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after the formidable achievement of bose - einstein condensation in spin - polarized alkali gases @xcite , the next challenge that experimentalists have already set themselves is to realize quantum degenerate conditions also in fermionic alkali vapors . one particular motivation in this respect is the prediction that a gas of spin - polarized atomic @xmath0li becomes superfluid at densities and temperatures comparable with those at which the bose - einstein experiments are performed @xcite . as a result of this experimental interest , the first theoretical studies of an ideal fermi gas trapped in a harmonic external potential have recently appeared @xcite . moreover , the effects of an interatomic interaction have also been considered @xcite . it is interesting to note that oliva s calculations for atomic deuterium were already performed a decade ago , even though magnetically trapped deuterium had not been observed at that time . in fact , it has still not been observed , because the loading of the trap can not be accomplished in the same way as for atomic hydrogen @xcite . this is presumably caused by the fact that deuterium binds more strongly to a superfluid helium film , that the surface recombination rate is much larger , and that the sample is contaminated with atomic hydrogen @xcite . fortunately , such problems do not arise for experiments with alkali gases and both @xmath0li and @xmath1k have indeed been trapped already @xcite . the most important qualitative feature of a trapped fermi gas is that its density profile ` freezes ' at low temperatures . this is a result of the pauli exclusion principle and can be easily understood by considering an ideal gas in the trapping potential @xmath2 . in general the extent of the gas cloud is determined by the classical turning point of the most energetic particles . if the gas is fully classical , i.e. , it obeys maxwell - boltzmann statistics , these particles have an energy of a few @xmath3 and the size of the cloud @xmath4 follows from @xmath5 , implying that @xmath6 . we see that , as we lower the temperature , the size of the cloud shrinks . moreover , if we keep the number of particles @xmath7 fixed , the density increases . this process gradually continues until we reach zero temperature and the density profile becomes equal to @xmath8 . if the gas obeys fermi - dirac statistics , however , the most energetic particles have in the degenerate ( nonclassical ) regime an energy that is equal to the fermi energy @xmath9 and the size of the gas cloud is always given by @xmath10 for temperatures @xmath11 . comparing this also with the density profile for an ideal bose gas , which in the degenerate regime consists of a large and narrow condensate peak with a width of about @xmath12 on top of a broad thermal background of size @xmath13 , we conclude that the density profile of a degenerate fermi gas is indeed ` frozen ' . in contrast to the ideal bose gas , the ideal fermi gas also does not have a phase transition . from a point of view of condensed matter physics , an atomic fermi gas thus appears much less interesting than a bose gas . the main objective of this contribution is to argue that this is no longer true if there are interactions between the atoms . for atomic alkali gases the most important interatomic interaction is the so - called central interaction @xmath14 , which consists of a sum of the usual singlet and triplet interactions . more precisely , we have that @xmath15 where @xmath16 and @xmath17 denote the projection operators on the subspace of singlet and triplet states , respectively . besides this interaction that is the net result of the coulomb attractions and repulsions between the electrons and nuclei of the atoms , we also have to consider the weak magnetic dipole - dipole interactions . of these , the electron - electron magnetic dipole interaction is most important and obeys @xcite @xmath18 here @xmath19 is the electron magnetic moment and the tensor operator @xmath20 is obtained from coupling the pauli spin matrices @xmath21@xmath22 and @xmath21@xmath23 of the two valence electrons to a tensor of rank 2 .    