# Trapping and transmission of matter-wave solitons in a collisionally inhomogeneous environment

**ABSTRACT** We investigate bright matter-wave solitons in the presence of a spatially varying scattering length. It is demonstrated that, even in the absence of any external trapping potential, a soliton can be confined due to the inhomogeneous collisional interactions. Moreover we observe the enhanced transmission of matter-wave solitons through potential barriers for suitably chosen spatial variations of the scattering length. The results indicate that the manipulation of atomic interactions can become a versatile tool to control the dynamics of matter waves. Comment: 4 pages, 4 figures

**0**Bookmarks

**·**

**80**Views

- [Show abstract] [Hide abstract]

**ABSTRACT:**We have experimentally realized the self-generation of dense trains of spin-wave solitons in active magnetic-film ring resonators. The principle of the soliton auto-oscillator was based on the nonlinear interaction of two copropagating spin waves having the ratio between their group velocities equal to 2 and the transmission bands well separated in the frequency space. We show that under these conditions up to six dark solitons can simultaneously circulate in the active ring. The solitonic nature of the generated pulses is proven by the comparison of the obtained waveforms with those calculated on the basis of the nonlinear Schrödinger equation model.Physical Review B 08/2009; 80(5). · 3.66 Impact Factor - SourceAvailable from: de.arxiv.org[Show abstract] [Hide abstract]

**ABSTRACT:**The aim of this introductory article is two-fold. First, we aim to offer a general introduction to the theme of Bose-Einstein condensates, and briefly discuss the evolution of a number of relevant research directions during the last two decades. Second, we introduce and present the articles that appear in this Special Volume of Romanian Reports in Physics celebrating the conclusion of the second decade since the experimental creation of Bose-Einstein condensation in ultracold gases of alkali-metal atoms.02/2015; - SourceAvailable from: D. J. Frantzeskakis
##### Article: TOPICAL REVIEW: Dark solitons in atomic Bose-Einstein condensates: from theory to experiments

[Show abstract] [Hide abstract]

**ABSTRACT:**This review paper presents an overview of the theoretical and experimental progress on the study of matter-wave dark solitons in atomic Bose-Einstein condensates. Upon introducing the general framework, we discuss the statics and dynamics of single and multiple matter-wave dark solitons in the quasi one-dimensional setting, in higher dimensional settings, as well as in the dimensionality crossover regime. Special attention is paid to the connection between theoretical results, obtained by various analytical approaches, and relevant experimental observations.Journal of Physics A Mathematical and Theoretical 05/2010; 43. · 1.77 Impact Factor

Page 1

arXiv:cond-mat/0509471v1 [cond-mat.soft] 18 Sep 2005

Trapping and transmission of matter-wave solitons in a collisionally inhomogeneous

environment

G. Theocharis1, P. Schmelcher2,3, P. G. Kevrekidis4and D. J. Frantzeskakis1

1Department of Physics, University of Athens, Panepistimiopolis,Zografos, Athens 157 84, Greece

2Theoretische Chemie, Physikalisch-Chemisches Institut,

Im Neuenheimer Feld 229, Universit¨ at Heidelberg, 69120 Heidelberg, Germany

3Physikalisches Institut, Philosophenweg 12, Universit¨ at Heidelberg, 69120 Heidelberg, Germany

4Department of Mathematics and Statistics,University of Massachusetts, Amherst MA 01003-4515, USA

We investigate bright matter-wave solitons in the presence of a spatially varying scattering length.

It is demonstrated that, even in the absence of any external trapping potential, a soliton can be

confined due to the inhomogeneous collisional interactions.

transmission of matter-wave solitons through potential barriers for suitably chosen spatial variations

of the scattering length. The results indicate that the manipulation of atomic interactions can

become a versatile tool to control the dynamics of matter waves.

Moreover we observe the enhanced

Introduction.

of atomic Bose-Einstein condensates (BECs) [1] has in-

spired, among others, many studies on their nonlinear

excitations. Especially, as far as matter-wave solitons are

concerned, dark [2], bright [3, 4] and gap [5] solitons have

been observed experimentally and studied theoretically.

Atom optical devices such as the atom chip [6] offer the

possibility to control and manipulate matter-wave soli-

tons. Their formal similarities with optical solitons indi-

cate that they may be used in applications similarly to

their optical counterparts [7].

Typically bright (dark) matter-wave solitons are

formed in BECs with attractive (repulsive) interatomic

interactions, i.e., for atomic species with negative (pos-

itive) scattering length a.

induced Feshbach resonances both the magnitude and

sign of the scattering length can be changed by tuning

the external magnetic field (see e.g. [8] and also [3, 4]

where the Feshbach resonance in7Li condensates was

used for the formation of bright matter-wave solitons).

