A Theoretical Study Of BCS-BEC Crossover

The field of ultracold Fermi atoms has been extensively studied with its experi-mental realization of degenerate Fermi gases in1999. Owing to their easy tunabilities, ultracold fermi gases provide model systems to study many-body physics. Especially the tunability of the strength of the attractive interaction between fermions via Fesh-bach resonance allows the realization of the crossover from a weakly paired Bardeen-Cooper-Schrieffer (BCS)(It is the microscopic theory put forward by Bardeen, Cooper and Schrieffer in1957to explain the phenomenon of superconductivity) superfluidity to a strongly paired diatomic molecular Bose-Einstein condensate (BEC). Furthermore, there is a general belief that the study of ultracold fermi gases is expected to be helpful for the understanding of pseudogap phenomenon of high-Tc cuprates.A sensible theoretical treatments of the BCS-BEC crossover is to set up a theory which can describe the entire regime from the BCS to the BEC limit. In the crossover regime, the fluctuation is important. After the pioneering work of Eagles and Leggett, Nozieres and Schmitt-Rink (NSR) took into account pairing fluctuation for the first time and used a diagrammatic method to deal with the case at finite temperature in the normal phase in1985. In addition to NSR-type fluctuation from the particle-particle channel, there is another type of fluctuation from the particle-hole channel which was first considered by Gorkov-Melik-Barkhudarov (GMB). The GMB correction is al-so called induced interaction. Most of studies about induced interaction in ultracold Fermi gases were confined to the lowest-order. Yu and co-workers considered the ex-tended GMB approximation which was beyond the lowest-order correction. However, they considered the extended GMB approximation in the NSR framework which is e-quivalent to expanding the fermion propagator to the lowest order in self-energy when one calculates the number equation via the fermion propagator. If the fluctuation cor-rections are important as in the BCS-BEC crossover, a better treatment might be to solve the Dyson equation including the self-energy to all orders when one calculates the number equation via the Green’s function. This is exactly the non-self-consistent T-matrix approximation (nTMA) framework. In this thesis, we generalize the ex-tended GMB approximation from NSR framework to nTMA framework and calculate physical quantities on the basis of this approximation.We calculate the transition temperature for the entire BCS-BEC crossover and compare our results with those in the literature. Our result is between that of Yu et al. and that of Tsuchiya et al. based on nTMA framework without considering the induced interaction. At unitarity where the scattering length between the fermionic atoms di-verges the transition temperature obtained by us is about90%of that of Tsuchiya et al. without the effect of the extended GMB approximation. In the Bardeen-Cooper-Schrieffer (BCS) limit, the critical temperature is about0.45TcBCS and it is the same as the result of the usual Gorkov and Melik-Barkhudarov (GMB) correction, whereas in the Bose-Einstein condensate (BEC) limit, the effect of the induced interaction can be neglected and the critical temperature turns to be the transition temperature of ideal Bose gases.In ultracold Fermi gases, the study of the behavior of single-particle spectral func-tion through which one can calculate almost all other physical quantities is of cen-tral importance. Theorists have anticipated that the single-particle spectral function exhibits pseudogap phenomenon above Tc. However, there exists confliction about the existence of pseudogap in strongly interacting Fermi gases in experiment recent-ly. In this thesis, we present the spectral function and hence density of states in the normal phase at unitarity including the extended GMB approximation in the nTMA framework. From the numerical results of the spectral weight function in the energy-momentum plane, we find two energy branches near Tc and they merge into a single upward branch with increasing temperature. From the results of A(k,ω) vs ω for a given set of wave vectors near kF at Tc, double-peak structures are found. From the numerical result of density of states which is the momentum summation of spectral function, we find a clear depletion in DOS around ω=0. All these indicate the exis-tence of pseudogap phenomenon including the induced interaction.The recent radio frequency (rf) spectroscopy experiments on ultracold Fermi gases provide valuable information on the spectral function. In these experiments, a radio frequency pulse drives atoms from one of two paired spin states into an unoccupied spin state, where they are counted to yield a spectrum of counts versus frequency. The obtained rf spectra are associated with the spectral function. Up to now, the effect of the induced interaction has not been taken into account in the theoretical studies of rf spectra. In this thesis, we give the rf spectra with the extended GMB approximation. The obtained rf spectra for a homogeneous and balanced system show a single peak, which is consistent with experiments. Furthermore, compared with experiments, we obtain a more satisfactory result without slightly overestimating the width of the peak.The early rf experiment was done in a trapping potential without momentum res-olution, and hence the measured rf spectra involve integration over momentum and an average over the density profile of trapped atoms. Later tomographic rf spectroscopy experiment provided spatially resolved rf spectroscopy which involved integration over the entire momentum only and the recent momentum-resolved rf spectroscopy which involved integration over the entire trap. In the near future, tomographic momentum-resolved rf spectroscopy could eliminate the complexity coming from the integration of the entire momentum and the entire trap. In this thesis, we also calculate the momentum-resolved rf spectra of homogeneous Fermi gases at unitarity above Tc. We find there exists upward and downward dispersion branches and the downward branch is dominant near Tc. The downward branch indicates the existence of the pseudogap contribution. With increasing temperature, there is only an upward branch left indicat-ing the disappear of pseudogap phenomenon.