| The Bose-Einstein condensation in a dilute atomic gas has been realized in experiment,which leads the ultracold atomic system become a hot research subject in physics and paves the way for exploring quantum many-body physics.In recent years,based on the development of the experimental techniques,the Fechbach resonance and the spin-orbit coupling have been observed,and the optical lattice has been constructed,and so on.The ultracold atom in an optical lattice is highly controlled and extremely clean,and has become a power tool for quantum simulation of many-body physics.However,the experimental reaserches are mostly based on the short-range interaction between the atoms.In order to explore novel many-body physics which are induced by the long-range interaction,the strong coupling of the ultracold atoms to an ultrahigh-finesses optical cavity has been proposed based on the cavity quantum electrodynamics system.This approach realizes a infinitely long-range interaction between atoms mediated by the cavity field,which is only dependent on the effective coupling strength between the atoms and the cavity field.On the other hand,this coupling system can also induce many novel dynamical properties.In particularly,the Bose-Einstein condensation was first loaded into an optical cavity in 2007 and the superradiant phase transition was observed in 2010.And quantum phases induced by the competing between the short-and long-range interactions was observed in 2016.With the progress of the experiment for the coupling system,more and more researchers in different fields begin to pay attention to this coupling system.Moreover,unlike bosons,fermions obey the Pauli exclusion principle,a quantum state can only occupy a identical fermion.And two fermions with weak attractive interaction can form Cooper pairs which are responsible for superfluid.Then,based on the current experimental progress and the unique property for the degenerate Fermi gas,we propose a scheme for the couplingof the degenerate Fermi gas to an ultrahigh-finesses optical cavity and mainly analyze significant and novel many-body phenomena in this thesis.The specific studies are as follows:1.Quantum mixed phase in a two dimensional polarized degenerate Fermi gasWe consider a strong coupling between the two-dimensional degenerate Fermi gas and an optical cavity,and propose a Fermi-Dicke model in the cavity quantum electrodynamics(CQED)system.Based on the mean-field theory,we find that there exists a quantum phase transition which is quite different from bosons.Specifically,when the coupling strength is smaller than a certain critical value and the effective Zeeman field strength is small,the system is in a normal phase with two Fermi surfaces.When the superradiance occurs with increasing the coupling strength,the phase transition between the normal phase and the superradiant phase is first order.Moreover,there exists a mixed phase where the normal phase and the superradiant phase coexist.Finally,we plot a whole phase diagram.2.The superfluid-superradiant mixed phase in an optical cavity with the interacting degenerate Fermi gasAs we all know,there exists a BCS superfluid phase when the interaction between fermions is negative.Then in the system of fermions coupling with an optical cavity,we consider that there exists an attractively interacting in our system.Based on the mean-field approximation,we find that there exists a new mixed phase in which the superfluid phase and superradiant phase coexist when the intermediate interaction between fermions and atom-cavity field coupling strengths exist,and a first-order quantum phase transition from the superradiant phase to the superfluid phase occurs.Moreover,in this mixed phase,the ratio of the scaled atomic polarization to the dimensionless superfluid order parameter is relatively large.Finally,we plot phase diagrams in the effective Zeeman field-effective coupling strength plane.3.The magnetic properties of Fermi gases in a high-finesses optical cavityThere is lack deep researches about the magnetic properties in an optical cavity in recent years.Then we study the atomic magnetic properties in different situations.Specifically,we consider a one-dimensional degenerate Fermi gas coupled to an optical cavity.When the detuning between the atomic frequency and the pump laser frequency is blue,the interacting between the atoms and the cavity field causes a atomic coupling between the different spins and sites.In this case,the magnetic correlation of the system is changed when the system is in a superradiant phase,i.e.,the system is transformed from the antiferromagnetic superradiant phase to the ferromagnetic superradiant phase.Moreover,we also analyze the effection of different filling factor to magnetic properties.While when the detuning between the atomic frequency and the pump laser frequency is red,the interacting between the atoms and the cavity field causes a atomic coupling between the different spins and same site.In this case,the system is always in a antiferromagnetic superradiant phase.Moreover,the different filling factor has little impact on this magnetic property.4.The topological superradiance of one dimensional Fermi gases in a shaken dynamical optical latticeWe provide a new way for exploring novel many-body physics.We consider the quasi-one dimensional degenerate Fermi atoms coupled to a shaken cavity field,which is induced by the two electro-optic modulators placed in the optical cavity.The pump and cavity fields couple with each other mediated by the cavity field.Then we can obtain a time-dependent dynamical optical lattice.This shaken optical lattice affects the atoms in turn.Based on the mean-field and high-frequency approximations,the system contains the cavity-dependent nearest-,next-nearest-,and cavity-dependent next-next-nearest-neighbor hoppings.These hoppings change the band structures and thus induce nontrivial topological superradiant phase,whichhas four degenerate edge states and a large winding number.Finally,we also analyze the symmetries for the system under different situations. |
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