טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
Ph.D Thesis
Ph.D StudentBakman Alexandra
SubjectThe Impact of Periodicity and its Breaking on
Wavepacket Dynamics
DepartmentDepartment of Physics
Supervisor Professor Emeritus Nimrod Moiseyev
Full Thesis textFull thesis text - English Version


Abstract

Periodicity and its breaking are responsible for many fascinating phenomena in physics. In this work, we analyze wavepacket dynamics in systems where periodicity plays an important role.

One such phenomenon is the collapse and revival of observables, created by an anharmonic term in an otherwise harmonic Hamiltonian. This simple model can help understand more complex scenarios, since the local potential minima can be approximated as a harmonic potential with a small anharmonic correction. In a harmonic potential, a coherent state oscillates periodically in position and momentum without changing its shape. However, when we introduce a weak anharmonic term, this periodic behavior changes and, in addition, the envelope of the state, as well as the envelope of the momentum and position observables, collapses and revives periodically, with a characteristic revival time inversely proportional to the strength of the anharmonic term.

In another phenomenon, namely the Talbot effect, the periodicity of initial conditions determines the system behavior. A plane wave incident on a periodic grating, creates after it an intricate intensity pattern, which recreates the wavefront just after the grating at integer multiples of twice the Talbot distance. The Talbot effect is a coherent wave interference effect, and thus is not confined to light waves. We recreated this effect with surface water waves analytically, numerically and experimentally, and saw good agreement between the results. The effect can be observed in the naked eye with water waves, which offer several unique advantages in such experiments. Due to their slow velocity and macroscopic wavelengths, their propagation can be observed with standard cameras. In this experiment, part of the wavelengths we used did not satisfy the paraxial condition, a fact that introduced corrections to the Talbot pattern, namely splitting of foci, and broke the periodicity of the paraxial Talbot effect. For smaller wavelengths, the splitting disappeared, and the intensity pattern of the Talbot effect was restored.

Lastly, an interesting phenomenon studied here is the creation of edge states in the three dimensional kicked rotor system, analogous to surface states formed in crystals, where periodicity breaking creates a new family of states. In this work, we study the edge states in angular momentum space in a three-dimensional kicked rotor, a paradigm system for studying quantum effects in a classically chaotic system. We study the system in a regime of fractional quantum resonance, which has no analog in classical physics. In this regime, the energy of the rotor grows quadratically with the number of kicks, since an initial wavefunction explores higher angular momentum states as time progresses. However, since the angular momentum quantum number cannot be negative for a three-dimensional kicked rotor, it creates a quasienergy edge state, which decays exponentially away from the edge at zero angular momentum. In order to study the mechanism of the creation of this state, we used a tight binding analog from solid state physics. In the derivation of the edge state energy we used perturbation theory, assuming weak kicking strength. However, we believe that this mechanism is relevant also in the non-perturbative regime.