Averaged collision and reaction rates in a two-species gas of ultracold fermions
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] N ov Averaged collision and reaction rates in a two-species gas of ultracold fermions
Alexander Pikovski
Institut f¨ur Theoretische Physik, Leibniz Universit¨at Hannover, Appelstr. 2, 30167 Hannover, Germany
Reactive or elastic two-body collisions in an ultracold gas are affected by quantum statistics.In this paper, we study ensemble-averaged collision rates for a two-species fermionic gas. The twospecies may have different masses, densities and temperatures. We investigate how averaged collisionrates are affected by the presence of Fermi spheres in the initial states (Pauli blocking of final statesis not considered). It is shown that, independently on the details of the collision, Fermi-averagedcollision rates deviate from Boltzmann-averaged ones, particularly for a gas with strong imbalanceof masses or densities.
I. INTRODUCTION
Experiments with ultracold atoms and molecules probecollisional physics at very low temperatures. Gases ofquantum degenerate fermionic atoms are being stud-ied [1], and experiments are progressing towards the cre-ation of degenerate gases of fermionic molecules [2]. Atvery low temperatures, quantum statistics plays an im-portant role in interparticle collisions. One way to ob-serve an effect which depends on the symmetry of theparticles is to look at low-energy collisions which obeyquantum threshold laws; this was measured in elasticcollisions in fermionic atomic gases [11] and in reactivecollisions of ultracold molecules [7]. Another effect ofquantum statistics is Pauli blocking of collisions, as wasobserved in Ref. [12]: collisions are suppressed if the finalstate cannot enter a filled Fermi sphere. In contrast tothe quantum threshold behavior, the observation of thiseffect requires the fermions to be quantum degenerate.In the present work we investigate how averaged col-lisional rates depend on the temperature, density, andmasses in a two-species Fermi gas. The usual experimen-tal observable in a cloud of ultracold gas is a collision ratewhich is averaged over the distribution of velocities in thegas. Usually this ensemble-averaging of collision rates isdone using a Boltzmann distribution, however as experi-ments move deeper into the quantum degenerate regime,quantum statistics starts playing a role. Particular atten-tion here will be focused on the imbalanced Fermi gas,i.e. a two-component gas where the two species have dif-ferent densities or masses. The effect of mass or densityimbalance on the averaged collision rates at low temper-atures is most pronounced in this setting, the rates candiffer significantly from a Boltzmann-averaged rate.Averaged collision rates for a two-component gas offermions are studied, for general two-body collisions withan arbitrary cross-section. The particles are assumedto be in equilibrium; effects of Pauli-blocking are notconsidered. The two types of fermions may have dif-ferent masses, different densities, and different tempera-tures. First, it is shown that an averaged collision rateis expressed as an average over the distribution of rela-tive energies (Sec. II). This distribution is calculated forfermions analytically at T = 0, and it is shown that thisfunction has a simple geometric interpretation (Sec. III). At finite temperatures, the relative energy distributionis calculated numerically and a simple approximation forall temperatures is proposed (Sec. III). In Sec. V we dis-cuss the threshold laws for ensemble-averaged collisionrates. Some details of the calculations are presented inthe Appendices. In Sec. IV, we discuss how the resultscan be transferred to a situation where the particles areconfined to a two-dimensional geometry. II. AVERAGED COLLISION RATE
