The excitations of trapped atoms with a Bose-Einstein condensate in a trap are determined by the conservation of common phonon and roton numbers of atomic motion, and these properties depend on the presence of gravitational field

Baranov, V.S.

Yarunin, D.B.

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

Fizika Nizkikh Temperatur, 2003, v. 29, No. 6, p. 674–677Phonons and rotons of trapped atoms in gravitational fieldD.B. Baranov and V.S. YaruninBogoliubov Laboratory of Theoretical Physics Joint Institute for Nuclear Research, Dubna 141980, RussiaE-mail: yarunin@thsun1.jinr.ruReceived December 19, 2002The excitations of trapped atoms with a Bose-Einstein condensate in a trap are determined bythe conservation of common phonon and roton numbers of atomic motion, and these properties de-pend on the presence of gravitational field.PACS: 05.70.JkIntroductionPhonon-like excitations of atoms in traps are ob-served experimentally. If we rotate a trap, the newform of atomic motion—rotons—may be expected. In4He the same maxon–roton excitations are followedby vortexes. They are defined as a singularity points ofa velocity field rot V r( )  0, connected to the condi-tion for an amplitude  to become zero ( )r  0 [1,2].Here we are going to look for the properties of atomicBose-Einstein condensate (BEC) in traps in the rota-tional (potential) stage of non-singular excitations.We show, that roton properties of a gas in a trap fol-lows from the conservation of a common number ofphotons and rotons, found in our model. The addi-tional potential energy of trapped atoms is promotedby coming of atoms outside of BEC via the increasingof phonon and roton energy due to the gravitationalfield, and these contributions are estimated below.Phonons and rotonsThe energies of translations H ph and orbital rota-tions H rot of atoms can be separated if the origin ofcoordinates is taken as the point of «equilibrium»,such as the centres of a circle (2D) or a sphere (3D).Each of N atoms in a trap contributes to phonon mo-tion, provided by an interaction between them, and torotations as well. So we represent the Hamiltonian ofN atoms as the sum of translational H ph, rotationalH rot energies and the rotation-translation interactionH ph rotH ph   ( )maxkkkk k,H rot  	iziiNmL U r21( ),H ph rot   1NL LkkkiiNk kimax( )  ,[ , ]' '  k k kk , [ , ]L L Li j z ij   2  , [ , ] ,L L Lizj iij  [ , .]L L Li zj iij   (1)Here  is a rotational energy; kmax is an upper levelof a trap with a potentialU; L are the operators of anorbital momentum; k is phonon–roton interaction;( )k is the energy of phonon with the momentum k ofan atom in a rectangular trap, expressed by the wellknown equation [3]ka nkmU  22 0arcsin,where U0 is a potential of a trap and a is a trapwidth. Really, phonons and rotons transform to eachother, and a connection between them is given by theintegral of motion M[ , ] ,maxH M M Lkkkk ziiN   01  ,[ , ] ,H K K LiiN 0 21,H H H H  rot ph ph rot(2)of a system (1), represented by Hamiltonian H . It ispossible to use the mapping of rotational variables toa couple of Fermi operators for each of i N 1 2, ...momentum operators by the use of substitution© D.B. Baranov and V.S. Yarunin, 20032L a a b b L a b L b az     , , ,{ , } { , } { , }a a a b b b   0,{ , } { , } , { , }a a b b a b    1 0(we omit the i numeration and show a very simpleone-particle H  h version of a model in the nextfew formulas) . The latter definitions lead to the rela-tions[ , ]a a b b a a b b      0, [ , ] ,a a b b a b    0so that the initial h may be written in boson-fermionvariables with L a a b b2    for each atom in ourmappingh a a b b a b b aa b             2 2( ),[ , ( )] [ , ] ,h a a b b h a a b b         120where the energies  a b, of fermions contain their ki-netic energies. These relations are truth for a largekmax, so that a number m0 of traped levels satisfy anequation m0 1 , just like it happens in experiment.If we turn back to the total Hamiltonian H , thethermodynamical