1.1 Preface
The Bose-Einstein condensation (BEC) was first predicted out in the 1925 when Einstein devoted to the statistical description of the quanta of light. He based on the paper of the Indian physicist S.N. Bose and considered the Bose-Einstein condensation as the condensation of atoms in the state of the lowest energy associated with a phase transition. For the gas of non-interacting massive bosons he concluded that below a certain temperature, a finite fraction of the total number of particles would occupy the lowest-energy single-particle state [1].
In 1938, Fritz London suggested the connection between the superfluidity of liquid 4He and the Bose-Einstein condensate. However, the interaction between helium atoms is strong, and this reduces the number of atoms in the zero-momentum state even at absolute zero. The fact that interactions in liquid helium reduce dramatically the occupancy of the lowest single-particle state led to the search for weakly interacting Bose gases with a higher condensate fraction. The difficulty with most substances is that at low temperatures they do not remain gaseous, but form solids or liquid and the effects of interaction thus become large.
The experimental studies on the dilute atomic gases were much later until the 1970 when the new techniques about magnetic trapping and advanced cooling mechanisms were developed in atomic physics. In a series of experiments hydrogen atoms were coming very close to BEC by being first cooled in a dilute refrigerator
and further cooled by evaporation.
In 1980s the cooling and trapping techniques based on laser such as laser cooling and magneto-optical trapping were developed to cool and trap neutral atoms. Alkali atoms are also to be cooled by such method because their optical transition can be excited by available lasers and also have the energy-level structure which can be cooled to very low temperature.
BEC in dilute alkali gas was then first observed by the team of Cornell and Wieman at Boulder and of Ketterle at MIT in 1995. This great achievement also won the Nobel Prize in physics in 2001. Following the successful experimental observations of BECs, more physical properties of BECs had been investigated such as loading BECs in optical lattices [2] generated by interference of laser beams. The first experiment involving the dynamics of BECs in periodic potentials carried out by Anderson and Kasevich was demonstrating a mode-locked atom laser to observe atomic Josephson oscillations [3,4]. In addition, the other properties of coherent macroscopic matter waves in a lattice have been explored [5,6].
An optical lattice is practically perfect periodic potential for atoms, produced by the interference of two or more laser beams. A Bose–Einstein condensate is the ultimate coherent atom source: collective atoms, all in the same state, and with an extremely narrow momentum spread. Combining the BEC and optical lattice gives an opportunity for exploring an analog of electrons in a solid-state crystal but with unprecedented control over both lattice and the particles.
The time-dependent behavior of Bose-Einstein condensed clouds such as
collective excitation modes and the expansion of a cloud releasing from a trap is an important source of information about the physical nature of the condensate.
1.2 Motivation
The study of elementary excitations is an important subject in the physics of quantum many-body system. In superfluid helium, Landau, Bogoliubov and Feynman all had pioneering theoretical work on this subject. After the experimental realization of Bose-Einstein condensation in trapped atomic gases, the study of collective excitations has become a popular subject of research in ultracold gases [7, 8]. From the theoretical side, new challenging features emerge for the nonuniform nature of these systems. From the experimental side the high precision of the measurement of collective frequencies provides a unique opportunity for a detailed comparison with theory to point out the role of the interactions and of quantum correlations.
On the other hand, Bose-Einstein condensate is a macroscopic quantum system that is easy to be exactly controlled experimentally. Therefore, many phenomena studied in solid-state system can be re-examined in a more direct and dramatic way. It is now possible to experimentally realize those model systems that had previously been studied theoretically but were impossible to test experimentally. One example of these is the experimental realization of the Hubbard model leading to the demonstration of the superfluid to Mott insulator transition in a Bose condensate of
87Rb atoms [9].
Although the theory and experiment about the excitation and free expansion have been studied for a long time and many results were known, the studies about dynamics of BEC in optical lattice are still not been well discussed. The properties of elementary excitations can be investigated by considering small deviations of the state of the gas from equilibrium and finding periodic solutions of the time-dependent Gross-Pitaevskii equation.
Bose-Einstein condensate in the optical lattice can be considered as the giant matter wave moving in the periodic potential. Therefore it will have group velocity, effective mass and the band structure. We put the BEC into the one dimension optical lattice to see its excitation mode and expansion behavior. Furthermore we want to see how the group velocity υg and effective mass m* affect these dynamic behaviors to have better understanding the dynamic of the BEC in optical lattice.
1.3 Organization of the thesis
In the thesis, we first introduce the Gross-Pitaevskii equation and method of effective mass (K• P perturbation) method to apply to the BEC in optical lattice and the basic theory about the excitation and free expansion in hydrodynamic approach in the Chapter 2. After the discussion we get the 1D G-P equation of BEC in optical lattice and then get the differential equation for the density fluctuation and velocity.
Then in Chapter 3, we try to get the analytic solution of excitation modes. The current density and velocity in free expansion of the condensate are also solved in this Chapter. Finally we get some conclusion and further discuss the results.