Experiments with clouds of ultracold gases measurecollision rates, e.g. using photoassociation spectra [13] ortrap loss rates due to chemical reactions [8]. Here one ac-tually measures rates which are averaged over the distri-bution of velocities in the gas. This ensemble-averagingis the topic of this section.Consider a collision between two particles, A and B ,which results in some final state f : A + B → f. (1)This may be an elastic collision, an inelastic collision,or a reaction. The particles A , B have masses m A , m B and velocities v A , v B in the laboratory frame. The totalscattering cross-section for the process (1) in the center-of-mass frame will be denoted σ ( p ), it depends on therelative momentum p of the colliding particles.In a gas cloud, where many collisions take place, thenumber density n α of particles in state α = f, A, B sat-isfies the rate equation dn f dt = h vσ ( p ) i n A n B (2)where h vσ i is the two-particle collision rate averaged overthe distribution of velocities in the system. The rateequation (2) refers only to the process (1). It is assumedthat, in the case of fermions, the final state f is not oc-cupied, thus there is no effect of Pauli blocking. If theparticles A and B are indistinguishable, a small modifi-cation to Eq. (2) may be necessary, see [6].The relevant quantity for many experiments is theensemble-averaged collision rate K = h vσ i , it has dimen-sions (length) / time. If velocities of particles of type A , B are distributed according to the distributions ¯ f A , ¯ f B ,the averaged collision rate K is [9] K = h vσ ( p ) i = Z Z vσ ( p ) ¯ f A ( v A ) ¯ f B ( v B ) d v A d v B (3)We will consider the case where the velocity distributionsand the cross-section depend only on the magnitude ofthe velocities, i.e. they depend only on the energy of theparticles. Changing to the center-of-mass frame with theformulas of Appendix A brings the integral to the follow-ing form: K = r m Z ∞ E rel σ ( E rel ) F ( E rel ) dE rel (4)with F ( E rel ) = 12 Z π dθ sin θ Z ∞ f A ( E A ) f B ( E B ) E / dE cm . (5)Here f A,B are the energy distribution functions. Thenormalization for the velocity distribution functionsis R ¯ f ( v ) d v = 1, the energy distribution functionsare normalized to R ∞ f ( E ) √ EdE = 1, and f A,B =2 π (2 /m A,B ) / ¯ f A,B . The expressions for E A and E B interms of E cm , E rel and θ , which are needed to evaluatethe integral (5), are given in Eq. (A3).The ensemble-averaged collision rate is given by thecross-section σ averaged with the function F , as shownby Eqs. (4), (5). The function F ( E rel ), which we calldistribution of relative energies, contains all the depen-dence on the densities and temperatures of the particlesinvolved in the collision. The cross-section σ ( p ) containsall the information about the two-body scattering.The expression for the distribution function of rela-tive energies, Eq. (5), can be brought to a different formwhich will be of use later. Transforming the integral inEq. (5) to the coordinate system ( x, y, z ), as described inAppendix A, we have the representation F ( E rel ) = 12 πη Z Z Z f A ( E A ) f B ( E B ) dxdydz. (6)The integration runs over the whole space, E A and E B are expressed through x, y, z using Eqs. (A3), (A5), andthe mass factor η is defined in Eq. (A6). III. DISTRIBUTION OF RELATIVE ENERGIESFOR FERMIONS
Now we discuss the distribution function of relativeenergies F ( E rel ), which determines the averaged colli-sion rate [cf. Eq. (4)] for a two-species gas of fermionsin equilibrium. We will consider a Fermi gas where thetwo species may have different densities or masses. Suchimbalanced systems have attracted recent experimental[19] and theoretical [18] attention. dr r FIG. 1: Intersection of two spheres with radii r and r , cen-ters separated by distance d . A. Zero temperature
First, the case of zero temperature is considered. Thedistribution functions for the particles A and B are nor-malized T = 0 Fermi functions f ( E ) = 32 µ / θ ( µ − E ) , (7)where θ ( x ) is the step function and µ is the chemicalpotential. The particles of type A , B may have differ-ent chemical potentials µ A , µ B . It will be convenient tointroduce the “reduced” chemical potentials ν A = 2 m B m A + m B µ A , ν B = 2 m A m A + m B µ B , (8)and for equal masses we have ν A,B = µ A,B . When theFermi functions, Eq. (7), are inserted in the represen-tation (6) for the distribution of relative energies, oneencounters the volume integral Z θ ( µ A − E A ) θ ( µ B − E B ) d x. (9)Once E A and E B are expressed in terms of x, y, z , it seenthat this integral has the following geometric meaning:it is the volume of intersection of two spheres. Let usbriefly discuss this volume.The