Dec 22, 2008 - dimensional 1D prolate âcigar-shapedâ traps, and the for- mation of solitons in a post-collapse state, have been reported 2â4 . In the mean-field ...
PHYSICAL REVIEW A 78, 063615 共2008兲
Bright solitons in the one-dimensional discrete Gross-Pitaevskii equation with dipole-dipole interactions Goran Gligorić,1 Aleksandra Maluckov,2 Ljupčo Hadžievski,1 and Boris A. Malomed3 1
Vinča Institute of Nuclear Sciences, P.O. Box 522, 11001 Belgrade, Serbia Faculty of Sciences and Mathematics, University of Niš, P.O. Box 224, 18001 Niš, Serbia 3 Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel 共Received 4 September 2008; published 22 December 2008兲 2
A model of the Bose-Einstein condensate of dipolar atoms, confined in a combination of a cigar-shaped trap and deep optical lattice acting in the axial direction, is introduced, taking into regard the dipole-dipole 共DD兲 and contact interactions. The model is based on the discrete nonlinear Schrödinger equation with an additional nonlocal term accounting for the DD interactions. The existence and stability of fundamental unstaggered solitons are studied for attractive and repulsive signs of both the local and nonlocal interactions. The DD forces strongly affect the shape and stability of on-site and intersite discrete solitons. The corresponding existence and stability regions in the parametric space are summarized in the form of diagrams, which feature a multiple stability exchange between the on-site and intersite families; in the limit of the dominating DD attraction, the on-site solitons are stable, while their intersite counterparts are not. We also demonstrate that the DD interactions reduce the Peierls-Nabarro barrier and enhance the mobility of the discrete solitons. DOI: 10.1103/PhysRevA.78.063615
PACS number共s兲: 03.75.Lm, 05.45.Yv
I. INTRODUCTION
Experimental studies of Bose-Einstein condensation 共BEC兲 in dilute gases of alkali metals and hydrogen, initiated in renowned works 关1兴, clearly demonstrate that interactions between condensed atoms is the origin of most phenomena observed in BEC. Usually, the interaction is determined by the contact potential, which is characterized by the s-wave scattering length, as. In the case of attraction between atoms 共as ⬍ 0兲, spectacular effects, such as the modulational instability 共MI兲, robust solitons and soliton trains in nearly onedimensional 共1D兲 prolate 共“cigar-shaped”兲 traps, and the formation of solitons in a post-collapse state, have been reported 关2–4兴. In the mean-field approximation, the condensate’s dynamics obeys the 3D Gross-Pitaevskii equation 共GPE兲, from which an effective 1D equation can be derived, in various settings, for the condensate trapped in a prolate trap 关5–10兴. The derivation relies on the approximate factorization of the 3D wave function into a product of a “frozen” 2D function, which corresponds to the ground state of the harmonic-oscillator potential responsible for the transverse confinement, and a freely varying axial 共1D兲 wave function. In the limit of a very deep optical-lattice 共OL兲 potential acting in the axial direction, the respective one-dimensional GPE with cubic nonlinearity may be reduced to the discrete nonlinear Schrödinger 共DNLS兲 equation 关11–13兴. It was predicted 关14兴 and eventually demonstrated experimentally 关15,16兴 that a condensate featuring magnetic dipole-dipole 共DD兲 interactions can be created in a vapor of chromium atoms 共it was also proposed that electric DD interactions may be induced by the application of a strong polarizing electric field to the condensate 关17兴兲. The DD attraction and repulsion between atoms may greatly enrich the variety of phenomena in dilute quantum gases, due to the long-range and anisotropic character of these interactions 关18–20兴. In particular, the DD interaction gives rise to a 1050-2947/2008/78共6兲/063615共10兲
d-wave mode of the collapse, different from the usual isotropic collapse driven by contact attractive interactions 关21兴. Stable isotropic 关22兴 and anisotropic 关23兴 solitons, supported by the competition between the attractive DD interactions and local repulsion, have been predicted too, in effectively two-dimensional 共“pancake-shaped”兲 settings. Nonlinear band-gap structure in the 1D dipolar condensate trapped in the OL was recently considered in Ref. 关24兴, and phase solitons of the sine-Gordon type were predicted in a spinor dipolar condensate placed in the same setting 关25兴. The objective of this work is to analyze the formation and dynamics of discrete bright solitons in the effectively 1D condensate of dipolar atoms trapped in the strong OL potential, taking into regard both the contact and DD interactions. This setting is modeled by the DNLS equation with an additional nonlocal term accounting for the DD interactions. In this work, we focus on unstaggered positive discrete solitons, while their staggered counterparts, featuring the alternation of the sign of the local discrete field, will be considered elsewhere. The paper is structured as follows. The model is formulated in Sec. II. Families of on-site and intersite solitons are presented in Sec. III. To this end, regions where unstaggered discrete solitons may be expected are predicted by means of analysis of the modulation instability. Then, the stability of the solitons in the presence of the DD attraction or repulsion is analyzed. In Sec. IV, we investigate the influence of the DD interactions on mobility of the unstaggered solitons. The paper is concluded by Sec. V. II. THE MODEL A. General analysis
The continuous Gross-Pitaevskii equation 共GPE兲, which includes both contact and DD interactions in the 3D geometry, is
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©2008 The American Physical Society
PHYSICAL REVIEW A 78, 063615 共2008兲
GLIGORIĆ et al.
