Sympathetic Cooling Route to Bose-Einstein Condensate and Fermi-Liquid Mixtures

Home , Fermi liquid theory

Dartmouth College Dartmouth Digital Commons

Open Dartmouth: Published works by Dartmouth faculty Faculty Work

10-27-2005

Robin Côté University of Connecticut, Storrs

Roberto Onofrio Dartmouth College

Eddy Timmermans Los Alamos National Laboratory

Follow this and additional works at: https://digitalcommons.dartmouth.edu/facoa

Part of the Physics Commons

Dartmouth Digital Commons Citation Côté, Robin; Onofrio, Roberto; and Timmermans, Eddy, "Sympathetic Cooling Route to Bose-Einstein Condensate and Fermi-Liquid Mixtures" (2005). Open Dartmouth: Published works by Dartmouth faculty. 3306. https://digitalcommons.dartmouth.edu/facoa/3306

This Article is brought to you for free and open access by the Faculty Work at Dartmouth Digital Commons. It has been accepted for inclusion in Open Dartmouth: Published works by Dartmouth faculty by an authorized administrator of Dartmouth Digital Commons. For more information, please contact [email protected]. arXiv:cond-mat/0510810v1 [cond-mat.stat-mech] 29 Oct 2005 ul vial nyi h omo condensed of previ- form were the that in mixtures only liquid available for quantum venue ously new of a study suggest the [1] (BEC) condensates Einstein Introduction. becueo h bevdnnFrilqi eairin behavior non-Fermi-liquid prob- observed the high the as identified of been cause also able (quantum have transition [5]) phase see criticality, quantum their a and near interactions prominence Mediated unaccessible. currently a,i esi eo e ecn fteFritempera- Fermi the of percent ten below fermion in ture dilute sets a experi- it In linearly as gas, behavior. varies serves Fermi-liquid often that of that test capacity mental feature heat a num- a temperature, occupation implies with quasi-particle This in en- the order bers. expanding second from in to follows behave ergy that fermions manner the universal which the a in into [2] mixtures regime fermion-boson Fermi-liquid physi- cooling atom in cold succeed when cists environment accessible more and ex- of ambiguity the data. by perimental impeded clear a again however, is systems, understanding these In itself. perconductivity t 9,adbcueteuaodbels ffrin leads refriger- fermions to of loss ability unavoidable its the drop because impairs and temperature capacity [9], low ate heat the BEC en- because the not [8] in have to regime cooling) bosons this (sympathetic use tered atoms that experiments fermion far, cool So superfluidity. of euetebsnmdae neatosta a Cooper- the can [3] also that pair interactions These mediated boson [4]. the effects reduce interaction strong is quantities by first- helium complicated other a and [3], this interactions of mediated description by principle sep- caused phase be helium to the understand aration we While [2]. mixtures sa rca fase scoigbelow cooling as step a of the crucial regime of as Fermi-liquid the that is into from crossing parameter [7], different mixtures a fundamentally helium in is mixtures that fermion-boson regime access also can 3 elqi.Frcl tmqatmlqi tde,which studies, liquid quantum atom cold For liquid. He h bv sus[]cnb tde namc cleaner much a in studied be can [6] issues above The yptei oln ot oBs-isencnest and condensate Bose-Einstein to route cooling Sympathetic < T T 2 c eateto hsc n srnm,DrmuhClee 61 College, Dartmouth Astronomy, and Physics of Department uecnutr 5 n,pras fhigh of perhaps, and, [5] superconductors 0 . 3 1 efrin,gvn rtcltemperature critical a giving fermions, He ihwihw td pcfi n rmsn itr composed mixture promising and specific t a formalism study energy-based we an which introduce We with temperatur the Fermi-liquid. access a and as mixture Fermi-Bose atomic dilute ASnmes 37.s 53.p 28.j 67.90.