Table Of ContentAn Analytical Window into the World of Ultracold Atoms
R. Radha∗ and P. S. Vinayagam
Centre for Nonlinear Science, PG and Research Department of Physics,
Government College for Women (Autonomous), Kumbakonam 612001, India.
In this paper, we review the recent developments which had taken place in the domain of quasi
one dimensional Bose-Einstein Condensates (BECs) from the viewpoint of integrability. To start
with, we consider the dynamics of scalar BECs in a time independent harmonic trap and observe
that the scattering length can be suitably manipulated either to compress the bright solitons to
attain peak matter wave density without causing their explosion or to broaden the width of the
5 condensates without diluting them. When the harmonic trap frequency becomes time dependent,
1 we notice that one can stabilize the condensates in the confining domain while the density of the
0 condensates continue to increase in the expulsive region. We also observe that the trap frequency
2 andthetemporalscatteringlengthcanbemanoeuvredtogeneratematterwaveinterferencepatterns
n indicatingthecoherentnatureoftheatomsinthecondensates. Wealsonoticethatasmallrepulsive
a three body interaction when reinforced with attractive binary interaction can extend the region of
J stability of the condensates in the quasi-one dimensional regime.
0 Ontheotherhand,theinvestigationoftwocomponentBECsinatimedependentharmonictrap
2 suggeststhatitispossibletoswitchmatterwaveenergyfromonemodetotheotherconfirmingthe
factthatvectorBECsarelonglivedcomparedtoscalarBECs. TheFeshbachresonancemanagement
] ofvectorBECsindicatesthatthetwocomponentBECsinatimedependentharmonictraparemore
s
stable compared to the condensates in a time independent trap. The introduction of weak (linear)
a
time dependent Rabi coupling rapidly compresses the bright solitons which however can be again
g
- stabilizedthroughFeshbachresonanceorbyfinetuningtheRabicouplingwhilethespatialcoupling
t of vector BECs introduces a phase difference between the condensates which subsequently can be
n
exploited to generate interference pattern in the bright or dark solitons.
a
u
q PACSnumbers:
.
t
a
I. INTRODUCTION
m
-
d
Weareawarethateventhoughmatterpervadestheen-
n
tire universe, it is found in just a few admissible forms
o
such as solid, liquid and gas. It is obvious that one can
c
[ initiateaphasetransitionbetweendifferentstatesofmat-
terbyeitherincreasingthetemperatureorpressure. This
1
understanding of generating new states of matter by in-
v
creasing the temperature was exploited in 1879 by Sir
1
6 WilliamCrookes[1]tocreate“plasma,”agascontaining
7 nonnegligiblenumberofchargecarriers. Itmustbemen-
4 tioned that the physical states of matter change in going
0 from one phase to another while the chemical composi-
. FIG. 1: The energy levels of different physical states of mat-
1 tions of matter remains the same. Can one go down the
ter.
0 temperature scale and generate a new state of matter at
5 ultracoldtemperatures? Thisquestionhasbeenplague-
1
ingthemindsofscientistsanditwasin1995,Cornelland
:
v Wieman[2]createda“Bose-Einsteincondensate”(BEC) At such low temperatures, a large fraction of the atoms
i at super low temperatures. Eventhough envisioned first
X getpiledupeitherinthegroundstateorinthelonglived
by Albert Einstein and a young Indian physicist named
metastable state. In other words, the atoms merge to-
r
a Satyendra Nath Bose in the 1920s [3, 4], it took more getherlosingtheirindividualidentitiesandbehavelikea
than seven decades to realize this singular state of mat-
giantmatterwave. Thisphenomenonisknownas“Bose-
ter. Incontrasttoplasmascontainingsuperhotandsuper
Einstein condensation.”
excited atoms, BECs were created at colder and colder
temperatures near absolute zero and they are composed
of supercold and super unexcited atoms (see Fig. (1)).
II. GROSS-PITAEVSKII (GP) EQUATION
ToinvestigatethedynamicsofaBose-Einsteinconden-
∗Electronicaddress: [email protected] sate, we now employ a Hartree or mean field approach
2
and assume that the wave function is a symmetrized ensuresconstancyoftheparticlesandthevariationsofψ
product of single particle wave functions. In the fully and ψ∗ may thus be taken to be arbitrary. Equating to
condensed state, all bosons (atoms with integral spin) zero the variation of E−µN with respect to ψ∗(r) gives
are in the same single-particle state, φ(r) and therefore the following evolution equation [5–7]
one can write down the wave function of the N-particle
system as (cid:126)2
− ∇2ψ(r)+V(r)ψ(r)+U |ψ(r)|2ψ(r)=µψ(r). (9)
2m 0
N
(cid:89)
Ψ(r1,r2,...,rN)= φ(ri). (1) We call equation (9) as the “time-independent Gross-
i=1 Pitaevskii equation.” This has the form of a time-
independentSchr¨odingerequationinwhichthepotential
The single-particle wave function φ(r ) is normalized
i acting on particles is the sum of the external potential
as
V(r) and a nonlinear term U |ψ(r)|2 that takes into ac-
0
(cid:90) count the mean field produced by the other bosons.
dr|φ(r)|2 =1. (2)
Ifonehastolookforthedynamicsofcondensates,itis
natural to use the time-dependent generalization of the
This wave function does not contain the correlations Schr¨odingerequationwiththesamenonlinearinteraction
producedbytheinteractionwhentwoatomsarecloseto term and obtain the time-dependent GP equation
eachother. Theseeffectsaretakenintoaccountbyusing
the effective interaction U δ(r−r(cid:48)). According to mean
0
field theory, the effective Hamiltonian may be written as − (cid:126)2 ∇2ψ(r)+V(r)ψ(r)+U |ψ(r)|2ψ(r)=i(cid:126)∂ψ(r,t).