volume of intersection of two spheres, which haveradii r and r and whose centers are separated by adistance d (see Fig. 1), is given by the expression V L ( d ; r , r ) = π d ( r + r − d ) { d +2 d ( r + r ) − r − r ) } . (10)This formula, as it stands, is only valid as long as thevolume of intersection has the shape of an asymmetriclens, as indicated in Fig. 1. However, if the distancebetween the spheres d is large, there is no intersection,and if d is small, one sphere is located completely insideof the other. We can write for the volume of intersection V ( d ; r , r ) = d > ( r + r ) min( πr , πr ) if d < ( r − r ) V L ( d ; r , r ) otherwise (11) a) E rel b) E rel FIG. 2: F ( E rel ) for a) ν A = 0 . ν B = 1 . ν A = ν B =1. The dashed lines are at ν A + ν B − √ ν A ν B and ν A + ν B + √ ν A ν B , the dotted line is at µ ∗ = ν A + ν B . In this form the expression is valid for all values of d , r ,and r .Returning to the distribution function of relative en-ergies, it is seen by using (7) in Eq. (6) that the resultis F ( E rel ; ν A , ν B ) = 92 √ π ν A ν b ) / V ( √ E rel ; √ ν A , √ ν B ) . (12)This is the volume of intersection of two spheres withradii √ ν A and √ ν B , the centers of the spheres being sep-arated by √ E rel , up to normalization. It is interestingto note that the masses of the particles do not appearexplicitly, but enter only through the reduced chemicalpotentials ν A , ν B .Let us discuss the general form of the function F ( E rel ).A plot is shown in Fig. 2a for the strongly imbalancedcase ν ≪ ν . This case occurs either if the particles havedifferent masses or if the particle densities are different,cf. Eq. (8). The function has the form of a smoothedstep. From the cases in Eq. (11), it is seen that F ( E rel )is a decreasing function in the range (shown by dashedlines in Fig. 2) ν A + ν B − √ ν A ν B < E rel < ν A + ν B √ ν A ν B . (13)For smaller values of E rel the function is constant andfor larger values of E rel it is identically zero. The step iscentered at µ ∗ = ( ν A + ν B ) /
2, but it is not symmetricaround this point.In the balanced case µ = ν A = ν B , the expression for F ( E rel ) simplifies considerably. This case is shown inFig. 2b. The distribution function has no flat region, it E rel T = = = = = = FIG. 3: F ( E rel ) at different temperatures, for ν A = 0 . ν B =1 . T = 0. is now a strictly decreasing function between E rel = 0and E rel = 2 µ . B. Finite temperatures.
For finite temperatures, the distribution functions ofthe particles are normalized Fermi–Dirac distributions f T ( E ) = N µ,T
11 + e ( E − µ ) /T , (14)the normalization constant N is given in Appendix B.We allow the particles A , B to be at different temper-atures T A , T B . The corresponding distribution of rel-ative energies F T ( E rel ) becomes, geometrically, the vol-ume of overlap of two spherical distributions whose radiiare broadened by temperature. One can introduce thereduced temperatures τ A = 2 m B m A + m B T A , τ B = 2 m A m A + m B T B . (15)As shown in Appendix B, the distribution of relative en-ergies can be written as F T ( E rel ) = N ν A ,τ A N ν B ,τ B τ A τ B × Z ∞ Z ∞ ( ǫ A ǫ B ) / F ( E rel ; ǫ A , ǫ B )cosh (cid:0) ǫ A − ν A τ A (cid:1) cosh (cid:0) ǫ B − ν B τ B (cid:1) dǫ A dǫ B . (16)Note that the masses of the particles enter only throughthe reduced chemical potentials ν α and temperatures τ α .In the limit of high temperatures, both particles aredistributed according to the Boltzmann distribution f ∞ ( E ) = 2 √ πT / e − E/T . (17)In this case the distribution of relative energies may beevaluated directly from Eq. (5). Then we obtain, asknown [9], that the distribution of relative energies isa Boltzmann distribution F ∞ ( E rel ) = 2 √ πT / ∗ e − E rel /T ∗ , (18)with the effective temperature T ∗ = τ A + τ B m A T B + m B T A m A + m B . (19)The distribution function of relative energies may becalculated numerically from Eq. (5) or from Eq. (16); forlow temperatures the form in Eq. (16) is more conve-nient. Figure 3 shows the result of such a calculation.The chemical potentials ν A , ν B were fixed at T = 0,and at T > F ∞ ( E rel ) from Eq. (18)] limit. It is seen that the dis-tribution function approaches the Boltzmann form when T ≈ max( ν A , ν B ). C. Approximation for all T Since the distribution of relative energies has a com-plicated form at finite temperatures, an approximationmight be helpful. The distribution function of relativeenergies may be approximated, at all temperatures, by aFermi distribution F T, appr ( E rel ) = N µ ∗ ,T ∗