+z 32.80.Pj, 05.30.Jp, 03.75.Ss, fe two numbers: favourable the PACS for the frequencies that trapping different find using we by enhanced mixture, trapped harmonically 3 T 4 iatmnod iia“.Glli,Uiestad Padova Università di Galilei”, “G. Fisica di Dipartimento odao itrso emosadBose- and fermions of mixtures atom Cold etrfrSaitclMcaisadCmlxt,IF,Uni INFM, Complexity, and Mechanics Statistical for Center F edsusasmahtccoigsrtg htcnsuccessf can that strategy cooling sympathetic a discuss We whereas , 5 -,Ter iiin o lmsNtoa aoaoy o A Los Laboratory, National Alamos Los Division, Theory T-4, 1 eateto hsc,Uiest fCnetct Storrs, Connecticut, of University Physics, of Department < T oi Côté,Robin 0 . 02 T F T srqie nthe in required is 1 c sfrtestudy the for is oet Onofrio, Roberto 3 He- T c 4 su- He T c ,3 4 3, 2, type energy. of number same same the the has with particles system equilibrium an which at rate, eann tm,tepoesas dsa vrg ki- average an the adds with also collide energy trap, process netic collision the the the of atoms, If out remaining way particles. their lost on the products, of energy the removes os eke rc fteeeg-aac n en the define and system energy-balance the particle of the temperature to of effective track response keep In- many-body we the loss, recombination. describing three-body of of (in stead and processes a relaxation traps), as back- spin magnetic trap temperature two-body the of room leave atoms, with ground Atoms collisions of lifetime. atom finite consequence true trapped a and (the have solid) metastable condensed gases a are being interest state ground of states many-body Dynamics. Temperature Loss-induced [11]. investigated experimentally cee eilsrt hs onsfrtepoiigcan- a promising the of for didate points these illustrate trapping We bichromatic fermion a scheme. the in adjusting frequencies by trapping met boson and be fermion- can efficient exchange ensure heat to BEC high hole- reduce sufficiently to but low heating, sufficiently keeping overlap between balance boson delicate help- fermion a the are strike to rates need The recombination ful. three-body low signifi- capacity heat heat crucial; BEC they is in increase before corresponding colli- fermions The many cantly. lighter require the bosons from sions Heavy regime. param- temperature cooling trapping proper sympathetic the with eters, and ratio mass fermion rate. heating recombinations) Fermi-hole (three-body the further increasing system introduces thereby Fermi BEC the The in losses [10]. heating significant to a qiiru nryo eprtr-eae system, temperature-relaxed a dE of energy equilibrium tal h bv olsosrdc h ubro atce of particles of number the reduce collisions above The nti ai,w hwta ihalreboson-to- large a with that show we Rapid, this In ( N n dyTimmermans Eddy and 7Wle aoaoy aoe,N 35,USA 03755, NH Hanover, Laboratory, Wilder 27 dE/dt j j , T , i azl ,Pdv 53,Italy 35131, Padova 8, Marzolo Via , N eiei hc h emosbehave fermions the which in regime e lymtgt emo-oehaigi a in heating fermion-hole mitigate ully species. ecietetmeauedynamics temperature the describe o ) j /dt swl sitra energy internal as well as , trso hsmxueaefurther are mixture this of atures ` iRm ,Rm 08,Italy 00185, Roma 1, Roma tà di 6 − Li– ǫ of j coll ( = P 6 87 ao,N 74,USA 87545, NM lamos, j iand Li eeut h oa nrybalance energy total the equate We . ∂E/∂T ǫ bmxue hc a enrecently been has which mixture, Rb T029 USA 06269, CT j coll dN em-iudmixtures Fermi-liquid 87 j ) b nlzn the Analyzing Rb. /dt dT/dt can 5 otevraino h to- the of variation the to cesteFermi-liquid the access + T P stetemperature the as j ( h odatom cold The - ∂E/∂N E sec loss each as , j ) dN j /dt 2

0.5 3 5 EF, where EF is the Fermi energy. At the same temperature the remaining degenerate system of NF−1 fermions 0.4 Fermi gas would have an energy EF less than the initial Fermi system of NF fermions. The net result is an eﬀective energy 2 0.3 increase of 5 EF per fermion lost (assuming, optimisti- 40 23 coll F K- Na cally, ǫj = 0). We apply Eq. (1) to the homogeneous PSfrag replacements mixture of fermions of mass m , and an ideal BEC of

