the casemodels,

ndencely 30%.

Physics Letters A 317 (2003) 513–522

www.elsevier.com/locate/pla

Guidance of matter waves through Y-junctions

P.G. Kevrekidisa,∗, D.J. Frantzeskakisb, G. Theocharisb, I.G. Kevrekidisc

a Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003-4515, USAb Department of Physics, University of Athens, Panepistimiopolis, Zografos, 15784 Athens, Greece

c Department of Chemical Engineering, Princeton University, 6 Olden Str., Princeton, NJ 08544, USA

Received 4 April 2003; received in revised form 29 July 2003; accepted 29 August 2003

Communicated by A.R. Bishop

Abstract

We discuss the concept of Y-junctions for matter waves and investigate their guidance properties, focusing onof Bose–Einstein condensates with attractive interatomic interactions. We examine both discrete and continuousrespectively relevant to Y-junctions formed in optical lattices (OL) and magnetic traps (MT). We find a strong depeof the transmitted fraction of the condensate (in the OL case) on the angle with a maximum (per arm) of approximateIn the MT case, we show that the presence of the junction’s wedge can prevent collapse of the original condensate. 2003 Elsevier B.V. All rights reserved.

atesat-eareri-icalas

n-ac-nu-ics,lin-in-es,

imi-le

s).

[7],aceires,s ofum

atear-heun-

des,ic

ics,split-

tiveuide

1. Introduction

Since the realization of Bose–Einstein condens(BECs) of dilute atomic alkali gases [1,2], muchtention has been paid to the study of their nonlinexcitations. Thus, among others, recent BEC expments have demonstrated the creation of topologstructures such as vortices [3], vortex lattices [4],well as dark [5] and bright [6] solitons, in condesates with repulsive and attractive interatomic intertions respectively. This has raised the possibility ofmerous applications in nonlinear matter-wave physreminiscent of the situation encountered in nonear optics in the last two decades. Of particularterest is the realization of matter-wave optics devicwhich may be used to manipulate such waves slarly to “conventional” light waves. Such an examp

* Corresponding author.E-mail address: kevrekid@math.umass.edu (P.G. Kevrekidi

0375-9601/$ – see front matter 2003 Elsevier B.V. All rights reserveddoi:10.1016/j.physleta.2003.08.069

of a matter wave device is the so-called atom chipconsisting of a microfabricated semiconductor surfaccommodating atom-optics elements such as wresonators, etc. One of the long term perspectivesuch devices is their potential application to quantinformation processing.

A relevant and interesting issue is to investighow to control the motion of matter waves and, in pticular, how to guide them through junctions: as in tsituation encountered, e.g., in the field of optics, a fdamental junction is the so-calledY-junction, consist-ing of a single input port, the so-called “trunk”, antwo (or more) output ports, the so-called branchwhich form a “wedge”. Y-junctions have been a topof intense theoretical interest in the context of optas they have been proposed as the basis for beamters [8], switches [9] and logical gates [10].1 Similar

1 In the studies cited in Ref. [10], a more engineering perspecwas taken in changing the properties of a single optical waveg

.

514 P.G. Kevrekidis et al. / Physics Letters A 317 (2003) 513–522

ican”-

nestlyi-

14].staveion,n-ub-ur-epte

p-on

nceg-

u-eslds

n-tun-heasingndf aum

OLss

re-ua-.p-x-terursed) is

tovelible

Smelyew.ex-es., thens-ten-,piedxact

riveeg-

gth

illl

(ind ine

de-) is

Csheid-illldeen

hemringingc-ofhe

ideas have also appeared in the contexts of chemkinetics and catalysis, ranging from the “synchrotroof [11], to the “steering wheel” of [12] and the chemical logic gates in Belousov–Zhabotinsky membraof [13]. Moreover such configurations have recenbecome of relevance to other frontier fields of mcroscopic physics such as carbon nanotubes [Apparently, Y-junctions should be one of the mofundamental elementary components of matter-woptics devices as well, since, in idealized operatthey could be able to “divide” in a controllable maner the matter waves to (at least) two parts and ssequently guide them in specified directions. The ppose of this Letter is to examine the above concof Y-junctions for matter waves and to provide som“proof of principle” results on their guidance proerties. In particular, the present study is focusedBECs with attractive interactions, both in the preseof an optical lattice (OL) and in the presence of manetic traps (MTs).