iប
冋
共r兲 ប 2 2 4 ប 2a s = − ⵜ + 兩共r兲兩2 t 2m r m +g
冕
i
册
冋
共z,t兲 1 2 = − − V0 cos共2qz兲 + 兩共z,t兲兩2 2 z2 t
1 − 3 cos2 兩共r⬘兲兩2dr⬘ + U共r兲 共r兲, 兩r − r⬘兩3
+g 共1兲
where m is the atomic mass, as ⬎ 0 共as ⬍ 0兲 is the s-wave scattering length accounting for the repulsive 共attractive兲 contact interactions, is the angle between vector 共r − r⬘兲 and the orientation of dipoles, and U共r兲 is the external potential. The coefficient g, which accounts for the DD interaction, is positive 共g ⬍ 0 was considered in Ref. 关22兴, where the negative sign was necessary to predict an isotropic 2D soliton; the effective sign of g can be reversed, in principle, by means of a rapidly rotating magnetic field 关26兴兲. It is assumed that all dipoles are strictly parallel, i.e., they are polarized by a strong external field, while the polarization direction itself may be selected by adjusting the orientation of the external field. In the effectively 1D case, the reduced and properly rescaled GPE, which includes the OL potential with strength −V0 and period / q, is
冋
共z,t兲 1 2 = − − V0 cos共2qz兲 + 兩共z,t兲兩2 i 2 z2 t +g
冕
+⬁
−⬁
册
1 − cos2 兩共z⬘兲兩2dz⬘ 共z兲. 兩z − z⬘兩3
L=
冕
冋
⬁
−⬁
+
1 兩共z兲兩4 + Gdd兩共z兲兩2 2 2
共2兲
冕
共3兲 −
+⬁
2n共z兲dz ⬅ 0,
冕
+⬁
册
冕
⬘ 共z兲dz ⬅ ⬘ , n⬘共z兲n+1
共7兲
共8兲
+⬁
兩n⬘共z兲兩2dz ⬅ 1 ,
冕
+⬁
兩n共z兲兩4dz ⬅ 2 , 共9兲
−⬁
冕 冕 +⬁
共4兲
共z,t兲 1 2 = − − V0 cos共2qz兲 + 兩共z,t兲兩2 2 z2 t
+⬁
−⬁
dz⬘兩n共z兲兩2兩n⬘共z⬘兲兩2 ⬇ 20
−⬁
共10兲
关in Eq. 共10兲, direct overlap between separated local wave functions is neglected兴. Substituting expansion 共8兲 in Lagrangian 共7兲 and performing the integration of tightly localized expression with regard to definitions 共9兲 and approximation 共10兲, we arrive at the following effective discrete Lagrangian, which, unlike its counterpart derived in the local setting 关12兴, includes terms accounting for the long-range DD interactions: Leff = 兺 n
冋冉
冊冉
−
⬘ * + F*F 兲 + 兩F 兩4 共FnFn+1 n n n+1 2 2
while the latter case corresponds to repulsive DD interaction, with Eq. 共2兲 taking the following form:
+
兩Fn⬘兩2 Gdd20 2 兩F 兩 . n 兺 兩n − n⬘兩3 2共/q兲3 n ⫽n
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冊
dFn* i0 dFn 1 − Fn* + 兩Fn兩2 − V00 兩Fn兩2 Fn 2 2 dt dt
共5兲
− 2g
册
−⬁
dz
where N is the number of atoms in the condensate, and ⬜ is the transverse-trapping frequency 关27兴. Two distinct cases are of basic interest, as concerns the 1D geometry: when the dipoles are polarized parallel to the z axis, with = 0, or perpendicular to it, with = / 2. The former case corresponds to attractive DD interaction, and Eq. 共2兲 becomes
兩共z⬘兲兩2 共z兲, 兩z − z⬘兩3
−⬁
兩共z⬘兲兩2 dz, 兩z − z⬘兩3
Fn being the corresponding complex amplitude. We fix the normalizations by imposing the condition that the amplitude 共largest value兲 of each local mode n共z兲 is 1, cf. Ref. 关12兴. Then, we define their norms and associated overlap integrals as
and the coefficient of the contact interaction is given by the well-known expression
冕
dz⬘
n
−⬁
冋
⬁
共z,t兲 = 兺 Fn共t兲n共z兲,
−⬁
i
冕
where Gdd = −2g and Gdd = g pertain to the attractive and repulsive DD interactions, respectively, and the asterisk denotes the complex conjugation. Assuming deep OL potential and following the derivation of the DNLS equations from the continuous ones in local models 关12,28兴, we approximate the wave function by a superposition of modes n共z兲 共such as Wannier functions兲, which are tightly confined to a vicinity of respective local potential minima 共zn = n / q兲,
+⬁
= 2兩as兩N/冑ប/m⬜ ,
共6兲
1 i 共t* − *t兲 + 兩z兩2 − V0 cos共2qz兲兩共z兲兩2 2 2
−⬁
兩共z兲兩2dz = 1,
册
兩共z⬘兲兩2 共z兲. 兩z − z⬘兩3
Both equations can be derived from the Lagrangian,
Here, the 1D 共axial兲 wave function is subject to normalization
冕
冕
⬘
册
共11兲
BRIGHT SOLITONS IN THE ONE-DIMENSIONAL …
PHYSICAL REVIEW A 78, 063615 共2008兲
Next, defining an offset of the chemical potential, Fn共t兲 ⬅ f n共t兲exp兵i关V0 − 1 / 共20兲 + ⬘ / 0兴t其, we derive the evolution equations for discrete amplitudes f n, as the Euler-Lagrange equations generated by Lagrangian 共11兲,
eigenvalue ⍀ has to be found from linearized equations for the small perturbations. Straightforward calculations yield the following dispersion relations for ⍀:
兩f n⬘兩2 df n 2 i = − C共f n+1 + f n−1 − 2f n兲 + ␥兩f n兩 f n + gdd f n 兺 , dt 兩n − n⬘兩3 n⫽n
⍀2 = 16C2 sin4
⬘
共12兲 with additional definitions
⬘ , C⬅ 20
2 ␥⬅ , 0
冉冊
1 q gdd ⬅ 2
3
1 − Ꭽ⌫ Gdd0 .
