Statics, Dynamics and Manipulations of Bright Matter-Wave Solitons in Optical Lattices

0.73 MB·English

by  P. G. Kevrekidis

Statics, Dynamics and Manipulations of Bright Matter-Wave Solitons in Optical
Lattices
P.G. Kevrekidis1, D.J. Frantzeskakis2, R. Carretero-Gonza´lez3, B.A. Malomed4, G. Herring1 and A.R. Bishop5
1 Department of Mathematics and Statistics, University of Massachusetts, Amherst MA 01003-4515, USA
2 Department of Physics, University of Athens, Panepistimiopolis, Zografos, Athens 15784, Greece
3 Nonlinear Dynamical Systems Group, Department of Mathematics and Statistics,
San Diego State University, San Diego CA, 92182-7720, USA, http://nlds.sdsu.edu/
5
4 Department of Interdisciplinary Studies, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
0
5 Center for Nonlinear Studies and Theoretical Division,
0
2 Los Alamos National Laboratory, Los Alamos, NM 87545, USA
(Dated: In Press Phys.Rev.A, 2005)
n
a Motivated by recent experimental achievement in the work with Bose-Einstein condensates
J (BECs), we consider bright matter-wave solitons, in the presence of a parabolic magnetic trap

3 and a spatially periodic optical lattice (OL), in the attractive BEC. We examine pinned states of
thesolitonandtheirstabilitybymeansofperturbationtheory. Theanalyticalpredictionsarefound

] to be in good agreement with numerical simulations. We then explore possibilities to use a time-
r
modulatedOLasameansofstoppingandtrappingamovingsoliton,andoftransferringaninitially
e
h stationarysolitontoaprescribedposition byamovingOL.Wealsostudytheemission ofradiation
t fromthesolitonmovingacrossthecombinedmagnetictrapandOL.Wefindthatthesolitonmoves
o
freely (without radiation) across a weak lattice, but suffers strong loss for stronger OLs.
.
t
a
m
I. INTRODUCTION [19]andquantum[13]superfluid-insulatortransitions. A
- large amount of theoretical work has been already done
d
for nonlinearMWs trapped in OLs(see Refs. [20, 21] for
n The recent progress in experimental and theoretical
o studies of Bose-Einstein condensates (BECs) [1] has led recent reviews).
c to an increase of interest in matter-wave (MW) solitons. The objective of this work is to systematically study
[ the statics and dynamics of one-dimensional(1D) bright
One-dimensional (1D) dark [2] and bright [3] solitons
1 havebeenobservedinexperimentswithrepulsiveandat- MWsolitonsconfinedinthecombinationoftheparabolic
v tractiveBECs,respectively. Veryrecently,brightsolitons magnetic trap (MT) and OL. Additionally, we examine
1 the possibility to control the motion of the soliton by
of the gap type, predicted in repulsive condensates [4],
3 means of a time-dependent OL potential (the latter is
havebeencreatedintheexperiment[5]. Theoreticalpre-
0 availablefortheexperiment). Inparticular,wewillshow
dictionsconcerningapossibilityoftheexistenceofstable
1
that,inthecasewhentheOLperiodiscomparabletothe
0 multi-dimensional solitons supported by a full [4, 6] or
characteristic spatial width of the soliton, it is possible
5 low-dimensional [7] optical lattice (OL) have also been
0 reported. The OL is created as a standing-wave inter- to: (a) snare and immobilize an originally moving soli-
/ ference pattern between mutually coherent laser beams ton in a local potential well, by adiabatically switching
t
a [8, 9, 10, 11, 12, 13]. theOLon,and(b)graspanddraganinitiallystationary
m soliton by a slowly moving OL, delivering it to a desired
ThestudyoftheMWsolitons,apartfrombeingafun-
- damentally interesting topic, may have important appli- location. Note that bright MW solitons may travel long
d distances in the real experiment, up to several millime-
cations. In particular, a soliton may be transferred and
n
ters [3], and are truly robust objects, being themselves
o manipulated similarly to what has been recently demon-
coherent condensates. Thus, the manipulation of bright
c strated, experimentally and theoretically, for BECs in
: magnetic waveguides [14] and atom chips [15]. More MW solitons is a very relevant issue for the physics of
v
BECs.
