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bragg spectroscopy , using two laser beams , has proven to be a very successful tool for probing the structure factor of trapped atomic bec s  @xcite . in this paper , we examine the analogous experiment on two - component fermi gases in both the normal and superfluid phases . we calculate the bragg scattering rate for varying interaction strengths in both the weak coupling bcs and bec limits and in the strong coupling unitarity limit . previously , we proposed measuring the inelastic ( stokes / anti - stokes ) scattering of an off - resonant laser beam on a two - component fermi gas  @xcite . in the bcs limit , the intensity of the scattering in this stokes experiment exhibits a large maximum just below the critical temperature , @xmath0 , at which the gas becomes superfluid . this effect is the light scattering analog of the hebel - slichter effect in conventional superconductors , a hallmark experimental test of bcs theory  @xcite . here , we demonstrate that the same effect can be observed in a bragg scattering experiment . we furthermore show that the pairing induced by the interactions can be readily detected in the bragg spectrum . in the strong coupling regime , we discuss how the presence of a pseudogap in the normal phase has significant effects on the bragg spectrum . the advantage of the bragg experiment as compared to the stokes experiment is that one varies in a controlled way the energy and momentum imparted to the atoms . compared with recent radio frequency ( rf ) experiments which involve a third unoccupied hyperfine state  @xcite , the bragg experiment involves only the two hyperfine states already present in the trap . the bragg experiment , avoiding complications due to non - trivial interaction effects between the third hyperfine state present in the rf experiment @xcite and the trapped gas , is thus simpler to interpret . we consider a homogeneous gas of fermionic atoms occupying two hyperfine states , denoted by @xmath1 and @xmath2 . we assume that the state @xmath2 has energy @xmath3 ( @xmath4 and @xmath5 in this paper ) above the state @xmath1 , and that the population of the two states is equal , @xmath6 . the two atomic states interact with a coupling which can be characterized by a scattering length @xmath7 in a vacuum . we ignore the interaction between atoms in the same hyperfine state since @xmath8-wave scattering is suppressed between two identical fermions . bragg scattering is realized by a pair of laser beams with wave vectors @xmath9 and @xmath10 and frequencies @xmath11 and @xmath12 illuminating the gas . an atom absorbing a photon from beam 1 and emitting a photon in beam 2 changes its energy - momentum by @xmath13 . bragg scattering can be used to probe both the density and spin correlation functions . pairing is predicted to have several effects on the frequency and momentum dependence of these correlation functions for @xmath14  @xcite . to probe the spin - flip correlation function , one tunes the two laser frequencies so that @xmath15 as illustrated in fig . [ braggfig ] . if the detuning , @xmath16 , from the optical transition to or from the upper level @xmath17 is large , population of the upper level can be neglected , and the effective light matter hamiltonian describing spin - flip bragg scattering can be written as @xmath18 ,   \label{braggterm}\ ] ] where @xmath19 , and @xmath20 describes the spatial profile of the laser beams and the dependence on the atomic states involved and their coupling to the electromagnetic field  @xcite . we neglect finite waist effects and take @xmath21 with @xmath22 .    to measure the effects of pairing we focus on the spin - flip rate @xmath23 induced by the bragg lasers , where @xmath24 is here the number of particles in the level @xmath2 , and @xmath25 denotes a thermal average . the heisenberg equation of motion for the operator @xmath24 is @xmath26=    \int d^3 r\left[{\mathcal{i}}({\mathbf{r}},t )    \psi_\downarrow^\dagger({\mathbf{r}},t )    \psi_\uparrow({\mathbf{r}},t)-\rm{h.c.}\right].\ ] ] for sufficiently weak laser beams , @xmath27 can by calculated by linear response theory : @xmath28\rangle({\mathbf{q}},\omega ) }   { \omega-\omega'}-c.c.