These studies paved the way for important experimen-

tal discoveries, such as the formation of molecular BECs

[9] and the revelation of the BEC-BCS crossover [10].

From the theoretical viewpoint, it was predicted that a

time-dependent modulation of the scattering length can

be used to prevent collapse in higher-dimensional attrac-

tive BECs [11], or to create robust matter-wave breathers

[12]. Adding to a constant bias magnetic field a gradient

in the vicinity of a Feshbach resonance allows for a spa-

tial variation of the scattering length over the ensemble

of cold atoms thereby yielding a collisionally inhomoge-

neous condensate. Due to the availability of magnetic

and optical (laser-) fields the external trapping poten-

tial and the spatial variation of the scattering length can

be adjusted independently. Moreover hyperfine species

with the magnetic quantum number MF = 0 do not

feel a potential due to the magnetic field but experience

magnetically-induced Feshbach resonances (see e.g. [13]).

In this case the external potential is formed exclusively

by an optical dipole potential and the magnetic field con-

The recent developments in the field

Employing magnetically-

figuration is responsible for the spatially dependent scat-

tering length. Recently this has been exploited to study

the properties of cold atomic gases in a collisionally in-

homogeneous environment (CIE) [14, 15].

Here we investigate the dynamics of bright matter-

wave solitons of a quasi one-dimensional (1D)7Li BEC

[3, 4] in a CIE demonstrating the appearance of unex-

pected phenomena which make the spatial manipulation

of the scattering length a versatile tool. Firstly we show

that the inhomogeneity of a induces an effective confin-

ing potential felt by a bright matter-wave soliton, even

in the absence of any external trapping potential. The

proposed scheme leads to a collision-induced breathing

soliton, which oscillates due to the effective confinement

and is periodically compressed when passing through the

region of a large scattering length. As a second prototyp-

ical situation for a CIE, we consider the transmission of

a soliton through a potential barrier underneath which a

suitably chosen spatially varying scattering length a(x)

is present. This setup allows to enhance the transmission

of the soliton, i.e., the barrier becomes more transparent

compared to the case of a spatially independent atom-

atom interaction. More specifically, we use the partic-

ular form a(B) for a7Li condensate near a correspond-

ing Feshbach resonance (see below). This species has al-

ready been used in order to prepare bright matter-wave

solitons. We believe that the collision-induced trapping

and transmission of matter-wave solitons reported here,

are two generic phenomena illustrating that collisionally

inhomogeneous matter-waves exhibit a number of inter-

esting and fundamentally new features that could also be

relevant for future applications.

Collision-induced trapping. To specify our setup we

choose the magnetic field dependence of the scattering

length a of a7Li condensate provided in Ref.[4] (see also

Ref. [3]). However, we emphasize that this case serves

only as a typical example of such a curve, used for con-

creteness. Let us focus on the regime 0 ≤ B ≤ 590 G,

which is far left from the Feshbach resonance at 720 G,

where the scattering length is small and the inelastic col-

Page 2

2

??

??

?

?

?

?

??

FIG. 1: The spatial variation of the scattering length for B =

450+ǫx (G) and magnetic field gradients ǫ = 0 (dotted line),

ǫ = 5 G/µm (solid line) and ǫ = 25 G/µm (dashed line).

lisional loss of atoms is practically negligible. We have

a(B) < 0 for 150 G < B < 520 G and a(B) > 0 else-

where. At B ≈ 352 G, the scattering length reaches a

minimum with a ≈ −0.23 nm. The experimentally ob-

served quasi-1D bright matter-wave solitons [3, 4] have

been created in the above-given regime of negative scat-

tering length.

Let us assume a magnetic field configuration B =

B0+ǫx (G), where ǫ is the field gradient and B0= 450 G,

being far from the position of the resonance. The gradi-

ent chosen is of the order of a few tens of G/µm which can

be experimentally realized in microscopic matter-wave

devices such as the atom-chip [6]. Other resonances and

species might require much smaller values of the gradient

to implement a significant change of the scattering length

on the scale of the condensate. To extract the spatial de-

pendence of the scattering length we apply a fifth-order

polynomial fit to the data for a(B) given in Ref. [4].

The result is shown in Fig. 1 for two nonzero values of

ǫ. The function a(x) possesses a single minimum and the

dependence on x is, as expected, much more dramatic for

a larger value of ǫ. For ǫ = 0 we have a = −0.182 nm.

