modulational instability of a bose–einstein condensate beyond the fermi pseudopotential with a...

September 26, 2012 12:2 WSPC/Guidelines-IJMPB S0217979212501640

International Journal of Modern Physics BVol. 26, No. 30 (2012) 1250164 (17 pages)c© World Scientific Publishing Company

DOI: 10.1142/S0217979212501640

MODULATIONAL INSTABILITY OF A BOSE EINSTEIN

CONDENSATE BEYOND THE FERMI PSEUDOPOTENTIAL

WITH A TIME-DEPENDENT COMPLEX POTENTIAL

DIDIER BELOBO BELOBO∗,§ , GERMAIN HUBERT BEN-BOLIE∗,¶,THIERRY BLANCHARD EKOGO†,‖, C. G. LATCHIO TIOFACK‡,∗∗ and

TIMOLEON CREPIN KOFANE‡,††

∗Laboratory of Nuclear Physics, Department of Physics,

Faculty of Science, University of Yaounde I,

P. O. Box 812, Yaounde, Cameroon†Departement de Physique de l’Universite des Sciences et Techniques de Masuku,

B. P. 943, Franceville, Gabon‡Laboratory of Mechanics, Department of Physics,

Faculty of Science, University of Yaounde I,

P. O. Box 812, Yaounde, Cameroon§[email protected]

¶[email protected]‖[email protected]

∗∗[email protected]††[email protected]

Received 29 March 2012Revised 12 June 2012Accepted 6 July 2012

Published 30 September 2012

The modulational instability (MI) of Bose–Einstein condensates based on a modifiedGross–Pitaevskii equation (GPE) which takes into account quantum fluctuations anda shape-dependent term, trapped in an external time-dependent complex potential isinvestigated. The external potential consists of an expulsive parabolic background witha complex potential and a gravitational field. The theoretical analysis uses a modifiedlens-type transformation which converts the modified GPE into a modified form withoutan explicit spatial dependence. A MI criterion and a growth rate are explicitly derived,both taking into account quantum fluctuations and the parameter related to the feed-ing or loss of atoms in the condensate which significantly affect the gain of instabilityof the condensate. Direct numerical simulations of the modified GPE show convincingagreements with analytical predictions. In addition, our numerical results also revealthat the gravitational field has three effects on the MI: (i) the deviation backwardor forward of solitons trains, (ii) the enhancement of the appearance of the MI and(iii) the reduction of the lifetime of pulses. Moreover, numerical simulations proved thatit is possible to control the propagation of the generated solitons trains by a proper choiceof parameters characterizing both the loss or feeding of atoms and the gravitational field,respectively.

§Corresponding author.

1250164-1

http://dx.doi.org/10.1142/S0217979212501640

mailto:[email protected]






D. B. Belobo et al.

Keywords: Bose–Einstein condensates; modified Gross–Pitaevskii equation; modifiedlens-type transformation; atoms feeding or loss; gravitational field; modulationalinstability criterion.

PACS numbers: 03.75.Lm, 03.75.Kk, 05.45.Yv, 34.20.Cf

1. Introduction

The study of nonlinear matter-wave phenomena has recently attracted much at-

tention since the experimental realization of Bose–Einstein condensates (BECs)

in ultra cold atomic clouds.1–3 After the production of condensates, many experi-

mental and theoretical works have been devoted to the investigation of the BECs

properties such as dark solitons,4–9 bright solitons,10,11 the generation and evolu-

tion of coherent structures via the MI12–17 and so on. The phenomenon of MI is a

general feature that appears in many nonlinear systems as a result of an interplay

between the nonlinear and the dispersive effects and was studied during the 1960s

in diverse fields, such as fluid dynamics, nonlinear optics and plasma physics.18

Recently, the phenomenon of MI was reported in the context of BECs both in ex-

perimental and theoretical works,12–17,19–22 and it has been considered as being

responsible for dephasing and localization phenomena in BECs.12–17,21,22

Condensate gases of very dilute alkali atomic species are well described by the

mean-field Gross–Pitaevskii equation (GPE) for the condensate wavefunction.23,24

The GPE is obtained by making some assumptions, such as the shape-independent

pseudopotential approximation.23,24 It has been a quite useful tool for the under-

standing of measurements in BECs experiments where both the gaseous parameter

na3s (n is the density of the gas, and as the scattering length) and the parameter

describing the strength of the trapping have been very small. Some recent experi-

mental observations of BECs show that the scattering length, as, can be tuned to

the desired value by using the Feshback resonance technique.25–29 Willing to obtain

more theoretical accurate predictions, some authors have recently proposed some

variant forms of the GPE that integrate quantum fluctuations around the mean-

field.30–34 In the light of these works, Haixiang et al.35 introduced a modified form of

the GPE which takes into account quantum fluctuations and the shape-dependent

confinement correction term that lead to better agreements with diffusion Monte

Carlo results.35

The generation and propagation of matter-wave solitons in BECs via the MI

have been paid increasing interests in the recent past years.12–17,21–24 Most of the

previous works devoted to study the MI in condensates do not take into account

neither the effects of quantum fluctuations nor the effects of the shape-dependent

correction term, on the MI process in condensates. However, Tiofack et al.36 have

just considered the latter effects on the MI of cigar-shaped BECs confined in an ex-

ternal harmonic potential trap. For atoms in the nK–mK temperature regime, the

effects of the Earth’s gravitational field cannot be neglected, especially in the case

of magnetic confining. Furthermore, during the evolution of a condensate soliton,

