Degenerate Fermi Gases with p-wave interactions

Transcription

1 Degenerate Fermi Gases with p-wave interactions Victor Gurarie Lectures given at the Jerusalem Winter School on Condensed Matter Physics and Quantum Information with Cold Atoms

2 Most of these notes follow the reference V. Gurarie and L. Radzihovsy, Ann Phys 3, (007), arxiv:cond-mat/060. Lecture : Narrow s-wave resonances. -channel model The two channel model is defined by its Hamiltonian Ĥ =,σ=, m â,σâ,σ+ p ( ɛ 0 + p 4m ) ˆb pˆbp + g V,p (ˆbp â + p, â + p, + ˆb ) p â + p, â + p,, (.) where V is the volume. It describes a physically appealing picture: fermions with spin which can glue together to form a boson with energy ɛ 0. g is the interconversion rate (or interaction strength). The limit g 0 is very attractive theoretically. Indeed, these are now noninteractive bosons and fermions. Let us understand this limit at zero temperature. Even though these are noninteracting particles, the fermion and boson numbers are not separately conserved. Only the combination which is conserved. ˆN =,σ â,σâ,σ + p In the spirit of the grand canonical ensemble we need to minimize Ĥ µ ˆN =,σ ( m µ ) ˆb pˆbp (.) â,σâ,σ + ( ) ɛ 0 µ + p ˆb p 4m pˆbp (.3) Define ɛ F to be the Fermi energy of the fermions if bosons are not present. Then we find three regimes: ) ɛ 0 > ɛ F. No bosons, all particles are fermions. µ = ɛ F. ) 0 < ɛ 0 < ɛ F. Some particles are bosons, and some particles are fermions. The number of fermions is such that the fermion Fermi energy is ɛ 0 /. Consequently, µ = ɛ 0 /. 3) ɛ 0 < 0. All particles are bosons. µ = ɛ 0 / < 0. It goes without saying that the bosons, when present, form a noninteracting Bose condensate. This is summarized on pictures shown on Figs. and.

3 " µ="! 0 F (a) " µ " F! 0 (b) " " F (c) µ! 0 Figure : An illustration of the BCS-BEC crossover in the limit of a vanishing resonance width g 0. The evolution with detuning ɛ 0 is illustrated, with (a) the BCS regime of ɛ 0 > ɛ F, where particles are all free atoms forming a Fermi sea, (b) the crossover regime of 0 < ɛ 0 < ɛ F, where a fraction of atoms between ɛ 0 and ɛ F have converted into BEC of bosonic molecules, and (c) the BEC regime of ɛ 0 < 0, where only Bose-condensed molecules are present.. The meaning of the model The model, at nonzero g, loos transparent, but to understand the meaning of the interactions in this model, one needs to study how fermions in vacuum would scatter. We compute the -body scattering amplitude. This is an exact calculation, not an approximation. We find where f(p) = a + r 0 p ip, (.4) a = mg 4πω 0, r 0 = 8π m g, ω 0 = ɛ 0 g mλ π. (.5) Here Λ is the momentum cutoff (the upper limit of the summation in the interaction term of (.)), or in other words, the maximum momenta and two fermions would have in

