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Europhysics Letters PREPRINT
Shape oscillation of a rotating Bose-Einstein condensate
Sabine Stock, Vincent Bretin, Fr´
ed´
eric Chevy and Jean Dalibard
Laboratoire Kastler Brossel(∗), 24 rue Lhomond, F-75231 Paris Cedex 05, France
PACS. 03.75.Fi – Phase coherent atomic ensembles; quantum condensation phenomena.
PACS. 32.80.Pj – Optical cooling of atoms; trapping.
Abstract. – We present a theoretical and experimental analysis of the transverse monopole
mode of a fast rotating Bose-Einstein condensate. The condensate’s rotation frequency is
similar to the trapping frequency and the effective confinement is only ensured by a weak quartic
potential. We show that the non-harmonic character of the potential has a clear influence on
the mode frequency, thus making the monopole mode a precise tool for the investigation of the
fast rotation regime.
The investigation of rotating gases or liquids is a central issue in the study of superflu-
ids [1,2]. During the recent years, several experiments using rotating atomic Bose-Einstein con-
densates have provided a spectacular illustration of the notion of quantized vortices [3,4,5, 6].
Depending on the rotation frequency of the gas, a single vortex or a regular array of vor-
tices can be observed experimentally. When the rotation frequency is increased to a very
large value, a new class of phenomena is predicted, in connection with quantum Hall physics
[7, 8, 9, 10, 11, 12, 13, 14 ]. For a gas confined in a harmonic potential, the fast rotation domain
corresponds to stirring frequencies Ω of the order of the trapping frequency ω⊥in the plane
perpendicular to the rotation axis (hereafter denoted z). From a classical point of view, the
transverse trapping and centrifugal forces compensate each other for this stirring frequency,
and the motion of the particles in the xy plane is only driven by Coriolis and interatomic
forces. This situation is similar to that of an electron gas in a magnetic field, since Lorentz
and Coriolis forces have the same mathematical structure.
In order to approach the regime of fast rotation two paths are currently being explored.
The first approach is implemented in a pure harmonic potential and is based on evaporative
spin-up, i.e. the selective removal of particles with low angular momentum [15]. The stirring
frequency Ω can then be raised close to ω⊥(Ω = 0.993 ω⊥was reached in [16]). The second
approach, which is followed here, consists in adding to the quadratic confinement a small
positive quartic potential, which ensures that the particles will remain confined even when Ω
exceeds ω⊥[17,18,19,20,21,22]. We have recently proven that this method can be successfully
implemented and we have mechanically stirred a rubidium Bose-Einstein condensate up to
Ω≃1.05 ω⊥[23].
All experiments performed so far (including the present one) are still deeply within the
mean field regime, characterized by a number of vortices Nvwell below the number of particles
(∗) LKB is a unit´e de recherche of Ecole Normale Sup´erieure and Universit´e Paris 6, associated to the CNRS.
c
EDP Sciences
2EUROPHYSICS LETTERS
N. Nevertheless in order to prepare for future investigations of possible quantum-Hall-like
states, one must design proper tools of investigation, such as the study of the eigenmodes
of the rotating gas. Possible examples are the transverse quadrupole modes, which allow
for a measurement of the zcomponent of the angular momentum [23], and the Tkachenko
oscillations of the vortex lattice, recently observed in [15]. Here we report on the experimental
and theoretical study of the lowest transverse monopole mode of an ultra-cold gas of rubidium
atoms in the fast rotation regime.
In a trap with axial symmetry along the z-axis, excitations can be characterized by their
angular momentum along z(quantum number mz) [24]. Here we are interested in excitations
carrying no angular momentum (mz= 0). In the limit ωz≪ω⊥which is of interest here,
it is possible to identify mz= 0 “transverse monopole” modes, which mostly affect the atom
distribution in the xy plane. For a non-rotating condensate the lowest frequency for these
m= 0 transverse modes is ωmp ≃2ω⊥(transverse breathing mode) [25, 26]. In the present
work we show experimentally that the relation ωmp ≃2ω⊥remains valid for large rotation
frequencies (as predicted by [27]), as long as the quartic term in the confinement is not
significant. We also explore the region Ω ∼ω⊥, where the quartic term plays an essential role.
We compare the measured monopole frequency with that derived from a simple hydrodynamic
model of an infinite, cylindrical condensate. Finally we discuss the structure of the mode,
which exhibits a very particular behavior in the range Ω &ω⊥.
