Acceleration and localization of matter in a ring trap
Yu. V. Bludov1 and V. V. Konotop1,2
Centro de F́ısica Teórica e Computacional, Universidade de Lisboa,
Complexo Interdisciplinar, Avenida Professor Gama Pinto 2, Lisboa 1649-003, Portugal
Departamento de F́ısica, Faculdade de Ciências, Universidade de Lisboa,
Campo Grande, Ed. C8, Piso 6, Lisboa 1749-016,
Portugal and Departamento de Matemáticas, E. T. S. de Ingenieros Industriales,
Universidad de Castilla-La Mancha 13071 Ciudad Real, Spain
A toroidal trap combined with external time-dependent electric field can be used for implementing
different dynamical regimes of matter waves. In particular, we show that dynamical and stochastic
acceleration, localization and implementation of the Kapitza pendulum can be originated by means
of proper choice of the external force.
PACS numbers: 03.75.Lm, 03.75.Kk, 03.75.Ss
I. INTRODUCTION
Exploring different geometries of potentials trapping
cold condensed atoms is of both fundamental and practi-
cal importance. Toroidal traps play a special role allow-
ing for ”infinite” atomic trajectories and for realization
of quasi-one-dimensional (quasi-1D) regimes. These ad-
vantages are relevant for designing highly precise sensors
based on matter wave interferometry [1, 2] as well as for
accurate study of such phenomena as superfluid currents,
stability of sound waves, solitons and vortices in Bose-
Einstein condensates (BEC’s) [3, 4]. Traps with circular
geometry are also believed to be conceptually important
for implementation of the main ideas of the accelerator
physics at ultra-low temperatures [2] and, in particular,
for acceleration of ultracold atoms [5]. In this last con-
text existence of well localized wave packets, and thus
attenuation of the dispersion, the latter being the intrin-
sic property of a quantum systems, is of primary impor-
tance. In the first experimental studies [2] it was shown
that the dispersive spreading out [1] can be compensated
by using betatron resonances in a storage ring. An al-
ternative way of contra-balancing dispersion is also well
known - it is nonlinearity, leading in quasi-1D regime to
existence of bright and dark matter solitons (see e.g. [6, 7]
and [8, 9, 10], respectively). This issue has already been
explored [11] from the point of view of acceleration of
matter waves in a toroidal trap with help of a modulated
optical lattice, which is known to be an efficient tool for
acceleration of matter waves [12].
In this paper we propose two alternative ways of ac-
celerating matter wave solitons – either by time varying
or by stochastic external electric field. These new ways
of soliton acceleration are especially relevant in view of
radiative losses [13] and distortions [12] of solitons mov-
ing in optical lattices (the effects acquiring significance
for long trajectories). At the same time, it turns out
that the toroidal geometry of a trap confining a BEC al-
lows one to realize a number of other dynamical regimes,
like dynamical localization of solitons and solitonic im-
plementation of the celebrated Kapitza pendulum. The-
oretical description of all mentioned phenomena can be
put witching the unique framework, based on the pertur-
bation theory for solitons, what is done in the present pa-
per. More specifically, in Sec. II we formulate the model
and the main physical constraints determining its valid-
ity. In sections III and IV we describe how by applying
external time-dependent electric field matter solitons can
be accelerated in the usual sense and in the sense of the
time increase of the velocity variance (the stochastic ac-
celeration), respectively. In Sec. V we describe localized
states of the matter in circular trap subject to external
field, and in Sec. VI we show that a matter soliton af-
fected by rapidly varying force represents an example of
the Kapitza pendulum [14]. Summary and discussion of
the results are given in the Conclusion.
II. SCALING AND THE EVOLUTION
EQUATION
We assume that a BEC is loaded in a circular trap,
which in cylindrical coordinates r = (ρ, ϕ, z) is described
by V = Vc(ρ) +mω
2/2, where ωz is the frequency of
the magnetic trap in the z–direction, Vc(ρ) is the poten-
tial in the radial direction, forming the trap circular in
the (x, y)–plane, and m is the mass of an atom. We also
suppose that the BEC is subject to external electric field
with amplitude E0, which produces an additional poten-
tial Vext = −α′E20/4, where α′ is the polarizability of the
atoms (see e.g. [15]). If the amplitude E0 or direction of
the field vary along some direction, say, along the x-axis,
smoothly on the scale of the trap radius R, the potential
energy Vext van be expanded in the Taylor series and, af-
ter neglecting nonessential constant, be rewritten in the
form Vext = −αx, where α = (α′/4)∂(E0)2/∂x|x=0 and
we restricted the consideration only to the first term of
the expansion. In order to realize one-dimensional geom-
etry we require torus radius to be much larger than the
core radius rc, what allows us to define a small param-
eter ǫ = rc/R ≪ 1. In order to introduce quantitative
characteristics, we consider the normalized ground state
http://arxiv.org/abs/0704.0733v1
φ of the eigenvalue problem
φ+ Vc(ρ)φ = εrφ ,
φ2ρdρ = 1 (1)
and define R1 =
φ2ρ2dρ, R2 =
φ2dρ/ρ
)−1/2
and λ =
φ4ρdρ
)−1/2
. In the case at hand λ ∼√
Rrc ∼ ǫ1/2R and thus λ≪ R1 ∼ R2 ∼ R.
In the present paper we are interested in the dynam-
ics of matter waves which spatial extension is much less
than the trap perimeter, allowing treat them similarly to
the matter solitons in an infinite one-dimensional trap.
This in particular the case where the spatial size of the
BEC excitations along the trap are of order of λ, which
is the well defined parameter and thus convenient for for-
mulation the constraints of the theory. Indeed, now we
can estimate the kinetic energy of the longitudinal ex-
citations as ε‖ = h̄
2/(2mλ2) and require it to be much
less than the kinetic energy of the transverse excitations,
εr ∼ h̄2/(2mr2c) (for the sake of simplicity here we assume
that the size of the trap in z-direction is of order of the
core radius: az =
h̄/mωz ∼ rc). Adding the require-
ment for the energy of the two-body interactions, which
is estimated as |g|n (where g = 4πh̄2as/m, as is the scat-
tering length, n ∼ N/V is a mean density, N is the total
number of atoms and V is the effective volume occupied
by the atoms and estimated as V ∼ πλrcaz), to be of or-
der of ε‖ and to be much less than εr (or more precisely,
requiring |g|n/εr ∼ ǫ), we can neglect in the leading or-
der the transitions between the transverse energy lev-
els [9, 10], and employ the multiple scale expansion [6, 10]
for description of the quasi-one-dimensional evolution of
the BEC. We also notice that subject to the assumptions
introduced, one has the estimate N ∼ ǫ3/2R/(8|as|).
