PHYSICAL REVIEW A 66, 053608 ~2002!

Interaction of dark solitons with localized impurities in Bose-Einstein condensates

Dimitri J. Frantzeskakis,1 G. Theocharis,1 F. K. Diakonos,1 Peter Schmelcher,2 and Yuri S. Kivshar31Department of Physics, University of Athens, Panepistimiopolis, Zografos, Athens 15784, Greece

2Theoretische Chemie, Physikalisch-Chemisches Institut, INF 229, 69120 Heidelberg, Germany3Nonlinear Physics Group, Research School of Physical Sciences and Engineering, Australian National University,

Canberra ACT 0200, Australia~Received 28 June 2002; published 12 November 2002!

We study the interaction of dark solitons with a localized impurity in Bose-Einstein condensates. We applythe soliton perturbation theory developed earlier in optics for describing the soliton dynamics and soliton-impurity interaction analytically, and then verify the results by direct numerical simulations of the Gross-Pitaevskii equation. We find that a dark soliton can be reflected from or transmitted through a repulsiveimpurity in a controllable manner, while near the critical point the soliton can be quasitrapped by the impurity.Additionally, we demonstrate that an immobile soliton may be captured and dragged by an adiabaticallymoving attractive impurity.

DOI: 10.1103/PhysRevA.66.053608 PACS number~s!: 03.75.Fi, 05.45.Yv, 05.30.Jp, 02.30.Jr

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I. INTRODUCTION

Since the experimental discovery of Bose-Einstein cdensation in dilute atomic alkali-metal gases@1# an enor-mous development of our knowledge on the properties ofcondensed phase has taken place. Among others, exments on the Bose-Einstein condensates~BECs! have dem-onstrated superfluidity of the condensed phase@2#, the pos-sibility of four-wave mixing @3#, the amplification of lightand atoms via a condensate@4# as well as the creation otopological structures such as vortices@5#, vortex lattices@6#,as well as dark@7# and bright@8# solitons. This opens thepromising perspective for numerous applications of the nlinear matter-wave physics and, in particular, it is very mureminiscent of the situation encountered for light waves aoptics many decades ago. One of the first and recentlyvented example of a coherent matter-wave device is thecalled atom chip@9#, that consists of a microfabricated semconductor surface accommodating atom-optics elemsuch as current or charge-carrying wires, resonators, etc.latter allow to trap and guide or, more generally speakingcontrol the motion of the matter waves. A long-term persptive of such coherent matter-wave devices is quantum inmation processing on the nanometer scale.

A relevant interesting issue in the above context is to lehow to control the motion of nonlinear excitations of thcondensate, in general, and of different types of solitonsparticular. Experimentally there exist several quantuphase-engineering techniques to generate dark solitonsBose-Einstein condensate~see, e.g., Denschlaget al. @7#!.The question then arises how one could influence or eguide their motion. The present paper makes a step indirection by investigating the interaction of a dark solittrapped in a confining potential with a localized inhomogneity or an impurity. The perspective hereby is that imputies could be used as elements of a matter-wave devicecontrols the motion of a network of nonlinear excitations

Dark solitons can be excited in BECs with repulsiveteractions; such a soliton is characterized by a notch inBEC density profile and a phase jump across its locali

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region. Their dynamics in a trapping potential has been stied theoretically by several authors@10–13#. Beyond theabove motivation, the dynamics of BECs in the presenceimpurities has recently become a subject of growing inter@14#. One process that has been studied is a response ocondensate to the propagation of significantly heavier imrities. The travelling impurity induces a BEC dynamics thleads, due to energy reasons, to the expulsion of the impufrom BEC. In this context, the investigation of the interactiof a soliton with a single impurity appears to be an importaissue. It is of fundamental interest in nonlinear wave theand has been studied in the framework of almost all nonear evolution equations possessing soliton solutions@15#.However, in the context of BECs where the relevant evotion equation is the Gross-Pitaevskii~GP! equation with aconfining potential@16#, the interaction of dark solitons withimpurities has not been studied in detail yet.