T/T F 0.2 40 87 K- Rb NB bosons of mass mB. The total heat capacity of an 6Li-23Na an ideal or near-deal gas mixture is the sum of the BEC 2 0.1 and fermion heat capacities, C = kBNF(π /2)(T/TF)[1+ 6 87 3/2 3/2 Li- Rb (45/8π )ζ5/2(1)(mB/mF) T/TF], where kB denotes the Boltzmann constant,p TF is the Fermi- 0 0 0.2 0.4 0.6 0.8 1 t/τ temperature, TF = EF/kB and ζ5/2(1) = 1.342. The time scale of the dynamics is the lifetime τ of the fermion −1 2 FIG. 1: Hole heating in homogeneous Fermi-Bose mixtures. system, τ = αnB, The temperature, T , scaled by the Fermi temperature, TF , is −1 depicted for four Fermi-Bose mixtures consisting of fermionic 4 T 6 40 23 87 2 Li or K atoms with Na or Rb as bosonic partners, d(T/TF) 5π TF −2 = . (3) starting at T/TF = 10 . For comparison, the hole heating 3/2 1/2 d(t/τ) 45ζ5/2(1) mB T 1+ 3 2 curve for pure fermions, assuming the same loss-rate, is also 8π / mF TF plotted (lighter continuous curve). The term ∝ T/TF in the denominator stems from the BEC heat capacityp and represents its capability to ab- and we identify the usual thermodynamic derivatives, sorb heat released from fermion-hole heating. Note the (∂E/∂T )= C, where C denotes the heat capacity at con- sensitive dependence on mass-ratio illustrated by Fig. stant number of particles, and (∂E/∂Nj) = µj , where 1, where the temperature of mixtures that start out at µj represents the chemical potential of the j-particles. T = 0.01TF is shown for four possible Fermi-Bose mix- Equating the energy rates and solving for the tempera- tures with stable alkalis (6Li or 40K for the fermionic ture derivative, we obtain the central equation of tem- species, 23Na or 87Rb for the bosonic species). For the perature dynamics sake of comparison, we recover the fermion-hole heating rate for a pure fermion system obtained in Ref. [10] by dT 1 dF dNj setting mB → 0, corresponding to a BEC-refrigerator of =  − ǫcoll  , (1) dt C dt j dt vanishing heat capacity (shown by the thin full line). Xj   Trapped interacting mixtures. - In addition to the mass- where we have defined F as the free energy that occurs in ratio dependence, the spatial density profiles in the realistic case of trapped gases affect the temperature dy- Fermi-liquid theory: F = E − µj Nj (here dependent j namics. The fermion-boson overlap can be reduced by on Nj rather than µj ). P Homogeneous ideal gas Fermi-Bose mixtures. - The ho- a repulsive inter-species interaction or by the occurrence mogeneous mixture of a single component degenerate of a true phase separation. In the non-phase separated fermion gas and a BEC allows a transparent description systems, the mutual overlap can be controlled by vary- of hole heating and its mitigation by sympathetic cooling. ing the ratio of the fermion to boson trapping frequencies Apart from border effects, the homogeneous mixture can (using, for instance, a bichromatic trap [9]). A decrease be realized by using blue-detuned laser light sheets [12]. in overlap reduces fermion loss but also reduces the ef- Atoms in the mixture undergo various three-body loss ficiency of sympathetic cooling. Which effect wins out, processes. For instance, a fermion can recombine with a depends sensitively on the density profiles and, hence, on the interaction parameters. boson into a highly energetic molecule while ejecting a 6 second boson, corresponding to To determine the trap densities in the case of a Li- 87Rb mixture, the most promising candidate shown in 2 2 Fig. 1, we have evaluated the interspecies elastic scat- n˙ B = −2αnBnF, n˙ F = −αnBnF (2) tering lengths for various Li and Rb isotopes. Although where nF and nB represent the fermion and boson densi- spectroscopic data for LiRb potential curves are lacking, ties respectively, and α denotes the three-body loss rate the excellent agreement between ab initio [13] and our coefficient. Processes involving three-boson collisions do parametrized potentials (see Table I) at large separation not lead to significant heating and those involving two or attest to the accuracy of the curves in that region. How- three indistinguishable fermions are Pauli-inhibited. If, ever, there is a discrepancy between both sets of curves as in Eq. (2), the rate is energy-insensitive, the loss of one near the equilibrium distance. Compared to [13] our po- out of NF fermions removes, on average, a kinetic energy tential curves are deeper by about 3% for the singlet and 3

