One of the fundamental assertions of statistical mechanics is that the time average of a physical observable is equivalent to the average over phase space, with microcanonical measure. A system for which this is true is said to be ergodic and dynamical properties can be calculated from static phase-space averages. Dynamics of a system which is fully integrable, that is has as many conserved quantities as degrees of freedom, is constrained to a reduced phase space and thus not ergodic, although it may relax to a modified equilibrium. In this thesis, we present a comprehensive study of chaos and thermalization of the one-dimensional Bose-Hubbard Model BHM) within the classical field approximation. This model describes the dynamics of quantum degenerate gases in a lattice for sufficient occupation of every momentum mode and weak two-body scattering, and is of interest because of experimental advances of cooling and trapping alkali atoms in the quantum degenerate regime. We study the chaos and its relation to thermalization. Two quantitative measures are compared: the ensemble-averaged Finite-time Maximal Lyapunov exponent, a measures of chaos and the normalized spectral entropy, a measure of the distance between the numerical time-averaged momentum distribution and the one predicted by thermodynamics. A threshold for chaos is found, which depends on two parameters, the nonlinearity and the total energy-per-particle. Below the threshold, the dynamics are regular, while far above the threshold, complete thermalization is observed, as measured by the normalized spectral entropy. We study individual resonances in the Bose-Hubbard model to determine the criterion for chaos. The criterion based on Chirikovs method of overlapping resonances diverges in the thermodynamic limit, in contrast to the criterion parameters inferred from numerical calculations, signifying the failure of the standard Chirikovs approach. The Ablowitz-Ladik lattice is one of several integrable models that are close to the BHM. We outline the method of Inverse Scattering Transform and generate the integrals of motion of the Ablowitz-Ladik lattice. Furthermore, we discuss the possible role of these quantities in the relaxation dynamics of the BHM.

Theoretical studies using the Kohn-Sham density functional formalism have been carried out to identify and investigate the stability of a variety of atomic clusters for their use in cluster assembled materials. The stable behavior found in a cluster system provides a way to classify inorganic clusters. The clusters in this study can be categorized in one of the following, jellium, all-metal aromatic, Zintl analogue or as a covalent metal-carbide. By understanding the electronic structure and ultimately the stable nature of a cluster first, it is proposed one can construct assemblies based on the stable cluster. The methodology presented is a viable way to design future nanomaterials with a variety of architectures and precise control over properties based on stable cluster motifs.

Hyperpolarized noble gases are used in a variety of applications including medical diagnostic lung imaging, tests of fundamental symmetries, spin filters, atomic gyroscopes, and atomic magnetometers. Typically 3He is utilized because large 3He polarizations on the order of 80% can be achieved. This is accomplished by optically pumping an alkali vapour which polarizes a noble gas nucleus via spin exchange optical pumping. One hyperpolarized noble gas application of particular importance is the K-3He co-magnetometer. Here, the alkali atoms optically pump a diamagnetic noble gas. The magnetic holding field for the alkali and noble gas is reduced until both species are brought into hybrid magnetic resonance. The co-magnetometer exhibits many useful attributes which make it ideal for tests of fundamental physics, such as insensitivity to magnetic fields. The co-magnetometer would demonstrate increased sensitivity by replacing 3He with polarized 21Ne gas. Tests of CPT violation using co-magnetometers would be greatly improved if one utilizes polarized 21Ne gas. The sensitivity of the nuclear spin gyroscope is inversely proportional to the gyromagnetic ratio of the noble gas. Switching to neon would instigate an order of magnitude gain in sensitivity over 3He. In order to realize these applications the interaction parameters of 21Ne with alkali metals must be measured. The spin-exchange cross section sigmase, and magnetic field enhancement factor kappa0 are unknown, and have only been theoretically calculated. There are no quantitative predictions of the neon-neon quadrupolar relaxation rate Gammaquad. In this thesis I test the application of a K-3He co-magnetometer as a navigational gyroscope. I discuss the advantages of switching the buffer gas to 21Ne. I discuss the feasibility of utilizing polarized 21Ne for operation in a co-magnetometer, and construct a prototype 21Ne co-magnetometer. I investigate polarizing 21Ne with optical pumping via spin exchange collisions and measure the spin exchange rate coefficient of K and Rb with Ne to be 2.9 x 10-20cm 3/s and 0.81 x 10-19cm3/s. We measure the magnetic field enhancement factor kappa0 to be 30.8 ＋/- 2.7, and 35.7 ＋/- 3.7 for the K-Ne, and the Rb-Ne pair. We measure the quadrupolar relaxation coefficient to be 214 ＋/- 10 Amagat&dot；s. Furthermore the spin destruction cross section of Rb, and K with 21 Ne is measured to be 1.9 x 10-23cm2 and 1.1 x 10-23cm2.