2m 0 ∂t
(cid:88)N (cid:20)p2 (cid:21) (cid:88) (10)
H = i +V(r ) +U δ(r −r ), (3) To ensure consistency between the time-dependent GP
2m i 0 i j
i=1 i<j equation(10)andthetime-independentGPequation(9),
under stationary conditions ψ(r,t) must evolve in time
V(ri) being the external potential. The energy of the as exp(−iµt/(cid:126)).
state given by equation (1) is written as
In the above equation (10), ψ(r,t), r = (x,y,z) rep-
(cid:90) (cid:20) (cid:126)2 N −1 (cid:21)resents the condensate wave function, ∇2 denotes the
E =N dr |∇φ(r)|2+V(r)|φ(r)|2+ U |φ(r)|4 .Laplacian operator, V(r) is the trapping potential as-
2m 2 0 sumedtobeV(r)=m(ω2r2+ω2x2)wherer2 =y2+z2,
(4) r x
ω aretheconfinementfrequenciesintheradialandax-
r,x
FromthemacroscopictheoryfortheuniformBosegas, ialdirectionsrespectively, U =4π(cid:126)2a/mcorrespondsto
0
the relative reduction of the number of particles in the
thestrengthofinteratomicinteractionbetweentheatoms
condensate is of the order of (na3)1/2, where n is the
characterizedbytheshort-ranges-wavescatteringlength
particledensity. Ifweintroducethewavefunctionofthe
a, and m is the atom mass.
condensed state as
From equation (10) it is obvious that the GP equa-
tion is an inhomogeneous (3+1) dimensional nonlinear
ψ(r)=N1/2φ(r), (5)
Schr¨odinger (NLS) equation. The inhomogeneity origi-
nates from the potential V(r) that traps the atoms in
then the density of particles is given by
thegroundstateandthenonlinearitycoefficientU which
0
n(r)=|ψ(r)|2 (6) representstheinteratomicinteractionrelatedtothescat-
tering length a. The scattering length can have either
and for N (cid:29) 1, the energy of the system may therefore positive or negative values which means the interaction
be written as can be either repulsive or attractive. It should be men-
tioned that it should be possible to vary the scattering
(cid:90) (cid:20) (cid:126)2 1 (cid:21) length periodically with time a(t) employing Feshbach
E = dr |∇ψ(r)|2+V(r)|ψ(r)|2+ U |ψ(r)|4 . resonance [8]. This means that understanding the dy-
2m 2 0
namics of BECs boils down to solving a variable coeffi-
(7)
cient(3+1)NLSequationforsuitablechoicesoftrapping
To find the optimal form for ψ, we minimize the energy
potentials V(r) and temporal scattering lengths a(t).
with respect to independent variations of ψ(r) and its
complex conjugate ψ∗(r), subject to the condition that
the total number of particles
III. INTEGRABILITY OF GP EQUATION-
(cid:90) REVIEW OF ANALYTICAL METHODS
N = dr|ψ(r)|2 (8)
Looking back at the (3+1) dimensional Gross-
be constant. For this, one writes δE −µδN = 0 where Pitaevskii equation (10), it is obvious that it is in gen-
the chemical potential µ is the Lagrange multiplier that eralnonintegrableforanarbitrarytrappingpotentialand
3
interatomic interaction. Hence, one has to investigate
whether the GP equation would admit integrability in
lower spatial dimensions for specific choices of trapping
potentialsandscatteringlengths. Inotherwords,onehas
to look for the associated nonlinear excitations in (1+1)
and(2+1)dimensionalGPequationswhichwouldreflect
upon the integrability of the associated dynamical sys-
tem under consideration. In a three dimensional BEC,
when the transverse trapping frequency ω (r = y,z)
r
is very high compared to the longitudinal trapping fre-
quencyω ,thenthetransverseconfinementistootightto
x
allowscatteringofatomstotheexcitedstatesofthehar- FIG.2: Schematicdiagramoftheinversescatteringtransform
monic trap in the transverse direction. Under this con- method.
dition, one obtains “cigar” shaped BECs and the three
dimensionalGPequationbecomesquasione-dimensional
in nature. Again, it should be mentioned that the quasi nonlinear PDE one has started with. Once the lineariza-
one-dimensional GP equation can be shown to be inte- tionisperformedintheabovesenseforagivennonlinear
grableonlyforsuitablechoicesoftrappingpotentialV(x) dispersive system q = K(q), where K(q) is a nonlinear
t
and interatomic interaction U0(x). functionalofqanditsspatialderivatives,theCauchyini-
tial value problem corresponding to the boundary condi-
A. Analytical Methods
tion q → 0 as |x| → ∞ can be solved by a three step
Eventhough one cannot precisely define the concept of process indicated schematically in Fig. 2.
