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Superfluidity of spin-polarized 6Li

a r X i v :c o n d -m a t /9508079v 1 18 A u g 1995

Super?uidity of spin-polarized 6Li

H.T.C.Stoof,1M.Houbiers,1C.A.Sackett,2and R.G.Hulet 2

1

Institute for Theoretical Physics,University of Utrecht,Princetonplein 5,

P.O.Box 80.006,3508TA Utrecht,The Netherlands

2

Physics Department and Rice Quantum Institute,Rice University,

P.O.Box 1892,Houston,Texas 77251

We study the prospects for observing super?uidity in a spin-polarized atomic gas of 6Li atoms,using state-of-the-art interatomic potentials.We determine the spinodal line and show that a BCS transition to the super?uid state can indeed occur in the (meta)stable region of the phase diagram if the densities are su?ciently low.Moreover,for a total density of 1012cm ?3,which still ful?lls this requirement,we ?nd a critical temperature of only 29nK .We also discuss the stability of the gas due to exchange and dipolar relaxation and conclude that the prospects for observing super?uidity in a magnetically trapped atomic 6Li gas are particularly promising for magnetic bias ?elds larger than 10T .

PACS number(s):03.75.Fi,67.40.-w,32.80.Pj,42.50.Vk

Ultracold atomic gases have received much attention in recent years,because of their novel properties.For instance,these gases are well suited for high-precision measurements of single-atom properties and for the ob-servation of collisional and optical phenomena that re?ect the (Bose or Fermi)statistics of the constituent particles.Moreover,a large variety of experimental techniques are available to manipulate the atomic gas samples by means of electromagnetic ?elds [1],which o?ers the exciting pos-sibility to achieve the required conditions for quantum degeneracy and to study macroscopic quantum e?ects in their purest form.

At present,most experimental attempts towards quan-tum degeneracy have been performed with bosonic gases and have been aimed at the achievement of Bose-Einstein condensation.In particular,most of the earlier experi-ments used atomic hydrogen [2,3].These experiments provided crucial ingredients for the recent attempts with alkali vapors,for which the experimental advances to-wards the degeneracy regime were so rapid that Bose-Einstein condensation has actually been reported now for the isotopes 87Rb [4]and 7Li [5].

In view of these exciting developments it seems timely to investigate theoretically also the properties of spin-polarized atomic 6Li,since 6Li is a stable fermionic iso-tope of lithium that can be trapped and cooled in much the same way as its bosonic counterpart.Therefore,mag-netically trapped 6Li promises to be an ideal system to study degeneracy e?ects in a weakly-interacting Fermi gas,thus providing valuable complementary information on the workings of quantum mechanics at the macro-scopic level.Moreover,using a combination of theoret-ical [6,7]and experimental [8]results,accurate knowl-edge of the interparticle (singlet and triplet)potential curves of lithium have recently been obtained which lead to the prediction of a large and negative s-wave scatter-ing length a of ?4.6·103a 0(a 0is the Bohr radius)for a spin-polarized 6Li gas.

This is important for two reasons:First,the fact that the scattering length is negative implies that at the low temperatures of interest (Λ?r V ,where Λ=(2πˉh 2/mk B T )1/2is the thermal de Broglie wavelength of the atoms and r V is the range of the interaction)the e?ective interaction between the lithium atoms is attrac-tive and we expect a BCS-like phase transition to a su-per?uid state at a critical temperature

T c ?5?F

2k F |a |

?1

,(1)

with ?F =ˉh 2k 2

F /2m the Fermi energy of the gas [9].Secondly,we see that the critical temperature depends exponentially on 1/k F |a |which usually,when the mag-nitude of a is of the order of the range of the interaction r V ,is very large in the dilute limit k F r V ?1.There-fore,it was previously concluded that the BCS transition in a dilute fermionic system (in particular spin-polarized deuterium)is experimentally unattainable [10].However,with the anomalously large scattering length of 6Li this conclusion needs revision as we will see now in more de-tail.

