We use classical statistical mechanics and thermodynamics to describe the response of a trapped multi-species atomic sample to a local deformation in the confining potential. An adiabatic deformation may not only increase the peak phase-space density, but also lower the temperature and spin-polarize the atoms.Peer reviewe

Javanainen, Juha

Kaivola, Matti

Shevchenko, Andriy

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Physical Review A

Thermodynamics of a multicomponent atom sample in a tightly compressed atom trap

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Thermodynamics of a multicomponent-atom sample in a tightly compressed atom trap

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This is an electronic reprint of the original article.This reprint may differ from the original in pagination and typographic detail.Author(s): Shevchenko, Andriy & Kaivola, Matti & Javanainen, JuhaTitle: Thermodynamics of a multicomponent-atom sample in a tightlycompressed atom trapYear: 2004Version: Final published versionPlease cite the original version:Shevchenko, Andriy & Kaivola, Matti & Javanainen, Juha. 2004. Thermodynamics of amulticomponent-atom sample in a tightly compressed atom trap. Physical Review A.Volume 70, Issue 1. ISSN 1094-1622 (electronic). ISSN 1050-2947 (printed). DOI:10.1103/physreva.70.011403.Rights: © 2004 American Physical Society (APS). http://www.aps.orgAll material supplied via Aaltodoc is protected by copyright and other intellectual property rights, andduplication or sale of all or part of any of the repository collections is not permitted, except that material maybe duplicated by you for your research use or educational purposes in electronic or print form. You mustobtain permission for any other use. Electronic or print copies may not be offered, whether for sale orotherwise to anyone who is not an authorised user.Powered by TCPDF (www.tcpdf.org)Thermodynamics of a multicomponent-atom sample in a tightly compressed atom trapA. Shevchenko* and M. KaivolaDepartment of Engineering Physics and Mathematics, Helsinki University of Technology, P. O. Box 2200, FIN-02015 HUT, FinlandJ. JavanainenDepartment of Physics, University of Connecticut, Storrs, Connecticut 06269-3046, USA(Received 23 February 2004; revised manuscript received 30 April 2004; published 28 July 2004)We use classical statistical mechanics and thermodynamics to describe the response of a trapped multi-species atomic sample to a local deformation in the confining potential. An adiabatic deformation may not onlyincrease the peak phase-space density, but also lower the temperature and spin-polarize the atoms.DOI: 10.1103/PhysRevA.70.011403 PACS number(s): 32.80.Pj, 05.70.Ce, 39.25.1kThe manipulation of atoms in microscopic potential wellsis the technology presently regarded as the most likely foun-dation for practical applications of atom optics and Bose-Einstein condensation (BEC) [1,2]. Well-localized magnetic[2–4] or electric [5,6] fields above the surface of a solidsubstrate have been successfully used to control the motionof microscopic atom samples [2,5] and to essentially sim-plify the creation of BEC [2,7,8].In regard to BEC, phase-space density is the key param-eter. It cannot be increased by scaling the trapping potential[9,10], but may be substantially adjusted by adiabaticallydeforming the trap [10–13]. This is also the case in evapora-tive cooling, where the trap is deformed in order to releaseatoms with a high kinetic energy. A similar increase of thephase space density is achieved by adding a tight potential toa wide trap. This method together with subsequent evapora-tion has been used to reach Bose-Einstein condensation ofcesium [14] and, very recently, to create a two-dimensionalBEC of cesium directly in a surface trap [15].Phase space compression by a local change in the trap-ping potential has been studied theoretically for a singleatomic species [11]. However, a given trap may carry morethan one species, each of which may respond differently tothe deformation. The present paper is based on our realiza-tion that allowing transitions between the Zeeman