Theory of an optical dipole trap for cold atoms

Transcription

1 Theory of an optical ipole trap for col atoms Article (Publishe Version) Garraway B M an Minogin V G () Theory of an optical ipole trap for col atoms. Physical Review A 6 (4) ISSN This version is available from Sussex Research Online: This ocument is mae available in accorance with publisher policies an may iffer from the publishe version or from the version of recor. If you wish to cite this item you are avise to consult the publisher s version. Please see the URL above for etails on accessing the publishe version. Copyright an reuse: Sussex Research Online is a igital repository of the research output of the University. Copyright an all moral rights to the version of the paper presente here belong to the iniviual author(s) an/or other copyright owners. To the extent reasonable an practicable the material mae available in SRO has been checke for eligibility before being mae available. Copies of full text items generally can be reprouce isplaye or performe an given to thir parties in any format or meium for personal research or stuy eucational or not-for-profit purposes without prior permission or charge provie that the authors title an full bibliographic etails are creite a hyperlink an/or URL is given for the original metaata page an the content is not change in any way.

2 Theory of an optical ipole trap for col atoms B. M. Garraway Sussex Centre for Optical an Atomic Physics School of Chemistry Physics an Environmental Sciences University of Sussex Falmer Brighton BN 9QJ Englan V. G. Minogin Institute of Spectroscopy Russian Acaemy of Sciences 49 Troitsk Moscow Region Russia Receive 6 April ; publishe 5 September The theory of an atom ipole trap compose of a focuse far re-etune trapping laser beam an a pair of re-etune counterpropagating cooling beams is evelope for the simplest realistic multilevel ipole interaction scheme base on a moel of a 35-level atom. The escription of atomic motion in the trap is base on the quantum kinetic equations for the atomic ensity matrix an the reuce quasiclassical kinetic equation for atomic istribution function. It is shown that when the etuning of the trapping fiel is much larger than the etuning of the cooling fiel an with low saturation the one-photon absorption emission processes responsible for the trapping potential can be well separate from the two-photon processes responsible for sub-doppler cooling atoms in the trap. Two conitions are erive that are necessary an sufficient for stable atomic trapping. The conitions show that stable atomic trapping in the optical ipole trap can be achieve when the trapping fiel has no effect on the two-photon cooling process an when the cooling fiel oes not change the structure of the trapping potential but changes only the numerical value of the trapping potential well. It is conclue that the separation of the trapping an cooling processes in a pure optical ipole trap allows one to cool trappe atoms own to a minimum temperature close to the recoil temperature keeping simultaneously a eep potential well. PACS numbers: 3.8.Pj 4.5.Vk PHYSICAL REVIEW A VOLUME I. INTRODUCTION In recent years there has been a growing interest in eveloping traps for col atoms with applications to funamental physical problems nanotechnology high-resolution microwave an optical spectroscopy frequency stanars an atom optics 5. Among ifferent types of atom traps stuie in recent years of practical importance is a far-offresonance optical ipole trap FORT base on a single focuse re-etune laser beam 6. The FORT prouces a nearly conservative potential well for atoms but incorporates the inevitable heating ue to the photon recoil associate with the scattere laser light 89. Although the heating rate may be very small at very large etuning from the resonance the photon recoil heating inevitably introuces an upper limit on the lifetime of atoms in the trap. It was previously propose that the heating mechanism in the FORT might be suppresse by aing the cooling laser fiels to a focuse trapping laser beam 78. Latest experiments with ifferent types of cooling laser fiels have shown that the