Effects of atomic interactions on Quantum Accelerator Modes
Effects of atomic interactions on Quantum Accelerator Modes.
Laura Rebuzzini,1, 2, ∗ Roberto Artuso,1, 3, 4 Shmuel Fishman,5 and Italo Guarneri1, 2, 3
Center for Nonlinear and Complex Systems and Dipartimento di Fisica e Matematica,
Università dell’Insubria, Via Valleggio 11, 22100 Como, Italy.
Istituto Nazionale di Fisica Nucleare, Sezione di Pavia, Via Ugo Bassi 6, 27100 Pavia, Italy.
Istituto Nazionale di Fisica della Materia, Unità di Como, Via Valleggio 11, 22100 Como, Italy.
Istituto Nazionale di Fisica Nucleare, Sezione di Milano, Via Celoria 16, 20133 Milano, Italy.
Physics Department, Technion, Haifa 32000, Israel.
We consider the influence of the inclusion of interatomic interactions on the δ-kicked accelerator
model. Our analysis concerns in particular quantum accelerator modes, namely quantum ballistic
transport near quantal resonances. The atomic interaction is modelled by a Gross-Pitaevskii cubic
nonlinearity, and we address both attractive (focusing) and repulsive (defocusing) cases. The most
remarkable effect is enhancement or damping of the accelerator modes, depending on the sign of
the nonlinear parameter. We provide arguments showing that the effect persists beyond mean-field
description, and lies within the experimentally accessible parameter range.
PACS numbers: 05.45.Mt, 03.75.-b, 42.50.Vk
Keywords:
Quantum Accelerator Modes (QAMs) are a manifesta-
tion of a novel type of quantum ballistic transport (in mo-
mentum), that has been recently observed in cold atom
optics [1]. In these experiments, ensembles of about 107
cold alkali atoms are cooled in a magnetic-optical trap
to a temperature of a few microkelvin. After releasing
the cloud, the atoms are subjected to the joint action of
the gravity acceleration and a pulsed potential periodic
in space, generated by a standing electromagnetic wave,
far-detuned from any atomic transitions. The external
optical potential is switched on periodically in time and
the period is much longer than the duration of each pulse.
For values of the pulse period near to a resonant integer
multiple of half of a characteristic time TB (the Talbot
time [2]), typical of the kind of atoms used, a consider-
able fraction of the atoms undergo a constant accelera-
tion with respect to the main cloud, which falls freely
under gravity and spreads diffusively.
The non-interacting model is a variant of the well-
known quantum kicked rotor (KR) [3], in which the ef-
fects of a static force, produced by the earth gravitational
field, are taken into account. The linear potential term
breaks invariance of the KR hamiltonian under space
translations. Such an invariance may be recovered by
moving to a temporal gauge, where momentum is mea-
sured w.r.t. the free fall: this transformation gets rid of
the linear term and the new hamiltonian, expressed in
dimensionless units, reads
Ĥ(t′) =
(p̂+ gt′)2 + k cos(x̂)
δ(t′ − tτ). (1)
where p̂ and x̂ are the momentum and position opera-
tor, k and τ are the strength and the temporal period of
the external kicking potential, g is the gravity accelera-
tion. The relationship between the rescaled parameters
and the physical ones, denoted by primes, is k = k′/~,
0.005
0.015
0.025
0.035
0.045
-500 0 500 1000
0.005
0.015
0.025
0.035
0.045
600 800 1000
0.005
0.015
0.025
-1000 0 1000 2000 3000
0.005
0.015
0.025
2200 2400 2600 2800 3000
FIG. 1: (Color online) The probability distribution at times
t = 25 (1st row) and 45 (2nd row). ((Red) line: lin-
ear case (u = 0), (purple) triangles/(green) circles: focus-
ing/defocusing nonlinearity (u = ∓1.25)). In the right col-
umn enlargements of mode are shown; the position of the
mode, predicted by (2) is marked by the (blue) vertical dot-
ted line.
τ = ~τ ′G2/M = 4πτ ′/TB, η = Mg
′τ ′/~G and g = η/τ ,
where η is the momentum gain over one period, G is
twice the angular wavenumber of the standing wave of
the driving potential and M is the mass of the atom.
Symmetry recovery allows to decompose the
wavepacket into a bundle of independent rotors
(whose space coordinate is topologically an angle): this
Bloch-Wannier fibration plays an important role in the
http://arxiv.org/abs/0704.1382v1
2000 2500 3000
2000 2500 3000
2000 2500 3000
FIG. 2: (Color online) The Husimi function of the QAM at
time t = 45, in the repulsive (a), linear (b) and attractive case
theory of QAMs [4].