integrabilityofadynamicalsystemgovernedbyanonlin- This method involves the following three steps:
ear partial differential equation (PDE), one can look for
the possible signatures of integrability namely, Painleve
(P-)property [9], Lax-pair [10] soliton solutions etc. A
1. Direct scattering transform analysis
given nonlinear dynamical system governed by a nonlin-
earPDEissaidtoadmitP-propertyifthecorresponding Considering the initial condition q(x,0) as the po-
solution can be locally given in terms of a Laurent se- tential,ananalysisofthelineareigenvalueproblem
riesexpansionintheneighborhoodofamovablesingular (11) is carried out to obtain the scattering data
point/manifold. The existence of a Lax-pair of a given S(0). For example, for the KdV equation we have
nonlinearpdeimpliesthatonecansomehowlinearizethe
S(0)=λ (0),n=1,2,...,N,C (0),R(x,0),−∞<x<∞
nonlinear dynamical system and subsequently exploit it n n
(13)
togeneratesolitonsolutions,therebyconsolidatingitsin-
tegrability. In this section, we dwell upon the analytical whereN isthenumberofboundstateswitheigen-
techniques like Inverse Scattering transform [11], Gauge values λn, Cn(0) is the normalization constant of
transformationmethod[12],Darbouxtransformationap- the bound state eigenfunctions and R(x,0) is the
proach [13–15], Hirota’s direct method [16, 17] besides reflection coefficient for the scattering data.
the approximation method like variational approach.
2. Time evolution of scattering data
I. Inverse Scattering Transform
Using the asymptotic form of the time evolution
The Inverse scattering Transform is a nonlinear equation (12) for the eigenfunctions, the time evo-
analogue of the Fourier transform which has been em- lution of the scattering data S(t) can be deter-
ployed to solve several linear partial differential equa- mined.
tions. Given the initial value of the potential q(x,0) and
3. Inverse scattering transform (IST) analysis
the boundary conditions, one has to identify two linear
differential operators L and B so that one can convert The set of Gelfand − Levitan − Marchenko in-
a (1+1) dimensional nonlinear pde into two linear equa- tegral equations corresponding to the scattering
tions, namely a linear eigenvalue problem data S(t) is constructed and solved. The resulting
solution consists typically of N number of local-
LΦ=λΦ, (11) ized, exponentially decaying solutions asymptoti-
cally (t → ±∞). In this way, one can success-
and a linear time evolution equation fullysolvetheinitialvalueproblemofthenonlinear
PDE.
Φ =BΦ, (12)
t
From the above, it is obvious that solving the initial
such that the compatability condition of the above two value problem of the given nonlinear PDE boils down to
equations (11) and (12), i.e., L = [B,L] generates the solving an integral equation.
t
4
II. Gauge Transformation Approach
This is an iterative method enabling one to gen- P = (I−P)σ U(µ )σ P −Pσ U(λ )σ (I−P)(,23)
x 3 1 3 3 1 3
erate soliton solutions starting from a seed solution.
P = (I−P)σ V(µ )σ P −Pσ V(λ )σ (I−P)(,24)
t 3 1 3 3 1 3
In this method, one again begins with the Zakharov-
Shabat(ZS)-Ablowitz-Kaup-Newell-Segur(AKNS)linear where
systems [18, 19] given by the following equations:
(cid:18) (cid:19)
1 0
σ = . (25)
Φx =UΦ (14) 3 0 −1
Φ =VΦ (15)
t
To generate a new solution from a given solution (vac-
where uum),forexampleq0 andr0 associatedwithmatricesU0
(cid:18) (cid:19) (cid:18) (cid:19) andV0,theeigenvalueproblemtakesthefollowingform
−iλ q A B
U = , V = . (16)
r iλ C D Φ0 =U0Φ0; Φ0 =V0Φ0. (26)
x t
In the above equation, that is equation (16), λ repre- where U| =U0 and V| =U0. Now, one can solve
seed seed
sents the spectral parameter, (q,r) the potential/field
equations (23) and (24) using the vacuum eigen function
variable of the given nonlinear PDE, A,B,C and D cor- Φ0 such that
respond to the undetermined functions of λ,x,t,q,r and
their spatial/time derivatives. The compatibility condi- P =σ P˜σ , (27)
3 3
tion (Φ ) = (Φ ) yields U − V + [U,V] = 0 which
x t t x t x
is equivalent to the given nonlinear PDE. One generally where
callsthematricesU and V as“Lax-pair”whichcontains
information about the linearization of the given dynam- P˜ =M(1)/[traceM(1)]. (28)
ical system. It should be mentioned that for scalar non-
and M(1) is a 2×2 matrix defined by
linear PDEs, U and V are 2×2 matrices and the eigen-
function Φ is 2×1 column vector while for vector (two (cid:18) (cid:19)
m 1/n
component in general) nonlinear PDEs, the eigenfunc- M(1) ≡Φ0(x,t;µ ) 1 1 Φ0(x,t;λ )−1, (29)
1 n 1/m 1
tion Φ is a 3×1 column vector and U and V are 3×3 1 1
matrices.
where m and n are arbitrary complex constants and
1 1
Now,gaugetransformingtheeigenfunctionΦ,wehave Φ0 isasolutionofthevacuumlinearsystemgovernedby
an iterated eigen function Φ(1)=g Φ where g = g(x,t,λ)
equations (26).
is a 2×2 matrix function so that
Now,substitutingthemeromorphicfunctiongwiththe
projectionmatrixP givenbyequation(27)intoequation
Φ(1) =U(1)Φ (17)
x (19), we get
Φ(1) =V(1)Φ (18)
t (cid:18)−iλ −q0 (cid:19) (cid:18) 0 P˜ (cid:19)
where, U(cid:48)(λ)= −r0 iλ −2i(λ1−µ1) −P˜ 012 .