11 + e ( E rel − µ ∗ ) /T ∗ (20)with the effective temperature T ∗ from Eq. (19) and theeffective chemical potential µ ∗ = ν A + ν B m A µ B + m B µ A m A + m B . (21)This gives a reasonable approximation for the shapeof F T ( E rel ) at low temperatures. For high tempera-tures, it approaches the exact distribution F ∞ ( E rel ) fromEq. (18). IV. TWO DIMENSIONS
Gases of ultracold atoms or molecules can be con-fined to thin layers. For example, recent experi-ments at JILA [8] investigated a gas of KRb moleculesin a pancake-shaped geometry and measured the rateof chemical reactions; related theoretical work consid-ered ensemble-averaged rates (Boltzmann distribution inRef. [14] and Fermi–Dirac distribution in Ref. [17]) for atwo-dimensional geometry. When the gas is confined toa thin layer, one can describe collisions as if the parti-cles were moving in a two-dimensional plane. The col-lisional cross-section, however, depends on the details of the interparticle interaction and the confinement. Inthis Section we will consider how the preceding results(Sec. II and Sec. III) can be adapted to the case of two-dimensional scattering.For two dimensions, what was said at the beginningof Sec. II remains valid, but now in Eq. (2) one shoulduse the two-dimensional number densities (dimensions1 / length ), and the two-dimensional scattering cross-section σ ′ (dimensions 1 / length). We will add a prime totwo-dimensional quantities to avoid confusion.The averaged two-dimensional collision rate K ′ , havingdimensions (length) / time, is K ′ = h vσ ′ ( p ) i = Z Z vσ ′ ( p ) ¯ f ′ A ( v A ) ¯ f ′ B ( v B ) d v A d v B . (22)For velocity distributions which depend only on the en-ergy of the particles, changing to the center-of-mass (seeAppendix A) brings the integral to the following form: K ′ = r m Z ∞ E / σ ′ ( E rel ) F ′ ( E rel ) dE rel (23)with F ′ ( E rel ) = Z π dθ π Z ∞ f ′ A ( E A ) f ′ B ( E B ) dE cm . (24)Here f ′ A,B are the energy distribution functions, the nor-malizations are R ¯ f ′ ( v ) d v = 1 and R ∞ f ′ ( E ) dE = 1, so f ′ A,B = (2 π/m
A,B ) ¯ f ′ A,B . The expressions for E A and E B in terms of E cm , E rel and θ , which are needed to evaluatethe integral (24), are given in Eq. (A3).The double integral in Eq. (24) can be transformedto the coordinate system ( x ′ , y ′ ), see Appendix A, whichresults in F ′ ( E rel ) = 1 πη Z Z f ′ A ( E A ) f ′ B ( E B ) dx ′ dy ′ . (25)The integration runs over the whole space, E A and E B are expressed through x ′ , y ′ using Eqs. (A3) and (A8),and η is given in Eq. (A6).So far the discussion has been quite general, now weconsider the case where the particles A , B are fermions.At zero temperature the distribution functions for theparticles A , B are f ′ ( E ) = 1 µ θ ( µ − E ) , (26)with possibly different chemical potentials µ A , µ B . Theintegral that results when (26) is inserted in Eq. (25) hasthe geometric interpretation of the area of overlap of twocircles. For finite temperatures, the boundary of thesecircles becomes broadened by temperature, as before.The area of intersection of two circles, with radii r and r separated by distance d (see Fig. 1), is A L = r arccos (cid:16) d + r − r dr (cid:17) + r arccos (cid:16) d + r − r dr (cid:17) − p ( r + r + d ) − r + r + d ) . (27)To include the cases where the circles are completely dis-joint and where one circle is inside the other, we write A ( d ; r , r ) = d > ( r + r ) min( πr , πr ) if d < ( r − r ) A L ( d ; r , r ) otherwise (28)It follows that the distribution function of relative en-ergies at T = 0 is F ′ ( E rel ) = 2 πν A ν B A ( √ E rel ; √ ν A , √ ν B ) , (29)where the ν α are given in Eq. (8). This function hasa similar shape as the corresponding function in threedimensions.In the high-temperature case, the energies of the par-ticles A , B follow the Boltzmann distribution f ′∞ ( E ) = 1 T e − E/T (30)with possibly different temperatures T A , T B . Then thedistribution of relative energies, calculated from Eq. (24),becomes a Boltzmann distribution in the relative energy F ′∞ ( E rel ) = 1 T ∗ e − E rel /T ∗ (31)with the effective temperature T ∗ as in Eq. (19). Thisresult has the same form as in three dimensions. V. THRESHOLD LAWS
