Observation exp´erimentale - Du sillage des insectes aux gaz de Fermi ultra-froids : dynamique

Dans l’h´elium superfluide, les modes de Kelvin ont pour la premi`ere fois

été observés par Packard en les excitant indirectement à l’aide d’un champ

radio-fréquenceviales électrons piégés au cœur des tourbillons. Dans le cas

de condensats de Bose-Einstein gazeux, l’excitation est toute aussi d´elicate

car elle n´ecessite de perturber le nuage sur des petites ´echelles de distance,

typiquement de l’ordre de la taille du cœur du vortex. Or, la m´ethode la

plus directe d’excitation d’un condensat est la modification des fr´equences

de piégeage qui se couple préférentiellement aux modes de grande longueur

d’onde, de l’ordre de la taille du nuage. Afin de contourner cette difficult´e,

nous avons décidé de procéder comme dans l’expérience de Packard à une

excitation indirecte des modes de Kelvin, non plus `a l’aide d’´electrons, mais

en utilisant un couplage non-lin´eaire entre le mode quadrupolaire

d’oscilla-tion de l’ellipticité - que nous avions déjà utilisé dans la mesure du moment

cin´etique du condensat [15] - et le mode de vibration de la ligne

tourbillo-naire. En effet, Beliaev a montr´e l’existence de processus non-lin´eaires que

l’on peut représenter schématiquement par une équation bilan

γ→γ

+γ

,

o`u γ, γ

et γ

sont trois quanta d’excitation du nuage satisfaisant les

rela-tion de conservarela-tion de l’´energie, de l’impulsion (dans l’espace libre) ou du

moment cin´etique (dans un pi`ege invariant par rotation). Si l’on fait le

pa-rallèle avec l’optique non linéaire, ce mécanisme est l’équivalent acoustique

du mélange non-linéaire induit par l’effet Pockels. Le mécanisme de Beliaev

étant responsable de l’amortissement du mode γ à basse température, on

remarque que plus le nombre de modes γ

et γ

accessibles est important

plus le taux d’amortissement augmente : un mode coupl´e aux kelvons verra

donc sa durée de vie réduite par rapport à un autre pour lequel d’éventuels

r`egles de s´election interdisent un tel couplage lorsque le nuage est mis en

rotation. Du fait de la chiralit´em

=−1 des modes de vibration d’une ligne

de vortex, une telle asym´etrie existe justement entre les deux modes

qua-drupolairesm

=±2. En effet la conservation du moment cin´etique indique

que la formations de deux kelvons ne peut proc´eder que du couplage avec

un mode de moment cin´etique m

=−2. Le mode quadrupolaire m

= +2

ne peut donc se désexciter dans ce canal et on s’attend par conséquent à ce

1.5. CONCLUSION 117

Fig. 1.3 – Mise en ´evidence exp´erimentale des excitations de Kelvin d’un

vortex quantique. À gauche, réponse de l’éllipticité du nuage à une

aniso-tropie tournant dans le même sens ou en sens opposé à celui de rotation. La

présence du réseau de vortex induit une levée de dégénérescence des deux

modes et le couplage aux modes de Kelvin introduit un ´elargissement du

modem

=−2. `A droite : imagerie d’absorption de la ligne de vorticit´e. Les

modulations dans le profil de densité après excitation résonnante du mode

m

=−2 traduit la pr´esence d’un mode de Kelvin.

