1-1 Preface
Bose-Einstein condensates were predicted in 1924 by S. N. Bose and A. Einstein.
S. N. Bose, an Indian physicist, worked out the statistics for photons. A. Einstein applied Bose’s statistical model to predict that almost all of the particles in a Bosonic
system would congregate in the ground state at an ultra-low temperature.
Particles, in this case, are referred to atoms or molecules. They are bosonic (fermionic) if they have integer (half-integer) spin, or equivalently, if the total number of electrons, protons, and neutrons they contain is even (odd). For fermions, the Pauli exclusion principle prevents two particles from occupying the same quantum state, whereas, for bosons, the probability of finding particles in the same quantum state increases dramatically and satisfy the distribution of Bose-Einstein statistics given by
( ) /
( ) 1
1
E k TB
N E = e −µ
−
where k is Boltzmann’s constant, B µ is the chemical potential, is temperature and is the energy of particles. As a gas of bosonic particles is cooled to below a critical temperature and are all in ground state, De Broglie’s matter waves are comparable to the distance between particles, the individual wavepackets start to overlap and then these waves start to oscillate in coherent. These bosonic particles are called Bose-Einstein condensates (BECs). Since all atoms of BECs occupy the same ground state energy, the many-body wave function is then the product of identical single-particle where is the number of condensate atoms. This single-particle wave function is therefore called the condensate wave function or
T E
N N
macroscopic wave function. If the number of atoms is high, BECs exhibit cubic nonlinearity which is equivalent to Kerr nonlinearity in optics due to interatomic forces of condensate atoms, characterized by the s-wave scattering length a s (typically 1 to 5 nm for alkali atoms). In a gas, the separation between atoms n−1 3 is much larger than the effective range of the interatomic forces, i.e. the quantity
. This inequality expresses that binary collisions are much more frequent than three-body collisions. It is in this limit that the theory of the weakly interacting Bose gas applies. Under weak atomic interaction condition, condensate wave functions of BECs in optical lattices can be described by the Gross-Pitaevskii (G-P) equation, or equivalently, the nonlinear Schrödinger equation with periodic potentials.
3 1
nas
BECs could not be observed until cooling techniques were developed to reach such a low temperature, because creation of BECs accompany with ultra-low temperature (around billionths of a degree above absolute zero). S. Chu, C.
Cohen-Tannoudji and W.D. Phillips who won the Nobel Prize in Physics in 1997 developed methods to cool and trap atoms with laser light. This achievement accomplishes the first observation of BEC in dilute alkali gas in 1995 by E.A. Cornell, W. Ketterle and C.E. Wieman who won the Nobel Prize in Physics in 2001. With successful experimental observation of BECs, many physical properties of BECs can be investigated further such as loading BECs in optical lattices [1] generated by interference of laser beams. The first experiments involving the dynamics of BECs in periodic potentials were carried out by Anderson and Kasevich, who used this approach to demonstrate a mode-locked atom laser [2], and observe atomic Josephson oscillations [2,3]. In addition, the properties of coherent macroscopic matter waves in a lattice, such as the Bloch-band structure [4], macroscopic interference effects [2],
Bloch oscillations and Landau–Zener tunnelling [5], have been explored in a number of experiments.
Due to the potential wells being separated by a finite distance, atoms can tunnel between adjacent wells. BECs in optical lattices are affected by the structures of optical lattices. The BEC spectrum has a band-gap structure [6], no BEC states can exist within the band gap in a linear regime, where the number of atoms is low. If the number of atoms is high, BECs behave nonlinearly. As the nonlinear term in the wave equation exactly compensates for wavepacket dispersion, solitons occur.
There are spatially localized nonlinear BEC states, called gap, or equivalently bright, solitons which exist within the band gaps [7]. As the energy is in band, there exist dips on condensate density and a sharp phase gradient of the wave function at the position of the minimum and are called dark solitons. Without optical lattices, bright solitons of BECs can only exist with attractive nonlinearity (as< ) below a certain 0 number of atoms [8]. On the contrary, dark solitons can only exist under repulsive nonlinearity ( ). However, bright and dark matter wave solitons is stable under both attractive and repulsive nonlinearity and have no restriction of number of atoms in BECs due to the periodicity of the optical lattice which leads to the effective dispersion of the BEC wavepakets deduced from the band structure.
s 0 a >
1-2 Motivation
Since the governing equation of BECs in optical lattices, G-P equation, is nonintegrable, several different theories whose accuracy depends heavily on the nature of the underlying problem are used to find the approximated solutions. The
tight binding approximation [9] is only accurate when the potentials are deep and well separated. The coupled-mode theory is valid when the energies are close to the gap and shallow potentials. An accurate solution of solitons can only be obtained by exactly solving the full nonlinear Schrödinger equation with a periodic potential.
Louis et al. [10] analyzed numerically the existence and stability of spatially extended and localized states of BEC loaded into an optical lattice. They demonstrated the existence of families of spatially localized matter-wave solitons existing at gaps.
Efremidis et al. [11] studied numerically the properties of gap solitons in BECs with either attractive or repulsive atom interaction. They found families of gap solitons, which are characterized by the position of the energy eigenvalue within the associated band structure. However, numerical simulations of BECs in optical lattices are highly computationally intensive, we describe Bose-Einstein condensates in one-dimensional optical lattices by introducing the effective-mass theory which has been extended to study semiconductor superlattices successfully without the need for full-scale numerical calculations. Pu et al. [12] have obtained an effective equation of motion governing the time evolution of the envelope of the condensate wave function in which the periodic external potential appears in the form of an effective mass. Numerical calculation confirmed that this envelope function approach provides us with useful qualitative insight into the condensate dynamics.
In the Thesis, we study the nonlinear effect of BECs in one-dimensional optical lattices, numerically and analytically. With a little modification, we add a term, band energy at a specific wave-vector, to the effective mass equation [12] of the envelope wave function of BECs in which the periodic external potential appears in the form of an effective mass. Accordingly, we obtain analytical solution of the wave function
of soliton solution which is a Bloch function from periodicity times the envelope function of the nonlinear effective-mass Schrödinger equation. We will theoretically observe the appearance of bright and dark solitons correspond to energy in and out-of band gap under different sign of the s-wave scattering length a . The analytic s solutions of BECs are compared with the numerical results of the G-P equation. We find that BECs in optical lattices can be described, qualitatively and quantitatively, by the effective-mass theory.
1-3 Organization of the Thesis
This Thesis is organized as follows. Chapter 2 gives a brief review of effective mass theory and a deduction of the effective mass of BECs in optical lattices. The plane wave method is applied to study the linear regime of BECs in optical lattices to obtain the band structure. In Chapter 3, the one-band effective mass theory of a G-P equation is introduced which gives analytic bright/dark soliton solution. We give the comparisons between analytic and numerical results. Comparisons between analytic and experimental results reported by Eiermann et al. are also included. We give a brief conclusion in Chapter 4.