Coupling mechanisms for damped vortex motion in superfluids

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1 PHYSICAL REVIEW B VOLUME 56, NUMBER 13 1 OCTOBER 1997-I Coupling mechanisms fo damped voex moion in supefluids H. M. Caaldo, M. A. Despósio, E. S. Henández, and D. M. Jezek Depaameno de Física, Faculad de Ciencias Exacas y Nauales, Univesidad de Buenos Aies, RA-18 Buenos Aies, Agenina and Consejo Nacional de Invesigaciones Cieníficas y Técnicas, Agenina Received 3 Decembe 1996 We invesigae diffeen dissipaive dynamics fo a voex line immesed ino a supefluid whee densiy flucuaions have been excied. Fo his sake, we conside vaious linea coupling models whee he voex ineacs wih he quasipaicles of he nomal fluid hough is coodinae o momenum. We can unambiguously show ha one and only one combinaion of hese vaiables leads o damped evoluion in ageemen wih he phenomenological descipions. S I. INTRODUCTION Quanum dissipaion has been a vivid banch of eseach fo seveal decades 1, in view of boh he fundamenal heoeical aspecs of he subjec and he vas applicaion fields, ha include quanum opics, physical chemisy, nuclea physics, and condensed-mae physics. The bes known models eso o a sysem-plus-esevoi descipion, whee he ievesible appoach o hemal equilibium of a quanum objec is examined using pojecion and/o educion echniques. The key ingedien o saing such a sudy is he Hamilonian, which consiss of hee ems, especively, coesponding o he fee sysem, o he esevoi, and o hei muual ineacion. Among he mos popula choices fo he lae, hose being linea in opeaos epesening he coodinae o he momenum of he quanum paicle ae pefeed fo applicaions in he fame of sandad nonequilibium saisical mechanics. In paicula, hose ineacion models known as he oaing wave appoximaion RWA and he full coupling FC model, have been favoed by many auhos. 3 6 In spie of he fac ha he RWA has song foundaions, especially concening applicaions o quanum opics, some of is dawbacks have been poined ou by seveal auhos. In paicula, being a velociy-dependen ineacion, i beaks he equivalence beween velociy and momenum mediaed by he ineia paamee. 7 This bings some undesiable consequences, mainly he fac ha one may no ecove he classical limi of a semiclassical evoluion. The FC model is, in conas, well behaved in he classical limi of he semiclassical descipion of quanum dissipaive moion. 7 Howeve, i is well known ha such ineacion mechanisms induce poenial ems which localize he Bownian paicle, which ough o be aificially emoved adding a couneem in he oiginal, unpeubed Hamilonian. 1, We have invesigaed hese aspecs of he RWA and he FC model focusing upon quanum hamonic moion 7 and spin elaxaion. 8 1 Howeve, he ealm of physical sysems which can be mapped ono a simple quanal Bownian moion model is much wide; ecenly, we have shown ha a single voex moving in a supefluid conaining quana of densiy flucuaions can be egaded as a quanum Bownian paicle ineacing wih a esevoi. 11 The descipion of he subsequen damped moion pesens a subsanial ageemen wih he phenomenological appoaches. The mos impoan aspec of he model is he ineacion mechanism adoped, which involves he voex velociy in a linea appoximaion. In his wok we invesigae o a deepe exen he diffeen dissipaive dynamics ha coupling models of he FC fom assign o poin voices immesed ino a supefluid which conains some exciaions. In paicula, we show ha FC Hamilonians expessed in ems of eihe he coodinae o he momenum of he voex give ise o localizaion phenomena in phase space. In ode o suppess phase-space localizaion, we seach he kind of mixed couplings ha involve linea combinaions of coodinae and momenum in he spii of Ref. 11. We find ha hee is a unique ineacion mechanism ha leads o damped evoluion in ageemen wih he phenomenological descipions. This mechanism is pecisely he one invesigaed in Ref. 11, whee i appeaed as a naual choice in view of he fac ha he fee moion of a poin voex in a fluid is elecomagneic, in ohe wods, he diving foce acs on he voex velociy similaly o he Loenz foce on a chaged paicle. This pape is oganized as follows. In Sec. II we eview he descipion of he moion of a fee voex filamen wih cylindical symmey in a supefluid a zeo empeaue and se ou noaion. In Sec. III, diffeen choices fo he FC ineacion Hamilonian ae discussed and i is shown ha if he coupling involves only he coodinae o he momenum of he poin paicle, is subsequen moion is no physically accepable. The soluion is pesened in Sec. IV, whee we show ha he only Hamilonian leading o ealisic evoluion of he voex involves boh he coodinae and he momenum of he paicle linealy combined as in he velociy of he fee moion. Finally, some concluding emaks ae pesened in Sec. V. II. CYCLOTRON VORTEX MOTION The cycloon moion of a cylindical voex paallel o he z axis in a supefluid a zeo empeaue is povoked by he Magnus foce, which povides he lif upon a cylinde ha moves wih velociy v v s elaive o he fluid and exhibis ciculaion aound he z axis, v s being he supefluid velociy and q v 1 is he sign of he voiciy accoding o he igh-handed convenion. The sucue of his foce is /97/56 13 /88 7 /$ The Ameican Physical Sociey