1250164-2


Modulational Instability of a Bose–Einstein Condensate

atoms in the BEC interact with the thermal cloud. This may lead to expulsion

(atoms loss) or absorption (atoms feeding) of atoms by the condensate. The mech-

anism of atoms gain or loss is related to a complex potential which has a parameter

related to the feeding or loss of atoms. The process of atomic loss is believed to

be important for the description of resonances, transport and localization effects

in condensates.37 Besides, feeding the condensate with atoms increases its density.

So, a question arises as how far one can increase the density of the condensate by a

feeding process without explosion of the condensate system. The atoms loss/feeding

is expected to destabilize the condensate. However, it is important to find out re-

gions of parameters where the transition, from stable to unstable results occurs. In

particular, the parameter related to the exchange of atoms may deeply modify the

unstable regions of the condensate modeled by the modified GPE. Some theoreti-

cal works have been devoted for the quest of exact bright solitons solutions of the

GPE in a time-dependent potential.38–40 The MI of a cubic-quintic 1D GPE in a

time-dependent complex potential has just been reported by Mohamadou et al.17

To the best of our knowledge, there has not been any report of the study of the MI

of a modified GPE in a time-dependent complex potential yet.

The purpose of this work is to investigate, both theoretically and numeri-

cally, the MI of the modified cigar-shaped GPE introduced by Haixiang et al.35

in a time-dependent potential which contains parabolic, linear and complex terms.

After a lens-type transformation, we will derive a time-dependent MI criterion that

takes into account quantum fluctuations, the shape-dependent correction term, the

strength of the parabolic background and the parameter related to the atoms feed-

ing or loss of the external potential. We will show that quantum fluctuations, the

strength of the external potential and the parameter of feeding or loss of atoms

significantly affect the MI growth rate of the condensate. The paper is organized as

follows. In Sec. II, we present the model and, by performing a linear stability analy-

sis, we show that the strength of the parabolic potential and the exchange of atoms

significantly change the MI of the BEC system. In Sec. III, we numerically integrate

the modified GPE35 and compare our numerical results with those obtained by the

linear stability analysis. Further, numerical findings prove that the gravitational

field has a profound impact on the MI of the condensate. Finally, Sec. IV concludes

the paper.

2. Theoretical Model

In order to investigate the MI in BECs, our starting point is the modified 3D GPE

introduced by Haixiang et al.35

ı~∂ψ(x, y, z; t )

∂t= − ~

2

2m∇2ψ(x, y, z; t ) + V (x, y, z)ψ(x, y, z; t )

+ g|ψ(x, y, z; t )|2ψ(x, y, z; t ) + g1|ψ(x, y, z; t )|3ψ(x, y, z; t )

+ g2∇2(|ψ(x, y, z; t )|2)ψ(x, y, z; t ) , (1)

1250164-3


D. B. Belobo et al.

where g = 4π~2as/m (as > 0) is the attractive two-body interaction, m being the

mass of atoms. The fourth term of the right-hand side of Eq. (1) represents the in-

clusion of quantum fluctuations with g1 = g(32/3√π)a

3

2

s and the last term accounts

for the shape-dependent confinement correction on the interaction potential with

g2 = (2/3)a2sg. The external potential is:

V (x, y, z; t ) =1

2m(ω2

xα21(t )x

2 + ω2yα

22(t )y

2 + ω2zα

23(t )z

2)

+1

2ω√m~ω(λ1(t )x+ λ2(t )y + λ3(t )z) +

ı~ω

2γ(t ) , (2)

where α1(t ), α2(t ), α3(t ) represent the strength of the parabolic background; λ1(t ),

λ2(t ), λ3(t ) characterize the Earth’s gravitational field or some linear potentials,

and ωx ≡ α1(t )ω, ωy ≡ α2(t )ω, ωz ≡ α3(t )ω are the angular frequencies, in the

x−, y−, z− directions, respectively, ω being the angular frequency. The parameter

related to feeding or loss of atoms is γ(t). It is convenient to manipulate Eq. (1)

in its dimensionless form. Thus, in terms of the dimensionless variables x = x/a,

y = y/a, z = z/a, t = (ω/2)t, φ(x, y, z; t) =√

2a3/Nψ (N =∫

|ψ|2dxdydz is the

number of atoms in the condensate), a =√

~/mω (harmonic oscillator length),

Eq. (1) becomes:

ı∂φ

∂t= [−∇2 + α2

1(t)x2 + α2

2(t)y2 + α2

3(t)z2 + λ1(t)x+ λ2(t)y + λ3(t)z + ıγ(t)

+ g|φ|2 + g1|φ|3 + g2∇2(|φ|2)]φ , (3)

with g = 4πN(as/a), g1 = (128√π/3

√2)N

3

2 (as/a)5

2 and g2 = (8πN/3)(as/a)3.