In order to get an insight on practical numbers, let us
consider 7Li atoms (as = −2 nm) in a trap with R =
100µm, rc = 5µm and az = 10µm. Then ǫ = 0.05, the
characteristic size of solitonic excitations is λ ≈ 22µm
and the number of particles is estimated as N ≈ 140. We
emphasize, that these estimates indicate only an order of
the parameters. Thus, for example, a condensate of 102÷
103 lithium atoms satisfy the conditions of the theory.
We will be interested in managing soliton dynamics
by means of weak (i.e not destroying solitons) electro-
magnetic field varying in time. Respectively, we con-
sider α time-dependent and characterized by the esti-
mate α ∼ h̄2/(mR1λ2). Then, starting with the Gross-
Pitaevskii equation, in which the external potential in
cylindrical coordinates has the form Vext = −αρ cos(ϕ) ≈
−αR1 cos(ϕ), and using the multiple-scale expansion one
ensures that the BEC macroscopic wave function in the
leading order allows factorization
Ψ = π−1/4a−1/2z e
−i(ωr+ωz)t/2e−z
2/2a2
zφ(r)ψ(t, ϕ), (2)
where ωr = 2εr/h̄ and ψ(t, ϕ) solves the nonlinear
Schrödinger equation, which we write in terms of A =
|g|m/
2πh̄2azψ, ζ = R2ϕ/λ, and τ = h̄t/mλ
− cos(κζ)f(τ)A + σ|A|2A . (3)
Here σ =signas, f(τ) ≡ mR1λ2α(t)/h̄2 and κ = λ/R2 ∼√
ǫ. We choose the scaling in such a way that all terms
in (3) are of the unity order, and in particular A = O(1).
This can be done, taken into account the normalization
|A|2dζ = 2
|as|N
, (4)
L = 2π/κ, which follows from the normalization condi-
tion for the order parameter
|Ψ|2d3r = N , and consid-
ering N ∼ az/|as|, what is of order of 103, in a typical
experimental setting.
Eq. (3) is subject to periodic boundary conditions
A(ζ, τ) = A(ζ + L, τ).
III. ACCELERATION OF BRIGHT MATTER
SOLITONS BY TIME-DEPENDENT EXTERNAL
FORCE
First we consider the acceleration, γ, which can be
achieved due to the potential Ve properly dependent on
time. An order of magnitude of γ can be estimated by
taking into account that Eq. (3) makes sense provided
that all terms are of the unity order. In the physical
units this gives γ ∼ h̄2/(m2λ3). Then, recalling the
above example of the lithium condensate we estimates
γ ∼ 7mm/s2, what is of order of the acceleration an-
nounced in [11]. It however does not provide the best es-
timate in our case, because it is based on the 1D model,
while lowering dimensionality imposes constrains on the
atomic density and consequently on the amplitude of the
applied force.
To describe the physics of the phenomenon we consider
a BEC with a negative scattering length (σ = −1). Then
a ”bright soliton” solution of Eq. (3) at f(τ) ≡ 0 (or more
precisely a periodic solution mimicking a bright soliton
in an infinite 1D system) which moves with a constant
velocity vn can be written down as follows [8]
As = e
−i(ω(k)+v2
/2)τ+ivnζη(k) dn (η(k)(ζ − vnτ), k) . (5)
Here dn(x, k) is the Jacobi elliptic function [16], k is
the elliptic modulus parameterizing the solution. The
frequency and the amplitude are given by: ω(k) =
(k2/2 − 1)η2(k) and η(k) = 2K(k)/L [K(k) is the com-
plete elliptic integral of the first kind]. The velocity of
the soliton is quantized vn = 2πn/L with n being integer.
To ensure that the solution As satisfies the scaling
relations imposed above, we notice that the size of
the soliton can be estimated as π/K(k) and its small-
ness implies that k is close to unity. In that case we
obtain the estimates 1 − k2 ∼ 16 exp(−2π/
ǫ) and
dn (η(k)(ζ − vnτ), k) ≈ 1/ cosh(η(k)(ζ − vnτ)).
In the limit k → 1 quantization of the velocity does not
play any significant role. We verified this numerically.
For example, for L = 10 deviation of the initial velocity
from the quantized one produces appreciable effect on
dynamics during intervals τ <∼ 100 only if k <∼ 0.99.
When external force is applied, f(τ) 6= 0, the velocity is
not preserved any more, what manifests itself in evolution
of the momentum P = (1/2i)
AζĀ−AĀζ
(here Ā
stands for complex conjugation of A) according to the
= −f(τ)
cos(κζ)
∂|A|2
dζ . (6)
The external field, however, does not affect the norm:
|A|2dζ=const. It follows from (5) that in the
adiabatic approximation the solution of the perturbed
equation (3) can be searched in the form
A = e−i(ω(k)+V (τ)
2/2)τ+iV (τ)ζη dn (η(ζ −X(τ)), k) (7)
where V (τ) = dX(τ)/dτ is the time-dependent velocity
of the soliton and X(τ) is the coordinate of the soliton
center. Substituting (7) in (6) and taking into account
the parity of the functions in the integrand as well as the
fact that all of them are periodic with the same period
L, we obtain the equation for the soliton coordinate
= −κC(k)f(τ) sin(κX) . (8)
C(k) =
2πE(k)
cos(θ)dn2
dθ (9)
and it is taken into account that N = 2ηE(k), where
E(k) is the complete elliptic integral of the second kind.
Depending on the choice of the function f(τ), Eq. (8)
describes different dynamical regimes. Now we are inter-
ested in acceleration which occurs during the rotational
movement of the soliton in the trap (i.e. X is a growing
function). We illustrate this acceleration using an exam-
ple of the simplest step-like dependence f(τ). To this
end we assume that initially the soliton is centered at
X(0) = 0 and require f(τ) to be a constant f0 for time
intervals such that the soliton coordinates X(τ) ∈ Ip and
to be zero for X(τ) /∈ Ip where the intervals Ip are given
by Ip = [(p +
)L, (p + 1)L] with p = 0, 1, .... Then, as
it is clear, the acceleration of the soliton, which is given
by the right hand side of (8), is positive for all times.
The above requirement introduces natural splitting of
the temporal axis in the set of intervals Tl = [τl, τl+1]
(l = 0, 1, ...), with τ0 = 0, such that f(τ) = 0 for τ ∈ T2p
and f(τ) = f0 for τ ∈ T2p+1 (here X(τl) = lL/2). Thus,
our task is to find τl. This can be done by taking into ac-
count that during each of the ”odd” intervals T2p+1 Eq.