states ofthe atoms in conjunction with adjustments of the shape of anatom trap adds flexibility to the preparation of an atomicsample in a microtrap that could be part of an “atom chip”[1]. Such a chip could contain not only magnetic microtraps,but also static electric or all-optical traps. If, for instance, thechip substrate is made of material which is transparent tolight, a gravito-optical surface trap [16–18] could serve as acontainer of cold atoms and as a trap to be modified locally.In the present paper we use classical statistical mechanicsand thermodynamics to investigate in general the propertiesof a multicomponent atom sample in a trap with locally de-formed potential, allowing for both thermal and chemicalequilibration. We demonstrate how an adiabatically drivendeformation may decrease the temperature, increase thephase-space density of atoms in a particular magnetic state,and spin-polarize atoms in an initially spin-independent atomtrap.Even though our method works for an arbitrary 3D atomtrap, for illustration we consider a model for a surface trap sosimple that the ensuing numerics can be handled with unso-phisticated MATHEMATICA programs. Thus, the atoms arebound to a potential of the form Usxd+ULsyd+Uhszd, wherex ,y, and z are Cartesian coordinates. The dependence on z isassumed to be harmonic, Uhszd=mw02z2 /2, with m being theatomic mass and w0 the trapping frequency in the directionperpendicular to the surface. In the x and y directions theatoms are confined to a slab of length l and width L. We setULsyd;0, and Usxd is a step function with the value L addedto a constant background in an interval of x of length hl,with h,1. Overall, the potential inside the slab reads asUtsx ,y ,zd=mw02z2 /2+Usxd. Figure 1 shows the trap bound-aries and the potential Utsx ,y ,zd in a plane sx ,zd at a fixedvalue of y. Simplistic as the model is, we believe that itproduces qualitatively correct results for the generic casewhen a potential is modified by an amount L in a fraction hof the trap volume.We start by considering an ensemble of identical, but, inthe sense of classical physics, distinguishable particles in thepotential described above. As customary in kinetic theory fordilute gases, we ignore the effect of the interactions on theequilibrium. For N particles the canonical partition functionisZN =1N!S 1s2p"d3 E d3r d3p e−H/kBTDN, s1dwhere kB is Boltzmann’s constant, T is the absolute tempera-ture, and H=p2 /2m+Ut is the one-particle Hamiltonian.*Electronic address: andrej@focus.hut.fiFIG. 1. (a) The trap boundaries in the x and y directions. (b) Thetrapping potential, Utsx ,y ,zd, in a plane sx ,zd at a fixed y from theinterval 0,y,L. In the sketch, L is negative.PHYSICAL REVIEW A 70, 011403(R) (2004)RAPID COMMUNICATIONS1050-2947/2004/70(1)/011403(4)/$22.50 ©2004 The American Physical Society70 011403-1Helmholtz free energy F=−kBT ln ZN is found to beF = NkBT1ln3 rsTdE0lsdx/lde−Usxd/kBT4 − 12 , s2dwherersTd =2p"3w0NkB2T2mlLs3dis the usual phase space density at the bottom of the confin-ing potential in the absence of the deformation, andnsTd =˛ m2pkBTw0NlLs4dis the corresponding density. The final −1 in Eq. (2) origi-nates from the Stirling expansion as in ln N ! =N ln N−N.The entropy S=−]F /]T isS = NkB13 − ln3 rsTdE0lsdx/lde−Usxd/kBT4 + kUxlkBT 2 , s5dwhere kUxl is the thermal expectation value of the potentialUsxd. The chemical potential is found as m=]F /]N. For thestepwise shape of Usxd, we derive the following analyticalexpressions for one-particle dimensionless entropy s=S /NkB, mean energy per particle expressed in units of tem-perature e= sF+TSd /NkB, and chemical potential j=m /kBsimilarly in units of temperature,s = 3 − lnrsTd1 − h + he−D/T+he−D/TDTs1 − h + he−D/Td, s6de = e0 + 2T +he−D/TD1 − h + he−D/T, s7dj = e0 + TSln rsTd1 − h + he−D/T − 1D . s8dHere D=L /kB is the local change of the potential and e0 thepotential at the bottom of the nondeformed trap, both also inunits of temperature. The phase space density at the potentialminimum within the deformation and the corresponding den-sity are obtained fromrsT,D,hd =rsTdh + s1 − hdeD/T, s9dnsT,D,hd =nsTdh + s1 − hdeD/T, s10dwith rsTd and nsTd defined in Eqs. (3) and (4).In the case of a single atomic species in the trap, an adia-batic change of the potential is described byssTi,Di,hid = ssTf,D f,h fd , s11dwhich is the requirement for conservation of entropy. Theindices i and f denote the initial and final values of the vari-ables, respectively. By solving Eq. (11), one can find Tf, andsubsequently, all other thermodynamic parameters of thesystem.We proceed to the case in which two or more atomicspecies k, each in thermal equilibrium in its own locally de-formed potential well, are allowed to exchange energy butnot particles. The final equilibrium temperature Tf of thewhole system is then calculated by equating the initial andfinal total energies of the system. On the other hand, if thedeformations are varied adiabatically starting from a com-mon temperature, the final temperature is found by equatingthe entropy of the whole system before and after thetransformation,oksksTi,Dki,hkidNk = oksksTf,Dkf,hkfdNk. s12dNext we think of atoms in a particular quantum state as aspecies that can exchange particles with the other species.Zeeman states in an angular-momentum degenerate energylevel serve as an example. Equilibration between the speciescan be controlled with rf driving fields, or with laser drivenRaman transitions. Inasmuch as the rotating wave approxi-mation of quantum optics is applicable, the detunings set therelative energies of the states for the purposes ofthermodynamics.Again we start by considering the case when each speciesis initially in its own equilibrium at a temperature Tki, where-after the exchanges of energy and particles equilibrate thewhole system. Since the system as a whole is closed, thetotal energy is conserved. We therefore writeokeksTki,Dk,hkdNki = okeksTf,Dk,hkdNkf . s13dThe final state will contain species with equal chemical po-tentials. We havejksTf,Dk,hkd = jk8sTf,Dk8,hk8d , s14dokNkf = N , s15dwhere k and k8 denote different species, and N is the con-served total number of atoms. The final temperature Tf andthe atom numbers for each species Nkf are found by solvingthe coupled equations (13)–(15).Yet another situation occurs if the local deformations ofthe states evolve adiabatically in an already equilibrated sys-tem. Then the chemical potentials of the subensembles willbe equal to each other at each moment of the transformationand the entropy of the whole system will be conserved.Hence, the equations for calculation of Tf and Nkf arejksTf,Dkf,hkfd = jk8sTf,Dk8f,hk8fd , s16dokNkf = Nt, s17dSHEVCHENKO, KAIVOLA, AND JAVANAINEN PHYSICAL REVIEW A 70, 011403(R) (2004)RAPID COMMUNICATIONS011403-2oksksTi,Dki,hki,NkidNki = oksksTf,Dkf,hkfNkfdNkf .s18dNow that the procedure for solving Tf and Nkf in differentthermodynamic processes has been laid down, we proceed toapplications. We first consider the possibility to reach alower temperature in laser cooling by introducing a smallpotential well in the trap during the cooling stage and adia-batically removing the well after the cooling is complete.This method can be applied if the cooling efficiency does notchange in the presence of the subtrap and if the cooledsample can reach thermal equilibrium before the coolingmechanisms are switched off. These conditions can be satis-fied, e.g., in evanescent-wave cooling of atoms in a GOST[12,15–18]. In fact, a “dimple” subtrap created in a GOSTwith an infrared focused laser beam is frequently employed[12,15]. Figure 2(a) shows the final temperature, TfsDi ,hd,that can be achieved by using a subtrap of initial depthuDiu sDi,0d and effective volume characterized by h. Thecalculations are done with the aid of Eq. (11), where D f is setto 0 and both hi and h f are replaced with the same parameterh. For generality, we use the dimensionless variables h anduDiu /T0, where T0 is the initial temperature fT0;Ti;TfsDi=0dg. The calculations show that after cooling of the atomsin the presence of a small subtrap, the temperature may bedecreased by an additional order of magnitude by slowlyswitching the subtrap off. The method allows reduction ofthe temperature below the recoil