In the case where the condensate is strongly confined in the transverse direction

defined here by the plane (y, z), Eq. (3) reduces to a quasi-one dimensionless form.

This can be achieved in experiments by making the radial frequency much larger

than the axial frequency and by strongly confining the radial motion. In order to

derive the 1D form of Eq. (3), we follow Muruganandam et al.41 Thus, we assume

that the BEC remains confined to the ground state in the transverse direction and

consider the ansatz41

φ(x, y, z; t) = φ(x; t)φ0(y)φ0(z) exp[−ı(α2 + α3)] , (4)

with φ0(y) = (α2/π)1

4 exp(α2y2/2), φ0(z) = (α3/π)

1

4 exp(α3z2/2), φ0(y) and φ0(z)

being normalized to unity. Introducing Eq. (4) into Eq. (3), then multiplying by the

left by φ0(y) and φ0(z) and integrating over y and z in space yield the dimensionless

1D form of Eq. (1):

ı∂φ

∂t(x; t) =

[

− ∂2

∂x2+ α2

1(t)x2 + λ1(t)x + ıγ(t) + g′|φ(x; t)|2 + g′1|φ(x; t)|3

+ g′2∂2

∂x2(|φ(x; t)|2)

]

φ(x; t) , (5)

1250164-4



where

g′ = 2Nasa

√α2α3

[

1− 2

3

(asa

)2

(α2 + α3)

]

g′1 =128

√2

15πN

3

2

(asa

)5

2

(α2α3)3

4

g′2 =4N

3

(asa

)3 √α2α3 .

(6)

Making the changes α21(t) = α(t) and λ1(t) = λ(t), Eq. (4) can be rewritten as:

ı∂φ

∂t(x; t) =

[

− ∂2

∂x2+ α(t)x2 + λ(t)x+ ıγ(t) + g′|φ(x; t)|2 + g′1|φ|3

+ g′2∂2

∂x2(|φ(x; t)|2)

]

φ(x; t) . (7)

Now, we examine the MI of the modified 1D GPE in a time dependent com-

plex potential described by Eq. (7). To this end, we use the modified lens-type

transformation17,42

φ(x; t) =1

√

l(t)Φ(X ;T ) exp[η(t) + ıf(t)Z(x, t)], (8)

where T , η and f are functions of time, X = x/l(t), and Z(x, t) = x2 + σ(t)x. We

demand that:

dT

dt=

1

l2(t), (9)

dl

dt= 4f(t)l(t) , (10)

df

dt= −4f2(t)− α(t) , (11)

dσ

dt=

−λ(t) + σ(t)α(t)

f(t), (12)

dη

dt= γ(t) , (13)

where Eq. (9) preserves the scaling. Thus, Eq. (7), in terms of the new rescaled

variables X and T becomes:

ı∂Φ

∂T= − ∂2Φ

∂X2−2ık(t)

∂Φ

∂X+k2(t)Φ+g′(t)|Φ|2Φ+g′1(t)|Φ|3Φ+g′2(t)|Φ|2XXΦ , (14)

with k(t) = f(t)l(t)σ(t), g′(t) = g′l(t) exp(2η(t)), g′1(t) = g′1√

l(t) exp(3η(t)) and

g′2(t) = (g′2/l(t)) exp(2η(t)).

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D. B. Belobo et al.

3. Linear Stability Analysis

The purpose of the linear stability analysis is to slightly perturb the system and

then study how this small perturbation increases or reduces during the propagation.

So, in order to examine the MI of the modified 1D GPE with the time-dependent

complex potential, we use the following ansatz:

Φ = (Φ0 + δΦ) exp

[

−ı∫ T

0

Θ(T ′)dT ′

]

, (15)

where Θ(T ) is the time depending nonlinear frequency shift, Φ0 is a real constant

and δΦ accounts for a small complex perturbation on the wavefunction envelope.

Inserting Eq. (15) into Eq. (14), keeping only terms of order one in δΦ and its

complex conjugate δΦ∗, and taking into account the fact that

Θ(T ) = k2(t) + Φ20[g

′(t) + Φ0g′

1(t)] , (16)

lead to the equation describing the dynamics of the perturbation

ı∂δΦ

∂T= −∂

2δΦ

∂X2− 2ık(t)

∂Φ

∂X+∆(t)(δΦ + δΦ∗) + g′2(t)Φ

20

(

∂δΦ

∂X2+∂δΦ∗∂X2

)

, (17)

where ∆(t) = Φ20[g

′(t) + (3/2)g′1(t)Φ0]. Now, we consider that the perturbation

obeys the relation δΦ = U + ıV , U and V being two real functions. Thus, Eq. (17)

yields the following system of two coupled equations for U and V

∂U

∂T= − ∂2V

∂X2− 2k(t)