(8) describes conservative nonlinear oscillator, the solu-
tion for which is well known. During ”even” intervals T2p
the motion is free (with a constant velocity) what means
FIG. 1: (Color online) The soliton velocity vs time (panels
a,b) for the parameters k = 0.99999, L = 10.0, f0 = 0.3,
n = 0.43 (panel a, the non-quantized velocity), n = 1.0 (panel
b, the quantized velocity), and the forms of the soliton (panels
c,d,e) in the instants of time indicated in the panel a. In
the panels a,b solid and dashed lines represent the velocity
numerically computed from Eq.(3), and Eq.(8), respectively.
that the time T2p necessary for the soliton to cross an
interval [pL, (p+ 1
)L] is
T2p = τ2p+1 − τ2p = L/(2v2p), (10)
where v2p is the velocity in the point pL. During the
time interval T2p+1 the soliton has to cross the interval
Ip. From this condition we obtain:
T2p+1 = τ2p+2 − τ2p+1 =
2E0/(H2p+1 + E0)
H2p+1 + E0
where H2p+1 = v
2p+1/2 + E0 is the energy of the soli-
ton in the point (p + 1/2)L, E0 = C(k)f0, and v2p+1 =
v2p is the soliton velocity in the same point. At the
end of the interval T2p+1 the soliton velocity is given
by v2(p+1) =
2(H2p+1 + E0). Thus one computes
that after p rotations the soliton acquires the velocity
v2(p+1), which can be obtained from the recurrent rela-
tion: v2(p+1) =
v22p + 4C(k)f0.
In Fig.1 a,b we compare the solution, obtained from
the perturbation theory, Eq.(8), with numerical simula-
tion of Eq.(3) for f0 = 0.3. Nevertheless during the nu-
merical simulation we used the values for T2p and T2p+1
(Eqs.(10)–(11)) obtained for the case of adiabatic approx-
imation. It follows form the results presented that the
dashed and solid lines perfectly match until τ ≈ 50.0. At
larger times appreciable discrepancy appears. It occurs
due to failure of the adiabatic approximation and can be
removed by introducing temporal corrections to T2p and
T2p+1. This naturally leads to an optimization problem,
which requires numerical approach and goes beyond the
scope of the present work. Finally we notice, that for the
above example of 7Li condensate the obtained accelera-
tion is 0.36mm/s2.
Comparison of the panels a and b in Fig.1 shows that
for k ≈ 1 quantization of the velocity is not important,
what is also confirmed by the evolution of the solitonic
forms depicted in the panels Fig.1 c-e.
IV. STOCHASTIC ACCELERATION OF
MATTER SOLITONS
Now we concentrate on another dynamical regime – on
the stochastic acceleration – where increase of the veloc-
ity of a matter soliton in a toroidal trap is achieved by
applying a fluctuating external field. To this end hold-
ing all conditions of the applicability of the model (3),
we consider the case of a stochastic force f(τ), which
is delta-correlated Gaussian process with characteristics
〈f(τ)〉 = 0 and 〈f(τ)f(τ ′)〉 = Dδ(τ − τ ′) (the angular
brackets stand for the stochastic averaging and D is the
dispersion). Now the dynamics can be described in terms
of the distribution function
P(V,Φ, τ) = 〈δ(Φ− Φ(τ))δ(V − V (τ))〉, (12)
where Φ(τ) ≡ κX is the angular coordinate of the soli-
ton, Φ(τ) and V (τ) with explicit time dependence stand
for the soliton coordinates obtained from the dynamical
equations while Φ and V are considered as independent
variables. The distribution function solves the Fokker-
Planck equation, which is obtained by the standard pro-
cedure (see e.g. [17]):
= −V ∂P
+ D̃ sin2(Φ)
. (13)
Here D̃ = κ4C2(k)D is the diffusion coefficient. Due to
the circular geometry of the trap Eq. (13) is considered
on the interval −π < Φ < π with the periodic boundary
conditions P(V,Φ − π, τ) = P(V,Φ + π, τ) with respect
to Φ and zero boundary conditions with respect V : P →
0 as V → ±∞. The normalization condition for the
probability density reads:
dΦP = 1.
Multiplying Eq.(13) by V and Φ and integrating over
V and Φ one readily obtains that the averaged velocity
and angular position of the soliton are constants, which
for the sake of simplicity will be considered zeros, i.e.
〈V 〉 = 0 and 〈Φ〉 = 0. Next, multiplying (13) by V 2, Φ2
and V Φ and performing the integration one obtains the
equations of the second momenta. They are not closed
and can be written down as follows:
〈V 2〉 = 2D̃〈sin2 Φ〉 , (14)
〈Φ2〉 = 2〈V Φ〉 , (15)
〈V Φ〉 = −2π
P (π, V, τ)V 2dV + 〈V 2〉 .(16)
Eq. (14) means that the average square velocity is grow-
ing with time, i.e. the soliton undergoes the stochastic
acceleration. The law of the growth of the velocity in-
variance deviates form the linear, as it would happen for
FIG. 2: (Color online) The mean square velocity (panel a) and
the stochastic acceleration γ̃ (panel b) of the soliton vs time
for different values of the dispersion, obtained by numerical
integration of Eq. (13) with parameters L = 10.0 and k =
0.99999. In panel b dashed lines depict the approximation of
numerical data by the law γ̃ ∝ τ−1/2. All axes in panels a,b
are represented in logarithmic scale.
the Brownian diffusion in the momentum space, what
happens because the diffusion coefficient in the Fokker-
Planck equation (13) is not a constant, but depends on
the angular variable. However, due to the diffusion one
can expect that the phase distribution will tend to ho-
mogeneous, i.e. that P → 1/(2π) as τ → ∞. In this
formal limit one obtains that 〈V Φ〉 → 0, 〈sin2 Φ〉 → 1/2
and hence 〈V 2〉 → D̃τ . In other words, the system (14)-
(16), describes random walk which in the limit of large
time, approaches the Brownian diffusion in the velocity
space. In that limit the stochastic acceleration, which
can be defined as γ̃ = d
〈V 2〉/dτ , would tend to zero
according to the law γ̃ ∝ τ−1/2.
In order to check the above predictions and reveal other
features of the stochastic dynamics of a soliton in a ring
trap we solved numerically Eq. (13) subject to the ini-
tial condition P(V,Φ, 0) = 〈δ(Φ)δ(V )〉. The results is
summarized in Fig. 2. In the panel a one observes the
predicted monotonic growth of the mean velocity with
time, which slightly different from the linear law. In the
panel b one can see that the stochastic acceleration γ̃
is a monotonically decreasing function, which at suffi-
ciently large times tends to zero. In particular, at τ >∼ 15
the decreasing of the acceleration with time can be well
approximated by the predicted law γ̃ ∝ τ−1/2, as it is
shown by dashed curves in the panel b of Fig. 2 (it was
verified that in at the same times 〈sin2 Φ〉 ≈ 1/2, what
is in agreement with the analytical predictions). Also
Fig. 2 b shows that the stochastic acceleration is larger
for larger D. The physical explanation of this last fact
is simple: the acceleration is generated by the stochastic
force, whose intensity is determined by the dispersion D.
