapped in the vortex soliton). In the adjacent interval, (1/3)Nmax

(S=1) (S=1) (S=1)

< N <

0.43Nmax , the unstable vortex does not collapse, but rather demonstrates stable quasiperiodic dynamics, featuring cycles of splitting into two fragments and recombination, which is a novel dynamical regime. For N > 0.43Nmax , the two segments produced by the splitting fail to recombine, as they collapse intrinsically. The stable vortex soliton created in the pancake-shaped condensate of 7 Li may contain up to 10, 000 atoms. The results were checked against direct simulations of the 2D Gross-Pitaevskii equation, and, which is crucially important for the veri?cation of their physical relevance, they were compared with the stability analysis of localized vortices in the full 3D equation. It was √ concluded that, for the moderately anisotropic 3D con?guration with the aspect ratio 10 ≈ 3.2, the stability interval of the S = 1 vortices occupies ≈ 40% of their existence region (hence, the 2D limit provides for quite a reasonable approximation). In the 3D isotropic limit, the stability interval expands to 65% of the existence domain. The enhancement factor 15

(S=1)

for the collapse threshold, caused by the introduction of vorticity S = 1, was found too; in the 3D isotropic con?guration it attains the value of 2. All higher-order localized vortices, with S ≥ 2, are completely unstable, in the 2D limit and in the full 3D Gross-Pitaevskii equation alike. Support from Deutsche Forschungsgemeinschaft (DFG), Bonn, is acknowledged. The work of B.A.M. was partially supported by the Israel Science Foundation through the Centerof-Excellence grant No. 8006/03.

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FIG. 1: Chemical potential ? (a) and energy E (b) versus normalized number of atoms N for solutions with vorticities S = 0, 1, and 2, as found from Eq. (15) with g ≡ ?1. Solid and dashed lines show stable and unstable branches of the solutions (see below). The arrow indicates the point where the S = 1 vortices loose their stability. FIG. 2: (a) The real part of the perturbation growth rate λ for di?erent azimuthal indices L. (b) The real and imaginary parts (solid and dashed lines, respectively) of the eigenvalues responsible for the Hamiltonian-Hopf bifurcation which destabilizes the vortices with S = 1 at the critical point, ? = ?cr ≈ 1.276. This value and the bifurcation point are marked by arrows. FIG. 3: Grey-scale plots illustrating recovery of the perturbed stable vortex with S = 1 for ? = 1.4. (a,b): Intensity and phase distributions in the initial con?guration including random noise. (c,d): The same in the self-cleaned vortex soliton at t = 120. The norms of the input and output solitons are N = 6.641 and N = 6.629, respectively (the small loss of the norm is due to absorbers placed at borders of the integration domain). FIG. 4: Formation of a stable vortex with S = 1 from a Gaussian input. (a,b): The intensity and phase distributions in the initial Gaussian with a nested vortex. (c,d): The intensity and phase distributions in the output (t = 700). The input and output con?gurations have the norm, respectively, N = 6.406 and N = 6.392. FIG. 5: Quasi-periodic evolution of the amplitude of the unstable S = 1 vortex for ? = 1.2. The labels a, b, c and d point at values of the amplitude at t = 0, 100, 140, and 180 (see Fig. 6). FIG. 6: Regular evolution of the unstable vortex with S = 1 for ? = 1.2 initiated by random noise, which shows periodic splitting into two fragments and their subsequent recombination: (a) t = 0, (b) t = 100, (c) t = 140, and (d) t = 180. Here, values of the input and output norm are N = 8.476 (at t = 0) and N = 8.460 (at t = 240). FIG. 7: The chemical potential (a) and energy (b) for families of three-dimensional solitons with the embedded vorticity S = 1 vs. the normalized number of atoms, with the moderately anisotropic (? = 10) and isotropic (? = 1) con?ning potentials. The continuous and dashed parts of the curves show, respectively, stable and unstable parts of the families.

20

FIG. 8: The same as in Fig. 2(a), but for vortex solitons with S = 1 in the full three-dimensional model: (a) ? = 10; (b) ? = 1. The arrows show the edge of the existence regions of the solitons on the scale of the chemical potential, as per Eq. (28).

21

5 0 -5 -10 -15 -20 0

(a) S=2 S=1 E S=0

60

(b)

40

?

S=2

20

S=1 S=0

10 20 30 40

10

20

30

40

0 0

N

N

6

(a)

3

(b) S=1 L=2

Re λ, Im λ

Re λ

S=2, L=2 S=1, L=1 S=1, L=2 S=1, L=3

0

0 -6

-3

0

3

-3 -2

-1

0

1

2

?

?

7.5

(a)

(a)

7.5

(b)

0.0

0.0

Y

-7.5 -7.5 7.5

0.0 X

7.5

Y

-7.5 -7.5 7.5

0.0 X

7.5

(c)

(d)

0.0

(c)

0.0

Y

-7.5 -7.5

0.0 X

7.5

Y

-7.5 -7.5

0.0 X

7.5

5

(a)

(a)

5

(b)

0

0

Y

-5 -5 5

Y 0 X 5

-5 -5 5

0 X

5

(c)

(d)

0

0

Y

-5 -5

Y 0 X 5

-5 -5

0 X

5

2.0

b

d

amplitude

1.5

a

1.0 0 40 80

c

120 160 200 240 time

D

E

F

G

<

<

<

<

;

;

;

;

10

(a)

?=10 ?=1

150

(b) S=1

S=1

100

0

?

-10

E

50 ?=10

?=1

-20 0 0 0

10

20

30

10

20

30

N

N

40

(a)

?=10

20

(b)

?=1

30

Re λ

20

Re λ

10

10

L=1

L=3 L=2

L=1

L=3 L=2

0 -5

0

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10

0 -3

-2

-1

0

1

2

3

?

?