aition of the cooling fiel can increase the lifetime of atoms an even the atomic ensity in the trap. The aition of the cooling fiel changes not only the Boltzmann factor that efines the lifetime of atoms in the FORT but may also have a profoun effect on all the basic parameters of the trap since the cooling fiel may strongly influence the atomic populations an coherences. Typically any cooling laser fiel operates at a etuning less than the etuning of the trapping laser beam. The cooling fiel may thus be responsible not only for perturbation of the steaystate internal atomic state but the perturbation of the trapping potential as well. Up to now the FORT with an aitional cooling laser fiel was theoretically iscusse for the simplest moel of a two-level ipole interaction scheme 73. A two-level moel has however a very limite connection with real experimental techniques that typically explore multilevel ipole interaction schemes. Physically there is a major ifference between the moels of a two-level atom an a multilevel atom in applications relate to the trapping an cooling atoms. In a two-level moel both trapping an cooling fiels excite atoms on the same atomic transition. As a result for a two-level atomic scheme in the FORT the epth of the potential well an the cooling limit have generally the same orer of magnitue 73 efine by a so-calle Doppler temperature 4. In multilevel atomic schemes the trapping an cooling laser fiels can prouce principally ifferent atomic transitions. The trapping fiel can prouce a potential well ue to the one-photon transitions while the cooling laser fiel can cool atoms own to the sub-doppler temperatures 5 8 ue to the two-photon transitions 9. It is thus important that in multilevel atomic schemes the optical processes use for trapping atoms in the FORT an those for sub-doppler cooling can have the ifferent physical origin. The use of the ifferent optical processes for trapping an cooling multilevel atoms thus raises new questions on basic parameters of the trap an the lifetime of atoms achievable in the FORT with a superimpose sub-doppler cooling process. In this paper we present a theoretical analysis of the FORT compose of a single re-etune trapping laser beam an a re-etune cooling fiel compose of the counterpropagating circular-polarize laser waves. To be specific we evelop an analytical theory for a simplest realistic moel of a 35-level atom interacting with one-imensional 5-947//64/4346/$ The American Physical Society

3 B. M. GARRAWAY AND V. G. MINOGIN PHYSICAL REVIEW A processes while a less re-etune - laser fiel is responsible for laser cooling ue to the two-photon optical processes. In the chosen interaction scheme the fiel of a linearly polarize trapping laser beam E t e E t rcosky t t is efine by a unit polarization vector e e z spatially nonuniform amplitue E t (r) an the wave vector k t ke y (k t /c). The cooling laser fiel is assume to be compose of two counterpropagating left circularly polarize waves E c E ce e i(kz c t) e e i(kz c t) E ce e i(kz c t) e e i(kz c t) FIG.. One-imensional scheme of the FORT compose of a re-etune trapping laser beam tr an the re-etune cooling laser fiel c in the - configuration a an the ipole interaction scheme of a 35-level atom with the trapping an cooling laser fiels b. The magnetic sublevels for the groun an excite states of the atom are enote as: j g mg m j e m e m. counterpropagating laser waves in a - configuration. In this moel the ipole potential is cause by the one-photon processes while the friction force comes from the twophoton processes. Our analysis shows that uner specific conitions formulate below the one-photon processes proucing the optical ipole potential can have a negligible influence on the twophoton processes responsible for sub-doppler cooling of atoms. On the other han the perturbation of the atomic internal state prouce by the two-photon cooling processes may not change the structure of the potential well but only the value of the potential well epth. The separation of the trapping an cooling processes achievable uner specifie conitions allows one to maintain the epth of the potential well in the FORT