V. LOCALIZATION OF MATTER INDUCED BY
THE EXTERNAL FIELD
Let us now turn to localized states of a matter in a
toroidal trap and concentrate on the states generated by
the constant external electric field, i.e. by f(τ) ≡ f0.
Respectively, we look for stationary solutions of Eq. (3)
in the form A = e−iωτA(ζ) and obtain for A(ζ) the equa-
tion:
− f0 cos(κζ)A + σ|A|2A = ωA (17)
which is subject to periodic boundary conditions
A(ζ, τ) = A(ζ + L, τ).
Several lowest branches of the numerically obtained
solutions of Eq. (17) are shown in Fig. 3. The lowest
branch approaches zero at the frequency ω0 ≈ −0.143
(it is interesting to mention that this frequency coincides
with the lowest gap edge of the spectrum of the Mathieu
equation (17) considered on the whole axis), where the
amplitude of the nonlinear periodic mode is small and
it transforms into the linear periodic Bloch mode at the
lowest gap edge. Such a behavior of the branch is sim-
ilar to that of the strongly localized modes in a BECs
in the optical lattice [18]. The lowest mode A is local-
ized in the vicinity of ϕ = 0 (Fig.3, b), i.e. around the
minimum of the effective potential and that is why such
a mode is stable and can exists even in the linear case,
where the two-body interactions are negligible (here it
is important that we are dealing with periodic boundary
conditions). The modes of the upper branches – B and
D (their typical examples are shown in Fig.3 b) – bifur-
cate at ω∗ ≈ −0.345. They are localized at ϕ = ±π, i.e.
near the points where the potential has maxima. As it
is clear, this is pure nonlinear effect and occurs due to
delicate balance between the attractive interactions and
repulsive forces of the external potential. Such balance
can easily be destroyed even by an infinitesimal pertur-
bation, what allows us to expect instability of the modes.
The mode C represents two local maxima of the atomic
distributions at ϕ = 0, π. Similar to the modes B and D
we one can expect it to be unstable, what physically can
be explained by existence of local atomic maxima at the
maxima of the potential. By direct numerical solution of
Eq. (3) (more specifically by perturbing the mode pro-
files by the factor 1 + 0.1 cos (21 ζ) and computing the
dynamics until τ = 1000) we have verified that, indeed,
only the mode A on Fig. 3 is dynamically stable, while
the modes B, C, and D are unstable.
VI. MATTER SOLITON AS A KAPITZA
PENDULUM
As the final example of nontrivial dynamics of a mat-
ter soliton in a toroidal trap we consider dynamical lo-
calization induced by a rapidly oscillating force f(τ) =
f0 [ν + cos(Ωτ)]. In this case the solitonic motion mimics
FIG. 3: (Color online) The number of bosons N (for the ex-
ample of the lithium condensate described in the text) vs
frequency ω (panel a) and shapes of the localized modes at
ω = −0.4 (panel b) for the case where L = 10.0, f0 = 0.3,
σ = −1.
the famous Kapitza pendulum, which acquires an addi-
tional stable point due rapid oscillation of the pivot [14].
Assuming that the physical conditions of the validity of
the quasi-1D approximation (3) holds and that the fre-
quency Ω is large enough, i.e. Ω2 ≫ κ2C(k)f0, one can
perform the standard analysis (see e.g. [14]), i.e. look for
a solution of (3) in a form X(τ) + ξ(τ) where ξ is small,
|ξ| ≪ |X |, and rapidly varying, and provide averaging
over rapid oscillations. Then one arrives at the equation
d2X/dτ2 = −∂U/∂X with the effective potential
U = −C(k)f0
ν cos(κX) + κ2C(k)
cos(2κX)
.(18)
If the condition κ2C(k)f0/(2Ω
2ν) > 1 is met, the ef-
fective potential U possesses two stable points: X = 0
(Φ = 0) and X = L/2 (Φ = π). So, it opens the pos-
sibility for the new type of soliton moving around the
new stable point. Two typical trajectories of the soli-
ton, obtained by numerical integration of Eq. (3), are
presented in Fig.4. One of the trajectories displays oscil-
lations around the new equilibrium point, while the other
one shows the large oscillations around the equilibrium
point Φ = 2π started with the same initial data but in the
case where Φ = π is not an equilibrium any more. The
amplitude of large oscillations decay with time because
of energy losses of the soliton in the nonconservative sys-
VII. CONCLUSIONS
In the present paper we have shown that dynamics of
a matter soliton in a toroidal trap, well reproducing one-
dimensional geometry, can be very efficiently governed by
time varying external electric field. In particular, such
regimes like dynamical acceleration, stochastic acceler-
ation, localization and implementation of the Kapitza
pendulum can be realized by proper choices of the time
dependence of the external force.
Experimental detection of the acceleration can be im-
plemented either by direct imaging of the atomic cloud,
FIG. 4: (Color online) The angular coordinate of the soliton
center vs time for the soliton motion affected by the rapidly
oscillating external force, obtained numerically from Eq.(3)
with parameters L = 10.0, n = 0.01, σ = −1, f0 = 0.15,
Ω = 2.0, and k = 0.99999.
which is well localized in space and has well specified tra-
jectory, or by measurement of the atomic distribution in
the momentum space displaying shift of the maximum
towards higher kinetic energies. Alternatively, one can
study the evolution of the atomic cloud releasing from
the trap (by switching the trap off) after some period of
accelerating motion. The respective dynamics will be a
spreading out cloud whose center of mass is moving with
the acquired velocity.
The obtained results were based on the one-
dimensional model, although deduced using the multiple-
scale method and thus mathematically controllable. This
means that a number of problem, are still left open. One
of them is the limitation on the soliton velocity, and thus
acceleration, introduced by lowering the space dimension,
which appears when the solitonic kinetic energy becomes
comparable with the transverse kinetic energy. Another
limitation on the soliton acceleration emerges from the
fact of the velocity quantization, when the radius of the
ring trap is not large enough. We also left for further
studies the diversity of localized stationary atomic distri-
butions supported by the external filed, indicating only
the lowest modes. We thus believe that the richness of
the phenomena which can be observed by simple com-
bination of the trap geometry and varying external field
will stimulate new experimental and theoretical studies.
Acknowledgments
The authors are indebted to H. Michinel for providing
additional data on Ref. [11]. YVB was supported by the
FCT grant SFRH/PD/20292/2004. VVK was supported
by the Secretaria de Stado de Universidades e Investi-
gaci?n (Spain) under the grant SAB2005-0195. The work
was supported by the FCT (Portugal) and European pro-
gram FEDER under the grant POCI/FIS/56237/2004.
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http://arxiv.org/abs/cond-mat/0610047

  A toroidal trap combined with external time-dependent electric field can be
used for implementing different dynamical regimes of matter waves. In
particular, we show that dynamical and stochastic acceleration, localization
and implementation of the Kapitza pendulum can be originated by means of proper
choice of the external force.