Since 6Li has an electron spin of s =1/2and a nuclear spin of i =1,the 1s groundstate of 6Li consists of six hyper?ne levels which are labeled by |1 through |6 in such a way that their energy increases at small magnetic ?elds.At zero magnetic ?eld these correspond exactly to the states |f,m f with a total spin f of either 1/2or 3/2and a hyper?ne splitting of 3a hf /2?10.95mK .In a conventional trapping experiment one tries to achieve both electron and nuclear spin-polarization by trapping only atoms in the doubly spin-polarized state |6 ≡|m s =1/2,m i =1 .In the case of 6Li,however,such a proce-dure creates a gas in which the atoms only interact ex-tremely weakly because the Pauli principle now forbids s-wave scattering.Therefore,we cannot take advantage of the large negative value of the triplet scattering length and the BCS-transition temperature is unattainable.

1

To avoid this problem we need to trap two hyper?ne states.This can be achieved most easily by using rela-

tively large magnetic?elds B?a hf/μe?0.011T(μe is the electron magnetron),because then the electron

and nuclear spins are almost decoupled and the states |4 ?|m s=1/2,m i=?1 ,|5 ?|m s=1/2,m i=0 , and|6 can all be trapped.Moreover,the trap can,for ex-ample,be loaded by creating?rst a doubly spin-polarized gas in the conventional manner and subsequently apply-ing a microwave pulse to populate one of the other trap-ping states.Notice that because the gas is now only electron(and not nuclear)spin polarized,there is a large cross-section4πa2for the thermalizing collisions required for evaporative cooling.In addition,this implies that the gas is in thermal equilibrium in the spatial degrees of freedom,even though the spin degrees of freedom are not.Notice also that the electron spin polarization is not complete because the states|4 and|5 have a small admixture(of O(a hf/μe B))of|m s=?1/2,m i=0 and |m s=?1/2,m i=1 ,respectively.Therefore,two atoms do not interact solely via the triplet interaction.When we consider the lifetime of the gas,however,we?nd that the in?uence of this can be neglected if the magnetic?eld is larger than1T.

Although it is possible to study any combination of the three trapping states,we will consider here only a gas in which the atoms are in a mixture of state|5 or state|6 because this minimizes the number of decay processes. Furthermore,we analyze here?rst the homogeneous case. The in?uence of the trapping potential will be discussed in a separate publication.Taking only s-wave scattering into account and following Ref.[11]to include all two-body processes,we can determine the thermodynamic properties of this gas by considering the hamiltonian

H= d x 6 α=5ψ?α( x) ?ˉh2?2

V0?4πaˉh2

(2π)3

ln 1+e?β(?( k)?μ′α)

+

4πaˉh2

(2π)3

Nα( k),(4)

introducing the notation Nα( k)for the occupation num-

bers which in our case are equal to the Fermi distribution

function(eβx+1)?1evaluated at?( k)?μ′α.

In the degenerate regime,neglecting corrections of

O((k B T/?F)2),the equation of state Eq.(4)can be easily

inverted and we?nd for the pressure

p= αnαk B T1πm n5n6.(5)

For the mechanical stability of the gas we must require

that?p/?nα≥https://www.sodocs.net/doc/cf16446058.html,ing the above result this leads to

two conditions on the densities in the two hyper?ne lev-

els,namely

n5≤1π6|a| 3n5 2 2/3,

which have to be ful?lled simultaneously.The line where

one of the equalities holds is called the spinodal line and

it is shown for6Li in Fig.1.Note that for the highest

metastable densities we have k F|a|=π/2.Therefore,

the ratio k B T c/?F is at most0.23and our determination

of the spinodal line is self consistent for the temperatures

of interest.