it should be noted that both these interactions do not commute with the electron spin operators @xmath24@xmath21@xmath25 and therefore also do not commute with the atomic hamiltonian , which in a magnetic field contains both a hyperfine and a zeeman term . as a consequence the central and dipole - dipole interactions are not fully diagonal in the basis in which the atoms are in definite hyperfine states . this is important in principle , because it implies that two atoms can also collide inelastically , i.e. , their hyperfine states can change during the collision . together with three - body recombination events , these inelastic collisions in fact always seriously limit the lifetime of a trapped alkali gas . nevertheless , we will in the following mostly neglect the nondiagonal parts of the interatomic interaction by restricting ourselves to ( doubly ) spin - polarized gases for which the ` good ' elastic collisions dominate the ` bad ' inelastic ones . clearly such a restriction is a minimum requirement for our discussion to be also of some experimental interest . as we will see in detail below , the most common phase transition in a weakly - interacting fermi gas is due to the formation of so - called cooper pairs . the main idea behind the famous bardeen - cooper - schrieffer theory for this phenomenon is in fact a bose - einstein condensation of these pairs @xcite . to see how we can arrive at a mean - field theory for this phase transition , let us first recapitulate the mean - field ( hartree ) theory for bose - einstein condensation in an atomic bose gas . at zero temperature we can then use a variational many - body wave function in which all the atoms are in the same one - particle state @xmath26 , so @xmath27 in the language of second quantization this essentially reads @xmath28 where @xmath29 is the creation operator for an atom at postion @xmath30 and @xmath31 is the ` vacuum ' state in which there are no atoms present in the trap . calculating the average of the hamiltonian @xmath32 in this variational wave function and minimizing with respect to @xmath26 leads of course to the gross - pitaevskii equation @xcite if @xmath33 . the above variational wave function has a definite number of particles . for practical calculations , in particular if we want to consider also nonzero temperatures or corrections to the mean - field theory , it is much more convenient to consider a variational wave function in which not the number of particles @xmath7 , but instead the phase @xmath34 is fixed . this state is given by @xmath35 as can be seen from the fact that now the matrix element @xmath36 has a definite phase in contrast to the matrix element @xmath37 . it is not difficult to show that by minimizing the average energy in the state @xmath38 we again recover the gross - pitaevskii equation . more important for our purposes is that we can easily show by fourier analysis that @xmath39 which shows that the number of particles @xmath7 and the phase @xmath34 are conjugate variables and obey the commutation relation @xmath40 = i$ ] . the heisenberg equation of motion for the average phase thus becomes @xmath41 \right\rangle     = -i \left\langle \frac{\partial h}{\partial n } \right\rangle \equiv -i \mu~,\ ] ] with @xmath42 the chemical potential of the gas . we therefore recover the important josephson relation @xmath43 which is also well known from the hydrodynamic formulation of the gross - pitaevskii equation @xcite . indeed , taking the gradient of this equation and using the definition of the superfluid velocity , i.e. , @xmath44 , we obtain the desired result that @xmath45    let us now return to cooper - pair formation in degenerate fermi gases . in analogy with bose - einstein condensation we can now use at zero temperature the variational wave function @xmath46 a few remarks are in order . first , we have denoted the various hyperfine states of the atoms by @xmath47 . in our previous discussion of bose - einstein condensation we should in principle also have indicated the hyperfine state of the atoms and used @xmath48 instead of @xmath29 in the variational wave function . however , as long as all the atoms are in the same hyperfine state , the precise atomic state which is trapped is unimportant from a theoretical point of view and only influences the interatomic interaction , i.e. , the particular value of the scattering length , that should be used in the gross - pitaevskii equation . for clarity we therefore suppressed the spin degrees of freedom in that case . second , the fermionic creation operators anticommute . as a result the cooper - pair wave function must obey @xmath49 , reflecting the pauli exclusion principle . there are essentially two ways to fulfill this antisymmetrization requirement . if all the atoms are in the same hyperfine state we have @xmath50 and the orbital part of the cooper - pair wave function is antisymmetric with respect to an exchange of the atoms . the relative angular momentum of the pairs must therefore be odd . in particular , we can have @xmath51-wave pairing . this situation occurs in doubly spin - polarized fermi gases and also in liquid @xmath52he @xcite . if we have an equal number of atoms in two hyperfine states , which is implicitly assumed in the above variational wave function , the spin part of the cooper - pair wave function can already be antisymmetric and the relative angular momentum of the pairs must then be even . we now can have @xmath53-wave pairing as in ordinary