4 n b Ε 0 Tc Ε 0 3 FIG. 9: The normalized density of bosonic molecules ˆn b = n b/n vs the normalized detuning ˆω 0 = ω 0/ɛ F in the limit of a vanishing resonance width, γ s 0. FIG. 0: The normalized critical temperature ˆT c = T c/t BEC as a function of the normalized detuning ˆω 0 = ω 0/ɛ F in the limit of a vanishing resonance width γ s 0. Figure : The normalized density of bosonic molecules ˆn b = n b /n, vs the normalized detuning ˆɛ 0 = ɛ 0 /ɛ F in the limit of a vanishing resonance width, g 0. These equations can then be used to determine the Because of the φ nonlinearities in S s [φ], Eq. (6.4), the normal-superfluid transition temperature T c (ω 0 ), defined functional integral Eq. (6.3) in general cannot be evaluated exactly. momenta However, the as discussed fermions in the will Introduction pass through order as a temperature to still at form a given a molecule. detuning ω 0 atat whichigher n 0 firstrelative vanishes. Setting µ = ω 0 and n 0 = 0, we find an implicit and in Sec. V A, for small g s (γ s ) the theory can each equation other without interacting. be analyzed by a controlled perturbative expansion in d 3 d 3 (π) 3 + e mtc ω 0 Tc + (π) 3 = n powers of γ s around the saddle-point (mean-field) approximation into infinity of Z s. To when this end, ω 0 we isloo 0. for This the spatially is the value We see that the scattering length a e 4mTc, turns uniform field configuration φ(r) = B that minimizes the of detuning at which bound states of(6.) fermions action(molecules) S s [φ]. We find the arefollowing aboutsaddle-point to startequation to form. It that uniquely gives T c (ω 0 ). The numerical solution of corresponds Eq. (6.) is presented to δs in Fig. 0. s [φ] The limiting behavior of T c (ω 0 ) is easy to deduce. B δ φ = φ=b Deep in the BEC regime, for ω 0 ɛ F, the a = first in- tegral is exponentially small, reflecting the fact that in π. (.6) ɛ 0 = g mλ this regime the fermion chemical potential µ is large and ɛ 0 µ gst d 3 (π) 3 ω negative and a number of thermally created fermionic n ωn + ( m µ) = 0. + g Thus the detuning is still very positive when bound states of fermions start s to BB form. At (6.3) atoms is strongly suppressed. The second integral then lower value of ɛ gives the critical temperature, 0 there are real bound states. where This ω that in this regime coincides with absent the BECintransition the absence temperature of interactions, closed suchform, as leading in (.3). to the so-called BCS-BEC gap equa- n = πt distinction (n + ) are the between fermion Matsubara ω 0 andfre- quencies. The sum over the frequencies can be done in a ɛ 0 is of course T c (ω 0 ɛ F ) T BEC = π ( ) /3 n tion for the mean field B(T, µ, ɛ 0 ) (and the corresponding Notice that at wea interactions,, (6.) g 0, condensate r 0. density B m ζ (3/) ) thatto is indeed further on the understand order of ɛ F. the As ωmeaning 0 is increased of the interactions, it is ɛ 0 µ = g s useful d 3 tanh to study the poles of E T through the BEC and crossover regimes, T c (ω 0 ) decreases, as the contribution of thermally created free where E is given by (π) 3, (6.4) E this scattering amplitude. The analysis has been thoroughly discussed in the Ann. Phys. paper atoms from referred the firsttointegral on the increases. frontwhen pagedetuning of the notes. reaches ω 0 = ɛ F, the solution of Eq. (6.) drops down ( ) to T c (ɛ F ) = 0. Beyond this point, for ω 0 > ɛ F in the E = m µ + gs BB (6.5) BCS regime, the bosons are completely converted into.3 free fermions Solution forming a Fermi atsea nonzero and T c (ω 0 ) stics but at 0. small The integralg on the right hand side of the BCS-BEC gap equation is formally divergent, scaling linearly with the uv momentum cutoff Λ. However, expressing the bare Small g means B. Narrow-resonance the problem limit can be attaceddetuning by theparameter mean field ɛ 0 intheory. terms of the Wephysical, definerenor- malized detuning ω 0 using Eq. (5.6), we can completely a parameter We extend our study of the s-wave resonant Fermi gas, eliminate the appearance of the microscopic uv scale described by the Bose-Fermi mixture action to the limit Λ in all physical quantities, and thereby obtain a uvconvergent g form of the BCS-BEC gap equation (.7) where γ s is small but nonzero. The overall qualitativeγ = ɛf picture is quite similar to the g s 0 limit discussed in [ ] the previous subsection and summarized in Fig.8, with ω 0 µ = g s d 3 tanh E T which only a few has newtofeatures. be small for g to be small. This is equivalent (π) to 3 demanding m E that. (6.6) r 0 g ɛf l, (.8) 3