共13兲
Equation 共12兲 conserves two dynamical invariants: norm N = 兺n兩f n兩2 and the Hamiltonian,
冋
册
兩f n⬘兩2兩f n兩2 ␥ H = 兺 C兩f n − f n+1兩2 + 兩f n兩4 + gdd 兺 . 2 兩n − n⬘兩3 n n⫽n ⬘
共14兲 Normalization condition 共3兲 in the framework of the present approximation takes the form of 兺n兩f n兩2 = 1 / 0, hence the norm of the stationary function is P ⬅ 兺 兩un兩2 = ␥/0 .
共15兲
⫻
冉冊 q 2
兺
n⫽n⬘
冤
1+
2C sin2
冉冊 q 2
兩n⬘ − n兩−3 exp关iq共n⬘ − n兲兴 1 − 2共3兲Ꭽ⌫
冥
,
共18兲
⬁ m−3 ⬇ 1.20 is the respective value of Riewhere 共3兲⬅兺m=1 mann’s zeta function. Unstaggered solitons can exist only when the uniform solution is modulationally unstable, i.e., when ⍀2 ⬍ 0. This condition is necessary but not sufficient for the existence of the solitons. Analysis of expression 共18兲 for the case of the local attraction shows that the MI takes place, i.e., the unstaggered solitons may exist, provided that the chemical potential of the uniform solution is negative, ⬍ 0, and the ratio of the DD and contact interaction strengths, ⌫, satisfies the following condition:
n
Taking into consideration definitions 共4兲 and 共13兲, the total number of atoms can be expressed in terms of norm 共15兲, cf. Ref. 关12兴, N=
20 P
2兩as兩2冑m⬜/ប
Our aim is to construct families of fundamental unstaggered soliton solutions to Eq. 共12兲, in the cases of attractive and repulsive contact interactions 关with Ꭽ ⬅ sgn共␥兲 ⬍ 0 and Ꭽ ⬎ 0, respectively兴, and in the presence of attractive 共gdd ⬍ 0兲 or repulsive 共gdd ⬎ 0兲 DD interactions. Replacing f n by ˜f ⬅ f / 冑兩␥兩, we cast Eq. 共12兲 in the following form: n n 兩f˜n⬘兩2 df˜n = − C共f˜n+1 + ˜f n−1 − 2f˜n兲 + Ꭽ兩f˜n兩2˜f n − ⌫f˜n 兺 , dt 兩n − n⬘兩3 n⫽n
⌫⬎
共19兲
1 . 2共3兲
共20兲
Thus, with the attractive contact interaction, unstaggered solitons can exist if the DD interaction is attractive too, ⌫ ⬎ 0, or if it is repulsive but not too strong, as demanded by condition 共19兲: 0 ⬍ −⌫ ⬍ 1 / 关2共3兲兴. On the other hand, in the case of the local repulsion, solitons can be expected only if the DD interaction is attractive and sufficiently strong, to satisfy condition 共20兲 (or −gdd ⬅ 兩␥兩⌫ ⬎ 兩␥兩 / 关2共3兲兴, in terms of Eq. 共17兲). Stationary solutions to Eq. 共17兲, with chemical potential , are sought as ˜f n共t兲 = un exp共−it兲, with real discrete function un satisfying a stationary equation, 2
⬘
un = − C共un+1 + un−1 − 2un兲 +
共17兲 where ⌫ ⬅ −gdd / 兩␥兩 measures the strength of the DD interaction versus the contact interaction. To find the parameter regions where fundamental unstaggered solitons can exist, we first consider the MI of real uniform solutions. Following the standard procedure, a small complex perturbation is added to the uniform solution, Ue−it, thus replacing it by 共U + ␦Un兲e−it, where 兩␦Un兩 Ⰶ U. The complex perturbation eigenmode is sought as ␦Un = 共␦an + i␦bn兲exp共iqn兲exp共i⍀t兲, with real ␦an, ␦bn, and real perturbation wave number q, while a 共generally complex兲
1 . 2共3兲
In the case of the local repulsion, condition 共19兲 is replaced by
共16兲
.