i generally, the similarity between bosonic MWs and light
X The paper is organizedasfollows. InSec. II, we intro-
waves suggests that numerous results known for opti-
r cal solitons [16], along with the possibility of manipu- ducethemodelandpresentanalyticalresults. InSec.III,
a we numerically investigate static and dynamical proper-
lation of atomic states (by means of resonant electro-
ties of the solitons,and study possibilities to manipulate
magnetic waves governing transitions between different
themasoutlinedabove. Theresultsoftheworkaresum-
states), may have impact on the rapidly evolving field of
marized in Sec. IV.
quantum atom optics (see, e.g., Ref. [17]).
A context where the dynamics of MW solitons is par-
ticularly interesting is that of BECs trapped in a peri-
odicpotentialinducedbytheabove-mentionedOLs. The II. THE MODEL AND ITS ANALYTICAL
CONSIDERATION
possibility to control the OL has led to the realization
of many interesting phenomena, including Bloch oscilla-
tions[10,18],Landau-Zenertunneling[8](inthepresence The Gross-Pitaevskii equation (GPE), which governs
of an additional linear external potential), and classical the evolution of the single-atom wave function in the
2
mean-field approximation,takes its fundamental form in Stationary positions of the soliton correspond to local
the 3D case. A number of works analyze its reduction extrema of the effective potential (4). This well-known
to an effective 1D equation in the case of strongly elon- heuristicresultcanberigorouslysubstantiatedbymeans
gated cigar-shaped BECs [22, 23, 24]. In particular, the of the Lyapunov-Schmidt theory applied to the under-
derivationinRef. [23]assumedthatthepotentialenergy lying nonlinear Schr¨odinger equation [28]. The effective
ismuchlargerthanthe transversekineticenergy. Agen- potentialcorrespondingtotheexternalpotential(2),act-
eral conclusion is that the effective equation reduces to ingonthestationarysoliton(3),canbeeasilyevaluated:
the straightforward 1D version of the GPE. In the nor-
malized form, it reads [20]
kπ
1 V (x )=ηΩ2x2 πV kcos(2kx )csch . (5)
iu = u +g u2u+V(x)u, (1) eff 0 0− 0 0 (cid:18) η (cid:19)
t xx
−2 | |
where u(x,t) is the 1D mean-field wave function (al- Notice that, depending on values of the parameters,this
though a different version of the 1D GPE, with a non- potentialmayhaveasingleextremumatx =0,ormul-
0
polynomial nonlinearity, is known too [24]). The combi- tiple ones.
nation of the MT and OL potential corresponds to The stability of the soliton resting at a stationary po-
sition can also be analyzed in terms of the effective po-
1
V(x)= Ω2x2+V0sin2(kx). (2) tential (4): the position is stable if it corresponds to a
2
potential minimum. This well-known result can be rig-
In Eq. (1), x is measured in units of the fluid heal-
orously derived using the theory of Ref. [29] and refor-
ing length ξ = ~/√n0g1Dm, where n0 is the peak den- mulated in Ref. [30] (see also Refs. [31] and [32]). In
sity and g1D g3D/(2πl⊥2) is the effective 1D interac- particular, the curvature of the potential at the station-
≡
tion strength, obtained upon integrating the 3D interac- ary position determines the key linearization eigenvalue
tion strength g3D = 4π~2a/m in the transverse direc- λ, that may cause aninstability, bifurcating throughthe
tions (a is the scattering length, m the atomic mass, originofthe correspondingspectralplane(thisfeatureis
and l⊥ = ~/mω⊥ is the transverse harmonic oscil- revealedby the heuristic [26] and rigorous[30] analysis).
lator lengthp, with ω⊥ being the transverse-confinement The eigenvalue is easily found to be
frequency). Additionally, t is measured in units of ξ/c
(where c = n g /m is the Bogoliubov speed of λ2 = η−1/2V′′(x ), (6)
0 1D − eff 0
sound), the atopmic density is rescaled by the peak den-
sity n , and energy is measured in units of the chemical confirming that minima and maxima of the effective po-
0
potential of the system µ = g n . Accordingly, the tential (4) give rise, respectively, to stable (λ2 < 0) and
1D 0
dimensionless parameter Ω ~ω /g n (ω is the con- unstable (λ2 >0) equilibria.