The formation of K＋&middot；&middot；&middot；Cl- heavy-Rydberg ion-pair states through dissociative electron attachment in K(np)/CCl4 collisions is investigated. The product ion-pair states are detected directly by electric field-induced dissociation. A Monte Carlo collision code is used to analyze the results that models both the initial Rydberg electron attachment and the subsequent evolution of the ion pairs. The data and the calculations demonstrate the production of long-lived bound K＋&middot；&middot；&middot；Cl- heavy-Rydberg ion pairs. They also suggest that collinear collisions can generate heavy-Rydberg states of low angular momentum.

Two methods of locating resonances of the e — Ps (e- — e – — e＋) atomic three-body system are presented. In the first method the homogeneous Faddeev-Merkuriev integral equations are solved by applying a separable expansion approximation on the potential terms in the Coulomb-Sturmian basis. This approximation transforms the integral equations into a matrix equation. The Coulomb-Sturmian matrix elements of the three-body Coulomb Green’s operator are then calculated as a contour integral of the two-body Coulomb’s Green matrices. The calculation of the Coulomb Green’s matrices are considerably simplified by the use of continued fractions that result from a tri-diagonal Jacobi matrix. The complex energies are searched for as the complex zeros of the Fredholm determinant. In the second method the scattering matrix S is computed by means of resolvent Green’s operators. The eigenphase sum is then calculated as a function of energy and the resonances appear as singularities of the plot of the eigenphase sum formula.

Nonlinear magneto-optical processes are a rich source of interesting and useful phenomena, with both practical and fundamental-physics applications. Theoretical modeling is helpful for understanding and visualizing the mechanisms for nonlinear magneto-optical effects (NMOE), and for analyzing and optimizing devices based on these effects. Part I of this Thesis describes Bloch-equation methods and visualization techniques that can be used to model a wide variety of NMOE in atomic vapors. Part II presents several applications of the methods, including the investigation and visualization of a specific effect involving radio-frequency fields, a study of the general consequences of hyperfine structure on NMOE, and modeling and optimization of systems for laser guide stars. Appendices present additional mathematical material and describe a Mathematica package used for density-matrix calculations.

The experimental success in ultra-cold atomic gases, both bosonic and fermionic have boosted the theoretical studies, and especially the a lot of numerical techniques have been developed and used to describe them. In this thesis, we introduce two numerical experiments in our group on ultra-cold atomic gases. The first concerns the scalar dipolar condensate. We have developed and implemented a Split-Step Fourier scheme in imaginary time, which enable us to seek the ground state of the dipolar condensate. The second part is focused on our ongoing efforts to investigate the trapped spin polarized Fermi gas using self-consistent Bogoliubov-de Gennes (BdG) calculation.