III. UNSTAGGERED SOLITONS
i
⌫⬎−
Ꭽu3n
− ⌫un
兺
n⬘⫽n
u n⬘
兩n − n⬘兩3
.
共21兲 Families of fundamental unstaggered solitons of on-site and intersite types were constructed as numerical solutions to Eq. 共21兲, by means of an algorithm based on a modified Powell minimization method, which uses a finite-difference expression for the underlying Jacobian. Initial Ansätze used to construct on-site and intersite discrete solitons were taken, respectively, as 兵u共0兲 n 其 = 共. . . , 0 , A , 0 , . . . 兲 and 共. . . , 0 , A , A , 0 , . . . 兲, where A is a real constant obtained from Eq. 共21兲 in the
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GLIGORIĆ et al. 0.9
0.9
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10 5
ev
ev 50
n
60
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FIG. 1. Typical examples of on-site 共a兲 and intersite 共b兲 unstaggered solitons with equal norms P found with zero 共⌫ = 0兲 共open squares兲, attractive 共⌫ = 1兲 共open circles兲, and repulsive 共⌫ = −0.2兲 共open triangles兲 dipole-dipole interactions. The intersite coupling constant is C = 0.8.
corresponding approximation. Results reported below were obtained in the lattice composed of 101 or 100 sites for the on-site and intersite configurations, respectively. It was checked that the results did not alter if an essentially larger lattice was used. A. Unstaggered solitons for the case of the contact attraction, in the presence of the dipole interactions
Fixing the contact interaction to be attractive, examples of on-site and intersite solitons with equal norms P are displayed in Fig. 1 for the cases of zero, attractive, and repulsive DD interaction, with ⌫ = 0, 1, and −0.2, respectively. In agreement with predictions of the above analysis, for the attractive DD interaction we have found no constraint on its strength ⌫ necessary for the existence of unstaggered solitons, but for the repulsive DD interaction, no solitons could be found for ⌫ ⬍ −0.4. Numerical results demonstrate an effect of the long-range interaction on the shape of the solitons: they are narrower and feature a higher amplitude if the DD interaction is attractive; in the opposite case, the solitons are broader and have a smaller amplitude, in comparison with their counterparts in the local model 共with ⌫ = 0兲; see Fig. 1. These conclusions comply with previously published findings 关20兴. The linearization of Eq. 共17兲 with Ꭽ = −1 for small perturbations around the solitons, and numerical computation of the corresponding eigenvalues, yield stability spectra for the on-site fundamental solitons, which are displayed in Figs. 2 and 3 for two values of the intersite coupling constant, C = 0.2 and 1.2, respectively. Each figure presents the results for zero 共a兲, repulsive 共b兲, and attractive 共c,d兲 DD interactions. It is observed that, in the absence of the long-range interactions, the on-site solitons are stable at all values of the parameters, as they should be 关Figs. 2共a兲 and 3共a兲兴. In the presence of the repulsive DD interaction, the on-site solitons remain stable, as seen from Figs. 2共b兲 and 3共b兲. On the other hand, in the presence of the attractive DD interaction, whose strength is equal to that of the local attraction contact interaction, i.e., ⌫ = 1, the on-site solitons are unstable in the entire parameter space 关Figs. 2共c兲 and 3共c兲兴. However, if the DD attraction is stronger than the contact attractive interaction, the on-site solitons are stable again; see Figs. 2共d兲 and 3共d兲.
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-4
-2
15
0
-8
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-4
-2
15
(c)
10
10
5
5
0
(d)
0
0
-5
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ev
40
-10
-8
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0
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0
FIG. 2. The eigenvalue 共“ev”兲 spectrum for on-site unstaggered solitons, as found with zero, ⌫ = 0 共a兲, repulsive, ⌫ = −0.2 共b兲, moderately strong attractive, ⌫ = 1 共c兲, and very strong attractive, ⌫ = 7 共d兲, dipole-dipole interactions, for C = 0.2. In this and similar figures below, curves depict pure imaginary 共neutrally stable兲 eigenvalues with largest absolute values, while symbols represent the pure real 共unstable兲 eigenvalues.
The stability properties of intersite solitons are opposite. Namely, without the DD interaction, all intersite solitons are unstable, as they must be 关Figs. 4共a兲 and 5共a兲兴, and they remain unstable in the presence of the repulsive DD interaction; see Figs. 4共b兲 and 5共b兲. However, Figs. 4共c兲 and 5共c兲 demonstrate that the DD attraction makes the intersite solitons stable—precisely when their on-site counterparts are unstable. With further growth of the attractive DD interaction, the intersite solitons are destabilized again, as seen in Figs. 4共d兲 and 5共d兲. A better insight into the exchange of stability between the on-site and intersite solitons, in the presence of the attractive 15
15
(a)
10
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ev
n
60
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(c)
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ev
50
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ev
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0.3
0.0
0.0
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ev
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(b)
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ev
ampl.
ampl.