que sa durée de vie soit supérieure à celle du m

=−2, ce que l’on observe

exp´erimentalement sur la Fig .1.3. Nous avons pu aussi observer

directe-ment l’excitation de la ligne de vorticit´e par imagerie d’absorption dans la

direction orthogonale `a l’axe de rotation puisque les oscillations spatiales du

champ de vitesse se traduisent apr`es temps de vol en modulations de densit´e

1.5 Conclusion

Dans ce chapitre, nous avons abordé certaines propriétés dynamiques

des condensats en rotation et nous avons montr´e que l’´etude du spectre

d’énergie du nuage permettait d’acquérir de précieuses informations sur les

propriétés du réseau de vortex. Une première extension de ce travail consiste

à généraliser cette étude aux modes de haute énergie de la ligne de

vorti-cité, associés à une dynamique du cœur du vortex qui n’ont pas encore été

étudiés expérimentalement. Un autre prolongement intéressant de ces

tra-vaux concerne le r´egime de rotation rapide. Dans ce cas, on peut montrer

que le condensat passe dans un r´egime de fortes corr´elations analogue aux

états de Laughlin utilisés pour décrire l’effet Hall Quantique Fractionnaire,

dont l’observation reste un défi posé à la communauté des atomes froids.

118 CHAPITRE 1. MODES DE VIBRATION...

Fig.1.4 – Observation des modes de Tkatchenko par le groupe d’E. Cornell

au JILA. Les lignes blanches indiquent la déformation du réseau associée au

mode de Tkatchenko.

Entre ces deux r´egimes existe une situation interm´ediaire, dite LLL-champ

moyen (pourLowest Landau Level) dans laquelle l’´energie cin´etique de

rota-tion l’emporte sur l’´energie d’interacrota-tion sans que les corr´elarota-tions quantiques

soient suffisantes pour d´etruire la condensation de Bose-Einstein. Ce r´egime

a été observé par le groupe d’Eric Cornell au JILA, qui l’a caractérisé par

spectroscopie des modes de Tkatchenko du r´eseau de vortex (Fig. 1.4 extraite

de [20]).

Bibliographie

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J. Gorter (North-Holland, Amsterdam, 1955), Vol. 1, Chap. 2.

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[4] M. R. Matthews et al., Phys. Rev. Lett.83, 2498 (1999).

[5] W.F. Vinen, Nature181, 1524 (1958).

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Lett.84, 806 (2000).

[7] M. W. Zwierlein, J. R. Abo-Shaeer, A. Schirotzek, C. H. Schunck et W.

Ketterle, Nature (London)435, 1047 (2005).

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[9] L.P. Pitaevskii, Zh. Eksp. Teor. Fiz.40, 646 (1961) ; Sov. Phys. JETP

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[13] G. Baym et E. Chandler, J. Low Temp. Phys. 50, 57 (1983)

[14] F. Zambelli et S. Stringari, Phys. Rev. Lett. 81, 1754 (1998)

[15] F. Chevy, K.W. Madison et J. Dalibard, Phys. Rev. Lett. 85, 2223

(2000) ;

[16] N. L. Smith, W. H. Heathcote, J. M. Krueger et C. J. Foot Phys. Rev.

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[18] K. E. Heikes et T. Maxworthy, J. Fluid Mech.125 319 (1982).

[19] M. Lewenstein et L. You, Phys. Rev. Lett.77, 3489 (1996) ; Y. Castin

and R. Dum, Phys. Rev. A 57, 3008 (1998).

[20] V. Schweikhard, I. Coddington, P. Engels, V. P. Mogendorff et E. A.

Cornell, Phys. Rev. Lett. 92, 040404 (2004)

BIBLIOGRAPHIE 121

Kelvin modes of a fast rotating Bose-Einstein

condensate

Kelvin modes of a fast rotating Bose-Einstein condensate

F. Chevy*

Laboratoire Kastler Brossel,^†24 rue Lhomond, 75005 Paris, France

S. Stringari

Dipartimento di Fisica, Universita` di Trento and BEC-INFM, I-38050 Povo, Italy

共Received 23 May 2003; published 3 November 2003兲

Using the concept of diffused vorticity and the formalism of rotational hydrodynamics we calculate the eigenmodes of a harmonically trapped Bose-Einstein condensate containing an array of quantized vortices. We predict the occurrence of a new branch of anomalous excitations, analogous to the Kelvin modes of the single vortex dynamics. Special attention is devoted to the excitation of the anomalous scissors mode.

