Transmission and Reflection of Bose-Einstein Condensates Incident on a Gaussian Potential Barrier

Abstract

We investigate how Bose-Einstein condensates, whose initial state is either irrotational or contains a single vortex, scatter off a one-dimensional Gaussian potential barrier. We find that for low atom densities the vortex structure within the condensate is maintained during scattering, whereas at medium and high densities, multiple additional vortices can be created by the scattering process, resulting in complex dynamics and disruption of the atom cloud. This disruption originates from two different mechanisms associated respectively with the initial rotation of the atom cloud and the interference between the incident and reflected matter waves. We investigate how the reflection probability depends on the vorticity of the initial state and on the incident velocity of the Bose-Einstein condensate. To interpret our results, we derive a general analytical expression for the reflection coefficient of a rotating Bose-Einstein condensate that scatters off a spatially-varying one-dimensional potential. Comment: 9 pages, 9 figures

Full text

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arXiv:cond-mat/0610053v1 [cond-mat.other] 3 Oct 2006
Transmission and Reflection of Bose-Einstein Condensates Incident on a Gaussian
Potential Barrier
A.M. Martin1, R.G. Scott2,3and T.M. Fromhold3
1School of Physics, University of Melbourne, Parkville, Victoria 3010, Australia
2Department of Physics, University of Otago, P.O. Box 56, Dunedin, New Zealand
3School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom
(Dated: February 6, 2008)
We investigate how Bose-Einstein condensates, whose initial state is either irrotational or contains
a single vortex, scatter off a one-dimensional Gaussian potential barrier. We find that for low atom
densities the vortex structure within the condensate is maintained during scattering, whereas at
medium and high densities, multiple additional vortices can be created by the scattering process,
resulting in complex dynamics and disruption of the atom cloud. This disruption originates from
two different mechanisms associated respectively with the initial rotation of the atom cloud and
the interference between the incident and reflected matter waves. We investigate how the reflection
probability depends on the vorticity of the initial state and on the incident velocity of the Bose-
Einstein condensate. To interpret our results, we derive a general analytical expression for the
reflection coefficient of a rotating Bose-Einstein condensate that scatters off a spatially-varying
one-dimensional potential.
PACS numbers: 03.75.Kk, 05.30.Jp, 67.40Vs
I.INTRODUCTION
Recently, several experiments have investigated the
scattering of Bose-Einstein condensates (BECs). These
experiments have included Bragg reflection in an opti-
cal lattice [1], reflection from optical [2] and magnetic [3]
mirrors, diffraction from a grating [4] and quantum reflec-
tion from a silicon surface [5]. In each case, interest has
focused on the reflected component of the BEC. For ex-
ample, investigations of quantum reflection from a silicon
surface have revealed that inter-atomic interactions have
a dramatic effect on the internal structure of the atom
cloud [5, 6].So far, reflection experiments have been
restricted to condensates whose initial state contains no
dynamical excitations.However, the methodology for
creating and observing vortices in BECs is well estab-
lished [7, 8, 9, 10] and, in the case of Bragg reflection,
numerical simulations predict that vortices and solitons
in the BEC’s initial state strongly influence the subse-
quent dynamics [11]. Previous theoretical work has also
shown that the presence of a vortex in the initial state
can have a pronounced effect on the internal structure
of a BEC that undergoes classical reflection from a hard
wall atom mirror [12, 13]. To our knowledge, however,
there has been no consideration of the effect of vortices
on BECs approaching a potential barrier of finite width
and height, where quantum-mechanical tunneling is pos-
sible as well as reflection. Quantum tunneling of BECs
can be studied experimentally by using sheets of laser
light to create the potential barrier [14] and plays a cru-
cial role in the dynamics and macroscopic coherence of
cold atoms in optical lattices [15].
In this paper, we investigate how BECs scatter off a
Gaussian potential barrier, which allows both reflection
and quantum-mechanical tunneling of the atoms. We use
numerical simulations of the Gross-Pitaevskii equation to
make a detailed study of how the strength of the inter-
atomic interactions, the energy of the incident BEC, and
the vorticity of the initial state affect the scattering prop-
erties of the condensate. Our simulations reveal regimes
in which dynamical excitations disrupt both the reflected
and transmitted atom clouds. In one regime, which we
call rotational disruption, the excitations originate from
the effect of the initial vortex on the scattering dynam-
ics. By contrast, the regime of interferential disruption
[16] occurs both in the presence and absence of a vor-
tex in the initial state with the excitations being cre-
ated from interference between the incident and reflected
matter waves. We find that rotational disruption arises
when the time taken for the BEC to scatter is compara-
ble with, or exceeds, the rotational period of the vortex.
Interferential disruption occurs when the scattering time
is greater than the correlation time of the BEC, which is
a measure of how quickly the atom cloud responds to a
perturbation. To interpret our results, we derive a gen-
eral expression for the reflection probability of a BEC
containing a vortex in terms of the reflection probability
of an irrotational BEC. We find that the vortex changes
the reflection probability by altering the distribution of
incident velocities at the barrier, due to the increase in
the physical size of the cloud and, more importantly, the
circulation of the atoms.
