Quantum dynamics of a Bose-Einstein condensate in a 1D optical lattice

In this work we employ the truncated Wigner approximation (TWA) to numerically investigate the non-equilibrium quantum dynamics of Bose-Einstein condensates (BECs) in an optical lattice. First, the basic concepts of laser cooling and condensation in ultra-cold alkalii-metal atoms are introduced.
We revise the main trends in the research on condensates trapped by light and illustrate the physics of tightly confined 1D bosonic systems. We discuss Bogoliubov theory, employed to generate the initial state of the BEC in thermal equilibrium both at zero and at finite temperature, and we describe the application of the TWA to the interacting condensates. Emphasis is then given to the numerical techniques used to solve Gross-Pitaevskii and Bogoliubov equations, sample the Wigner-distributed initial states and unravel the quantum dynamics onto stochastic trajectories.
We study the dynamical splitting of a BEC initially in the harmonic trap by raising an optical potential. We discuss the difficulties one meets in studying the properties of the whole matter fields and their overtaking by means of extracting information for the condensate mode only. The TWA is
used to investigate the phase diffusion along the lattice and the fluctuations in the atom number in individual sites. We propose an explanation to the experimental findings about the saturation of the number squeezing observed by Orzel et. al. [Science 291, 2386 (2001)] based on the non-adiabaticity of the splitting. We study the effect of the lattice potential on the thermal properties of the condensate before rethermalization takes place. We also evolve an ideal gas in the harmonic trap raising first the non-linear interaction and subsequently the optical potential and we compare the evolution of this system to the case of an initially interacting condensate.
We then turn to the numerical study of the dipolar motion of bosonic atoms in a very shallow, strongly confined 1D optical lattice using the parameters of the recent experiment [Fertig et al., Phys. Rev. Lett. 94, 120403 (2005)]. We find that, due to momentum uncertainty, a small, but
non-negligible, atom population occupies the unstable velocity region of the corresponding classical dynamics, resulting in the observed dissipative atom transport. This population is generated even in a static vapor, due to quantum fluctuations which are enhanced by the lattice and the confinement, and is not notably affected by the motion of atoms or finite temperature.