Dissipative dynamics of matter-wave solitons in a nonlinear optical lattice
F. Kh. Abdullaev,1 A. Gammal,2 H. L. F. da Luz,2and Lauro Tomio1,*1
Instituto de Física Teórica, Universidade Estadual Paulista, 01405-900, São Paulo, Brazil 2
Instituto de Física, Universidade de São Paulo, 05315-970, São Paulo, Brazil 共Received 1 July 2007; published 11 October 2007兲
Dynamics and stability of solitons in two-dimensional共2D兲 Bose-Einstein condensates共BEC兲, with one-dimensional 共1D兲 conservative plus dissipative nonlinear optical lattices, are investigated. In the case of focusing media共with attractive atomic systems兲, the collapse of the wave packet is arrested by the dissipative periodic nonlinearity. The adiabatic variation of the background scattering length leads to metastable matter-wave solitons. When the atom feeding mechanism is used, a dissipative soliton can exist in focusing 2D media with 1D periodic nonlinearity. In the defocusing media 共repulsive BEC case兲 with harmonic trap in one direction and nonlinear optical lattice in the other direction, the stable soliton can exist. Variational approach simulations are confirmed by full numerical results for the 2D Gross-Pitaevskii equation.
DOI:10.1103/PhysRevA.76.043611 PACS number共s兲: 03.75.Lm, 05.30.Jp
I. INTRODUCTION
The dynamics of optical and matter-wave solitons with different type of management of system parameters has been under intensive investigations in the past years 关1,2兴. Two types of modulations have been considered: Dispersion and nonlinearity management, which can both occur in time and space. Temporal strong and rapid modulations of the disper-sion are more interesting in optical fibers due to many ad-vantages of dispersion managed solitons for optical commu-nications and for storage of information 关3–5兴. Temporal modulations of the nonlinearity are promising in fiber ring lasers and Bose-Einstein condensates 共BEC兲 关6–8兴. In the latter case, several recent works have been done discussing the suppression of collapse, leading to the existence of stable multidimensional solitons in attractive BEC, and with the generation of periodic patterns of matter waves 关9–17兴. In optics, nonlinearity managed solitons have also been ob-served, as described in Refs.关18–21兴.
In view of the actual experimental possibilities, recent attention has also been devoted to periodic spatial manage-ment in nonlinear optics and Bose-Einstein condensates. In optical media, the nonlinear Kerr coefficient can be periodi-cally modulated in space, leading to the problem of an opti-cal beam in a two-dimensional共2D兲medium with nonlinear-ity management. In BEC, the spatial variation of scattering length is possible关22–28兴, for example, by using optically induced Feshbach resonance关29,30兴. In the 2D case, the situ-ation is less clear. The study of a one-dimensional共1D兲 non-linear periodic potential in a 2D nonnon-linear Schrödinger
equa-tion 共NLSE兲 shows that broad solitons are unstable. As
verified in Ref.关31兴, narrow solitons centered on the maxi-mum of the lattice potential can be stable, but the stability region is so narrow that they are physically unstable. Stable gap solitons can exist in BEC under combination of linear and nonlinear periodic potentials关32–34兴. However, in gen-eral the models considered till now are strongly idealized. In particular, using the optically induced Feshbach resonances
we can generate a mixture of conservative and dissipative nonlinear optical lattices. In view of that, around the Fesh-bach resonance one can observe nonvanishing contributions of the imaginary part of the scattering length.
In the present work, after an analysis of a conservative system with nonlinear optical lattice, we consider the influ-ence of nonlinear dissipation on the dynamics and the stabil-ity of solitons. In particular, we note that the role of such kind of dissipation can be crucial for the existence of solitons in multidimensional nonlinear optical lattices. Such hope is supported by the well-known fact that homogeneous nonlin-ear dissipations can arrest collapse in the cubic focusing mul-tidimensional NLSE关35兴. The possibility of existence of dis-sipative solitons is investigated considering compression effects and atom feeding. For the stability analysis of the solutions we consider the Vakhitov-Kolokolov共VK兲criterion 关36兴, which has been discussed in Refs. 关37,38兴, comple-mented by numerical simulations.