2 56 COUPLING MECHANISMS FOR DAMPED VORTEX FIG. 1. Tajecoies of mean values of Heisenbeg opeaos in he coodinae a and velociy b planes fo he cycloon moion in Eq..8. The coodinaes in a and he velociies in b ae, especively, given in unis of v s / and v s. We have assumed a voex iniially a es a he oigin of he coodinae plane. Noe ha he cene of he cicle in b coesponds o he supefluid velociy. F M q v h s lẑx v v s..1 In wha follows we shall assume a unifom supefluid velociy poining along he x axis. This moion has been descibed in Ref. 1 and coesponds o he Hamilonian dynamics geneaed by H 1 M p q va M v s y wih he veco poenial. A h sl y, x,.3 which gives ise o he componen of he lif popoional o he voex velociy. The componen of his foce depending upon he supefluid velociy is minus he gadien of he scala poenial M v s y. Hee M is he dynamical mass of a voex elemen of lengh l and s is he numbe densiy of he supefluid, which a zeo empeaue coincides wih he oal densiy pe uni mass /m, m being he mass of a single aom. Fuhemoe, q vh s l. M is he unpeubed cycloon fequency, whee h is Planck s consan. In ems of he complex posiion and momenum opeaos q x iy; p p x ip y,.5 he Heisenbeg equaions of moion ae q p M i q,.6 whee he igh-hand side hs displays he complex fom of he classical Magnus foce. Inegaion of Eqs..6 and.7 gives q f q i M f p v s i f,.9 p i M f q f p i s Mv i f,.1 whee q,p ae he iniial posiion and momenum and f ei The above equaions descibe he simple cycloon moion whose ajecoies fo he mean values of he Heisenbeg opeaos in he coodinae and velociy planes ae depiced in Figs. 1 a and 1 b. III. DISSIPATIVE DYNAMICS If he supefluid conains elemenay exciaions, hese can behave as a esevoi o which he voex may couple. This ineacion povides a dissipaion mechanism ha damps he cycloon moion. If we denoe by he densiy opeao of he voex, we can deive, in he weak coupling non-makovian limi, 9,1 a genealized mase equaion GME wih ime-dependen coefficiens; his is achieved combining he sandad educion-pojecion pocedue of nonequilibium saisical mechanics 13 wih he ime convoluionless mehod developed by Chauvedi and Shibaa. 1 Fo an ineacion of he fom o, equivalenly ṗ i M p q im v s, q iq v s,.7.8 H in i i S i B i, 3.1 whee S i and B i ae voex and esevoi opeaos, especively, he GME eads