A context where nonlinear excitations are particlarly relevant for BECs is the one of optical lattic(OL) that are generated by standing wave laser fie[15–20], in which the BEC is trapped in the antiodes of the standing waves and can coherentlynel between them through the interwell barriers. Ttunability and experimental control over the OLs hallowed for the observation of numerous interestphenomena including Bloch oscillations [17,21] aZener tunnelling [15] (in the additional presence olinear external potential) or classical [22] and quant[20] superfluid–insulator transitions among others.

One of the most advantageous features of theis the possibility (for sufficiently low temperaturewhere the mean field approximation is valid) toduce them to a discrete nonlinear Schrödinger eqtion (DNLS) in the so-called tight-binding limit [23]Even though this approximation is not always aplicable (but its applicability can be systematically eamined [24]), there are physically relevant parameranges where it can indeed be justified. This occwhen the chemical potential (which is directly relatto the number of atoms mounted on the OL [2,23]

to minimize the losses in a bend, or to install a blockerminimize losses in a junction. We should note that a similar leof engineering control is not presently (to our knowledge) feasin BEC.

lmuch lower than the height of the wells. The DNLreduction is convenient because of the large voluof earlier works in this system (in relation particularto nonlinear optics); see, e.g., [25] for a recent reviHence, BECs in the presence of OLs become anperimental laboratory where intrinsic localized modand envelope solitons can be investigated; see, e.grelevant theoretical works of [23,26]. Normally, wheapplying the tight binding approximation in the preence of the external harmonic trap, a quadratic potial is “inherited” by the resulting DNLS lattice. Ifhowever, one assumes that the size of the occusites is much smaller than the size of the trap, the eDNLS equation is retrieved:

iu̇mn = −k�2umn − 2|U |N |umn|2umn

(1)+ η(m,n)umn.

We will loosely follow the notation of [27]; howevenotice that, in our case, the nonlinearity is attractand hence we are referring to Li atoms with a (native) scattering length of≈ −1.4 nm; additionally,|U | is the absolute value of the interaction stren(see [27]). The nonlinearity prefactor will be ofO(1)

(measured in neV) which roughly means that we wconsiderN = 5 × 104 atoms loaded in the opticalattice. This number is experimentally realizablefact much higher numbers of atoms can be uselithium experiments [28]). The tunneling amplitudk will also be set to unity in our study.�2umn =um+1,n + um,n+1 + um−1,n + um,n−1 − 4um,n is thediscrete Laplacian with unit spacing. The overdotnotes temporal derivative where the time of Eq. (1multiplied by a factor of 0.6582 to convert it to mi-croseconds in physical units [27].

The additional termηu in Eq. (1) is what distin-guishes the present work from earlier studies in BE(and in particular in OLs). This term introduces teffect of the junction and is responsible for the guing of localized (solitary wave) excitations that we wbe studying in the context of OLs in BECs. It shoube noted that recently magnetic waveguides have bconstructed in [29], condensates were loaded on tusing optical tweezers and their propagation/scatte(and transmission/reflection) properties in this settwere monitored as a function of their initial veloity. One can easily envision a similar constructionY-junctions. If such an experiment is performed in t

P.G. Kevrekidis et al. / Physics Letters A 317 (2003) 513–522 515

ial,rip-

a-es-lsol-

an

i-aree

in

lnd

r ofthe

inL

erap)ill

f

in.l ofin-theme

ave

n-forinthein

ad-on-um

po-hebil-

ot-of

de.ionrion,on-hlyge,hero-

nts,dng

g.,e,thetelyn.othbe

ingichi-

t inte

eding

entng

umn,e

umlly

hehich

presence of two-dimensional optical lattice potentthen Eq. (1) becomes a relevant tight binding desction of this setup.