The organization of the paper is as follows. The model is described in the next section. In Sec. III, we investigate the properties of localized states in the cases of attractive and repulsive 2D condensates in 1D nonlinear optical lattice, with or without harmonic trap in one of the dimensions. In Sec. IV, we perform an analysis of the evolution of a 2D soliton under 1D periodic nonlinearity and dissipation, using the variational approach and by direct numerical simulation of the GP equation.
II. MODEL
Recently, the generation of nonlinear optical lattices in BEC by two counterpropagating laser beams near the optical induced Feshbach resonance has been suggested关25,26兴. The spatial variation of the optical intensity leads to a spatial periodic variation of the atomic scattering length. Such struc-ture can support new types of localized nonlinear states. Considering the minimum requirements of having spatial pe-riodic variation in the nonlinear cubic term of a 2D GP equa-tion, for the wave function ⬅共x1,x2,t兲 we can have the following expression:
iប
t = −
ប2
2m
冉
2x 1 2+
2
x 2
2
冊
−g共x1,x2兲兩兩2, 共1兲with
g共x1,x2兲 ⬅ g0
2 +共g1+ig2兲关cos
2共kx
1兲+␦0cos2共kx2兲兴. 共2兲
In the above equations= 0 共1兲for the 1D共2D兲case with␦0
parametrizing the anisotropy of the nonlinear optical lattice in the 2D space共␦0= 0, if the optical lattice is given only in
the x1 direction, and =1 when the lattice geometry is the
same in both directions兲.g0is related to thes-wave two-body
scattering length as, g0⬅共4ប2/m兲as, with g0⬎0 共g0⬍0兲
for attractive共repulsive兲condensates; g1 共⬎0兲 is related to
the optical intensity; andg2parametrizes dissipative effects.
The optically induced scattering length and the inelastic collision rate coefficientKinel 共imaginary part ofas兲 are de-scribed by关29兴
Re共as兲=as0+
1 2ki
冋
⌫stim共x兲⌬
⌬2
+共⌫spon/2兲2
册
, 共3兲
Kinel⬅Im共as兲= 2ប
m
1
ki
冋
⌫stim共x兲⌫spon
⌬2
+共⌫spon/2兲2
册
, 共4兲
whereas0is the background two-body scattering length,⌬is
the detuning from the photoassociated resonance, andបkiis the relative momentum of the collision.⌫stimis the resonant
transition rate between the continuum and the molecular state关proportional to the laser intensityI共x兲兴, with⌫spon
be-ing the spontaneous decay rate from the excited molecular state. Far from the resonance, the imaginary part of the scat-tering length is small, such that Im共a兲≪Re共a兲. In Ref.关30兴, it was shown that in the experiment with87Rb one can obtain optically induced large variations of the scattering length. The laser intensity was 460 W / cm2 and the variations a
s occurred from 10a0 to 190a0共withas0= 100a0,a0 being the
Bohr radius兲.