∂U

∂X

∂V

∂T= ∆′(t)

∂2U

∂X2− 2k(t)

∂V

∂X− 2∆(t)U ,

(18)

with ∆′(t) = 1− 2g′2(t)Φ20. Assuming that the functions U and V take the forms

U = Re

U0 exp

[

−ı(KX −∫ T

0

Ω(T ′)dT ′)

]

V = Im

V0 exp

[

−ı(KX −∫ T

0

Ω(T ′)dT ′)

]

,

(19)

whereKX−∫ T

0 Ω(T ′)dT ′ represents the phase of modulation,K is the wave number

and Ω(T ) is the frequency of the modulation. Introducing Eq. (19) into the coupled

system given by Eq. (18), we obtain:

Ω2 − 4k(t)KΩ+ 4k2(t)K2 −K4∆′(t)− 2K2∆(t) = 0 . (20)

Relation (20) is the time-dependent dispersion relation. The modified 1D GPE is

unstable under modulation if the reduced discriminant is negative. Hence, the MI

criterion is:

K2[K2∆′(t) + 2∆(t)] < 0 , (21)

1250164-6



and the gain of MI G is

G = |Im(√

K2[K2∆′(t) + 2∆(t)])| . (22)

Until now, both the MI criterion Eq. (21) and the gain of MI Eq. (22) are implicit

relations. In order to easily derive explicit expressions of both the MI criterion

and that of G, we consider that the strength of the external potential is a non-

vanishing constant12–16 (this true in most of current experiments). So, we choose,

as solutions of Eqs. (9)–(11) (in the case of an attractive parabolic background),

l(t) = | cos(2√αt)|, f(t) = −(√α/2) tan(2

√αt) and T (t) = (1/2) tan(2

√αt). These

relations have some singularities at any time t = tn = (2n+ 1)π/4√α, (n being

a positive integer). We restrict our study to the case where t varies from zero to

t(n=0) = π/4√α in order to avoid negative values of T .

Solutions of Eqs. (12) and (13) can be rather complicated, so we consider the

simplest case and assume that σ(t) is a constant (this means that the complex

potential is a constant), to easily solve them. We obtain σ = λ/α and η(t) = γt.

Thus, in the next, our study focuses in the case where the external potential is

time-independent. The explicit forms of both the MI criterion and that of the gain

of MI are obtained by introducing the above explicit expressions of f(t), l(t), T (t)

and η(t) into Eqs. (21) and (22), where the time-dependent nonlinear coefficients

are:

g′(t) = 2Nasa

√α2α3

[

1− 2

3

(asa

)2

(α2 + α3)

]

(1 + 4αT 2)−1

2 exp(2γt)

g′1(t) =128

√2

15πN

3

2

(asa

)5

2

(α2α3)3

4 (1 + 4αT 2)−1

4 exp(3γt)

g′2(t) =4N

3

(asa

)3 √α2α3(1 + 4αT 2)

1

2 exp(2γt) . (23)

It is obvious, from expressions of ∆(t), ∆′(t) and Eq. (21) that our model fails to

show the influence of the gravitational field on the occurrence of the MI. However,

we will show in the next section that, the gravitational field significantly affects the

dynamics of the condensate.

The MI criterion is characterized by the strength of the parabolic trap as well

as the feeding or loss of atoms. The case where the number of atoms in the BEC

is constant (γ = 0) and in the absence of the gravitational field (λ = 0) has been

studied in details in Ref. 36. So, in this section, we focus our attention in the two

cases where the condensate loses (γ < 0) and gains (γ > 0) atoms, with α 6= 0,

respectively. The loss or feeding of atoms has a huge impact on the MI of the

condensate.

In the case where the condensate loses atoms (γ < 0), the MI growth rate

increases with increasing γ, for a fixed value of α. This means that the atoms loss

enhances the MI of the condensate. One can see this behavior on Fig. 1 where the

MI growth rate is plotted as a function of the wave number K, for three negative

values of γ. In both the cases of weak confinement [Fig. 1(a)] and strong confinement

1250164-7


D. B. Belobo et al.

6 4 2 0 2 4 60

5

10

15

20

25

30

K

Gain

(a)

5 0 50

10

20

30

40

50

60

K

Gain

(b)

3 2 1 0 1 2 30

5

10

15

K

Gain

(c)

5 0 50

5

10

15

20

25

K

Gain

(d)

Fig. 1. MI growth rate as a function of the wave number K, for three different values of γ, inthe case where the condensate loses atoms. Panels (a) and (c) α = 0.00001 (weak confinement)for small values of γ: γ = −0.01 dotted line, γ = −0.012 dash-dotted line, γ = −0.015 solid line,

N = 5, Φ0 = 10. Panels (b) and (d) α = 0.001 (strong confinement): γ = −0.019 dotted line,γ = −0.015 dash-dotted line, γ = −0.012 solid line, N = 20, Φ0 = 40. Panel (a) t = 30; panel(c) t = 60; Panel (b) Φ0 = 40, t = 30; Panel (d) t = 60. The other parameters are as = −2.75 nm,a = 1.576 · 103 nm.