Within this density region(i.e.also for n5=n6)we

can now consider the critical temperature of the gas.For

that we need to derive the BCS gap equation.This can

be achieved most easily by diagonalizing the hamiltonian

by means of a Bogoliubov transformation and then cal-

culating the equilibrium value of?0=V0 ψ5( x)ψ6( x) .

In the limit of vanishing?0,or equivalently T→T c,this

procedure leads to the linearized BCS gap equation

1

(2π)3

1?N5( k)?N6( k)

the triplet potential in the hamiltonian.However,from the Lippmann-Schwinger equation for the two-body T-matrix[12]we?nd that this divergence is cancelled by a renormalization of1/V0to1/T2B( 0, 0;0)=m/4πaˉh2. Therefore,the critical temperature is determined by the condition

m

(2π)3

N5( k)+N6( k)

8?F? 1+2π2δ?F 2 exp ?π

2.Therefore,at su?ciently

high magnetic?elds the coupling becomes small and the

T-matrix element in Eq.(9)can be calculated in the

distorted-wave Born approximation leading to

G ex?π3ˉh2a2hf m

theless,it is so weak that it can always be treated by ?rst-order perturbation theory [13].Thus,if B ?a hf /μe we can neglect the hyper?ne coupling between the elec-tron and nuclear spins and the total dipolar rate consist of the sum of a one spin-?ip and a two spin-?ip con-tribution,i.e.G d =G 1sf +G 2sf .Due to the di?erent spin-matrix elements and the di?erence in the energy re-leased in the transition,we ?nd the convenient relation G 2sf (B )=2G 1sf (2B ).Hence,we need to consider only the one spin-?ip process,which we again treat in the distorted-wave Born approximation.For magnetic ?elds larger than 1T this leads to

G

1sf

?

6

2mμe B

m (μ0μ2e )

2

2

r

i ?Y ?m (?r ).

(12)

The rate constant for the one spin-?ip process is also

shown in Fig.2.Again,it is much larger than the rate constant of a similar proces in atomic hydrogen.Notice that for magnetic ?elds larger than 10T the electron-electron dipolar rate dominates the decay and then also leads to a lifetime of the order of seconds for a den-sity of 1012cm ?3.The decay rates due to the electron-nucleus dipolar interaction are even smaller by a factor of (μN /μe )2?20·10?6and are completely negligible.This proves that our assumption of a non-equilibrium distribu-tion in the spin-degrees of freedom is justi?ed,because relaxation between the hyper?ne levels |5 and |6 re-quires a nuclear spin-?ip and will therefore take place on a timescale set by the electron-nucleus dipolar rate which is much longer than the lifetime of the gas.

10

10

B (T)

10

-14

10

-12

10

-10

G (c m 3

/s )

1

2

FIG.2.The decay rate constants as a function of magnetic ?eld.Curve 1shows G ex ,whereas curve 2gives G 1sf .

In summary,we have shown that due to the large and negative triplet scattering length the BCS transition to a

super?uid state occurs at experimentally accessible den-sities and temperatures in spin-polarized 6Li.It should

be pointed out however that the scattering length is ex-tremely sensitive to the interatomic potential,so the ex-act conditions required may be somewhat di?erent than that given here.A better estimate of the scattering length can be obtained by repeating the experiment of Ref.[8]for 6Li.Moreover,as is well-known from liquid 3

He,the phase below the critical temperature is truly su-per?uid because it costs a ?nite amount of (free)energy to have gradients in the phase of the order parameter ?0.As a result we have a macroscopic (free)energy barrier for the decay of super?ow and the gas can sustain persis-tent mass currents.By calculating the various collisional decay rates in the gas,we have also shown that reason-able lifetimes can be achieved even for densities as high as 1012cm ?3if the bias magnetic ?eld is larger than 10T .We hope that our work will stimulate new experiments with this interesting quantum gas.

We are grateful to Eric Abraham for his help with the construction of the interatomic potentials and to Michel Bijlsma for useful discussions.The work at Rice is supported by the National Science Foundation and the Welch Foundation.

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