DOI: 10.1103/PhysRevA.68.053601 PACS number共s兲: 03.75.Kk

I. INTRODUCTION

The existence of the macroscopic wave function

describ-ing a quantum fluid imposes a velocity flow curl free

every-where except on singularity lines known as vortices. Around

these lines, the circulation of the velocity is nonzero and is

quantized in unit of h/m where m is the mass of the particles

of the fluid. Recent experiments have demonstrated the

nucleation of such quantized vortices in stirred gaseous

Bose-Einstein condensates关1– 4兴.

Multiple quantized vortices are energetically unstable in

harmonic traps so that for large rotation frequencies,

Bose-Einstein condensates nucleate several singly quantized

vorti-ces that were observed to form regular triangular lattivorti-ces

known as Abrikosov lattices关4 – 6兴. In this configuration, the

circulation of the velocity field over a circle orthogonal to

the rotation axis and of radius R much larger than the vortex

interspacing is simply␲R

n

h/m, where n

is the density of

the vortex lines. This is the same formula as that of the

velocity field of a rigid body rotating at the angular velocity

⍀

⫽n

^h

2m^. 共1兲

The dynamical properties of a single vortex line were first

studied by Lord Kelvin关7兴and his results were transposed to

quantum fluids 关8 –10兴. For an excitation of wave vector k

propagating along the rotation axis, the dispersion relation

␻

of these modes共Kelvin modes or kelvons兲is

␻

⬃^ប^k

2m^ln共1/k␰兲, 共2兲

where␰⫽(8␲␳a)

⫺1/2

is the healing length giving the vortex

core diameter, a is the scattering length characterizing atom

binary interactions, and ␳ is the density of the gas. These

modes present the very peculiar feature that they only exist

with a single helicity as recently proved in the experiments

of Ref.关11兴. Indeed, the Kelvin-Helmholtz theorem that

con-strains the vorticity to move along with the fluid imposes an

angular momentum equal to ⫺ប to the Kelvin modes.

Although the problem of studying the dynamics of a

vor-tex array seems more involved at first sight, it is considerably

simplified for long wavelength perturbations. Indeed, in this

case a coarse grain averaging method permits to smooth the

discrete nature of vortices. In the case of a homogeneous

condensate, it was shown that excitations of wave vector k

propagating transversally to the rotation axis satisfy the

Tkatchenko dispersion relation关12,13兴

␻

⫽^ប⍀_4m

⁰

k

,

where⍀

is the effective angular frequency of the

conden-sate defined by Eq. 共1兲. The Tkatchenko modes are elastic

excitations of the lattice and have been recently investigated

theoretically also in the presence of harmonic traps 关14兴.

First experimental evidence for such modes has been

re-ported in Ref. 关15兴.

In this paper we use the coarse grain method to study the

eigenmodes of a rotating Bose-Einstein condensate confined

by a harmonic trap using a fully hydrodynamic approach

including vorticity 关16兴. In addition to the usual collective

modes exhibited by the condensate in the absence of

rota-tion, we identify an additional branch, analogous to the

Kelvin modes exhibited by a single vortex line. With respect

to the Tkatchenko modes, whose frequencies vanish in the

Thomas-Fermi limit 关14兴, the Kelvin excitations emerging

from our hydrodynamic picture approach a finite value in the

Thomas-Fermi limit. In the case of elongated traps their

fre-quencies actually scale like ␻

/␻

_⬜

for a fixed value of

⍀

/␻

_⬜

.

The paper is organized as follows. In Sec. II we develop

the formalism of rotational hydrodynamics in the presence of

harmonic trapping and derive the general equations共10兲for

the dispersion law of the linearized excitations. In Sec. III we

briefly summarize the results for the surface excitations

which represent a natural generalization of the modes

exhib-ited by nonrotating Bose-Einstein condensates. Section IV is

*

Also at Laboratoire de la Matie`re Condense´e, Colle`ge de France,

Paris, France.