To investigate the regimes of rotational and interfer-
ential disruption we consider three BECs whose initial
state contains a single vortex: BEC A with low atom
density, BEC B with medium density and BEC C with
high density. We identify the effect of the initial vortex
by comparing the dynamics of these atom clouds with
irrotational counterparts labeled BECs Ai, Biand Ci.
The layout of the paper is as follows: in section II we
specify the parameters for each of the three BECs and
describe the theoretical model and computational tech-

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niques. In section III, we present our numerical results,
which show how low, medium, and high density BECs
scatter off the Gaussian potential barrier, both with and
without a vortex in the initial state. In section IV, we
derive a general expression for the reflection probability
of an irrotational BEC. We then use this expression to
derive an approximate analytical formula for the reflec-
tion probability of a condensate whose initial state does
contain a vortex, impinging on the same scattering po-
tential. We compare this formula with reflection proba-
bilities obtained directly from the numerical simulations
presented in section III and use it to provide physical
insight into the effect of vortices on the scattering pro-
cess. Finally, in section V, we summarize our results and
propose experiments to test them.
II.SYSTEM PARAMETERS, THEORETICAL
MODEL, AND COMPUTATIONAL TECHNIQUES
Each BEC contains N
is initially confined by a harmonic trapping potential,
VT(x,y,z) = m[ω2
at (−∆x,0,0). We consider trap frequencies ωx= ωy=
50 × 2π rad s−1≪ ωz so that the spatial width of the
BEC is much smaller along the z-direction than along the
x- and y-directions. Consequently, the dynamics reduce
to two-dimensional motion in the x − y plane. At time
t = 0, we create an additional Gaussian potential barrier
[14]
23Na atoms of mass m and
x(x + ∆x)2+ ω2
yy2+ ωzz2]/2 centered
VL(x) = V0e−x2
σ2,(1)
of width σ = 1µm along the x-direction, by switching
on a far blue-detuned laser beam that travels along the
y-direction and creates a sheet of laser light. The in-
tensity of the laser beam determines the barrier height,
V0, which we take to be 6.2peV, similar to recent ex-
periments [14]. Simultaneously, we accelerate the BEC
towards the Gaussian potential by abruptly displacing
the harmonic trap through a distance ∆x along the x-
direction [4, 5, 6]. After the displacement, the center
of the trap coincides with the Gaussian potential energy
barrier and the total potential energy of the trap and
laser beam, for motion in the x − y plane, is given by
V (x,y) = m(ω2
xx2+ ω2
yy2)/2 + VL(x).(2)
The solid curve in Fig 1(a) shows the form of V (x,y = 0).
After the displacement of the harmonic trap, the conden-
sate moves away from its initial state [shown schemati-
cally by the dashed curve in Fig. 1(a)], reaches the Gaus-
sian potential with a mean incident velocity vx≈ ωx∆x,
and is then scattered by the potential barrier.
The time-dependent Gross-Pitaevskii equation for the
system is
i¯ h∂Ψ(x,y,t)
∂t
= −¯ h2
2m
?∂2
∂x2+
∂2
∂y2
?
Ψ(x,y,t)
?x
(a)
0
x
(b)(c)
20 m
?
x
y
FIG. 1: (a) Solid curve: A schematic of the potential energy
V (x,y = 0) created by the harmonic trap and laser beam
when t ≥ 0; Dashed curve: a schematic of the initial proba-
bility density |Ψ(x,y = 0,t = 0)|2for BEC B. (b) The initial
density |Ψ(x,y,t = 0)|2(black=high density, white=zero) of
BEC B, where the white arrow shows the direction of the
condensate’s circulation and the horizontal bar denotes the
scale. (c) The equivalent phase plot, φ(x,y,t = 0), (white= 0,
black= 2π).
+
?V (x,y) + U0| Ψ(x,y,t) |2?Ψ(x,y,t)
(3)
where Ψ(x,y,t) is the wavefunction for motion in the
x − y plane at time t ≥ 0 and U0= 4π¯ h2a/m where the
s-wave scattering length a = 2.9nm.
We determine the initial BEC wave function by solv-
ing Eq. (3) for t ≤ 0 using an imaginary time algorithm
[17]. When the initial state contains a vortex, we im-
pose the requirement that there is a 2π-phase change
in the condensate wavefunction around the trap center
at (x,y) = (−∆x,0), which corresponds to a quantized
angular momentum of −¯ h about the z-axis. The wave-
function is normalized according to
?
| Ψ(x,y,t) |2dxdy =N
Lz,(4)
where Lz is the confinement length in the z-direction.
For BECs A, B and C we choose N/Lz as 2.5 × 108
m−1, 5×109m−1and 2.5×1011m−1respectively, which
gives corresponding peak atom densities n0= 2.8 × 1018
m−3, 2.1 × 1019m−3and 1.6 × 1020m−3. The density,
| Ψ(x,y,t = 0) |2, and phase, φ(x,y,t = 0), of the ini-
tial state for BEC B are shown in Figs. 1(b,c). Having
obtained the initial state of the BEC, we determine its
motion by solving the Gross-Pitaevskii equation (3) nu-
merically using the Crank-Nicolson method [18].