\right)\nonumber\\   = -2{\mathcal{i}}^2{\rm im}\mathcal{d}({\mathbf{q}},\omega+i\eta ) , \label{braggterm}\end{gathered}\ ] ] with @xmath29 . bragg scattering thus probes the fourier transform of the spin - flip correlation function @xmath30\rangle$ ] . the response of the gas is usually detected experimentally by measuring the total momentum imparted on the gas  @xcite ; the rate of momentum transfer is straightforwardly related to the scattering rate by @xmath31 where @xmath32 is the total momentum of the gas in the two hyperfine states . in the weak coupling limit @xmath33 , the gas can be described by bcs theory . the correlation function in ( [ braggterm ] ) is then a sum of contributions from normal and anomalous green s functions  @xcite . we obtain after some algebra , @xmath34 ,   \label{uvexpression}\end{gathered}\ ] ] with @xmath35 . here , @xmath36 and @xmath37 , etc . , with @xmath38 and @xmath39 as usual in bcs theory . the quasiparticle energies are @xmath40 and @xmath41 with @xmath42 , where @xmath43 , @xmath44 is the chemical potential , and @xmath45 the bcs pairing gap . the fermi functions are @xmath46 and @xmath47 , with @xmath48 . the first term of ( [ uvexpression ] ) , describes scattering of quasiparticles while the last two terms describe the creation and annihilation of two quasiparticles pair breaking processes .    for an ideal gas in the normal phase , ( [ uvexpression ] ) yields in the limit @xmath49 , @xmath50 ( @xmath51 is the fermi temperature ) , and @xmath52 , @xmath53 where @xmath54 is the fermi energy , @xmath55 is the total density of the gas , and @xmath56 is the fermi momentum .    the spin flip induced by bragg scattering can excite collective spin waves . these modes are undamped in the normal phase for an attractive interaction  @xcite . for a repulsive interaction in the normal phase , the spin waves are damped and they change the spectral shape of the particle - hole continuum . the single particle approximation ( [ uvexpression ] ) does not include such collective modes . in order to include them , we sum bubbles to obtain the random phase ( rpa ) approximation  @xcite , @xmath57 with @xmath58 the scattering matrix for @xmath59 . note the @xmath60 sign in the denominator due to the spin flip . we first consider the temperature dependence of the scattering rate . an attractive feature of the stokes experiment is that one can observe a strikingly large maximum in the scattered signal just below the critical temperature @xmath0  @xcite . we now demonstrate that the same effect is present for the bragg scattering experiment in the experimentally realizable parameter range , @xmath61 , @xmath62 , and @xmath63 . figure  [ tdepfig ] shows the scattering rate ( [ braggterm ] ) as a function of @xmath64 in the weak coupling limit , in units of the normal phase rate , which is constant for @xmath50 . for illustration , we have taken @xmath65 and @xmath66 . solving the bcs equations as a function of @xmath64 , we obtain the scattering rate from eqs . ( [ uvexpression ] ) and ( [ rpa ] ) . the critical temperature is @xmath67 . we plot the results for frequency differences between the two laser beams @xmath68 , @xmath69 , and @xmath70 . the numerical normal phase results agree with ( [ nexact ] ) , confirming the numerical accuracy of the calculations .    as fig . [ tdepfig ] shows , the scattering rate in the superfluid state has a large maximum below @xmath0 for @xmath71 . this maximum is due to a large quasiparticle scattering rate arising from the increased density of states at the fermi energy in the superfluid phase just below @xmath0 , @xmath72 . for @xmath71 , both scattering states @xmath73 and @xmath74 in eq . ( [ uvexpression ] ) can be in the region around the fermi surface of a large density of states yielding an increased scattering rate . the maximum in the scattering rate depends crucially on the coherence factors for quasiparticle scattering in ( [ uvexpression ] ) adding , viz . , @xmath75 , for a spin flip . for bragg scattering into the same hyperfine state , the coherence factor is @xmath76 at the fermi surface and the increased density of states would not lead to a maximum in the scattering rate . for @xmath77 , the scattering rate is suppressed by the increasing gap @xmath78 since the density of quasiparticles available for scattering scales as @xmath79 . for @xmath80 the lasers can not break cooper pairs . the physical reason for the coherence factors adding ( @xmath75 ) for spin - flip scattering is that the quasiparticles carry the same spin in the superfluid as in the normal phase . on the other hand , in scattering without spin flip , which couples to the particle density , the particle number carried by the quasiparticles is suppressed at the fermi surface , since a quasiparticle is a superposition of a hole and a particle . in this case the coherence factors subtract ( @xmath81 ) .    