that excees consierably the kinetic energy of trappe atoms. As a result the lifetime of atoms associate with the photon scattering in the FORT can be extremely long thus leaving other processes such as atom collisions an the reabsorption of laser light 4 to be responsible for the atom storage time. II. MULTILEVEL INTERACTION SCHEME The simplest scheme for a FORT may be escribe by the ipole interaction of a 35-level atom with a linearly polarize focuse trapping laser beam an one-imensional cooling - laser-fiel configuration as shown in Fig.. In the scheme a far re-etune laser beam is assume to be responsible for atomic trapping ue to the one-photon optical where e (/)(e x ie y ) are spherical unit vectors c is the frequency of the cooling laser waves an k c /c is the magnitue of the wave vector. With respect to the quantization axis Oz the first wave in Eq. is a polarize wave an the secon one is a polarize wave. Note that in a nonrelativistic approach the magnitue of the wave vector for the counterpropagating cooling laser fiels is consiere to be the same. For the above interaction scheme the atomic Hamiltonian has a stanar form HH M D E where the Hamiltonian H escribes the internal atomic states with energy levels E g E g an E e E e E e an the last term escribes the ipole interaction between the atom an the total laser fiel EE t E c. III. UNPERTURBED TRAPPING POTENTIAL Consier first the trapping potential for the above interaction scheme in the absence of the cooling fiel. The quasiclassical atomic-ensity-matrix equations escribing the ipole interaction of a 35-level atom with the fiel in a rotating wave approximation RWA are liste in Appenix A. The set inclues only equations for the ensity matrix elements that o not ecay to zero for an interaction time t. The ensity matrix elements entering the equations of Appenix A are efine with respect to the timeepenent atomic wave functions. The equations inclue on the left-han sie the total convective time erivatives t t v r. The Rabi frequency for the trapping laser beam an the etuning are: 3 4 r E tr 3 t

4 THEORY OF AN OPTICAL DIPOLE TRAP FOR COLD ATOMS PHYSICAL REVIEW A where is the reuce ipole matrix element an is the atomic transition frequency. The Rabi frequency is efine with respect to the strongest -type ipole transition j g m g j e m e. The spontaneous ecay rate W sp is efine in the usual way as W sp c 3. 6 Eliminating an explicit time an position epenence in the equations for the ensity matrix elements with the substitutions g e g e e i(kyt) g e g e e i(kyt) g e g e e i(kyt) an solving next the equations for a steay state one can fin the ipole raiation force on a 35-level atom with the use of a well-justifie quasiclassical formula F D"E where DD ab ba is an expectation value of the atomic ipole moment an the graient is assume to act on the electric fiel E only. The raiation force on a 35-level atom in a laser beam inclues as usual two partial forces i.e. the graient force an the raiation pressure force FF gr F rp F gr rre g e 3 g e g e F rp ke y Im e g 3 e g e g The case of large negative etuning is of particular interest in this paper i.e. we will let (r) an for a motionless atom the forces are then F gr 3 7 r F rp 3 7 ke r y. The graient force prouces the potential well for col atoms that iffers from that for a two-level atom 7 by a numerical factor only that is U F gr r 3 r 7. 3 The raiation pressure force prouces an aitional asymmetric potential well that can be neglecte at large etuning. IV. QUANTUM KINETIC EQUATIONS In general to escribe the atomic motion we can use the approach base on the quantum kinetic equations for the atomic ensity matrix in the Wigner representation. This is relatively straightforwar for a scheme such as the one consiere in Fig. where the atom interacts with both laser fiels an. Since we wish to analyze the operation of the trap at large etunings it is sufficient for our purpose to consier the quantum kinetic equations at small optical saturation. We will take into account one- an two-photon optical processes an neglect higher-orer optical processes. To simplify the consieration of the equations of motion we write