21
U(1) =gUg−1+g g−1, (19) (30)
x
A comparison of the eigenvalue problems expressed
V(1) =gVg−1+gtg−1. (20) in terms of the vacuum eigenfunction Φ0 and the new
(transformed) eigenfunction Φ(1) would enable us to re-
Thetransformationfunctiong mustbeadoptedfromthe
latethevacuumsolutionoftheassociatednonlinearPDE
solutions of certain Riemann problems in the complex λ
with the new solution. Thus, we get an explicit solution
plane and it must be a meromorphic (regular) function.
as [12]
The simplest form of g satisfying the above criteria can
be written as
q(1) =−q0−2i(λ −µ )P˜ , (31)
1 1 12
(cid:20) (cid:21)(cid:18) (cid:19)
g = I+ λ1−µ1P(x,t) 1 0 , (21) r(1) =−r0+2i(λ1−µ1)P˜21, (32)
λ−λ 0 −1
1
where
whereλ andµ aretwoarbitrarycomplexnumbersand
1 1
P is an undetermined 2×2 projection matrix (P2 =P). M(1) M(1)
Hence, g−1 is now given by P˜12 = M(1)+12M(1), P˜21 = M(1)+21M(1). (33)
11 22 11 22
(cid:18) (cid:19) (cid:20) (cid:21)
1 0 λ −µ
g−1 = 0 −1 · I− λ1−µ1P(x,t) . (22) Once we get q(1) and r(1) from the input solution q0
1 and r0, one can repeat the same procedure to obtain yet
Thus,thevanishingoftheapparentresiduesatλ=λ another new solution q(2) and r(2)) using q(1) and r(1))
1
and λ=µ imposes the following constraints on P as as the input solution. For example, to construct second
1
5
iterated solution q(2), r(2), one needs to find a solution Now,introducingtheDarbouxtransformationintothe
of the linear system given by equations (17) and (18) known eigen function, we now obtain the transformed
where the matrices U1 and V1 are associated with the eigenfunction as
input solutions q(1) and r(1). However, one can find the
solution of above eigen value problem in terms of Φ0 as Φ(1)(x,t,λ)=D(x,t,λ)Φ(x,t,λ), (43)
Φ(1) =gΦ0. (34) whereD(x,t,λ)isthe“Darbouxmatrix”whichisequiv-
alenttoλI−S. WhileI istheidentitymatrix,thematrix
Thus, the new iterated solution q(2) and r(2) (in anal- S can be generated as
ogy with equations (31) and (32)) can be written as
S =HΛH−1, detH(cid:54)=0, (44)
q(2) =−q(1)−2i(λ −µ )P˜(1), (35)
2 2 12
where the matrix H is defined as H = (h ,...,h ),
r(2) =−r(1)+2i(λ −µ )P˜(1). (36) 1 N
2 2 21 where h represents the column solution of the linear
i
eigenvalue problem given by equations (14) and (15),
Thus, one can repeat the same procedure N-times to
Λ=diag(λ ,...,λ ). TheneweigenfunctionΦ(1) again
obtain the Nth iterated solution as 1 N
satisfiesequations(17)and(18)andtheDarbouxmatrix
D plays the role of the transformation function g. Ac-
qN =−q(N−1)−2i(λ −µ )P˜(N−1), (37) cordingly, we have
N N 12
rN =−r(N−1)−2i(λ −µ )P˜(N−1). (38)
N N 21
U(1) =DUD−1+D D−1, (45)
The above iteration could become extremely handy, x
particularly in the context of the generation of multisoli- V(1) =DUD−1+DtD−1. (46)
ton solutions as one can obtain N-soliton solution from
the vacuum eigen function Φ0 of the linear system. To check the form of U(1) given by equation (45), we
substitute eqns. (14), (17) into the x derivative of equa-
III. Darboux Transformation Approach tion(43) to obtain
In 1882 G. Darboux studied the eigen value problem of
U(1)Φ(1) =D Φ+DUΦ. (47)
the one dimensional Schr¨odinger equation x
Then, making use of equation (43), one obtains equa-
tion(45). Similarly,onecanchecktheformofV(1) given
−Φ −q(x)Φ=λΦ, (39)
xx by (46).
Thus,startingfromaseedsolutionU ofthegivennon-
where q(x) is a potential function and λ is a constant
linear pde, one can generate the iterated U(1). This pro-
spectral parameter. He postulated that if q(x) and
cedurecanberepeatedtogeneratemultisolitonsolutions.
Φ(x,λ) are two functions satisfying equation (39) and
f(x) = Φ(x,λ ) is a solution of equation of (39) for
0 IV. Hirota Bilinear Method
λ = λ where λ is a fixed constant, the functions q(cid:48)
0 0
and Φ(cid:48) are defined by
Althoughtheinversescatteringformalismwasthefirst
analytical technique that has been developed to solve
f
q(cid:48) =q+2(lnf) , Φ(cid:48)(x,λ)=Φ (x,λ)− xΦ(x,λ), the initial value problem of nonlinear pdes, it involves
xx x f
sophisticatedmathematicalconceptslikesolvinganinte-
(40)
gral equation and hence quite complicated and intricate.