x 1D 0 x
≡
fining frequency in the axial direction) determines the We note in passing (this will be important in what
magnetic trapstrength,V is the OLstrength, while k is follows) that the minima of the effective potential (4)
0
the wavenumber of the OL; the latter, can be controlled differ from the minima of the external potential V(x)
by varying the angle θ between the counter-propagating trapping the atoms. For instance, for η = √2, V =
0
laser beams, so that λ 2π/k =(λlaser/2) sin(θ/2) [25]. 0.25 and Ω = 0.1, the first three minima of V(x) (apart
≡
Finally, g = 1 is the renormalizednonlinear coefficient, from the one at x = 0) are located at the points x =
±
which is positive (negative) for a repulsive (attractive) 3.0789,6.1587,9.2356,whiletheminimaofV arefound
eff
condensate. As we are interested in the ordinary bright at x =3.0166,6.0247,9.0089.
0
MW solitons, which exist in case of attraction, we will We now turn to numerical results, aiming to examine
fix g = 1. the validity of the theoretical predictions, as well as to
−
Without the external potential (Ω = V0 = 0), Eq. (1) perform dynamical experiments using the OL to guide
supports bright soliton solutions of the form the soliton motion.
1
u (x x )=η sech[η(x x )]exp iη2t , (3)
s 0 0
− − (cid:18)2 (cid:19)
III. NUMERICAL RESULTS
where η is the soliton’s amplitude, and x is the coordi-
0
nateofitscenter. Movingsolitonscanbegeneratedfrom
A. Stability of the solitons
the zero-velocity one by a Galilean boost.
Inthepresenceoftheexternalpotential,thefirstissue
Webeginthenumericalpartbyexaminingthesteady-
is to identify stationary positions for the soliton. This
state soliton solutions and their stability in the context
issue can be addressed, using an effective potential for
of Eq. (1). Such solutions are sought for in the form
the soliton’s central coordinate (see, e.g., Refs. [26] and
u(x,t) = exp(iΛt)w(x), which results in the stationary
[27]), which is defined by the integral
equation,
+∞
V (x )= V(x)u (x x )2dx. (4)
eff 0 Z−∞ | s − 0 | Λw =(1/2)wxx+w3 V(x)w. (7)
−
3
To examine the linear stability of the solitons, we take a 2.805 1.6
2.8 1.4
perturbed solution as P2.795 1.2
u(x)=eiΛthw+ǫ(cid:16)a(x)e−iωt+b(x)eiω∗t(cid:17)i, (8) 20.7.49 0.05 0.1 V0 0.15 0.2 0.25 u(x)00..681
0.38
whereǫandω areaninfinitesimalamplitudeand(gener- W0.36 0.4
ally speaking, complex) eigenfrequency of the perturba- 0.34 0.2
tion, and linearize Eq. (1). 0.05 0.1 V0 0.15 0.2 0.25 0−10 −8 −6 −4 −2 0x2 4 6 8 10
2.87 1.6
Equations (1) and (2), with g = 1 and arbitrary co-
− 1.4
efficients V0,Ω andk, possessa scalinginvariance,which P2.86 1.2
allows us to fix Λ = 1 (hence η = √2). It should be 2.85 1
noted that, in the absence of the MT (Ω = 0), the soli- 0.60.060.08 0.1 0.120.14V00.160.18 0.2 0.220.24 u(x)0.8
ton’s frequency should be chosen so that it belongs to a 0.55 0.6
bandgap in the spectrum of the linearized Eq. (1) with W00.4.55 0.4
0.4 0.2
tBhleocpherwioadviecs.poHteonwteiavler(,2)t,hteoMavToidporteesnotniaanlcweitwhitfihnliitneeaΩr 03..33550.060.08 0.1 0.120.14V00.160.18 0.2 0.220.24 02−10 −8 −6 −4 −2 0x2 4 6 8 10
makes this condition irrelevant. In principle, it might be 1.8
P3.34 1.6
interestingtoinvestigatehowtheincreaseofΩfromzero 1.4
gbruatdtuhaisllymloifrtesftohremcaolnisdsiuteioinsolefftthbeeyroesnodntahnecescaovpoeidoafntchee, 30.3.830.15 0.2 V0 0.25 u(x)01..821
0.7
present work. 0.6 0.6
W0.5 0.4
Toestimateactualphysicalquantitiescorrespondingto 0.4 0.2
the above normalized values of the parameters, we con- 0.30.15 0.2 V0 0.25 0−5 0 5 x 10 15
sider a cigar-shaped 7Li condensate containing N 103