Acceleration and localization of matter in a ring trap
Yu. V. Bludov1 and V. V. Konotop1,2
Centro de F́ısica Teórica e Computacional, Universidade de Lisboa,
Complexo Interdisciplinar, Avenida Professor Gama Pinto 2, Lisboa 1649-003, Portugal
Departamento de F́ısica, Faculdade de Ciências, Universidade de Lisboa,
Campo Grande, Ed. C8, Piso 6, Lisboa 1749-016,
Portugal and Departamento de Matemáticas, E. T. S. de Ingenieros Industriales,
Universidad de Castilla-La Mancha 13071 Ciudad Real, Spain
A toroidal trap combined with external time-dependent electric field can be used for implementing
different dynamical regimes of matter waves. In particular, we show that dynamical and stochastic
acceleration, localization and implementation of the Kapitza pendulum can be originated by means
of proper choice of the external force.
PACS numbers: 03.75.Lm, 03.75.Kk, 03.75.Ss
I. INTRODUCTION
Exploring different geometries of potentials trapping
cold condensed atoms is of both fundamental and practi-
cal importance. Toroidal traps play a special role allow-
ing for ”infinite” atomic trajectories and for realization
of quasi-one-dimensional (quasi-1D) regimes. These ad-
vantages are relevant for designing highly precise sensors
based on matter wave interferometry [1, 2] as well as for
accurate study of such phenomena as superfluid currents,
stability of sound waves, solitons and vortices in Bose-
Einstein condensates (BEC’s) [3, 4]. Traps with circular
geometry are also believed to be conceptually important
for implementation of the main ideas of the accelerator
physics at ultra-low temperatures [2] and, in particular,
for acceleration of ultracold atoms [5]. In this last con-
text existence of well localized wave packets, and thus
attenuation of the dispersion, the latter being the intrin-
sic property of a quantum systems, is of primary impor-
tance. In the first experimental studies [2] it was shown
that the dispersive spreading out [1] can be compensated
by using betatron resonances in a storage ring. An al-
ternative way of contra-balancing dispersion is also well
known - it is nonlinearity, leading in quasi-1D regime to
existence of bright and dark matter solitons (see e.g. [6, 7]
and [8, 9, 10], respectively). This issue has already been
explored [11] from the point of view of acceleration of
matter waves in a toroidal trap with help of a modulated
optical lattice, which is known to be an efficient tool for
acceleration of matter waves [12].
In this paper we propose two alternative ways of ac-
celerating matter wave solitons – either by time varying
or by stochastic external electric field. These new ways
of soliton acceleration are especially relevant in view of
radiative losses [13] and distortions [12] of solitons mov-
ing in optical lattices (the effects acquiring significance
for long trajectories). At the same time, it turns out
that the toroidal geometry of a trap confining a BEC al-
lows one to realize a number of other dynamical regimes,
like dynamical localization of solitons and solitonic im-
plementation of the celebrated Kapitza pendulum. The-
oretical description of all mentioned phenomena can be
put witching the unique framework, based on the pertur-
bation theory for solitons, what is done in the present pa-
per. More specifically, in Sec. II we formulate the model
and the main physical constraints determining its valid-
ity. In sections III and IV we describe how by applying
external time-dependent electric field matter solitons can
be accelerated in the usual sense and in the sense of the
time increase of the velocity variance (the stochastic ac-
celeration), respectively. In Sec. V we describe localized
states of the matter in circular trap subject to external
field, and in Sec. VI we show that a matter soliton af-
fected by rapidly varying force represents an example of
the Kapitza pendulum [14]. Summary and discussion of
the results are given in the Conclusion.
II. SCALING AND THE EVOLUTION
EQUATION
We assume that a BEC is loaded in a circular trap,
which in cylindrical coordinates r = (ρ, ϕ, z) is described
by V = Vc(ρ) +mω
2/2, where ωz is the frequency of
the magnetic trap in the z–direction, Vc(ρ) is the poten-
tial in the radial direction, forming the trap circular in
the (x, y)–plane, and m is the mass of an atom. We also
suppose that the BEC is subject to external electric field
with amplitude E0, which produces an additional poten-
tial Vext = −α′E20/4, where α′ is the polarizability of the
atoms (see e.g. [15]). If the amplitude E0 or direction of
the field vary along some direction, say, along the x-axis,
smoothly on the scale of the trap radius R, the potential
energy Vext van be expanded in the Taylor series and, af-
ter neglecting nonessential constant, be rewritten in the
form Vext = −αx, where α = (α′/4)∂(E0)2/∂x|x=0 and
we restricted the consideration only to the first term of
the expansion. In order to realize one-dimensional geom-
etry we require torus radius to be much larger than the
core radius rc, what allows us to define a small param-
eter ǫ = rc/R ≪ 1. In order to introduce quantitative
characteristics, we consider the normalized ground state
http://arxiv.org/abs/0704.0733v1
φ of the eigenvalue problem
φ+ Vc(ρ)φ = εrφ ,
φ2ρdρ = 1 (1)
and define R1 =
φ2ρ2dρ, R2 =
φ2dρ/ρ
)−1/2
and λ =
φ4ρdρ
)−1/2
. In the case at hand λ ∼√
Rrc ∼ ǫ1/2R and thus λ≪ R1 ∼ R2 ∼ R.
In the present paper we are interested in the dynam-
ics of matter waves which spatial extension is much less
than the trap perimeter, allowing treat them similarly to
the matter solitons in an infinite one-dimensional trap.
This in particular the case where the spatial size of the
BEC excitations along the trap are of order of λ, which
is the well defined parameter and thus convenient for for-
mulation the constraints of the theory. Indeed, now we
can estimate the kinetic energy of the longitudinal ex-
citations as ε‖ = h̄
2/(2mλ2) and require it to be much
less than the kinetic energy of the transverse excitations,
εr ∼ h̄2/(2mr2c) (for the sake of simplicity here we assume
that the size of the trap in z-direction is of order of the
core radius: az =
h̄/mωz ∼ rc). Adding the require-
ment for the energy of the two-body interactions, which
is estimated as |g|n (where g = 4πh̄2as/m, as is the scat-
tering length, n ∼ N/V is a mean density, N is the total
number of atoms and V is the effective volume occupied
by the atoms and estimated as V ∼ πλrcaz), to be of or-
der of ε‖ and to be much less than εr (or more precisely,
requiring |g|n/εr ∼ ǫ), we can neglect in the leading or-
der the transitions between the transverse energy lev-
els [9, 10], and employ the multiple scale expansion [6, 10]
for description of the quasi-one-dimensional evolution of
the BEC. We also notice that subject to the assumptions
introduced, one has the estimate N ∼ ǫ3/2R/(8|as|).
