d. guery-odelin- spinning up and down a boltzmann gas

8/3/2019 D. Guery-Odelin- Spinning up and down a Boltzmann gas

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Spinning up and down a Boltzmann gas

D. Guery-OdelinLaboratoire Kastler Brossel, Departement de Physique de lEcole Normale Superieure, 24 Rue Lhomond,

F-75231 Paris Cedex 05, France

Received 2 March 2000; revised manuscript received 27 April 2000; published 14 August 2000

Using the average method, we derive a closed set of linear equations that describes the spinning up of an

harmonically trapped gas by a rotating anisotropy. We find explicit expressions for the time needed to transferangular momentum as well as the decay time induced by a static residual anisotropy. These different time

scales are compared with the measured nucleation time and lifetime of vortices by Madison et al. We find a

good agreement that may emphasize the role played by the noncondensed component in those experiments.

PACS numbers: 03.75.Fi, 05.30.Jp, 32.80.Pj, 67.40.Db

I. INTRODUCTION

Superfluidity of a Bose-Einstein condensate is naturallyinvestigated by its rotational properties 1,2. So far, twoexperimental schemes have been successfully applied to gas-eous condensates in order to generate quantized vortices

3,4. The first one uses phase engineering by means of laserbeams whereas the second one is the analog of the rotatingbucket experiment, initially suggested by Stringari 5,6.

In the latter, atoms are first confined in a static, axiallysymmetric Ioffe-Pritchard magnetic trap upon which a non-axisymmetric attractive dipole potential is superimposed bymeans of a strirring laser beam. In this paper, we address thequestion of the behavior of the noncondensed component forthis experiment. We investigate the transfer of angular mo-mentum to an ultracold harmonically confined gas by such atime-dependent potential. We also study possible mecha-nisms for its dissipation. First Sec. II we recall the Lagra-gian and the Hamiltonian formalism for a single particle in a

rotating frame. In Sec. III, we give the expression for therotating potential that has been used in our paper. We brieflyexpose analytical results for the single-particle trajectory inthis potential in Sec. IV. The rest of the paper deals with thecrucial role played by collisions. A classical gas that evolvesin this potential thermalizes in the rotating frame, leading toa finite value of its mean angular momentum. We investigatethis equilibrium state in Sec. V. In Sec. VI, we derive theanalytical expression for the time needed to spin up a clas-sical gas with an approach based on the average method 7.We evaluate the time needed to transfer angular momentumand thus we deduce the characteristic time for vortex nucle-ation via angular momentum transfer from the uncondensedto the condensed component. In Sec. VII, we consider a re-lated problem: what is the time needed to dissipate a givenangular momentum by a static residual anisotropy? We havein mind the role of the axial asymmetry of a magnetic trapIoffe-Pritchard or time-orbiting potential traps induced bythe presence of gravity. The connection with decay of quan-tum vortices is made in Sec. VIII.

II. A REMINDER ON THE ROTATING FRAME

In this section, we recall the Hamiltonian for a classicalparticle in a rotating frame characterized by the fixed rotation

vector . Without loss of generality, we choose a rotationaround the z axis ez and the same origin for the rotat-ing frame R as the one of the laboratory frame R see Fig.1. In the following, quantities with a prime are evaluated inthe rotating frame. Coordinates in R are linked to coordi-nates in R just by a rotation of angle t:

xy

cos t sin tsin t cos t

xy . 1

The Lagrangian for the single-particle movement in the ro-tating frame reads 8

L12 mv

2

12 mr

2mvrVext , 2

where Vext is the potential energy that describes the role ofexternal forces. The correspondance between the laboratoryand rotating frames for momentum, angular momentum, andthe Hamiltonian is given by

pL

vmvmrp,

LrprpL,

HpvLHL. 3

Note that the angular momentum as well as the momentumare the same in both frames, but the link between the mo-

FIG. 1. Rotating frame with respect to laboratory frame.

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mentum and the velocity differs. Using the Hamiltonian for-malism, one can easily extract the equation of motion in therotating frame:

mdv

dtFextFcenFcor , 4

where Fext refers to the external force derived from the po-

tential Vext , Fcen is the centrifugal force:

FcenmrVcen 5

with Vcenm2(x 2y 2)/2, and Fcor is the Coriolis force:

Fcor2mv. 6

Note that all formulas of the system 3 are still valid evenfor a time-dependent rotation vector .