The purpose of this paper is twofold. First, we apply tsoliton perturbation theory earlier developed for dark sotons in optics~see Ref.@17# for a comprehensive review! andderive an effective equation for the motion of a dark solitin a trapping potential, and in the presence of either repulsor attractive impurities. Second, we study, both analyticaand numerically, the dynamics of the condensate and a dsoliton in the presence of a static~nonpropagating! impurity.This analysis is rather general, and it may be used to descthe interaction of a dark soliton with ‘‘artificial’’ impuritiesinduced by sharply focused laser beams used to engineedensity of the BEC in experiments@7#. Various interactioneffects such as reflection, transmission, and quasitrappinthe dark soliton by a repulsive impurity are described averified by direct simulations. The quasitrapping effect fouis characterized by a relatively large time for the solitoimpurity interaction, and is followed by subsequent solitoscillation in a limited spatial domain. Additionally, the posibility of the dragging of a dark soliton by an adiabaticamoving attractive impurity is demonstrated.

The paper is organized as follows. In Sec. II, we descrour analytical results based on the perturbation theoryvented for optical dark solitons. Section III is devoted to

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FRANTZESKAKIS et al. PHYSICAL REVIEW A 66, 053608 ~2002!

detailed comparison between our analytical predictionsnumerical results obtained by the direct integration of theequation. Finally, Sec. IV provides a summary and concsions.

II. MODEL AND ANALYTICAL RESULTS

A. The effective Thomas-Fermi wave function

A convenient model to study the mean-field dynamicsBECs is the GP equation which has the form of(311)-dimensional nonlinear Schro¨dinger ~NLS! equationwith an external trapping potential~see, e.g., a review pape@16#, and references therein!. In the case when the confinement for two of the three spatial dimensions is much stronthan in the third dimension, the GP equation can be reduto an effective quasi-(111)-dimensional GP equation@18#~see also the relevant experimental works in Refs.@7,19#!.For repulsive interatomic interactions, the latter equationbe expressed in the following dimensionless form:

iut11

2uxx2uuu2u5V~x!u, ~1!

where the spatial coordinatex and timet are normalized tothe harmonic-oscillator lengtha'5A\/mv' and oscillationperiod, 1/v' , respectively. The frequencyv' belongs to thetwo dimensions with strong confinement. The normalizfield u describes the macroscopic wave functionc of thecondensate, according to the following scaling relation:

c~x,t !5S mv'

4pa\ D 1/2

u~x,t !,

wherea is the scattering length.In order to study the interaction of a BEC dark solito

with a localized impurity in the framework of Eq.~1!, it isconvenient to decompose the external potentialV(x) as fol-lows:

V~x!5U tr~x!1bd~x!, ~2!

where Utr(x) is the ~conventional parabolic! time-independent trapping potential, which is assumed tosmooth and slowly varying on the soliton scale, and theditional sharp potentiald(x) accounts for an impurity localized in space at the pointx50, and it is described by a Dirad function. The parameterb in Eq. ~2! which measures theimpurity strength is assumed to be small and may take eipositive or negative values for repulsive or attractive imprities, respectively. The impurity potential causes a deformtion of the condensate wave function@14#.

In order to treat analytically Eq.~1!, first we look for the

profile of the background field oscillations,u5ub(x)e2 iu02t,

whereu02 is the normalized density of the BEC cloud, in th

presence of the potentialV(x) that is given by the real equation

u02ub1

1

2

d2ub

dx22ub

35V~x!ub . ~3!

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When the amplitude maxuub(x)u is small, the nonlinearterm in Eq.~3! can be neglected and, assuming that inabsence of the potentialV(x) the background amplitude iu0, we look for a solution of Eq.~3! in the form

ub~x!5u01 f ~x!. ~4!

Substituting Eq.~4! into Eq. ~3!, neglecting all nonlinearterms with respect tof and keeping only the leading-ordeterms with respect to the parametersb andk ~see below!, weobtain the following linear equation forf (x):

1

2

d2f

dx222u0

2f 5u0@U tr~x!1bd~x!#. ~5!

A physically relevant solution of Eq.~5! may be obtainedas follows: First, in the absence of the impurity~i.e., in thecase b50), we assume an anisotropic cigar-shaped hmonic trap, described by the effective one-dimensionaltential U tr(x)5k2x2, where k25(vx

2/2v'2 )!1, vx being

the frequency of the trap in the axial direction. In this cathe spatial derivatived2f /dx2 is small and can be neglectedand, as a result, an approximate solution of Eq.~5! readsf (x)52(2u0)21U tr(x), resembling the well-knownThomas-Fermi~TF! approximation for the wave functionwhere the densityuuu2 is quadratic in the termu02U tr(x).Second, for a homogeneous BEC~i.e., in the caseU tr50),Eq. ~5! is equivalent to its homogeneous counterpart andmatching condition at the position of the impurity reads

d f

dxU01

2d f

dxU02

52bu0 .