5 where l is the mean separation between the particles and r 0 was defined in (.5). Thus r 0 is large and negative. To use mean field theory we assume that the bosons Bose condensed. We replace ˆbp V Bδ p,0 (.9) where B is the condensate density. This gives us a reduced Hamiltonian Ĥ reduced = ( ),σ m µ â,σâ,σ + (ɛ 0 µ) B B + g ( B â, â, + ) B â, â,, (.0) One needs to minimize this at given B over all possible fermion configurations, and calculate the ground state energy E(B). Then one needs to minimize E(B) with respect to B. This is a standard procedure because (.0) loos lie the usual BCS Hamiltonian studied in superconductivity. Once E(B) is found, the condition de/db = 0 becomes ɛ 0 µ = g d 3 (π) 3 ( p m µ) + g B B (.) This equation (called the gap equation in the theory of superconductivity) has two unnowns, µ and B (assuming that ɛ 0 is given). The second equation which relates them is the equation that ˆN = N, where the operator ˆN is introduced in (.), and N is the number of initial fermions in the systems. It can be shown that this equation taes the form N V = B B + d 3 (π) 3 ( p m µ). (.) + g B B µ m Solving these two equations gives the boson density n = B B as a function of ω 0. The solution loos very much lie the one shown on Fig, except the corners have been rounded and ɛ 0 got replaced by ω 0. Now even if ω 0 > ɛ F, there are some (very small) number of bosons present, while remaining fermions form a BCS superconductor, described by (.0). Even at ω 0 < 0, there are still some fermions present which also form a superconductor, however at negative chemical potential. µ approximately follows ω 0 /, if ω 0 ɛ F, and µ = ɛ F if ω 0 > ɛ F. This is just lie in the noninteracting case, except µ follows ω 0 instead of ɛ 0 and it does so approximately (deviating from it by an amount which goes to 0 if g goes to 0) and not exactly. Notice that this solution wors for all ω 0, as long as g is small (or r 0 l). It wors even when a. 4

6 .4 Large g or small r 0 limit. If g is large, it can be shown that mean field theory still wors at very large ω 0 where (.) describes a BCS superconductor, and at very large negative ω 0 where it describes a BEC of wealy interacting molecules. When ω 0 is close to zero, this problem is not solvable except numerically. To elucidate further this limit, we observe that physically large g means that g, while a is ept fixed. In other words, according to (.5), ɛ 0 should also be taen into infinity at the same time so that the ratio ɛ 0 /g remains fixed. In this limit, the two channel model becomes equivalent to a one channel model. This can be shown, for example, in the following way. Let s write the equations of motion for ˆb p which follow from (.) i ˆbp = [Ĥ, ˆb p ] = ɛ 0ˆbp + g â p V, â p +, (.3) At large g and large ɛ 0, the time derivate can be neglected, to give Substituting it bac into the Hamiltonian, we find Here Ĥ =,σ=, ˆbp = g â p ɛ 0 V, â p +,. (.4) m â,σâ,σ λ V p, â p +, â p, â p, â p +, (.5) λ = g ɛ 0, (.6) which remains constant in this limit, and can be expressed in terms of the scattering length a (given by (.5)). (.5) is the Hamiltonian for a one channel model, describing the fermions with short ranged featureless attractive interactions. It describes the crossover between BCS and BEC, as the attractive interactions strength gets stronger. Thus this Hamiltonian can be solved in the far BCS limit (small λ), and far BEC limit (large λ), but hard to study in the intermediate regime. It is completely equivalent to the two-channel problem if g is large. This can be summarized by the diagram shown on Fig 3. It shows when, depending on the values of the scattering length a, r 0 and the particle separation l n /3, the problem can be solved perturbatively and when it cannot. Finally, we remar that the regime of large g is usually called broad resonance, while small g is narrow resonance. The origin of this term is the following: g controls the ratio 5