III. THE ROTATING TRAP

In Bose-Einstein condensation experiments, the magnetic

confinement is axially symmetric. In order to spin up thesystem, one possibility consists of breaking this symmetry bysuperimposing a small rotating anisotropy as initially sug-gested by Stringari 5. This breaking of the rotational invari-ance of the external potential can be carried out experimen-tally by adding a rotating stirring dipolar beam to themagnetic field of the trap 4. This combination of light andmagnetic trapping induces the following harmonic externalpotential see Fig. 1 9:

Vextm0

2

2 1x2 1y22z2 , 7

where we have defined the geometric parameter z

/0

,which is responsible for the shape of the cloud. For instance,if we set 0, the trap is isotropic for 1, cigar-shapedfor 1, and disk-shaped for 1. The potential 7 is timedependent if expressed with laboratory coordinates (x,y ,z),but static in the rotating frame, i.e., in terms of (x,y,z).

IV. SINGLE-PARTICLE TRAJECTORY

Let us first investigate the single-particle trajectory. Equa-tions in the laboratory frame and for the transverse coordi-nates are given by

x

y0

2

x

y

02

cos 2t sin 2t

sin 2t cos 2t x

y . 8In the rotating frame, the corresponding equations are timeindependent. They can be rewritten by means of the complexquantity xiy:

2i 022 0

20, 9

where denotes the complex conjugate of . The secondterm in Eq. 9 accounts for the Coriolis force, the centrifu-gal force leads to a reduction of the harmonic strength third

term, and the last term is the contribution of the asymmetry.The stability of the single-particle movement is extractedfrom the dispersion relation of Eq. 9. We find a window ofinstability around the value 0, i.e., for

01;01 . For 01, the stability isessentially ensured by the harmonic trapping even if reduced

by the centrifugal force, whereas for 01 the Co-riolis force plays a crucial role in the stabilization of thetrajectory. The latter effect is similar to magnetron stabiliza-

tion in a Penning trap for ions 10. Note that for an inter-acting classical gas the motion of the center of mass is thesame as that for a single particle.

V. EQUILIBRIUM STATE OF THE GAS

In practice, experiments are carried out in the so-calledcollisionless regime, since on average an atom undergoesless than a collision during a transverse oscillation period.However, collisions are of course essential to explain thedynamics of the gas induced by the potential 7. We con-sider the situation in which a gas at a given temperature T0 isinitially at rest in the lab frame, and at t0 the axial asym-metry is spinned up at a constant angular velocity as

explained above. If 01 one expects that elasticcollisions will ensure the thermalization of the gas in therotating frame. This equilibrium state is defined since a mini-mum of the effective potential VeffVextVcen always existsin this range of values for . In Fig. 2, we compare thestability of a single particle upper part of the diagram withthat of the interacting gas. In the rotating frame, one cancompute equilibrium quantities by means of the Gibbs dis-tribution that reads 11

r,veH(r,v)/kBT0, 10

where H is given by

Hmv2

2Vextx,y,z

m2

2x2y2 . 11

As regards statistical properties of the gas, the rotation isequivalent to a reduction of the effective transverse frequen-cies of the trap due to the contribution of the centrifugalforce. The Coriolis force plays no role for the equilibriumstate. During the thermalization, the mean angular momen-tum per particle increases from zero to its asymptotic value

Lz. This last quantity is easily derived from the Gibbs dis-tribution 10:

FIG. 2. Stability diagram for the single particle and for an in-

teracting gas in the potential with a rotating anisotropy.

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Lzmx2y 2

2kBT0022

022 220

4. 12

For typical experimental parameters 0/2200 Hz, 0/2, 0.05, and T01 K, the angular momentum perparticle Lz/150, unlike the superfluid part for which,for instance, the angular momentum is equal to per particle

in the presence of one vortex.To check that the gas undergoes a full rotation, one can

search for a displacement of the critical temperature inducedby centrifugal forces 6, or perform a time-of-flight mea-surement. In the latter case, one expects that the ratio be-

tween x and z size scales as (12/02)1/2 for long-time

expansion.

