Investigations On Coherent Property And Nonlinear Effect Of Superfluid Atomic Fermi Gases In The BCS-BEC Crossover

Bose-Einstein condensation(BEC) is a phenomenon of macroscopic occupa-tion in one or several quantum states by a large number of identical bosons when the system temperature is below to a critical temperature. BEC is one of the most important conclusions in quantum statistical physics, and the origin of supercon-ductivity and superfluidity. The Pauli exclusion principle prevents multiple oc-cupancy of identical fermions to share the same quantum state. However, system of fermions with weak attractive interaction is known to form large overlapping Cooper pairs, which behave like (quasi) composite bosons and can condense into the pair state with zero center of mass momentum giving rise to superconduc-tivity and superfluidity. It is revealed that as the interaction between particles changes from attractive to repulsive in Fermi systems, a continuous transition from a Bardeen-Cooper-Schrieffer(BCS) type superfluid of loosely bound pairs to BEC of molecules occurs, while BCS and BEC are the two-limiting cases of the BCS-BEC crossover.With the spectacular development of laser cooling and trapping techniques, experimenters chill dilute gases of certain atoms to nanoKelvin temperatures, which allows the realizations of BEC in weakly interacting bosonic atomic(i.e. 87Rb,23Na) gases and quantum degenerate fermionic atomic (i.e.6Li,40K) gases, and using a Feshbach resonance they explore the superfluidity as well as BCS-BCS crossover in quantum degenerate Fermi gases. Ultracold atomic physics has so far been in essence a hot boundary field between atomic and molecular, nonlin-ear and quantum optics and condensed matter physics, which not only serves as a test-bed for theories developed in matter-wave optics, quantum manipulation, high temperature superconductors, Quark-gluon plasma, and neutron stars, but also leads to new achievements in atom laser, atom interferometers, atom clocks, and light clocks.A complete description of ultracold fermionic atomic gases in the BCS-BEC crossover based on quantum many-body theory is still not clear. Because ul-tracold atoms are trapped in external trapping potentials, the imhomogeneous and mesoscopic features of the system make a microscopic approach of nonlinear collective excitations and non-equilibrium problems even more difficult. There-fore, in this dissertation we use an order-parameter equation, which is based on extended Thomas-Fermi functional theory and generalized superfluid hydrody-namics equations, to investigate coherent properties and nonlinear dynamical be-haviors of superfluid Fermi gases in the BCS-BEC crossover in details, including the following aspects:An important topic in the study of ultracold Fermi gases is a direct obser-vation of superfluid properties in the BCS-BEC crossover. One expects that in-terference patterns appear after superfliud Fermi gases released from combined harmonic oscillator and optical lattice potentials due to the long-range phase co-herence. We first start from superfluid hydrodynamics equations and include a proper quantum pressure term, then obtain an order-parameter equation. We solve the order-parameter equation valid for the crossover from BCS superfluid to BEC to obtain an initial distribution of subcondensates formed in an optical lattice. Then we investigate the coherent evolution of the subcondensates when both harmonic oscillator and optical lattice potentials are switched off. The inter-ference patterns of the superfluid Fermi gas along the BCS-BEC crossover dur-ing a nearly ballistic expansion are calculated by means of Feynman propagator method combined with numerical simulations. The result obtained agrees with the experimental observation[ J. K. Chin et al., Nature 443,961 (2006)].The existence of macroscopic coherent behavior in superfluid fermionic atomic gases resembles other macroscopic coherent states found in the field of nonlinear and quantum optics, and hence one would expect to observe macroscopic non-linear collective phenomena, of which a soliton is a striking example. We study, both analytically and numerically, the formation and propagation of dark solitons in a quasi-one dimensional superfluid Fermi gas in the crossover from BCS su-perfluid to BEC. Starting from a superfluid order-parameter equation we derive a Korteweg-de Vries(KdV) equation for weak nonlinear excitations under quasi-one-dimensional and long wavelength approximations. We present dark soliton solutions valid for both BCS and BEC limits and also for the crossover, and show that dark solitons in different superfluid regimes possess different features. Par-ticularly, a dark soliton in the BCS (BEC) regime has larger (smaller) propagating velocity and smaller (larger) spatial width. Upon moving to the boundary of the condensate, it generally decelerates and generates small radiations, which dis-play different behavior in different superfluid regimes. We study also a head-on collision between two dark solitons and demonstrate that the phase shift due to the collision changes non-monotonically along the BCS-BEC crossover. All an-alytical results are checked by numerical simulations and good agreements be-tween them are found.Quasi-one-dimensional condition requires that atomic numbers of systems must be very small and confinements in transverse directions be very strong, however, which are hard to realize experimentally in superfluid fermionic atomic gases. The investigation of the linear and nonlinear sound propagation of cigar-shaped superfluid Fermi gas with a large particle number is hence of particular interest. We first solve analytically the eigenvalue problem of linear collective ex-citations and provide explicit expressions of all eigenvalues and eigenfunctions, which are valid for all superfluid regimes in the BCS-BEC crossover. The lin-ear sound speed obtained agrees well with that of a recent experimental mea-surement. We then consider a weak nonlinear excitation and show that the time evolution of the excitation obeys a KdV equation. Different from the result ob-tained in quasi-one-dimensional case studied previously, where subsonic dark solitons are obtained via the balance between quantum pressure and nonlinear effect, we demonstrate that bright solitons with supersonic propagating velocity can be generated in the present three-dimensional system through the balance between a waveguide-like dispersion and the inter-particle interaction. The su-personic bright solitons obtained display different physical properties in different superfluid regimes and hence can be used to characterize superfluid features of the BCS-BEC crossover.The results presented here may be useful for understanding coherent and nonlinear properties of superfluid fermionic atomic gases in the BCS-BEC crossover, and guiding experimental and theoretical investigations of ultracold fermi gases more extensively.