7 5 d below ɛ F, se-condensed es, stabilized cular (closedngly-coupled pairs, that, ith it by the distinct from y anisotropic, inct (molecually decouple the Feshbach over to BEC g, where conooper pairs) plete. In the a true bound to a positive in the open- Feshbach resstem is taen trong unitary ent scattering ccessible in a r 0 ) plays the teracting via wo-body) pootential bared) resonant eter γ s (pror 0 ) is effece systems, a nsionless pa- 3 a bg ) a s parameter) e resonance, the gas (no he resonance, intractable in rtant distincch resonances ssible regions resonance are implicity and erimental sysems (exhibit- [9]) were a [3 5, 8] that of the BCS- EC superfluying enlight- otwithstand- -field descripde of the BCS the Fermi ena mean-field broad FR narrow FR ~ n F r 0 pertubatively inaccessible 0 /3 n /3 a = a FIG. 3: Illustration of perturbatively accessible and inaccessible (grey) regions in the inverse particle spacing vs inverse scattering length, n /3 a plane around a Feshbach resonace, where a diverges. Note that outside the grey region, even for a broad Feshbach resonance there is a small parameter that is either the gas parameter or Feshbach resonance coupling, or both, and hence the system can be analyzed perturbative.. Figure 3: Illustration of perturbatively accessible and inaccessible (grey) regions in the inverse particle spacing vs inverse scattering length, n /3 a plane around a Feshbach resonace, where a diverges. Note that outside the grey region, even for a broad Feshbach resonance there is a small parameter that is either the gas parameter or Feshbach resonance coupling, or both, and hence the system can be analyzed perturbatively.. of a to ω 0, according to (.5). The experimentalists tune ω 0 by changing magnetic field treatment and outside of the BEC regime where, although mean-field techniques brea down, a treatment perturbative in n /3 a is still possible [6]. The inability to quantitatively treat the crossover regime for generic (non-resonant) interactions is not an uncommon situation in physics, where quantitative analysis of the intermediate coupling regime requires an exact or numerical solution [34]. By integrating out the virtual molecular state, systems interacting through a broad (large γ) resonance can be reduced to a nonresonant two-body interaction of effectively infinite γ, and are therefore, not surprisingly, also do not allow a quantitatively accurate perturbative analysis outside of the BCS wea-coupling regime [3]. The study of a fermionic gas interacting via a broad resonance reveals the following results. If a < 0 (the interactions are attractive but too wea to support a bound state) and n /3 a, such a superfluid is the standard BCS superconductor described accurately by the mean-field BCS theory. If a > 0 (the interactions are attractive and strong enough to support a bound state) and n /3 a, the fermions pair up to form molecular bosons which then Bose condense. The resulting molecular Bose condensate can be studied using n /3 a as a small parameter. In particular, in a very interesting regime where a r 0 (even though a n /3 ) the scattering 6 length of the bosons becomes approximately a b 0.6a [35], and the Bose condensate behaves as a wealy inter- H, so that ω 0 µ B H. Here µ B is the Bohr magneton, and H is the deviation of the magnetic field from the resonance. They then plot /a as a function of H. The bigger g, the more slowly a changes with changing H or ω 0. Thus the plot of a vs H becomes more broad. Typically most of the experiments are carried out in the regime of a broad resonance. One reason is that reducing the particle density always taes us into the regime of a broad resonance, corresponding to l r 0. It is easy experimentally to reduce the density (while increasing it may lead to instabilities and is problematic). Lecture : p-wave resonances. p-wave -channel model If all fermions are identical (in the identical atomic or spin state), they cannot interact via s-wave interactions. The minimal coupling is p-wave, and the corresponding resonant

