Phonons and rotons of trapped atoms in gravitational field

Phonons and rotons of trapped atoms in gravitational field

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.

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Datum:2003
Hauptverfasser: Baranov, V.S., Yarunin, D.B.
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Sprache:English
Veröffentlicht: Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2003
Schriftenreihe:Физика низких температур
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3-й Международный семинар по физике низких температур в условиях микрогравитации
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Zitieren:Phonons and rotons of trapped atoms in gravitational field / D.B. Baranov V.S. Yarunin // Физика низких температур. — 2003. — Т. 29, № 6. — С. 674-677. — Бібліогр.: 7 назв. — англ.

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spelling irk-123456789-1288582018-01-15T03:02:53Z Phonons and rotons of trapped atoms in gravitational field Baranov, V.S. Yarunin, D.B. 3-й Международный семинар по физике низких температур в условиях микрогравитации 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. 2003 Article Phonons and rotons of trapped atoms in gravitational field / D.B. Baranov V.S. Yarunin // Физика низких температур. — 2003. — Т. 29, № 6. — С. 674-677. — Бібліогр.: 7 назв. — англ. 0132-6414 PACS: 05.70.Jk http://dspace.nbuv.gov.ua/handle/123456789/128858 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
topic 3-й Международный семинар по физике низких температур в условиях микрогравитации
3-й Международный семинар по физике низких температур в условиях микрогравитации
spellingShingle 3-й Международный семинар по физике низких температур в условиях микрогравитации
3-й Международный семинар по физике низких температур в условиях микрогравитации
Baranov, V.S.
Yarunin, D.B.
Phonons and rotons of trapped atoms in gravitational field
Физика низких температур
description 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.
format Article
author Baranov, V.S.
Yarunin, D.B.
author_facet Baranov, V.S.
Yarunin, D.B.
author_sort Baranov, V.S.
title Phonons and rotons of trapped atoms in gravitational field
title_short Phonons and rotons of trapped atoms in gravitational field
title_full Phonons and rotons of trapped atoms in gravitational field
title_fullStr Phonons and rotons of trapped atoms in gravitational field
title_full_unstemmed Phonons and rotons of trapped atoms in gravitational field
title_sort phonons and rotons of trapped atoms in gravitational field
publisher Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate 2003
topic_facet 3-й Международный семинар по физике низких температур в условиях микрогравитации
url http://dspace.nbuv.gov.ua/handle/123456789/128858
citation_txt Phonons and rotons of trapped atoms in gravitational field / D.B. Baranov V.S. Yarunin // Физика низких температур. — 2003. — Т. 29, № 6. — С. 674-677. — Бібліогр.: 7 назв. — англ.
series Физика низких температур
work_keys_str_mv AT baranovvs phononsandrotonsoftrappedatomsingravitationalfield
AT yarunindb phononsandrotonsoftrappedatomsingravitationalfield
first_indexed 2025-07-09T10:02:51Z
last_indexed 2025-07-09T10:02:51Z
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fulltext Fizika Nizkikh Temperatur, 2003, v. 29, No. 6, p. 674–677 Phonons and rotons of trapped atoms in gravitational field D.B. Baranov and V.S. Yarunin Bogoliubov Laboratory of Theoretical Physics Joint Institute for Nuclear Research, Dubna 141980, Russia E-mail: yarunin@thsun1.jinr.ru Received December 19, 2002 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 de- pend on the presence of gravitational field. PACS: 05.70.Jk Introduction Phonon-like excitations of atoms in traps are ob- served experimentally. If we rotate a trap, the new form of atomic motion—rotons—may be expected. In 4He the same maxon–roton excitations are followed by vortexes. They are defined as a singularity points of a 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 atomic Bose-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 of photons and rotons, found in our model. The addi- tional potential energy of trapped atoms is promoted by coming of atoms outside of BEC via the increasing of phonon and roton energy due to the gravitational field, and these contributions are estimated below. Phonons and rotons The energies of translations H ph and orbital rota- tions H rot of atoms can be separated if the origin of coordinates 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 to rotations as