VI. TIME NEEDED TO REACH EQUILIBRIUM

In this section we derive the expression for the time tupneeded to reach the equilibrium state in the rotating coordi-nate system. In other words, tup corresponds to the time

needed to build correlations between x, vy , y, and vx . Toestimate this time, we use the classical Boltzmann equation.Our analysis relies on the use of the average method as ex-plained in Ref. 7. For instance, the equation for xvyyvx involves xy , which itself is coupled to xvyyvx and so on. Terms that do not correspond to a con-served quantity in a binary elastic collision lead to a nonzerocontribution of the collisional integral. For instance, a meanvalue such as vxvy involves the occurence of quadrupoledeformations in the velocity distribution that make the con-tribution vxvyIcoll0. Here, Icoll stands for the collisionalkernel of the Boltzmann equation 12. At this stage, weperform a Gaussian ansatz for the distribution function. After

linearization this quadrupolar contribution results in the so-called relaxation-time approximation 12,13:

vxvyIcollvxvy

.

This method leads to a closed set of 1313 linear equationssee the Appendix, and furthermore provides an explicit linkbetween the relaxation time and the collisional rate in thesample.

It is worth emphasizing that the dynamic transfer of an-gular momentum to a classical gas by rotating a superim-posed axial anisotropy involves a coupling between all quad-rupole modes. The average method is fruitful in the sensethat it yields a closed set of equations when one deals withonly quadratic moments, namely, monopole mode, scissorsmode, quadrupole modes, etc. Noninertial forces are linear inposition or velocity, and thus give rise only to quadratic mo-ments using the average method.

Although unimportant for equilibrium properties, the Co-riolis force plays a crucial role for reaching equilibrium. Fig-ure 3 depicts a typical thermalization of the gas in the rotat-ing frame, leading to the equilibrium value 12 of theangular momentum, obtained by a numerical integration ofthe 1313 set of equations. After a tedious but straightfor-

ward expansion of the dispersion relation, one extracts thesmallest eigenvalue that drives the relaxation for a given col-lisional regime. First we focus on the regime in which ex-

periments are performed: collisionless (0

10) and with22/0

2. Denoting tupCL as the time needed to spin up the

gas in this regime, we find

tupCL

8

2 0

2

. 13

The result is independent of the geometrical aspect ratio ofthe trap as physically expected. Using the same numericalvalues as in the preceding section, we deduce that this time isvery long tup

CL15 s. Note that to transfer just of angular

momentum per particle, one needs only 100 ms. This time ison the order of the nucleation time for vortices that has beenexperimentally observed by the ENS group 14. One should

nevertheless be careful since the noncondensed component isactually a Bose gas rather than a classical one that evolves ina nonharmonic potential because of the mean-field potentialdue to the condensed component.

In the hydrodynamic limit, the characteristic time forspinning up is

tupHD1/ 220

2.

So far this regime is not accessible for ultracold atom experi-ments since inelastic collisions prevent the formation of veryhigh-density samples. We recover here the special feature ofthe hydrodynamic regime, i.e., the time needed to reach equi-librium increases with the collisional rate. In Fig. 4 we have

reported the evolution of tup as a function of 0 from anumerical integration of the 1313 system. The smallestvalue is obtained between the collisionless and hydrody-namic regimes, as is usual for the relaxation of a thermal gas7.

VII. TIME NEEDED TO DISSIPATE A GIVEN ANGULAR

MOMENTUM

In this section, we consider a gas with a given angularmomentum, obtained for instance as explained before, in astatic harmonic trap. If this gas evolves in an axially sym-metric trap, the angular momentum is a conserved quantity.

FIG. 3. Mean angular momentum per particle as a function of

time.