out the initial microscopic equations but we neglect first the spatial variation in the trapping laser beam amplitue E t (r). The effect of the spatially inhomogeneous trapping laser beam will be taken into account in Sec. VI. A set of the atomic ensity matrix equations in the Wigner representation an a rotating wave approximation RWA escribing an interaction of a 35-level atom with laser waves of constant amplitue at weak optical saturation is liste in Appenix B. The set only inclues the equations for the ensity matrix elements escribing one- an two-photon processes. As before the ensity matrix elements entering the equations of Appenix B are efine with respect to the time-epenent atomic wave functions. The upper inices for the momentum-shifte ensity matrix elements are chosen to have ifferent meaning for the terms coming from the interaction with a trapping an cooling fiel i.e. () ab a rp ke y tb () ab a rp ke z tb ab arpnktb 4 where n is a unit vector that efines the irection of the spontaneous photon emission. The total time erivative on the left-han sie of the equations is efine as before by Eq. 4. The Rabi frequency for the cooling laser fiel efine with respect to the strongest -type ipole transition an the etuning of the cooling fiel are E c 5 c. 5 Functions with efine the angular anisotropy of the spontaneous photon emission n 3 6 n z n 3 8 n z 6 where n z is the projection of the unit vector n on the quantization axis Oz. Numerical factors entering the ensity matrix equations are ue to the Clebsch-Goran coefficients efining the ipole matrix elements for specific atomic transitions. Our choice of sign of the reuce ipole moment can be seen from the values of the matrix elements for the strongest -type an -type atomic transitions

5 B. M. GARRAWAY AND V. G. MINOGIN PHYSICAL REVIEW A e g g e 5 e g g e e g g e 5. Note that when the cooling laser fiel an photon recoil are neglecte the equations of Appenix B reuce to the quasiclassical equations of Appenix A. The quantum kinetic equations of Appenix B are use below to erive the quasiclassical kinetic equation of atomic motion that efines the ipole raiation force on the atom in the total laser fiel the momentum iffusion tensor an the potential well for the motionless atom. V. QUASICLASSICAL KINETIC EQUATION The initial quantum kinetic equations for the atomic ensity matrix elements can be first simplifie by the stanar proceure of eliminating the explicit time an position epenence with proper substitutions. Since the initial equations inclue only terms escribing one- an two-photon processes we introuce substitutions that also take into account only one- an two-photon processes: g e g e e ikzit g e g e e ikyit g e g e e ikzit g e g e e ikzit g e g e e ikyit g e g e e ikzit g e g e e ikzit g e g e e ikyit g g g g g g g g e e e e e e e e g e g e e ikzit e ik(yz)i()t g g e ik(yz)i()t g g g g e ikz e ik(yz)i()t g g e ik(yz)i()t e e e e e ik(yz)i()t e e e e e ikz e ik(yz)i()t e e e ik(yz)i()t e e e e e ikz e ik(yz)i()t e e e ik(yz)i()t e e e e e ikz e e e e e ik(yz)i()t. 7 Having mae the above substitutions we neglect in the microscopic equations all terms that escribe three-photon an higher-orer multiphoton processes. This proceure finally results in a set of equations that oes not inclue explicit time an position epenence. An analysis of the ensity matrix equations that have no explicit time an coorinate epenence can be one in the usual way 3. Uner the conition that atom-laser fiel interaction time excees the characteristic relaxation times of the atomic-ensity-matrix the momentum with of the atomic ensity-matrix elements can be assume to excee the photon momentum k. This principal assumption which will be checke below allows one to expan the ensitymatrix elements in powers of the photon momentum k. Consiering next the time evolution of the equations expane in successively increasing orers of the photon momentum k one can conclue that the iagonal elements aa an the off-iagonal elements ab are functionals of the Wigner quasiprobability istribution function ww(rpt) w g g e e. ; 8 The structure of the functional epenence for the iagonal an off-iagonal ensity