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3
(d)
(c)
(a)
(b)
(e)
(f)
(g)
(h)
10 m
?
x
y
FIG. 2: Evolution of BEC A: plots of |Ψ(x,y,t)|2(black=high
density, white=zero) for vx = 6.3mms−1at t = 3.6ms (a),
4.0ms (b), 4.4ms (c), 4.8ms (d), 5.2ms (e), 5.6ms (f), 6.0ms
(g) and 6.4ms (h). Coordinate axes are inset and the hori-
zontal bar indicates the scale. The arrows in (a) and (h) show
the direction of rotation.
III. NUMERICAL RESULTS FOR SCATTERING
OFF A GAUSSIAN POTENTIAL BARRIER
In Fig. 2, we show the density profile of BEC A, with
∆x = 20µm (vx= 6.3mms−1) at t = 3.6ms (a), 4.0ms
(b), 4.4ms (c), 4.8ms (d), 5.2ms (e), 5.6ms (f), 6.0ms
(g) and 6.4ms (h). Figs. 2(a-e) show that as the BEC
approaches the Gaussian scattering potential at x = 0
(dashed lines in Fig. 2), a standing wave forms due to
interference between the incoming and reflected matter-
waves. In Figs. 2(d,e) the standing wave undergoes a π
phase shift between the upper (large y) and lower (small
y) edges of the BEC. This is due to the non-uniform ini-
tial phase of the BEC, shown in Fig. 1(c). After scatter-
ing, the BEC splits into reflected (x < 0) and transmitted
(x > 0) components [Figs. 2(f-h)]. Each of the compo-
nents contains a vortex [enclosed by the arrows in Fig.
2(h)]. For the transmitted cloud, the quantized circula-
tion is in the same direction as in the incident cloud. By
contrast, the reflected cloud rotates in the opposite direc-
tion [12, 13]. Physically, this is because atoms towards
x
y
LR=30 m
?
LT=30 m
?
LR=23 m
?
LT=24 m
?
LR=19 m
?
T
L =22 m
?
(c)
(a)
(b)
FIG. 3: BEC A: plots of |Ψ(x,y,t = 10ms)|2for vx =
9.4mms−1(a), 7.5mms−1(b) and 6.3mms−1(c). The circu-
lar arrows indicate the direction of rotation and the horizontal
arrows show the values of LR and LT, defined in the text and
footnote [19]. Co-ordinate axes are inset.
the top of the rotating incident cloud, in Fig. 2, approach
the barrier with a higher velocity component along the
x-direction (vx) than those at the bottom. But because
vxis reversed after reflection, the direction of rotation is
also reversed in the reflected cloud. This is in contrast to
the transmitted atoms, which emerge through the barrier
with little change in their velocity and so the direction
of rotation is preserved. Since the trap remains switched
on throughout the scattering process, we do not observe
the splitting of the reflected component of the BEC that
was reported in Ref. [13].
We now consider how changing vxaffects the dynam-
ics of the atom cloud. Figure 3 shows the density profile
of BEC A at t = π/ωx= 10ms after trap displacements
of 30µm (a), 24µm (b) and 20µm (c), corresponding
to vx= 9.4mms−1, 7.5mms−1and 6.3mms−1respec-
tively. Since this time is half the period of oscillation
of the trap, both the transmitted and reflected portions
of the condensate will approximately be at their turning
points, and hence nearly stationary. In Fig. 3, the ver-
tical dashed line marks the trap center at x = 0. When
vx= 9.4mms−1, the average kinetic energy of the atoms
incident upon the barrier is 10peV, which exceeds the
Gaussian barrier height. Hence, most of the condensate is
transmitted [Fig. 3(a)]. As we decrease vx[Figs. 3(b,c)],

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4
(a)
(b)
(c)
20 m
?
(d)
(e)
(f)
x
y
FIG. 4: Evolution of BEC B: plots of |Ψ(x,y,t)|2(black=high
density, white=zero) for vx = 6.3mms−1at t = 3ms (a), 4ms
(b), 5ms (c), 6ms (d), 7ms (e), 8ms (f). The dashed line at
x = 0 marks the point where the laser potential is maximal.
Co-ordinate axes are inset and the horizontal bar indicates
scale. Lower plot: phase φ(x,y,t = 8ms) [white= 0, black=
2π] within the region enclosed by the box in (f).
indicate the direction of circulation.
Arrows
the average energy of the incident atoms decreases and
more of the condensate is reflected. For vx= 6.3mms−1
[Fig. 3(c)], most of the atoms are reflected by the barrier
because the average kinetic energy of the incident atoms
is 4.6peV, which is less than the barrier height. Since
the BEC has a finite spatial width along the x-direction
of lx= 13µm, atoms that are towards the left-hand side
of the BEC’s initial state travel further before reaching
the barrier and therefore have a higher incident veloc-
ity. Such atoms have a higher transmission probability
and therefore form a large fraction of the transmitted
atom cloud. We therefore expect that the distance (LT)
that the higher velocity transmitted cloud travels past
the scattering potential before coming to rest in the har-
monic trap will be greater than the distance (LR) that
the slower reflected cloud retreats from the scatterer be-
fore reaching the turning point of the harmonic trap [19].