Therefore, the spatially localized solution of Eq.~5! is f (x)52(b/2)exp(22u0uxu). Combining these particular solutions, we express the solution of Eq.~5! as follows:

f ~x!521

2u0U tr~x!2

b

2e22u0uxu. ~6!

The background field densityub2(x) given by Eqs.~4! and

~6! actually describes an effective TF-like condensate wfunction modified by a localized impurity. This densityillustrated in Fig. 1~solid line! for u051 and for the har-monic trapping potentialU tr(x)5(kx)2 ~dashed line!, withk50.05; this value is approximately twice as large the oused in the experimental studies of BEC dark solitons in R@7#. We remark that both for the illustrations@see Fig. 1# aswell as the numerical calculations thed function for the lo-calized impurity atx50 has been replaced by the steep funtion 10b cosh21(x/0.05) with b560.15 for repulsive or at-tractive impurities, respectively. It is interesting to obserthat the repulsive impurity@Fig. 1~a!# creates a hole on thecondensate wave function, in accordance with the earlierdictions @14#, while the attractive one@Fig. 1~b!# creates ahump. In any case, it is clear that the condensate is shalike the inverted harmonic trapping potential with a dip orhump, having the size of the healing length~which is,

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INTERACTION OF DARK SOLITONS WITH LOCALIZED . . . PHYSICAL REVIEW A66, 053608 ~2002!

roughly speaking, equal to 1/56 for the values of the paraeters used! around the impurity.

B. Dynamics of dark solitons

To describe the dynamics of a dark soliton on top ofinhomogeneous background described by the effectivelike cloud, we seek for a solution of Eq.~1! in the form

u5ub~x!e2 iu02ty~x,t !, ~7!

where ub obeys Eq.~3! and the unknown complex fieldy(x,t) represents a dark soliton, which is governed byfollowing effective equation:

i y t11

2yxx2ub

2~ uyu221!y521

ub

dub

dxyx . ~8!

Apparently, the right-hand side and part of the nonlineterms of Eq.~8! can be treated as a perturbation. To obtthe contribution of the nonlinear terms within perturbatitheory, we may use Eqs.~4! and ~6! and approximateub

2 asub

2'u0212u0f , based on the smallness of the functionf (x)

~due to the slowly varying properties of the trapping potetial and the smallness of the parameterb characterizing theimpurity!. In this way, and upon introducing the transformtions t→u0

2t, x→u0x, we obtain the following perturbedNLS equation for the dark soliton:

i y t11

2yxx2~ uyu221!y5P~y!, ~9!

where the total perturbationP(y) has the form

FIG. 1. The ground-state condensate intensityub2(x) ~solid!, de-

scribing an effective Thomas-Fermi cloud, for a harmonic trapppotentialU tr(x)5(kx)2 ~dotted! with k50.05. The potential due tothe localized impurity atx50 is approximated by the steep functio10b cosh22(x/0.05) withb560.15. Both cases of repulsive~a! andattractive~b! impurities are, respectively, shown.

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u0e22u0uxu@~12uyu2!y2u0~x/uxu!yx#. ~10!

In the absence of the perturbationP(y), Eq.~9! representsa conventional defocusing NLS equation which has a dsoliton solution of the form@22#

y~x,t !5cosw tanhj1 i sinw, j5cosw@x2~sinw!t#,~11!

wherew is the soliton phase angle (uwu,p/2) describing thedarkness of the soliton through the relation,uyu2512cos2w/cosh2j ~note that the limiting casesw50 and cosw!1 correspond to the so-calledblack and gray solitons, re-spectively@17#!. To treat analytically the effect of the pertubation ~10! on the dark soliton, we employ the adiabatperturbation theory developed in Ref.@20# ~see also the re-view @17#!. According to this approach, the parameters ofdark soliton~11! become slowly varying functions oft, butthe functional form remains unchanged. Thus, the soli‘‘phase angle’’ becomesw→w(t) and, as a result, the solitocoordinate becomesj→j5cosw(t)@x2x0(t)#, where

x0~ t !5E0

t

sinw~ t8!dt8 ~12!

is the soliton center. As has been shown in Ref.@20#, theevolution of the parameterw is then defined by the equation

dw

dt5

1

2 cos2w sinwReH E

2`

1`

P~y!y t* dxJ . ~13!