Similarly, to investigate if the same idea is applicble in the absence of the optical lattice, but in the prence of a configuration of magnetic trap(s), we aconsider the following continuum model, which is reevant to the continuum limit of Eq. (1). The latter cbe rewritten in a rescaled form as follows:

(2)iu̇ = −�u − |u|2u + η(x, y)u,

whereη(x, y) in Eq. (2) is used in the form of quasone-dimensional magnetic trap potentials whichdirectly relevant to BEC (see, e.g., [2] for details). Wwill thus study numerically the behavior of matterthe setting of a Y-junction formed byη(m,n) (for acondensate in an OL) or byη(x, y) (for a condensatein a MT). Especially, we will study the potentiaof these junctions for transmission or trapping areflection, the relevant effects of asymmetry and/othe angle of the wedge composed by the arms ofjunction.

Our presentation will be structured as follows:Section 2, we will present the results on the Oof Eq. (1), while in Section 3, we will give somrepresentative results of the parabolic (magnetic tchannel wedge of Eq. (2). Finally in Section 4, we wbriefly summarize our findings and conclude.

2. Y-junctions in the presence of the optical lattice

In the case of the OL, our initial condition will be othe formumn(t = 0) ∼ Asech(rmn)exp(iα(n − n0)),where rmn =

√(m − m0)2 + (n − n0)2 and 2A =

α = 1 in the numerical experiments reported hereThe results however have been found to be typicasmall amplitude (and hence higher mobility) nonlear wave excitations. The phase factor provideswave with an initial speed (cf. with the continuulimit [30]), along the guiding direction of the negativx-semiaxis (so as to study the interaction of the wwith the junction).

The first “experiment” performed in the two-dimesional DNLS setting (where collapse is arrested“sufficient” discreteness [25]) is the one reportedFig. 1. The case of a waveguiding wedge alongprincipal diagonal of the square lattice is reported

this case. The top panel shows an appropriatelyjusted contour plot (in such a way as to show the cdensate as dark in the grayscale in a white vacubackground). It also shows the contour plot of thetential η(m,n) in this case of 45 degree angle. Ttop subplot in the upper panel indicates the probaity (norm ratio R) of finding the particle in the topright quadrant of the figure (solid line) and in the btom right one (dashed line). Given the symmetrythe configuration in this case, the two lines coinciIt is worth noting in this case that the transmissreaches a maximum rateR (of ≈ 21% in this case, fot ≈ 13) when the condensate arrives at the junctbut thereafter decreases falling to an approximate cstant value of 4.5% in each quadrant. Hence, rougonly 10% of the total norm passes through the wedwhile the rest is reflected. This is mirrored also in tlower panel showing the two-dimensional spatial pfile of η and of the field’s norm fort = 25 µs.

We have also performed numerical experimeshown in Fig. 2, in which the wedge is formebetween the diagonal and the principal axis alowhich the original waveguiding is performed (e.between thex-axis and a 45 degree tilt). In this casalmost all the condensate moves in the direction oforiginal axis. Approximately 79% of the condensais transmitted along the principal axis, while on5.6% (att = 20 µs) moves along the tilted directioThe remaining power is reflected. Notice that bthis and all other numerical experiments shouldcompared with the case of straight line waveguidin the 2d case (for the present setting), in wh92.1% of the norm is “transmitted” to the right semplane. This comparison is necessary, given thathe 1d DNLS problem there only exist approximatravelling solutions in the lattice setting (comparto exact travelling solutions where the correspondtransmission percentage would have been 100%).

