We calculate the critical temperature for Bose-Einstein condensation in a gas of bosonic atoms across a Feshbach resonance and show how medium effects at negative scattering lengths give rise to pairs reminiscent of the ones responsible for fermionic superfluidity. We find that the formation of pairs leads to a large suppression of the critical temperature. Within the formalism developed by Nozi\`eres and Schmitt-Rink, the gas appears mechanically stable throughout the entire crossover region, but when interactions between pairs are taken into account we show that the gas becomes unstable close to the critical temperature. We discuss prospects of observing these effects in a gas of ultracold $^{133}\text{C}\text{s}$ atoms where recent measurements indicate that the gas may be sufficiently long lived to explore the many-body physics around a Feshbach resonance.
We calculate the critical temperature for Bose-Einstein condensation in a gas of bosonic atoms across a Feshbach resonance and show how medium effects at negative scattering lengths give rise to pairs reminiscent of the ones responsible for fermionic superfluidity. We find that the formation of pairs leads to a large suppression of the critical temperature. Within the formalism developed by Nozi`eres and Schmitt-Rink, the gas appears mechanically stable throughout the entire crossover region, but when interactions between pairs are taken into account we show that the gas becomes unstable close to the critical temperature. We discuss prospects of observing these effects in a gas of ultracold 133Cs atoms where recent measurements indicate that the gas may be sufficiently long lived to explore the many-body physics around a Feshbach resonance.
A self-consistent theory is derived to describe the BCS--Bose-Einstein-condensate crossover for a strongly interacting Fermi gas with a Feshbach resonance. In the theory the fluctuation of the dressed molecules, consisting of both preformed Cooper pairs and ``bare'' Feshbach molecules, has been included within a self-consistent T-matrix approximation, beyond the Nozi\`eres and Schmitt-Rink strategy considered by Ohashi and Griffin. The resulting self-consistent equations are solved numerically to investigate the normal-state properties of the crossover at various resonance widths. It is found that the superfluid transition temperature Tc increases monotonically at all widths as the effective interaction between atoms becomes more attractive. Furthermore, a residue factor Zm of the molecule's Green function and a complex effective mass have been determined to characterize the fraction and lifetime of Feshbach molecules at Tc. Our many-body calculations of Zm agree qualitatively well with recent measurments of the gas of 6Li atoms near the broad resonance at 834G. The crossover from narrow to broad resonances has also been studied.
We investigate strong-coupling effects on the superfluid phase transition in a gas of Fermi atoms with a Feshbach resonance. The Feshbach resonance describes a composite quasiboson that can give rise to an additional pairing interaction between the Fermi atoms. This attractive interaction becomes stronger as the threshold energy $2\ensuremath{\nu}$ of the Feshbach resonance two-particle bound state is lowered. In a recent paper, we showed that in the uniform Fermi gas, this tunable pairing interaction naturally leads to a crossover from a BCS state to a Bose-Einstein condensate (BEC) of the Nozi\`eres and Schmitt-Rink kind, in which the BCS-type superfluid phase transition continuously changes into the BEC type as the threshold energy is decreased. In this paper, we extend our previous work by including the effect of a harmonic trap potential, treated within the local-density approximation. We also give results for both weak and strong coupling to the Feshbach resonance. We show that the BCS-BEC crossover phenomenon strongly modifies the shape of the atomic density profile at the superfluid phase-transition temperature Tc, reflecting the change of the dominant particles going from Fermi atoms to composite bosons. In the BEC regime, these composite bosons are shown to first appear well above Tc. We also discuss the ``phase diagram'' above Tc as a function of the tunable threshold energy $2\ensuremath{\nu}.$ We introduce a characteristic temperature ${T}^{*}(2\ensuremath{\nu})$ describing the effective crossover in the normal phase from a Fermi gas of atoms to a gas of stable molecules.
The crossover between the Cooper-pair condensation and the Bose-Einstein condensation of ``di-electronic'' molecules in two-dimensional superconductors is investigated in detail on the basis of the Nozi\`eres and Schmitt-Rink formalism. It is shown that temperature dependence of the chemical potential \ensuremath{\mu} so calculated is classified into two classes as decreasing temperatures; i.e., class (a) where \ensuremath{\mu} approaches the point of Bose-Einstein condensation of two-dimensional ideal Bose gas of ``di-electronic'' molecules, and class (b) where \ensuremath{\mu} diverges positively along the line of BCS-type mean-field pair condensation. This feature is rather universal irrespective of strength V of the attractive interaction of the s-wave type. While the former class (a) has been found by Schmitt-Rink, Varma, and Ruckenstein, existence of the latter class (b) is recognized here. In the case where V is fixed, class (a) is realized for electron number density N smaller than Ncr, which is an increasing function of V, and class (b) is realized for N larger than Ncr. If N\ensuremath{\gg}Ncr in particular, there exists a regime, where the Fermi-liquid-like description is valid, between the BCS-type mean-field transition temperature and the Fermi temperature. In the situation where V is changed with N being fixed, low-temperature states for the strong-coupling case belong to class (a) while those for the weak-coupling case belong to class (b). Therefore, with decreasing V, the chemical potential \ensuremath{\mu}(T), at temperatures far below the Fermi temperature, shows a discontinuous jump at V=Vcr(N) corresponding to the transition from class (a) to (b). However, this is in contradiction to a physical picture that the chemical potential should smoothly cross over between the above two limits unless the liquid-gas transition occurs. This shows in turn a necessity of improving the Nozi\`eres and Schmitt-Rink formalism itself especially in two dimensions. A preliminary approach beyond their formalism is briefly discussed.