for increasing @xmath82 , the peak in the rate below @xmath0 decreases and eventually disappears for @xmath83 . from fig . [ tdepfig ] , we see that the scattering rate in the superfluid state is always smaller than in the normal state when @xmath84 ( green dashed and black dash - dotted lines ) . this is because both scattering states can not be in the region around the fermi surface where the single particle density of states is increased . bragg scattering does not probe the increased density of states for the superfluid state for higher frequency as effectively as for @xmath85 , and the resulting rate is decreased compared to the normal state . note that the kink in scattering rate just below @xmath0 for @xmath86 , at which point @xmath87 , is physical . below this temperature , the gap is too large to allow the creation of two quasiparticles and the second term in ( [ uvexpression ] ) vanishes . finally , fig . [ tdepfig ] , for @xmath88 illustrates the case of large energy transfer , @xmath89 . contrary to the case @xmath90 , the rate is finite for @xmath14 as the energy transfer is large enought to break pairs and the second term in ( [ uvexpression ] ) is finite . for @xmath91 , the scattering rate approaches that of the normal state , as expected . we conclude that there is a peak in the scattering rate for @xmath92 and @xmath71 . also , the peak disappears unless @xmath93 and @xmath94 ; otherwise , both scattering states can not be in the region of increased density of states . we now consider the frequency dependence of the bragg scattering rate . figure  [ wdepfig ] shows the rate from ( [ rpa ] ) as a function of @xmath82 for the gas in the weak coupling limit with @xmath65 and @xmath95 . the scattering rate is depicted for various temperatures below and just above @xmath0 where the gas is normal . for comparison , we also plot the result of the single particle approximation ( [ uvexpression ] ) in the normal phase . the large peak in the normal phase response just above the single particle continuum arises from the spin wave present above the particle - hole continuum for an attractive interaction . the finite width of the peak is an artifact of our having included a small imaginary part in the frequency @xmath82 for computational reasons . we see from fig . [ wdepfig ] that when the gas becomes superfluid , the spin wave vanishes . this is because only potential flow , i.e. , density fluctuation modes , is possible in the superfluid  @xcite . other modes such as spin waves require a normal component to be present . in a trapped system this effect leads to the damping of the spin - dipole mode in the superfluid phase  @xcite . in the superfluid phase , the particle - hole spectrum disappears and is replaced by a low frequency gap , which reflects the pair breaking energy described by the second term in ( [ uvexpression ] ) . at zero temperature the gas does not respont for @xmath80 , the minimum energy required to break a pair . the pairing gap can thus be detected directly as a gap of @xmath96 in the bragg spectrum . by contrast , in the rf experiment of @xcite the shift in the spectrum scales as @xmath97 , since one transfers particles to a third hyperfine state which in general interacts differently with the two paired hyperfine states  @xcite . we now consider bragg scattering in the weak coupling bec regime @xmath98 . in this limit , the molecules are tightly bound with a radius much smaller than the average interparticle distance , and are weakly interacting both with each other and with unbound atoms  @xcite . the system can therefore to a good approximation be regarded as an ideal gas of point molecules and atoms in thermal equilibrium . the molecules bose condense below the critical temperature @xmath99 . the bragg scattering induced by ( [ braggterm ] ) then either breaks a molecule by flipping the spin of a bound @xmath100 atom , thereby creating two free @xmath101 atoms , or it simply flips the spin of a free @xmath100 atom . the fermi golden rule result for bragg scattering rate into the free atomic @xmath101 states for such a system is then a sum of three terms , @xmath102 the scattering from the unbound atoms is given by @xmath103 where @xmath104 , @xmath105 , @xmath106 , and @xmath107 . the rate from a bec of molecules with density @xmath108 is @xmath109 with @xmath110 the energy of a molecule with zero momentum relative to the bottom of the free particle continuum of a pair of @xmath100 and @xmath101 atoms , and @xmath111 the matrix element of ( [ braggterm ] ) ( excluding @xmath112 ) between a molecule at rest and a pair of free @xmath101 atoms with momenta @xmath113 and @xmath114 . the scattering rate from thermal molecules is @xmath1152\pi\delta(\omega+e_b+\frac{p^2}{4m}-\epsilon-\epsilon ' ) ,   \label{bragtherm}\end{gathered}\ ] ] with @xmath116 the bose distribution , and @xmath117 the energy of molecules with momentum @xmath118 . the molecule breaks into two @xmath101 atoms with momenta @xmath119 and @xmath120 and energies @xmath121 and @xmath122 . the matrix element in ( [ bragtherm ] ) is , from galilean invariance , independent of @xmath123 .    