Substituting Eq.~10! into Eq.~13! and taking into accounthat for spatially slowly varying trapping potentialU tr thehigher-order derivatives may be omitted, we obtain the flowing result~for u051):

dw

dt52

1

2

dUtr

dx

13

4bE

0

1`

dxexp~22x!

@cosh4~x2x0!2cosh4~x1x0!#,

~14!

where we assume additionally that the dark soliton is closa black one, i.e.,w is sufficiently small. Evaluating the integrals in Eq.~14! and using the definition of Eq.~12!, weobtain the following effective equation for the soliton cente

d2x0

dt252

dW

dx0. ~15!

It is readily seen that Eq.~15! represents an equation omotion for a classical particle with the coordinatex0 movingin the effective potential,

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FRANTZESKAKIS et al. PHYSICAL REVIEW A 66, 053608 ~2002!

W~x0!51

2 H U tr~x0!1b

2 cosh2x0J . ~16!

It is important to notice that, in the absence of the impur(b50), Eq.~15! describes the motion of a dark soliton in thpresence of a trapping potential, and it predicts that the ston oscillates in a harmonic trap. An equation similar to E~15!, ~16! has first been stated without derivation in Re@10#, but without the factor of 1/2. The same equation fouin Ref. @10# is derived in Ref.@11# by considering the dipolemode of a condensate carrying a dark soliton, and thusnot directly related to Eq.~15!. The correct result has beeobtained by a multiple time scale boundary layer theorycently developed by Busch and Anglin@12# who assumedthat the potential, the background density, and velocity vslowly on the soliton scale.

In the general caseb5” 0, we do not assume that the impurity potential varies slow on the soliton scale, however,take into account the modification of the condensate grostate in the presence of the impurity. It is important to metion that the term proportional tob in the potential~16! doesnot possess the appearance one might expect by inspeEqs. ~5! and ~6!. This reflects the pointlike character of thimpurity: the healing length is the shortest length availablethe condensate with respect to density variations and thefective pointlike impurity generates a disturbance of tbackground cloud on the same length scale as the darkton.

The resulting equation~16! shows that the character of theffective potential is changed in the vicinity of the impuri~i.e., in a localized region aroundx50): In particular, itbecomes repulsive~attractive! for b.0 (b,0) for the darksoliton due to the presence of the impurity localized atcenter of the trapping potential.

III. NUMERICAL RESULTS

Adopting the particle picture of the dark soliton describabove, we now analyze the soliton dynamics in the framwork of Eqs.~15! and~16!. In particular, we assume that theffective potential is given by W(x0)5(1/2)(kx0)2

1(b/4)cosh22(x0) ~the values of the parameters arek50.05 andb560.15, see Fig. 1!.

At first, in the case of a repulsive impurity (b.0) @seeW(xo) in Fig. 2~a!#, it is clear that the dynamical systemhand is characterized by three fixed points, namely, twoliptic ~stable! located atx0'62 and one hyperbolic~un-stable! located atx050. Figure 2~b! shows the associatephase plane (x0 ,dx0 /dt) including several phase curveThe separatrix passing through the hyperbolic fixed poin(0,0) possesses the appearance of an intersecting doubleand, by definition, separates two types of motion taking plin a single well or above both wells. As a result, differetypes of the soliton dynamics and soliton-impurity interation are expected, depending on the initial conditions@recallthat we consider almost black solitons with cosw'1, i.e., theinitial soliton velocity is sinw'0, and, as a result, the initiaconditions refer solely to the initial soliton positionx0(0)]:

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~a! For ux0(0)u,5.48 ~corresponding to the phase curvsurrounding the elliptic fixed points! the dark soliton isre-flectednearly elastically by the impurity, and then it oscilates in a relatively small spatial region@see Fig. 1~b!#, ascompared to the entire length of the condensate~the latter isL'56).

~b! For ux0(0)u.5.48 ~corresponding to the phase curvsurrounding the separatrix! the dark soliton istransmittednearly elastically through the impurity, and then it oscillatin a relatively large region inx, as compared to the entirlength of the condensate.

~c! The point ux0(0)u55.48 corresponds to a critical regime described by the figure-eight separatrix; in this caseinteraction time is very large.