We investigate the non-universal behavior of the BCS-BEC crossover model at the normal to superfluid transition. By using a finite temperature quantum field theoretical approach due to Nozieres and Schmitt-Rink and by making the effective range expansion of the effective two-body interaction we numerically calculate the crossover transition temperature as a function of the scattering length, aF, and the effective range parameter, re. In an ultracold two-component atomic Fermi gas BCS-BEC crossover physics is expected to appear near magnetic-field-induced Feshbach resonances. By matching two-body scattering properties near a Feshbach resonance to a simple renormalized model potential, a broad resonance is characterized by a small effective range and the gas displays a universal BCS-BEC crossover behavior. On the other hand, for a narrow resonance thermodynamic quantities depend the effective range which may be large and negative at resonance. For increasing values of −re we find the transition temperature to be suppressed in the crossover region. We furthermore argue for existence of a lower bound of Tc at fixed coupling for increasing −re in the crossover region.
In this thesis, we investigated the physics of two- and three-dimensional ultra cold Bose gases in the strongly interacting regime at zero temperature. This regime can be experimentally accessed using a Feshbach resonance. We applied a self-consistent diagrammatic approach to determine the chemical potential of three-dimensional Bose gases for a wide range of interaction values. We showed that such strongly interacting Bose gases become unstable towards the formation of molecules at a finite positive scattering length. In fact, the interaction between atoms becomes effectively attractive and the system looses its metastability before reaching the unitary limit. We also found that such systems are nearly fermionized close to the instability point. Near this critical point, the chemical potential reaches a maximum and the contribution to the system energy due to three-body forces is estimated to be only a few percent. We also studied the same system using a self-consistent renormalization group method. This approach confirms the existence of an instability point towards the formation of molecules as well as fermionization. We showed that the instability and accompanying maximum are precursors of the sign change of the effective two-body interaction strength from repulsive to attractive near resonance. In addition, we examined the physics of two-dimensional Bose gases near resonance using a similar self-consistent diagrammatic approach as the one introduced for three-dimensional Bose gases. We demonstrated that a competition between three-body attractive interactions and two-body repulsive forces results in the chemical potential of two-dimensional Bose gases to exhibit a maximum at a critical scattering length beyond which these quantum gases possess a negative compressibility. For larger scattering lengths, the increasingly prominent role played by three-body attractive interactions leads to an onset instability at a second critical value. The three-body effects studied for these systems are universal, fully characterized by the effective two-dimensional scattering length and are, in comparison to the three-dimensional case, independent of three-body ultraviolet physics.
Since the realization of Bose-Einstein condensation in 1995, experimentalists in ultracold atomic gas community have developed a set of entirely new methods to study quantum gases. In ultracold atomic gases, implementation of synthetic gauge fields, the spin-orbit coupling, optical lattices and Feshbach resonances, provides a clean and highly controllable platform to tune the parameters of the system, which are typically fixed for real materials. Motivated by the rapid developments of experimental progresses, I investigate the ultracold atomic gases with synthetic gauge fields. \n \nIn this thesis, a bosonic gas in optical lattices in the presence of an effective staggered magnetic flux will be studied, using a generalized Bose-Hubbard model. Phase transition from the Mott insulating phase to the superfluid phase will be discussed. Novel superfluid phases including the stripe phase and the plane wave phase will be discussed. Furthermore, the existence of a tricritical point in the phase diagram is predicted. \n \nTo study Fermi systems with synthetic gauge fields, a spin-1=2 Fermi gas with Rashba spin-orbit coupling is investigated by studying the structure of vortex line through the Bardeen-Cooper-Schrieffer to Bose-Einstein condensation (BCSBEC) crossover. Due to the spin-orbit coupling, the s-wave interacting Fermi gas behaves like an effective p-wave paired system. A discussion about the vortex line structure will be provided by calculating the spatial distribution of the gap function, the density, the particle current, and also the elementary excitations. It has been found that the most robust superfluid occurs around the unitary point, indicated by the smallest healing length and the largest critical current. \n \nFinally, motivated by the recent progress of p-wave resonantly Fermi gas in experiment, an extension of Nozières-Schmitt-Rink method to the p-wave case will be given. An investigation of the normal state properties will be given by calculating the equation of state and the p-wave contacts, which have been measured by the experiment. Furthermore, the superfluid transition temperature with realistic parameters appropriate for experiment is calculated and is found to be within experimental reach.