QAMs appear when the time gap between kicks ap-
proaches a principal quantum resonance, i.e. τ = 2πl+ǫ,
with l integer and |ǫ| small. The key theoretical step
is that in this case the quantum propagator may be
viewed as the quantization of a classical map, with |ǫ|
playing the role of an effective Planck’s constant [4]:
QAMs are in correspondence with stable periodic orbits
of such pseudo-classical area-preserving map. We refer
the reader to the original papers for a full account of the
theory, we just mention a few remarkable points: stable
periodic orbits are labelled by their action winding num-
ber w = j/q, which determines the acceleration of the
QAM w.r.t. the center of mass distribution
. (2)
The modes are sensitive to the quasimomentum (Bloch
index induced by spatial periodicity), being enhanced at
specific, predictable values [4]; also the size of the elliptic
island around the pseudoclassical stable orbit plays an
important role (if the size is small compared to |ǫ| the
mode is not significant [4]).
We consider in this letter the role of atomic interactions
in such a system; namely evolution is determined by a
nonlinear Schrödinger equation with a cubic nonlinearity:
iψ̇(x, t′) =
Ĥ(t′) + u|ψ(x, t′)|2
ψ(x, t′), (3)
where u is the rescaled nonlinear parameter, whose sign
describes an attractive (negative)/repulsive (positive)
atomic interaction. We will come back to its connec-
tion with physical units in the end of the paper. The
0.005
0.015
0.025
-2.5 0 2.5
-2.5 0 2.5
0 0.5 1 1.5 2 2.5 3 3.5 4
FIG. 3: (Color online) Maximum height reached by the mode
at time t = 45 as a function of u in position (a) and momen-
tum (b) representation. In the inset the exponential decrease
of hmax for positive u in shown in a semi-logarithmic plot.
condensate wave function is normalized to unity. The
dynamics does not only acquire in this way a qualitative
novel form, but, due to the nonlinear term, Bloch de-
composition into independent rotors breaks down. The
main scope of this letter will be to numerically scrutinize
how QAMs are still present in the modified system, and
explore how nonlinearity modifies their features. In the
end we will briefly comment upon some stability issues,
by showing that a more refined description, including loss
of thermalized particles, does not destroy the scenario we
get from a mean field description.
Our analysis will be restricted to QAMs correspond-
ing to fixed points of period q = 1 of the pseudoclassical
map; the numerical analysis of nonlinear evolution has
been performed by using standard time-splitting spec-
tral methods [5]. There are several physical parameters
characterizing the system: g, τ , k and u. Here we mainly
address the role of nonlinearity u: we fix k = 1.4, l = 1,
ǫ = −1, τη ≃ 0.4173, and choose as the initial state
a symmetric coherent state centered in the stable fixed
point of the pseudoclassical map (x0 ≃ 0.3027, p0 = 0),
whose corresponding winding number is zero.
A quite remarkable feature appears when we com-
pare results for opposite nonlinearity signs (keeping the
strength |u| fixed), see fig.(1). As in the linear system, the
wave packet splits into two well-separated components:
the accelerator mode (whose acceleration is still compati-
ble with (2)) and the remaining part, which moves under
two competitive contributions, the free fall in the gravi-
tation field, and the recoil against the accelerating part.
Note that for the present choice of the parameters, the
former contribution is negligible compared to the second.
We remark some features, that are common to what
we observed for a choice of other parameter values: the
distribution around the accelerator mode is more peaked
and narrower in the presence of attractive nonlinearity;
the opposite happens in the case of a repulsive interac-
0.005
0.015
0.025
0.035
0.045
2 (b)
-3000 -2000 -1000 0 1000 2000 3000
-4 -3 -2 -1 0 1 2 3 4
-4 -3 -2 -1 0 1 2 3 4
FIG. 4: (Color online) The probability distribution at t = 45
for strong nonlinearity (u = ±3); the initial states (see text)
are shown in the insets (line and symbols as in fig.1).
tion. This can also be appreciated from a Husimi repre-
sentation of the modes (see fig.(2)).
While for repulsive interactions the spreading of the
distribution, together with peak damping, seems to de-
pend monotonically on the nonlinearity strength, the
attractive case exhibits more complicated features (see
fig.(3)). Enhancement of the accelerator mode is only
observed for small nonlinearities, while a striking feature
appears at larger values of |u|, namely the accelerator
mode is suppressed (see fig.(4a)). The intuitive expla-
nation of this result is that strong focusing nonlinearity
opposes to the separation of the wave packet into two
parts; indeed, in the case of exact resonance (namely
τ = 2π), the mode is absent, so the whole wave freely
falls without splitting and then the maximum height of
the wave, plotted vs u as in fig.(3a), is found to mono-
tonically increase to the left towards a saturation value.
While the behavior shown in fig.(3) has been observed
for a variety of other parameter choices, we mention that
more complex, strongly fluctuating behaviour was some-
times observed at large focusing nonlinearities. In all
such cases a bad correspondence between the quantum
and the pseudoclassical dynamics was also observed, al-
ready in the linear case.