6 87 6 85 7 87 7 85 r s Li- Rb Li- Rb Li- Rb Li- Rb and nF( ) in a Thomas-Fermi approximation are deter- aT 1.0 −43.6 −36.6 −151 −118 mined by (see Molmer reference in [6]) 0.99280 −17.0 −12.9 −60.0 −48.7 0.97404 +17.0 +18.8 +5.43 +8.43 µ − V (r) − λ n (r) n (r) = B B FB F aS 1.0 +153 +215 −16.3 −1.11 B λB 0.99280 −152 −77.4 +27.1 +33.8 [2m ]3/2 0.97404 +45.5 +50.4 +167 +280 n (r) = F {µ − V (r) − λ n (r)]}3/2 (4) F 6π2~3 F F FB B

TABLE I: Singlet and triplet elastic scattering lenghts (shifted where λB and λFB denote the boson-boson and fermion- and unshifted) for the mixtures of Li and Rb isotopes, in Bohr boson interaction strengths proportional to the corre- radii. The values of s are found so that aT = ±17 a0, and sponding s-wave scattering lengths, aB and aF B: λB = the same scaling was assumed for the singlet. The scattering ~2 ~2 lengths were obtained from potential curves with ab initio 4π aB/mB and λFB =2π aFB (1/mF +1/mB). In Eq. data from [15], joined smoothly to an exponential wall of the (4), we tacitly assume temperature independent density − form ce bR at short separations R, and to the long-range form profiles, a good approximation in the regime of interest. 6 8 10 α −βR −C6/R −C8/R −C10/R ∓AR e at R=13.5 a0. Here, Solving Eq. (4) numerically, we find that a decrease in ∓ stands for the singlet X1Σ+ and triplet a3Σ+ molecular confinement strength for the bosons generally increases states, respectively. The values for C6 = 2545 a.u. from [16], 5 7 the overlap with the Fermi species. Controlling the rel- and C8 = 2.34 × 10 a.u. and C10 = 2.61 × 10 a.u. from ative trapping frequencies allows for a variation of the [17], were used. The parameters of the exchange energy are α = 4.9417 a.u. and β = 1.1836 a.u. [18], while the constant fermion-boson overlap and, hence, the rate of Fermi-hole A = 0.0058 a.u. was found by fitting the ab initio data. heating and the efficiency of sympathetic cooling. The effect of inter-species interactions is marginal on the Fermi gas but is very pronounced for the BEC, and the mutual 7% for the triplet channel. By scaling the whole potential overlap is very sensitive to the interaction parameters. curves so that Vs(R)= sV (R) with s ≤ 1, we can explore The temperature trajectories shown in Fig. 2 are calcu- 3 −1 the effect of having shallower potential curves; the values lated for ωf = 10 s , while varying the Bose trap fre- of the scattering lengths change significantly as s varies quency as shown in the caption. We choose the chemical accordingly. In the singlet case, one bound level dis- potentials in the initial state to be 1 µK for the Fermi appears, and the scattering length varies between ±∞, gas and 100 nK for the Bose gas. The scattering lengths while the triplet scattering length becomes positive. Re- are chosen as aB=5.8 nm for Rb and aFB=-0.90 nm for cently, Zimmermann and co-workers [11] determined the the Li-Rb interactions. magnitude of the triplet scattering length of 6Li-87Rb We consider the fermions to be subject to loss by back- +9 to be |aT | = 17−6 a0. In Table I, we present values ground scattering,n ˙ F(x) = −γnF(x), as well as by the for the singlet and triplet scattering lengths for the un- 3-body recombination of Eq. (2). While these loss-rates shifted (s = 1) and shifted potentials (so that aT agrees are unknown, we have chosen values that corresponded −1 2 −1 with ±17 a0). These values as well as those calculated in to γ = 10 Hz and τ3 = αnB = 4s at the peak [14] are tentative and future measurements are required densities of case ωf /ωb = 1 of Fig. 2. The behavior to specify the potentials more accurately. Such experi- shown in Fig. 2 is quite robust with respect to various mental feedback can be obtained from either one of the choices of the relevant parameters. In addition, we as- isotopes so that we list the results for all combinations, sumed that a continued evaporative cooling of the bosons regardless of their fermionic or bosonic nature. drains away energy from the system at a realistic rate. In We now study the temperature of a 6Li – 87Rb mix- the calculations with different fermion and boson trap- ture trapped in an idealized trap: both species are con- ping frequencies, we took the evaporative cooling rates tained by cylindrical trapping potentials VF(r) and VB(r) to be equal and accounted for the low overlap drop in that are of the hard wall type in the transverse direction evaporative cooling efficiency by introducing a figure of with the same radius RT, where RT is much larger than merit as described in Ref. [19]. This allows to intro- the BEC healing length and the average fermion-fermion duce an effective evaporative cooling rate −Q˙ eff propor- distance. In the longitudinal direction, the fermions and tional to the overlap of the two clouds. The rate of free bosons experience harmonic trapping potentials with dif- energy change is then identified as sum of three terms: ˙ ferent trapping frequencies ωf and ωb respectively. Apart dF/dt = (dF/dt)3b + (dF/dt)Bkgnd − Qeff , where the free from its simplicity, such a situation reproduces the rele- energy rates have to be calculated from the resulting den- vant heating features of a generic three-dimensional cold sity profiles, atom mixture with anisotropic confinement. From Table 6 87 2 2/3~2 I, we find that some hyperfine states of the Li and Rb dF (6π ) α 3 5/3 2 = d r [nF(r)] [nB(r)] , isotopes that can be trapped magnetically give positive dt 3b 5mF Z valued inter-species scattering lengths, thereby favoring (5) spatial separation in the trap. The density profiles nB(r) whereas as in [10], the background scattering free energy 4