It is interesting to compare the above experimwith the case in which the waveguiding is done alothey-principal axis (as opposed to thex-principal axisin the previous example). In that case the maximtransmission rateR is 2% in the 45 degree directiowhile it is 15% in they-axis direction. However, oncagain the rate decreases after reaching the maximvalue (recall that the maximum value essentiaindicates the “arrival” of the discrete soliton at twedge). The rate finally reaches a steady state in w

516 P.G. Kevrekidis et al. / Physics Letters A 317 (2003) 513–522

Fig. 1. The upper panel shows the contour plot of norm|umn|2 for t = 25 µs (top subplot) and the corresponding contour plot of the wedge(middle subplot). Notice that, for reasons of clarity, in the two-dimensional, grayscale contour plots, theopposite of the norm is shown (cf. withthe corresponding colour scales to the right of the plot). The time shown is selected so that the transmitted ratioR has reached a constant value(so as to reflect the fraction of particles that have indeed “tunnelled” through the junction). The bottom subplot shows the ratio of the powerthat is in the top right quadrant and in the bottom right quadrant of the figure (in the two components of the wedge). The 3d representation ofη

and of the norm|umn|2 is shown in the lower panel of the figure.

P.G. Kevrekidis et al. / Physics Letters A 317 (2003) 513–522 517

Fig. 2. The same as the upper panel of Fig. 1, but for the case in which one of the two guiding directions of the wedge is thex-principal axis.The solid line of the bottom panel indicates the bottom left quadrant particle percentage.

Fig. 3. The same as the upper panel of Fig. 1, but for the case in which one of the two guiding directions of the wedge is they-principal axis.The dashed line in the bottom panel corresponds to the bottom right quadrant (including the guidingy-semiaxis), while the solid correspondsto the wedge arm of 45 degrees (top right quadrant).

518 P.G. Kevrekidis et al. / Physics Letters A 317 (2003) 513–522

his case.

Fig. 4. The same as in the upper panel of Fig. 1, but for a wedge angle of 51 degrees. Notice that the norm ratio is much higher in t

ndn

bein

s.do

uchin

s anofiresisfornday

gles

hisnaltureionlsolece)

ularepartthe

dgeone1 (atriche

iresreeitmalhe

as

of

eryng)

ofhes”tureThis

only 0.5% of the total power is along the diagonal, aonly 3% along they-axis. This situation is shown iFig. 3 for t = 20 µs.

From the above findings, the impression mayformed that the wedge waveguiding in OLs resultsvery low probability yield along the wedge directionHowever, this is not always true. Even though wenot report a systematic study ofR as a function ofthe angle here, we give a case example with mhigher yield than the previous ones. In particularFig. 4 we consider the case where the wedge haangle of≈ 51 degrees with the incoming directionthe pulse. Here each of the two directions acquabout 30% of the original incoming pulse (whichclose to the maximum that we have observedsymmetric wedges; cf. also below). Hence, we fithat the fraction of the transmitted condensate mstrongly depend on the angle, but there are anof considerably high transmission factor (≈ 60% istransmitted along both arms of the wedge in tcase). As a partial explanation of the strong directiodependence of the results, we refer to the lattice naof the substrate. In particular, the transmitted fractdoes not depend only on the junction angle but acritically on how “close” the direction of a given angis to the lattice points of the underlying (square latti

substrate. Hence the delicate interplay of the angdirection of the junction ports with the “effectivdiscreteness” (how many lattice sites and how far afrom each other the port “sees”) is responsible forrich phenomenology reported herein.2

Finally, we also considered cases where the wehad a small asymmetry, such as a wedge in whichof the angles was 45 degrees while the other was 5combination of the cases reported above for symmewedges). In this case the result is shown in Fig. 5. T51 degree “arm” of the wedge in this case acqumore than 47% of the total norm, while the 45 degone only receives≈ 4.3% of the total norm. Hence,is clear that asymmetric wedges may be the optiway to maximize the power in a single arm. Tremaining fraction of the condensate is reflected

2 One more issue that can critically affect the fractiontransmitted atoms is how “wide” the potentialη is, i.e., howmuch it deviates from one-dimensionality. Herein, we examine vnarrow (of width much smaller than the optical lattice spacisuch potentials. Naturally the results will be different, if insteadnarrow one-dimensional ports, one uses finite-width “road patcto construct a river delta type of junction. In the latter case, curvarelated phenomena of the type observed in Ref. [31] may arise.topic will be considered in future investigations.