By considering the following variable changes and defi-nitions in Eq.共1兲,
x= 2kx1, y= 2kx2, =
4wRt
2 , 共5兲
wR=
ER
ប, ER= ប2k2
2m , ␥i=0,1,2= gi
2兩g0兩
, 共6兲
we obtain the dimensionless equation
i u
+ 2u
x2+ 2u
y2+␥共x,y兲兩u兩 2u
= 0, 共7兲
where
␥共x,y兲 ⬅␥0+共␥1+i␥2兲关1 +␦0+ cos共x兲+␦0cos共y兲兴,
共8兲
and the wave function was redefined such that
u⬅u共x,y,兲=
冑
2兩g 0兩
4ER
. 共9兲
From Eq. 共6兲 to Eq.共9兲, we should note that␥0 is fixed to
−1 / 2 for attractive condensates; and +1 / 2 for repulsive con-densates. Different cases can be considered: Full 1D geom-etry is realized when =␦0= 0. The anisotropic 2D case is
realized for= 1 and␦0= 0. And the 2D isotropic case can be
achieved with=␦0= 1. Next, we consider more explicitly in
our study the anisotropic 2D case, with= 1 and ␦0= 0. As
the soliton is completely free in the y direction, we also examine the possibility of having a harmonic trapm22x
2 2
/ 2. Following the transformations 共5兲, a dimensionless fre-quencyis also defined, such that
⬅2 2
8wR
, m
22
2
x22=
冉
4ER2
冊
2y2. 共10兲
In order to extend our study of the stability conditions to a few realistic cases, the effect of a compression is also veri-fied, which can be achieved by a time variation of the back-ground value of the scattering length关22兴, leading to a time-dependent␥0. To achieve this purpose, we use the following time-dependent expression for␥0:
␥0→␥0共兲=␥0exp关2␣共−c兲共−c兲兴, 共11兲 where is the Heaviside function 关共x兲= 0共1兲 for x⬍0 共x
⬎0兲兴 controlling the initial time when the compression is switched on. Compression effect, achieved by a feeding pro-cess, can be described by an additional linear termi␣fu in the GP equation 关39兴. If the modulation of nonlinearity in time is induced by increasing the transverse frequency of the trap, then we should multiply the full nonlinear term by exp共2␣兲. With these considerations, Eq.共7兲can be written as
iu = −
2u
x2− 2u
y2−␥共x,兲兩u兩 2u
+2y2u+i␣fu, 共12兲
where
␥共x,兲 ⬅␥0共兲+共␥1+i␥2兲关1 + cos共x兲兴. 共13兲
In the above, one should take␣f= 0 when␣⫽0 in Eq. 共11兲, as such parameters have a similar role in the formalism.
III. CONSERVATIVE SYSTEM
It is useful to describe shortly the solitons and their sta-bility in the conservative case共␥2= 0兲. The one-dimensional
conservative case has been considered by using a variational approach共VA兲in关26兴. Using an exact approach, the 2D case with 1D nonlinear optical lattice was studied in关31兴, where the authors have considered the case with attractive back-ground nonlinearity共␥1⬎0兲. Looking for perspective
appli-cations to BEC, we will consider here the 2D problem with a 1D nonlinear optical lattice. Following Ref. 关26兴, we start our analysis using the VA formalism.
v= − 2v
x2− 2v
y2−关˜␥0+␥1cos共x兲兴v
3+2y2v, 共14兲
where we are redefining ␥0 to˜␥0⬅␥0+␥1. In view of our
definitions in Eq. 共5兲, this implies that for attractive con-densed systems we have˜␥0=␥1+ 1 / 2 and, for repulsive ones, ␥
˜0=␥1− 1 / 2. The sign of␥˜0gives the sign of the background
field. But, we should note that we can have situations where the same˜␥0ⱖ0 can refer to attractive or repulsive conden-sates. For example:˜␥0= 1 with␥1= 1 / 2 and␥0= 1 / 2
共attrac-tive兲;␥˜0= 1 with␥1= 3 / 2 and ␥0= −1 / 2 共repulsive兲.
These two situations differ in Eq. 共14兲, because the strength of the oscillatory term is different. However, as the results are similar, we prefer to analyze separately the cases of repulsive condensates with negative background field, which occur when 0⬍␥1⬍1 / 2共0⬎˜␥0ⱖ−1 / 2兲.
The corresponding averaged Lagrangian L is obtained from the densityL as
L=
冕
−⬁ ⬁
dx
冕
−⬁⬁
dyL, 共15兲
2L=v2−
冏
vx
冏
2−
冏
v
y
冏
2+˜␥0+␥1cos共x兲
2 v
4
−2y2v2
.
