Small-amplitude normal modes of a vortex in a trapped Bose-Einstein condensate

a r X i v :c o n d -m a t /9907045v 1 [c o n d -m a t .s t a t -m e c h ] 2 J u l 1999

Small-amplitude normal modes of a vortex in a trapped Bose-Einstein condensate

Marion Linn 1,2and Alexander L.Fetter 1

1

Department of Physics,Stanford University,Stanford,CA 94305-40602

Physikalisches Institut,Universit¨a t Bonn,Nu?allee 12,D-53115Bonn,Germany

(February 1,2008)

We consider a cylindrically symmetric trap containing a small Bose-Einstein condensate with a singly quantized vortex on the axis of symmetry.A time-dependent variational Lagrangian analysis yields the small-amplitude dynamics of the vortex and the condensate,directly determining the equations of motion of the coupled normal modes.As found previously from the Bogoliubov equations,there are two rigid dipole modes and one anomalous mode with a negative frequency when seen in the laboratory frame.

PACS number(s):03.75.F,05.30.Jp,32.80.Pj,67.40.Db

I.INTRODUCTION

After the achievement of Bose-Einstein condensates in trapped atomic gases [1–3],interest in these systems has increased dramatically.An especially intriguing ques-tion concerns condensates containing a vortex.The low-energy normal modes are important in characterizing the behavior of such a system.They have been studied exten-sively [5–11],mostly with the Bogoliubov equations [12]that involve quantum-mechanical normal-mode ampli-tudes.Consequently,the associated dynamical motion of the vortex core and the condensate center of mass can only be obtained indirectly.Here,we use a variational Lagrangian procedure to provide a more direct and intu-itive treatment of the small-amplitude normal modes of a small (weakly interacting)Bose-Einstein condensate con-taining a singly quantized vortex.As in previous applica-tions [8,13],this approach yields a clear physical picture of the dynamics.We show that it reproduces the three lowest modes as found from the more intricate Bogoli-ubov analysis,thereby clarifying the physical interpreta-tion.This description is particularly interesting for the anomalous mode that causes the local instability of the vortex in a trap at rest [5,6,10].

The article is organized as follows:in Sec.II we derive the e?ective Lagrangian for a slightly o?-center vortex in a small condensate.In the next section we solve the corresponding equations of motion for the vortex and the condensate and discuss the physical interpretation.Fi-nally we summarize our results in the conclusion.

II.EFFECTIVE LAGRANGIAN

We consider a small Bose-Einstein condensate with N particles containing a singly quantized vortex on the axis of symmetry,and observed in a frame rotating with an angular velocity ?.The harmonic trap is axisymmetric,with radial and axial frequencies ω⊥and ωz (the ratio λ=ωz /ω⊥characterizes the axial asymmetry).The s -wave scattering length a (assumed positive)characterizes

the interparticle interactions.For a general condensate wave function Ψ,the Lagrangian for such a system is given by

L (Ψ)= dV

i

2

??2x ??2y +(x 2+y 2)+λ ??2

z +z 2

,(2)

and L z =?i (x?y ?y?x )is the z -component of the angular-momentum operator.All quantities are ex-pressed in dimensionless units (the radial and axial co-ordinates are scaled with the radial and axial oscillator lengths d ⊥=

ˉh /Mωz ,and fre-quencies are scaled with the radial trap frequency ω⊥).The condensate wave function Ψis normalized to unity and γ≡Na/d z is the small interaction parameter.

Next we construct a trial condensate wave function.The condensate wave function is assumed to be un-changed with respect to the noninteracting case along the axis of symmetry,which allows us to use the ground-state gaussian ?0(z )=π?1/4exp (?z 2/2).The ra-dial wave function is based on the vortex ground state,namely a two-dimensional gaussian times the vortex fac-tor x +iy =re i?.We are interested in the relative motion of the vortex which requires introducing time-dependent parameters for the vortex position.In the weak-coupling limit,the radius of the vortex core is comparable to the radius of the condensate,so that the displacement r 0(t )=[x 0(t ),y 0(t )]of the vortex,the displacement r 1(t )=[x 1(t ),y 1(t )]of the condensate,and the induced velocity of the condensate α(t )=[αx (t ),αy (t )]must all be included.Hence we use the following trial function Ψv (x,y,z,t )=

C

2

[(x ?x 1)2+(y ?y 1)2+z 2]e i (αx x +αy y ),(3)

1

where C?2=1+|r1?r0|2.Since we focus on the dynam-ical motion of the vortex and the condensate,we ignore the possibility of monopole or quadrupole modes,and the dimensionless gaussian widths are taken as one.The same trial function,Eq.(3),also serves to characterize the stability of the vortex in a small condensate[14].