with
Moreoverinthismethod, oneshouldhaveapriorknowl-
−Φ(cid:48) −q(cid:48)Φ(cid:48) =λΦ(cid:48). (41) edge of the potential u(x,t) at t = 0, namely the ini-
xx tial data u(x,0) and the boundary conditions imposed
From equations (39) and (41), it is obvious that they on it. On the other hand, eventhough Darboux and
areofthesameform. Therefore,thetransformation(40) gauge transformation approaches are iterative in nature
which converts the functions (q,Φ) to (q(cid:48),Φ(cid:48)) satisfying and are purely algebraic without involving highly com-
thesamepartialdifferentialequationistheoriginalDar- plex mathematics, they warrant the identification of the
boux transformation which is valid for f (cid:54)=0. Lax-pair of the associated dynamical system. Hence, it
In this method, one again begins with the ZS-AKNS is imperative to look for an alternative method to gen-
linear eigen value problem given by equations (14) and erate localized solutions (soliton solutions) of nonlinear
(15), where the eigen function Φ is a 2×2 matrix of the pdes and in this context, Hirota’s direct method comes
following form to our rescue. In this method, neither does one need any
prior information about the potential (or physical field)
(cid:18) (cid:19)
Φ (x,t,λ) Φ (x,t,λ) of the associated nonlinear pde, nor the Lax-pair of the
Φ(x,t,λ)= 11 12 . (42)
Φ (x,t,λ) Φ (x,t,λ) associateddynamicalsystem. Thismethodwhichhasan
21 22
6
inbuiltdeepalgebraicandgeometricstructureismoreel- alizable potentials. To start with, we consider the dy-
egant and straightforward and can be directly employed namics of BECs in a time independent harmonic trap
to generate soliton solutions of nonlinear pdes. with exponentially varying scattering length. We gener-
The salient features of the Hirota method are the ate the associated bright solitons and study their colli-
following: sionaldynamics. Wethenintroducesuitabletimedepen-
i) The given nonlinear partial differential equation denceintheharmonictrapandinvestigatethedynamics
has to be converted into a bilinear equation through a of BECs. We then show that how the interplay between
transformationwhichcanbeidentifiedfromthePainlev´e trap frequency and temporal scattering length can gen-
analysis. Each term of the bilinear equation has the erate matter wave interference pattern in the collision of
degree two. bright solitons. We then reinforce the binary attraction
with a repulsive three body interaction to enhance the
The Hirota bilinear operators are defined as stability of BECs.
(cid:18)∂ ∂ (cid:19)m(cid:18) ∂ ∂ (cid:19)n A. Dynamics of quasi one dimensional BECs in a
DmDn(G·F)= − −
t x ∂t ∂t(cid:48) ∂x ∂x(cid:48) time independent harmonic trap
G(t,x)F(t(cid:48),x(cid:48))| .
t(cid:48)=t,x(cid:48)=x For a time independent harmonic oscillator potential
ii)ThedependentvariablesGandF inthebilinearform and exponentially varying scattering lengths, the GP
have to be expanded in the form of a power series in equation (10) for cigar shaped BECs takes the following
terms of a small parameter ε as form [20–24]
G = εg(1)+ε3g(3)+ε5g(5)+··· , (48)
∂ψ ∂2ψ 1
F = 1+ε2f(2)+ε4f(4)+··· . (49) i∂t + ∂x2 +2a(t) |ψ|2ψ+ 4λ2x2ψ =0, (50)
iii) After substituting the above functions into the bilin-
where time t and coordinate x are measured in units
earformandequatingdifferentpowersofε,asetoflinear 2/ω and a , where a = ((cid:126)/mω )1/2 and a =
pdes can be generated. ((cid:126)/m⊥ω )1/2 a⊥relinearosc⊥illatorlengths⊥inthetransv0erse
iv)Finally,solvingthelinearpdes,onecangeneratesoli- 0
and cigar-axis direction respectively. ω and ω are
ton solutions. ⊥ 0
corresponding harmonic oscillator frequencies, m is the
It should be mentioned that the key to the success of
atomic mass and the trap frequency λ = 2|ω |/ω <<
the method lies in the identification of the dependent 0 ⊥
1. The Feshbach resonance managed nonlinear coeffi-
variable transformation as well as in choosing an opti-
cient which represents the scattering length reads a(t)=
mum power series to linearise the given nonlinear pde.
a˜ exp(λt).
0
Equation (50) admits the following Lax-pair [23]
A. Approximation method
(cid:18) (cid:19)
iζ Q
Φ = UΦ, U = , (51a)
x −Q∗ −iζ
I. Variational Approach
Φ = VΦ, (51b)
t
Variational approach is a qualitative semi-analytical ap-
(cid:18) −2iζ2+iλxζ+i|Q|2 [(λx−2ζ)Q+iQ ] (cid:19)
proach. By using the variational approximation, one V = x ,
can study the dynamics of Bose-Einstein condensates [−(λx−2ζ)Q∗+iQ∗x] 2iζ2−iλxζ−i|Q|2
described by the mean-field Gross-Pitaevskii equation.
For the purpouse of variational analysis, Larangian den- where we have slightly modified the Lax-pair (given in
sity is calculated for the corresponding time-dependent ref. [23]) by allowing the nonisospectral parameter ζ to
Gross-Pitaevskii equation and the effective Lagrangian be complex keeping the initial scattering length unity.
can be obtained by integrating the initial trial wave Thenonisospectralcomplexparameterobeysthefirstor-
function with variational parameters over space. The der ordinary differential equation of the form
numerical value of each variational parameter can be
obtained from the numerical solution of corresponding ζt =λζ, ζ(t)=α(t)+iβ(t), (52)
Euler-Lagrangian equations.
and the macroscopic wave function ψ is related to Q by
the transformation
IV. BRIGHT MATTER WAVE SOLITONS AND
THEIR COLLISION IN SCALAR BECS IN (cid:18)λt λx2(cid:19)
Q=exp +i ψ(x,t). (53)
CERTAIN SIMPLE POTENTIALS 2 4
In this section, we investigate the dynamics of scalar Now, to generate the bright soliton solutions of equa-
Bose-Einsteincondensatesincertainsimplephysicallyre- tion (50), we consider a trivial vacuum solution Q(0) =0
7
to give the following vacuum linear systems
(cid:18) (cid:19)
iζ 0
Φ(0) = Φ(0) =U(0)Φ(0), (54a)
x 0 −iζ
(cid:18)−2iζ2+iλxζ 0 (cid:19)
Φ(0) = Φ(0)
t 0 2iζ2−iλxζ FIG. 3: The dynamics of bright solitons for the parametric
choice λ= 0.02, β =2, α =0.1, δ =0.5, φ =0.1.