[Fig. 1(b)], we realize that the gain increases as γ increases. Furthermore, for small

enough values of γ, the amplitude of the gain of the MI globally decreases with

time as shown in Fig. 1(c) (weak confinement) and Fig. 1(d) (strong confinement).

But, for large enough values of γ, our model, as it will be shown in the next section,

fails to fit the numerical results and the MI growth rate is deeply affected by the

atoms loss.

In the case where the BEC gains atoms (γ > 0), the feeding of atoms enhances

the MI of the condensate. This behavior is depicted in Fig. 2 where the MI gain

is drawn as a function of the wave number K, for three different values of γ, and

α being fixed. For both strong trapping [Fig. 2(a)] and weak trapping [Fig. 2(b)],

1250164-8



5 0 50

10

20

30

40

50

60

70

K

Gain

(a)

5 0 50

10

20

30

40

50

60

K

Gain

(b)

5 0 50

10

20

30

40

50

60

70

80

90

K

Gain

(c)

6 4 2 0 2 4 60

20

40

60

80

100

120

K

Gain

(d)

Fig. 2. MI growth rate as a function of the wave number K, in the case where the condensategains atoms, for three different values of γ: γ = 0.008 solid line, γ = 0.004 dash-dotted line,γ = 0.001 dotted line. Panels (a) and (c): α = 0.00001 (weak confinement), Φ0 = 10, N = 5.Panels (b) and (d) α = 0.001 (strong confinement), Φ0 = 40, N = 20. Panel (a) t = 10; panel(c) t = 30. Panel (b) t = 30; panel (d): t = 35. The other parameters are as = −2.75 nm,a = 1.576 · 103 nm.

the gain increases with increasing values of γ. As time increases, the gain globally

increases as confirmed in Fig. 2(c) (weak trapping) and Fig. 2(d) (strong trapping),

and the amplitude of the MI growth rate is higher than in Figs. 2(a) and 2(b).

A comparison of Figs. 1 and 2 shows that the amplitude of the gain is larger for

a condensate in the feeding regime than when the BEC loses atoms. This implies

that adding atoms to the condensate enhances the instability, while removing atoms

from the condensate alleviates the instability.

The exchange of atoms (feeding or loss) has a profound impact in the occurrence

of the MI of the BEC system. Furthermore, quantum fluctuations extend the MI

region of the condensate to the whole spectrum of the wave number K. The shape-

confinement term is very small and does not have any effect on the MI. A similar

result has been obtained in Ref. 36.

1250164-9


D. B. Belobo et al.

4. Numerical Simulations

In order to compare our previous analytical results, we also integrate numerically

the MI of the full 1D modified GPE [Eq. (7)] with the split-step Fourier method.

We choose as initial condition the function

φ(x; 0) = φTF [φ0 + ε cos(Kx)] , (24)

where φTF ≃ 1 − (1/2)αx2 is the background wavefunction in the Thomas–

Fermi approximation.12–17 In most of our numerical simulations, we have set

as = −2.75 nm, a = 1.576 · 103 nm,41 ε = 0.001,12–16 Φ0 = 10, K = 1.5.

Our numerical simulations start by injecting the initial condition Eq. (24)

through the system. First of all, we examine the effects of the exchange of atoms

on the dynamics of the BEC, so we switch off the gravitational field (λ = 0). It

has been shown in Refs. 12–17 that the detection of the occurrence of instability

can be done by investigating the maxima amplitude of the initial plane wave. Let

us first consider the case where the condensate loses atoms. Figures 3 (parame-

ters correspond with those in Fig. 1) represent temporal evolution of the maxima

amplitude, Maxx|Φ(x; t)|2, for three different values of γ. In Fig. 3(a) (weak confine-

ment), the density of the condensate increases with γ and it globally reduces with

time increasing. The amplitude of the condensate first gradually varies and then

starts oscillating at random. This result is in good agreement with the analytical

predictions. The MI sets in at a critical time t = Tcrit = 18. For large enough values

of γ, Fig. 3(b) (weak confinement) shows that the density of the BEC increases

with increasing values of γ. A result that does not corroborate its theoretical coun-

terpart, as mentioned above. Thus, the linear stability analysis is not sufficient for

the description of the MI growth rate for larger values of γ. For a too large value

of γ, the generated pulses are destroyed after a shorter time of propagation. This

means that for too large values of the parameter characterizing the loss of atoms,

the generated solitons will propagate with shorter lifetimes. Hence, one can infer

from Fig. 3(b) that the atoms loss does not reduce the number of atoms in the

condensate for too large values of γ. This may be due to the fact that quantum

fluctuations are more important since in the absence of quantum fluctuations, it

was proved in Ref. 17 that the amplitude of the condensate reduces with time when

