Investigations On Collective Excitations And Their Interactions Of Superfluid Atomic Fermi Gases

Bose-Einstein condensation(BEC) is a phenomenon of macroscopic occupation 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.Due to the Pauli's exclusion principle,fermions can only form BEC through the formation of Cooper pairs based on the Barden-Cooper-Schrieffer(BCS) theory.BEC is the origin of many kinds of macroscopic quantum phenomena,which also relates to superconductivity and superfluidity.It is revealed that BEC and BCS is two limits of BCS-BEC crossover theory.With the development of laser cooling and trapping techniques in recent years,BEC in weak interaction bosonic atom gases was finally realized.Subsequently, contributing to the achievement of quantum degeneracy of fermionic atom gases and condensates,the possibility of BEC from fermionic atom pairs was also achieved.The crossover of fermionic atom gases between BCS superfluid and BEC could be realized through Feshbach resonance,which allowed control on the magnitude of the interactions between two atoms even its sign. The research of ultracold quantum gases is the cross point of many disciplines, including atomic,molecular physics,nonlinear and quantum optics,statistical and condensed matter physics,and hence has significant importance from both the theoretical viewpoint(including novel quantum properties and manipulations of photon-atom interactions,modulations of strongly correlated systems) and technological viewpoint(including atom laser,atom interferometers,atom chips,atom clocks and quantum computers).The study of elementary excitations is a basic concept in quantum manybody theory and one of the basic topics in statistical physics and condensed mat- ter physics.Collective excitations are the fundamental excited modes in superfluid atom gases which describe the collective oscillation behaviors of many-body system in the external trapping potential.A lot of studies on the collective excitations of superfluid gases have been done.Techniques of precision measurement of oscillation frequencies have been developed.A deeper understanding of collective excitations is of much importance not only in helping us to reveal novel characteristics of the interaction properties of these nonuniform quantum systems, but also to explore the superfluidity properties of ultracold quantum gases.The collective excitations of the superfluid Fermi gases in BCS-BEC crossover could be studied in microscopic description in principle.However,the microscopic mechanisms of the BCS-BEC crossover are still not clear.The external trapping potential makes the problem more complicated.Simultaneously,it is not easy to handle the issues of nonlinearity and nonequilibrium in time-dependent evolutions of the collective excitations and their interactions.Because of the ultralow temperature,a macroscopic wave function(order parameter) could be used to describe the superfluid Fermi gases and its time-evolution.The equation of the wave function could be obtained through adding a proper quantum pressure term in hydrodynamical equations.Superfluid order parameter equation could be used to describe the dynamics of collective excitations and their interactions in BCS-BEC crossover when considering the equation of state obtained from quantum Monte-Carlo simulations.The main results of the collective excitations and their interactions of superfluid Fermi gases in external trapping potentials based on the macroscopic order parameter description are as follows:1.We studied different collective excitation behaviors of superfluid Fermi gases in BCS-BEC crossover in different trapping potentials.Using variational methods we calculated some typical collective modes in detail which were studied by some world-famous experimental groups.We studied the relations between collective frequencies and scattering length under the assumption of in- cluding(excluding) the kinetic terms,and demonstrate that when the anisotropic index is large(cigar-or disc-shape) the kinetic term is important and hence the Thomas-Fermi approximation could not be used.We calculated the particle numbers of condensate with different anisotropic index to show the dependence on the different superfluid regimes and to guide the future experiments.2.For technical reasons,physical properties of the ultracold quantum gases are obtained mainly through the detection of time-of-flight.Here,we studied the anisotropic free expansion of the superfluid Fermi gases when switching off the external trapping potential and obtained the relation between the expansion velocity and interaction parameters of the gases in BCS-BEC crossover regime within cigar-and disc-shape trapping potentials.3.If the amplitude of the external perturbations is large enough,collective excitations will locate in nonlinear regime even at zero-temperature.We studied the nonlinear responses of the superfluid Fermi gases in trapping potential with external perturbations.Especially,we calculated nonlinear frequency shifts of three low-energy collective modes in detail which were mostly detected in labs. Divergences will appear when choosing some typical parameters,which corresponding to the resonant interactions between those collective modes.Based on the Lindstedt-Poincar(?) perturbation method,we studied the resonant interactions and second-harmonic generation when the phase-matching condition is fulfilled.All analytical results have been checked by numerical simulations.4.We discussed Josephson effects of superfluid Fermi gases.Our results show that nonlinear Rabi oscillations of superfluid Fermi gases could happen due to the coherent properties of double-well potential.Macroscopic self-trapping (MST) phenomenon will also occur with some typical coupling parameters due to the interaction between atoms.We show the Josephson effect and MST phenomenon in different superfluid regimes of BCS-BEC crossover,give the relation between the coupling strength and the oscillation of particle number,and predict the range of the system parameters for guiding an experiment. The results in this dissertation will help us with a better understanding of the superfluidity properties of the Fermi gases in BCS-BEC crossover,and will favor to the experiments in future.