Our numerical simulations confirm this: for example, in
Fig. 3, LT≥ LRfor all the values of vx.
We now consider how the higher density BEC B re-
flects off the potential barrier. Figure 4 shows the den-
sity profile of BEC B at t = 3ms (a), 4ms (b), 5ms (c),
(b)
(a)
(c)
(e)
(d)
(f)
20 m
?
x
y
FIG. 5:
(black=high density, white=zero) for vx = 6.3mms−1at t =
3ms (a), 4ms (b), 5ms (c), 6ms (d), 7ms (e), 8ms (f). The
dashed line at x = 0 marks the point where the laser potential
is maximal. Co-ordinate axes are inset and the horizontal bar
indicates scale.
Evolution of BEC Bi:plots of |Ψ(x,y,t)|2
6ms (d), 7ms (e) and 8ms (f), after a trap displacement
of 20µm (vx= 6.3mms−1). As the BEC impinges upon
the Gaussian potential barrier, a standing wave forms be-
tween the incoming and reflected matter waves. Figure
4(b) shows the first stage of the standing wave forma-
tion in which maxima (black) and nodal lines (white)
appear at the leading edge of the atom cloud. In con-
trast to BEC A, the reflected component of the BEC is
significantly disrupted, as shown in Figs. 4(e,f). This
disruption is accompanied by the formation of new vor-
tices within the boxed region in Fig. 4(f). The phase
variation, φ(x,y,t = 8ms), within this region is shown
in the lower part of Fig. 4(f). Around each vortex, φ in-
creases from 0 to 2π in the direction of circulation shown
by the arrows in Fig. 4(f).
To explain the disruption shown in Fig. 4, we first con-
sider the dynamics of BEC Bi. Figure 5 shows density
profiles for this initially irrotational condensate at t =
3ms (a), 4ms (b), 5ms (c), 6ms (d), 7ms (e) and 8ms
(f) after a trap displacement 20µm (vx= 6.3mms−1).
No dynamical excitations are produced in the transmit-
ted or reflected clouds. We therefore conclude that the
disruption observed for BEC B (Fig. 4) is related to the
rotation of the cloud.
To gain further insights into this disruption, we now
examine how changing vxaffects the transmission and re-

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5
(a)
(b)
(c)
x
y
LR=17 m
?
LT=23 m
?
LR=30 m
?
LT=30 m
?
LR=22 m
?
T
L =24 m
?
FIG. 6: BEC B: plots of |Ψ(x,y,t = 10ms)|2for vx =
9.4mms−1(a), vx = 7.5mms−1(b) and vx = 6.3mms−1
(c). The dashed line at x = 0 marks the point where the laser
potential is maximal. The circular arrows indicate the direc-
tion of rotation and the horizontal arrows show the values of
LRand LT, defined in the text and footnote [19]. Co-ordinate
axes are inset.
flection of BEC B. Figure 6 shows the condensate density
at t = 10ms after trap displacements of ∆x = 30µm (a),
24µm (b), and 20µm (c), corresponding to mean incident
velocities of vx= 9.4mms−1, 7.5mms−1and 6.3mms−1
respectively. For vx= 9.4mms−1, the behavior of BEC
B [Fig. 6(a)] is qualitatively the same as BEC A [Fig.
3(a)]: the transmitted cloud retains the vortex structure
of the incident cloud and only a small fraction of atoms
are reflected. As vxdecreases [Figs. 6(b,c)], the reflec-
tion probability rises, but the structure of the reflected
cloud becomes increasingly disrupted. This disruption
contrasts with BEC A, where for all vx values the re-
flected cloud retains a well defined vortex structure (Fig.
3). For BEC B, we find that LT≥ LR(see values in Fig.
6) just as for BEC A. However, for vx= 7.5mms−1and
vx = 6.3mms−1, LR is lower for BEC B [Fig. 6(b,c)]
than for BEC A [Fig. 3(b,c)] due to the fragmentation
of the atom cloud that occurs for BEC B only. This
fragmentation transfers kinetic energy from the center-of-
mass motion of the reflected cloud into internal vorticity,
thus damping the oscillation.
The effect of the vortex on the scattering process de-
pends on the scattering time of the vortex core (tsv) rel-
ative to its rotation time (tr).
the vortex core is approximately the healing length ξ =
1/√8πn0a, it follows that tsv≈ 2ξ/vxand tr= πξ2m/h.
If the ratio
Since the diameter of
tr
tsv
=m
2hπξ∆xωx∝
∆x
√n0
(5)
is ≫ 1, so that there is insufficient time for the vortex to
rotate during the scattering process, the vortex will have
little effect on the dynamics of the transmitted and re-
flected atom clouds. Physically, this is because all parts
of the BEC are incident on the Gaussian potential with
a variation in the incident velocity, due to the rotation,
that is small compared to vx. Conversely, if tr/tsv<∼1 we
expect that the rotation of the BEC will disrupt the re-
flected component of the BEC. We refer to such a regime
as rotational disruption. For the lower-density BEC A,
tr/tsv ≫ 1 for all three of the vx values considered in
Fig. 3, and so the reflected atom cloud is not signifi-
cantly fragmented. By contrast, for the higher density
BEC B tr/tsv < 1, and hence rotational disruption oc-
curs in the reflected atom cloud. Note that reducing vx
from 9.4mms−1[Fig. 6(a)] to 6.3mms−1[Fig. 6(c)]
causes tr/tsv to decrease, and thus increases the frag-
mentation of the reflected cloud.