The dependence of the averaged collision rate on theparameters of the two-species gas (mass, temperature,chemical potential) can be found using the results ofthe preceding sections, if the scattering cross-sectionis known. Some conclusions can be drawn, however,without specific knowledge of the cross-section in thelow-energy collisional regime where quantum thresholdlaws apply. According to the Wigner threshold laws(see [4, 10] for a review), the cross-section at low energiesscales with the relative energy as σ ( E ) ∝ E p , (32)where the exponent p depends on the type of collision.For elastic collisions with short-range forces in the ℓ -thpartial wave channel, we have p = 2 ℓ . If molecules un-dergo chemical reactions upon collisions, the quantumthreshold laws hold only if the chemical reaction barrieris absent or much lower than the centrifugal barrier ofscattering [3, 15]. For these exothermic barrierless re-actions in the ℓ -th partial wave, p = ℓ − /
2. In gasesof ultracold atoms or molecules, one expects collisionsto take place in the lowest partial wave channel that isallowed by exchange symmetry.The averaged rate constant at zero temperatures forcollisions obeying Eq. (32) can be obtained in closed form from Eqs. (4) and (12). Here we only note the followingproperty of this rate K . If the chemical potentials arescaled as ν A → λν A , ν B → λν B , then K → λ p +1 / K .In particular, for p = − / K is con-stant.For finite temperatures, the exact dependence of theaveraged rate constant on the reduced temperatures τ A , τ B and chemical potentials ν A , ν B can be determinedfrom the representation (16). Inserting Eqs. (16) and (32)into Eq. (4) and performing the integration over E rel first,one obtains expressions for K . For p = − /
2, one findsagain that K is constant, and for other values of p morecomplicated expressions result which involve combina-tions of Fermi–Dirac integrals (see Appendix B for theirproperties). In the strongly quantum degenerate regime( µ/T ≫ F ( E rel ) with a Fermifunction with chemical potential µ ∗ and temperature T ∗ ,cf. Eq. (20). Then, the averaged collision rate for cross-sections obeying Eq. (32) has a simple form, it scales forall temperatures as K appr ∝ T p +1 / ∗ F p +1 ( µ ∗ /T ∗ ) F / ( µ ∗ /T ∗ ) (33)where the F are the Fermi-Dirac integrals.For high temperatures, the distribution of relative en-ergies approaches a Boltzmann distribution. The thresh-old law (32) then leads to the following well-known scal-ing of the collision rate K ∝ T p +1 / ∗ , (34)which also follows from Eq. (33). VI. CONCLUSIONS
Two-body collision or reaction rates in an ultracoldgas are affected by quantum statistics. The partial wavechannel of the collision depends on the symmetry of theparticles involved; the density of final states may be mod-ified (e.g. Pauli blocking for fermions); and, finally, theavailable initial states depend on the ensemble. In thispaper, we have investigated initial-state effects for a gasof two-species fermions with possibly different masses,particle densities, and temperatures. Independently ofthe details of the collision, the ensemble-averaged colli-sion rate is the cross-section averaged with the distribu-tion of relative energies. This distribution function wascalculated and its form was discussed in detail. An ap-proximation was proposed, which becomes exact in thehigh-temperature limit. It was also shown how to trans-fer these results to particles confined to two dimensions.As an application of the results, we considered the scal-ing of the averaged collision rates for the case where thecross-section follows the Wigner threshold law.The results will be of use for the analysis of experi-ments which measure collision or reaction rates in gasesof ultracold atoms or molecules. In particular, they willserve to determine how the measured averaged rates inquantum degenerate two-species gases depend on thetemperature, the masses and densities of the particles.I would like to acknowledge helpful discussions withL. Santos, M. Klawunn and C. Salomon.
