Equilibrium Dynamics Of Multi-Component Ultracold Bosonic Gases

This essay focuses on the static and dynamical properties of multi-component Bose-Einstein condensates, including the discussion about double-well system and atom-to-molecule conversion process. In the first two chapters, we briefly review the experimental and theoretical backgrounds of Bose-Einstein condensa-tion, and also introduce the relevant investigations of multi-component conden-sates. Our research achievements correspond to the content from Chapter 3 to Chapter 6 which are listed as follows:In Chapter 3, we investigate the fixed-point transitions of two-component Bose-Einstein condensates in double-well trapping potentials. Based on the sym-metry between the two kinds of atoms, we classify the fixed-point solutions into four categories:symmetrical, anti-symmetrical, isotropical and asymmetrical. We discuss the existence of each kind of solution and the corresponding dynam-ical stability, and give the stability diagrams in parameter spaces. We apply a high-frequency periodic modulation to the energy bias between the two wells and vary the value of the amplitude to tune the effective tunneling strength. As a result, during the evolution the final state may belong to a different fixed-point solution from the initial one. This is the so-called fixed-point transitions which reply on the parameter values sensitively. We believe this phenomena could be used to transfer condensates between wells.In Chapter 4, we discuss the phase separation of atomic and molecular Bose-Einstein condensates trapped in isotropic three-dimensional harmonic potential. Under Feshbach resonance, atoms can be converted into molecules. By find-ing the energy minimum, we determine the ground state phase diagram. The possible ground states are vacuum state, pure molecular superfluid state and mixed atomic-molecular superfluid state. The asymmetry between atoms and molecules in the Feshbach term inhibits the existence of pure atomic superfluid state. Hessian matrix is introduced to determine the stability of each phase. Under Thomas-Fermi approximation, we plot the phase diagram. Due to the correspondence between the parameter space and coordinate space, we plot the density distribution of atomic and molecular condensates. Both prove the ex-istence of phase separation. We also notice the discontinuous transition about some critical interaction strength.In Chapter 5, we give an experimental proposal to show another approach to increase the atom-molecule conversion efficiency by manipulating the Ramsey type of magnetic field pulses. Ignoring the effect of trapping potential, we de-scribe the system with a pendulum-like model under mean-field approximation. We find the fixed-point solutions, and determine the periodicity of Rabi oscilla-tion for one constant magnetic field. The periodicity diverges when the molecular energy approaches the atomic energy. By applying the Ramsey type of magnetic field pulses, we find that if the separation between the two pulses is appropriately tuned, the atom-molecule conversion efficiency could reach the maximum. The basic idea is about the transition of the system between Rabi oscillation orbitals given by different energy contours. We give a simple and easy-understanding picture to explain how our proposal would work.In Chapter 6, we put Bose-Einstein condensates in a high-finesse optical cavity, and investigate the influence of photons on the atom-molecule conver-sion efficiency under Feshbach resonance. We consider the dispersive interaction between photons and atoms or molecules. Since photons decay very fast, the corresponding evolution adiabatically follows that of atomic and molecular con-densates. Thus we can get the effective Hamiltonian describing pure atoms and molecules. Based on mean-field approximation, we find the existence of nonlin-ear terms would yield more possible solutions. When the pumping strength is above a critical value, we may observe the bistability phenomena which could be proposed as a way to tune the molecular production rate with photons in cavity. We also give the quantum model and the comparison with the mean-field model is expected.