On the other hand, in the case of an attractive impu(b,0), as shown in Fig. 2~a!, it is clear that the dynamicasystem in hand is characterized by a single fixed ponamely, an elliptic~stable! one, located atx050. Figure 2~c!shows the associated phase plane including several pcurves. Apparently all trajectories are periodic and the spadomain where the soliton oscillates is the same as in theb50. Thus, the presence of the attractive impurity doesqualitatively affect the soliton motion, but rather modifielocally ~i.e., in the vicinity of the site of the impurity! thesoliton’s kinetic energy.

The above theoretical predictions have been checkeddirect numerical simulations of the GP equation. In particlar, we have used a split-step Fourier method@21# to inte-

FIG. 2. ~a! The effective potentialW(x0)5(1/2)(0.05x0)2

1(b/4)cosh22(x0) for a repulsive impurity,b50.15~solid line!, andan attractive impurity,b520.15 ~dotted line!. ~b! The associatedphase plane forb510.15, where several trajectories are showDots, small circles, diamonds, and stars correspond to the numeresults obtained by direct integration of the GP equation for diffent initial positions of the soliton taken inside, outside, or onleft-hand separatrix, respectively. The corresponding types ofsoliton dynamics are shown in Figs. 3, 4, 5, and 6, respectively.~c!The associated phase plane forb520.15, where several trajectories are shown. Stars correspond to the numerical results obtaby the GP equation fort50,5,8,15,20,30, for a black soliton initially placed atx0521.

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INTERACTION OF DARK SOLITONS WITH LOCALIZED . . . PHYSICAL REVIEW A66, 053608 ~2002!

grate the GP equation~1! with an initial condition of theform

u~x,0!5ub~x!tanh@x2x0~0!#, ~17!

whereub(x)511 f (x) is the effective TF-like wave functionmodified by the localized impurity@ f (x) is given in Eq.~6!with u051] and x0(0) is the initial soliton position. Thisinitial condition represents a dark soliton with cosw51 @seeEq. ~11!#, i.e., a black soliton with zero initial velocity. Notice that the particular choice of the initial condition~17!implies that the characteristic lengthk21 of the trapping po-tential is much larger than the soliton width, i.e.,k21@1~recall that we have takenk50.05), a fact indicating that inthis regime the validity of the perturbation theory presenin the preceding section is guaranteed.

Finally, it is important to notice that since the initial condition used@see Eq.~17!#, is obviously only an approximation and not an exact solution of the GP equation withimpurity, it does not describe the pure soliton state, but aincludes various phonon modes. Consequently, in themerical results presented below, changes in the backgrodensity of the condensate, even ‘‘far’’ away from the solitoare observed. Also in an experiment, the prepared soldoes not perfectly coincide with the theoretical analyticaor numerically obtained soliton. We expect that our analycal predictions and numerical results~which, as we will seebelow, are in a fairly good agreement! can be verified in acorresponding experiment.

A. Repulsive impurity

The three different types~a!, ~b!, and ~c! of the solitonevolution and interaction with a repulsive impurity predictabove, have been investigated numerically upon considethe initial soliton positionsx0(0)523, 28, 25.48, and25.5, respectively.

In particular, Fig. 3 shows the evolution of the solitodensityuuu2, corresponding to Eq.~17! with x0(0)523, ontop of the effective TF cloud~solid line!, with a potentialV(x)5(0.05x)211.5 cosh22(x/0.05) ~dotted line!. Noticethat both the effective TF wave function and potential aresame as the ones illustrated in Fig. 1~a!. As can be seen inFig. 3, the dark soliton is reflected by the impurity~at t'28; see third panel! nearly elastically, and it oscillates ithe interval23,x,21. The oscillations of the soliton takplace in a spatial domain significantly smaller than thetension of the BEC trap~here the corresponding fraction2/56). According to the particle picture of the solitoadopted in the preceding section, this behavior is effectivdescribed by a periodic trajectory inside the separatrix loon the left-hand side of Fig. 2~b!. To illustrate this, the nu-merically obtained soliton positions and velociti(x0 , dx0 /dt) corresponding to the instants of the six sufigures of Fig. 3 are shown by six dots in Fig. 2~b!, startingfrom (23,0) and taking place clockwise. It is clearly sethat the relevant analytical prediction, according to whichsoliton center follows the periodic trajectory is in fairly gooagreement with the numerical result obtained by the dirintegration of the GP equation. This agreement is not o

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qualitative but quantitative, since the maximum relative erfound in the soliton velocity does not exceed 20%.