共16兲 Here, it is interesting to observe that a scaling given by is applied to the observables obtained from the above equations as the root-mean-square radius inxandydirections, chemi-cal potentials, frequencies, and energies. In order to see that, we can redefine all the observables using the variable trans-formation,¯x⬅xand¯y⬅y, such that we have no depen-dence in a new set of observables共represented with a “bar”兲 that are being calculated. This scaling essentially implies considering⬅1 in all the equations. At the end, the physi-cal observables will be given by the relations 共5兲 共with
= 1兲. For example, in the case of mean-square radius we will have
具x12典=具x
2典
4k2, 具x2 2
典=具y
2典
4k2. 共17兲
Next, in our VA we consider the Gaussian ansatz
v共x,y兲=Aexp
冉
− x22a12− y2
2a22
冊
, 共18兲whereA is the amplitude andai共i= 1 , 2兲the corresponding widths in thexandydirections, respectively.N=a1a2A2is
the normalization of Eq.共18兲, with the corresponding aver-aged Lagrangian given by
L=
冕
−⬁ ⬁
dx
冕
−⬁⬁
dyL=N 2
冋
−冉
1 2a1
2+
1 2a2
2
冊
− 2a2 2
2
+ N
4a1a2
共␥˜0+␥1e− 2a
1 2/8
兲
册
. 共19兲From the Euler-Lagrange equations for the parameters,
L/N= 0 andL/ai=1,2= 0, we obtain
2= 1
a12+
1
a22− N a1a2
共˜␥0+␥1e−2a1 2/8
兲+2a22, 共20兲
N= 4a2
a1
关
˜␥0+␥1e− 2a1 2/8
共
1 + 2a1 2
4
兲兴
, 共21兲
2a 2 4
+a2
2
a12
共˜␥0+␥1e−2a12/8兲
关
˜␥0+␥1e−2a12/8共
1 + 2a
1 2
4
兲兴
− 1 = 0. 共22兲
In the case that= 0, this set of equations, for,N, anda2,
can be expressed in terms ofa1, as
0= −
1
a12
冉
␥˜0+␥1e− 2a
1 2
/8
关
1 −2a12 8兴
␥
˜0+␥1e−2a1 2/8
关
1 + 2a1 2
4
兴
冊
, 共23兲
a2,0⬅a1
冑
␥˜0+␥1e− 2a
1 2/8
共
1 + 2a1 2
4
兲
␥
˜0+␥1e− 2a
1
2/8 , 共24兲
N0=
4
冑
关
˜␥0+␥1e− 2a1 2/8
共
1 + 2a1 2
4
兲兴
关˜␥0+␥1e− 2a1 2/8
兴 . 共25兲
For the case that⫽0, the relation fora2in terms ofa1can
be obtained from Eqs.共22兲and共24兲:
a2=
1
a2,0
冑
冋
冑
1 4+
2a 2,0 4
−1
2
册
. 共26兲Equations共26兲,共20兲, and共21兲form the set of equations for
⫽0.
Next, we consider separately the cases of attractive sys-tems, with ␥˜0=␥1+21⬎0, and repulsive ones, with ␥˜0=␥1
−12⬍0.
A. Attractive condensate„␥˜0=␥1+ 1 / 2…
This case, which corresponds to␥0= 1 / 2 and ␥1⬎0, has
been investigated recently in关31兴. With= 0, it is applied to the set of Eqs.共23兲,共25兲, and共24兲. In the general case with
⫽0, we should consider Eqs.共20兲,共21兲, and共26兲.
Let us first verify the analytic limiting cases of the VA expressions for= 0,
a2,0→a1 for a1≪1 and a1≫1,
0→− 1/a1 2
for a1≪1 and a1≫1,
N0→
4 ␥ ˜0+␥1
= 4
2␥1+ 1/2 for a1= 0,
N0→4
␥ ˜0 =
4 ␥1+ 1/2
for a1→⬁,
and the limiting cases of the VA expressions for⫽0,
a2→
再
a1 for a1≪1,
→
再
− 1/a12
for a1≪1,
for a1≫1,
冎
N→ 4
␥ ˜0+␥1
for a1= 0,
N→4a2
␥ ˜0a1
→0 for a1→⬁.
In Fig.1, we plot the corresponding results for the chemical potential as a function of N 共upper frame兲 and N as a function of a1 共lower frame兲. Numerical solutions to PDE
results were done with the algorithm presented in scheme C of Ref.关40兴. Considering the Vakhitov-Kolokolov共VK兲 cri-terion关36兴expressing the necessary condition for the soliton stability 共given by d/dN⬍0兲, we observe that the results given in the upper frame of Fig.1 provide evidence that the solitons are not stable. The soliton is orbitally stable if the orbit near the ground state remains close to this state for all times. In view of some limitations on the applicability of this criterion, as discussed in Sec. V of Ref.关38兴, the actual con-clusions on the stability were also verified by numerical simulations. These results are in agreement with the predic-tion of Ref.关31兴.