matrix elements can be irectly etermine from the structure of the expane equations: aa R aa kr aa w kq aa w p z ab S ab ks ab w kt ab w p z 9 where R aa S ab R aa S ab...q aa T ab... are functions of atomic momentum p or atomic velocity vp/m that have to be etermine by the solution proceure. In accorance with the efinition of the quasi-probability istribution function 8 the unknown iagonal functions satisfy the normalization conitions R aa R aa Q aa.... Substituting next the general solution 9 into the expane equations for the atomic ensity-matrix elements one can see that the Wigner function ww(rpt) satisfies an infinite Fokker-Planck equation. To the secon orer in the photon momentum k the equation for the quasiprobability istribution function ww(rpt) is then escribe by a secon-orer Fokker-Planck kinetic equation w t w v r p Fw p D iiw i where ixyz F is the ipole raiation force on the atom an D ii is the momentum iffusion tensor

6 THEORY OF AN OPTICAL DIPOLE TRAP FOR COLD ATOMS PHYSICAL REVIEW A In this section we consier only homogeneous laser waves where the ipole raiation force has the clear meaning of a raiation pressure force FF rp e z F rp. This force an the momentum iffusion tensor entering Eq. are F rp k ImS e g S e g 6 S e g S e g D ii k ii R e e iz k R e e ii ImT e g S e g S e g R e e 3 R e e R e e R e e ImT g e T g e 3 R e e R e e T e g 3 ImT g e T g e. 3 In the above equations the coefficients ii ii etermine the angular anisotropy of the spontaneous photon emission ii nn i n xx yy 3 zz 5 ; ii nn i n xx yy 5 4 zz 5. 5 The values of the force an momentum iffusion coefficients F rp an D ii which govern the time evolution of the quasiclassical istribution function can be explicitly etermine by solving the steay-state equations that follow from the expane equations for the atomic-ensity-matrix elements. These are consiere separately in zero an first orer in photon momentum k. Explicit equations for the quantities F rp an D ii are given in Sec. VIII. VI. DIPOLE GRADIENT FORCE We next take into consieration the ipole graient force F gr associate with the graient of the trapping laser beam amplitue E t E t (r). The erivation of the graient force can be one in a way that generalizes the proceure consiere in the preceing section. Representing the trapping laser beam amplitue in the form of a Fourier expansion ir ab p 3/ iqe iq"r ab p q 3 q. 7 For a laser beam amplitue that varies in space on a scale that is large compare to the size of the atomic wave packet we use an expansion of the ensity-matrix elements to first orer in the small momentum q ab p q ab p q p abp. This transforms the terms in equations of Appenix B into ir ab p ir ab p r p abp. 8 9 When the substitutions 9 are introuce into the equations of Appenix B the Fokker-Planck equation inclues in the above approximation the total raiation force as a sum of the ipole graient force an the raiation pressure force FF gr F rp. 3 The ipole graient force is then etermine by the steaystate optical coherences as F gr rre S g e 3 S g e S g e. 3 Note that formal mathematical expression for this part of the total raiation force coincies with that given for the graient force by Eq. 9. The explicit expressions given by Eqs. 3 an 9 are generally ifferent since the functions S ab escribe the interaction of the atom with the total laser fiel efine by the equations of Appenix B while the functions ab escribe the interaction of the atom with only the trapping laser beam efine by the equations of Appenix A. We will see in Sec. VIII that Eq. 3 gives the graient force 4 that iffers by a numerical factor from the graient force given by Eq. 9. VII. LOW-SATURATION SOLUTION We will etermine the steay-state atomic ensity matrix elements which efine the partial forces an the momentum iffusion tensor in a regime characterize by two important limits. That is we consier large etunings where an we consier low optical saturation 3 E t r 3/ E t qe iq"r 3 q 6 s t s c 33 one shoul introuce into the equations of Appenix B the following substitutions: when the one-photon an two-photon processes play a ominant role in the time evolution of the atomic-ensity-matrix