Appendix A: Transformation of coordinates
The coordinate transformation between the laboratoryframe and the center-of-mass frame is given by the fol-lowing expressions. Two particles A , B have masses m A , m B and velocities v A , v B in the laboratory frame. Thevelocities in the center-of-mass frame are v = v A − v B , ( m A + m B ) V = m A v A + m B v B , (A1)The corresponding momenta are p = m ( v A − v B ) and P = ( m A + m B ) V , where m = m A m B / ( m A + m B ). Therelated energies are E A = 12 m A v A , E B = 12 m B v B ,E cm = 12 ( m A + m B ) V , E rel = 12 mv . (A2)The angle between v and V is denoted by θ . The en-ergies in the laboratory frame and center-of-mass frameare related by E A = mm B E cm + mm A E rel + r mE rel E cm m A + m B cos θE B = mm A E cm + mm B E rel − r mE rel E cm m A + m B cos θ (A3)and also E cm + E rel = E A + E B . (A4)The Jacobian of the transformation from ( v A , v B ) to( v , V ) is unity.To discuss the integral over the center-of-mass energy E cm and the angle θ in Eq. (5) (here E rel is a regarded asa constant parameter), we view θ as the azimuthal anglein spherical coordinates, and introduce the coordinates( x, y, z ) as x = η p E cm cos φ sin θ, y = η p E cm sin φ sin θ,z = η p E cm cos θ, (A5)with η = √ m A m B m A + m B . (A6)The Jacobian is ∂ ( x, y, z ) ∂ ( E cm , θ, φ ) = 12 η sin θ p E cm . (A7) The coordinates ( x, y, z ) are proportional to the Carte-sian components of the center-of-mass velocity V or thecenter-of-mass momentum P .For the two-dimensional integral over the center-of-mass energy, one can use the coordinates ( x ′ , y ′ ) givenby x ′ = η p E cm cos θ, y ′ = η p E cm sin θ (A8)with the Jacobian ∂ ( x ′ , y ′ ) ∂ ( E cm , θ ) = 12 η . (A9) Appendix B: Distribution function at finitetemperature
The distribution function of the particles in a Fermigas is the normalized Fermi–Dirac distribution f T ( E ) = N µ,T
11 + e ( E − µ ) /T , (B1)with the normalization constant N µ,T = 2 √ πT / F / ( µ/T ) , (B2)such that R ∞ f ( E ) √ E dE = 1. Here F j ( x ) = 1Γ(1 + j ) Z ∞ t j e t − x dt (B3)is the Fermi-Dirac integral of order j [16]. The expan-sion of this function for the strongly degenerate, low-temperature regime ( x ≫
1) has the form F j ( x ) = x j +1 Γ( j + 2) (cid:2) C x − + . . . (cid:3) (B4)for half-integer j , see Ref. [16] for more details. For thenon-degenerate, high-temperature regime ( x ≪ −
1) wehave F j ( x ) = e x .The chemical potential of a Fermi gas, for a fixed par-ticle density, depends on the temperature. It is relatedto the Fermi energy E F = µ ( T = 0) by F / ( µ/T ) = 43 √ π (cid:16) E F T (cid:17) / . (B5)In order to derive a representation for the integral (5)when f A and f B are Fermi-Dirac distributions, write forthe Fermi function n F ( ξ ) = (1 + e ξ/T ) − : n F ( ξ ) = − Z ∞ ξ ∂n F ∂ξ ( u ) du = − Z ∞−∞ θ ( u − ξ ) ∂n F ∂ξ ( u ) du (B6)Here ∂n F ∂ξ = − (cid:20) T cosh (cid:18) ξ T (cid:19)(cid:21) − (B7)Inserting Eq. (B6) into Eq. (5) and changing the order ofintegration, we encounter the T = 0 distribution function of relative energies which was discussed in Sec III. Theresult is given in Eq. (16). [1] W. Ketterle and M. W. Zwierlein, Making, probing andunderstanding ultracold Fermi gases , in