superconductors . in principle , we can of course also have @xmath54-wave pairing as in the high - temperature superconductors , but this turns out to be extremely unlikely for dilute gases . third , in actual applications it is again more convenient to use a variational wave function with a well defined phase . in this case it is obtained by multiplying the cooper - pair wave function in the right - hand side of eq . ( [ fixedn ] ) with @xmath55 and summing over all even values of @xmath7 . we then find @xmath56 and exactly the same relation between the states @xmath57 and @xmath38 as for a condensate of single atoms . the josephson relation given in eq . ( [ jr ] ) is therefore also valid in this case . moreover , in the state @xmath38 we have a nonvanishing expectation value @xmath58 which suggest that @xmath59 is the appropriate order parameter for the bardeen - cooper - schrieffer transition , just like @xmath60 is the order parameter for bose - einstein condensation . although this identification of the order parameter is correct , it turns out that it is more convenient in practice to work with the so - called bcs gap parameter @xmath61 where @xmath62 is a shorthand notation for the elastic part of the interatomic interaction , and @xmath63 and @xmath64 are the relative and center - of - mass coordinates of the cooper pair , respectively .    knowing the order parameter of the phase transition of interest , it is straightforward to obtain the corresponding mean - field theory . the main idea is first to write in the interaction part of the hamiltonian @xmath65 the operators @xmath66 and @xmath67 as a sum of a mean value and fluctuations , and then to neglect terms that are quadratic in the fluctuations . in this manner we arrive at the mean - field hamiltonian @xmath68 where the renormalized chemical potentials are in a good approximation ( but see below ) given by @xmath69 and the density profile of atoms in spin state @xmath47 obeys @xmath70 . it is important to note that in trapped alkali gases it is indeed appropriate to have a chemical potential for each spin state , because the time scale for relaxation towards equilibrium in spin space is always much larger than the equilibration time for the spatial degrees of freedom . this is for example quite dramatically demonstrated by the two overlapping condensate experiments by myatt @xcite .    to complete the mean - field theory we should now calculate the mean values of the operators @xmath66 and @xmath67 in a thermal ensemble with the hamiltonian @xmath71 . this clearly makes the theory selfconsistent . to perform the calculation we write the annihilation operators at the positions @xmath30 and @xmath72 as @xmath73 respectively . substituting this in the mean - field hamiltonian and neglecting gradients in the center - of - mass coordinate @xmath74 , we find that @xmath75 diagonalizing the above hamiltonian by means of a bogoliubov transformation , the spin - density profiles can then be calculated from @xmath76 and , most importantly , the bcs gap parameter from @xmath77 notice that the averages in the right - hand side of eqs . ( [ spin ] ) and ( [ gap ] ) depend on the position @xmath74 , since the mean - field hamiltonian @xmath71 depends parametrically on this position . to arrive at such a simplyfied ( thomas - fermi ) description of the inhomogeneity of the gas we have to be able to neglect gradients of the densities @xmath78 and the bcs gap parameter @xmath79 . this is indeed true for present experiments with fermionic alkali gases , because the number of trapped atoms is so large that both the correlation length and the size of the cooper pairs are small compared the typical length scale on which the trapping potential varies @xcite . we refer for the details of the diagonalization to our previous work @xcite . for our present purposes it is however important to mention that in the optimal case of equal spin densities , the final result of eq . ( [ gap ] ) is the famous bcs gap equation @xcite @xmath80 where the so - called bogliubov dispersion @xmath81 obeys @xmath82 in principle , one can solve the bcs equation for any interatomic potential @xmath83 . however , for alkali gases the complete central interaction is usually not very well known and we have only information on the two - body scattering length @xmath84 . we therefore would like to reformulate the gap equation in such a way that only this scattering length enters . this is achieved by noting that the gap equation is very similar to the lippmann - schwinger equation for the two - body t(ransition ) matrix . indeed the latter reads @xcite @xmath85 using the lippman - schwinger equation with @xmath86 , we can after some algebraic manipulation show that the gap equation is equivalent to @xmath87 this result serves our purposes since the two - body t