Using the same initial configuration for a dark solitoinitially placed atx0(0)528, in Fig. 4 we show the evolution of the soliton density which, in this case, is transmittnearly elastically through the impurity~no emission of radia-tion is observed!, i.e., it is oscillating in the interval28,x,7.5. Thus, the soliton motion takes place in a relativelarge spatial region compared to the condensate length~here

FIG. 3. Evolution of a black soliton densityuuu2 on top of theeffective TF cloud~solid!, with the potentialV(x) ~dotted! beingthe same as the one in Fig. 1~a!. The dark soliton is initially placedatx0(0)523 and, in accordance with the analytical prediction, itreflected by the impurity~at t'28) and then oscillates in the interval 23,x,21. This behavior is effectively described by the priodic trajectory surrounded by the left part of the separatrix shoin Fig. 2~b!.

FIG. 4. The same as in Fig. 3 but for the dark soliton initiaplaced atx0(0)528. According to the analytical results, the soton is transmitted through the impurity and then oscillates ininterval 28,x,7.5. This behavior is effectively described by thperiodic trajectory enclosing the separatrix in Fig. 2~b!.

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FRANTZESKAKIS et al. PHYSICAL REVIEW A 66, 053608 ~2002!

the fraction is'15.5/56). This behavior is effectively described by the periodic trajectory enclosing the separashown in Fig. 2~b!. As done above the soliton’s position iphase space corresponding to the instants of the six suures of Fig. 4 are indicated~small circles! in Fig. 2~b!. Theyfollow the corresponding periodic trajectory and a goagreement between the analytical prediction and the numcal results is found again~the maximum relative error in thesoliton velocity is found to be slightly smaller in this casi.e., of the order of 15%).

We will now investigate the third possible type of solitoimpurity interaction corresponding to a critical regime.Fig. 5, we show the evolution of a dark soliton with an initiposition corresponding to the extremal valuexmax of the left-hand part of the separatrix in Fig. 2~b!. According to theprevious discussion, it is expected that the dark soliton winteract infinitely long with the impurity approaching thpoint x50. As can be seen in Fig. 5, in direct simulations tdark soliton reaches the impurity site~at t'30) and it re-mains ‘‘quasitrapped’’ by the impurity tillt'38. Neverthe-less, the numerical simulations show that fort.38 the soli-ton is released by the impurity, i.e., the soliton center, instof remaining on the separatrix, escapes to a periodic tratory inside the separatrix and eventually performs an osction. This behavior can be seen in Fig. 2~b!, where the firstfour of the soliton’s phase space points~corresponding to thesix subfigures of Fig. 5! are indicated by stars and follow thseparatrix~the maximum relative error in the soliton velocifound does not exceed 10%), while the last two phase sppoints follow a periodic trajectory inside the separatrix.

A slightly different initial soliton position, but still closeto the separatrix, i.e.,x0(0)525.5 ~note: this value isslightly smaller than the extremal valuexmax) produced simi-lar results for the soliton-impurity interaction, but insteadleads to the soliton traversal through the impurity. This resis illustrated in Fig. 6, where the dark soliton, after reach

FIG. 5. The same as in Fig. 3 but for the dark soliton initiaplaced atx0(0)525.48. The soliton reaches the pointx50, then itis quasitrapped by the impurity for some finite time, and eventuoscillates following a periodic trajectory enclosed by the left partthe separatrix in Fig. 2~b!.

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the impurity ~at t'30) first is trapped untilt'38, as in theprevious case. However, fort.38 the soliton escapes tonearby periodic trajectory surrounding the separatrix, evtually performing an oscillation in a relatively large spatidomain. This behavior can also be seen in Fig. 2~b! indicatedby triangles that follow the separatrix~the maximum relativeerror in the soliton velocity found does not exceed 10%while the last two triangles follow a periodic trajectory surounding the separatrix. It is worth mentioning that, in bocases mentioned above, the trapping time is estimated tof order t tr'10.