limit, a problematic task forlaser cooling alone. We note that in general the use of asmaller but deeper subtrap will allow the temperature to de-crease to a lower level. However, depending on the totalnumber of atoms, this can lead to an inappropriately highloss rate of the trapped atoms due to inelastic two- and three-body collisions. The calculated curves for Tf /T0 help to findoptimum values for the parameters h and Di.In the next example we calculate the peak phase spacedensity of atoms in a trap with an adiabatically inserted sub-trap [11]. The peak phase space density rsD f ,hd related to itsoriginal value r0=rsD f =0d is shown in Fig. 2(b) as a func-tion of h, volume ratio, and uD fu /T0, the final depth of thesubtrap over the initial temperature. The calculations are car-ried out using Eq. (11) with Di=0 and hi=h f ;h to find Tf,and Eq. (9) to find rsTf ,D f ,hd;rsD f ,hd. The results agreewell with the experiments of Ref. [12], where the phasespace density was locally increased by two orders of magni-tude using a “dimple” trap within a GOST. It can be seen thatat each fixed value of h, there is an optimum value ofuD fu /T0, at which the phase-space density reaches its maxi-mum. Above this value, the number of atoms transferred intothe subtrap becomes too large, so that the rest of the atoms inthe trap can no longer keep the temperature at a low enoughlevel.We proceed to the case when a spin-independent atomtrap is locally deformed with a magnetic field. In a dipoletrap with, e.g., spin-unpolarized 87Rb in the lower hyperfineground state, the magnetic field splits up the sample intothree species characterized by the magnetic quantum num-bers mF= h−1,0 , +1j. When the trap is deformed with a lo-cally high magnetic field, the atoms with mF= +1 are at-tracted to the deformation, while atoms with mF=−1 arerepelled from it. The deformation can also be created byusing a magnetic-field minimum which will attract mF=−1atoms. Independently of the deformation type, there will bethree species sk=1,2 ,3d with Dki=0 and Dkf = h−D f ,0 , +D fj,where now D f is chosen to be positive. All the parameters hkiand hkf are set to the same value h.Let us denote the initial temperature by T0, and the initialphase-space density, equal for all species, by r0. We calculatethe enhancement of the phase-space density, rsTf ,−D f ,hd /r0, for the species with Dkf =−D f, considering threedifferent cases. In the first case we assume that the exchangeof energy between the species occurs much faster than par-ticle exchange, and ignore the latter altogether. We simplyapply the conservation of entropy in Eq. (12). In the secondFIG. 2. (a) A decrease of temperature in an atom trap caused bythe adiabatic removal of a subtrap. (b) An increase of the peakphase-space density of trapped atoms caused by adiabatic insertionof a subtrap. (c) An increase of the peak phase-space density ofatoms in a particular magnetic state due to adiabatic deformation ofthe trap with a static magnetic field. (d) The spin polarization ofatoms in a dipole trap due to adiabatic deformation of the trappingpotential with a static magnetic field.THERMODYNAMICS OF A MULTICOMPONENT-ATOM PHYSICAL REVIEW A 70, 011403(R) (2004)RAPID COMMUNICATIONS011403-3case, we allow the equilibration of atom numbers to followthe equilibration of temperature. We solve Eq. (12), and thenreplace all Tki in Eqs. (13) and (14) with the obtained Tf.Then, solving Eqs. (13)–(15), we find a new Tf and newatom numbers Nkf. In the third case we allow the system tobe in total equilibrium (thermal and chemical) at each mo-ment of the transformation, and use Eqs. (16)–(18) directly.When the phase space density increases substantially for thespecies attracted toward the deformation, rsTf ,−D f ,hd@r0,the results for the three scenarios turn out to be almost indis-tinguishable. In Fig. 2(c) we show a plot corresponding tothe first case. The dependence of rsTf ,−D f ,hd /r0 on theparameters h and D f /T0 is similar to the dependence illus-trated in Fig. 2(b) for a single species. However, here thefactor by which the phase space density increases is aboutthree times