†Unite´ de Recherche de l’Ecole normale supe´rieure et de

Kelvin nature is predicted. The properties of the scissors

os-cillations for a rotating condensate are discussed in detail

using the formalism of linear-response theory. Finally, in

Sec. V we obtain a general dispersion relation for the Kelvin

modes in elongated traps.

II. ROTATIONAL HYDRODYNAMICS AND ELEMENTARY EXCITATIONS

It is well known that in the so-called Thomas-Fermi

re-gime, where the mean-field interaction dominates over the

quantum pressure, the dynamics of a nonrotating condensate

can be described by the classical equations of

hydrodynam-ics:

⳵

␳⫽⫺“•共␳v兲, 共3兲

m⳵

v⫽⫺“共U

t⫹

g␳⫹mv

/2兲, 共4兲

where ␳ is the local particle density, g⫽4␲ប

a/m is the

coupling constant characterizing the interatomic force, and v

is the velocity field satisfying the irrotational condition

“⫻v⫽0. In the case of cylindrical harmonic trapping,

U

t⫽

^m₂ 关␻

_⬜2

共x

⫹y

兲⫹␻

z

兴,

the hydrodynamic equations共3兲and共4兲admit a class of

ana-lytic solutions 关17兴whose frequencies have been confirmed

experimentally with high accuracy.

In the presence of vortex lines the hydrodynamical

for-malism must be modified. Here we will employ a simplified

procedure by assuming that the characteristic wavelength of

excitations is large enough so that we can average all the

physical quantities over domains containing several vortices.

In this case the average lab frame velocity field v¯ is no longer

curl free. Since each vortex carries a flux h/m, the average

vorticity of the flow is

⍀¯⫽“⫻^¯v

2 ⫽n

^h

2m^u, 共5兲

where n

is the vortex surface density and the unit vector u is

the local direction of the vortex lines. According to Eq.共5兲,

the average vorticity⍀¯ characterizes the local vortex

distri-bution: its direction indicates the orientation of the vortex

lines while its modulus is proportional to the vortex density.

As in Refs. 关16,18兴, we shall simply assume that the

av-erage velocity field and density satisfy classical

hydrody-namical equations, including rotational terms, namely:

⳵

¯␳⫽⫺“•共¯ v␳¯兲,

m⳵

^¯v⫽⫺“冉U

t⫹

g¯␳⫹m^¯v

2冊⫺m共“⫻^¯v兲⫻¯.v 共6兲

This set of equations omits terms arising from the

shear-ing of the vortex lattice and leadshear-ing to the Tkatchenko

modes. Nevertheless, it can be shown that these are

vanish-ingly small in the Thomas-Fermi approximation关13兴. Thus,

as long as we restrict ourselves to modes whose frequencies

do not vanish in this regime, the effects of the lattice

defor-mation can be neglected.

A stationary solutions of Eqs.共6兲is given by

v

¯

⫽⍀

⫻r,

g¯␳

/m⫽␮⫺共␻

_⬜2

⫺⍀

0 2

兲共x

⫹y

兲/2⫺␻

z 2

z

/2, 共7兲

where␮ is the chemical potential and⍀

⫽⍀

u

character-izes the vortex density and direction according to Eq.共5兲. As

expected, the stationary velocity field is equivalent to that of

a rigid body rotating at angular velocity⍀

, while the

den-sity is described by the usual Thomas-Fermi profile with the

trapping potential U

corrected by the centrifugal potential

U

c⫽⫺

m⍀

0 2

(x

⫹y

)/2. The static behavior described above

has been verified experimentally. In particular, the

modifica-tion of the transverse trapping is used to measure

experimen-tally the effective angular velocity⍀

关4,6兴. More interesting

effects concern the study of the dynamics of the condensate.