Our 87Rb Bose-Einstein condensate contains 3 ×105atoms. It is produced by radio-
frequency evaporation in a Ioffe-Pritchard magnetic trap, to which we superimpose a far-blue
detuned laser beam. The magnetic trap provides a harmonic confinement with cylindrical
symmetry along the zaxis, with ωz/2π= 11.0 Hz and ω(0)
⊥/2π= 75.5 Hz. The laser beam
adds a negative quadratic and a positive quartic potential in the xy plane. The quadratic
term decreases the trap frequency ω⊥by ∼15%. The quartic term allows to explore rotation
frequencies around and slightly above the trapping frequency ω⊥. It reads kr4/4, with r2=
x2+y2and k= 2.6(3) ×10−11 J/m4. The oscillation frequency of the condensate center-
of-mass (dipole motion) in the xy plane is ωdp /2π= 65.6 (±0.3) Hz. Because of the quartic
component of the trapping potential, ωdp is slightly larger than ω⊥, even for an arbitrarily
small amplitude of the dipole motion. The condensate has indeed a finite radius Rc≃6.5µm
and therefore explores regions of space where the contribution of the quartic term is significant.
Using a perturbative treatment of the quartic term we infer that ωdp −ω⊥≃2kR2/(7 mω2
⊥),
so that ω⊥/(2π) = 64.8 (±0.3) Hz.
The procedure for setting the gas in rotation at a frequency Ω ∼ω⊥has been described
in detail in [23]. We apply an additional laser beam creating a rotating anisotropy in the xy
plane for two successive phases, with respective durations of 300 ms and 600 ms. During the
first phase the rotation at frequency Ω ∼ω⊥/√2 resonantly excites the transverse quadrupole
mode, and vortices subsequently penetrate the condensate. After a 400 ms relaxation period
we apply the second stirring phase at the final desired frequency. This phase is followed by
a 500 ms relaxation period. By measuring the transverse size of the condensate, we have
checked that it is then rotating at a frequency close to the stirrer frequency Ω (within 2%) for
the regime of interest Ω ≤1.05 ω⊥. In particular, for Ω .ω⊥we observe large vortex arrays
involving up to ∼50 vortices. During the stirring and relaxation periods, we apply radio
frequency (r.f.) evaporative cooling. The r.f. is set 24 kHz above the value which removes
all particles from the trap. Assuming Ω = ω⊥, so that the transverse confinement is purely
quartic, this gives a trap depth U0∼40 nK, hence a temperature T.10 nK.
We then excite the transverse monopole mode(s) of the gas by changing for a period τ0the
intensity of the laser creating the quartic potential from Ito I′. We choose τ0= 2 ms, which
S. Stock et al.:Shape oscillation of a rotating Bose-Einstein condensate 3
27 ms25 ms 26 ms 28 ms 29 ms 30 ms 31 ms 32 ms
(a)
(b)
(c)
Fig. 1 – Series of images of the condensate for various waiting times τafter excitation. Row a)
corresponds to a rotation frequency Ω/(2π) = 62 Hz, row b) to Ω/(2π) = 64 Hz and row c) to
Ω/(2π) = 66 Hz.
is short compared to the oscillation period of the monopole oscillation 2π/ωmp = 7.5 ms.
We then wait for an adjustable duration τbefore performing a time-of-flight expansion and
absorption imaging. The images are taken along the rotation axis zso that we have access to
the column density in the xy plane.
Typical results are shown in figure 1 for Ω/2π= 62, 64, 66 Hz. Each image corresponds to
a destructive measurement of the atom density after a duration τ= 25,...,32 ms and a 18 ms
time-of-flight. In order to check that the oscillation is in the linear regime, we explored values
between 0 and 4 for the ratio I′/I: we found that the frequency ω0is (within experimental
uncertainties) independent of the excitation strength |I−I′|/I, and that the amplitude of the
oscillation varies linearly with this strength. We have also varied the duration of the time-of-
flight between 6 and 18 ms. We thus verify that all features reported here are independent of
this duration, which corresponds to a mere scaling of the initial spatial density distribution.
We have taken similar sequences of images for various stirring frequencies Ω, with τvarying
from 0 to 40 ms by steps of 1 ms. We extract from these images the transverse size R(τ) of
the gas. The variations of R(τ) are well fitted by a single sinusoidal function, with frequency
ωmp. The characteristic lifetime of the excitation is larger than 100 ms, so that its decay is
negligible during the 40 ms period. The variations of ωmp as a function of Ω are plotted in
figure 2. An obvious feature of this plot is that ωmp varies very weakly for Ω except in the
vicinity of ω⊥. We now discuss in more details three relevant domains of this graph.
(1) For a non rotating gas (Ω = 0) we find ωmp/(2π) = 130.6 (±1.5) Hz, which is close to
the well known result ωmp ≃2ω⊥= 2π×129.6 (±0.6) Hz [28].