0.6
15
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FIG. 3. The same as in Fig. 2, but for C = 1.2 and with ⌫ = 6 in panel 共d兲.
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BRIGHT SOLITONS IN THE ONE-DIMENSIONAL … 15
(a)
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(b)
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0
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FIG. 4. The same as in Fig. 2, but for intersite unstaggered solitons.
15 10
10
DD interaction, is provided by looking at the picture with the growth of the DD interaction. Figure 6 displays the eigenvalue spectra for the on-site and intersite solitons, obtained at different values of the relative strength of the DD attraction, ⌫. It is seen that, with the increase of ⌫, there appears a region of instability of the on-site solitons. This region emerges close to = 0; see the right-hand part in Fig. 6共c兲. At the same values of ⌫, there exists a stability “window” for the intersite solitons, which turns out to be identical to the instability region for the on-site solitons; see Fig. 6共d兲. With the further growth of the DD attraction, this region quickly expands toward higher value of 兩兩, making all the on-site solitons unstable and their intersite counterparts stable—in particular, at ⌫ = 2; see Figs. 6共e兲 and 6共f兲. However, this is not the eventual situation, because subsequent increase of ⌫ again initiates the stability exchange
5
5
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15 10
5
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(b)
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ev
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0
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ev
(c)
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0
0
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FIG. 5. The same as in Fig. 3, but for intersite unstaggered solitons.
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PHYSICAL REVIEW A 78, 063615 共2008兲
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(f)
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FIG. 6. The eigenvalue spectra for the on-site 共a兲, 共c兲, 共e兲 and intersite 共b兲, 共d兲, 共f兲 unstaggered solitons, for C = 0.8 and different values of the relative strength of the dipole-dipole attraction: ⌫ = 0 共a兲, 共b兲; ⌫ = 0.9 共c兲, 共d兲; ⌫ = 2 共e兲, 共f兲.
between the intersite and on-site solitons. Now, a stability window for the on-site solitons, and the corresponding instability window for the ones of the intersite type, emerge close to = 0; see Figs. 7共a兲 and 7共b兲. Eventually, all the on-site solitons become stable again, while all their intersite counterparts get unstable—in particular, at ⌫ = 6 in Figs. 7共c兲 and 7共d兲. Unlike the opposite situation observed in Figs. 6共e兲 and 6共f兲, this one represents the eventual picture, as no changes of the stability occur at still larger values of ⌫. In particular, in the model with the DD attraction and no contact interactions 共␥ = 0, i.e., ⌫ = ⬁兲, the situation remains the same as at ⌫ = 6; see Figs. 7共e兲 and 7共f兲. In fact, the contact interactions may indeed be eliminated, by means of the Feshbachresonance technique, in the condensate of chromium atoms, thus leaving the DD forces as a single nonlinear factor affecting the dynamics of the condensate 共if the nonlinear loss induced by the Feshbach resonance may be neglected兲 关16兴. The effect of the DD interaction on the stability for both types of the unstaggered solitons, at different values of intersite coupling constant C, can be summarized in stability diagrams based on the systematic computation of the eigenvalue spectra. The stability regions in the plane of ⌫ and norm P are presented in Figs. 8–10. These diagrams make it obvious that the stability exchange between the on-site and intersite modes proceeds in essentially the same way at all values of C, a difference being observed only in values of ⌫ at which
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FIG. 7. The same as in Fig. 6, but for the stronger dipole-dipole attraction: ⌫ = 4 共a兲, 共b兲; ⌫ = 6 共c兲, 共d兲; and ⌫ = ⬁ 共e兲, 共f兲 共the latter case implies ␥ = 0, i.e., the absence of the contact interactions兲.
the exchange commences and finishes. Namely, at C = 0.2, Fig. 8 demonstrates that the stability window for the on-site solitons, which simultaneously is the instability region for their intersite counterparts, emerges at ⌫ = 3, and all on-site solitons become stable for ⌫ ⬎ 8. For C = 0.8 and 1.2, the respective values of ⌫ are lower, as seen in Figs. 9 and 10. It can be concluded that, as C increases, the values of ⌫ at which the contact interaction becomes negligible 共versus the DD attraction兲 become smaller.