The evolution of an untrapped, quasi-1D bright matter-

wave soliton in a CIE is described by the following nor-

malized Gross-Pitaevskii (GP) equation:

i∂tψ = −1

2∂2

xψ − g(x)|ψ|2ψ.(1)

Here, ψ is the mean field wavefunction (with the den-

sity |ψ|2measured in units of the peak density n0), x

is given in units of the healing length ξ = ?/√n0g0m

(where g0 = 2?ω⊥a(B0) and ω⊥ is the confining fre-

quency in the transverse direction), and the time unit

is ξ/c (where c =

?n0g0/m is the Bogoliubov speed of

sound). Finally, the spatially dependent nonlinearity is

given by g(x) = a(x)/a(B0), with a(x) ≡ a(B0+ ǫξx)

(note g(x = 0) = g(ǫ = 0) = 1). Typically, for a quasi-

1D7Li condensate with ω⊥= 2π×1000Hz and n0= 109

m−1, the healing length and speed of sound amount to

t(ms)

x(µm)

0204060

−40

−30

−20

−10

0

10

0.2

0.4

0.6

0.8

1

1.2

t(ms)

x(µm)

0204060

−20

−15

−10

−5

0

5

10

0.2

0.4

0.6

0.8

1

1.2

FIG. 2: (Color online) Spatio-temporal contour plot of the

density of a bright matter wave soliton for B0 = 450G

and magnetic field gradients ǫ = 5G/µm (left panel) and

ǫ = 25G/µm (right panel). The dashed line in the left panel

corresponds to the analytical prediction according to Eq. (4).

ξ = 2µm and c = 4.6mm/s, respectively.

Introducing the transformation ψ = u/√g in the re-

gion g(x) > 0 we reduce Eq. (1) to the following per-

turbed nonlinear Schr¨ odinger (NLS) equation,

i∂tu +1

2∂2

xu + |u|2u = R(u),

dxln(√g)∂xu+O?L2

(2)

with a perturbation R(u) ≡

The first term of R is of the order O(LS/LB), where

LS and LB are the characteristic spatial scales of the

soliton and of the inhomogeneity due to the magnetic

field gradient, respectively. In the case R = 0 (⇔ ǫ = 0),

Eq. (2) has a bright soliton solution of the form [16],

d

S/L2

B

?.

u(x,t) = ηsech[η(x − x0)]exp[i(kx − φ(t)],

where η is the amplitude and inverse width of the soliton,

and x0is the soliton center. The parameter k = dx0/dt

defines both the soliton velocity and wavenumber, and

φ(t) = (1/2)(k2−η2)t is the soliton phase. For the above

mentioned typical values of the parameters, and for η = 1

(a soliton with 103atoms and width 2ξ = 4µm), it is clear

that LS/LB= 2ξǫ/B0. This means that for sufficiently

small values of the magnetic field gradient, e.g., for ǫ =

5 G/µm, the perturbation R is of order O(10−2).

such a case, we may employ the adiabatic perturbation

theory for solitons [17], to obtain the following equation

of motion for the soliton center,

(3)

In

d2x0

dt2= −∂Veff

∂x0

,Veff(x0) ≡ −1

6g2(x0)(4)

With g(x) being proportional to a(x), it is obvious

(see Fig. 1) that the soliton “feels” a collision-induced

effective confinement potential Veff, although there is no

external trapping potential [see Eq. (1)]. It is then natu-

ral to expect that the collision-induced confinement leads

to oscillations of the soliton if it is displaced from the

minimum of the confinement potential. This has been

verified by direct numerical integration of the GP Eq.

(1) for ǫ = 5 G/µm and initial condition ψ = sech(x)

(i.e., the soliton is initially at x0(0) = 0 where B = B0).

Page 3

3

The result is shown in the left panel of Fig. 2, where

the spatio-temporal contour plot of the soliton density

is directly compared to the analytical prediction of Eq.

(4) (dashed line); the agreement between the two is ex-

cellent. We observe that the matter-wave is periodically

compressed whenever it reaches the region of large scat-

tering lengths (x ≈ −18µm with a ≈ −0.23nm), thus ex-

hibiting a pronounced breathing behavior. This results

in a spontaneous and robust breathing behavior of the

matter-wave soliton in the collisionally inhomogeneous

environment.

For significantly larger field gradients, e.g., ǫ = 25

G/µm, the soliton width becomes comparable to the

magnetic length scale LB. This means that the perturba-

tion R in Eq. (2) is now of the order LS/LB= O(10−1)

and nonadiabatic effects are expected to be significant.

The numerical integration of the GPE confirms this ex-

pectation and reveals that although the soliton is still

confined and performs corresponding oscillations, its evo-

lution is nonadiabatic, i.e., emission of small amplitude

wave radiation is observed. The corresponding results are

shown in the right panel of Fig. 2: Larger field gradients

lead to oscillations with a higher frequency and smaller

amplitude.