matrix is directly related to the two - body scattering length as we will see next . in doubly spin - polarized fermi gases all the atoms are in the same hyperfine state . although it is straightforward to generalize the following to an arbitrary state @xmath47 , we will in first instance restrict ourselves to the fully stretched state @xmath88 in which both the electron as well as the nuclear spin have a maximal projection on the magnetic field axis . the relevant interaction matrix element is then equal to @xmath89 with @xmath90 the angle between the interatomic separation @xmath91 and the magnetic field @xmath92 . for this potential we have to calculate the two - body t matrix . treating the weak electron - electron magnetic dipole interaction in born approximation @xcite and making use of the fact that for the central interaction only @xmath51-waves contribute at low momenta , we find @xmath93 here @xmath84 is the triplet @xmath51-wave scattering length and @xmath94 denotes the angle between the momentum transfer @xmath95 and the magnetic field axis . we now have two cases to consider . generically we expect the @xmath51-wave scattering length to be of the order of the range of the triplet potential and therefore @xmath96 to be much smaller than one . in that case the contribution of the triplet potential to the two - body t matrix is negligible and the effective interaction between the atoms is dominated by the long - range dipole - dipole interaction . due to the complicated angular dependence of this interaction it is not possible to solve the bcs gap equation analytically . however , we can nevertheless make progress by noting that the magnetic dipole - dipole interaction is most attractive when @xmath91 is directed along the magnetic field . we thus expect that if cooper pairs are formed their wavefunction @xmath97 is most likely proportional to @xmath98 , which implies that @xmath99 . since this gap is anisotropic in the relative wave vector @xmath100 , the gas is below the critical temperature an anisotropic superfluid , just like @xmath52he in the so - called a phases @xcite .    to obtain an estimate for the critical temperature we explicitly consider only the @xmath51-wave part of the dipole - dipole interaction , which results in the approximation @xmath101 this explicitly confirms that the dipole - dipole interaction is only attractive in the channel @xmath102 . substituting the above two - body t matrix into the bcs gap equation and linearizing with respect to @xmath103 to obtain an equation for the critical temperature , we find @xmath104 here we used the notations @xmath105 for the cauchy principle value part of the integral and @xmath106 for the fermi distribution function @xmath107 evaluated at @xmath108 . note that we have also used that the bcs gap equation will have a nontrivial solution in the center of the trap first , since the density of the gas is highest there . introducing a ` scattering length ' @xmath109 for the electron - electron magnetic dipole interaction by means of @xmath110 , i.e. , @xmath111 a@xmath112 with @xmath113 the mass of a hydrogen atom and a@xmath112 its bohr radius , the equation for the critical temperature becomes identical to the linearized bcs gap equation that has been studied previously in the context of @xmath53-wave superconductors by s de melo @xcite . using their result , we have for the critical temperature @xmath114 with @xmath115 euler s constant . unfortunately , the bcs transition to an anisotropic superfluid thus occurs at extremely low temperatures in this case and appears to be out of reach experimentally . for example for @xmath0li at a density of @xmath116 @xmath117 , we have @xmath118 nk and @xmath119 .    the second case to consider appears to be more promising . as mentioned before , the triplet @xmath51-wave scattering length is in general too small to be able to dominate over the dipole - diple interaction . for @xmath0li it is only @xmath120 a@xmath112 , for instance . however , we can imagine that it is possible , in the same way as in the recent experiment by inouye @xcite , to ( optically ) trap a hyperfine state that has a @xmath51-wave feshbach resonance @xcite and tune the external bias magnetic field such that the @xmath51-wave scattering length becomes large and negative . the two - body t matrix is then well approximated by @xmath121 substituting this into the bcs gap equation , it is not difficult to show that it is solved by the _ ansatz _ @xmath122 where @xmath123 are the spherical components of a vector @xmath124 that is normalized as @xmath125 . furthermore , linearizing the bcs gap equation with respect to @xmath103 , we find that the critical temperature is now determined by @xmath126 the solution can be obtained by the same methods as before and reads @xmath127    it is important to realize