In order to get an insight on practical numbers, let us
consider 7Li atoms (as = −2 nm) in a trap with R =
100µm, rc = 5µm and az = 10µm. Then ǫ = 0.05, the
characteristic size of solitonic excitations is λ ≈ 22µm
and the number of particles is estimated as N ≈ 140. We
emphasize, that these estimates indicate only an order of
the parameters. Thus, for example, a condensate of 102÷
103 lithium atoms satisfy the conditions of the theory.
We will be interested in managing soliton dynamics
by means of weak (i.e not destroying solitons) electro-
magnetic field varying in time. Respectively, we con-
sider α time-dependent and characterized by the esti-
mate α ∼ h̄2/(mR1λ2). Then, starting with the Gross-
Pitaevskii equation, in which the external potential in
cylindrical coordinates has the form Vext = −αρ cos(ϕ) ≈
−αR1 cos(ϕ), and using the multiple-scale expansion one
ensures that the BEC macroscopic wave function in the
leading order allows factorization
Ψ = π−1/4a−1/2z e
−i(ωr+ωz)t/2e−z
2/2a2
zφ(r)ψ(t, ϕ), (2)
where ωr = 2εr/h̄ and ψ(t, ϕ) solves the nonlinear
Schrödinger equation, which we write in terms of A =
|g|m/
2πh̄2azψ, ζ = R2ϕ/λ, and τ = h̄t/mλ
− cos(κζ)f(τ)A + σ|A|2A . (3)
Here σ =signas, f(τ) ≡ mR1λ2α(t)/h̄2 and κ = λ/R2 ∼√
ǫ. We choose the scaling in such a way that all terms
in (3) are of the unity order, and in particular A = O(1).
This can be done, taken into account the normalization
|A|2dζ = 2
|as|N
, (4)
L = 2π/κ, which follows from the normalization condi-
tion for the order parameter
|Ψ|2d3r = N , and consid-
ering N ∼ az/|as|, what is of order of 103, in a typical
experimental setting.
Eq. (3) is subject to periodic boundary conditions
A(ζ, τ) = A(ζ + L, τ).
III. ACCELERATION OF BRIGHT MATTER
SOLITONS BY TIME-DEPENDENT EXTERNAL
FORCE
First we consider the acceleration, γ, which can be
achieved due to the potential Ve properly dependent on
time. An order of magnitude of γ can be estimated by
taking into account that Eq. (3) makes sense provided
that all terms are of the unity order. In the physical
units this gives γ ∼ h̄2/(m2λ3). Then, recalling the
above example of the lithium condensate we estimates
γ ∼ 7mm/s2, what is of order of the acceleration an-
nounced in [11]. It however does not provide the best es-
timate in our case, because it is based on the 1D model,
while lowering dimensionality imposes constrains on the
atomic density and consequently on the amplitude of the
applied force.
To describe the physics of the phenomenon we consider
a BEC with a negative scattering length (σ = −1). Then
a ”bright soliton” solution of Eq. (3) at f(τ) ≡ 0 (or more
precisely a periodic solution mimicking a bright soliton
in an infinite 1D system) which moves with a constant
velocity vn can be written down as follows [8]
As = e
−i(ω(k)+v2
/2)τ+ivnζη(k) dn (η(k)(ζ − vnτ), k) . (5)
Here dn(x, k) is the Jacobi elliptic function [16], k is
the elliptic modulus parameterizing the solution. The
frequency and the amplitude are given by: ω(k) =
(k2/2 − 1)η2(k) and η(k) = 2K(k)/L [K(k) is the com-
plete elliptic integral of the first kind]. The velocity of
the soliton is quantized vn = 2πn/L with n being integer.
To ensure that the solution As satisfies the scaling
relations imposed above, we notice that the size of
the soliton can be estimated as π/K(k) and its small-
ness implies that k is close to unity. In that case we
obtain the estimates 1 − k2 ∼ 16 exp(−2π/
ǫ) and
dn (η(k)(ζ − vnτ), k) ≈ 1/ cosh(η(k)(ζ − vnτ)).
In the limit k → 1 quantization of the velocity does not
play any significant role. We verified this numerically.
For example, for L = 10 deviation of the initial velocity
from the quantized one produces appreciable effect on
dynamics during intervals τ <∼ 100 only if k <∼ 0.99.
When external force is applied, f(τ) 6= 0, the velocity is
not preserved any more, what manifests itself in evolution
of the momentum P = (1/2i)
AζĀ−AĀζ
(here Ā
stands for complex conjugation of A) according to the
= −f(τ)
cos(κζ)
∂|A|2
dζ . (6)
The external field, however, does not affect the norm:
|A|2dζ=const. It follows from (5) that in the
adiabatic approximation the solution of the perturbed
equation (3) can be searched in the form
A = e−i(ω(k)+V (τ)
2/2)τ+iV (τ)ζη dn (η(ζ −X(τ)), k) (7)
where V (τ) = dX(τ)/dτ is the time-dependent velocity
of the soliton and X(τ) is the coordinate of the soliton
center. Substituting (7) in (6) and taking into account
the parity of the functions in the integrand as well as the
fact that all of them are periodic with the same period
L, we obtain the equation for the soliton coordinate
= −κC(k)f(τ) sin(κX) . (8)
C(k) =
2πE(k)
cos(θ)dn2
dθ (9)
and it is taken into account that N = 2ηE(k), where
E(k) is the complete elliptic integral of the second kind.
Depending on the choice of the function f(τ), Eq. (8)
describes different dynamical regimes. Now we are inter-
ested in acceleration which occurs during the rotational
movement of the soliton in the trap (i.e. X is a growing
function). We illustrate this acceleration using an exam-
ple of the simplest step-like dependence f(τ). To this
end we assume that initially the soliton is centered at
X(0) = 0 and require f(τ) to be a constant f0 for time
intervals such that the soliton coordinates X(τ) ∈ Ip and
to be zero for X(τ) /∈ Ip where the intervals Ip are given
by Ip = [(p +
)L, (p + 1)L] with p = 0, 1, .... Then, as
it is clear, the acceleration of the soliton, which is given
by the right hand side of (8), is positive for all times.
The above requirement introduces natural splitting of
the temporal axis in the set of intervals Tl = [τl, τl+1]
(l = 0, 1, ...), with τ0 = 0, such that f(τ) = 0 for τ ∈ T2p
and f(τ) = f0 for τ ∈ T2p+1 (here X(τl) = lL/2). Thus,
our task is to find τl. This can be done by taking into ac-
count that during each of the ”odd” intervals T2p+1 Eq.