atoms in a trap with ωx = 2π 25 Hz and ω⊥ =≃70ωx. FIG. 1: For each of the three sets of the pictures, the left
Then, for a 1D peak density n×= 108 m−1, the param- panel shows the continuation of the soliton branch to val-
0 ues near V(cr), at which it disappears (for soliton solutions
eter Ω in Eq. (2) assumes the value Ω = 0.1, while the 0
trapped at different wells). The right panel shows the solu-
time and space units correspond to 0.3 ms and 1.64µm,
tionattheinitialandfinalpointsofthecontinuation(andthe
respectively. These units remainvalidfor othervaluesof
corresponding potentials). The left panels show the norm of
Ω, as one may vary ω⊥ and change ωx accordingly; in the soliton solution (proportional to the number of atoms in
this case, other quantities, such as N, also change. the condensate), P = +∞|u(x)|2dx, and its squared width,
Figure1summarizesournumericalfindingsforthesta- W =P−1 +∞x2|u(x)R|−2d∞x,as a function of the OL strength
bility problem. As expected, the (zeroth-well) solution −∞
forthe solitonpinnedatx =0exists anditisstable for V0. ThetoRpsetofthepanelspertainstothezeroth-wellsolu-
0 tion(thesolitonpinnedinthecentralpotentialwell);thesolu-
all values of the potential’s parameters. We have typi-
tionintherightpanelisshownbythesolidlineforV =0.25,
0
cally chosen to fix Ω = 0.1 and k = 1 (i.e., λ = 2π) and and by the dash-dotted line for V = 0. The corresponding
0
vary V0; however, it has been checked that the results potential is shown by the dotted line for V0 = 0.25, and by
presented below adequately represent the phenomenol- the dashed line for V = 0. Similarly, in the middle set, the
0
ogy for other values of (Ω,k) as well. solid line (and the dashed one for the potential) correspond
Thenext(first-well)solution,correspondingtothepo- to V0 = 0.25, and the dashed-dotted line, together with the
tentialminimumclosesttox0 =0,existsforvaluesofthe dotted one for the potential, correspond to V0 = 0.06 for
MTstrengthV smallerthanacriticaloneV(cr). Within the first-well solution [notice that this branch terminates at
0 0 V ≈ 0.045]. Finally, in the bottom set of the panels, the
theaccuracyof0.0025,wehavefounditsnumericalvalue 0
solid line (and the dashed one for the potential) again cor-
tobeV0(cr)|num =0.045,inverygoodagreementwiththe respond to V0 = 0.25, while the dashed-dotted line (and the
prediction following from the analytical approximation dotted one for thepotential) correspond toV =0.15 for the
0
(5) for the effective potential, which shows that the cor- third-well solution [this branch terminates at V0≈0.1425].
respondingpotentialminimumdisappears,mergingwith
a maximum, at V(cr) 0.048. The corresponding
0 |anal ≈
pinned-soliton solution is indeed stable prior to its dis-
appearance, in agreement with the analytical prediction
based on Eq. (6).
It is quite natural that the discrepancy between the
Similarly, the subsequent (second-well) solution, as-
theoretical and the numerical results increases for the
sociated with the next potential minimum (if it ex-
higher-well solutions, given that the numerically exact
ists), is found to disappear (for the same parameters)
profile of the pinned soliton gets more distorted under
at V0(cr)|num = 0.1 ± 0.0025, while the analytical ap- the actionofthe MT. Notice,for example, the difference
proximation (5) yields V(cr) 0.112. Finally, a intheamplitudebetweenthesolitoninthetoppaneland
0 |anal ≈
similar result was obtained for the third-well solution: in the one in the bottom panel in Fig. 1, which clearly
V(cr) =0.1425 0.0025, and V(cr) 0.176. illustrates this effect.