7 B. M. GARRAWAY AND V. G. MINOGIN PHYSICAL REVIEW A elements. In aition to the above approximations we restrict our treatment to the case of slowly moving atoms where where the steay-state groun-state populations N R g g N R g g an N R g g connecte to the groun-state two-photon coherence g g are y kv y z kv z. 34 Uner all these conitions the sets of equations for the functions R aa S ab an T ab which follow from the equations of Appenix B expane to the first orer in the photon momentum have a simple analytical solution. When solving the last equations one can note that the efficiencies of the twophoton optical processes between the groun-state magnetic sublevels epen crucially on two parameters an N D 3 N D N D 6 k v z an the common enominator is kv z9k v z kv z9k v z D 7 4 k v z which efine the frequency withs of the two-photon resonance structures relate to the groun-state coherences g g an g g g g. Physically the origin of the twophoton frequency withs can be unerstoo in the following way. The laser fiel connects the groun-state probability amplitues with the upper-state probability amplitues through one-photon excitation processes. These one-photon absorption processes perturb the groun-state probability amplitues an then lea to ecay rates for the groun-state coherences of the orer of the rates of the one-photon transitions i.e. of the orer of an. These quantities play accoringly the role of the frequency withs for the twophoton resonance processes relate to the groun-state coherences. Since the two-photon processes inuce by the cooling laser fiel are only important for eep sub-doppler cooling of atoms in the trap 9 the physical meaning of parameters an allows us to introuce two conitions necessary for separating the cooling an trapping processes: i.e.. 36 The left inequality guarantees that the trapping fiel oes not influence the two-photon cooling process while the right inequality is neee for the two-photon friction coefficient to be greater than the one-photon friction coefficient. Uner the conition 36 an in lowest orer of the small parameters s t s c y an z the low-saturation lowvelocity solutions give the one-photon coherences entering Eq. 3 as S g e i3/ i N S g e i i N S g e i3/ i N 37 The other steay-state coherences an populations entering Eqs. an 3 are efine by similar equations S g e i i N S g e i/6 i N S g e i/ i N... 4 Qualitatively the behavior of the groun-state populations at low velocities is efine by the groun-state coherence g g that is inuce by the cooling fiel. The velocity position of the narrow two-photon structures in the grounstate populations can be seen from the energy conservation law. In an atom rest frame the absorption of a photon from one traveling wave an emission of a photon into the other traveling wave results in a two-photon transition between groun-state sublevels g g that oes not change the atom energy (kv)(kv). Energy conservation thus irectly shows that the two-photon resonance structures which are ue to the cooling fiel are locate at zero velocity kv. VIII. MODIFIED OPTICAL POTENTIAL DEPTH AND KINETIC ENERGY The atomic coherences that we have erive Eqs. 37 together with the values of the groun-state populations give a new value of the graient force at zero velocity an negative etuning F gr r. 4 In a corresponing way they give a new value of the potential Ur F gr r 55 r r

8 THEORY OF AN OPTICAL DIPOLE TRAP FOR COLD ATOMS PHYSICAL REVIEW A we can erive an explicit expression for the friction coefficient ue to the two-photon processes 6: 7 r 44 where r k /M is a recoil frequency. The last expression shows that at low effective saturation the friction ue to the two-photon processes oes not epen on intensity. The twophoton friction is however much higher than that ue to the one-photon processes. This higher friction is naturally explaine by the smaller value of the velocity with of the two-photon structure /k compare to the with of the onephoton structure /k see the right inequality 36. In a way that is similar to the raiation force the momentum iffusion tensor D ii also inclues a narrow two-photon velocity structure locate at zero velocity Fig.. The contribution