well. So we represent the Hamiltonian of N atoms as the sum of translational H ph, rotational H rot energies and the rotation-translation interaction H ph rot� H ph � � �� � �( ) max k k k k k, H rot � � � � � � � � � �i z i i N m L U r 2 1 ( ), H ph rot� � � �� �� �1 N L Lk k k i i N k k i� max ( )� � , [ , ]' '� � �k k kk � � , [ , ]L L Li j z ij� � � 2 � , [ , ] ,L L Li z j i ij� �� � [ , .]L L Li z j i ij� �� � � (1) Here � is a rotational energy; kmax is an upper level of a trap with a potentialU; L are the operators of an orbital momentum; �k is phonon–roton interaction; �( )k is the energy of phonon with the momentum k of an atom in a rectangular trap, expressed by the well known equation [3] ka n k mU � �� 2 2 0 arcsin � , where U0 is a potential of a trap and a is a trap width. Really, phonons and rotons transform to each other, and a connection between them is given by the integral of motion M [ , ] , max H M M Lk k k k z i i N � � �� � � �0 1 � � , [ , ] ,H K K Li i N � � � �0 2 1 , H H H H� � � �rot ph ph rot (2) of a system (1), represented by Hamiltonian H . It is possible to use the mapping of rotational variables to a 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, 2003 2L 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 simple one-particle H � h version of a model in the next few formulas) . The latter definitions lead to the rela- tions [ , ]a a b b a a b b� � � �� � � 0, [ , ] ,a a b b a b� � �� � 0 so that the initial h may be written in boson-fermion variables with L a a b b2 � �� � for each atom in our mapping h a a b b a b b aa b� � � � �� � � � � ��� � � � � � � 2 2 ( ), [ , ( )] [ , ] ,h a a b b h a a b b� �� � � � �� � � � � 1 2 0 where the energies � a b, of fermions contain their ki- netic energies. These relations are truth for a large kmax, so that a number m0 of traped levels satisfy an equation m0 1�� , just like it happens in experiment. If we turn back to the total Hamiltonian H , the thermodynamical parameters � and � will correspond the integrals of motion K and M H H� � � � � �� ��� � � � � �K M kk k k kph rot max [ ( ) ] � �� � � � � �[ ]� �a i i b i i i N a a b b 1 , � �� � �� � � �a b a b , , ,� � � � � 2 , { } { , },a a b bi j i j ij � �� � � for the partition function of a system. The latter may be represented as the path integral for a large Gibbs distribution Q K M� � � � �Sp exp [ ( )]� � �H � �� � � D D Da Da Db Db S k k k i i N i i i� �* * * exp 1 , 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 of freedom of the system (1) in an ordinary way. Inte- gral over all fermionic (a b, ) fields in Q may be calculated exactly as Det ( )�2S [4,5], so we get a path integral over the variables � k, � k * with an effec- tive action Sef for every k Q D D S k k k� �� � � �* exp ( , )ef 0 , S S N Rkef ph Det( , ) ln0 � � � , S d dt dtk k kph � � � �� � � ��� � � � � * 0 , R L / N / N L k k k k k � � � � � � � � � � � � * , L d/dt L d/dta b � � � �� � � � � �� �, . The quasiclassical equations of motion � �� � �� S S k k ef ef� � * 0 (3) are used for the extremal trajectories ( )*� k 0 and � k 0 that describe an effective translational modes in the field of roton motion. In the N �� 1 limit the transfor- mation � �k kN� and � �k kN* *� leads to the re-normalized action Seff S S N S Rk Nef eff ph Det� � � � ( ln )| | | | |� � , so that the equation of motion (3) in a new scale of boson trajectories looks like � �� � �� S S R k k t k eff Sp * * ( ln )� � � � � �� � � � � � � � � � ��d dt t R R t k k k k k � � � � �� ( ) ( )* Sp 1 0, (4) where a formula ln lnDet SpR Rk k� was used. As K is an integral of motion for (1), the first bond rela- tion for a thermodynamical parameters are 1 1 1 � � � � � � � � � � � � � ��� �S K R R k k keff Sp . (5) In the same way the second bond relation is found as 1 1 2 1 2 0 � � � � � � � � � � � � � � � � � � � � � � � ��S M R R dt k k k k eff Sp | | � ! ! ! . (6) The calculation of Sp in (4)–(6) is carried out fol- lowing [4,5] both in the matrix and Path Integral sense Phonons and rotons of trapped atoms in gravitational field Fizika Nizkikh Temperatur, 2003, v. 29, No. 6 675 Sp Spln lnR L L L L k k k� � � � �� � � � � � � � � � � � � 0 0 1 0 0 01 1 � � �k k� * 0 � � � � � � � � � � ! ! " # $ %$ & ' $ ($ � = Sp ln L L m L Lm k� � � ) � � � � � � � �� � � � � � ��0 0 1 2 0 0 0 1 1 1 � � k k k m � � * 0 2 � � � � � � � � � � ! ! " # $ %$ & ' $ ($ � � � � �� � � � � �1 2 2 1 2 1Sp ln , ( )( ).