P.G. Kevrekidis et al. / Physics Letters A 317 (2003) 513–522 519

Fig. 5. The same as in Fig. 1, but for one arm of the wedge at 51 degrees, when the other is at 45. The dashed line shows the fraction of thecondensate along the 51 degree arm, while the solid along the 45 degree one.

520 P.G. Kevrekidis et al. / Physics Letters A 317 (2003) 513–522

on

n-e,

lly)the

le

edric

d

h),t)else

ot

of

topn;ithlyt isilelot

ere,

tert isngtelyhe

erege).

ut%%of

tric

kethely

nal

edndeirthethextg-

wetion

Thisof

d tortedate

tricthence

pal

small amplitude radiation along the original directi(of the negativex-semiaxis).

3. Y-junctions formed by magnetic trap channels

In the case of the continuum limit of Eq. (2), we evision that the Y-junction, and particularly the wedgcan be formed by an appropriately (geometricaplaced array of magnetic traps. Thus, we considerfollowing form of the external potential,

η(x, y) =

y2, x � 0,

(y − a1x)2, x > 0, y � 0,

(y + a2x)2, x > 0, y < 0,

wherea1 anda2 are then directly related to the angof the wedge directions with the positivex-semiaxisaccording toφ1,2 = arctan(a1,2).

For reasons of generality, as no exact localizsolution of Eq. (2) is known, we considered a gene(Gaussian) initial condition of the form:

(3)u(x, t) = Aexp

[−

(r

R

)2]

exp(ikx)

with amplitudeA = 1, wavenumberk = 1, widthR =1 andxc = −5,yc = 0, the coordinates of the localizewavepacket center inr = √

c(x − xc)2 + (y − yc)2.The factorc, among other things (such as the widtdetermines whether wave focusing will (or will nooccur in this 2-dimensional version of the modwhich, as is well known, is susceptible to collap[30].

We examine two cases:

• c = 1 where localization is weaker and does nlead to collapse (case (a)) and

• c = 0.1 (case (b)). The latter, in the absencethe wedge (i.e., forη(x, y) = y2,∀x, y) leads tocollapse in finite time.

Fig. 6 shows two subcases of case (a). Thepanel shows the contour plot of the initial conditiothe middle panel exhibits the symmetric case wa1 = a2 = 0.35 (i.e., for an angle of approximate19 deg). In this subcase, 40% of the wavepacketransmitted in each of the wedge directions, whapproximately 20% is reflected. The contour p

is shown for t = 15. On the other hand, for thasymmetric case of the bottom panel of the figua1 = 0.2 (angle of approximately 11 deg), whilea2 =0.5 (i.e., angle of approximately 26.5 deg). In the latcase, the transmitted fraction of the wavepacke45% along the 11 deg angle, while it is 35% alothe 26.5 deg angle. In this case also, approxima20% of the initial wavepacket is reflected from twedge.

Fig. 7 shows two subcases of case (b) (whcollapse would occur in the absence of the wedWhat we observe is that the wedgetears the localizedinitial condition into two equal pieces almost withoreflection (the reflected fraction is approximately 0.5in the symmetric case, while it is approximately 2in the asymmetric one). The symmetric subcasethe middle panel of the figure results in a symmetransmission of 49.75% of the initial packet fora1 =a2 = 0.2. The asymmetric case witha1 = 0.2 anda2 = 0.5 yields transmission of 62% of the wavepacin the former direction, while 36% is transmitted in tlatter (bottom panel of the figure). More interestingin both of the above cases the splitting of the origiwavepacketprevents collapse from occurring.