8 model taes form Ĥ = m â â+ ( ) ɛ 0 + p ˆb p,α 4m p,αˆbp,α + g V,p α (ˆbp,α â + â p + p + ˆb p,α â + p â+ p (.) Now the molecuesl have structure, expressed by the index α in the creation operator ˆb. They have angular momentum (spin). is The interactions are wea when g is small. The corresponding dimensionless parameter This parameter is small at low density. γ = g ɛ F. (.) Thus experiments (when they are done) will typically be in the regime of wea interactions, which can be understood using mean field theory. Why would one want to study these p-wave systems?. The transition between BCS and BEC is a transition, not a crossover. Indeed, in the mean field theory we will replace ˆb p,α V B α δ p,0 and plug it bac into the Hamiltonian. The resulting superconductor Hamiltonian has a spectrum (this is derived in the BCS theory) ( ) p E = m µ + B α Bβ p αp β. (.3) If µ > 0, we will have gapless excitations with the appropriate choice of p α (such that p /(m) = µ and B α p α = 0). If µ < 0, there are no gapless excitations. Conclusion: there is a phase transition at µ = 0. As we saw in the previous lecture, varying ɛ 0 changes µ and maes it go through zero, thus allowing to observe the transition.. The condensate density B α is a vector. Choosing different directions of this vector chooses different phases of such p-wave condensate. Thus there is a possibility of observing different phases and transitions between them in addition to the BCS-BEC transition. 3. If we are in dimensional space, and if B α = u α + iv α where v α u α = 0, then this leads to a superfluid which is called a chiral p-wave D superconductor. Such a superfluid, if µ > 0, has excitations which obey nonabelian statistics. Observing such excitations would be a milestone achievement. Notice that in D, g is dimensionless, so in D it is not as straightforward to see if mean field theory wors. It depends on how large g is regardless of particle separation. 7 ).

9 . The meaning of the model As before, we need to compute the -body (two fermion) scattering amplitude to understand the meaning of the model. This is given by Here Here f(p) = p v + 0 p ip3. (.4) v = 6π mg (ɛ 0 c ), 0 = π m g ( + c ). (.5) c = m 9π Λ3 g, c = m 3π g Λ. (.6) Here v is called the scattering volume and can be controlled via changing detuning ɛ 0 maing it go through infinity in a manner similar to the scattering length in case of s-wave resonances. 0 is a parameter equivalent to r 0 in the case of s-wave, but having dimensions of momentum. We see that at small momenta ip 3 can be completely neglected, and the poles of the scattering amplitude are given by p m = m 0 v. (.7) At positive v, this is the bound state (with negative energy) (remember that 0 < 0!). At negative v, this is a resonance (a long lived bound state with positive energy, which eventually decays bac into nonbound fermions). v ɛ 0, thus we tune between these two regimes with the detuning, as before..3 Phases of the condensates Once the condensation taes place, ˆb p,α V B α δ p,0. B α is a complex vector and can tae two different forms. We can always write B α = u α + iv α (.8) where u α and v α are two real vectors. Moreover, they can always be chosen orthogonal to each other (changing the angle between them is equivalent to multiplying B by a phase, which is not obvious, but can be checed by explicit algebra). If v = 0, then B α = u α (.9) 8