to proceed , we model the molecule - free atom pair matrix element @xmath124 as follows . we write state of a molecule at rest as @xmath125 , with @xmath126 the vacuum and @xmath127 creating an atom with momentum @xmath128 in hyperfine state @xmath129 . the fourier transform of the molecular wave function is @xmath130 . the spin - flip process illustrated in fig.[braggfig ] breaking the molecule and adding momentum @xmath131 creating two free @xmath101 atoms with momenta @xmath113 and @xmath114 corresponds to @xmath132 . however , the process @xmath133 connects to the same final state , and the two matrix element must be added coherently in the golden rule expression . we thus find the matrix element @xmath134 ( where the factor @xmath135 avoids double counting ) . the bcs wave function correctly describes a condensate of non - overlapping bosons ( cooper pairs ) with binding energy @xmath110 in the bec limit @xmath136 . to connect this wave function to the above analysis , we note that in the bcs expression the bragg scattering ( [ uvexpression ] ) , it is the second term that describes the breaking of cooper pairs . thus , bcs theory yields @xmath137 . in the bec limit , we have @xmath138 and @xmath139 with @xmath140 . bcs theory therefore yields @xmath141 for the fourier transform of the molecular wave function and @xmath142 in the bec limit . this is the asymptotic bound state wave function for a potential with scattering length @xmath7 as expected . we therefore use the bec limit of the bcs result for the matrix element @xmath143 . note that we neglect any closed channel components deep in the bec regime . the populations of the molecular and atomic states are found by solving the atom - molecule equilibrium problem ; the density of the thermal molecules , the molecular bec and the thermal atoms are shown in fig . [ densfig ] for varying @xmath64 .    in fig . [ molbraggfig ] , we plot the scattering rate as a function of @xmath82 obtained from ( [ molbragg ] ) for a molecule energy @xmath144 ( corresponding to @xmath145 ) and for momentum transfer @xmath66 . the scattering rate is shown in fig . [ molbraggfig ] for a pure molecular bec at @xmath14 ( a ) and for thermal molecules at @xmath146 above the bec critical temperature @xmath99 . for @xmath14 , the rate is given by ( [ bragbec ] ) and the threshold for the scattering is given by @xmath147  the minimum energy required to break a @xmath148 molecule and create two @xmath101 atoms with momenta @xmath149 and @xmath150 . the rate from the thermal molecules at @xmath146 obtained from ( [ bragtherm ] ) does not differ significantly from that of the molecular bec at @xmath14 . this is because both temperatures are so low that there are essentially no atoms present , as one sees in fig . [ densfig ] . the only effect of finite temperature is then a small shift @xmath151 in the threshold for scattering , from ( [ bragtherm ] ) . thus , the onset of bose - einstein condensation of molecules has only minor effects on the bragg scattering rate since @xmath152 in the bec limit .    in fig . [ molbraggfig ] ( b ) we plot the scattering rate for @xmath153 , at which most of the atoms are still bound as molecules but some free atoms are present , as can be seen from fig .  [ densfig ] . the bragg scattering consists of a signal down to @xmath154 from the free atoms , given by ( [ atombragg ] ) , and a signal from the thermal molecules for @xmath155 , given by ( [ bragtherm ] ) . note that even though the atomic signal at small @xmath156 is much larger than the molecular signal at high @xmath82 , the contribution to the f - sum rule from the molecules is much larger than that from the free atoms . for higher temperatures , the molecular signal of course disappears with the vanishing molecule density , and the bragg scattering is given by that of the free atoms only . the scattering length diverges , @xmath157 , at the feshbach resonance . in this unitarity limit , the interactions are strong and considerably modify the properties of the gas . an important characteristic of the fermi gas at unitarity is the presence of a pseudogap in the normal phase close to @xmath0 ; i.e. , the suppression