3 88 CATALDO, DESPÓSITO, HERNÁNDEZ, AND JEZEK 56 i H, 1 i,j i j d Si, S j, ij i S i, S j, ij, 3. whee a,b denoes an anicommuao. The opeaos S i appeaing in his expession will be eihe he coodinae o he momenum of he fee cycloon moion displayed in Eqs..9 and.1. In spie of is inegodiffeenial equaion sucue, he GME is a diffeenial one, since he unknown unde he inegal sign is aken a ime ; he imedependen funcions ij () and ij () ae he eal and imaginay pas, especively, of he coelaion beween hea bah opeaos 6 Bi B j iji ij. 3.3 We shall deive equaions of moion fo he expecaion values of he posiion and momenum componens of he voex. Taking ino accoun vey geneal elaions involving opeaos a, b, and c, namely T a b, c, T a,b,c a,b,c, 3. T a b, c, T a,b,c a,b,c, 3.5 we eadily obain fo an obsevable O Ȯ i O,H 1 i,j i j d O,Si,S j ij i O,S i,s j ij. 3.6 Since he geneal coupling 3.1 is linea in q and/o p, he moion equaions fo he expecaion values ae of he fom q a qq qa qp pc q, ṗa pq qa pp pc p. 3.7 In geneal, he coefficiens in Eq. 3.7 ae ime dependen and involve ime inegals of he coelaion funcion of he esevoi. The Makovian egime suppesses hese ime dependences, since in ha case all uppe inegaion limis become infinie. In such a siuaion, similaly o he fee cycloon moion, one can mege Eqs. 3.7 ino a single foce equaion, namely whee q i1q qa, i1a qq a pp is a complex ficion consan, a qp a pq a qq a pp is he deeminan of he linea sysem a he hs of Eq. 3.7 and a a qp c p a pp c q, 3.11 is a efeence acceleaion. In ode o appeciae he effec of he esevoi consiued by he exciaions of he supefluid, a his poin i will be insucive o compae he expecaion value of he Magnus foce equaion.8 and he foce equaion 3.8. In fac, we obseve ha he esevoi induces boh a dissipaive and a consevaive coupling, especively, measued by he paamees Im( ) and. We also ealize ha he sengh and diecion of he oiginal Magnus foce hs of Eq..8 will be modified hough he paamees Re( ) and a. In ohe wods, in addiion o a damping foce, an hamonic esoing foce plus a gaviylike componen appea. This means ha, in geneal, he moion akes place aound he minimum of he induced consevaive poenial, which povides hen a fixed poin o he dynamics, namely q F a. 3.1 Howeve, i is impoan o emak ha such a fixed poin disappeas wheneve he hamonic foce in Eq. 3.8 vanishes, i.e., when he deeminan of he linea sysem a he hs of Eq. 3.7 is idenically zeo. A. Coodinae-dependen coupling We fis conside a coupling model ha descibes he damping mechanism by means of an ineacion Hamilonian of he fom H in B 3.13 wih he posiion veco of he poin voex and B a veco funcion of opeaos ha ceae densiy flucuaions in he liquid. In he fame of he cuen descipion, his funcion is elaed o he Hemiian pa of he Feynman-Cohen opeao Ô k ha ceaes a phonon o a oon wih momenum k Ref. 15 Ô k k 1 k k Nkk k k k k, 3.1 whee N is he numbe of aoms in he liquid and kk. Noice ha if, fo example, he Hemiian opeao B is chosen as popoional o he funcion B k Ô k Ô k, 3.15 B k B k k i k 3.16 is elaed o he Fouie ansfom of he dynamical sucue faco S(k, ) of he supefluid, which fo he case of liquidhelium isoopes is expeimenally known fo a wide ange of ansfeed momena and enegy. 16,17 Fom Eq. 3.6 we eadily ge

4 56 COUPLING MECHANISMS FOR DAMPED VORTEX FIG.. Same as Fig. 1 fo ajecoies aising fom Eq. 3.8 fo a ficion coefficien Im ( ).5. The dashed line ajecoies coespond o he iniial condiions of Fig. 1, while he coninuous line ones assume he same iniial velociy bu an iniial posiion q (, 1). ṗi q p M i q, M p qim v s dq Inseing Eq..9 in Eq we obain he coefficiens coesponding o Eq. 3.7 as a qq i, a qp 1 M, c q, a pq M 1 M, a pp i 1 M, c p im v s 1 i M i, whee he involved ime-dependen coefficiens ead d f, d wih f ( ) given by Eq..11. Specifying now he Makovian limi whee any ime dependence oiginaed in a ime inegal is suppessed, we obain he hamonic esoing sengh and he second-ode equaion fo he acceleaion is q i 1 M q v s M qv s M. 3.8 In ode o illusae wih some numeical esuls, we have consideed a Gaussian esevoi. 18 In his model he imaginay pa of he coelaion funcion 3.3 is given by he exponenial decay law ( )Aue u, whee u is he invese of he esevoi elaxaion ime and A is a posiive consan depending on he specific fom of he B opeaos. Then all paamees of he foce equaion 3.8 ae deemined by a unique dimensionless ficion coefficien Im( ) cf. Eq. 3.8: ImIm M A Mu, 3.9 in addiion o he dimensional paamees and v s. Theefoe Eq. 3.8 can be saighfowadly inegaed and some of he ajecoies ae depiced in Fig., whee we se u 1 in ode o enfoce he Makovian hypohesis. One expecs ha in a supefluid wih velociy v s, he only iniial condiion which deemines he dynamics a lae imes is he voex velociy. Howeve, Fig. shows ha fo a given iniial velociy, diffeen choices of he iniial voex posiion give ise o significan depaues beween he coesponding velociies a lae imes. This eflecs anohe undesied consequence of he pesence of he fixed poin Eq B. Momenum-dependen coupling A simila behavio akes place if one assumes a coupling ha depends upon he voex momenum, i.e., if he ineacion Hamilonian is H in p B p p. 3.3 In his case, he coesponding equaions of moion fo he expecaion values of complex posiion and momenum ae M 3.7 q p M i q p dp, 3.31