that up to this point the direction of the vector @xmath124 is arbitrary . this reflects the rotational symmetry of the problem . however , as we have seen , the magnetic dipole - dipole interaction breaks this symmetry and will cause * d * to lie either parallel or perpendicular to the magnetic field , depending on precisely which hyperfine state @xmath47 is trapped . because we are dealing with an attractive interaction we also have to make sure that the gas is mechanically stable . this leads to the restriction that @xmath128 , or equivalently @xmath129 . unfortunately , the latter again severely limits the feasibility of experimentally observing the in principle interesting possibility of a transition to an anisotropic superfluid . we now turn our attention to the case of fermionic gases that are a mixture of two hyperfine states . in such a gas we can have @xmath53-wave collisions between atoms in different hyperfine states , which has as an advantage that interacting effects are expected to be much more important . moreover , it has an additional advantage that it is now in principle possible to use evaporative cooling to cool the gas to low temperatures . a fermionic gas consisting of three hyperfine states has recently been considered as well @xcite , but since it does not lead to any qualitative different physics we restrict ourselves here to mixtures of only two spin states . we denote the two hypefine states involved from now on by @xmath130 and @xmath131 . experimentally , there are two ways to realize such a system . we can either magnetically trap two low - field seeking states , or use an optical trap . in the latter case it is presumably necessary to load the trap by precooling the gas in a magnetic trap using sympathetic cooling .      if the @xmath53-wave scattering length @xmath84 between two atoms in different spin states is positive , a first many - body effect that we have to consider is the phase separation of the gas into two phases with opposite ` magnetization ' . roughly speaking , this implies that instead of having overlapping spin densities @xmath132 and @xmath133 , the gas prefers to separate into two phases in which ( almost ) all the atoms are either in the state @xmath130 or in the state @xmath131 . the driving force behind this instability is that although the phase separation increases the kinetic energy of the gas , this increase is more than compensated by the decrease in interaction energy . more precisely , the gas is stable if the free - energy surface @xmath134 $ ] has a positive curvature in all directions . using that @xmath135 the free energy in the degenerate regime is given by @xmath136 = \int d{\bf r}~    \left\ {    ( 6\pi^2)^{2/3 } \frac{3\hbar^2}{10 m }        \left ( n_{\uparrow}^{5/3}({\bf r } ) + n_{\downarrow}^{5/3}({\bf r } ) \right )           + \frac{4\pi a \hbar^2}{m } n_{\uparrow}({\bf r } ) n_{\downarrow}({\bf r } )   \right\}\ ] ] and stability of the gas requires that the spin densities obey @xmath137 . in the particular case of equal spin densities this reduces to the condition that @xmath138 . note that above we have used the t - matrix approximation to evaluate the average interaction energy . because the chemical potential @xmath139 is equal to the derivative @xmath140 , we must for consistency use the same approximation to determine the renormalized chemical potentials @xmath141 . therefore , we use in the following always that @xmath142 instead of the less accurate ( only born approximation ) relation given in eq . ( [ chem ] ) . even though the interaction between the atoms is effectively repulsive , there can nevertheless occur a bcs pairing transition to a superfluid state due to the so - called luttinger - kohn instability @xcite . physically , the instability is a result of the fact that two atoms in the same hyperfine state can exchange a fluctuation ( phonon ) in the density of the other hyperfine state , leading to an effectively attractive interaction between the atoms involved . the @xmath51-wave transition associated with the luttinger - kohn effect has been studied by baranov @xcite . these authors obtain for the critical temperature the estimate @xmath143 where notably @xmath84 is the @xmath53-wave scattering length . it should , however , be kept in mind that to be able to observe the transition we must require that the gas is mechanically stable and does not phase separate . as we have already seen , this implies for equal spin densities that @xmath138 . moreover , for generic values of the scattering length we will even have that @xmath144 at the low densities of interest . it therefore seem once again practically impossible to observe the above transition in a real atomic gas . our last chance of achieving a superfluid phase in a fermionic gas thus appears to be a spin - polarized gas with a