(8) describes conservative nonlinear oscillator, the solu-
tion for which is well known. During ”even” intervals T2p
the motion is free (with a constant velocity) what means
FIG. 1: (Color online) The soliton velocity vs time (panels
a,b) for the parameters k = 0.99999, L = 10.0, f0 = 0.3,
n = 0.43 (panel a, the non-quantized velocity), n = 1.0 (panel
b, the quantized velocity), and the forms of the soliton (panels
c,d,e) in the instants of time indicated in the panel a. In
the panels a,b solid and dashed lines represent the velocity
numerically computed from Eq.(3), and Eq.(8), respectively.
that the time T2p necessary for the soliton to cross an
interval [pL, (p+ 1
)L] is
T2p = τ2p+1 − τ2p = L/(2v2p), (10)
where v2p is the velocity in the point pL. During the
time interval T2p+1 the soliton has to cross the interval
Ip. From this condition we obtain:
T2p+1 = τ2p+2 − τ2p+1 =
2E0/(H2p+1 + E0)
H2p+1 + E0
where H2p+1 = v
2p+1/2 + E0 is the energy of the soli-
ton in the point (p + 1/2)L, E0 = C(k)f0, and v2p+1 =
v2p is the soliton velocity in the same point. At the
end of the interval T2p+1 the soliton velocity is given
by v2(p+1) =
2(H2p+1 + E0). Thus one computes
that after p rotations the soliton acquires the velocity
v2(p+1), which can be obtained from the recurrent rela-
tion: v2(p+1) =
v22p + 4C(k)f0.
In Fig.1 a,b we compare the solution, obtained from
the perturbation theory, Eq.(8), with numerical simula-
tion of Eq.(3) for f0 = 0.3. Nevertheless during the nu-
merical simulation we used the values for T2p and T2p+1
(Eqs.(10)–(11)) obtained for the case of adiabatic approx-
imation. It follows form the results presented that the
dashed and solid lines perfectly match until τ ≈ 50.0. At
larger times appreciable discrepancy appears. It occurs
due to failure of the adiabatic approximation and can be
removed by introducing temporal corrections to T2p and
T2p+1. This naturally leads to an optimization problem,
which requires numerical approach and goes beyond the
scope of the present work. Finally we notice, that for the
above example of 7Li condensate the obtained accelera-
tion is 0.36mm/s2.
Comparison of the panels a and b in Fig.1 shows that
for k ≈ 1 quantization of the velocity is not important,
what is also confirmed by the evolution of the solitonic
forms depicted in the panels Fig.1 c-e.
IV. STOCHASTIC ACCELERATION OF
MATTER SOLITONS
Now we concentrate on another dynamical regime – on
the stochastic acceleration – where increase of the veloc-
ity of a matter soliton in a toroidal trap is achieved by
applying a fluctuating external field. To this end hold-
ing all conditions of the applicability of the model (3),
we consider the case of a stochastic force f(τ), which
is delta-correlated Gaussian process with characteristics
〈f(τ)〉 = 0 and 〈f(τ)f(τ ′)〉 = Dδ(τ − τ ′) (the angular
brackets stand for the stochastic averaging and D is the
dispersion). Now the dynamics can be described in terms
of the distribution function
P(V,Φ, τ) = 〈δ(Φ− Φ(τ))δ(V − V (τ))〉, (12)
where Φ(τ) ≡ κX is the angular coordinate of the soli-
ton, Φ(τ) and V (τ) with explicit time dependence stand
for the soliton coordinates obtained from the dynamical
equations while Φ and V are considered as independent
variables. The distribution function solves the Fokker-
Planck equation, which is obtained by the standard pro-
cedure (see e.g. [17]):
= −V ∂P
+ D̃ sin2(Φ)
. (13)
Here D̃ = κ4C2(k)D is the diffusion coefficient. Due to
the circular geometry of the trap Eq. (13) is considered
on the interval −π < Φ < π with the periodic boundary
conditions P(V,Φ − π, τ) = P(V,Φ + π, τ) with respect
to Φ and zero boundary conditions with respect V : P →
0 as V → ±∞. The normalization condition for the
probability density reads:
dΦP = 1.
Multiplying Eq.(13) by V and Φ and integrating over
V and Φ one readily obtains that the averaged velocity
and angular position of the soliton are constants, which
for the sake of simplicity will be considered zeros, i.e.
〈V 〉 = 0 and 〈Φ〉 = 0. Next, multiplying (13) by V 2, Φ2
and V Φ and performing the integration one obtains the
equations of the second momenta. They are not closed
and can be written down as follows:
〈V 2〉 = 2D̃〈sin2 Φ〉 , (14)
〈Φ2〉 = 2〈V Φ〉 , (15)
〈V Φ〉 = −2π
P (π, V, τ)V 2dV + 〈V 2〉 .(16)
Eq. (14) means that the average square velocity is grow-
ing with time, i.e. the soliton undergoes the stochastic
acceleration. The law of the growth of the velocity in-
variance deviates form the linear, as it would happen for
FIG. 2: (Color online) The mean square velocity (panel a) and
the stochastic acceleration γ̃ (panel b) of the soliton vs time
for different values of the dispersion, obtained by numerical
integration of Eq. (13) with parameters L = 10.0 and k =
0.99999. In panel b dashed lines depict the approximation of
numerical data by the law γ̃ ∝ τ−1/2. All axes in panels a,b
are represented in logarithmic scale.
the Brownian diffusion in the momentum space, what
happens because the diffusion coefficient in the Fokker-
Planck equation (13) is not a constant, but depends on
the angular variable. However, due to the diffusion one
can expect that the phase distribution will tend to ho-
mogeneous, i.e. that P → 1/(2π) as τ → ∞. In this
formal limit one obtains that 〈V Φ〉 → 0, 〈sin2 Φ〉 → 1/2
and hence 〈V 2〉 → D̃τ . In other words, the system (14)-
(16), describes random walk which in the limit of large
time, approaches the Brownian diffusion in the velocity
space. In that limit the stochastic acceleration, which
can be defined as γ̃ = d
〈V 2〉/dτ , would tend to zero
according to the law γ̃ ∝ τ−1/2.
In order to check the above predictions and reveal other
features of the stochastic dynamics of a soliton in a ring
trap we solved numerically Eq. (13) subject to the ini-
tial condition P(V,Φ, 0) = 〈δ(Φ)δ(V )〉. The results is
summarized in Fig. 2. In the panel a one observes the
predicted monotonic growth of the mean velocity with
time, which slightly different from the linear law. In the
panel b one can see that the stochastic acceleration γ̃
is a monotonically decreasing function, which at suffi-
ciently large times tends to zero. In particular, at τ >∼ 15
the decreasing of the acceleration with time can be well
approximated by the predicted law γ̃ ∝ τ−1/2, as it is
shown by dashed curves in the panel b of Fig. 2 (it was
verified that in at the same times 〈sin2 Φ〉 ≈ 1/2, what
is in agreement with the analytical predictions). Also
Fig. 2 b shows that the stochastic acceleration is larger
for larger D. The physical explanation of this last fact
is simple: the acceleration is generated by the stochastic
force, whose intensity is determined by the dispersion D.