of the single particle dos around the fermi surface  @xcite . a pseudogap has been observed experimentally for high @xmath0 superconductors where it is presently subject to intense investigations  @xcite . since a gap in the spectrum of single particle excitations naturally leads to a gap in the low temperature bragg spectrum ( figs . [ wdepfig]-[molbraggfig ] ) , we now examine whether the pseudogap can be detected in bragg spectroscopy . we calculate the bragg spectrum at unitarity using a minimal many body theory that includes the correct two - particle physics leading to the resonance . for the single channel problem , such a theory including pair fluctuations in the ladder approximation was originally developed many years ago  @xcite and later examined in great detail by several authors  @xcite . this approach was then extended to the multichannel case relevant for atoms interacting via a feshbach resonance , treating the bound state as a point boson  @xcite , and taking into account that the bound state is a composite two - fermion object  @xcite . the atoms are described by the atomic propagator @xmath158 with @xmath159 . the atom self - energy is given in the ladder approximation by @xmath160\label{selfenergy}\ ] ] where @xmath161 , @xmath162 , the trace denotes a sum over matsubara frequencies @xmath163 and integration over @xmath128 , and @xmath164 is the many - body scattering matrix depending on the center of mass @xmath165 and relative momenta @xmath166 of the two scattering particles  @xcite . in the limit of a broad resonance , the effective interaction mediated by the feshbach molecule can be regarded as instantaneous with a resonant scattering length @xmath167 at unitarity  @xcite . neglecting the frequency and momentum dependence of the effective interaction coming from the molecular state , we recover the single channel theory scattering matrix @xmath168 the pair propagator is , as usual , @xmath169 here , @xmath170 , @xmath171 , @xmath106 , and @xmath107 . the structure of eqs . ( [ selfenergy])-([pairprop ] ) is indicated in fig . [ feynfig ] ( a ) .    to obtain the bragg spectrum , we need to evaluate the atomic propagator at frequency @xmath172 . ( from here on we do not write the infinitesimal positive imaginary part of the frequency explicitly . ) from ( [ selfenergy ] ) , we obtain @xmath173 . \label{rselfenergy}\end{gathered}\ ] ] here @xmath174 , @xmath175 , and @xmath176 and @xmath177 are the spectral functions for non - interacting atoms and the pair propagator , respectively . we evaluate the real and imaginary parts of @xmath178 directly from ( [ rselfenergy ] ) . we also calculate the real part from the imaginary part using a kramers - kronig relation , and find that the two calculations agree to a high accuracy , confirming the consistency of the numerical calculations .    in order to evaluate medium effects , we need to determine @xmath0 and the chemical potential , @xmath44 . we determine @xmath44 by calculating the thermodynamic potential @xmath179 at unitarity in the ladder approximation . the correction @xmath180 to the ideal gas result can be expressed as an integral over the phase of the on - shell scattering matrix @xmath164  @xcite : @xmath181 with @xmath182 , @xmath183 defined below ( [ pairprop ] ) . the angle @xmath184 is defined by writing @xmath185 . equation ( [ omega ] ) reduces to the virial expansion for @xmath179 at high temperatures . the chemical potential @xmath44 is then determined by @xmath186 .    in order to calculate the spin - flip correlation function consistently , one needs in principle to choose a functional @xmath187 $ ] , generate the self - energy as @xmath188 , and then generate the correlation functions from the kernel @xmath189  @xcite . the diagrams relevant for the ladder approximation are given in fig . [ feynfig ] ( b ) where all propagators are full green s functions , @xmath190 , including those in @xmath164 . the conserving spin - flip correlation function thus obeys an integral equation with @xmath164 and the full @xmath190 s . the imaginary part of the first term in this equation is @xmath191 with @xmath192 , @xmath193 , and @xmath194 and @xmath183 defined as in ( [ rselfenergy ] ) . the spectral function @xmath195 describes the excitations of the interacting atoms . we calculate the bragg scattering rate at the critical temperature @xmath0 where we expect the pseudogap effects to be most pronounced . from the thermodynamic potential ( [ omega ] ) , we obtain @xmath196 and can calculate the critical temperature from the thouless criterion @xmath197 . we then evaluate the atom self - energy from ( [ rselfenergy ] ) and from this obtain the spectral function .    