parameters  and  will correspondthe integrals of motion K and MH H          K M kkkkkph rotmax[ ( ) ] [ ] a i ibi iiNa a b b1,     a ba b,, ,     2 ,{ } { , },a a b bi j i j ij   for the partition function of a system. The latter maybe represented as the path integral for a large GibbsdistributionQ K M    Sp exp [ ( )]  HD D Da Da Db Db Skk k iiNi i i * * * exp1,S L dt0over all the boson  , *(and fermion a a, *, b b, *) tra-jectories, satisfying the periodical (and antiperio-dical) boundary conditions on [ , ]0  for each k (and i).The Lagrange function L represents all the degrees offreedom of the system (1) in an ordinary way. Inte-gral over all fermionic (a b, ) fields in Q may becalculated exactly as Det ( )2S [4,5], so we get apath integral over the variables  k,  k* with an effec-tive action Sef for every kQ D D Skk k  * exp ( , )ef 0 ,S S N Rkef ph Det( , ) ln0    ,Sddtdtk k kph    	   *0,RL / N/ N Lkk kk k		*,L d/dt L d/dta b         , .The quasiclassical equations of motionS Sk kef ef *0 (3)are used for the extremal trajectories ( )* k0 and  k0that describe an effective translational modes in thefield of roton motion. In the N  1 limit the transfor-mation  k kN and  k kN* * leads to there-normalized action SeffS S N S Rk Nef eff ph Det   ( ln )| | | | |  ,so that the equation of motion (3) in a new scale ofboson trajectories looks likeSS Rk kt keff Sp* *( ln )    			ddtt RRtk k kkk  ( )( )*Sp 1 0, (4)where a formula ln lnDet SpR Rk k was used. As Kis an integral of motion for (1), the first bond rela-tion for a thermodynamical parameters are1 1 1    		SK RRkkkeff Sp . (5)In the same way the second bond relation is found as1 121 20    	SM RRdtkkkkeff Sp | | !!!.(6)The calculation of Sp in (4)–(6) is carried out fol-lowing [4,5] both in the matrix and Path IntegralsensePhonons and rotons of trapped atoms in gravitational fieldFizika Nizkikh Temperatur, 2003, v. 29, No. 6 675Sp Spln lnRLLLLkk k				001 00011 k k* 0		 !!"#$%$&'$($= Sp lnLL mLLmk)				0012000111  kk km * 02		 !!"#$%$&'$($     122 1 2 1Sp ln , ( )( ).* *T T L L L Lk k k k k k    The particular time-independent ( )* * k00 , k00 solution of equations (4)–(6) is similar to«slow» trajectories in superfluid 4He theory [1], asso-ciated with the Bose-Einstein condensation (BEC) inquantum liquid. In the same way  0 0*, trajectoriescorrespond BEC of atoms in a trap, observed experi-mentally during the last years [6]. Using BEC as-sumption, we get the approachT T L L   0 0202 | |and see the variational equation (4) in the form00200 02 2 2 2 2 		 		 tanh tanh  !!,  020202 1 24 ( | | ) / , (7)while the equations for integrals of motion K, M ap-pears asK k k 				12 2 2 2 2tanh + tanh !!k2,M k k 		 		12 2 2 2 2tanh tanh !!k 2 2| |. k .These equations determine the BEC ordering in a trapvia the trajectories 0*, 0 and chemical potentials , in terms of energies ( )k , , and k.The right side of the equation (7) can be repre-sented  as00200 00 02  sinhcosh cosh( )A AA A,A002 2 2 		   ,and in the case of a small phonon-roton interaction 0 * the approximation0  210202| |, **is fulfilled, so that sinh ( )A A0 0 1  ** . Therefore,the inequality    020 0 0202024**   , | | (8)is valid. It means, that BEC trajectories exist only forlow phonon-roton interaction. It is worthy to note,that for an interaction  between polar atoms insegnetoelectric media, expressed by the same model as(1), the condition of dipol cooperation is expressedby the inequality [4] 2  ,where  and  stands as the radiation and two-levelsystem frequencies correspondingly. Just the oppo-site, the inequality (8) follows from the previous for-mulas as the condition for the equations (4)–(6) tohave the BEC solution, and the condition of a largeBEC density  0 0 0| |  is also satisfied. The com-plete line of relations between the parameters of atheory looks like00 02 0| | || * * ,  ** 1 (9)where both the BEC condition (8) and square rootdecomposition are taken into account. It can be seen,that the left side of (9) leads to formula (8).Trapped atoms in gravitational fieldNow we are going to look for the properties of atrapped atomic (BEC) in gravitational field. Let usconsider a pure phonon Hamiltonian H php (withoutthe influence of gravitational field) for finite trap po-tential. In this case the ordinary commutation rela-tions for phonon operators is broken and can be writ-ten in the form[ , ] ( , )+ + + +k k k kf  , (10)676 Fizika Nizkikh Temperatur, 2003, v. 29, No. 6D.B. Baranov and V.S. Yaruninwhere f k k( , )+ + is a polynomial function of opera-tors +k and +k . Now the Hamiltonian of gravita-tional field in coordinate space H q mgz can bewritten in initial double quantized particle operatorsb bk k,H g k kcb b , where c is constant.After diagonalizati on, H gcan be represented in termsof phonon operators +k, +kbbk k k k kk k k k k  "#$%$ , , + ++ +,,H g k k k k k kc c c     1 2 3+ + + + + +( ).For the case of finite trap (10) the commutator of to-tal Hamiltonian H H H H  ph rot ph rotp withthe Hamiltonian of gravitational field H g is notequal to zero.[ , ]H H H Hph rot ph rotpg  - 0.The deformation of the trap potential is shown onFig. 1. Following the paper [7] we can conclude thatthe gravitational field manifest itself in decreasing ofupper level kmax and in the shift of energy levels in thetrap. In particular, for ideal bose gas in the trap [7],the critical temperature and number of condensateparticles are compared for the cases of gravitationalfield and in microgravity environment. It was shown,that the presence of gravitational field leads to the de-creasing of condensate fraction and increasing of criti-cal temperature if we keep constant the total numberof particles in trap.ConclusionThe dilute gas of interacting bosons in magnetictrap under rotation was considered as a system withHamiltonian H (1), (2), which consists of three terms:phonon Hamiltonian H ph, Hamiltonian of roton exci-tations H rot and Hamiltonian of phonon-roton inter-action H ph rot . The partition function of the systemwas written as a path integral over boson and fermiontrajectories with integrals of motion (2). After inte-gration over fermions, the quasiclassical equations ofmotion for boson trajectory in the ground state withtheir integrals of motion were obtained. The conclu-sions on the equations (4)–(9) are as follows:a) for a given 0 the inequality (8) can be satisfiedfor large | |0 , so the presence of rotons diminishes theBEC density;b) the less amount of BEC bosons | |02 we have, thelarger will be a rotational momentum of atom for agiven trap.These conclusions seems to be in accordance withthe prioriting Popov [1] result, that the zero points ofboson trajectory   0 are starting points for a vor-texes in superfluid 4He. As to the influence of grav-ity, we note here the point (c) as follows: c) the gravi-tational field diminishes the BEC density in a trap.1. V.N. Popov, Functional Integrals and Collective Exci-tations, Cambridge Univ. Press (1987).2. H. Karatsuji, Phys. Rev. Lett. 68, 1746 (1992) .3. L.D. Landau and E.M. Lifshits, Quantum Mechanics,Nauka, Moscow (1985).4. V.B. Kiryanov and V.S. Yarunin, Teor. Mat. Phys.43, 91 (1980).5. V.N. Popov and V.S. Yarunin, Collective Effects inQuantum Statistics of Radiation and Matter, KluwerPubl. (1988).6. F. Dalfovo, S. Giorgini, L.P. Pitaevsky, and S. Strin-gari, Rev. Mod. Phys. 71, 463 (1999).7. D. Baranov and V. Yarunin, JETP Letters 71, 266(2000).Phonons and rotons of trapped atoms in gravitational fieldFizika Nizkikh Temperatur, 2003, v. 29, No. 6 677U(Z)UUU–h/2 h/2hU+0ggZFig. 1. The deformation of the trap potential. The dashedand solid lines for a parabolic trap potential in a free space( )U0 and in a gravitational field ( , )U U  .

Phonons and rotons of trapped atoms in gravitational field

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Phonons and rotons of trapped atoms in gravitational field

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Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)