Generally speaking, the initial conditions located in tneighborhood of the separatrix lead to a quasitrapping pnomenon characterized by a relatively large timet tr of inter-action between the dark soliton and impurity. This phenoenon is followed by either reflection or transmission aeventual oscillation of the soliton. Apparently, quasitrappicannot be understood quantitatively in terms of perturbattheory since the latter predicts that the soliton will remainfinitely long at the impurity. However, it is worth mentioning that the perturbation theory developed above is basedthe particlelike properties of the dark soliton, and it doestake into account the radiation component of the field, whdescribes the wave properties of the soliton. We stronsurmise that the quasitrapping effect is a signature ofwave nature of the soliton excitation, but its detailed invetigation is beyond the scope of this paper.

We have also carried out numerical simulations for neablack solitons characterized by a small initial velocity~e.g.,sinf'0.1); these results provide a qualitatively similar piture for both the soliton evolution and soliton-impurity inteaction. This can be understood in terms of the presenanalysis, since nearly black solitons follow the trajectoriesthe phase plane in the neighborhood of the correspondones followed by the black solitons. For example, an initiafinite soliton velocity may result in a change of reflectio

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FIG. 6. The same as in Fig. 3 but for the dark soliton initiaplaced atx0(0)525.5. The soliton reaches the pointx50, then itis quasitrapped by the impurity for some finite time, and eventuaoscillates following a periodic trajectory that encloses the right pof the separatrix in Fig. 2~b!.

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INTERACTION OF DARK SOLITONS WITH LOCALIZED . . . PHYSICAL REVIEW A66, 053608 ~2002!

into transmission, which can easily be understood, sincechange of the initial conditions drives the soliton~after inter-acting for some finite time with the impurity! to follow aperiodic orbit enclosing the separatrix.

B. Attractive impurity

Here we investigate numerically the dynamics of a neablack soliton in the presence of an attractive impurityb,0). As discussed above, in this case the effective potenbecomes attractive for the dark soliton in the vicinity of timpurity ~i.e., atx50) as shown in Fig. 2~a!. However, if adark ~black! soliton is initially located far away from theimpurity ~i.e., for ux0u.2), there is no way to be trappesince its initial potential energy~due to the trapping potential! is larger than the depth of the impurity-induced well. Othe other hand, ifux0u,2 the black soliton oscillates in thattractive impurity-induced well, occupying a comparativesmall region around the location of the impurity. Both theresults have been confirmed by direct numerical simulatiof the GP equation. Particularly, the oscillation of a blasoliton, initially placed atx0521,22,26, is illustrated inthe associated phase plane shown in Fig. 2~c! by the corre-sponding periodic trajectories~solid lines!. Again, the nu-merically found soliton positions and velocities@see pointsdepicted by stars in Fig. 2~c!# in the case of a black solitonwith x0521 ~i.e., initially located inside the well induceby the impurity!, are in a fairly good agreement with thanalytical predictions.

In this context, an interesting issue is to consider an abatically moving attractive impurity, which moves withsmall constant velocityv on top of the background TF wavfunction without inducing significant perturbations. In th

FIG. 7. Evolution of a black soliton densityuuu2 on top of theeffective TF cloud~solid!, with the potentialV(x) ~dotted! beingthe same as that in Fig. 1~b!. The dark soliton that is initially placedat x0(0)50 is captured and dragged by a steadily moving attrac(b520.15) impurity with a velocityv50.1. This occurs up tot'40, when the soliton is released by the impurity, and eventuoscillates following a periodic trajectory in the interval24<x<4.

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case, the question arises whether this steadily moving imrity may drag a black soliton, which is stationary if it iinitially placed at the bottom of the trapping potential (x50), or it slightly oscillates in the well induced by a statattractive impurity. Apparently this issue is important for aplications, since it would directly demonstrate the possibilof controlling the motion of black solitons in BECs, bmeans of artificial impurities induced by properly tunesharply focused laser beams.

In order to investigate this issue, we have integratedmerically the original GP equation with the modified potetial V(x;t)5(0.05x)221.5 cosh22@(x2vt)/0.05# and with aninitial condition u(x,0)5ub(x)tanh(x), i.e., a black solitoninitially placed at the location of the impurity. In the numercal simulations, varying the value of the velocityv of theimpurity, we observed that forv.vc'0.25 the black solitonremains immobile and the moving impurity causes strofluctuations in the condensate background wave function.the other hand, whenv is taken belowvc the soliton is ableto follow the impurity for a finite time, inversely proportional to the value ofv. Such a case is illustrated in Fig. 7where the evolution of a black soliton, dragged by a steadmoving impurity with v50.1, is shown. It is clearly seenthat the soliton is captured and pulled by the impurity upt'40, but later on it escapes and, having gained energstarts to oscillate in the interval (24,4) ~note that att540the soliton center is placed atx54). In order to get a quali-tative insight into this effect, in Fig. 8 we show the succesive forms of the relevant effective potentialW(x0) for thetime instants corresponding to the snapshots in Fig.

