טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentShmuel Osovski
SubjectFingerprints of Classical Chaos in Manipulation of Cold
Atoms in One-dimensional Optical Lattices
DepartmentDepartment of Chemistry
Supervisor Professor Emeritus Moiseyev Nimrod


Abstract

The recent pioneering experiments of the Phillips [Nature, 412, 52 (2001) ] and Raizen [Science, 293, 274 (2001) ] groups have demonstrated the dynamical tunnelling of cold atoms interacting with standing electromagnetic waves,creating a 1D optical lattice. Our calculations showed, that one can achieve a control of the tunneling period over orders of magnitude range, simply by changing the frequency difference of the waves by about 10% only. In this narrow parameter region, the mechanism of the tunnelling oscillations evolves from the two-state to the three-state one. It was demonstrated that the change in the underlying mechanism leads to a dramatic enhancement of the dynamical tunnelling. Moreover, a complete suppression of the dynamical tunnelling can also be achieved. In both of the experiments the effective ħ is large and the quantum system is far from its semi-classical limit. Therefore, one may wonder, whether there are fingerprints of the classical mixed phase space in the quantum dynamics.

It is shown that despite of the large effective ħ, the quantum Floquet-Bloch quasienergy states can still be classified as regular and chaotic states, by the calculations of Shannon’s entropy . Husimi distributions of the Floquet-Bloch states show that the regular states are exponentially localized at regular islands located around fixed points in the mixed phase space. In both experiments the quantum and the classical phase-space entropies are quite similar, although the classical phase space is a mixed regular-chaotic space. In both experiments a decay of the mean momentum amplitude, < P(t) > was measured. An explanation is provided through a study of the band structure of the Floquet-Bloch quasi-energy spectrum. It was found that the oscillations in the mean momentum, is very sensitive to the population of the k = 0 Bloch quasi-momentum state, and appear if and only if the k=0 Bloch quasi-momentum state is populated by more than 99%. This result is in complete harmony with the experimental observation. The random phase distribution of the population probability amplitudes of the k= 0 Bloch quasi-momentum states by the initial wavepacket causes the decay of the mean momentum amplitude. The random phase distribution of the probability amplitudes are in a way, a fingerprint of the classical chaos in the quantum dynamics. This kind of fingerprints of chaos in quantum-mechanics is novel and has been first indirectly measured in the NIST and AUSTIN experiments.