Disruption of the reflected cloud is even more pro-
nounced for the higher density BEC C. Figure 7 shows
the density of this BEC at t = 1ms (a), 2ms (b), 3ms (c),
4ms (d), 5ms (e), 6ms (f), 7ms (g) and 8ms (h) after
a trap displacement of 30µm (vx= 9.4mms−1) [20]. As
the condensate impinges upon the scattering potential, a
standing wave forms between the incident and reflected
matter waves, Figs. 7(b-d). This standing wave seeds
solitons [6, 21, 22], which decay via the snake instabil-
ity [24] into vortex-antivortex pairs [Figs. 7(e,f)], thus
strongly disrupting the internal structure of the cloud
[Figs. 7(g,h)]. Consequently, when incident atoms sub-
sequently pass through the barrier they produce some ir-
regularity in the transmitted atom cloud. Note, that this
irregularity is less pronounced towards the right hand
edge of the transmitted cloud, which contains the atoms
that passed through the barrier before solitons and vor-
tices formed at negative x (x = 0 is marked by the dotted
line in Fig. 7). We emphasize that although BEC C is
in a regime where rotational disruption occurs, the ini-
tial vortex does not (in contrast to BEC B) cause the
severe fragmentation shown in Fig. 7. To demonstrate
this, Fig. 8 shows BEC Ci(identical to BEC C, except
irrotational) scattering off a Gaussian potential at t =
1ms (a), 2ms (b), 3ms (c), 4ms (d), 5ms (e), 6ms (f),
7ms (g) and 8ms (h) after a trap displacement of 30µm
(vx= 9.4mms−1). Comparison of Figs. 7 and 8 reveals
that the dynamics are qualitatively the same irrespective
of whether or not the BEC contains an initial vortex.
To understand these results, we recall previous work
[11, 21, 22, 23] on the Bragg reflection of a BEC in an op-

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6
(d)
(c)
(a)
(b)
(e)
(f)
(g)
(h)
40 m
?
x
y
FIG. 7: Evolution of BEC C: plots of |Ψ(x,y,t)|2(black=high
density, white=zero) for vx = 9.4mms−1at t = 1ms (a),
2ms (b), 3ms (c), 4ms (d), 5ms (e), 6ms (f), 7ms (g) and
8ms (h). Co-ordinate axes are inset and the horizontal bar
indicates the scale.
tical lattice. In Refs. [21, 22] it was shown that at Bragg
reflection, fragmentation can arise from the density and
phase imprinting that accompanies standing wave forma-
tion. When the correlation time,
tc=
m
2√2hn0a,
(6)
is much less than the Bloch period (tB), this imprint-
ing leads to the formation of solitons and vortices, which
disrupt the atom cloud. For BECs C and Ci, a similar
disruption occurs when tc≪ ts, where ts= lx/vxis the
approximate duration of the reflection process. This ef-
fect is described as interferential disruption [16], since it
originates from the interference pattern, in this case pro-
duced by the superposition of the incident and reflected
matter waves. For BEC C this interferential disruption
dominates the dynamics and completely masks any ef-
fects due to the rotational disruption.
40 m
?
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
x
y
FIG. 8:
(black=high density, white=zero) for vx = 9.4mms−1at t =
1ms (a), 2ms (b), 3ms (c), 4ms (d), 5ms (e), 6ms (f), 7ms
(g) and 8ms (h). Co-ordinate axes are inset and the horizon-
tal bar indicates the scale.
Evolution of BEC Ci:plots of |Ψ(x,y,t)|2
IV.REFLECTION PROBABILITY OF THE BEC
In this section, we investigate how the reflection prob-
ability of the BEC varies with vx and depends on the
presence or absence of an initial vortex. Defining the
reflection probability of the condensate as
R(vx= ∆xωx) =
?
y
?
x<0| Ψ(x,y,t = π/ωx) |2dxdy
?
y
?
x| Ψ(x,y,t) |2dxdy
,
(7)
we use our numerical solution of the Gross-Pitaevskii
equation to quantify the reflection probabilities of BEC
A [R(vx), dotted curve in Fig. 9(a)] and its irrotational
equivalent [Ri(vx), dashed curve in Fig. 9(a)]. Figure
9(a) shows that as vxdecreases, both R(vx) and Ri(vx)
increase. For vx<∼7.3mm s−1, R(vx) < Ri(vx) and
conversely for vx>∼7.3mm s−1, R(vx) > Ri(vx). This
crossover is revealed more clearly by the dashed curve
in Fig. 9(b), which shows r(vx) = R(vx) − Ri(vx). For
vx<∼7.3mm s−1, r(vx) < 0, but at higher vx, r(vx) > 0.