We note, from the VA results, that in the limit of largea1
the system has a tendency to stabilize, indicating that with just a small trapping potential we can produce a stable re-gion. This behavior is shown by the VA results given in Fig.
2. The variational approach, besides an expected small quan-titative shift, provides a good qualitative picture of the re-sults when compared with full numerical predictions. If one is first concerned with the stability of the system共instead of the quantitative results of the observables兲, the VA provides a nice and reliable picture.
In our VA, when we keep fixed共zero or nonzero兲and increase the value of␥1, we observe that the general picture
in respect to stability of the system does not change. This leads us to conclude that we cannot improve the stability of the system by increasing the strength of the lattice periodic-ity for attractive condensates. In the following, we are going to analyze the cases with␥˜0⬍0.
B. Repulsive condensate with␥˜0⬍0„␥˜0=␥1− 1 / 2…
We should recall that by repulsive condensate we mean an atomic system where the particles have originally positive
7 9 11 13
N
−0.3 −0.2 −0.1 0.0
µ
variational exact
0 2 4 6 8 10 12
a1
7 8 9 10 11 12 13
N
variational exact (a)
(b)
FIG. 1. Attractive case, with˜␥0= 1 and␥1= 0.5. Results for the chemical potential, as a function ofN共upper frame兲andNvsa1 共lower frame兲, obtained using variational approach 共VA兲 and full numerical calculations. The variational parameter for the width,a1, and the root-mean-square radius,
冑
具x2典, are related bya1=
冑
2具x2典 关Actually, the physical observables depend onkas given by Eqs.共5兲 and共17兲兴.5 7 9 11 13
N
−0.08 −0.04 0.00 0.04
µ
ω=0 ω=0.01 ω=0.045
0 2 4 6 8 10
a 1 5
7 9 11 13
N
ω=0 ω=0.01 ω=0.045
(a)
(b)
two-body scattering length, such that in Eq.共2兲we haveg0
⬍0 or␥0= −1 / 2. So, given␥1 共parameter of the spatial
pe-riodic variation of the atomic scattering length兲, ␥˜0=␥1
− 1 / 2. And, if we also consider a negative background, such that˜␥0⬍0,␥1will be restricted to the interval 0⬍␥1⬍1 / 2. Some other limitations are applied in the parameters, con-sidering that the widths andN must be real positive quanti-ties. The relation between the widthsa2anda1, Eq.共24兲for = 0, implies a limitation to the values ofa1:
e−2a1 2/8
ⱖ 1 2␥1
− 1, →a1,max2 = 8
2ln
冉
␥1 1 2−␥1冊
. 共27兲
This limit,a1,max, is necessary in order to havea2andNreal
and positive quantities for any value of. It will also restrict the actual values of the parameter␥1 to 1 / 4⬍␥1⬍1 / 2. The
cases with ␥1⬎1 / 2 are also allowed, in principle, without
upper limit fora1. However, such cases will correspond to a
positive background field,␥˜0⬎0, that has already been con-sidered in the preceding subsection. In view of the above, let us also verify for this case the analytic VA limits.
For= 0
a2,0→
再
a1 for a1≪1,
⬁ for a1=a1,max,
冎
0→
再
− 1/a12 for a1≪1,
1/共2a12兲 for a1=a1,max,
冎
N0→
再
8/共4␥1− 1兲 for a1= 0,
⬁ for a1=a1,max
冎
and for⫽0
a2→
再
a1 for a1≪1,
1/
冑
for a1=a1,max,冎
→
再
− 1/a12
for a1≪1,
1/共2a1 2
兲+ for a1=a1,max,
冎
N→
再
8/共4␥1− 1兲 for a1= 0, 32/关共1 − 2␥1兲冑
2a1,max3
兴 for a1=a1,max.
