Theoretical Study On Several Problems Of Ultracold Molecules

The development of the technique of cooling, trapping and manipulating atoms and molecules with laser and magnetic field makes AMO (atom-molecule and optic-s) a hot and novel physical area.. This experimental platform has several wonderful features:accurate controlling, clean system, tunable interactiong strength and other pa-rameters. Therefore, it becomes a ideal system to simulate condensed matter physics, also provides great opportunities for super-accurate meaturement, quantum calculat-ing and other fields. Ultracold atomic and molecular system is an ideal experimental platform for people to explore the quantum world. In this area, ultracold molecules, due to their more kinetic degree of freedom, more inner energy level, novel interation,, therefore more applications, have attracted a lot of attentions, theoretical, experimen-tal efforts. Molecules have rotational and vibrating degree of freedom, It is therefore very difficult to directly cool molecules down to a very low temperature. People often associate cold atoms into molecules with optical and magnetic field.I use the title "theoretical study fo several problems of ultra cold molecules" for this thesis. Actually, we mainly address two problems:1, Atom-Molecule dark state, that plays an important role when associating bosonic atoms into molecules.2, the ground state of polar molecules.When associating bosonic atoms of one kind into molecules, people use a tech-nique called stimulated Raman adiabatic passage, that is developed in quantum op-tics. They tune the laser sequences, adiabatically fransfer particles from atomic state to molecular state. In this process, ideally, the system will stay in a state call atom-molecular coherent state which is a superposition of atomic sate and molecular state. Before our research, all the study is on the scope of mean-field approximation.Polar molecules are associated by atoms of two species. They have large elec-tric dipoles, therefore large dipole-dipole interaction. Dipole-dipole interaction is an anisotropic, long-range interaction, which is attractive in some direction, while repul-sive in other direction, therefore very interesting and novel. As the technique develops, Prof. Jun Ye’s group has successfully created polar molecules…. Someone has the-oretically predicted that the ground state of polar molecules will be a BCS super-fluid state due to their partially attractive interaction, while in the normal state, the Fermi sur-face will deform in the polarization direction. How to understand these two phenomena in one picture, and how they affect each other is an interesting problem.In this thesis, we study these two problems:1, In the experiment associating bosonic atoms into molecules, atomic state, ex-cited molecular state, and stable ground molecular state, form a three level system. The existence of a dark state plays a key role in the transferring process. Analytically, we constructed the wave function of the dark state under single-mode approximation. Numerical calculation has verified thatit is indeed the atom-molecule dark state. By comparing the quantum state with mean-field wave-function, we found that as particle number increases, mean-field result of particle population approaches quantum result. Also, we pointed some parameter area, that will break the adiabatic process, therefore should be avoided in experiment.2, We apply the Hartree-Fock-Bogoliubov theory to a system of uniform dipolar fermionic polar molecules.which recently has attracted much attention due to rapid ex-perimental progress in achieving such systems. By calculating the anisotropic superfluid-order parameter and the critical temperature Tc, we show that high-Tc superfluid can be achieved with a quite modest value of interaction strength for polar molecules. In addition, we also show that the presence of the Fock-exchange interaction enhances superfluid pairing. It has an interesting physical interpretation:The strength of BCS pairing is proportional to the density of state of fermions near Fermi surface. For3-d system, dos is proportional to Fermi momentum. For polar molecules, BCS pairing is dominated by p-wave pairing, and the order parameter is mainly in polarization direc-tion. Fock term induced Fermi-surface deformation just extend Fermi momentum in this direction. In anoter word, this deformation increase fermions that can pair, decease those that can not pair. So it enhances BCS pairing.