0 |num ± 0 |anal ≈
4
B. Soliton dynamics and manipulations 180 −2 2.2
6 0 12.8
4 2 1.6
Havingaddressedtheexistenceandstabilityofthesoli- 2 1.4
tons, we now proceed to study their possible dynamical x−20 x 4 11.2
manipulationbymeansofthe OL.First,weexaminethe −4 6 0.8
0.6
possibility to trap a soliton using the secondary minima −6 8 0.4
−8 0.2
in the OL potential. In particular, it is well known that, −10 10
0 100 200 300 400t500 600 700 800 0 100 200 300 400t500 600 700 800
in the absence of the OL, the soliton in the magnetic −2
14 2
trap, when displaced from the center, x0 = 0, executes 12 0
harmonic oscillations with the frequency Ω, as a conse- 10 2 1.5
quence of the Ehrenfest theorem (alias the Kohn’s the- 8 4
x 6 x 6 1
orem [33], which states that the motion of the center of 4
8
massofacloudofparticlestrappedinaparabolicpoten- 2 0.5
0 10
tial decouples from the internal excitations). This result
−2 12 0
canalsobeobtainedusingthevariationalapproximation 0 100 200 300 400t500 600 700 800 0 100 200 300 400t500 600 700 800
10 −10
[34] and, more generally, is one of the results obtained −8
8 2
from the moment equations for the condensate in the −6
6 −4
parabolic potential [35]. 4 −2 1.5
x x0
2 2 1
1680 −−1150 122..82 −02 468 0.5
x 024 x−50 1111...246 1−8040 100 200 300 400t500 600 700 800 −1−10080 100 200 300 400t500 600 700 800 022.2
−2 5 0.8 6 −6 1.8
−4 0.6 4 −4 1.6
−6 10 0.4 2 −2 1.4
−8 0.2 x 0 x 0 1.2
0 100 200 300 400t500 600 700 800 150 100 200 300 400t500 600 700 800 −2 2 01.8
−4 4 0.6
−6 6 0.4
FIG. 2: (color online) An example of snaring the originally −8 8 0.2
movingsoliton usingtheopticallattice. Theleftpanelshows −100 100 200 300 400t500 600 700 800 100 100 200 300 400t500 600 700 800
the motion of the soliton’s center of mass. The dashed line
showsthesituationwithouttheOL(butinthepresenceofthe FIG. 3: (color online) Panels have the same meaning as in
magnetictrap). IfweturnontheOLpotential,asthesoliton Fig. 2, but now for the case of a moving OL. The left panel
arrives at theturningpoint of its trajectory, it gets captured shows the soliton’s center of mass by the solid line and the
by the secondary minimum of the full potential, created in motion of the OL’s center by the dashed line. The potential
a vicinity of this point. The right panel shows the same, V(x,t = 0) is sketched by the dash-dotted line to illustrate
butthroughthespace-timecontourplotsofthelocaldensity, the structure and location of the potential wells. The right
|u(x,t)|2. panel again shows the space-time contour plot of |u(x,t)|2.
Thetopsetofthepanelsisgeneratedwitht =50andτ =5
0
in Eq. (9). The second set pertains to t = 100 and τ = 10
A new issue is whether one can capture the soliton 0
(both have V = 0.25). The situation for a shallower well,
performing such oscillations by turning on the OL. Fo- 0
with V = 0.17, is shown in the third and fourth panels. In
0
cusing,aspreviously,onthemostrelevantcasewhenthe
all cases, x =0 and x =3π.
ini fin
widthofthesolitoniscomparabletotheOLwavelength,
we display an example of the capture in Fig. 2. The
dashed and solid lines show, respectively, the harmonic
oscillationsintheabsenceoftheOL,andanumericalex- nowfully captured(for verylongtimes) by the potential
minimum newly generated by the optical trap.