冎
In Fig.3, we plotNvsa1and the chemical potentialvsN,
for˜␥0= −0.1 and␥1= 0.4, considering VA and four values of 共0, 0.07, 0.1, 0.3兲. In the case of = 0, we also include results obtained from exact PDE calculations. Following the VK criterion for stability,d/dN⬍0, we notice that stable regions start to appear with⬇0.1. With⬎0.32
the un-stable regions almost disappear. However, as one can ob-serve in the lower frame, the widtha1is quite limited due to the condition共27兲. The observablesandaidepend on the wave parameterk of to the spatial periodic variation of the atomic scattering length through the scaling relations共5兲and 共6兲 with = 1. However, contrary to some discussions and conclusions of Ref.关31兴, specific values of the parameterk
cannot affect the conclusions on stability. In such cases of conservative systems, the stability results from combined ef-fects given by the parameters˜␥0,␥1, and. Our main
con-clusion is that, without the trapping potential共included in the
ydirection兲, taking= 0, the optical lattice cannot stabilize the solutions, neither for repulsive nor for attractive conden-sates.
In order to further check the role of the optical lattice, for the repulsive case we also investigate the case with constant
and different values of␥1. From the results shown in Fig.
3, for ␥1= 0.4, we found appropriate to consider = 0.07,
which has a marginal stability near⬇0.05. The results are shown in Fig.4, where we first observe that a larger␥1can
help to allow the width a1 to increase, within the limiting
condition共27兲. However, the marginal stability remains for corresponding different values of the chemical potential. In
20 30 40 50 60
N
−0.5 −0.3 −0.1 0.1 0.3
µ
ω=0
ω=0.07
ω=0.1
ω=0.3
ω=0 (exact)
ω=0.3 (exact)
0 1 2 3
a
1
20 30 40 50 60 70
N
ω=0 ω=0.07 ω=0.1 ω=0.3
(a)
(b)
order to keep the plots of Fig.4for different values of␥1in
the same frames, we have normalized the numberNsuch that it is equal to one whena1is zero.
The plots of the profiles are presented in Fig.5 for= 0 and 0.3. Stability is verified by the numerical evolution con-firming the VK prediction. The profiles at t= 50 are practi-cally undistinguishable from the initial form, in good agree-ment with the VA.
IV. EVOLUTION OF 2D SOLITON UNDER 1D PERIODIC NONLINEARITY AND DISSIPATION
In this section, we will consider the case␥2⫽0 we have
in Eq.共12兲. To study the dynamics of a 2D soliton with 1D periodic nonlinearity and dissipation, we also apply a varia-tional approach and full numerical calculations. In the Gaussian ansatz共18兲, we should also include parameters re-lated to dissipative effects and initial conditions. As dissipa-tive solitons are known to be chirped for a bright soliton 关41兴, the ansatz can be taken in the following form:
u=Aexp
冋
−共x−x0兲2
2a12 − y2
2a22
册
exp关ib1共x−x0兲 2+ib2y2
+i共兲兴. 共28兲
In this case, we include the chirp parametersbi共i= 1 , 2兲, a coordinate parameter inxdirection given byx0, and a phase of the wave packet.
The Lagrangian density for Eq.共12兲is
L共x,兲= i 2共uu
ⴱ−u ⴱu兲−兩u
x兩2−兩uy兩2+
␥共x兲 2 兩u兩
4
, 共29兲
where␥共x兲is given by Eq.共13兲. Next, from the ansatz共28兲, we obtain the corresponding averaged Lagrangian:
L=
冕
−⬁ ⬁
dx
冕
−⬁⬁
dyL= − 2A
2a 1a2
冋
a12
共b1+ 4b1 2
兲+ 1
a12
+a22共b2+ 4b22兲+ 1
a22+ 2− A2
2 关˜␥0
+␥1cos共x0兲e− 2a
1 2/8
兴
册
. 共30兲The equations for the soliton parameters i=关A,a,b,兴 in the VA are derived from共see for example关42兴兲
L
i− d
d L
i,
=
冕
−⬁ ⬁
dx
冕
−⬁⬁ dy
冋
Ruⴱ
i+ c.c.