In Ref. 关18兴the hydrodynamic equations have been used to

study the time evolution of a condensate containing a vortex

array, following the sudden switch on of a static deformation

in the plane of rotation. This produces peculiar nonlinear

effects that have been experimentally observed in Ref.关6兴. In

the following we will focus on the behavior of the linearized

solutions which can be derived by looking for small

pertur-bations of the density and velocity field:

␳

¯⫽␳¯

₀

⫹␦␳,

v

¯⫽⍀

⫻r⫹␦v,

with respect to the equilibrium values共in this case,␦v can be

interpreted as the velocity in the rotating frame兲. If the

char-acteristic wavelength of the perturbation is much larger than

the vortex spacing, the ‘‘averaged’’ hydrodynamical

equa-tions共6兲can still be applied and, in the frame rotating at the

angular velocity⍀

, they read, in linear approximation,

⳵

␦␳⫽⫺“⬘•共¯␳

␦v兲, 共8兲

⳵

␦v⫽⫺“⬘冉g␦␳

m 冊⫺2⍀

⫻␦v, 共9兲

where“⬘ denotes the derivation with respect to the

coordi-nates in the rotating frame. These equations are strictly

iden-tical to the usual hydrodynamical linearized equations

lead-ing to the phonon spectrum, except for the 2⍀

⫻␦v term

which arises from the Coriolis force. The system of

equa-tions共8兲and共9兲constitutes the starting point of our analysis.

We will study its eigenmodes and show that the spectrum in

addition to the usual ‘‘phonon’’ modes displays a new class

of solutions carrying negative angular momentum and that

can be physically regarded as Kelvin excitations.

Let us set⳵

t⫽⫺

i␻⬘in Eqs.共8兲and共9兲. Using Eq.共9兲, we

can then express ␦¯ as a function ofv ␦␳ as

␦v⫽ ¹

␻⬘

⫺4⍀

关i␻⬘共f兲

_⬜

⫹2⍀

⫻f兴⫺ ¹

i␻⬘^共^f^兲

^储

^,

where f⫽⫺“⬘(g␦␳)/m and (f)

_储

and (f)

⬜

^{are, respectively,}

the projections of f on the directions parallel and orthogonal

to z.

Inserting this expression for the velocity field in the

lin-earized mass conservation equation 共8兲, we get a closed

equation for the density fluctuations that can be written as

i␻⬘共␻⬘

⫺4⍀

兲␦␳⫽A•␦␳, 共10兲

whereAdenotes the linear operator

A•␦␳⫽⫺ⵜ

_⬜

⬘•冋g␳

m 共i␻⬘ⵜ

_⬜

⬘␦␳⫹2⍀

⫻ⵜ

_⬜

⬘␦␳兲册

⫹ⵜ

_储

⬘冋g␳

m 冉␻⬘

⫺4⍀

0 2

i␻⬘ ^ⵜ

^储

^⬘^␦^␳冊册, 共11兲

and␳

is the equilibrium density profile共7兲. Equations共10兲

and共11兲are the main result of this paper. Simple solutions

can be found in the form of polynomials.

Let us stress again that the calculation of␻⬘ is performed

in the rotating frame. The corresponding frequency␻ in the

lab frame is obtained through the simple relation ␻⬘⫽␻

⫺m

z⍀0

, whereបm

is the angular momentum of the

exci-tation along the roexci-tation axis. Since both␦␳ and its complex

conjugate are solutions of Eq. 共11兲 we find that for any

so-lution with angular momentum m

and eigenfrequency␻⬘,

there should be another solution with angular momentum

⫺m

and frequency⫺␻⬘. Since the unperturbed state共

con-densate rotating at angular velocity ⍀

) is the ground state

of the system in the rotating frame, the physical solutions

carrying energyប␻⬘ should correspond to positive

frequen-cies␻⬘.