the condensate loses atoms. Moreover, Fig. 3(b) proves that the MI sets in at a

critical time t = Tcrit = 16, while for too large values of γ, t = Tcrit = 18 in

Fig. 3(a), confirming that the increase of γ enhances the MI. When the strength

of the external potential is strong, the condensate is compressed at the beginning

of the propagation, then, after a gradual variation, the maximal amplitude starts

oscillating randomly and globally decreases with time. Figure 3(c) depicts this be-

havior that is in good accordance with the theoretical analysis. Figures 3 confirm

the fact that an instability occurs in the BEC system due to atoms loss. In order

to investigate the dynamics of the condensate, one should plot the spatial-temporal

evolution of the original wavefunction given by Eq. (24). As assumed in the theoret-

ical study, the initial plane wave evolves into a solitons train. Figures 4(a) and 4(b),

1250164-10



0 20 40 60 80 1000.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

t

Ma

xx|P

si(x,

t)|2

(a)

0 20 40 60 80 1000

1

2

3

4

5

6

7

8

t

Ma

xx|P

si(x,

t)|2

(b)

0 20 40 60 80 1000

0.5

1

1.5

2

2.5

3

3.5

4

t

Max

x|P

si(x, t)

|2

(c)

Fig. 3. Temporal evolution of the maximal amplitude maxx |ψ(x; t)|2 for three different valuesof the parameter γ, when the BEC loses atoms. Panels (a) α = 0.00001 (weak confinement, withsmall values of γ) γ = −0.01 solid line, γ = −0.012 dash-dotted line, γ = −0.015 dash-dotted line,N = 5. Panel (b) α = 0.00001 (weak confinement with large enough values of γ) γ = −0.004 solidline, γ = −0.003 dotted line, γ = −0.002 dash-dotted line, N = 5. Panel (c) α = 0.001 (strongconfinement) γ = −0.019 solid line, γ = −0.015 dotted line, γ = −0.012 dash-dotted line, N = 20.The other parameters are λ = 0, as = −2.75 nm, a = 1.576 · 103 nm, K = 1.5, ε = 0.001.

where the condensate is submitted to weak confinement, display some samples of

chains of pulses. However, the magnitude of the pulses globally decreases with time

in Fig. 4(a), where γ is relatively small. The number of pulses increases with time,

due to collisions among pulses. The solitons train is symmetric around the center of

the spatial coordinate x = 0. In Fig. 4(b), γ is relatively large and we observe that

the magnitude of the wave increases with time. The train of solitons has a shorter

lifetime than in the case where γ is small. Hence, a proper choice of the value of

γ, that should be relatively small, may enhance the lifetime of a train of solitons.

Figure 4(c), also portrays the spatial-temporal evolution of the BEC submitted to

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D. B. Belobo et al.

(a) (b)

(c)

Fig. 4. Spatial-temporal evolution of modulated solitary plane waves, in the the case where theBEC loses atoms. Panel (a) α = 0.00001 (weak confinement, for small value of γ) γ = −0.01,N = 5. Panel (b) α = 0.00001 (weak confinement, for a large value of γ) γ = −0.001, N = 5.Panel (c) α = 0.001 (strong confinement) γ = −0.01, N = 20. The other parameters are λ = 0,

as = −2.75 nm, a = 1.576 · 103 nm, K = 1.5, ε = 0.001.

a strong confinement. The magnitude of the train of solitonic matter-waves induced

by modulation globally reduces with time. A result that well recovers the analytical

one. Solitons of the train evolves oscillating around the spatial axis’ center x = 0

and the number of pulses grows with time.

In the case where the condensate gains atoms (γ > 0), Figs. 5(a) and 5(b)

represent temporal evolution of the maxima amplitude of the initial wavefunction.

Maxima amplitude first gradually increase, and then begin oscillating at random.

Amplitudes increase with time and also increase as well as γ increases. The value

of γ does not have a significant impact on the occurrence of the MI. In Fig. 5(a),

the BEC is weakly trapped, the MI sets in at t = Tc = 16. Meanwhile, in Fig. 5(b),

the condensate is strongly trapped, the instability sets in at t = Tc = 10. Thus,

the MI sets in earlier when the condensate is strongly confined. Figures 5(c) (weak

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0 20 40 60 80 1000

2

4

6

8

10

12

14

t

Ma

xx|P

si(x,

t)|2

(a)

0 20 40 60 80 1002

4

6

8

10

12

14

16

t

Ma

xx|P

si(x,

t)|2

(b)

(c) (d)

(e)

Fig. 5. Temporal evolution of maxima amplitudes maxx |ψ(x; t)|2. Panels (a) and (b) for threedifferent values of γ =: γ = 0.001 solid line, γ = 0.004 dotted line, γ = 0.008 dash-dotted line.Panel (a) α = 0.00001 (weak confinement), N = 5. Panel (b) α = 0.001 (strong confinement),N = 20. Panels (c), (d) and (e) display spatial-temporal evolution of the wavefunction. Panel (c)α = 0.00001 (weak confinement), γ = 0.001, N = 5. Panel (d) α = 0.001 (strong confinement),γ = 0.001, N = 20. Panel (e) α = 0.00001 (weak confinement), γ = 0.01, N = 5. The otherparameters are λ = 0, as = −2.75 nm, a = 1.576 · 103 nm, K = 1.5, ε = 0.001.