0 0 1 1
= V(0)Φ(0). (54b)
Solving the above linear systems keeping in mind that
the spectral parameter ζ varies with time by virtue of
equation (52), we have
(cid:18)eixζ−2i(cid:82)ζ2dt 0 (cid:19)
Φ(0)(x,t,ζ)= . (55)
0 e−ixζ+2i(cid:82)ζ2dt
Now, effecting the gauge transformation
Φ(1)(x,t,ζ)=gΦ(0)(x,t,ζ), (56)
where “g” is a meromorphic solution of the associated
FIG. 4: Contour Plot of bright soliton for λ=0.02.
Riemann problem. The new linear eigenvalue problems
now take the following form
Φ(1) =U(1)Φ(1), Φ(1) =V(1)Φ(1), (57)
x t where m and n are arbitrary complex constants.
1 1
with Hence, choosing the complex parameters ζ = α (t)+
1 1
iβ (t)andµ =ζ∗ andemployingthegaugetransforma-
U(1) =gU(0)g−1+g g−1, (58a) 1 1 1
x tion approach [12], we arrive at the matter wave bright
V(1) =gV(0)g−1+gtg−1. (58b) soliton
We now choose g as (cid:18)λt iλx2(cid:19)
ψ(1)(x,t)=2β exp − sech(θ )exp(iξ ),
(cid:18) ζ −µ (cid:19)(cid:18)1 0 (cid:19) 0 2 4 1 1
g = 1+ ζ1−ζ1P1(x,t) 0 −1 (59) (63)
1
where
where ζ and µ are arbitrary complex parameters and
1 1 (cid:90)
P1 is a projection matrix (P12 = P1). Imposing the con- θ1 = 2β1x−8 (α1β1)dt+2δ1, (64a)
straint that U(1) and V(1) do not develop singularities
(cid:90)
around the poles ζ = ζ and ζ = µ , the choice of the
1 1 ξ = 2α x−4 (α2−β2)dt−2φ , (64b)
projection matrix P is governed by the solution of the 1 1 1 1 1
1
following set of partial differential equations α = α eλt, β =β eλt. (64c)
1 10 1 10
P = (1−P )σ U(0)(µ )σ P −P σ U(0)(ζ )
1x 1 3 1 3 1 1 3 1
The above bright soliton solution given by equations
σ (1−P ) (60a)
3 1 (63) and (64a-64c) is identical to the one given by Liang
P = (1−P )σ V(0)(µ )σ P −P σ V(0)(ζ ) et al. [22] using Darboux transformation. From the
1t 1 3 1 3 1 1 3 1
profile of the bright soliton trains shown in Fig. 3, we
σ (1−P ) (60b)
3 1
infer that matter wave density |ψ|2 increases with the
where increase of the absolute value of the scattering length
leading to the compression of the bright soliton trains of
(cid:18)1 0 (cid:19) BEC. The contour plot (shown in Fig. 4) which takes a
σ3 = 0 −1 . (61) cross-sectional view of Fig. 3 in the x−t plane shows
that the width of bright solitons decreases progressively.
Looking at the above system of equations, we under-
stand that P depends only on the trivial matrix eigen-
1
function Φ(0)(x,t,ζ), a diagonal matrix and has a com-
pact form given by
M(1)
P (x,t) = σ σ , (62a)
1 3[traceM(1)] 3
(cid:18)m 1/n (cid:19) FIG. 5: The dynamics of bright soliton for λ= -0.02.
M(1) = 1 1 Φ(0)(x,t,ζ )−1 (62b)
n 1/m 1
1 1
8
solution has been expressed in terms of doubly periodic
Jacobianellipticfunctions. However,theaboveapproach
cannot be employed to generate multisoliton solutions
analytically. The gauge transformation approach comes
in handy at this juncture as it offers the advantage of
constructing multisoliton solution from the solution of
the corresponding vacuum linear system.
Now, to generate the bright solitons of equation (65)
for both regular and expulsive potentials, we introduce
the following modified lens transformation [29–32]
FIG. 6: Contour plot of bright soliton for λ= -0.02.