Note also that Eq. 共10兲 is valid for ␻⬘⫽0 and ␻⬘

⫽4⍀

and that these two special cases must be treated

sepa-rately. The case ␻⬘

⫺4⍀

⫽0 does not lead to any new

mode. The case␻⬘⫽0 is instead more interesting since it is

related to the Tkatchenko’s modes, as stressed in the

Intro-duction. Starting the analysis back from Eqs.共8兲and共9兲, we

can show that the density perturbation␦␳ of a zero energy

modes depends only on the radial coordinate r

⬜

⫽冑x

⫹y

and that the velocity field is given by

v

z⫽

0,

v

⬜

⫽ ^g

2m⍀

⍀

⫻“␦␳.

In the case where␦␳ is parabolic, this zero energy mode

can be simply interpreted as a modification of ⍀

, i.e., a

change of the vortex density.

The rest of the paper is devoted to the solution of Eq.共11兲

for special cases of physical interest. We will first discuss the

surface oscillations and then the scissors modes, which will

derive the general dispersion law for the Kelvin modes

work-ing in the geometry of elongated traps.

III. SURFACE MODES

Surface modes are characterized by the form ␦␳⫽(x

⫾iy )

and carry angular momentum m

z⫽⫾

l. Inserting this

ansatz into the differential equation共10兲yields the following

equation for the eigenfrequencies␻

_⫾

⬘

in the rotating frame:

␻

_⫾

⬘

⫾2⍀

␻

_⫾

⬘

⫺l共␻

_⬜2

⫺⍀

0 2

兲⫽0.

The positive solution of this equation reads

␻

_⫾

⬘

⫽冑l␻

_⬜2

⫺共l⫺1兲⍀

0 2

⫿⍀

or, in the laboratory frame,

␻

_⫾l⫽冑

l␻

_⬜2

⫺共l⫺1兲⍀

⫾共l⫺1兲⍀

.

Contrary to what happens in the case of a nonrotating

condensate, the two modes are no longer degenerate, and the

degeneracy lift amounts to

⌬␻

l⫽

␻

_⫹l⫺

␻

_⫺l⫽

2共l⫺1兲⍀

. 共12兲

This result is valid for any l 关20兴but can be checked for

some special cases previously reported in the literature.

In the l⫽1 case共dipole motion兲one finds⌬␻⫽0. In this

case the eigenfrequencies are actually unaffected by the

ro-tation since␻

_⫾l⫽

␻

_⬜

irrespective of the value of⍀

. This is

not surprising since we know that the generalized Kohn

theo-rem 关19兴 implies that the center of mass motion is

deter-mined only by the frequency of the harmonic trap.

The case l⫽2 is also well documented, both theoretically

and experimentally关21,22兴. Using sum-rule approach, it can

be shown in particular that the degeneracy lift amounts to

⌬␻⫽2 具ᐉ

典

m具r

⬜²

典^, ^共¹³^兲

ᐉ

is the angular momentum per particle along z of the

un-perturbed condensate. Since the unun-perturbed velocity field is

rigidlike, the angular momentum is given by the classical

formula具ᐉ

典⫽m具r

⬜²

典⍀

. Inserting this relation into Eq.共13兲

yields result共12兲for l⫽2.

IV. SCISSORS MODES

The scissors modes are associated with a density

fluctua-tion ␦␳⫽(x⫾iy )z carrying angular momentum m

z⫽⫾

1 along the z axis. Using Eq.共11兲, one finds that the

eigenfre-quencies ␻⬘ must satisfy the nontrivial cubic equation:

␻⬘

⫾2⍀

␻⬘

⫺␻⬘共␻

⫹␻

_⬜2

⫺⍀

兲⫿2⍀

␻

⫽0. 共14兲

This equation is analogous to that found by the authors of

Dans le document Du sillage des insectes aux gaz de Fermi ultra-froids : dynamique des fluides classiques et quantiques (Page 117-200)