(2) When the gas rotates at a frequency Ω not too close to ω⊥, the effect of the quartic
potential remains small. In these conditions we find that the monopole frequency ω0stays
approximately constant. At first sight this may seem a surprising result. For a rotating gas
in a pure harmonic potential, the transverse confinement is reduced due to the centrifugal
force. In the calculation of the equilibrium shape of the gas, the trapping frequency ω⊥is
thus replaced by the weaker value (ω2
⊥−Ω2)1/2. One might have expected that a similar
replacement should be done also for the monopole frequency; this is clearly not the case. For
4EUROPHYSICS LETTERS
ωmp(Ω) / ωmp(Ω = 0)
Stirring Frequency Ω/2π [Hz]
1.00
1.10
1.08
1.06
1.04
1.02
010 70
6050403020
Fig. 2 – Transverse monopole frequency ωmp (Ω) normalized by ωmp(Ω = 0) as a function of the
stirring frequency Ω. For a non rotating condensate (Ω = 0) we measure ωmp(0)/(2π) = 130.6 Hz.
The vertical bars indicate the error given by the fitting routine.
a cigar-shaped condensate in the hydrodynamic regime, it was proven in [27] that ω0stays
equal to 2ω⊥when Ω varies. In the case of an ideal gas described by classical mechanics,
the equality ωmp = 2 ω⊥also holds for any Ω as a consequence of the combined action of
centrifugal and Coriolis forces.
(3) When the rotation frequency Ω approaches ω⊥, the contribution of the quartic term
becomes more important and the monopole frequency ωmp deviates significantly from 2ω⊥.
As shown in figure 2, this deviation reaches ∼8 % for Ω/(2π) = 68 Hz, i.e. Ω/ω⊥≃1.05. We
have also plotted in figure 2 the prediction of a theoretical treatment of the monopole mode
in the quartic + quadratic trap, neglecting the harmonic confinement along the z-direction.
The agreement between these predictions and the experimental data is satisfactory, given the
difference in geometry between the experiment and the model.
The starting point of our theoretical treatment is the set of equations for rotational hy-
drodynamics [27], which we write in the laboratory frame:
∂tρ=−∇·(ρv) (1)
∂tv=−∇((U+gρ)/m +v2/2) + v×(∇×v),(2)
where ρand vare the density and velocity field of the atom distribution. U=mω2
⊥r2/2+kr4/4
stands for the trapping potential and g= 4π~2a/m characterizes the strength of the atomic
interactions (ais the scattering length). As explained in [29, 27, 30] these equations are valid
when one is interested in a phenomenon whose characteristic length scale is larger than the
distance between vortices.
The eigenmodes are obtained by linearizing (1-2) around the rotating equilibrium solution
veq =Ω×rgρeq =µ−U−Ucen ,(3)
where µis the chemical potential and Ucen =−mΩ2r2/2 the centrifugal potential. We elim-
inate the velocity field to get a closed equation for the density variation δρ, assuming an
oscillation at frequency ω:
−∇·(ρeq ∇[δρ]) = m
g(ω2−4Ω2) [δρ].(4)
S. Stock et al.:Shape oscillation of a rotating Bose-Einstein condensate 5
In absence of a quartic term the eigenfrequencies for the m= 0 modes are
ω2
n= 4Ω2+ 2n(n+ 1)(ω2
⊥−Ω2), n positive integer, (5)
and the corresponding eigenmodes are polynomials of degree nwith respect to r2. For n= 1,
we recover in particular ω1≡ωmp = 2 ω⊥.
When Ω approaches ω⊥, all transverse modes m= 0 become degenerate. Such a macro-
scopic degeneracy is reminiscent of the degeneracy of the energy levels of a single particle in a
uniform magnetic field, leading to the well known Landau level structure [31]. An equivalent
degeneracy occurs, still at the single particle level, for an isotropic 2D harmonic oscillator of
frequency ω⊥, considered in a frame rotating at frequency Ω = ω⊥[32]. However we empha-
size that our result here holds not for a single particle spectrum, but for the eigenmodes of a
N−body system treated in the mean-field approximation. The occurrence of such a macro-
scopic degeneracy raises interesting questions concerning the linear response of the rotating
system to an arbitrary excitation.
When the quartic potential is present, the above degeneracy is lifted. We set x=r/R in
(4) with R= (4µ/k)1/4, to get the dimensionless eigenvalue equation for the m= 0 modes:
−1
x
d
dx x(1 −ǫx2−x4)d[δρ]
dx = Λ(ǫ) [δρ] where ǫ=mR2
2µ(ω2
⊥−Ω2).(6)
For each Ω, i.e. for each ǫ, we are interested in the lowest eigenvalue Λ0(ǫ) of the hermitian
operator in the left hand side of (6). We can then deduce the frequency of the lowest transverse
monopole mode which is plotted as a continuous line in figure 2:
ω2
mp = 4Ω2+µΛ0(ǫ)
mR2.(7)
The quantity Λ0(ǫ) is an increasing function of ǫwhich we calculate numerically using a
variational method with polynomial trial functions. Just at the critical rotation Ω = ω⊥,
ǫ= 0 and we get Λ(0) ≃11.5. The slow rotation limit corresponds to ǫ≫1, in which case
Λ(ǫ)≃8ǫ; we then recover the result ωmp = 2ω⊥.