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If the local interaction is repulsive, unstable unstaggered solitons may only be supported by the DD attraction. Accordingly, the numerical solution of Eq. 共21兲 with sign Ꭽ = + 1 demonstrates that the solitons exist for ⌫ ⲏ 0.4– 0.5, which agrees with the prediction of the MI analysis presented at the beginning of this section. Figure 11 displays examples of both on-site and intersite unstaggered solitons found with different values of ⌫. At smaller ⌫, the unstaggered solitons are broader, with a weakly pronounced singlesite maximum in the on-site modes, see Figs. 11共a兲 and 11共b兲. With the increase of ⌫, the DD attraction makes the solitons narrower, and single-site the peak in the on-site configurations more salient, as can be seen in Figs. 11共c兲 and 11共d兲. Figures 11共c兲 and 11共d兲 demonstrate that the further increase of ⌫ does not strongly affect the shapes of the discrete solitons. The stability analysis for the unstaggered localized modes, performed, as above, via the computation of the stability eigenvalues, leads to a conclusion that, as well as in the case of the attractive contact interaction which was considered in detail above, the on-site solitons are stable when the intersite solitons are unstable and vice versa, hence it is sufficient to summarize the results for one type of unstaggered soliton. The stability diagram for on-site unstaggered solitons is presented in Fig. 12. It demonstrates that the unstaggered solitons exist at ⌫ ⬎ 0.5. With the increase of ⌫ up to ⌫ ⬇ 1, the stability of the localized modes is switched more than once. However, due to the competition between repulsive contact and attractive DD interactions, the solitons found in this parameter region are very broad with a vague single-site maximum, and results of the stability analysis are very sensitive to parameters. It may be called an area of marginal stability. As ⌫ continues to grow, the on-site solitons be-
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come, first, definitely unstable, but then, at ⌫ ⬎ 3, all the on-site solitons are stable. The same conclusion remains valid in the limit of ⌫ = ⬁, i.e., in the medium dominated by the DD attraction, in the absence of any contact interactions.
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FIG. 11. The on-site 共a兲, 共c兲, 共e兲 and intersite 共b兲, 共d兲, 共f兲 unstaggered solitons found for the same value of parameter and different values of ratio ⌫ between strengths of the dipole-dipole attraction and local repulsion, respectively. ⌫ = 0.6 共a兲, 共b兲; ⌫ = 3 共c兲, 共d兲; and ⌫ = −0.2 共e兲, 共f兲. Open squares correspond to = −9.775, open circles to = −5.025, and open triangles to = −2.525.
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P FIG. 12. The stability diagram for on-site unstaggered solitons, as predicted by the computation of eigenvalue spectra for C = 0.8, in the case of the repulsive contact interaction. The solitons are stable in the white area, exponentially unstable in the gray one, and solitons do not exist in the light gray area.
FIG. 13. The P共兲 共norm versus the chemical potential兲 共a兲, 共c兲, 共e兲 and G共P兲 共free energy versus norm兲 共b兲, 共d兲, 共f兲 dependencies for the families of on-site and intersite unstaggered solitons in the model with the attractive contact interaction, while the dipoledipole interaction is, respectively, nil 共⌫ = 0兲, attractive 共⌫ = 1兲, and repulsive 共⌫ = −0.2兲. In observed parameter space, for the ⌫ = 0 and −0.2, all on-site solitons are stable and intersite unstable, and for ⌫ = 1 all on-site solitons are unstable and intersite stable. Here and in Figs. 14 and 15, the solid curve and symbols represent, respectively, the on-site and intersite families, and C = 0.8. C. Development of the instability of unstaggered bright solitons
In addition to the stability analysis developed above, usually a simpler approach to the same problem is used in a form of the slope 共Vakhitov-Kolokolov兲 criterion: a necessary condition for the stability of 共part of兲 a soliton family, described by the dependence of the soliton’s power on chemical potential , is the negative slope of the dependence, dP / d ⬍ 0 关29,30兴. However, the slope criterion is based on the assumption of the local character of the nonlinear interaction, which is violated here by the nonlocal character of the DD interaction. The comparison of these dependencies with the rigorous stability results presented above demonstrates that the slope criterion is nevertheless valid for the on-site family, but not for the intersite one. Actually, in the present model, stable on-site and unstable intersite modes, or stable intersite and unstable on-site ones, always appear in pairs, and the P共兲 dependencies for them are quite similar, see Figs. 13–15. However, the spectral criterion suggests that the instability of both types of unstaggered solitons may lead to their drift across the lattice 关30兴. Indeed, direct simulations demonstrate that slightly perturbed unstable on-site and intersite unstaggered solitons de-
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velop only the drift type of instability, as illustrated by Fig. 16. In other words, an unstaggered soliton that belongs, according to the computation of the eigenvalues, to the region of the exponential instability, is not destroyed but rather jumps to a neighboring lattice site, where it continues to exist as a trapped breather, with the initial or smaller value of the power, depending on the rate of radiation losses accompanying the jump. Thus, while the presence of the DD interaction gives rise to the stability exchange between the on-site and intersite unstaggered localized modes, it does not alter the scenario of the instability development. It is relevant to mention that, although our discrete model does not directly include the possibility of the onset of collapse in the self-attractive BEC 共which manifests itself through an instability of the amplitude type; in the framework of a discrete model, the collapse can be accounted for by a nonpolynomial on-site nonlinearity 关12兴兲, the stability exchange in the presence of the DD interaction, revealed by the above analysis, can offer a possibility to prevent the trend to collapse in 2D and 3D BEC settings, as suggested by recent experimental results 关16兴.