(cid:112)
ψ(x,t)= A(t)Q(x,t)exp(iΦ(x,t)), (66)
Figure 5 shows that the peak value of the matter wave where the phase has the following simple quadratic form
density |ψ|2 decreases with the decrease of the absolute
1
value of the scattering length leading to a broadening of Φ(x,t)=− c(t)x2. (67)
the bright soliton trains thereby enhancing their width 2
andthisisagainconfirmedbythecorrespondingcontour
Substitutingthemodifiedlenstransformationgivenby
plotinfigure6. Ourinvestigationshowsthatthescatter-
equation (66) in equation (65), we obtain the modified
ing length can be suitably manipulated to compress the
NLS equation
brightsolitonsofBECsintoanassumedpeakmatterden-
sity without causing their explosion while on the other 1
iQ + Q −ic(t)xQ −ic(t)Q+a(t)A(t)|Q|2Q=0,
hand, it can be manoeuvred judiciously to broaden the t 2 xx x
localizedsolitonswithoutallowingthedilutionofthecon- (68)
densates and this interpretation completely agrees with with
that of Liang et al. [22]. Investigations of the quasi one
dimensional GP equation [22, 23, 25–29] in the presence λ(t)=c(cid:48)(t)−c(t)2, (69)
ofanexpulsiveparabolicpotentialforpositivescattering
lengths also confirm our above observations. and
From the above, we observe that one can either com-
d
pressorbroadenthebrightsolitonsintheexpulsivetime c(t)=− lnA(t). (70)
dt
independent trap either by exponentially increasing or
decreasing the scattering lengths respectively. It must
Equation (68) admits the following linear eigenvalue
be mentioned that the exponentially varying scattering
problem
lengthwithatrapfrequencydependencewouldmakethe
exact solutions of the GP equation less interesting from (cid:18) iζ(t) Q (cid:19)
φ = Uφ, U = , (71)
an experimental point of view. Hence, it would be in- x −Q∗ −iζ(t)
teresting to investigate the impact of a general time de-
φ = Vφ, (72)
pendent scattering length and a time dependent trap on t
the condensates. The addition of time dependent trap V11 V12
frequency will facilitate us to tune the trap suitably and V = .
study its impact on the condensates. V21 V22
B. Impact of transient trap on BECs where
The introduction of time dependance in the trap en- i
V = −iζ(t)2+ic(t)xζ(t)+ a(t)A(t)|Q|2
suresthatthecondensatesarenowconfrontedwithboth 11 2
time dependent scattering length and time dependent i
trapping potential and accordingly equation (50) gets V12 = (c(t)x−ζ(t))Q+ 2Qx
modified as (in dimensionless units) [30]
i
V = −(c(t)x−ζ(t))Q∗+ Q∗
21 2 x
∂ψ 1∂2ψ λ(t) i
i + +a(t)|ψ|2ψ− x2ψ =0, (65) V = iζ(t)2−ic(t)xζ(t)− a(t)A(t)|Q|2 (73)
∂t 2∂x2 2 22 2
where a(t) = −2as(t)/aB, λ(t) = ω02(t)/ω⊥2, aB is the In the above linear eigenvalue problem, the spectral
Bohr radius, λ(t) describes time dependent harmonic parameter“ζ”whichiscomplexisnonisospectralobeying
trap which can be confining (λ(t) < 0) or expulsive the following equation
(λ(t) > 0). This equation has been mapped onto a lin-
ear Schr¨odinger eigenvalue problem [30] and one soliton ζ(cid:48)(t)=c(t)ζ(t), (74)
9
with a(t)=1/A(t). It is obvious that the compatability
condition (φ ) =(φ ) generates equation (68).
x t t x
Substitutingequation(70)witha(t)=1/A(t)inequa-
tion (69), we get
a(cid:48)(cid:48)(t)a(t)−2a(cid:48)(t)2−λ(t)a(t)2 =0. (75)
Fromtheabove,itisevidentthattheGPequation(65)
is completely integrable only if the trap frequency λ(t)
and the scattering length a(t) are connected by equation
(75)andtheaboveconditionisconsistentwithRef. [29].
Thus,themodifiedlenstransformationhasfacilitatedthe
identification of integrability of equation (65). It should
FIG.7: Twosolitoninteractionintheexpulsivetrap(λ(t)<
also be mentioned that equation (65) is completely inte-
0) with a(t) =a exp(−0.125t2), a = 0.5, α = 2.31, β =
grable only for certain suitable choices of trap frequency 0 0 10 10
1.5, α =3.12, β =1.2, φ =.005, δ =0.002, φ =0.002,
λ(t) depending on the solvability of equation (69). For 20 20 1 1 2
δ =0.001.
example, when λ(t)=constant=k and a(t)=eΛt, where 2
Λ is the trap frequency, equation (65) reduces to (50)
describingthedynamicsofBECsmovinginanexpulsive
parabolic potential and exponentially varying scattering
length [22, 24]. The above parametric choice is consis-
tent with equation (75) with k =−λ2 ensuring the inte-
grability of the model. The above model has also been
experimentally realized [33].
To generate the soliton solution of equation (68) (or
equation (65)), we consider the seed solution Q(0) = 0
and solve the linear systems given by equations (71) and
(72) keeping in mind equation (74) to obtain
φ(0)(x,t,ζ)=(cid:32)eixζ(t)−i(cid:82)0tζ(t)2dt 0 (cid:33).
0 e−ixζ(t)+i(cid:82)0tζ(t)2dt FIG.8: Twosolitoninteractionintheexpulsivetrap(λ(t)<
(76) 0) with a(t) =a0exp(−0.125t2), a0 = 0.5, α10 = 2.31, β10 =
Employing the gauge transformation approach and 1.5, α20 = −2.12, β20 = 1.2, φ1 = .05, δ1 = 0.02, φ2 = 0.02,
choosing ζ = α (t)+iβ (t) and µ = ζ∗, one obtains δ2 =0.01.
1 1 1 1 1
the one soliton solution of equation (65) [34]
(cid:115)
1
ψ1(x,t)= a(t)2β1(t) sechθ1 e−2ic(t)x2+iξ1, (77) Thus, it is obvious that one can obtain varieties of
soliton profiles depending on the choice of the scattering
length a(t) and the time dependent trap λ(t) consistent
where
with equation (75).