We remark that the mode damping is sensitive to the
choice of the initial state, as shown in fig.(4). While
a gaussian initial wave packet leads to the mentioned
QAM suppression, we may tailor a QAM enhancing ini-
tial condition as follows: we take the quasimomentum
β0 that in the linear case dominates the mode (here
β0 = π/τ−η/2 ≃ 0.5551 [4]) and we drop from the initial
0 0.5 1
0 0.5 1
0 0.5 1
0 0.5 1
0 0.5 1
0 0.5 1
FIG. 5: (Color online) The quasi-momentum distribution
function (thick line) for |u| = 0.5 (a) and |u| = 3 (b) at
different times (1)t = 5, (2)t = 25 and (3)t = 45. Pur-
ple(dark)/green(light) lines refer to attractive/repulsive in-
teractions.
gaussian all components with |β − β0| > 0.15. As quasi-
momentum is the fractional part of momentum, this leads
to the comb like state of fig.(4b). Even through quasimo-
mentum is not conserved due to nonlinearity, the QAM
is strongly enhanced with respect to the linear case and
the recoiling part is almost cancelled.
Another way of looking at the nonlinear evolution with
techniques that are proper in the linear setting is to con-
sider the distribution function over quasimomenta, de-
fined by
f(β, t) =
|〈n+ β|ψ(t)〉|
. (4)
This distribution is stationary under linear evolution, its
shape being determined by the choice of the initial state.
We consider the evolution of a gaussian wave packet
(for which the linear f is essentially a constant - the hor-
izontal red line of fig.(5)), and probe the effect of nonlin-
earities of both signs. Typical results are as in fig.(5): the
effect of attractive (repulsive) nonlinearity is to enhance
(lower) the distribution around a value β̄ ≃ 0.4. No devi-
ation occurs for quasimomentum β0 (marked by vertical
lines), whose wave function, according to fig.(4b), closely
follows the linear pseudo-classical island. Again the β̄
peak of the focusing case is suppressed for large focusing
nonlinearities.
To make sure that our findings may be experimentally
significant we discuss some stability issues: the first con-
cerns decay properties of the QAMs. It is known that
linear modes decay due to quantum tunnelling out of
pseudoclassical islands [6]: we checked that, on the avail-
0 20 40
0 10 20 30
FIG. 6: (Color online) (a) Probability inside the island for
|u|=3 (symbols and line as in fig.(1)). (b) The mean number of
non-condensed particles vs the number of kicks, for u equal to
0.1, 0.5, 0.75, 1, 2, 5, 7 and 10 (starting from below); 12 terms
in the sum (5) are considered.
able time scale, the nonlinear decay behaves in a similar
way. In fig.(6a) the probability inside the classical is-
land is shown as a function of time for the initial state
of fig.(4b); it has been calculated integrating the Husimi
distribution of each β-rotor fiber over the island area and
summing the contributions of different rotors.
However in the condensate regime there is another pos-
sible mechanism that might completely modify the for-
mer picture, namely depletion of the condensate due to
proliferation of noncondensed, thermal particles. A stan-
dard technique to estimate the growth of the number of
thermal particles is provided by the formalism of Castin
and Dum [7], which has been employed in similar con-
texts in [8]. To the lowest order in the perturbation ex-
pansion and in the limit of zero temperature T → 0, the
number of non-condensed particles is given by:
〈δN̂(t)〉 =
〈vk(t)|vk(t)〉 (5)
where vk(t) is one of the mode functions of the system.
The modal functions (uk(t), vk(t)) are pairs of functions
that represent the time-dependent coefficients of the de-
composition, in terms of annihilation and creation op-
erators, of the equation of motion for the field operator
describing the thermal excitations above the condensate.
They describe the spatial dependence of these excitations
and propagate by modified Bogoliubov equations.
Our findings (see fig.(6b)) are consistent with a poly-
nomial growth of noncondensed particles, namely in our
parameter region (and within the time scale we typically
consider) no exponential instability takes place. This is
consistent with recent experimental work [9], where 87Rb
atom condensate has been used to explore QAMs. In [9],
a condensate of 50000 Rb atoms with repulsive interac-
tions is realized. In the case of a ”cigar shaped” trap, the
relationship between the number of atoms in the conden-
sate N and the effective 1-d nonlinear coupling constant
u is, in our units, N = ua2
/2a0 [10], where a0 is the
3-dimensional scattering length and a⊥ ≫ a0 is the ra-
dial extension of the wave function. Using the parameter
values of the experiment [9], one finds N ≃ 105 ·u and so
N ∼ 50000 corresponds to u ∼ 0.5. Therefore our range
of parameters includes the experimental accessible one.
We have investigated effects of atomic interactions, in
the form of a cubic nonlinearity, on the problem of quan-
tum accelerator modes: in particular we have charac-
terized the consequences of both attractive and repul-
sive interaction; we have also provided evidences that
the modes are not strongly unstable when reasonable pa-
rameters are chosen.
We thank G. S. Summy for providing us with details
of his work. This work has been partially supported by
the MIUR-PRIN 2005 project ”Transport properties of
classical and quantum systems”. S.F. acknowledges sup-
port by the Israel Science Foundation (ISF), by the US-
Israel Binational Science Foundation (BSF), by the Min-
erva Center of Nonlinear Physics of Complex Systems,
by the Shlomo Kaplansky academic chair. I.G. acknowl-
edges hospitality by the Institute of Theoretical Physics
at the Technion where part of this work was done.
∗ Electronic address: laura.rebuzzini@uninsubria.it
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mailto:laura.rebuzzini@uninsubria.it