IV. THE MOBILITY OF UNSTAGGERED SOLITONS IN THE PRESENCE OF DD INTERACTION
The motion of discrete localized modes across the lattice is usually interpreted in terms of the Peierls-Nabarro 共PN兲 barrier 关31,32兴. In general, it is realized as an energy difference that must be overcome by a localized state to allow its progressive motion. Although the PN concept is intuitively 10
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FIG. 15. The same as in Fig. 14, but for ⌫ = 3. In observed parameter space, all on-site solitons are stable and intersite unstable.
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clear, its definition is not unique 关31,33,34兴. Taking into regard the structure of the present model, which includes the contact and DD interactions, the PN barrier is considered below in the framework of the free-energy concept 关33兴, which is defined as a combination of the Hamiltonian and norm, G = H + P.
共22兲
Typical G共兲 dependencies for the families of unstaggered solitons with the local attraction are presented, for the cases of attractive, repulsive, and zero DD interactions 共⌫ = 1, −0.2 and 0, respectively兲, in Figs. 13共d兲, 13共f兲, and 13共b兲. According to these figures and their counterpart showing the respective P共兲 curves, in the presence of the DD attraction the on-site and intersite soliton modes have small differences in their powers and free energies. The P共兲 and G共P兲 curves for the on-site and intersite solitons approach each other still closer with the increase in positive ⌫. Actually, this is a consequence of the fact that the attractive DD interaction makes the solitons narrower, and their amplitude higher, in comparison with the standard model including only the contact interaction. An opposite trend is observed in the case of the repulsive DD interaction, with the corresponding P共兲 and G共P兲 curves for the on-site and intersite solitons getting more separated in comparison to the standard model, as seen in Figs. 13共c兲 and 13共d兲. The PN barrier is defined as the difference between values of G for the on-site 共“on”兲 and intersite 共“in”兲 stationary localized modes with equal values of norm P 关32,33兴, 共⌬GPN兲 P = Gon − Gin = ⌬H + P⌬ ,
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where ⌬H = Hon − Hin and ⌬ = on − in. In the present model, in the absence of the DD interaction 共⌫ = 0兲, the on-site and intersite configurations with equal values of P are found in a limited region of parameter space 共C , 兲, as can be seen in Fig. 13共a兲. Therefore, the moving modes in the standard model, to which the present one reduces for ⌫ = 0, are expected and were indeed found 关35兴 in a small parameter region. However, the mobility region expands with the increase of ⌫ ⬎ 0, which corresponds to the attractive DD interaction, and, for ⌫ = 1, the G共P兲 curves for on-site and intersite modes become very close, which indi-
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Figs. 14 and 15. This fact suggests that the combination of the repulsive local interaction and long-range DD attraction makes the localized modes highly mobile. The latter prediction was readily confirmed by direct simulations 共details not shown here, as they do not reveal anything special兲.
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FIG. 17. The dynamics of unstaggered on-site soliton modes, initiated by the application of the kick with k = / 18, for fixed values of P = 4 and C = 1.2, and different relative strengths of the zero or attractive dipole-dipole interaction: 共a兲 ⌫ = 0, 共b兲 ⌫ = 0.5, and 共c兲 ⌫ = 1. The local interaction is attractive too.
cates the trend to vanishing of the PN barrier in nearly the entire parameter space, see Fig. 13共f兲. The repulsive DD interaction acts in the opposite direction, making the PN barrier larger, as seen in Fig. 13共d兲. In direct simulations, the motion of a discrete soliton was initiated, as usual, by applying a “kick” of size k, i.e., by multiplying the static soliton configuration with exp共ikn兲. In particular, the evolution of on-site unstaggered solitons with fixed power, P = 4, induced by the application of the kick with k = / 18 in the model with the local attraction, fixed intersite coupling constant, C = 1.2, and different values of ⌫ is displayed in Fig. 17. Without the DD interaction 共⌫ = 0兲, the on-site soliton remains pinned. In the case of a moderately strong DD attraction 共⌫ = 0.5兲, the kicked soliton passes some distance in the lattice and gets retrapped. With the stronger DD attraction 共⌫ = 1兲, the solitons demonstrate persistent motion. These observations are in full agreement with the expectations based on the evaluation of the PN barrier. Thus, a general conclusion is that the DD attraction enhances the soliton mobility, which is a natural effect of the interaction that makes the medium more nonlocal, hence less discrete, effectively. In the model with the contact repulsion, the P共兲 and G共P兲 curves for the on-site and intersite soliton modes are close to each other in the entire parameter region, as seen in
关1兴 M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Science 269, 198 共1995兲; K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, Phys. Rev. Lett. 75, 3969 共1995兲; D. G. Fried, T. C. Killian, L. Willmann, D. Landhuis,
This paper is an attempt to advance the understanding of the influence of the DD 共dipole-dipole兲 interaction on the formation and properties of localized dynamical structures in a cigar-shaped BEC trapped in a very deep axial optical lattice. The underlying model is based on the 1D discrete Gross-Pitaevskii equation with contact and DD nonlocal interaction terms. The cases of both attractive and repulsive signs of the contact and DD interactions were considered. Unstaggered on-site and intersite solitons have been found in the model with the local attraction, for either sign of the DD interaction. If the local interaction is repulsive, unstaggered solitons may be supported by a sufficiently strong long-range DD attraction. The phenomenon of the multiple stability exchange between the families of on-site and intersite unstaggered solitons with the increase of the strength of the DD attraction has been found for either sign of the local interaction. Eventually, when the DD attraction dominates over the local interactions 共in particular, if the local interactions are absent兲, the on-site solitons are stable while their intersite counterparts are not. The results suggest possibilities to prevent the collapse in dipolar BECs. It was also shown that the DD interaction affects the mobility of the discrete soliton modes. Namely, through the evaluation of the PN barrier and by means of direct simulations, we have demonstrated that, with the enhancement of the attractive DD interaction, the PN barrier decreases and the mobility improves strongly. A natural extension of the present analysis would be to apply it to 2D and 3D discrete models with the long-range DD interactions. In particular, of special interest may be the problem of the onset of collapse in such settings, when either the contact or DD interaction is attractive. Also interesting would be the consideration of higher-order solitons, such as antisymmetric bound states in the 1D lattice, and discrete localized vortices in the 2D geometry. ACKNOWLEDGMENTS
G.G., A.M., and Lj.H. acknowledge support from the Ministry of Science, Serbia 共Project 141034兲. B.A.M. appreciates the hospitality of the Vinča Institute of Nuclear Sciences in Belgrade, Serbia.