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D. B. Belobo et al.

(a) (b)

(c) (d)

Fig. 6. Effects of the gravitational field on the MI. The dynamics of modulated plane wavesare examined via the spatial-temporal evolution of the original wavefunction |ψ(x; t)|. Panel (a)α = 0.00001 (weak confinement), λ = 0.01, N = 5. Panel (b) α = 0.00001, λ = −0.01, N = 5.Panel (c) α = 0.001 (strong confinement), λ = 0.01, N = 20. Panel (c) α = 0.001 (strongconfinement), λ = −0.01, N = 20. The other parameters are γ = −0.01, as = −2.75 nm,a = 1.576 · 103 nm, K = 1.5, ε = 0.001.

confinement case), and 5(d) (strong confinement case) exhibit spatial-temporal evo-

lution of the condensate system. The lifetime of trains of solitons are shortened

compared to the cases where the BEC loses atoms. When the value of γ increases,

Fig. 5(e) shows that the MI occurs earlier (t = Tc = 14) and the lifetime of the

pulses is significantly reduced. This means that an appropriate choice of the pa-

rameter γ may help to control the propagation of trains of solitons. Furthermore,

in Figs. 5(c), 5(d) and 5(e), the magnitude of the wavefunction increases with time,

meaning that our numerical findings recover the analytical ones.

Now, let us present our numerical results for the situation where the conden-

sate is subjected to the gravitational field or some linear potentials. Figure 6 por-

tray spatial-temporal evolution of the initial wavefunction, the BEC being weakly

trapped. In Fig. 6(a), λ is positive, the trail of the solitons train is deviated back-

ward in space, to the x-direction. A comparison between Figs. 4(a) and 6(a) shows

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(a) (b)

(c) (d)

Fig. 7. Effect of the increase of the intensity of the gravitational field on the MI. Panel(a) λ = 0.01. Panel (b) λ = 0.1. Panel (c) λ = −0.01. Panel (d) λ = −0.1. The other parametersare γ = 0, α = 0, as = −2.75 nm, a = 1.576 · 103 nm, K = 1.5, ε = 0.001, N = 5.

a striking confirmation of this behavior. In Fig. 6(a), in addition to shifting back-

ward the trail of the chain of pulses, the spatial symmetry of the chain is destroyed

regarding that plotted in Fig. 4(a). Similar behaviors are obtained when the sign of

λ is changed from a positive value to a negative one, but the trail of the condensate

is deviated forward. A convincing example is obtained by comparing Figs. 4(a) and

6(b), where Fig. 6(b) depicts changes due to a negative value of λ. Moreover, in the

presence of the linear potential, the instability occurs earlier. This other effect of

the gravitational potential can be observed by comparing Figs. 6(a) and 6(b) with

Fig. 4(a). The trains of pulses emerge earlier in Figs. 6(a) and 6(b) (t = Tc ≈ 4)

than in Fig. 4(a) (t = Tc ≈ 18). For a condensate in a strong trapping, analogous

results are also obtained. Figures 6(c) (λ > 0) and 6(d) (λ < 0) display the spatial-

temporal evolution of a condensate strongly confined and should be compared to

Fig. 4(c) in order to notice the differences. So, the gravitational field brings two new

effects which are the deviation of the trail of the condensate and the enhancement of

the occurrence of the MI, respectively. Besides these two effects, the gravitational

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D. B. Belobo et al.

field also reduces the lifetime of trains of solitons when λ increases. In order to

make easy interpretations, we cancel out the parameter related to the exchange of

atoms (γ = 0) and the parabolic potential (α = 0). We have plotted in Fig. 7 the

spatial-temporal evolution of the wavefunction subjected only to the gravitational

field. A confrontation of Figs. 7(a) (λ = 0.01) and 7(b) (λ = 0.1) where, λ takes

positive values shows that the lifetime of the solitons’ train is significantly reduced

in Fig. 7(b). A result similar to this one is also obtained when λ takes negatives

values. We can realize that by comparing Figs. 7(c) (λ = 0.01) and 7(d) (λ = 0.1).

The gravitational field (or some linear potentials) brings three major effects on the

MI of the condensate. First, it profoundly changes the trail of chains of pulses.

Secondly, it enhances the occurrence of the MI in the BEC system, and it reduces

the lifetime of pulses when its intensity increases.