4. Conclusions

In conclusion, in this work we have discussthe concept of Y-junctions for matter waves awe have presented an initial exploration of thguidance properties. We focused particularly onpossibility of guiding of matter waves in BECs wiattractive interatomic interactions, both in the contof 2-dimensional optical lattices (the so-called “egcartons”) and magnetic trap channel wedges.

In the quasi-discrete case of the optical latticehave found a strong angle dependence in the fracof the transmitted condensate along the wedge.fraction can be maximized upon appropriate choicethe angle and the maximum per arm has been founbe approximately 30% in the relevant cases repoherein. Furthermore, the fraction of the condensin a single arm can be maximized in asymmewedge configurations and can be made to be oforder of 50%. We have also explored the dependeof the relevant fractions in cases where princi

P.G. Kevrekidis et al. / Physics Letters A 317 (2003) 513–522 521

Fig. 6. The top panel shows the initial condition contour plot for the magnetic trap channel wedge. The middle panel shows the propagation ofthis initial condition through the symmetric wedge (witha1 = a2 = 0.35), while the bottom panel through the asymmetric one witha1 = 0.2anda2 = 0.5. The time in Eq. (2) for which the middle and bottom panels are reported ist = 15.

Fig. 7. Same as Fig. 6. The top panel shows the initial condition. The middle panel show the symmetric wedge result fora1 = a2 = 0.2, whilethe bottom shows the asymmetric case fora1 = 0.2 anda2 = 0.5. The time reported in the latter two panels ist = 25.

522 P.G. Kevrekidis et al. / Physics Letters A 317 (2003) 513–522

dge

theveds aorethevedena

seza-reinetry

nein-ob-olicpe

Re

sitype-nsandfor

d.

. 88

3.

87

1)

pt.

100

.

od.

ev.

is/

rö-9,

62)

direction axes are included in the arms of the weconfiguration.

In the continuum case of the wedge formed byarray of the magnetic traps, we have also obserconsiderable variation in the transmission ratio afunction of the angle. We have also, perhaps minterestingly, demonstrated that the presence ofwedge can tear the initially localized matter wain half (more efficiently for more strongly localizewavepackets), preventing collapse type phenomdue to the splitting.

It would be very interesting to examine how theresults compare with potential experimental realitions of the geometric configurations presented heand more generally to examine the effects of geomon the dynamics of matter waves.

Furthermore, it would be of interest to examisimilar phenomena in other contexts, such as forstance synaptic bifurcations relevant to neural prlems, where the equations would be of the parab(rather than nonlinear wave), Fitz–Hugh Nagumo ty[32].

Acknowledgements

This work was partially supported by AFOSand NSF/ITR (I.G.K.), by NSF (DMS-0204585), thEppley Foundation for Research and the Univerof Massachusetts (P.G.K.), as well as by the Scial Research Account of the University of Athe(G.T., D.J.F.). P. Schmelcher and J. ShmiedmayerD. Christodoulides are gratefully acknowledgedvaluable discussions.

References

[1] M.H. Anderson, et al., Science 269 (1995) 198;K.B. Davis, et al., Phys. Rev. Lett. 75 (1995) 3969;C.C. Bradley, et al., Phys. Rev. Lett. 75 (1995) 1687.

[2] F. Dalfovo, S. Giorgini, L.P. Pitaevskii, S. Stringari, Rev. MoPhys. 71 (1999) 463.

[3] M.R. Matthews, et al., Phys. Rev. Lett. 83 (1999) 2498;K.W. Madison, et al., Phys. Rev. Lett. 84 (2000) 806;S. Inouye, et al., Phys. Rev. Lett. 87 (2001) 080402.

[4] J.R. Abo-Shaeer, et al., Science 292 (2001) 476;

J.R. Abo-Shaeer, C. Raman, W. Ketterle, Phys. Rev. Lett(2002) 070409;P. Engels, et al., Phys. Rev. Lett. 89 (2002) 100403.