10 and is real. This is called the p x phase of the condensate (or sometimes called the polar phase). This describes the molecules whose angular momentum has projection zero on the axis formed by u α. If v 0, let us consider u = v (their lengths coincide). Then B α = u α + iv α (.0) and can never be real. This is called the chiral, or ferromagnetic, or p x + ip y phase of the superconductor. This describes molecules whose angular momentum has projection + on the axis parallel to u v. By choosing u to point along the x-direction, and v along the y-direction, the two situations are described by B α = 0 : p x, B α = i : p x + ip y. (.) 0 0 The important question is which one is realized in the -channel model..4 Mean field theory and the choice of the phase To understand which situation is realized in a system governed by the -channel model, we implement the mean field scheme, just as in the s-wave case (except in this case this scheme is reliable as long as the system is dilute enough). We substitute ˆb p,α V B α δ p,0 to find by analogy with (.0) Ĥ reduced = ( ) m µ â â+ ( ) ɛ 0 µ + p B α 4m αb α + g ( α Bα â V â + ) B α â â. (.) Here the conserved particle number is now ˆN = a a + b p,αb p,α. (.3) p,α We now need to find the ground state energy E(B α ) and minimize it with respect to B α. In contrast with the s-wave case, there could be now different minima and maxima of E, so one needs to loo for a global minimum. A calculation to loo for the ground state energy of (.) gives the following expression E = ( u + v ) u 3 + v 3 [ (u [ω 0 µ + a ln {a 0 (u + v})]+a + c u + v +a + v ) ( + u v ) ]. (.4) 9

11 Here it is assumed that u α v α = 0, that is, the two vectors are perpendicular, and u, v are their respective lengths. a 0, a, and a are three coefficients which can be expressed in terms of g, ɛ F, and µ, and whose explicit form is unimportant. The only important information about it is that a is an increasing function of µ if µ > 0. If a < 0, then a vanishes. Now we distinguish two possible regimes. In the BCS regime, ω 0 is large positive. We expect that u and v are going to be small. If so, then the quartic terms with a coefficient in front can be neglected. The large ω 0 will be compensated by the large logarithm if u and v are small. We need to minimize the term proportional to a. It can be shown that the minimum is achieved if u = v. Thus in the BCS regime u = v and the phase is p x +ip y. As ω 0 is decreased, µ decreases with it (recall the discussion at the beginning of Lecture. Then we enter the BCS regime. In this regime, a becomes zero (when µ < 0). Then the ω 0 µ term is balanced by the quartic term, while the terms proportional to a vanish. Now we need to minimize the quartic terms proportional to a. Obviously this is achieved if u = v (since that s when u v = 0). Thus in the BEC regime the appropriate phase is also p x + ip y. One way to interpret it is to say that the interactions between the molecules, described by the quartic term, are ferromagnetic and their angular momenta want to align. This shows that the BCS-BEC condensate described by the -channel model is always in the chiral p x + ip y phase. This argument goes through in D as well (assuming that the mean field theory applies). Thus when confined to D, this BCS-BEC condensate will form a chiral D p-wave superconductor, which has non-abelian excitations at µ > 0. The idea is now to tune µ by tuning ɛ 0 so that µ > 0 but not too large so that to maximize the gap (and allow for stable quasiparticles with which to do quantum computations using their non-abelian statistics)..5 Stability of the p-wave superfluid This section follows the paper J. Levinsen, N. Cooper, V. Gurarie, PRL 99, 040 (007). Since it deals with estimates, this section features explicit h, which was set to previously. The experiments done to try to create p-wave molecules showed that they are unstable, with the lifetime of the order of ms. This so far prevented the experimentalists from creating a p-wave superfluid in the lab. Let us understand why this is so. The main process which leads to decay of the molecules is the following. All of the 0