册
, 共31兲where the perturbation termR is
R= −i␥2关1 + cos共x兲兴兩u兩2u+i␣fu. 共32兲
Here we are taking into account a linear amplification term 共␣f兲 describing the atoms feeding. Finally, from the above, we obtain the following system of five coupled ordinary dif-ferential equations共ODE兲to be solved for the parameters of our variational approach共VA兲:
1.0 1.1 1.2
N(4γ1−1)/(8π)
−0.2 −0.1 0.0 0.1
µ
γ1=0.50
γ1=0.45 γ1=0.40 γ1=0.35 ω=0.07
0 1 2 3 4
a1
1.0 1.1 1.2 1.3 1.4
N
(4
γ1
−1)/(8π)
γ1=0.5
γ1=0.45
γ1=0.40
γ1=0.35
ω=0.07
(a)
(b)
FIG. 4. VA results for the repulsive case, with˜␥0= −0.1 and
fixed to 0.07, considering␥1= 0.35, 0.4, 0.45, and 0.5. In the upper frame we havevsN/N共a1= 0兲and, in the lower frame,N/N共a1 = 0兲vsa1.
−4 −3 −2 −1 0 1 2 3 4
x,y 0
0.1 0.2 0.3 0.4 0.5
|
ψ|
2 /N
x−direction (y=0)
y−direction (x=0)
共A2a
1a2兲= −␥2A4a1a2e− 2a2/8
cos共x0兲−␥2A4a 1a2
+ 2␣fA2a1a2, 共33兲
共A2a13a2兲= 8A2a1 3a
2b1− ␥2
2A
4a 1 3a
2
冋
1 +1
4cos共x0兲共4
−2a 1 2
兲e−2a1 2/8
册
+ 2␣fA2a1 3
a2, 共34兲
共A2a1a2 3
兲= 8A2a1a2 3
b2− ␥2
2 A
4a 2 3
a1关1 +e− 2a
1 2/8
cos共x0兲兴
+ 2␣fA2a1a2 3
, 共35兲
b1= 1
a14− 4b1 2
−˜␥0A
2
4a12 − ␥1A2
4a12 cos共x0兲e
−2a12/8
冉
1 + 2a1 2
4
冊
, 共36兲b2= 1
a24− 4b2 2
−␥˜0A
2
4a22 − ␥1A2
4a22 cos共x0兲e −2a12/8
. 共37兲
By taking into account that the normN=A2a1a2, with i,j
= 1 , 2 共i⫽j兲, we have
N= −
␥2N2 a1a2
关1 +e−2a1 2/8
cos共x0兲兴+ 2␣fN, 共38兲
共ai2兲= 8ai
2 bi+
␥2Nai 2aj
冋
1 +e−2a12/8cos共x
0兲
冉
1 +␦i,1 2a1 2
4
冊
册
, 共39兲bi,= 1
ai4− 4bi 2
− N
4ai3aj
再
␥0+␥1冋
1 +e −2a1 2/8
cos共x0兲
⫻
冉
1 +␦i,1 2a1 2
4
冊
册
冎
. 共40兲Next, we present some of our results, when considering pe-riodic nonlinearity with dissipative effects. Considering the scaling of observables with, discussed for the conservative systems in Sec. III, which can also be verified in the present case, we have the corresponding transformationbi→bi/2.
In Fig.6, we have results for the full numerical simula-tions共PDE兲for the evolution of the matter-wave packet un-der combination of the conservative and dissipative nonlin-ear optical lattice in the case of the attractive condensate
␥0= 1 / 2. As we can see the collapse is arrested by the
dissi-pative nonlinear optical lattice. The results are compared with the prediction of the VA approach共ODE兲. We observe a good agreement of the VA with full numerical calculations.
We also have investigated the role of a deviation␦of the given norm from the critical norm, in the initial wave packet:
A→A共1 +␦兲. The results of the full numerical simulations are presented in Fig.7. Increasing the deviation ␦ from the critical norm, multiple peaks are observed, corresponding to revivals of the wave packet during the collapse. The number of peaks grows as␦ varies from 0.02 to 0.5. The
focusing-defocusing cycles connect the action of the periodically varying in the space with the inelastic three-body interac-tions. In a linear conservative optical lattice, with inelastic three-body interactions, the focusing-defocusing oscillations have been studied in Ref.关43兴. Such oscillations have been studied in Ref. 关44兴, using a harmonic trap with inelastic three-body interactions.