of the two-photon structure can be shown to ecrease the value of the iffusion coefficient DD zz at zero velocity. The reason is that the analysis of the equations liste in Appenix B shows that the secon part of the iffusion coefficient in Eq. 3 is much bigger than the first one. However this secon part of the iffusion coefficient D manifests a narrow velocity ip at velocity vv z. In our basic approximations the iffusion coefficient D at zero velocity D can be estimate as FIG.. Velocity epenence of the raiation pressure force F rp soli line an iffusion coefficient DD zz ashe line in a small velocity region kv z at negative etuning an saturation parameter G5. which is about. times less than the unperturbe potential 3. The reuce value of the graient force an the potential in the total laser fiel is naturally explaine by the ominant role of the cooling fiel in proucing atomic populations in accorance with the leftmost conition in Eq. 36. While the trapping fiel alone prouces the groun-state atomic populations at zero velocity as N N 3/34 N 4/7 the cooling fiel reistributes the populations to the values N N 9/ N 4/. This reistribution multiplie by the relative strengths of the ipole transitions gives the ecrease in the potential. However in a threeimensional cooling fiel the ecrease of the potential shoul be less profoun. Substitution of the steay-state atomic coherences an groun-state populations into Eq. gives an explicit expression for the raiation pressure force in a low-saturation low-velocity approximation. Since the above analytical expressions 38 an 4 neglect a weak velocity epenence ue to the one-photon Doppler processes the raiation pressure force shown in Fig. is only escribe in our analysis near zero velocity i.e. where v /k. In the case of a re-etune cooling fiel where the low-velocity part of the raiation pressure force reuces to the cooling force. If we represent the force at a re etuning an at low velocities in the stanar form of a friction force F rp Mv 43 D 46 7 k. 45 The iffusion coefficient an the friction coefficient jointly efine the kinetic energy of the trappe atoms an thus an effective temperature foun accoring to the Einstein relation as Ek B T D 3 M 3. IX. CONDITIONS FOR STABLE TRAPPING 46 Assuming now that kinetic energy 46 of col trappe atoms is much less than the epth U() of the potential well 4 one can get a sufficient conition for stable atomic trapping. 47 Comparing the last conition with the conition for eep laser cooling 36 one can see that both conitions can be satisfie when the etuning of the trapping fiel is much larger than that of the cooling fiel. 48 The above two conitions efine by Eqs. 36 an 47 thus justify the iea of a stable ipole trap. The stable atomic trapping in the optical ipole trap can thus be achieve when the trapping fiel has no effect on the two-photon cooling process an when the cooling fiel oes not change the

9 B. M. GARRAWAY AND V. G. MINOGIN PHYSICAL REVIEW A structure of the trapping potential but changes only the numerical value of the trapping potential well. The lifetime of atoms in the trap associate with the iffusive heating can finally be estimate as e U()/E 49 where is the oscillation frequency of an atom in the trap an the Boltzmann factor U()/E consierably excees unity. The above evaluations of the conitions necessary an sufficient for stable atomic trapping can be illustrate by a specific example. Assume that the Rabi frequency an the etuning for a cooling fiel are accoringly an an the Rabi frequency an the etuning for the trapping fiel are an 4. For these parameters the one-photon withs 4 an.5 3 satisfy conitions 36 for eep laser cooling. In their turn the kinetic energy 46 estimate as E.4 an the potential-well epth estimate accoring to Eq. 4 as U().8 satisfy the sufficient conition 47 for stable atomic trapping giving for the Boltzmann factor a sufficiently large value U()/E. APPENDIX A Below is liste a set of quasiclassical equations for the atomic-ensity-matrix elements in the RWA that escribe the ipole interaction of a 35-level atom with the trapping laser fiel efine by Eq.. Note that the set inclues only the equations that