arXiv:cond-mat/0509471v1 [cond-mat.soft] 18 Sep 2005
Trapping and transmission of matter-wave solitons in a collisionally inhomogeneous
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
of atomic Bose-Einstein condensates (BECs)  has in-
spired, among others, many studies on their nonlinear
excitations. Especially, as far as matter-wave solitons are
concerned, dark , bright [3, 4] and gap  solitons have
been observed experimentally and studied theoretically.
Atom optical devices such as the atom chip  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 .
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.  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
 and the revelation of the BEC-BCS crossover .
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 , or to create robust matter-wave breathers
. 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. ).
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
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. (see also
Ref. ). 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-
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-
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 . 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. .
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
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
0 2040 60
0 2040 60
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,
xu + |u|2u = R(u),
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 ,
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 , to obtain the following equation
of motion for the soliton center,
,Veff(x0) ≡ −1
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).
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
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
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,
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
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
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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 . 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  in comparison to the symmetric potential
case . 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-
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