to understand the bragg scattering rate at unitarity , we first analyze the effects of the interactions on the single particle spectrum , described by the spectral function , @xmath198 . in fig . [ spectralfig ] , we plot @xmath198 at @xmath0 for various momenta , measured in units of @xmath199 . we have chosen parameters corresponding to a resonant interaction with @xmath200 and a negligible effective range . for this set of parameters , we find @xmath201 with @xmath202 in good agreement with other bec - bcs crossover results based on a similar approximations  @xcite . the spectral functions obey the sum rule @xmath203 to a very good approximation , providing another important check of the numerical calculations .    in fig . [ spectralfig ] , we clearly see the double peak structure of @xmath198 with the spectral weight at @xmath154 significantly suppressed for all @xmath204 . this suppression is the characteristic feature of the pseudogap . we refer to the peak at @xmath205 as the _ hole peak _ at frequency @xmath206 , and the peak at @xmath207 as the _ particle peak _ at @xmath208 . the positions of the two peaks found by fitting to a double lorentzian ( normalized such that its total spectral weight is @xmath209 ) are plotted in fig . [ posfig ] ( a ) . we see that the distance of the maxima to @xmath154 reaches a minimum : @xmath210 at @xmath211 , providing a qualitative value for the pseudogap . the spectral weight of the hole peak decreases and that of the particle peak increases with increasing @xmath204 . this behavior is illustrated in fig . [ posfig ] ( b ) where we plot the spectral weights @xmath212 of the particle peak and the hole peak @xmath213 obtained by fitting to the double lorentzian . for large @xmath214 , the spectral function approaches that of an ideal gas with a single peak at @xmath215 as expected . the behavior of the spectral function at unitarity as described here is in general agreement with the analysis of ref . the spectral function behaves qualitatively as the bcs spectral function @xmath216 with two peaks located around @xmath154 . there are important differences from bcs theory though . the two peaks for the spectral function in the unitarity limit at @xmath0 have large widths ( which are unequal for the particle and hole peaks ) and the spectral density is suppressed but finite at @xmath154 . the pseudogap is a _ suppression _ of spectral density and not a real gap . also , the positions of the two peaks are not symmetric around @xmath154 and the minimum gap is not at @xmath217 but slightly above , as can be seen from fig . [ posfig ] . we now ask whether the pseudogap behavior of the spectral function at unitarity has consequences for the bragg spectrum . in fig.[braggstrong ] , we plot @xmath218 for @xmath219 and @xmath220 . the @xmath221 s are obtained from ( [ imd0uni ] ) using the spectral function calculated from ( [ rselfenergy ] ) . we evaluate the ( computationally demanding ) multidimensional integrations using a monte carlo routine . the solid line in fig . [ braggstrong ] is obtained by using the double lorentzian fit @xmath222 for the spectral function in ( [ imd0uni ] ) . here , the position of the hole @xmath223 and particle @xmath224 peaks , the width and the weight @xmath225 are calculated as functions of momentum @xmath204 using approximate functions obtained from fitting @xmath198 to the double lorentzian for various @xmath204 . the approximate functions for @xmath223 , @xmath224 , and @xmath225 are given as dashed lines in fig . [ posfig ] . once the approximate double lorentzian form @xmath222 is obtained , the integration in ( [ imd0uni ] ) is much faster than when using the spectral function @xmath198 obtained from the full ladder calculation . we see from fig . [ braggstrong ] that the double lorentzian model @xmath222 yields results for @xmath226 in good agreement with the full numerical calculation based on @xmath198 . from this , we conclude that the pseudogap spectrum close to @xmath0 can be well described by a simple double peak spectral function .    