5. Conclusion

In this paper, we have studied both analytically and numerically the MI of BECs

of a modified GPE, that takes into account the energy dependence of the two-

body scattering length through an effective-range expansion, trapped in an exter-

nal time-dependent complex potential. The external trapping consists of a repulsive

parabolic and complex potential considering the effects of the gravitational field or

some other linear potentials. In our theoretical study, we have used a modified lens-

type transformation, then derived a MI criterion and an explicit expression of the

gain of MI that take into account quantum fluctuations, the shape-dependence of

the external potential, the strength of the parabolic background and the exchange

(feeding or loss) of atoms with the thermal cloud. The loss or feeding of atoms

enhances the MI. However, the gain of MI increases with time when the condensate

gains atoms while it decreases when the condensate loses atoms. The exchange of

atoms significantly modifies the occurrence of the MI. We have integrated numer-

ically the modified GPE in the complex potential, in order to test the analytical

predictions, and good agreements are obtained. The lifetime of trains of solitons is

greater in the case where the BEC loses atoms than when it gains atoms since the

pulses are destroyed. Moreover, our numerical results showed that the gravitational

field (or some linear potentials) deviates the trail of trains of solitons, enhances the

appearance of these trains and reduces the lifetime of the pulses when the absolute

value of γ is increased. Hence, with a correct choice of parameters characterizing the

exchange of atoms and that of the gravitational field, it will be possible to control

the propagation of trains of solitons.

In future works, the model can be extended to other physically interesting

situations, such as the MI in BECs with dipole–dipole interactions.

References

1. M. H. Anderson et al., Science 269, 198 (1995).2. K. B. Davis et al., Phys. Rev. Lett. 75, 3969 (1995).

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3. C. C. Bradley et al., Phys. Rev. Lett. 75, 1687 (1995).4. S. Burger et al., Phys. Rev. Lett. 83, 5198 (1999).5. B. P. Anderson et al., Phys. Rev. Lett. 86, 2926 (2001).6. J. Denschlag et al., Science 287, 97 (2000).7. K. Bongs et al., C. R. Acad. Sci. Paris 2, 671 (2001).8. S. Stellmer et al., Phys. Rev. Lett. 101 120406 (2008).9. A. Weller et al., Phys. Rev. Lett. 101, 130401 (2008).

10. K. E. Strecker et al., Nature (London) 417, 150 (2002).11. L. Khaykovich et al., Science 296, 1290 (2002).12. G. Theocharis et al., Phys. Rev. A 67, 063610 (2003).13. J.-K. Xue, Phys. Lett. A 341, 527 (2005).14. L. Wu and J.-F. Zhang, Chin. Phys. Lett. 24, 1471 (2007).15. E. Wamba, A. Mohamadou and T. C. Kofane, Phys. Rev. E 77, 046216 (2008).16. E. Wamba, A. Mohamadou and T. C. Kofane, J. Phys. B, At. Mol. Opt. Phys. 41,

225403 (2008).17. A. Mohamadou et al., Phys. Rev. A 84, 023602 (2011).18. G. P. Agrawal, Nonlinear Fiber Optics (Rochester, New York, 2006).19. M. Kasevich and A. Tuchman, Private Communication.20. F. S. Cataliotti et al., New J. Phys. 5, 71 (2003).21. A. Smerzi et al., Phys. Rev. A 65, 021602 (2002).22. V. V. Konotop and M. Salerno, Phys. Rev. A 65, 021602 (2002).23. F. Dalfovo et al., Rev. Mod. Phys. 71, 463 (1999).24. A. J. Leggett, Rev. Mod. Phys. 73, 307 (2001).25. S. Inouye et al., Nature (London) 392, 151 (1998).26. Ph. Courteille et al., Phys. Rev. Lett. 81, 69 (1998).27. J. L. Roberts et al., Phys. Rev. Lett. 81, 5109 (1998).28. V. Vuletic et al., Phys. Rev. Lett. 82, 1406 (1999).29. S. L. Cornish et al., Phys. Rev. Lett. 85, 1795 (2000).30. E. Braaten and A. Nieto, Phys. Rev. B 56, 14745 (1997).31. A. Fabrocini and A. Polls, Phys. Rev. A 60, 2319 (1999).32. A. Fabrocini and A. Polls, Phys. Rev. A 64, 063610 (2001).33. A. Banerjee and M. P. Singh, Phys. Rev. A 64, 063604 (2001).34. D. Blume and C. H. Greene, Phys. Rev. A 63, 063601 (2001).35. H. Fu, Y. Wang and B. Gao, Phys. Rev. A 67, 053612 (2003).36. C. G. Latchio Tiofack et al., Appl. Math. in press.37. F. Trimborn, D. Witthaut and S. Wimberger, J. Phys. B, At. Mol. Opt. Phys. 41,

171001 (2008).38. J. Belmonte-Beitia et al., Phys. Rev. Lett. 100, 164102 (2008).39. L.-C. Zhao et al., Chin. Phys. Lett. 26, 120301 (2009).40. D. S. Wang et al., J. Phys. B, At. Mol. Opt. Phys. 42, 245303 (2009).41. P. Mururaganandam and S. K. Adhikari, Comp. Phys. Comm. 180, 1888 (2009).42. D. Schumayer and B. Apagyi, Phys. Rev. A. 65 053614 (2002).

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