large and negative @xmath53-wave scattering length . besides the possibility of using a feshbach resonance to tailor the scattering length , we can also directly make use of the anomalously large @xmath0li triplet scattering length of @xmath145 a@xmath112 by trapping a spin - polarized @xmath0li gas in a bias magnetic field of at least @xmath146 t @xcite . substituting eq . ( [ t2b ] ) into the bcs gap equation we see that the solution @xmath79 is now independent of the wave vector @xmath100 and therefore equal to @xmath147 . furthermore , a linearization in @xmath147 gives @xmath148 which results in the critical temperature @xmath149 since the mechanical stability of the gas requires also for a negative scattering length only that @xmath150 , this expression shows that the prospects of observing a bcs transition in this case are indeed most favorable .    in view of this encouraging situation we have studied in more detail the behavior of the spin density profile @xmath78 and the bcs gap parameter @xmath147 for a spin - polarized @xmath0li gas in the same magnetic trap that has been used for the bose - einstein condensation experiments with @xmath151li @xcite . the results for equal spin densities are shown in fig . [ profiles ] and lead to three important conclusions . first , we see explicitly that the density profile of a degenerate fermi gas is indeed completely ` frozen ' . clearly , even the bcs transition to a superfluid has essentially no effect on the density profile . from an experimental point of view this is somewhat unfortunate , because it implies that the appearance of a condensate of cooper pairs can not be observed by the same methods that have been so successful in the bose - einstein condensation experiments . we come back to this problem shortly . second , for a total density of @xmath152 @xmath117 the critical temperature is about @xmath153 nk . in view of the achievements with bosonic alkali gases , this appears to be a density - temperature combination which is certainly within reach experimentally . finally , we have also compared the density profile of @xmath0li with that of a noninteracting fermi gas with the same number of particles . the strong attractive interaction between the @xmath0li atoms evidently results in a substantial increase of the density in the center of the trap . experimentally , this is a favorable effect because for a fixed number of atoms in the trap it enhances the critical temperature as is shown quantitatively in fig . [ tc ] .    for future convenience we mention that in a good approximation the critical temperature obeys @xmath154 with @xmath155 the thermal de broglie wavelength and @xmath156 the size of the harmonic oscillator ground state . it must be kept in mind that the latter formula can only be used for values of @xmath157 that are less then about @xmath158 , since for larger values the gas is mechanically unstable and undergoes a spinodal decomposition first . we have argued that for condensed matter physics in trapped atomic fermi gases , a spin - polarized gas with a large and negative @xmath53-wave scattering length between the atoms in the two different hyperfine states appears to be most promising . in view of the large uncertainties in the interatomic interaction potential of @xmath1k , the most likely candidate for the achievement of a gaseous bcs superfluid is at present @xmath0li . however , before successful experiments with atomic @xmath0li can be performed , some experimental problems need to be resolved . one problem is that in spin - polarized atomic @xmath0li not only the @xmath53-wave scattering length but also the exchange and dipolar decay rates are anomalously large . as a result the gas has usually a very short lifetime . to enhance the lifetime to about @xmath159 s at the densities of interest , we can either apply a bias magnetic field of about @xmath160 t or use optical methods to trap two high - field seeking states . the latter solution seems to be most practical and is actively being persued at the moment .    assuming that we are able to achieve the necessary conditions for the bcs transition , the next problem that arises is the detection of the cooper pair condensate . as we have seen , the density profile shows essentially no sign of the phase transition . time - of - flight measurements , that were the ` smoking gun ' for the bose - einstein condensation experiments , are therefore not appropriate here . another possible signature that comes to mind are the frequencies of the collective modes . because of the large @xmath53-wave scattering length required for relatively high critical temperatures , the collective modes are always in the hydrodynamical regime . they are therefore described by the ( local ) conservation laws and the josephson relation . in the case of equal spin densities we