larger, as if the sample was spin-polarized at thetemperature T0 prior to applying the magnetic field.Our final example may be of interest to those who aresearching for ways to spin-polarize atoms without increasingthe temperature. Let atoms in a ground state with three mag-netic substates be stored in a dipole trap in the presence of anexternal magnetic field. The constant magnetic field splits theinitial degeneracy of the species. However, if a rf field inresonance with the splitting is used to accelerate the equili-bration between the species, the degeneracy is restored as faras it comes to thermodynamics. Next, a local magnetic-fielddeformation is inserted adiabatically. To calculate the finalatom numbers Nkf of the three magnetic states, we solve Eqs.(16)–(18) with hki=hkf ;h, Dki=0, and Dkf = h−D f ,0 , +D fj.The dependence of the normalized population N1f /Nt (k=1corresponds to the state deformed by −D f) on h and D f /T0 isplotted in Fig. 2(d). The result shows that the use of a defor-mation with a large relative volume, h, and/or a large relativestrength, D f /T0, can provide nearly complete spin polariza-tion of the atoms. By switching off the rf field and slowlyremoving the deformation, one can then return the trap to itsinitial shape and the sample to its initial temperature. Wenote that decoupling of the magnetic states by switching offthe rf field leads to an essential change of the system, be-cause the rest of the transformation proceeds without equili-bration of the magnetic-state populations. The disappearanceof the rf field does not affect the spatial profiles of the mag-netic energy levels and, therefore, it does not change thetemperature.In conclusion, we have proposed a simple theoreticalmodel based on classical statistical mechanics to describe atrapped sample of atoms in the presence of an adiabaticallyvarying local deformation of the confining potential. Such adeformation can substantially increase the peak phase spacedensity [10,11]. In regard to applying the model to integratedatom chips [1] we have noted here that a rich variety of newways to manipulate the sample opens up by making use ofthe magnetic substates of an atom. Examples are presentedon increasing the phase phase density, on atom cooling, andon polarizing the atomic spins.We acknowledge financial support from the Academy ofFinland, NSF, and NASA. One of us (J.J.) is particularlygrateful to the Helsinki University of Technology for thehospitality extended to him during the initial phases of thiswork.[1] R. Folman, P. Krüger, J. Schmiedmayer, J. Denschlag, and C.Henkel, Adv. At., Mol., Opt. Phys. 48, 263 (2002).[2] J. Reichel, Appl. Phys. B: Lasers Opt. 75, 469 (2002).[3] D. Cassettari et al., Appl. Phys. B: Lasers Opt. 70, 721 (2000).[4] M. Vengalattore, W. Rooijakkers, and M. Prentiss, Phys. Rev.A 66, 053403 (2002).[5] P. Krüger et al., Phys. Rev. Lett. 91, 233201 (2003).[6] S. K. Sekatskii, B. Riedo, and G. Dietler, Opt. Commun. 195,197 (2001).[7] W. Hänsel, P. Hommelhoff, T. W. Hänsch, and J. Reichel, Na-ture (London) 413, 498 (2001).[8] S. Schneider et al., Phys. Rev. A 67, 023612 (2003).[9] W. Ketterle and D. E. Pritchard, Phys. Rev. A 46, 4051 (1992).[10] P. W. H. Pinkse et al., Phys. Rev. Lett. 78, 990 (1997).[11] D. M. Stamper-Kurn et al., Phys. Rev. Lett. 81, 2194 (1998).[12] M. Hammes, D. Rychtarik, H.-C. Nägerl, and R. Grimm, Phys.Rev. A 66, 051401 (2002).[13] L. Viverit, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Phys.Rev. A 63, 033603 (2001).[14] T. Weber et al., Science 299, 232 (2003).[15] D. Rychtarik, B. Engeser, H.-C. Nägel, and R. Grimm, e–printcond-mat/0309536.[16] M. Hammes et al. J. Mod. Opt. 47, 2755 (2000).[17] Yu. B. Ovchinnikov, I. Manek, and R. Grimm, Phys. Rev. Lett.79, 2225 (1997).[18] J. Söding, R. Grimm, and Yu. B. Ovchinnikov, Opt. Commun.119, 652 (1995).SHEVCHENKO, KAIVOLA, AND JAVANAINEN PHYSICAL REVIEW A 70, 011403(R) (2004)RAPID COMMUNICATIONS011403-4

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Thermodynamics of a multicomponent-atom sample in a tightly compressed atom trap

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