periment,where,atthemomentwhenthesolitonarrives
at the turning point (it is x = 3π for this case, i.e., the Instead of being a means to snare for moving soliton,
third potential minimum), we abruptly turn on the OL, theOLmayalsobeusedasameansofmovingthesoliton
so that in a prescribed way, i.e., as a “robotic arm” depositing
thesolitonatadesiredlocation(see,e.g.,[36]). Thispos-
V(x,t)= 1Ω2x2+ 1V 1+tanh t−t0 sin2(kx). sibility is demonstrated (with varying levels of success)
2 2 0(cid:20) (cid:18) τ (cid:19)(cid:21) in Fig. 3. The top two sets of figures are performed for
(9) a strong OL (V =0.25), while the bottom two are used
0
Here t and τ are constants controlling,respectively, the for a weaker OL potential (with V = 0.17). In all the
0 0
switch-on time and duration of the process; in the simu- cases the potential used is
lations,weuset =31.7andτ =0.1. Weclearlyobserve
0
that, contrary to the large-amplitude oscillations of the 1
solitontakingplacewhenthe OLis absent,the solitonis V(x)= Ω2x2+V0sin2(k(x y(t))), (10)
2 −
5
where the position of the OL is translated according to 15 −5
2
1.8
10 0 1.6
1 t t0 1.4
y(t)=xini+ 2(xfin−xini)(cid:20)1+tanh(cid:18) −τ (cid:19)(cid:21). (11) x 5 x 5 11.2
0.8
0 10 0.6
Here xini and xfin are, respectively, the initial and final 0.4
(target)positionsofthesoliton. Inthecaseunderconsid- 0.2
−5 15
eration, x = 0 and x = 3π, i.e., the aim is to trans- 0 100 200 300 400t500 600 700 800 0 100 200 300 400t500 600 700 800
ini fin
port the MW soliton from the central well to the third 0 2.2
2
one,onthe rightofthe center. Inthe topsetofthe pan- 10 1.8
1.6
els with t = 50 and τ = 5, we observe what happens if 20 1.4
the motio0nof the potentialcenter is not sufficiently slow x30 11.2
toadiabaticallytransportthesolitontoitsfinalposition. 40 00..68
Inparticular,the solitongetstrappedinthesecondwell, 50 0.4
0.2
without being able to reachits destination. However,we 600 10 20 30 40 50t60 70 80 90 100
observethatthis difficulty canbe overcome,ifthe trans-
port is applied with a sufficient degree of adiabaticity; FIG. 4: (color online) The same as the previous figure, but
see, e.g., the middle panel with t = 100 and τ = 10, with Ω = 0 (i.e., in the absence of the magnetic trap). For
0
which succeeds in delivering the soliton at the desired t0 = 100 and τ = 10 (top panels) the soliton is delivered to
its final location of x =3π. However, the same is not true
position. Notice that the final position of the center of fin
for t =50 and τ =5 in thebottom panel, where the soliton
the OL is at y = 3π, which is different from the center 0
fails tostop butrathercontinuesitsmotion,losing moreand
of the third well of the effective potential, around which
more of its power through emission of radiation.
the soliton will oscillate, upon arrival. The theoretical
prediction that was presented above (for V = 0.25) for
0 −30 20
thiswellisx =9.0089,whileinthesimulationsthesoli- 3 10
0 −20 > 0
ton oscillates around 9.04 in very good agreement with 2.5 <x−10
−10 −20
the theory. The two lower sets of the panels are meant 2 −30
tgpoueraiilrmlaunesnttertaesitnaegrtehtphaeetrafeodffirimacibeeadnttifcotirrtayanisssphnoaorlltto.wthTeerhsepionntguelnmetceiaorilncwdailhtieeoxrne- x12000 011..55 2Max(|u(x,t)|)x12340 100 200 300 400t500 600 700 800
the relevantwell (to whichthe soliton is to be delivered) 30 0 0
0 100 200 300 400t500 600 700 800 0 100 200 300 400t500 600 700 800
isnearthethresholdofits existence. As aresult,neither −25 20
in the case with τ = 5 (the third set of panels), nor in −−2105 2 15
10
the onewithτ =10,isthe OLsuccessfulintransporting −10
−5 1.5 5
the soliton at the desired position. x 0 x 0
A similar numerical experiment in the absence of the 5 1 −5
10
magnetic trapis showninFig.4. The toppanels display 15 0.5 −10
the successfultransferofthe solitonby the OL ofa form 20 −15
25 0 −20
similar to that in Eq. (9), with Ω=0 and V =0.25, for 0 100 200 300 400t500 600 700 800 0 100 200 300 400t500 600 700 800
0
t =100andτ =10. Noticethat,inthepresentcase,the
0 FIG. 5: Motion of the soliton induced by the linear ramp in
final positions of the OL’s center and of the solitoncoin-
Eq. (13) with t = 100 and t = 120. The top panel shows
1 2
cide [as the atomic potential and the effective potential
thecase with strong radiation loss in a deep OL (V =0.25);
0
for the soliton have the same set of minima in this case, noticeapparentfrictioninthemotionofthesoliton’scenterof
cf. Eq. (5)]. However, once again, the same experiment, massinthetoprightpanel,andthecorrespondinglossofthe
if not performed with a sufficient degree of adiabatic- soliton’s norm in the panel below it. On the contrary, in the
ity (as in the bottom panel of Fig. 4, with t0 = 50 and case of the shallow OL, with V0 = 0.07 (the bottom panel),
τ = 5), is not successful in depositing the soliton at the themovingsolitondoesnotgenerateanyvisibleradiation. In
prescribedlocation. Instead, inthis casethe solitoncon- both cases, τ =1 was used.