Ultracold FermiGases , Proceedings of the International School of Physics“Enrico Fermi” Varenna 2006, edited by M. Inguscio etal (IOS Press, Amsterdam, 2008).[2] L. D. Carr et al , Cold and ultracold molecules: science,technology and applications , New J. Phys. , 055049(2009).[3] M. T. Bell and T. P. Softley, Ultracold molecules andultracold chemistry , Mol. Phys. , 99 (2009).[4] J. Weiner et al , Experiments and theory in cold and ul-tracold collisions , Rev. Mod. Phys. , 1 (1999).[5] R. V. Krems, Cold controlled chemistry , Phys. Chem.Chem. Phys. , 4079 (2008).[6] P. S. Julienne and F. H. Mies, Collisions of ultracoldtrapped atoms , J. Opt. Soc. Am. B. , 2257 (1989).[7] S. Ospelkaus et al , Quantum-state controlled chemical re-actions of ultracold potassium-rubidium molecules , Sci-ence et al , Controlling the quantumstereodynamics of ultracold bimolecular reactions , NaturePhysics , 502 (2011).[9] K. E. Shuler, Reaction Cross Sections, Rate Coefficientsand Nonequilibrium Kinetics , in
Chemische Elemen-tarprozesse , edited by H. Hartmann (Springer, Berlin,1968).[10] H. R. Sadeghpour et al , Collisions near threshold inatomic and molecular physics , J. Phys. B: At. Mol. Opt.Phys. , R93 (2000).[11] B. DeMarco et al , Measurement of p-Wave Threshold Law Using Evaporatively Cooled Fermionic Atoms , Phys.Rev. Lett. , 4208 (1999).[12] B. DeMarco et al , Pauli Blocking of Collisions in a Quan-tum Degenerate Atomic Fermi Gas , Phys. Rev. Lett. ,5409 (2001).[13] R. Napolitano et al , Line Shapes of High Resolution Pho-toassociation Spectra of Optically Cooled Atoms , Phys.Rev. Lett. , 1352 (1994).[14] G. Qu´em´ener and J. L. Bohn, Dynamics of ultracoldmolecules in confined geometry and electric field , Phys.Rev. A , 012705 (2011).[15] G. Qu´em´ener, N. Balakrishnan, and A. Dalgarno, In-elastic collisions and chemical reactions of molecules atultracold temperatures , in
Cold Molecules: Theory, Ex-periment, Applications , edited by R. Krems et al (CRCPress, 2009).[16] J. S. Blakemore,
Semiconductor statistics (PergamonPress, Oxford, 1962); R. Kim, M. Lundstrom,
Notes onFermi-Dirac Integrals , arXiv:0811.0116[17] A. Pikovski et al , Nonlocal state swapping of polarmolecules in bilayers , Phys. Rev. A , 061605(R) (2011).[18] F. Chevy and C. Mora, Ultra-cold polarized Fermi gases ,Rep. Progr. Phys. , 112401 (2010). L. Radzihovsky andD. E. Sheehy, Imbalanced Feshbach-resonant Fermi gases ,Rep. Prog. Phys. , 076501 (2010).[19] G. B. Partridge et al , Pairing and Phase Separation in aPolarized Fermi Gas , Science , 503 (2006). Y. Shin et al , Phase diagram of a two-component Fermi gas withresonant interactions , Nature451