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In contrast, if a small asymmetry exists between the x and

y spring constants, the angular momentum is no longer aconserved quantity and it is thus dissipated. We call tdown thetypical time for the relaxation of the angular momentum.This problem is very different from the one we faced previ-ously since the rotating frame is not an inertial frame. Wethus expect tdowntup . As in the previous treatment, thisproblem can be computed with the average method. The cor-responding equations are nothing but the scissors modeequations 2, i.e., a linear set of four equations involving

xy, xvyyvx , xvyyvx , and vxvy see the Appen-dix. Searching a solution of this system of the form exp(t), one finds

1

4 112/c2 , 14

where the critical anisotropy is related to and 0 by c1/(40) and where we assume c

21. In Fig. 5, we plot

different curves for the relaxation of angular momentum de-pending on the value of with respect to c . For clong-dashed line, one has a purely damped relaxation; in

the limiting case c , tdown1/(220

2). On the contrary

for c solid line, one has a damped oscillating behavior;in the limiting case c , tdown4, and the oscillatingfrequency is equal to 0. Moreover, as we use a linearanalysis, this decay time does not depend on the specific

initial value of the angular momentum.

VIII. CONNECTION WITH DECAY OF QUANTUM

VORTICES

We now apply the results derived above to a recent ex-periment 4 which determined the decay time of a quantumvortex when it is placed in a trap with a small fixed anisot-ropy. For this expriment the axial asymmetry induced bygravity is on the order of 1% and the collisional rate is suchthat 010. The corresponding decay time of the angularmomentum of the thermal component is evaluated from Eq.14 to be 500 ms. This value is of the same order as thetypical observed lifetime of a vortex.

Of course, the study we present so far is not directly rel-evant for the superfluid itself, but it can reasonably describethe thermal part. Indeed, one possible interpretation may bethat the thermal part acts as a reservoir of angular momentumand thus ensures the stability of the vortex as long as thisthermal part is itself rotating. Moreover, when the angularmomentum of the thermal part is zero, a dissipative couplingbetween the Bose Einstein condensate with a vortex and thethermal gas can take place.

In a recent paper 15, Fedichev and Shlyapnikov proposea scenario for such a dissipative mechanism. It relies on thescattering of thermal excitations by the vortex, leading to the

motion of the vortex core to the border of the condensate.The main assumption for this calculation is that the thermalpart is not rotating. Actually, our paper provides a justifica-tion of this assumption by stressing the key role played byresidual axial anisotropy in the confining trap.

IX. CONCLUSION

In this paper we have investigated the transfer and dissi-pation of angular momentum for a classical gas in all colli-sional regimes. We derive very different time scales for bothprocesses. The short time required to dissipate angular mo-mentum suggests a high sensitivity of the experiment de-

scribed in Ref. 4 to the presence of a residual static anisot-ropy see different time scales between Figs. 3 and 5. Inparticular, the competition between a rotating and a staticanisotropy explains why no evidence for rotation of the clas-sical gas was found in Ref. 4, and may give relevence tothe scenario proposed in Ref. 15.

ACKNOWLEDGMENTS

I acknowledge fruitful discussions with V. Bretin, F.Chevy, J. Dalibard, K. Madison, A. Recati, S. Stringari, and

FIG. 4. Time needed to reach equilibrium by rotation of a small

anisotropy. A minimum of tup is obtained in-between the collision-

less and the hydrodynamic regimes.

FIG. 5. Decay of angular momentum due to a small residual

anisotropy. We distinguish three different regimes: c damped

oscillations solid line, c small dashed line the frontier with

the region of purely damped decay (c , long dashed line.


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F. Zambelli. I am grateful to Trentos BEC group where partof this work was carried out.

APPENDIX

Hereafter, the 1313 closed set of equations that de-scribes the dynamic of a classical gas induced by the rotatingpotential 7:

dxy

dtxvy

yvx 0, A1

dx2y2

dt2xvx

yvy 0, A2

dx2y2z2

dt2xvxyvyzvz0,

A3

dx2y22z2

dt

2xvxyvy2zvz0,

A4

dxvyyvx

dt20

2xy

4

3xvxyvyzvz

2

3xvxyvy2zvz0, A5

dxvyyvx

dt

2022 xy

2xvxyvy2vxvy0, A6

dxvxyvy

dt0

22 x2y2

202

3x2y2z22xvyyvx

02

3x2y22z2vx

2vy

20, A7

dxvxyvyzvzdt

02x2y2

2222

30

2x2y2z2

122

30

2x2y22z2

vx2vy

2vz

22xvyyvx0,

A8

dxvxyvy2zvz

dt0

2x2y2

202

3 122 x2y2z2

02

3

1222 x2y22z2

vx2vy

22vz

22xvyyvx0,

A9

dvxvy

dt0

2xvyyvx2vx2vy

2

022 xvyyvx

vxvy

, A10

dvx2vy

2

dt20

22 xvxyvy8vxvy

402

3xvxyvyzvz

202

3xvxyvy2zvz

vx2vy

2

,

A11

dvx2vy

2vz

2

dt20

2xvxyvy

202

3 2222xvxyvyzvz

202

3 122 xvxyvy2zvz0,

A12

dvx2vy

22vz

2

dt20

2xvxyvy

402

3 122xvxyvyzvz

202

3 122

2

xvxy

vy2z

vz

vx2vy

22vz

2

. A13

One can show from the Gaussian ansatz that the relaxationtime is the same for all quadrupolar contributions. In thissystem, linear terms in account for the Coriolis forcewhereas quadratic terms in are due to centrifugal forcecontributions. All quadrupolar modes are involved in thissystem. In order to enlighten the physics of this system, let

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us consider some limiting cases. One can check the conser-

vation of energy in the rotating frame dH/dt0. Thestationary state, obtained by setting all time derivatives of

moments to zero, is nothing but the equipartition law see

Ref. 11, See 44: (12)x2(12)y2

2z2vx2

vy2

vz2 . If 0 and 0, the trap

is axially symmetric and one recovers the conservation of the

angular momentum. In addition the system gives rise to threeindependent linear systems on quadrupolar quantities: onefor the m2,xy mode, another one for m2,x 2y 2,and finally the m0 mode that describes the coupling be-tween monopole (r2) and quadrupole mode (m0,x 2

y 22z2) 7. Finally, one can recover the scissors modeequations for a classical gas by considering the case 0and 0 2: Eqs. A1, A5, A6, and A10.

1 G. Baym, in Mathematical Methods in Solid State and Super-

fluidity Theory, edited by R.C. Clark and E.H. Derrick Oliver

and Boyd, Edinburgh, 1969.

2 D. Guery-Odelin and S. Stringari, Phys. Rev. Lett. 83, 4452

1999; O.M. Marago, S.A. Hopkins, J. Arlt, E. Hodby, G.

Hechenblaikner, and C.J. Foot, ibid. 84, 2056 2000.

3 M.R. Matthews, B.P. Anderson, P.C. Haljan, O.S. Hall, C.E.

Wieman, and E.A. Cornell, Phys. Rev. Lett. 83, 2498 1999.

4 K. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, Phys.

Rev. Lett. 84, 806 2000.

5 S. Stringari, Phys. Rev. Lett. 76, 1405 1996.

6 S. Stringari, Phys. Rev. Lett. 82, 4371 1999.

7 D. Guery-Odelin, F. Zambelli, J. Dalibard, and S. Stringari,

Phys. Rev. A 60, 4851 1999.

8 L.D. Landau and E.M. Lifshitz, Mechanics, 3rd ed. Perga-

mon, Oxford, 1980, Vol. 1, Sec. 39.

9 Another proposal for such a potential based on the time-

orbiting potential is under investigation by the Oxford group,

see J. Arlt, O. Marago, E. Hodby, S.A. Hopkins, G. Hechen-

blaikner, S. Webster, and C.J. Foot, e-print cond-mat/9911201.

10 L.S. Brown and G. Gabrielse, Rev. Mod. Phys. 58, 233 1986.

11 L.D. Landau and E.M. Lifshitz, Statistical Physics, 3rd ed.

Pergamon, Oxford, 1980, Vol. 5, Secs. 26 and 34.

12 K. Huang, Statistical Mechanics, 2nd ed. Wiley, New York,

1987.

13 U. Al Khawaja, C.J. Pethick, and H. Smith, e-print,

cond-mat/9908043.

14 J. Dalibard Private communication.

15 P.O. Fedichev and G.V. Shlyapnikov, Phys. Rev. A 60, R1779

1999.


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