* *T T L L L Lk k k k k k� �� � � � The particular time-independent ( )* *� �k 0 0� , � �k 0 0� solution of equations (4)–(6) is similar to «slow» trajectories in superfluid 4He theory [1], asso- ciated with the Bose-Einstein condensation (BEC) in quantum liquid. In the same way � �0 0 *, trajectories correspond BEC of atoms in a trap, observed experi- mentally during the last years [6]. Using BEC as- sumption, we get the approach T T L L� � �� �0 0 2 0 2� | |� and see the variational equation (4) in the form � � � � �0 0 2 0 0 0 2 2 2 2 2 � � � � � � � � � � � � � � � �� � � � tanh tanh � � � � � � � ! ! , � � �0 2 0 2 0 2 1 24� �� �( | | )� / , (7) while the equations for integrals of motion K, M ap- pears as K k k� � � � � � � � � � � � � � � � � � � �1 2 2 2 2 2 tanh + tanh � � � � � � � � ! ! �� k 2, M k k� � � � � � � � � � � � � � � � � � �� �1 2 2 2 2 2 tanh tanh � � � � � � ! ! �� k � �2 2| |.� k . These equations determine the BEC ordering in a trap via 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 �0 0 2 0 0 0 0 0 2 � �� � � � � � sinh cosh cosh ( )A A A A , A0 0 2 2 2 � � � � � � � � � � � �� � � � � �� , and in the case of a small phonon-roton interaction � �0 * the approximation �0 � � � � � �� � � 2 10 2 0 2| | , � � ** is fulfilled, so that sinh ( )A A0 0 1� �� ** . Therefore, the inequality � � � � �0 2 0 0 0 2 0 2 0 24** � �� �, | | (8) is valid. It means, that BEC trajectories exist only for low phonon-roton interaction. It is worthy to note, that for an interaction � between polar atoms in segnetoelectric media, expressed by the same model as (1), the condition of dipol cooperation is expressed by the inequality [4] � �2 �� � , where � and � stands as the radiation and two-level system frequencies correspondingly. Just the oppo- site, the inequality (8) follows from the previous for- mulas as the condition for the equations (4)–(6) to have the BEC solution, and the condition of a large BEC density � �0 0 0| |�� � is also satisfied. The com- plete line of relations between the parameters of a theory looks like � �0 0 0 2 0 | | ||� � �* * , �� ** 1 (9) where both the BEC condition (8) and square root decomposition are taken into account. It can be seen, that the left side of (9) leads to formula (8). Trapped atoms in gravitational field Now we are going to look for the properties of a trapped atomic (BEC) in gravitational field. Let us consider a pure phonon Hamiltonian H ph p (without the 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. 6 D.B. Baranov and V.S. Yarunin where 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 be written in initial double quantized particle operators b bk k , � H g k kcb b� � , where c is constant. After diagonalizati on, H gcan be represented in terms of phonon operators +k, +� k b b k k k k k k 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 rot p with the Hamiltonian of gravitational field H g is not equal to zero. [ , ]H H H Hph rot ph rot p g� � -� 0. The deformation of the trap potential is shown on Fig. 1. Following the paper [7] we can conclude that the gravitational field manifest itself in decreasing of upper level kmax and in the shift of energy levels in the trap. In particular, for ideal bose gas in the trap [7], the critical temperature and number of condensate particles are compared for the cases of gravitational field 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 number of particles in trap. Conclusion The dilute gas of interacting bosons in magnetic trap under rotation was considered as a system with Hamiltonian 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 system was written as a path integral over boson and fermion trajectories with integrals of motion (2). After inte- gration over fermions, the quasiclassical equations of motion for boson trajectory in the ground state with their 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 satisfied for large | |�0 , so the presence of rotons diminishes the BEC density; b) the less amount of BEC bosons | |�0 2 we have, the larger will be a rotational momentum of atom for a given trap. These conclusions seems to be in accordance with the prioriting Popov [1] result, that the zero points of boson 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 in Quantum Statistics of Radiation and Matter, Kluwer Publ. (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 field Fizika Nizkikh Temperatur, 2003, v. 29, No. 6 677 U(Z) U U U –h/2 h/2h U + 0 g g Z Fig. 1. The deformation of the trap potential. The dashed and solid lines for a parabolic trap potential in a free space ( )U0 and in a gravitational field ( , )U U� � .

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