In Fig.9(a) we compare R(vx) and Ri(vx) to
the equivalent plane wave reflection probability RP(vx)
(solid curve). The difference between RP(vx) and Ri(vx)

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7
0
0.2
0.4
0.6
0.8
R
1.0(a)
4
6
vx(mm?s??)
8 10
-0.04
-0.06
-0.02
0
0.02
0.04
r
0.06
5.5 6.57.5 8.5
-1
9.5
vx(mm?s??)
(b)
4-420 -2
0
vx(mm?s???)v?-
x
fi
f,g, (arb.?units)
(c)
-1
-1
FIG. 9: (a) Dotted curve: reflection probability, R(vx), for
BEC A.Dashed curve: reflection probability, Ri(vx), for
BEC Ai. Solid curve: reflection probability, RP(vx), for a
plane wave. (b) Dashed curve: r(vx) obtained from numeri-
cal simulations. Solid curve: the expression for r(vx) obtained
from Eq. (18) with I = 0.007mm2s−2and J = 0.19mm2
s−2. (c) Solid curve: the local velocity distribution function,
fi(vx−vx,y = 0) for BEC Ai. Dashed curve: the local veloc-
ity distribution function, f(vx−vx,y = 0) for BEC A. Dotted
curve: g(vx− vx,y = 0).
is due to the finite width of the BEC along the x-
direction, which means that the atoms start from var-
ious positions in the harmonic trap and thus reach the
barrier with a range of incident velocities rather than
the single velocity of an incident plane wave. We can
model the spread of velocities by defining the probabil-
ity Fi(vx−vx)dvx=?
arrives at the barrier with an incident velocity between
vx and vx+ dvx.Here fi(vx− vx,y)dvx = |Ψ((vx−
vx)/ωx,y,t = 0)|2dvx/?
is the local probability that an atom starting from posi-
tion y arrives at the barrier with a normal velocity com-
ponent between vxand vx+ dvx. For each y, this local
probability is determined from the initial shape of the
BEC wavefunction, which defines the initial spatial dis-
tribution of the atoms and hence the resulting velocity
yfi(vx−vx,y)dydvxthat an atom
vx|Ψ((vx−vx)/ωx,y,t = 0)|2dvx
distribution as the atoms arrive at the barrier. For each
y, fi(vx− vx,y) is symmetrical about vx− vx= 0 owing
to the symmetry of the initial wavefunction. This can
be seen from the solid curve in Fig. 9(c), which shows
fi(vx−vx,y = 0) for BEC Ai. The reflection probability
of the BEC can be expressed in terms of its velocity dis-
tribution and the reflection probability of a plane wave
by
Ri(vx) =
?∞
0
Fi(vx− vx)RP(vx)dvx. (8)
Since Fi(vx− vx) is sharply peaked at vx = vx we can
use the Taylor expansion
RP(vx) ≈ RP(vx) +∂RP(vx)
∂2RP(vx)
2∂v2
∂vx
????
vx
(vx− vx)
+
x
????
vx
(vx− vx)2
(9)
to write Eq. (8) in the form
Ri(vx) ≈ RP(vx)
?∞
0
????
Fi(vx− vx)dvx
?∞
0
?∞
0
+
∂RP(vx)
∂vx
∂2RP(vx)
2∂v2
vx
????
Fi(vx− vx)(vx− vx)dvx
+
x
vx
Fi(vx− vx)(vx− vx)2dvx.
(10)
Since Fi(vx− vx) is symmetrical about 0 and its width
is less than vx for all the vx values considered in Figs.
9(a,b), Eq. (10) reduces to
Ri(vx) ≈ RP(vx) +I
2
∂2RP(vx)
∂v2
x
????
vx
,(11)
where [25]
I =
?∞
0
Fi(vx− vx)(vx− vx)2dvx> 0.(12)
It follows from Eqs. (11) and (12) that Ri(vx) > RP(vx)
if
∂2RP(vx)
∂v2
x
????
vx
> 0(13)
and Ri(vx) < RP(vx) if
∂2RP(vx)
∂v2
x
????
vx
< 0.(14)
Evaluating the integral in Eq. (12) numerically, we find
that for BEC Ai, I = 0.007mm2s−2.
From the shape of the solid curve in Fig. 9(a), we see
that for vx<∼7.3mm s−1,
∂2RP(vx)
∂v2
x
???
vx
< 0, whilst for

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8
vx>∼7.3mm s−1,
(11), we expect Ri(vx) > RP(vx), for vx>∼7.3mm s−1,
and Ri(vx) < RP(vx), for vx<∼7.3mm s−1, which is
confirmed by comparing the dashed [Ri(vx)] and solid
[RP(vx)] curves in Fig. 9(a).
We now consider a BEC that initially contains a single
vortex. In this case, Eq. (8) can be generalized to
∂2RP(vx)
∂v2
x
???
vx
> 0. Hence, from Eq.
R(vx) =
?∞
0
F(vx− vx)RP(vx)dvx
(15)
where the velocity distribution for the incident atoms in
the rotating BEC, F(vx−vx), equals the velocity distri-
bution function for an irrotational BEC plus a correction
G(vx− vx) that is
F(vx− vx) = Fi(vx− vx) + G(vx− vx),
where F(vx−vx) =?