V. LOCALIZATION OF MATTER INDUCED BY
THE EXTERNAL FIELD
Let us now turn to localized states of a matter in a
toroidal trap and concentrate on the states generated by
the constant external electric field, i.e. by f(τ) ≡ f0.
Respectively, we look for stationary solutions of Eq. (3)
in the form A = e−iωτA(ζ) and obtain for A(ζ) the equa-
tion:
− f0 cos(κζ)A + σ|A|2A = ωA (17)
which is subject to periodic boundary conditions
A(ζ, τ) = A(ζ + L, τ).
Several lowest branches of the numerically obtained
solutions of Eq. (17) are shown in Fig. 3. The lowest
branch approaches zero at the frequency ω0 ≈ −0.143
(it is interesting to mention that this frequency coincides
with the lowest gap edge of the spectrum of the Mathieu
equation (17) considered on the whole axis), where the
amplitude of the nonlinear periodic mode is small and
it transforms into the linear periodic Bloch mode at the
lowest gap edge. Such a behavior of the branch is sim-
ilar to that of the strongly localized modes in a BECs
in the optical lattice [18]. The lowest mode A is local-
ized in the vicinity of ϕ = 0 (Fig.3, b), i.e. around the
minimum of the effective potential and that is why such
a mode is stable and can exists even in the linear case,
where the two-body interactions are negligible (here it
is important that we are dealing with periodic boundary
conditions). The modes of the upper branches – B and
D (their typical examples are shown in Fig.3 b) – bifur-
cate at ω∗ ≈ −0.345. They are localized at ϕ = ±π, i.e.
near the points where the potential has maxima. As it
is clear, this is pure nonlinear effect and occurs due to
delicate balance between the attractive interactions and
repulsive forces of the external potential. Such balance
can easily be destroyed even by an infinitesimal pertur-
bation, what allows us to expect instability of the modes.
The mode C represents two local maxima of the atomic
distributions at ϕ = 0, π. Similar to the modes B and D
we one can expect it to be unstable, what physically can
be explained by existence of local atomic maxima at the
maxima of the potential. By direct numerical solution of
Eq. (3) (more specifically by perturbing the mode pro-
files by the factor 1 + 0.1 cos (21 ζ) and computing the
dynamics until τ = 1000) we have verified that, indeed,
only the mode A on Fig. 3 is dynamically stable, while
the modes B, C, and D are unstable.
VI. MATTER SOLITON AS A KAPITZA
PENDULUM
As the final example of nontrivial dynamics of a mat-
ter soliton in a toroidal trap we consider dynamical lo-
calization induced by a rapidly oscillating force f(τ) =
f0 [ν + cos(Ωτ)]. In this case the solitonic motion mimics
FIG. 3: (Color online) The number of bosons N (for the ex-
ample of the lithium condensate described in the text) vs
frequency ω (panel a) and shapes of the localized modes at
ω = −0.4 (panel b) for the case where L = 10.0, f0 = 0.3,
σ = −1.
the famous Kapitza pendulum, which acquires an addi-
tional stable point due rapid oscillation of the pivot [14].
Assuming that the physical conditions of the validity of
the quasi-1D approximation (3) holds and that the fre-
quency Ω is large enough, i.e. Ω2 ≫ κ2C(k)f0, one can
perform the standard analysis (see e.g. [14]), i.e. look for
a solution of (3) in a form X(τ) + ξ(τ) where ξ is small,
|ξ| ≪ |X |, and rapidly varying, and provide averaging
over rapid oscillations. Then one arrives at the equation
d2X/dτ2 = −∂U/∂X with the effective potential
U = −C(k)f0
ν cos(κX) + κ2C(k)
cos(2κX)
.(18)
If the condition κ2C(k)f0/(2Ω
2ν) > 1 is met, the ef-
fective potential U possesses two stable points: X = 0
(Φ = 0) and X = L/2 (Φ = π). So, it opens the pos-
sibility for the new type of soliton moving around the
new stable point. Two typical trajectories of the soli-
ton, obtained by numerical integration of Eq. (3), are
presented in Fig.4. One of the trajectories displays oscil-
lations around the new equilibrium point, while the other
one shows the large oscillations around the equilibrium
point Φ = 2π started with the same initial data but in the
case where Φ = π is not an equilibrium any more. The
amplitude of large oscillations decay with time because
of energy losses of the soliton in the nonconservative sys-
VII. CONCLUSIONS
In the present paper we have shown that dynamics of
a matter soliton in a toroidal trap, well reproducing one-
dimensional geometry, can be very efficiently governed by
time varying external electric field. In particular, such
regimes like dynamical acceleration, stochastic acceler-
ation, localization and implementation of the Kapitza
pendulum can be realized by proper choices of the time
dependence of the external force.
Experimental detection of the acceleration can be im-
plemented either by direct imaging of the atomic cloud,
FIG. 4: (Color online) The angular coordinate of the soliton
center vs time for the soliton motion affected by the rapidly
oscillating external force, obtained numerically from Eq.(3)
with parameters L = 10.0, n = 0.01, σ = −1, f0 = 0.15,
Ω = 2.0, and k = 0.99999.
which is well localized in space and has well specified tra-
jectory, or by measurement of the atomic distribution in
the momentum space displaying shift of the maximum
towards higher kinetic energies. Alternatively, one can
study the evolution of the atomic cloud releasing from
the trap (by switching the trap off) after some period of
accelerating motion. The respective dynamics will be a
spreading out cloud whose center of mass is moving with
the acquired velocity.
The obtained results were based on the one-
dimensional model, although deduced using the multiple-
scale method and thus mathematically controllable. This
means that a number of problem, are still left open. One
of them is the limitation on the soliton velocity, and thus
acceleration, introduced by lowering the space dimension,
which appears when the solitonic kinetic energy becomes
comparable with the transverse kinetic energy. Another
limitation on the soliton acceleration emerges from the
fact of the velocity quantization, when the radius of the
ring trap is not large enough. We also left for further
studies the diversity of localized stationary atomic distri-
butions supported by the external filed, indicating only
the lowest modes. We thus believe that the richness of
the phenomena which can be observed by simple com-
bination of the trap geometry and varying external field
will stimulate new experimental and theoretical studies.
Acknowledgments
The authors are indebted to H. Michinel for providing
additional data on Ref. [11]. YVB was supported by the
FCT grant SFRH/PD/20292/2004. VVK was supported
by the Secretaria de Stado de Universidades e Investi-
gaci?n (Spain) under the grant SAB2005-0195. The work
was supported by the FCT (Portugal) and European pro-
gram FEDER under the grant POCI/FIS/56237/2004.
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http://arxiv.org/abs/cond-mat/0610047