comparing to the ideal gas scattering rate for the same density and temperature , also plotted in fig . [ braggstrong ] , we see that interactions suppress the spectral weight of @xmath218 at low @xmath82 and push it to higher frequencies . this is a direct effect of the pseudogap . as opposed to the real gap in the @xmath14 bragg spectrum , the spectral density is however finite at small @xmath82 . this is as expected since the single particle spectral density is suppressed but finite for @xmath154 ( fig . [ spectralfig ] ) . also , @xmath227 is comparable to the pseudogap @xmath228 so thermal excitations are present . from ( [ imd0uni ] ) , we see that that both effects make @xmath226 finite for small @xmath82 . the approximation @xmath229 to the spin - flip correlation function is not conserving and does therefore not obey the f - sum rule . to obtain a conserving approximation , we need to solve the integral equation in fig . [ feynfig ] ( b ) . this will change the quantitative form for @xmath230 . the solution is however an iteration with the @xmath164 matrix of @xmath229 . we therefore expect @xmath231 to have the same qualitative characteristics as @xmath218 , i.e. , a suppressed but finite spectral weight at low @xmath82 . solving the fully self - consistent equation for the spin - flip correlation function is beyond the scope of this paper @xcite .    in the bcs limit , we see that the emergence of a gap in the single particle spectrum leads to a maximum in the bragg scattering rate just below @xmath0 ( fig .  [ tdepfig ] ) . one could therefore expect a similar maximum to appear at unitarity at a temperature @xmath232 defined as the temperature where the pseudogap emerges . however , the presence of a maximum requires the spectral function to have well - defined sharp quasiparticle peaks like in the bcs case ( [ abcs ] ) . the pseudogap spectral function at unitarity ( fig . [ spectralfig ] ) on the other hand has peaks of width @xmath233 . there will therefore be no maximum in the bragg scattering rate for @xmath234 due to the emergence of the pseudogap @xcite . we conclude that the pseudogap present in the normal phase close to @xmath0 at unitarity leads to a significant suppression of the low frequency bragg scattering rate . the rate remains however finite since the pseudogap is only a suppression of the single particle spectral density , and because @xmath0 is comparable to the value of the pseudogap . the emergence of a pseudogap with lowering @xmath64 does not give rise to a maximum in the scattering rate , as opposed to the bcs case . this is because the quasiparticles are strongly damped at unitarity . we briefly comment on the possible experimental observation of the effects described in this paper . a typical fermi energy for trapped atomic gases is @xmath235hz . thus , we require a frequency resolution of the bragg experiment @xmath236hz . likewise , a typical fermi momentum of the gas is @xmath237m@xmath238 which is the same order of magnitude as the wave numbers for optical lasers . this means that the resulting angle between two laser beams needed to obtain a momentum transfer of @xmath239 is reasonable . the results presented in this paper are for a homogenous system . from the thomas - fermi approximation , we expect the presence of the trapping potential to smooth out features like the gap in the bragg spectrum . to avoid such smoothening , one could focus the laser beams on a smaller section of the cloud where the gas can be regarded as homogeneous . in this case , the results presented in the present paper become directly applicable . we are grateful to cindy regal and chris pethick for helpful discussions . we thank the aspen center for physics where parts of this research were carried out . this research was also supported in part by nsf grants phy03 - 55014 and phy05 - 00914 . d.  m.  stamper - kurn , a.  p.  chikkatur , a.grlitz , s.  inouye , s.  gupta , d.  e.  pritchard , and w.  ketterle , phys.rev . lett .  * 83 * , 2876 ( 1999 ) ; j.  steinhauer , r.  ozeri , n.  katz , and n.  davidson , phys . lett .  * 88 * , 120407 ( 2002 ) . a possible approach is via quantum monte carlo . see , e.g. , calculations of the spin susceptibility in the pseudogap regime of the bcs - bec crossover by m. randeria , n. trivedi , a. moreo , and r. scalettar , phys . lett . * 69 * , 2001 ( 1992 ) , and n. trivedi and m. randeria , phys . . lett . * 75 * , 312 ( 1995 ) .
we propose a bragg spectroscopy experiment to measure the onset of superfluid pairing in ultracold trapped fermi gases . in particular , we study two component fermi gases in the weak coupling bcs and bec limits as well as in the strong coupling unitarity limit . 
 the low temperature bragg spectrum exhibits a gap directly related to the pair - breaking energy . 
 furthermore , the bragg spectrum has a large maximum just below the critical temperature when the gas is superfluid in the bcs limit . in the unitarity regime , we show how the pseudogap in the normal phase leads to a significant suppression of the low frequency bragg spectrum .