?
distribution functions f(vx− vx,y) and g(vx− vx,y) =
f(vx−vx,y)−fi(vx−vx,y), which are shown in Fig. 9(c),
for y = 0, by the dashed and dotted curves respectively.
Since g(vx−vx,y) and G(vx−vx) are both symmetrical
about vx− vx= 0, it follows from an analysis similar to
that presented in Eqs. (8-10) that
(16)
yf(vx−vx,y)dy and G(vx−vx) =
yg(vx−vx,y)dy are defined in terms of the local velocity
R(vx) ≈ Ri(vx)
+
∂2RP(vx)
2∂v2
x
????
vx
?∞
0
G(vx− vx)(vx− vx)2dvx,
(17)
which can be rewritten as
R(vx) ≈ Ri(vx) +J
2
∂2RP(vx)
∂v2
x
????
vx
≈ RP(vx) +(I + J)
2
∂2RP(vx)
∂v2
x
????
vx
(18)
where
J =
?∞
0
G(vx− vx)(vx− vx)2dvx.(19)
Unlike I, the sign of J can be either negative or posi-
tive, depending on the form of G(vx− vx). To highlight
this, we initially consider a simple analysis based on the
form of the local distribution function g(vx− vx,y = 0)
[Fig. 9(c)]. For vx ≈ vx (i.e. in the region of the vor-
tex core) f(vx− vx,y = 0) [dashed curve in Fig. 9(c)]
is close to zero, because the atom density falls to zero at
the vortex core, whereas fi(vx− vx,y = 0) [solid curve
in Fig. (c)] is maximal. Consequently, g(vx− vx,y = 0)
[dotted curve in Fig. 9(c)] is negative for vx≈ vx. Since
the atoms have moved away from the vortex core, to-
wards the edges of the BEC, we expect that away from
the vortex core (where |vx−vx| ≫ 0) f(vx−vx,y = 0) >
fi(vx− vx,y = 0) and hence g(vx− vx,y = 0) > 0, as
can be seen in Fig. 9(c) (dotted curve). Consequently,
the contribution to J is likely to be positive because the
integrand in Eq. (19) is largest when (vx−vx)2is large.
However, if y ?= 0, vx is also perturbed by the circu-
lation of the BEC around the vortex core. For y > 0,
vxis increased and for y < 0, vx is decreased. This ef-
fect further increases the spread of incident velocities at
the barrier and, due to the (vx− vx)2term in Eq. (19),
makes J more positive. From our full numerical analysis
we find that the dominant cause of the shift in the re-
flection probabilities is the circulation of atoms around
the vortex. Evaluating Eq. (19) numerically, we find for
BEC A that J = 0.19mm2s−2≫ I.
Since J and I are both positive for the BEC con-
sidered here, from Eqs. (11) and (18), we expect that
R(vx) − RP(vx) will have the same sign as, but a larger
magnitude than, Ri(vx) − RP(vx). This is confirmed by
the curves shown in Fig. 9(a) and highlights the fact
that the presence of the vortex increases the distribution
of incident velocities upon the scattering potential.
In Fig. 9(b), we compare the expression for r(vx) ≈
J
2
∂v2
x
the numerical values of r(vx) (dashed curve) obtained
from our solutions of the Gross-Pitaevskii equation. The
expression for r(vx) (solid curve) and the numerical sim-
ulations (dashed curve) are in good agreement because
the spread of incident velocities is narrow and hence the
Taylor expansions used in the derivation of Eqs. (11) and
(18) are reasonably accurate. For BEC B we find that
the discrepancy between the numerical r(vx) values and
those obtained from Eq. (18) becomes larger. This dis-
crepancy is not only due to an increase in the spread of
incident velocities, which introduces higher order terms
into the Taylor expansion, but also arises from the dis-
ruption of the BEC upon scattering, an effect which is not
described in our approximate analysis. Equations. (11)
and (18) are therefore only valid when no interferential
or rotational disruption is present in the BEC.
∂2RP(vx)
???
vx, obtained from Eq. (18) (solid curve), with
V.CONCLUSIONS
We have investigated how BECs with different atom
densities scatter off a Gaussian potential when the initial
state is either irrotational or contains a single vortex.
We find three distinct regimes for the formation of dy-
namical excitations and hence rotational or interferential
disruption: (i) at low densities there is no fragmentation
of the reflected or transmitted components of the BEC,
irrespective of whether or not there is a vortex in the ini-
tial state; (ii) at medium densities for which tr/tsr<∼1,
rotational disruption occurs in the reflected component
of a BEC with an initial vortex, but no disruption is ob-
served when the BEC is initially irrotational; (iii) at high
densities, there is strong interferential disruption in the
reflected atom cloud if tc<∼ts, both in the presence and
absence of an initial vortex.

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9
By considering the velocity distribution of the inci-
dent atoms, we have derived expressions for the reflection
probabilities of rotating and irrotational BECs in terms
of the reflection probability of a single plane wave inci-
dent on the scattering potential. This analytic approach
agrees well with our numerical calculations of reflection
probabilities for BECs scattering from a Gaussian bar-
rier. It shows that the velocity spread of an irrotational
BEC causes a positive or negative deviation from RP(vx),
depending on the curvature of RP(vx). When a vortex is
introduced, the circulation further increases the spread
of incident velocities, leading to an even larger deviation
from RP(vx).