0.2

ωf /ωb=1

0.15 [1] M. W. Zwierlein et al., Phys. Rev. Lett. 92, 120403 ωf /ωb=2 (2004); T. Bourdel et al., Phys. Rev. Lett. 93, 050401 (2004); G. Modugno et al., Science 297 2240 (2002); R. F Hulet, presentation in the Quantum Gas Conference at 0.1 Fermi gas

T/T the Kavli Institute for Theoretical Physics, Santa Bar- PSfrag replacements bara, CA, May 10-14 (2004).

ωf /ωb=10 [2] G. Baym and C. Pethick, Landau Fermi-liquid theory, 0.05 (Wiley, 1991). [3] J. Bardeen, G. Baym, and D. Pines, Phys. Rev. Lett. 17, 372 (1966). 0 [4] E. Krotscheck and M. Saarela, Phys. Rep. 232, 1 (1993). 0 5 10 15 20 73 t(ms) [5] G. R. Stewart, Rev. Mod. Phys. , 797 (2001). [6] Cold atom fermion-BEC mixtures are expected to un- 6 87 FIG. 2: Fermi hole heating of harmonically trapped Li- Rb dergo phase separation, as discussed in L. Viverit, C. mixtures. The dependence of the degeneracy parameter T/TF J. Pethick, and H. Smith, Phys. Rev. A 61, 053605 −2 (starting from an initial value of T/TF = 10 ) on time is (2000), and trapped phase separated systems can ex- depicted for three cases of ωf /ωb = 1 (dot-dashed curve), hibit a variety of spatial arrangements as shown by K. ωf /ωb = 2 (dashed), ωf /ωb = 10 (continuous). The dotted Molmer, Phys. Rev. Lett. 80, 1804 (1998). The prospect curve represents the intrinsic Fermi hole heating for the same of Cooper-pairing induced by BEC mediated interactions 87 loss-rate of the fermions but in the absence of the Rb atoms. was treated by H. Heiselberg et al., Phys. Rev. Lett. 85, 2418 (2000). If the attractive Yukawa interaction that follows from the static perturbation treatment (see, for increase rate is given by instance, M. J. Bijlsma, B. A. Heringa, and H. T. C. Stoof, Phys. Rev. A 61, 053601 (2000)), is valid then 2 2/3~2 dF 3(6π ) γ 3 r 5/3 the formation of Cooper pairs that are smaller than the = d r [nF( )] . (6) Yukawa range is similar to that of excitons in exciton dt 10mF Z Bkgnd BECs, see, for instance, P. Nozieres, and D. Saint James, 43 Finally, the temperature trajectory follows from Eq. (1), J. Physique , 1133 (1982). Magnetic instabilities of spinor BECs or phase separation of a BEC-mixture can in which we use Eqs. (5) and (6) provide a quantum phase transition in cold atom fermion- BEC mixtures. dT (dF/dt) + (dF/dt) − Q˙ = 3b Bkgnd eff , (7) [7] Cold atom single component fermion-BEC mixtures with dt CF(T )+ CB(T ) a Fermi-velocity vF