S. C. Moss, D. Kleppner, and T. J. Greytak, ibid. 81, 3811 共1998兲. 关2兴 K. E. Strecker, G. B. Partridge, A. G. Truscott, and R. G. Hulet, Nature 共London兲 417, 150 共2002兲; see also K. E. Strecker, G. B. Partridge, A. G. Truscott, and R. G. Hulet, New
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GLIGORIĆ et al. J. Phys. 5, 73 共2003兲. 关3兴 L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, and C. Salomon, Science 296, 1290 共2002兲. 关4兴 S. L. Cornish, S. T. Thompson, and C. E. Wieman, Phys. Rev. Lett. 96, 170401 共2006兲. 关5兴 V. M. Pérez-García, H. Michinel, and H. Herrero, Phys. Rev. A 57, 3837 共1998兲. 关6兴 A. E. Muryshev, G. V. Shlyapnikov, W. Ertmer, K. Sengstock, and M. Lewenstein, Phys. Rev. Lett. 89, 110401 共2002兲. 关7兴 L. Salasnich, Laser Phys. 12, 198 共2002兲; L. Salasnich, A. Parola, and L. Reatto, Phys. Rev. A 65, 043614 共2002兲; 66, 043603 共2002兲. 关8兴 S. Sinha, A. Y. Cherny, D. Kovrizhin, and J. Brand, Phys. Rev. Lett. 96, 030406 共2006兲. 关9兴 L. Khaykovich and B. A. Malomed, Phys. Rev. A 74, 023607 共2006兲. 关10兴 S. De Nicola, B. A. Malomed, and R. Fedele, Phys. Lett. A 360, 164 共2006兲; S. De Nicola, R. Fedele, D. Jovanovic, B. Malomed, M. A. Man’ko, V. I. Man’ko, and P. K. Shukla, Eur. Phys. J. B 54, 113 共2006兲. 关11兴 A. Trombettoni and A. Smerzi, Phys. Rev. Lett. 86, 2353 共2001兲; F. Kh. Abdullaev, B. B. Baizakov, S. A. Darmanyan, V. V. Konotop, and M. Salerno, Phys. Rev. A 64, 043606 共2001兲; G. L. Alfimov, P. G. Kevrekidis, V. V. Konotop, and M. Salerno, Phys. Rev. E 66, 046608 共2002兲; R. CarreteroGonzález and K. Promislow, Phys. Rev. A 66, 033610 共2002兲; N. K. Efremidis and D. N. Christodoulides, ibid. 67, 063608 共2003兲. 关12兴 A Maluckov, L. Hadžievski, B. A. Malomed, and L. Salasnich, Phys. Rev. A 78, 013616 共2008兲. 关13兴 M. A. Porter, R. Carretero-González, P. G. Kevrekidis, and B. A. Malomed, Chaos 15, 015115 共2005兲. 关14兴 K. Góral, K. Rzaźewski, and T. Pfau, Phys. Rev. A 61, 051601共R兲 共2000兲. 关15兴 A. Griesmaier, J. Werner, S. Hensler, J. Stuhler, and T. Pfau, Phys. Rev. Lett. 94, 160401 共2005兲; J. Stuhler, A. Griesmaier, T. Koch, M. Fattori, T. Pfau, S. Giovanazzi, P. Pedri, and L. Santos, ibid. 95, 150406 共2005兲; A. Griesmaier, J. Phys. B 40, R91 共2007兲. 关16兴 T. Koch, T. Lahaye, J. Metz, B. Frölich, A. Griesmaier, and T. Pfau, Nat. Phys. 4, 218 共2008兲; M. Ueda and K. Huang, Phys. Rev. A 60, 3317 共1999兲; L. Santos and G. V. Shlyapnikov,
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