### Abstract:

The invention of laser cooling and trapping techniques for neutral atoms
provides a platform to visualize quantum effects associated with matter. In
addition, this technique lends control and manipulation of the factors that
re flect changes in the dynamics of a system in quantum regime. One of such
examples is a kicked rotor system. Particles in a delta kicked spatially periodic
potential, mimics this system. The dynamical evolution of this system
in the quantum regime is very different from that in a classical regime. Average
energy of a classical system increases linearly as the number of kicks
increases whereas, for a quantum system, the average energy saturates to a
certain limit. This key feature of saturation in a quantum system is caused
by suppression of momentum diffusion. At this point, the system is analogous
to a crystal with defects that shows localization of electrons near lattice
sites, popularly known as Anderson localization. The energy-time relation of
a kicked rotor system can be mapped to the position-momentum relation of
such a defective crystal. This deviation in energy evolution can be considered
as a 'meter-scale' to understand the transition from quantum to classical
regime. Small amount of noise coupling to a quantum system leads to decoherence
in the evolution of the system. For a classical system the energy
gained by the system does not saturate, instead, it increases linearly with
increase in number of kicks. By increasing the amount of noise, the system
can undergo a complete transition to the classical regime. The noise could
be generated by various means such as noise in amplitude of each kicks, fluctuation
in the periodicity of the kicks or any other mechanism which causes
loss in coherence. It has been predicted that noise in the periodicity of kicks,
where the periodicity follows Lèvy's statistics the decoherence is suppressed.
In fact, in certain parameter space, the decoherence-time increases to in finity.
This is very important to prepare decoherence free system for experiments
such as quantum information processing and quantum emulation. The use
of ultra-cold atoms is very important because as the temperature goes down,
the momentum distribution confi nes to very small range. The corresponding
wavelength of the particles becomes large enough to observe the interference
effect and therefore the atoms' energy reaches a state where quantum behaviour
dominates. Red detuned counter propagating laser beams can be
used to create periodic optical potentials. Fast ashes of such field, given on
ultra-cold atoms exactly mimics a kicked rotor system. This system is very
simple to study the dynamical evolution with the additional noise. We trap
Rb atoms in Magneto-optic Trap and further cool them down by Sysiphus
cooling to prepare particles for the experiment. At temperature below 10
microK, ashes of standing wave are applied to the atoms which leads to redistribution
of momenta of the atoms. After certain time of free evolution,
atoms are imaged to observe the momentum distribution and consequently
the measurement of the energy. In this thesis, we report the progress towards
experiments to study the evolution of system in which periodicity of the kicks
follows Lèvy's statistics.