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FIG. 8. The effective potentialW(x0) in the case of a steadilymoving attractive impurity withb520.15 and velocityv50.1 fortime instants corresponding to the snapshots of Fig. 7. Fort,40 thesoliton performs oscillations in the deepest part of the potenthus being pulled by the impurity. Att'40 the formation of anadditional maximum is observed and the potential becomes slower, allowing the soliton to gain energy. This effect may qualitively explain the release of the soliton by the moving impurity flater times.

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FRANTZESKAKIS et al. PHYSICAL REVIEW A 66, 053608 ~2002!

For t,40 the potential is always purely attractive and tdark soliton gets trapped and is transported by the impuperforming, at the same time, small oscillations insidewell. However, at approximatelyt540 an additional maxi-mum forms and the potential becomes shallower. This allothe soliton to gain enough energy for performing largercillations ~ignoring the barrier that is going to be formed!,finally leading to an escape of the soliton from the moviimpurity.

IV. SUMMARY AND CONCLUSIONS

We have studied the interaction of a dark soliton withlocalized impurity in Bose-Einstein condensates. This sittion has been modeled by an external potential in theequation, which, apart from the conventional confining ptential of the trap, incorporates an additionald-like potential.The latter provides a phenomenological description ofpresence of a sufficiently narrow impurity compared toBEC’s length. Experimentally, such an impurity could bealized by a correspondingly focussed laser beam intersecthe condensate. The impurity potential modifies the baground Thomas-Fermi wave function by forcing a ho~hump! at the location of a repulsive~attractive! impurity.We have studied the dynamics of a dark soliton placed onof the effective TF cloud employing the perturbation theofor dark solitons earlier developed in optics@20#. Accordingto this approach, the soliton-impurity interaction is effetively described in terms of a collective variable, the solitcenter.

In the case of a repulsive impurity, we have demonstrathat the soliton behaves like a classical particle in the pence of an effective double-well potential, whose centralcal maximum is induced by the presence of the impurity.a result, the perturbation theory predicts, depending oninitial conditions, an elastic soliton reflection by the impuriand nearly elastic transmission through the impurity, bcases being accompanied by soliton oscillations in theevant spatial domains. The analytical predictions are tcompared to the results of the direct numerical integrationthe GP equation. We find that there is a fairly good agr

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ment between the analytical and numerical results forsoliton reflection and transmission through the impuriNear the critical regime we observe a quasitrapping phenenon, characterized by a relatively large interaction time asubsequent reflection or transmission of the dark soliton.

The dynamics of a dark soliton in the presence ofattractive impurity has been studied as well. In this case,have shown that the soliton motion is not qualitatively afected by the impurity, since the soliton oscillates in the saspatial domains as in the case of a purely harmonic confinpotential. Nevertheless, we have demonstrated that a bsoliton, initially placed at the bottom of the confining potetial, can be captured and dragged by an adiabatically movattractive impurity, up to a certain time inversely propotional to the dragging velocity. It is found that the impurityinduced convection of the soliton occurs for impurity velocties below a certain threshold. The release of the solitonthe impurity after finite time has qualitatively been explainby adopting the particle picture of the soliton and its motiin the relevant effective time-dependent potential.

The above results show that spatially fixed or adiabcally moving impurities represent an effective tool for mnipulating the soliton motion in the condensates: qualtively different processes can, to some extent, be controby, e.g., the coupling strength of the impurity. Higher dimesions than the effective one-dimensional situation considehere as well as further possible elements to control thetion of nonlinear excitations are relevant issues to be conered in future investigations.

ACKNOWLEDGMENTS

This work has been partially supported by the SpecResearch Account of the University of Athens. D.J.F. appciates the hospitality of the Department of Theoretical Cheistry at the University of Heidelberg. Valuable discussiowith H.E. Nistazakis~University of Athens! on numericalsimulations and with J. Schmiedmayer on matter-wave opare gratefully acknowledged. The work of Y.S.K. has bepartially supported by the Australian Research Council.

a,

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