Collision-induced transmission. In our second setup

we consider the scattering of a bright matter-wave soliton

(see Eq. (3) for t = 0 and η = 1) of an external potential

barrier thereby comparing the results for homogeneous

and inhomogeneous atomic interactions. An important

quantity in this context is the transmission coefficient T.

In order to compare the transmission in the above

mentioned cases, the incoming and outgoing scattering

environments should be identical. This means that the

scattering length should asymptotically, i.e. outside the

range of the barrier, take on the same values. We guar-

antee this by employing a localized inhomogeneous mag-

netic field of the form,

B(x) =1

2[(B1+ B2) + (B1− B2)tanh(w(x − xB))],(5)

where the parameters w and xBcharacterize the inverse

width and location of the region of inhomogeneity, re-

spectively, while the field values B1 and B2 are chosen

to obtain equal scattering lengths a(B1) = a(B2) suffi-

ciently far from the barrier. This value will also be used

for the homogeneous case. In the case of7Li it is straight-

forward to find the values B1and B2, due to the conve-

nient form of the function a(B) in the considered range

of field strengths (see also Fig.1). Here we use B1= 450

G and B2 = 265 G for which a = −0.182 nm (other

choices are, of course, equally possible and lead to sim-

ilar results). We finally note that the above mentioned

field configuration can be realized by a multi-wire setup

or a current density flowing in a half plane augmented by

a homogeneous bias field.

Our potential barrier is assumed to be of the form

?

?

?

?

?

?

?

?

?

FIG. 3:

placed at x = 0 off a barrier (shaded area) located at xB =

−20µm is shown; the transmitted and reflected parts of the

soliton (accordingly labeled) are shown for t = 6ms for the

inhomogeneous case. The spatial dependence of the scattering

length is also shown. Note that for the homogeneous case

a = −0.182nm.

(Color online) The scattering of a soliton initially

Vb(x) = V0sech2(α(x − xB)), where V0, α−1and xBare

the barrier’s amplitude, width and location, respectively

(note that the inhomogeneity is centered at the same po-

sition where the barrier is located, i.e., at x = xB). In

the following we assume V0= 1 and α−1= 1/2, i.e., the

width of the barrier is half the soliton width, so as to

avoid the classical Ehrenfest regime.

The setup is illustrated in Fig. 3: The initial (t = 0)

form of the soliton, as well as its transmitted and re-

flected parts (for the inhomogeneous case at t = 6ms)

are depicted and labeled respectively. At the location

of the barrier (shaded region), and for the collisionally

inhomogeneous case, there exists a local spatial change

of the scattering length, which is obtained by inserting

the magnetic field in Eq. (5) into the function a(B) (uti-

lizing the fifth-order polynomial fit used in the previous

section). The spatial dependence of the scattering length

is also shown in Fig. 3; as discussed above, the scattering

length takes on equal values far from the barrier for both

the homogeneous and inhomogeneous case.

We have numerically integrated the GPE incorporat-

ing the potential barrier term Vb(x)ψ(x,t) to determine

the transmission coefficient T.

T as a function of the width of the inhomogeneity (left

panel) and the soliton’s incident velocity (right panel)

are shown in Fig. 4. Generally, for a fixed soliton ve-

locity, or width of the inhomogeneity, the transmission

T in the inhomogeneous case is always larger than the

one in the homogeneous case. Particularly, for k = 1.1c

and wξ = 1.3, the relative difference of the transmis-

sion T for the two cases becomes maximal, being ≈ 15%.

This result clearly demonstrates that the transmission of

a matter-wave soliton may be enhanced in the presence

of a spatially dependent collisional interaction.

Conclusions. We have explored the dynamics of bright

matter-wave solitons subject to a spatially varying non-

The results presenting

Page 4

4

?

?

?

?

?

????

?

?

?

?

??