(cid:90) t
θ = 2β (t)x−4 (α (t(cid:48))β (t(cid:48)))dt(cid:48) +2δ , Figures (7) and (8) describe the evolution of the two
1 1 1 1 1
0 soliton solution for an expulsive trap (λ(t) < 0) for dif-
(cid:90) t ferent initial conditions evolving the scattering length of
ξ1 = 2α1(t)x−2 (α1(t(cid:48))2−β1(t(cid:48))2)dt(cid:48) −2φ1(,78) the form a(t)=0.5exp(−0.125t2). From the figures, one
0 observes that the matter wave density |ψ|2 of the con-
α1 = α10e(cid:82)0tc(t(cid:48))dt(cid:48), densatesdecreasesslowlybyvirtueofthedecreaseinthe
β1 = β10e(cid:82)0tc(t(cid:48))dt(cid:48), atobrsyolouftethvealsuoelitoofnthpeulssecsatitserdinicgtalteendgtbhyatnhde itnhietiatlracjoenc--
and φ , δ , α and β are arbitrary real constants. ditions. It should be mentioned that the identification
1 1 10 10
The striking feature of this bright soliton solution is of this critical parametric regime in which one observes
thatitsamplitudereliesstronglyonthescatteringlength the slow decay of the condensates enables one to avoid
a(t) and the time dependent trap λ(t) while the velocity this domain by operating the system under a safe range
is governed by the external trap λ(t) alone. of parameters.
Thisprocedurecanbeeasilyextendedtogeneratemul- Figure9(a)showstheinteractionofsolitonsfora(t)=
tisoliton solution and one can study the collisional dy- 0.5exp(0.0025t2). It can be observed from fig. (9b) that
namicsofbrightsolitonsforasuitablechoiceofλ(t)and the confining nature of the trap (λ(t) > 0) is preserved
a(t) consistent with equation (75). only for a finite length of time (t < 14). During this
10
FIG.10: Twosolitoninteractionintheconfiningtrap(λ(t)=
0.09) with a(t) =a exp(0.3it), a = 0.5, α = 0.09, β =
0 0 10 10
0.71, α = 0.031, β = 0.11, φ = 5.1, δ = 7.2, φ = 4.2,
20 20 1 1 2
δ =4.1.
2
[35].
It can also be observed that at t>14, the time depen-
dent trap becomes expulsive again for the same choice
of a(t) (i.e., a(t) = 0.5exp(0.0025t2)) (fig. 9b). In or-
der to sustain the confining nature of the trap (λ(t)>0),
the scattering length should become complex as it is be-
ing done in the case of cold alkaline earth metal atoms
FIG. 9: Two soliton interaction in the confining trap (λ(t)>
0) at different intervals of time with a(t) =a exp(0.0025t2), [36]. Under this condition, the matter wave density |ψ|2
0
a = 0.5, α = 0.01, β = 0.1, α = 0.28, β = 0.11, periodically changes with time by virtue of the periodic
0 10 10 20 20
φ =δ =0.1, φ =δ =0.2. modulation of scattering length [37, 38] and this is remi-
1 2 2 1
niscentoftherecentexperimentalobservationofFaraday
waves [39]. We also observe that the two soliton pulses
period, the two soliton pulses slide over each other like keep exchanging energy among themselves continuously
liquid balls as shown in figures 9 (c) and 9 (d). After during propagation as shown in Fig. 10
this critical period (t >14), the trap becomes expul- We observe from the above that the addition of time
sive again which sets in the compression of the soliton dependence in the trap enables one to stabilize the con-
pulses resulting in the increase of the matter wave den- densates for a longer period of time by selectively tun-
sity|ψ|2 ofthecondensates. Itcanalsobeobservedthat ing the trapping potential. It should be mentioned that
for a(t) = 0.5exp(−0.125t2), the absolute value of the one can also selectively choose λ(t) and a(t) and observe
scattering length decreases and hence there is a slow de- their interplay in the collisional dynamics of bright soli-
cayofthecondensateswhilefora(t)=0.5exp(0.0025t2), tons. Theinterplaybetweenλ(t)anda(t)consistentwith
eventhough the absolute value of the scattering length equation (75) results in the “matter-wave interference
increases, the soliton pulses begin to get compressed (or pattern” in the collisional dynamics of bright solitons.
thematterwavedensity|ψ|2 increases)afterafinitetime
delay. It can be easily understood that this delay is in- C. Matter wave interference pattern in the
troducedbythetimedependenttrap. Whenthetimede- collision of bright solitons
pendent trap λ(t) becomes a constant, equation (65) re-
ducestothedynamicsofBECsinanexpulsiveparabolic To generate the matter wave interference pattern, we
potential and time independent scattering length. Un- now allow the two bright solitons to collide with each
der this condition, the soliton trains begin getting com- other in the presence of a trap for suitable choices of
pressed and the matter wave density |ψ|2 increases as scattering length a(t) and trap frequency λ(t) (or c(t))
soon as the absolute value of the scattering length in- consistent with equation (75).
creases and one does not observe any time delay in the Case (i): When c(t) = −1, the scattering length
compression of soliton trains [22, 24]. It should also be evolves as a(t) = a e−t (shown in Fig. 12(b)) where a
0 0
mentionedthatthistimedelayinthecompressionofthe is an arbitrary real constant and the trap frequency λ(t)
soliton pulses can be suitably manipulated by changing becomesexpulsiveandisequaltoaconstant(λ(t)=−1).
the trapping coefficient λ(t). Thus, the bright solitons Under this condition, the collisional dynamics of two
canbecompressedintoadesiredwidthandamplitudein bright solitons which are initially separated as shown in
a controlled manner by suitably changing the trap and Fig. 11 (a) in the expulsive harmonic trap is shown in
our observation is consistent with the numerical results Fig. 11 (b) and the corresponding density evolution in