that exceeds the BEC sound velocity c, exhibit zero-sound as well as BEC-sound collective ex- where CF and CB denote the fermion and boson heat citations, whereas vF < c-mixtures (the regime of helium capacities. The resulting temperature dynamics is de- liquid mixtures) only exhibit BEC-sound, as shown in D. picted in Fig. 2 for three different trapping frequency H. Santamore, S. Gaudio, and E. Timmermans, Phys. 93 ratios. These results show that the heating rate is miti- Rev. Lett. , 250402 (2004). [8] A. G. Truscott et al., Science 291, 2570 (2001); F. gated best by using a larger ωf /ωb-ratio, at least until the Schreck et al., Phys. Rev. Lett. 87, 080403 (2001); Z. spatial overlap is decreased due to an excessive spreading Hadzibabic et al., Phys. Rev. Lett. 88, 160401 (2002); of the Bose cloud. 91, 160401 (2003); G. Roati et al., Phys. Rev. Lett. 89, Conclusion. - Our study of loss-induced heating, based 150403 (2002). on an effective equilibration model, reveals that a cold [9] R. Onofrio and C. Presilla, Phys. Rev. Lett. 89, 100401 115 atom Fermi-Bose mixture of high boson to fermion mass (2002); J. Stat. Phys. , 57 (2004). 87 ratio such as 6Li-87Rb can be maintained in the Fermi [10] E. Timmermans, Phys. Rev. Lett. , 240403 (2001). [11] C. Silber et al., e-print cond-mat/0506217 (9 June 2005). liquid temperature regime. This result, a consequence of [12] N. Davidson et al., Phys. Rev. Lett. 74, 1311 (1995). the increased BEC heat capacity due to the larger trap- [13] B. O. Roos (private communication). ping frequency ratio between fermions and bosons, could [14] H. Ouerdane and M. J. Jamieson, Phys. Rev. A 70, open up a new avenue for cold atom studies. We also 022712 (2004). discuss a specific cooling strategy which could be soon [15] M. Korek et al., Chem. Phys. 256, 1 (2000). implemented in the experimental investigations going on [16] A. Derevianko, J. F. Babb, and A. Dalgarno, Phys. Rev. 63 for the specific 6Li-87Rb mixture, for which anomalous A , 052704 (2001). [17] S. G. Porsev and A. Derevianko, J. Chem. Phys. 119, heating has been observed [11]. 844 (2003). R.C. acknowledges partial support from NSF, R.O. [18] B. M. Smirnov and M. I. Chibisov, Sov. Phys. JETP 21, from Cofinanziamento MIUR, and E.T. from the Los 624 (1965). Alamos Laboratory Directed Research and Development [19] M. Brown-Hayes and R. Onofrio, Phys. Rev. A 70, (LDRD) program. 063614 (2004).