FIG. 4: The transmission T as a function of the inverse width

w of the inhomogeneity, in units of ξ−1(left panel), or the

soliton’s incident velocity k, in units of c (right panel). Dashed

and solid lines correspond to the collisionally homogeneous

and inhomogeneous cases, respectively.

linearity, which can be realized by means of an external

inhomogeneous magnetic field on top of a bias field in or-

der to be close to a region of strong changes of the scatter-

ing length, such as a Feshbach resonance. It was demon-

strated that a confinement (trapping) of the matter-wave

can be achieved solely on basis of the spatial change of

the collisional interaction, i.e., without the presence of

an external trapping potential thereby creating a breath-

ing matter-wave state in the collisionally inhomogeneous

environment. In an adiabatic regime, such a state could

be well described within the realm of soliton perturba-

tion theory, while for abrupt variations of the scattering

length, radiative emissions are non-trivial resulting in en-

ergy losses and hence shorter period oscillations. Using

a localized spatial variation of the scattering length, we

have shown that the transmission of matter-wave solitons

through a barrier can be enhanced by suitably manipu-

lating the collisional properties of the condensate in the

vicinity of the potential barrier. Collisionally inhomoge-

neous environments therefore hold considerable promise

in the effort to control and manipulate matter-waves in

experimental applications. Interesting future directions

may involve combining this type of effective potential

with optical potentials in order to examine transmission

properties and symmetry breaking or nonlinear trapping

phenomena similarly to the recent works of [18]. An in-

teresting variation on these themes that the present set-

ting offers is, among others, that the effective potential

is asymmetric and can hence lead to a modified dynami-

cal picture [19] in comparison to the symmetric potential

case [18]. Such studies are currently in progress and will

be reported in future publications.

Acknowledgements. This work was supported by

the “A.S. Onasis” Public Benefit Foundation (GT), the

Special Research Account of Athens University (GT,

DJF), as well as NSF-DMS-0204585, NSF-CAREER,

NSF-DMS-0505063 and the Eppley Foundation for Re-

search (PGK).

[1] F. Dalfovo et al., Rev. Mod. Phys. 71, 463 (1999).

[2] S. Burger et al., Phys. Rev. Lett. 83, 5198 (1999); J.

Denschlag et al., Science 287, 97 (2000); B. P. Anderson

et al., Phys. Rev. Lett. 86, 2926 (2001); Z. Dutton, et

al., Science 293, 663 (2001).

[3] K. E. Strecker et al., Nature 417, 150 (2002).

[4] L. Khaykovich et al., Science 296, 1290 (2002).

[5] B. Eiermann et al., Phys. Rev. Lett. 92, 230401 (2004).

[6] R. Folman et al., Adv. Atom. Mol. Opt. Phys. 48, 263

(2002); J. Reichel, Appl. Phys. B 75, 469 (2002); J. Fort-

agh and C. Zimmermann, Science 307 860 (2005).

[7] Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From

Fibers to Photonic Crystals (Academic Press, San Diego,

2003).

[8] S. Inouye et al., Nature 392, 151 (1998); J. Stenger et

al., Phys. Rev. Lett. 82, 2422 (1999); J. L. Roberts et

al., Phys. Rev. Lett. 81, 5109 (1998); S. L. Cornish et

al., Phys. Rev. Lett. 85, 1795 (2000).

[9] J. Herbig et al., Science 301, 1510 (2003).

[10] M. Bartenstein et al., Phys. Rev. Lett. 92, 203201 (2004).

[11] F. Kh. Abdullaev et al., Phys. Rev. A 67, 013605 (2003);

H. Saito and M. Ueda, Phys. Rev. Lett. 90, 040403

(2003); G. D. Montesinos et al, Physica D 191 193

(2004).

[12] P. G. Kevrekidis et al., Phys. Rev. Lett. 90, 230401

(2003); D. E. Pelinovsky et al., Phys. Rev. Lett. 91,

240201 (2003); D. E. Pelinovsky et al., Phys. Rev. E 70,

047604 (2004); Z. X. Liang et al., Phys. Rev. Lett. 94,

050402 (2005); M. Matuszewski et al., Phys. Rev. Lett.

95, 050403 (2005).

[13] D. Blume et al., Phys. Rev. Lett. 89, 163402 (2002).

[14] F. Kh. Abdullaev and M. Salerno, J. Phys. B 36, 2851

(2003); H. Xiong et al., Phys. Rev. Lett. 95, 120401

(2005).

[15] G. Theocharis et al., preprint cond-mat/0505127 (Phys.

Rev. A, in press).

[16] V. E. Zakharov and A.B. Shabat, Zh. Eksp. Teor. Fiz.

61, 118 (1971) [Sov. Phys. JETP 34, 62 (1971)].

[17] Yu. S. Kivshar and B. A. Malomed, Rev. Mod. Phys. 61,

763 (1989).

[18] Th. Anker et al., Phys. Rev. Lett. 94, 020403 (2005); P.

G. Kevrekidis et al., Phys. Lett. A 340, 275 (2005).

[19] T. Kapitula and P. G. Kevrekidis, Nonlinearity 18, 2491

(2005).

#### View other sources

#### Hide other sources

- Available from D. J. Frantzeskakis · May 27, 2014
- Available from arxiv.org