In evaluating the integration in L(Ψv),we retain all terms up to second order in the time-dependent param-eters,yielding

L e?=?2?λ

2

√

2

(x21+y21)?x0(x1?x0)?y0(y1?y0)?1

√

2?γ

2π

+?

?˙αx?x?˙αy?y? ˙?x?2˙δx δy+ ˙?y?2˙δy δx ?1

2

?2+?δ·?+ ?2?+3√

2

√

2√

M

,(11)

where n′andΦ′are the complex?uctuations of the

particle density and the velocity potential respectively.

The density?uctuations are obtained by expanding

the density n=|Ψv|2=n0+δn,where n0=

(r2/π3/2)exp(?r2)exp(?z2)is the unperturbed density

for the condensate with a vortex.We?nd

δn≈2n0r· r1?r0

r

+r? +δy 2

r

+r? ?δx 2

2

√

√

where δ0is an in?nitesimal amplitude and the factor 1/

√r

(cos ?δx +sin ?δy ),(19)

and the complex density ?uctuation (with time depen-dence e ?iωa t )is

n ′a =?

δ0

2

n 0 r ?2r

+r

(cos ?δy ?sin ?δx ),

(21)

with the corresponding complex velocity potential

Φ′

a =

δ0ˉh 2iM

22δ0(?cos ωa t ,sin ωa t ),

(23)r 1=

1

2δ0(?cos ωa t ,sin ωa t );

(24)

they both execute small-amplitude in-phase motion with r 0twice

as

large

as

r

1

.

For

?=0,the anomalous frequency (17)is negative,and the motion is counter-clockwise in a positive direction (namely right-circular motion with positive helicity).With increasing rotation frequency ?,the negative anomalous frequency increases towards zero and vanishes at the metastable rotation fre-quency ?m =1?γ/(2

√r

(cos ??x ±+sin ??y ±),(27)

and hence the complex density ?uctuations are

n ′±=?±(0)n 0

r ?

1

r

+r (?±ω±)

(sin ??x ?cos ??y ),

(29)

and the resulting complex ?uctuations for the velocity

potential are

Φ′±=?±

(0)ˉh r

e ±i?,(30)independent o

f the external rotation frequency ?.With

these expressions,the normalization integral is one for both normal modes with frequencies ω±.

The ?nal expression for the motion of the vortex and the condensate in the dipole modes [r (t )≡r 0(t )=r 1(t )]is

r ±(t )=r 0(cos ω±t ,±sin ω±t ),

(31)

with an in?nitesimal amplitude r 0.We again ?nd cir-cular trajectories,and here they are the same for the vortex and the condensate motion,con?rming that the modes are indeed the rigid dipole oscillations of the vor-tex and the condensate.For stability reasons,only the region |?|<1is relevant;thus the motion is always counter-clockwise (positive)for ω+and clockwise (nega-tive)for ω?.Comparison with the Bogoliubov results [6]3

shows exact agreement for?=0.The rotation splits the dipole frequencies by twice the external rotation fre-quency.This result can be visualized by expressing the dipole modes(for?=0)in terms of circular quanta, namely as circular motions in the positive and negative sense.For zero external rotation,the angular velocities are degenerate.In a frame rotating with?in the positive sense,the positive mode rotates at the di?erence1??and hence slows down.The negative mode,however, speeds up by the same amount,leading to an e?ective splitting in the rotating frame of2?.

IV.CONCLUSION

We have shown that a time-dependent Lagrangian ap-proach provides physical insight into the nature of the low-energy modes of a vortex in a small trapped Bose-Einstein condensate.The anomalous mode is an in-phase circular motion of the vortex and the condensate with di?erent amplitudes.In a rotating frame,the sense of this mode reverses from positive to negative when the external rotation frequency exceeds the metastable fre-quency[14].The dipole modes are rigid rotations of the vortex and the condensate together.In terms of the den-sity and phase?uctuations,all these results agree exactly with those from the Bogoliubov approach[6].

ACKNOWLEDGMENTS

This work was supported in part by NSF Grant No.94-21888and by the DAAD(German Academic Exchange Service)“Doktorandenstipendium im Rahmen des gemeinsamen Hochschulsonderprogramms III von Bund und L¨a ndern”(M.L.). A.L.F.thanks the As-pen Center for Physics where part of this work was per-formed.