[5] S. Burger, et al., Phys. Rev. Lett. 83 (1999) 5198;J. Denschlag, et al., Science 287 (2000) 97;B.P. Anderson, et al., Phys. Rev. Lett. 86 (2001) 2926.

[6] K.E. Strecker, et al., Nature 417 (2002) 150;L. Khaykovich, et al., Science 296 (2002) 1290.

[7] D. Müller, et al., Phys. Rev. Lett. 83 (1999) 5194;N.H. Decker, et al., Phys. Rev. Lett. 84 (2000) 1124;D. Müller, et al., Opt. Lett. 25 (2000) 1382;D. Cassettari, et al., Phys. Rev. Lett. 85 (2000) 5483;J. Reichel, et al., Appl. Phys. B 72 (2001) 81;R. Folman, et al., Phys. Rev. Lett. 84 (2000) 4749;W. Hansel, et al., Nature 413 (2001) 498;R. Folman, et al., Adv. Atom. Mol. Opt. Phys. 48 (2002) 26

[8] G.-D. Chang, et al., J. Opt. Commun. 21 (2000) 4.[9] T.C. Huang, et al., Opt. Lett. 19 (1994) 2107.

[10] D.N. Christodoulides, E.D. Eugenieva, Phys. Rev. Lett.(2001) 233901;D.N. Christodoulides, E.D. Eugenieva, Opt. Lett. 26 (2001876;D.N. Christodoulides, N. Efremidis, E.D. Eugenieva, OLett. 26 (2001) 1978.

[11] M.D. Graham, et al., Science 264 (1994) 80.[12] X. Li, et al., Chaos 12 (2002) 190.[13] O. Steinbock, P. Kettunen, K. Showalter, J. Phys. Chem.

(1996) 18970.[14] A. Rochefort, Ph. Avouris, Nanoletters 2 (2002) 253.[15] B.A. Anderson, M.A. Kasevich, Science 282 (1998) 1686.[16] C. Orzel, et al., Science 291 (2001) 2386.[17] O. Morsch, et al., Phys. Rev. Lett. 87 (2001) 140402.[18] F.S. Cataliotti, et al., Science 293 (2001) 843.[19] M. Greiner, et al., Phys. Rev. Lett. 87 (2001) 160405.[20] M. Greiner, et al., Nature (London) 415 (2002) 39.[21] D.I. Choi, Q. Niu, Phys. Rev. Lett. 82 (1999) 2022.[22] F.S. Cataliotti, et al., cond-mat/0207139;

See also A. Smerzi, et al., cond-mat/0207172.[23] A. Trombettoni, A. Smerzi, Phys. Rev. Lett. 86 (2001) 2353[24] G.L. Alfimov, et al., Phys. Rev. E 66 (2002) 046608.[25] P.G. Kevrekidis, K.Ø. Rasmussen, A.R. Bishop, Int. J. M

Phys. B 15 (2001) 2833.[26] F.K. Abdullaev, et al., Phys. Rev. A 64 (2001) 043606.[27] G. Kalosakas, K.Ø. Rasmussen, A.R. Bishop, Phys. R

Lett. 89 (2002) 030402.[28] See, e.g., http://www.lkb.ens.fr/recherche/atfroids/angla

lithium.html and the original references mentioned therein.[29] A.E. Leanhardt, et al., Phys. Rev. Lett. 89 (2002) 040401.[30] See, e.g., C. Sulem, P.-L. Sulem, in: The Nonlinear Sch

dinger Equation, Vol. 139, Springer-Verlag, New York, 199p. 21.

[31] G. Haas, et al., Phys. Rev. Lett. 75 (1995) 3560.[32] R. FitzHugh, Biophys. J. 1 (1961) 445;

J.S. Nagumo, S. Arimoto, S. Yoshizawa, Proc. IREE 50 (192061.