tinues to move along the OL, emitting radiation waves
and decreasing its amplitude.
with
To better illustrate the emission of radiation and its
dependenceonthedepthoftheOL(sinceitisknownthat
1 t t t t
1 2
the emission is absentin the parabolic potential without α(t)=0.1 tanh − tanh − . (13)
× 2(cid:20) (cid:18) τ (cid:19)− (cid:18) τ (cid:19)(cid:21)
theOLingredient),wehavealsoperformedthefollowing
numericalexperiment. Wetookthepotentialoftheform
InEq.(13),t andt are,respectively,theinitialandfinal
1 2
momentoftime,betweenwhichthelinearrampisapplied
1 toacceleratethesolitontoafinitepropagationspeed. We
V(x)= Ω2x2+V0sin2(kx)+α(t)x, (12) display two such numerical simulations in Fig. 5. The
2
6
first is performed in a deep OL, with V = 0.25, taking ferent wells analytically, and verified it numerically, we
0
initially the soliton in the third well (t = 100 and t = then explored a possibility to use the OL as a tool to
1 2
120wereused). The secondsimulationwasperformedin manipulate the soliton. We were able to stop the soliton
a shallow OL, with V = 0.07, the soliton being initially at a prescribed location by turning on the OL, in an ap-
0
takeninthefirstwell(theonlyoneexistingatsuchvalues propriate fashion. We have also found the adiabaticity
ofthe parameters). Thetoppanelsclearlyshowthatthe condition necessary to secure the transfer of the soliton
emission of radiation leads to the gradual decay of the by a moving OL (with and without the magnetic trap).
soliton’s amplitude. On the contrary, when the OL is Finally, we have shown the absence of any visible emis-
weaker(inthebottompanels),thesolitonmovesthrough sion of radiation from the soliton moving across a weak
it practically without radiation loss. OL;however,thesolitonlosesalargefractionofitsnorm,
moving through a stronger lattice.
Given the recent prediction of solitons and vortices in
IV. CONCLUSION multi-dimensional OL potentials [4] (for recent experi-
mental work on a similar topic in nonlinear optics, see
Wehaveexaminedanumberofstaticanddynamicfea- Refs. [37, 38] and referencestherein), it would be of par-
tures of bright matter-wave (MW) solitons in the pres- ticular interest to implement similar dragging and ma-
ence of the magnetic trap and optical lattice (OL). We nipulation of solitons in higher dimensions. The consid-
usedtheperturbationtheorytopredicttheexistenceand eration of this case is currently in progress.
stability of the MW solitons trapped in the combined This work was partially supported by NSF-DMS-
potential. A sequence of saddle-node bifurcations of the 0204585,NSF-CAREER,andtheEppleyFoundationfor
effective potential, which lead to consecutive disappear- Research (PGK); the Israel Science Foundation grant
ance of the higher-well solitonic bound states with the No.8006/03(BAM);andtheSanDiegoStateUniversity
decrease of the OL strength was predicted, through the Foundation (RCG). PGK also gratefully acknowledges
disappearance of the potential wells in the effective po- the hospitalityofthe CenterforNonlinearStudies ofthe
tential. Los Alamos National Laboratory. Work at Los Alamos
Havingidentifiedthestabilitycharacteristicsofthedif- is supported by the US DoE.
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