5 886 CATALDO, DESPÓSITO, HERNÁNDEZ, AND JEZEK 56 ṗi pm qim v s. 3.3 q i 1 M j q v s M j q Using Eq..1 we obain he coefficiens a qq i 1 pm, 3.33 M j v s i jp M v s p 3.1 fo j,p. Then assuming p he fixed poin lies a a qp 1 M 1 pm, 3.3 q F p q F i v s, 3. c q M p v s i, 3.35 a pq M, 3.36 a pp i, c p im v s whee he supescips, p indicae he ype of coupling Hamilonian unde consideaion. IV. COMBINED COUPLING MODEL In view of he esuls pesened in he pevious secion, i appeas naual o inquie whehe a suiable linea combinaion of he pevious coupling Hamilonians could suppess he fixed poin ininsic o he coupling mechanisms hee discussed. Fo his sake, we now invesigae he phase-space dynamics ha he GME associaes o he ineacion Hamilonian In his case H in A p p B..1 pm 3.39 and he coesponding acceleaion in he Makovian egime is q i 1 pm q v s M p q M p v s i. 3. I should be noiced ha similaly o wha happens fo he coodinae-dependen coupling cf. Eq. 3.8, he ajecoies will develop aound a fixed poin, epesening a nonealisic dynamics. Moeove, if we se p p and /(M ) Eqs. 3.8 and 3. can be wien as Fom now on we assume ha he wo-dimensional veco opeaos A and B of he esevoi saisfy A i A j B i B j fo i j. and ha B R( )A, whee R( ) is he maix associaed wih a oaion in an angle. This hypohesis implies ha he physical opeao which epesens he elemenay exciaion is unique, and ha is diffeen manifesaions in ineacion Hamilonians involving eihe he coodinae o he momenum of he voex canno diffe bu in he oienaion of he veco in he diecion pependicula o he voex filamen. Accodingly cf. Eq. 3.3, we have Ai B i Bi A i cos, Ax B y By A x sin,.3. FIG. 3. Same as Fig. 1 fo he ajecoies aising fom Eq..18 fo a ficion coefficien Im( ).5. The dashed line in a coesponds o he iniial condiions of Fig. 1 while he coninuous line assumes he same iniial velociy bu an iniial posiion q (, 3). Boh cases yield in b he same ajecoy on he velociy plane, i.e., a spial cuve conveging o he supefluid velociy.

