טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
Ph.D Thesis
Ph.D StudentRahav Saar
SubjectBilliards in Atom Optics
DepartmentDepartment of Physics
Supervisor Professor Emeritus Shmuel Fishman (Deceased)


Abstract

Several topics related to the implementation of billiards, where the walls are formed by rapidly moving light beams and the particles are cold atoms, are studied.  The classical and quantum dynamics in a high frequency field are found to be described by an effective time independent Hamiltonian. It is calculated in a systematic expansion in the inverse of the frequency (ω) to order ω-4. The work is an extension of the classical result for the Kapitza pendulum, which was calculated in the past to order ω-2. The analysis makes use of an implementation of the method of separation of time scales and of a quantum gauge transformation in the framework of Floquet theory. The effective time independent Hamiltonian enables one to explore the dynamics in presence of rapidly oscillating fields, in the framework of theories that were developed for systems with time independent Hamiltonians. The spectral properties of integrable billiards with localized perturbations are studied. The statistics of energy levels of a rectangular billiard, which is perturbed by a strong localized potential, are studied analytically and numerically, when this perturbation is at the center or at a typical position. The form factor, that is the Fourier transform of the energy-energy correlation function, is calculated analytically, in the framework of the semiclassical geometrical theory of diffraction, and numerically. Contributions of classical orbits that are non diagonal are calculated and are found to be essential. The resulting statistics is intermediate between those of integrable and chaotic systems.