12 atoms interact with attractive but relatively short ranged van der Waals interactions which, at very short distances of R e.5nm, lead to formation of a strongly bound molecule with a very large binding energy (of the order of h /(mre ). Here R e h/λ, where Λ is the cutoff introduced earlier. These are not the same as the wealy bound molecules discussed here. Once these strongly bound molecules are formed, large amount of energy is released in the form of a nearby atom which flies out carrying away this energy. This leads to the destruction of the molecules nearby. We can estimate the rate for this process in the following way. Suppose two wealy bound molecules collide and one of them becomes a strongly bound molecule while the other decays and its constituent atoms fly out carrying away the large binding energy of the strongly bound molecule. The rate of this process is given by Γ = nvσ in. (.5) Here n is the density, v is the molecular velocity, and σ in is the inelastic cross section of such a collapse process. Suppose the wealy bound molecules have size l, and an average separation between the particles is also a (we are close to resonance where all distances are roughly the same). Then we can estimate the inelastic cross section by σ in l d l v h mr e P. (.6) Here l d is the elastic cross section, d is the dimensionality of space, l/v is the time the molecules spend close by, and h/(mre) is a characteristic rate of the collapse (the only parameter of dimension of rate which depends on R e only). Finally, P is the probability that as two molecules of size l collide, at least 3 of the four atoms which formed these two molecules will find themselves at a distance R e to be able to carry out the collapse process. Estimating P is not obvious since the atoms are all strongly interacting. If atoms were not interacting at all, then P (R e /l) d. Indeed, the probability that one atom approaches the other is (R e /l) d, and that has to be multiplied by the probability that the third atom approaches the first two. However, it was demonstrated by Petrov, Salomon, and Shlyapniov, that strongly interacting fermions have a wave function which goes as ψ r γ, where r represents coordinates of all three particles (rescaling their coordinate by a factor of λ rescales the wave function by a factor of λ γ ). γ is an exponent which has to be calculated separately for each case. Then P ( Re l ) d+γ. (.7)

13 This is obtained by integrating ψ over the coordinates of the two particles with the third one being ept fixed. Putting it all together using n /l d gives Γ h ml ( Re Noting that h/(ml ) is of the order of ɛ F / h, we find l ) d+γ. (.8) Γ ɛ F h ( Re l ) d+γ. (.9) In order to have enough time to observe the resonant superfluids before they decay, we need, at least, to have Γ ɛ F / h. Now we distinguish several cases:. 3D s-wave ferimons. γ was calculated by Petrov, Salomon, and Shlyapniov, to be γ 0.. This gives Γ ɛ F h Typically R e /l /00. This gives ( Re l Γ ɛ F h ) d+γ = ɛ F h ( Re l ) (.0) ( ) 3.56 ɛ F 00 h. (.) The decay is indeed very slow. The system has plenty of time to be created and observed before it decays.. 3D s-wave bosons. γ was calculated by Efimov, to be γ =. Then Γ ɛ F h. (.) This is very fast. The system will decay before anything can be observed with it. 3. 3D p-wave fermions. Unlie the s-wave case, this needs to be rethought. The p- wave molecules are very small because their wave function behaves as /r (in general, the angular momentum s wave function goes as /r s+ according to the standard results in quantum mechanics). Thus the probability of seeing atoms close by goes as /r 4, which is not integrable over d 3 r at small r. Thus the molecule has a typical size R e, and not l. This leads to σ in R e(r e /v) h/(mr e) and Γ = nvσ in = h R e ml l ɛ F h R e l. (.3)

14 This is better than 3D s-wave bosons, but not as good as 3D s-wave fermions. The decay rate can be estimated as (taing into account that ɛ F / h 0KHz in a typical experiment) Γ 0KHz 00 /(0ms). (.4) Thus the lifetime of the molecules is expected to be 0ms, which is about 0 times larger than observed. Yet it is nown that this formula tends to overestimate the lifetime, presumably due to some numerical factor of the order of 0 neglected in its derivation. The origin of the relatively short lifetime of the p-wave molecules is the following. In the s-wave case, a molecule decays if a third atom approaches it (to tae away its binding energy so that it could collapse into a deeply bound state). But the third atom is identical to one of the atoms which made up the molecule and the Pauli principle will prevent it from coming close. In case of p-wave, a molecule has an angular momentum, say +. A third atom will approach this molecule with the angular momentum, thus beating the Pauli principle. Then the molecule will collapse and the atom will carry away its energy. Therefore, new fresh ideas are needed if a stable p-wave superfluid is to be created. 3