6 56 COUPLING MECHANISMS FOR DAMPED VORTEX whee Ay B x Bx A y sin, Ax A x Bx B x..5.6 The evoluion of he mean values of he complex posiion and momenum ae given by q p M i q p d q e i p p,.7 H in pp x M y B x p y M x B y,.17 and his in un, indicaes ha he only ineacion mechanism which does no localize he voex in phase space, involves he paicula combinaion of coodinae and momenum giving he fee velociy of a poin voex moving unde he Magnus foce. The coupling Hamilonian hus obained coesponds o he case consideed in Ref. 11. The secondode equaion fo he acceleaion now eads q i1 M q v s,.18 whee one can idenify he ficion coefficien Im( ) of Eq. 3.8 as ṗi M p qim v s d q ImImM,.19 p p e i, and he coesponding coefficiens ae a qq i 1 im p e i M p,.8.9 which may be appoximaed o Im( ) 8A /Mu, unde he same assumpions leading o he hs of Eq Inegaion of Eq..18 yields he ajecoies depiced in Fig. 3 whee i is shown ha he shape of he obis does no depend on he iniial voex posiion, as expeced. c q Mv s p a qp 1 M 1 im p e i M p,.1 ei i p i,.11 a pq M 1 im p e i M,.1 a pp i 1 im p e i M, c p Mv s i M i M p e i i,.13.1 wih and p defined below Eq. 3.. The fixed poin is suppessed if he deeminan of he sysem vanishes. This condiion yields sin sin 1,.15 p and since and p mus be eal quaniies, his implies ha, p..16 One can ealize ha he above elaionship equies V. CONCLUDING REMARKS The damped moion of a voex in a supefluid was invesigaed unde diffeen linea couplings in he voex obsevables. We have shown ha a coupling popoional o he fee velociy epoduces he basic feaues of a ealisic damped moion. I is woh noing ha his Hamilonian coupling is equivalen o a Lagangian coupling popoional o he ue voex velociy. Moeove, we have shown ha any ohe linea combinaion of coodinae and momenum mus be discaded, since i poduces nonealisic ajecoies aound a fixed poin. Noe ha in conas o supefluids, he ineacion Hamilonian fo voices in supeconducos may be popoional o he voex coodinae 19 since in such a case, he pinning poenial gives ise o localizaion. This is equivalen o aking p in ou equaions. I is ineesing o discuss some feaues egading he simples dissipaive dynamics wihou localizaion, i.e., ha undegone by a fee Bownian paicle. In such a case, he coupling o he hea bah is usually modeled by a anslaionally invaian fom involving only coodinaes. 1 Howeve, an ineacion linea in he momenum, which in his case is popoional o he fee velociy, has been shown o be useful in siuaions whee boh he sysem and he esevoi vaiables ae funcions of he same se of degees of feedom. This is pecisely he case if he Bownian paicle epesens a collecive exciaion ineacing wih ininsic ones, and one can hen show ha he dynamics do no conain any fixed poin. 1 In fac, he limi in ou equaions should indeed coespond o such dynamics, wih, i.e., a coupling popoional o he momenum o fee velociy is he condiion fo nonappeaance of fixed poins Eq..15. Finally, we have shown ha in he pesence of exenal fields wih, ealisic dynamics ake place only when boh and p ae diffeen fom zeo and elaed accoding o p /M.

7 888 CATALDO, DESPÓSITO, HERNÁNDEZ, AND JEZEK 56 1 A. O. Caldeia and A. J. Legge, Physica A 11, ; Ann. Phys. N.Y. 19, U. Weiss, Quanum Dissipaive Sysems, Seies in Condensed Mae Physics Vol. Wold Scienific, Singapoe, G. S. Agawal, in Quanum Opics, edied by G. Höhle, Spinge Tacs in Moden Physics Vol. 7 Spinge-Velag, Belin, 197. K. Lindenbeg and B. Wes, Phys. Rev. A 3, J. Seke, Physica A 193, E. S. Henández and M. A. Despósio, Physica A 1, M. A. Despósio, S. M. Gaica, and E. S. Henández, Phys. Rev. A 6, H. M. Caaldo, Physica A 165, M. A. Despósio and E. S. Henández, J. Phys. A 8, L. D. Blanga and M. A. Despósio, Physica A 7, H. M. Caaldo, M. A. Despósio, E. S. Henández, and D. M. Jezek, Phys. Rev. B 55, R. J. Donnelly, Quanized Voices in Helium II Cambidge Univesiy, New Yok, E. S. Henández and C. O. Doso, Phys. Rev. C 9, S. Chauvedi and F. Shibaa, Z. Phys. B 35, D. Pines and P. Noziées, The Theoy of Quanum Liquids Addison-Wesley, Reading, MA, 1966, Vol. I. 16 A. Giffin, Exciaions in a Bose-Condensed Liquid Cambidge Univesiy, New Yok, H. R. Glyde, Exciaions in Liquid and Solid Helium Claendon, Oxfod, J. A. Whie and S. Velasco, Phys. Rev. A 6, P. Ao and D. J. Thouless, Phys. Rev. Le. 7, A. H. Caso Neo and A. O. Caldeia, Phys. Rev. E 8, M. A. Despósio, J. Phys. A 9,

KINEMATICS OF RIGID BODIES

KINEMATICS OF RIGID BODIES KINEMTICS OF RIGID ODIES In igid body kinemaics, we use he elaionships govening he displacemen, velociy and acceleaion, bu mus also accoun fo he oaional moion of he body. Descipion of he moion of igid