After the collapse is arrested, as observed in Fig. 6, the spreading out of the pulse can be compensated by an adia-batic variation of the background scattering length, consider-ing the time variation of ␥0. When we consider a time
de-pendent␥0, as given in Eq.共11兲, the feeding term parameter ␣fshould be zero, because one can show共with a redefinition of the wave function兲that it has a similar effect. In Fig.8, we show our full numerical results confirming the stabilization of the condensate after the collapse was arrested. The mecha-nism of this stabilization was given by an appropriate tuning of the parameters␣ andcof Eq.共11兲.
About the dissipative dynamics presented in this section, we should also add that effects of the lattice can also be observed in uniform systems. The variational approach can
0 2 4 6 8 10
τ
0 1 2 3 4 5
A
mp
litu
d
e
PDEγ2=0
PDEγ2=0.002
ODEγ2=0 ODEγ2=0.002
FIG. 6. Results for the amplitude共A兲 as a function of, using variational approach共VA兲and full numerical calculations, for the attractive case with ␥0=␥1= 0.5. The VA and the full numerical calculations have the value of= 1 fixed to the same value for
= 0, implying a small shift ofA共0兲, as shown by the results.
0 1 τ 2 3
0 5 10 15 20 25
A
mp
litu
d
e
δ=0.02 δ=0.03 δ=0.1 δ=0.5
γ2= 0.0025
describe dissipative dynamics between narrow and broad solitons relative to the lattice spacing 2/. In the limit of a narrow soliton, we have a cubic dissipative term. In the op-posite limit of a broad soliton, the averaged GPE contains an effective quintic imaginary term, providing a different dissi-pative dynamics. This effect appears with the replacement of
i␥2cos共x兲兩u兩2u by i⌫
3兩¯u兩4¯u, with ⌫3= 3␥2 2
/共22兲 and ¯u a
field averaged over rapid modulations. Thus renormalization of the parameter corresponding to effective three-body losses, ⌫3, can be controlled by the values of the period
2/ and amplitude ␥2 of the nonlinear modulations. The
number of cycles reduces as⌫3grows and for large values of
the damping, with the soliton decaying monotonically. These observations are also confirmed by the existence of focusing-defocusing cycles关44兴 in 3D BEC in a harmonic trap with two- and three-body inelastic processes.
V. CONCLUSION
Dynamics and stability of matter-wave solitons in the mixture of conservative and dissipative nonlinear optical
lat-tices are investigated, considering 2D BEC, with 1D conser-vative plus dissipative nonlinear optical lattices.
In the first part of this work, we analyzed conservative systems, with nonlinear optical lattices, for attractive and repulsive condensates. The role of the scales when calculat-ing the observables as the chemical potential and the widths was clarified. Our conclusion is that, in a 2D system, a non-linear periodic lattice in one direction by itself cannot give stable solutions, satisfying the VK criterion关36兴. Such a pe-riodic lattice in the x direction cannot compensate for the collapsing effect which results from the other dimension. We verify that stable solutions can be obtained by controlling the soliton with a harmonic trap in theydirection. For repulsive condensates, the 2D stable soliton can exist in the geometry with 1D nonlinear optical lattices in one direction and har-monic trap in the other direction.
In the second part of the work we analyze the dynamics of the above 2D system, with periodic nonlinearity in the x
direction and without trap in the y direction, when we add nonconservative nonlinear optical lattice terms. We show that the collapse of the condensate can be arrested by a dis-sipative periodic nonlinearity. To study the evolution of the 2D wave packet we apply the time-dependent variational ap-proach. To compensate the wave packet broadening, the adiabatic time variation of scattering length is used. It is shown that the metastable dissipative soliton can exist in a 2D condensate with 1D periodic nonlinearity. Analytical pre-dictions are confirmed by the numerical simulations of a full 2D GP equation.
ACKNOWLEDGMENTS
We thank Fundação de Amparo à Pesquisa do Estado de São Paulo for partial support. L.T. and A.G. also thank Con-selho Nacional de Desenvolvimento Científico e Tecnológico for partial support.
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