Finally, we note that with current techniques it should
be possible to perform experimental tests of our theoret-
ical predictions, relating to the existence of three distinct
dynamical regimes and to the effect of a vortex on the
reflection probability of a BEC impinging on a potential
barrier.
This work was supported by the ARC, the EPSRC, the
Royal Society (London) and the University of Melbourne.
[1] O. Morsch, J.H. M¨ uller, M. Cristiani and E. Arimondo,
Phys. Rev. Lett. 87, 140402 (2001).
[2] A. S. Arnold, C. MacCormick, and M. G. Boshier, Phys.
Rev. A 65, 031601 (2002).
[3] K. Bongs, S. Burger, G. Birkl, K. Sengstock, W. Ertmer,
K. Rzazewski, A. Sanpera, and M. Lewenstein, Phys.
Rev. Lett. 83, 3577 (1999).
[4] A. G¨ unther, S. Kraft, M. Kemmler, D. Koelle, R. Kleiner,
C. Zimmermann and J. Fort´ agh, Phys. Rev. Lett. 95,
170405 (2005).
[5] T.A. Pasquini, Y. Shin, C. Sanner, M. Saba, A. Schi-
rotzek, D.E. Pritchard, and W. Ketterle, Phys. Rev. Lett.
93, 223201 (2004); T.A. Pasquini, M. Saba, G.-B. Jo, Y.
Shin, W. Ketterle, D.E. Pritchard, T.A. Savas and N.
Mulders, Phys. Rev. Lett. 97, 093201 (2006).
[6] R.G. Scott, A.M. Martin, T.M. Fromhold and F.W.
Sheard, Phys. Rev. Lett. 95, 073201 (2005).
[7] M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S.
Hall, C. E. Wieman, and E. A. Cornell, Phys. Rev. Lett.
83, 2498 (1999).
[8] K. W. Madison, F. Chevy, W. Wohlleben, and J. Dal-
ibard, Phys. Rev. Lett. 84, 806 (2000).
[9] F. Chevy, K. W. Madison, and J. Dalibard, Phys. Rev.
Lett. 85, 2223 (2000).
[10] C. Raman, J. R. Abo-Shaeer, J. M. Vogels, K. Xu, and
W. Ketterle, Phys. Rev. Lett. 87, 210402 (2001).
[11] R.G. Scott, A.M. Martin and T.M. Fromhold, Phys. Rev.
A 69, 063607 (2004).
[12] J.J. Garca-Ripoll, G. Molina-Terriza, V. M. Prez-Garca
and L. Torner, Phys. Rev. Lett. 87, 140403 (2001).
[13] I. Josopait, L. Dobrek, L. Santos, A. Sanpera and M.
Lewenstein, Eur. Phys. J. D. 22, 385 (2003).
[14] Y. Shin, M. Saba, T.A. Pasquini, W. Ketterle, D.E.
Pritchard and A.E. Leanhardt, Phys. Rev. Lett. 92,
050405 (2004).
[15] M. Greiner, O. Mandel, T.W. H¨ ansch and I. Bloch, Na-
ture 419, 51 (2002).
[16] R.G Scott, D.A.W. Hutchinson and C.W. Gardiner,
cond-mat/0608135 (unpublished).
[17] M.L. Chiofalo, S. Succi and M.P. Tosi, Phys. Rev. E, 62,
7438 (2000).
[18] W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P.
Flannery, Numerical Recipes, the Art of Scientific Com-
puting (Cambridge University Press, Cambridge, 1994).
[19] We calculate the expectation values of the positions of the
transmitted and reflected atom clouds when they come
to rest at t = π/ωx as
LT =
?
y
?
x>0| Ψ(x,y;t = π/ωx) |2xdxdy
?
y
?
x| Ψ(x,y;t) |2dxdy
and
LR =
?
y
?
x<0| Ψ(x,y;t = π/ωx) |2xdxdy
?
y
?
x| Ψ(x,y;t) |2dxdy
.
[20] Since the interactions in BEC C play a dominant role,
the diameter of the condensate is 50µm and hence it is
only possible to consider ∆x > 25µm. For this reason we
only consider ∆x = 30µm for system C.
[21] R.G. Scott, A.M. Martin, T.M. Fromhold, S. Bujkiewicz,
F.W. Sheard and M. Leadbeater, Phys. Rev. Lett. 90,
110404 (2003).
[22] R.G. Scott, A.M. Martin, S. Bujkiewicz, T.M. Fromhold,
N. Malossi, O. Morsch, M. Cristiani and E. Arimondo,
Phys. Rev. A. 69, 033605 (2004).
[23] L. Fallani, L. De Sarlo, J.E. Lye, M. Modugno, R. Saers,
C. Fort and M. Inguscio, Phys. Rev. Lett. 93, 140406
(2004).
[24] Z. Dutton, M. Budde, C. Slowe and L.V. Hau, Science
293, 663 (2001).
[25] Clearly, for a symmetrical condensate,?
0.
dvxF(vx−vx) >