involve ensity-matrix elements not vanishing at the interaction time t. t g g i3 ei(kyt) e g c.c. e e 3 e e t g g ie i(kyt) e g c.c. e e 4 3 e e e e t g g i3 ei(kyt) e g c.c. 3 e e e e X. CONCLUSION The above analysis shows thus that in an optical ipole trap one-photon processes responsible for trapping atoms can be well separate from the two-photon processes responsible for the eep sub-doppler cooling of atoms. This separation of the trapping an cooling processes ensures in turn a stable operation of the trap accoring to Eq. 49. This principal conclusion is also vali for the three-imensional case since if one as new - cooling fiels along the axes Ox an Oy all the above conclusions remain vali with only the numerical coefficients changing. This justifies the possibility of separating the trapping an cooling processes in a three-imensional case. Finally note that the above quasiclassical analysis cannot give a quantitative limit for the temperature of laser-coole atoms since it consiers atomic velocities higher than the recoil velocity. The cooling limit can however be seen from a qualitative estimation. Accoring to the above analysis the two-photon cooling process can operate when the velocity with of the two-photon resonance excees the recoil velocity v/kv r. That is when the atomic energy excees the recoil energy EE r k /M. These limits give the physical justification for the key assumption that the atomicensity-matrix elements can be expane in powers of the photon momentum k. ACKNOWLEDGMENT This work was supporte in part by the Royal Society an the Russian Fun for Basic Research Grant No t e e i3 ei(kyt) g e c.c. e e t e e ie i(kyt) g e c.c. e e t e e i3 ei(kyt) g e c.c. e e t g e i3 ei(kyt) g g e e g e t g e ie i(kyt) g g e e g e t g e ie i(kyt) g g e e g e. APPENDIX B Below are liste the atomic-ensity-matrix equations in the Wigner representation an the RWA. They escribe the ipole interaction of a 35-level atom with trapping an cooling laser fiels efine by Eqs. an. The equations inclue only terms escribing one-photon an twophoton optical processes

10 THEORY OF AN OPTICAL DIPOLE TRAP FOR COLD ATOMS PHYSICAL REVIEW A t g g i3 ei(kyt) () e g ie ikzit () e g i 6 eikzit () e g c.c. n e e n e e 3 n e e n t g g ie i(kyt) () e g i eikzit () e g t g g i3 i eikzit () e g c.c. n e e 4 3 n e e ei(kyt) () e g n e e n ie ikzit () e g i 6 eikzit () e g c.c. 3 n e e n e e n e e n t e e ie ikzit () g e c.c. e e t e e i3 ei(kyt) () g e i eikzit () g e c.c. e e t e e ie i(kyt) () g e i 6 eikzit () g e e ikzit () g e c.c. e e t e e i3 ei(kyt) () g e i eikzit () g e c.c. e e t e e ie ikzit () g e c.c. e e t g e i3 ei(kyt) () e e ie ikzit () e e e ikzit () g g i 6 eikzit () e e g e t g e i3 ei(kyt) () g g () e e ie ikzit () e e i 6 eikzit () e e i eikzit () g g g e i(kyt) t () g e ie g g 3 e e () ie ikzit () e e i 6 eikzit () g g e ikzit () g g e ikzit () e e g e t g e i3 ei(kyt) g g (t) () e e i eikzit () g g e ikzit () e e e ikzit () e e g e t g e ie i(kyt) () g g e e () i eikzit (t) e e e ikzit () e e i 6 eikzit () g g e ikzit () g g g e t g e i3 ei(kyt) () g g () e e i eikzit () g g e ikzit () e e e ikzit () e e g e

11 B. M. GARRAWAY AND V. G. MINOGIN PHYSICAL REVIEW A () () 3 () i(kyt) t () g e ie g g 3 e e t g g i e i(kyt) e g ei(kyt) g e ie ikzit () e e i 6 eikzit () g g e ikzit () g g e ikzit () e e g e i 6 eikz e it () e g e it g e 3 n e e () 3 n e e t g e i3 t g e i3 ei(kyt) () g g () e e i 6 eikzit () e e ie ikzit () e e i eikzit () g g g e ei(kyt) () e e ie ikzit () g g e ikzit () e e i 6 eikzit () e e g e t g g i 3 t g g i ei(kyt) () e g e i(kyt) () g e i 6 eikz e it () e g e it () g e n e e 3 n e e 3 n e e n 6 eikzit () e g e ikzit () g e 3 n e e n e e 3 n e e n t e e i3 n e e n ei(kyt) () e g e e ie ikzit () g e t e e ie ikzit () g e i 6 eikzit () e g t e e i 3 t e e i e e ei(kyt) () g e e i(kyt) () e g i eikzit () g e i 6 eikzit () e g e e eikz e it () g e e it () e g e e t e e i e i(kyt) () g e 3 ei(kyt) () e g i eikzit () e g i 6 eikzit () g e e e t e e ie ikzit () e g i 6 eikzit () g e t e e i3 e e ei(kyt) () g e e e. ie ikzit () e g 4346-

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