thus obtain the following set of equations . the continuity equation for a superfluid is @xmath161 with the total density @xmath162 and the total current density @xmath163 consisting of a normal and superfluid contribution . in addition , newton s law gives @xmath164 where @xmath51 denotes the pressure in the gas . finally , we have also the josephson relation , which leads to ( cf . ( [ vs ] ) ) @xmath165    in principle , we must also take into account the conservation of energy . however , for a degenerate fermi gas the specific heats at constant pressure and volume are almost equal and the continuity equation for the total energy density essentially decouples from the previous ones . this has important consequences , because for density fluctuations the josephson relation just copies newton s law and we must conclude that the first sound modes are not affected by the bcs transition . of course , a measurement of second sound modes would be a clear signature of the transition , but this is presumably quite hard experimentally because second sound is primarily a temperature wave due to the fact that the energy fluctuations are almost decoupled . in analogy with sound attenuation in superconductors , it has been suggested by fetter @xcite that the damping of the first sound modes might be strongly influenced by the appearance of a cooper pair condensate , but more work is needed to make sure whether this interesting suggestion would work . a property of the gas that is certainly influenced by the bcs transition is the decay of the gas . qualitatively this can be easily understood from the correlator method devised by kagan @xcite . in this approach we can relate the decay rate constant @xmath166 for two - body decay to the rate constant in the normal phase @xmath167 by @xmath168 hence , the cooper pair condensate enhances the decay of the gas . a quantitative estimate of the effect is somewhat complicated by the fact that we can not use a pseudopotential to calculate the cooper pair wavefunction @xmath169 from our knowledge of the gap parameter @xmath147 . in any case , the increase in the two - body decay rate of the gas can only be used as a detection method if the gas is trapped in a magnetic trap , since in an optical trap the two - body decay will essentially be eliminated and the lifetime of the gas is determined by the rate at which photons scatter off the atoms in the gas . because this is a one - atom problem , it is also not affected by the bcs transition . a final detection method that we would like to mention is the scattering of a beam of @xmath0li atoms from the gas cloud . since such an experiment is quite similar to a tunneling experiment , it is directly sensitive to the existence of the gap parameter @xmath147 . therefore , a measurement of the angular distribution of the scattered atoms appears to be a promising way to get detailed information about the condensate of cooper pairs . of course , to be most sensitive we need a very cold beam . however , this is clearly not an impossible requirement , because the first results of such experiments with a condensate of @xmath170rb atoms have recently been reported @xcite . from a theoretical point of view , we are at present performing a study of the above scattering process to see how large the effects of the bcs transition are . it is interesting to note that it is also possible to detect vortices in this manner , because the atomic beam ( in contrast to a laser beam ) does not only see the core of the vortex but its complete velocity profile . this is in fact also true for a bose condensate . an experiment of this sort may therefore also be of interest in the context of ongoing research on the properties of trapped atomic bose gases . most of the work presented here has been performed in close collaboration with ian mcalexander , cass sackett , and randy hulet . we are very grateful for their continued encouragement and crucial contributions . we also thank yvan castin , eric cornell , jean dalibard , allan griffin , massimo inguscio , tony leggett , andrei ruckenstein , guglielmo tino and peter zoller for helpful discussions . we acknowledge support from the stichting fundamenteel onderzoek der materie ( fom ) , which is financially supported by the nederlandse organisatie voor wetenschappelijk onderzoek ( nwo ) . ( to be published , cond - mat/9807027 ) . . private communication . . . . . and . . ( to be published , cond - mat/9712262 ) . . private communication . . private communication .
we present an overview of the various phase transitions that we anticipate to occur in trapped fermionic alkali gases . 
 we also discuss the prospects of observing these transitions in ( doubly ) spin - polarized @xmath0li and @xmath1k gases , which are now actively being studied by various experimental groups around the world .