This analysis also explains the periodic apparition of a hole at the center of the density
profile for Ω ∼ω⊥, as observed in figure 1. The stationary density profile (3) varies as r4
around the origin, whereas the mode δρ varies as r2in this region. The curvature of the
density profile is therefore dominated by the phase of the perturbation.
For Ω ≥ω⊥and a relatively strong excitation (I′/I = 4), we observe that the time
evolution of the mode structure becomes asymmetric (see figure 3; this asymmetry is also
slightly visible in the last row of figure 1). It consists of a periodic entering positive density
wave, which starts on the edge of the condensate and gradually moves to the center in a time
period ∼π/ω⊥. A possible interpretation of this unusual structure is that two (or more)
transverse modes m= 0 are simultaneously excited with a non-zero relative phase. Since
these modes have similar frequencies (cf. Eq. (5)) the initial phase difference between them
stays nearly constant, and this can give rise to the observed phenomenon.
To summarize we have presented in this paper a detailed study of the transverse monopole
mode of a fast rotating degenerate Bose gas. We have shown that the non-harmonic character
of the potential (which is essential for the confinement of the gas) has a clear influence on
the mode frequency. We have also given a simple analysis of this mode for an infinitely long
condensate which is in good agreement with the experimental data.
Theoretical studies have shown that the addition of a quartic term in the harmonic po-
tential may lead to the formation of a “giant vortex”, i.e. a vortex with a circulation larger
6EUROPHYSICS LETTERS
26 ms 27 ms 28 ms 29 ms 30 ms 31 ms 32 ms25 ms
Fig. 3 – Time evolution of the density profile of the oscillating cloud, for Ω/2π= 68 Hz.
than the single quantum h/m [18, 19, 20, 21, 22, 33]. The large hole appearing at the center of
the condensate during the monopole oscillation when Ω &ω⊥should not be confused with
such a giant vortex. We are dealing here with a transient state of the condensate, while the
predicted giant vortex state is a stationary state of the system, for an appropriate angular
momentum. Another example for a transient large core in a condensate is provided by an
experiment recently performed in Boulder, in which a hole pierced in a rotating condensate
confined in a purely quadratic trap was shown to persist for a long time [34].
We now briefly discuss some perspectives opened by this work. First we recall that the
nature of the gas when the rotation frequency Ω is around or above ω⊥is still unclear. The
detailed study of [23] showed that the number of visible vortices is not sufficient to account for
the measured rotation frequency of the gas. Two classes of explanation have been proposed to
account for this finding. Either the fast rotating gas cannot be described anymore by a single
macroscopic wave function [35], or the vortices are still present but they are distorted and do
not show up clearly in the images of the condensate. We hope that the present experimental
measurement of the transverse mode frequency can be used to discriminate between these two
hypotheses. In principle a generalization of the above theoretical treatment to a condensate
in a 3D trap is possible, and the quantitative predictions of such a model could be compared
with the experimental findings.
From the experimental point of view a natural extension of the present work is to switch to
a two-dimension geometry, using a strong confining potential along the rotation axis. Vortices
then become point objects and the predicted properties of the system depend on the ratio
between the atom number Nand the number of vortices Nv. When Nis large, the Bose-
Einstein condensate presents a regular vortex array, as already observed in [16]. The array
is expected to melt when Ndecreases to a value of the order of Nv[36] (see also [11]). For
N < Nvthe quantum Hall regime for particles should emerge. It is an interesting problem
to determine the signature of these various regimes on the eigenmodes of the system. For the
monopole mode considered here and for a pure harmonic confinement, it has been predicted
in [37, 38] that the frequency ωmp of the breathing motion of a 2D gas remains strictly equal
to 2 ω⊥for any equilibrium state. This does not hold anymore in presence of the quartic term,
and thus the deviation of ωmp from 2 ω⊥can in principle be used to monitor the emergence
of new quantum phases in the rotating gas.
∗∗∗
We thank S. Stringari and the ENS group for useful discussions. S. Stock acknowledges
support from the European Union (CQG network HPRN-CT-2000-00125), the Studienstiftung
des deutschen Volkes and the DAAD. This work is partially supported by CNRS, Coll`ege de
France, R´egion Ile de France, and DRED.
S. Stock et al.:Shape oscillation of a rotating Bose-Einstein condensate 7
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