The broad objective of ultrafast strong-field studies is to be able to measure and control atomic and molecular dynamics on a femtosecond timescale. This thesis work has two major themes: 1) Study of high-energy photoelectron distributions from atomic targets. 2) Electron localization control in atomic and molecular reactions using shaped laser pulses. The first section focuses on the study of photoelectron diffraction patterns of simple atomic targets to understand the target structure. We measure the full vector momentum spectra of high energy photoelectrons from atomic targets Xe, Ar and Kr) generated by intense laser pulses. The target dependence of the angular distribution of the highest energy photoelectrons as predicted by Quantitative Rescattering Theory QRS) is explored. More recent developments show target structure information can be retrieved from photoelectrons over a range of energies, from 4Up up to 10Up, independent of the peak intensity at which the photoelectron spectra have been measured. Controlling the fragmentation pathways by manipulating the pulse shape is another major theme of ultrafast science today. In the second section we study the asymmetry of electron and ion) emission from atoms and molecules) by interaction with asymmetric pulses formed by the superposition of two colors 800 & 400 nm). Xe electron momentum spectra obtained as a function of the two-color phase exhibit a pronounced asymmetry. Using QRS theory we can analyze this asymmetric yield of the high energy photoelectrons to determine accurately the laser peak intensity and the absolute phase of the two-color electric field. This can be used as a standard pulse calibration method for all two-color studies. Experiments showing strong left-right asymmetry in D＋ ion yield from D2 molecules using two-color pulses is also investigated. The asymmetry effect is found to be very ion-energy dependent.

The collisional dynamics of the 62P levels in cesium have been studied utilizing steady state laser absorption and laser induced florescence techniques. In addition the production of a blue beam produced by two photon absorption has been observed in potassium. The collisional broadening rate for cesium, gammaL, for He, Ne, Ar, Kr, Xe, N2, H2, HD, D2, CH4 , C2H6, CF4, and 3He are 24.13, 10.85, 18.31, 17.82, 19.74, 16.64, 20.81, 20.06, 18.04, 29.00, 26.70, 18.84, and 26.00 MHz/torr, respectively for the 62 P1/2 &rarr； 62S1/2 transition and 20.59, 9.81, 16.47, 15.54, 18.41, 19.18, 27.13, 28.24, 22.84, 25.84, 26.14, 17.81, and 22.35 MHz/torr for the 62 P3/2 &rarr； 62S1/2 transition. The corresponding pressure-induced shift rates, delta, are 4.24, -1.60, -6.47, -5.46, -6.43, -7.76, 1.11, 0.47, 0.00, -9.28, -8.54, -6.06, and 6.01 MHz/torr for the 62P 1/2 &rarr； 62S1/2 transition and 0.69, -2.58, -6.18, -6.09, -6.75, -6.20, -4.83, -4.49, -4.54, -8.86, -9.38, -6.47, and 0.60 MHz/Torr for the 62P 3/2 &rarr； 62S1/2 transition. These values have been compared with the values of other alkalis and the inter-atomic difference potentials have been determined using the impact approximation. The energy exchange rates between the two excited states of cesium by collisions with N2, H2, HD, D2, CH4, C 2H6, CF4, and C2F6 have been measured and shown to correlate with both the rotational energy defect and the vibrational energy defect. And finally, while pumping from the ground state 42S to the excited 5 2D and 62S states of potassium, a blue beam corresponding to the transition 52 P &rarr； 42S was observed. The effects of input power and buffer gas pressure where observed.

In this thesis, we present work on coherent control of multilevel quantum systems in the strong field limit using shaped ultrafast laser pulses. In recent years there have been numerous multiphoton absorption experiments in two, three, and four-level atomic/molecular systems and many are performed in the limit of weak fields where perturbation theory is valid. Here, we describe a series of experiments aimed at exploring and understanding multiphoton transitions when the exciting field is strong and perturbation theory breaks down. Our approach to strong field control utilizes both parameterized scans of various pulse shapes and closed-loop learning control to identify a pulse shape that is optimal for populating a target quantum state. With this we will highlight the difference between sequential population transfer and adiabatic rapid passage in multilevel systems with multiphoton coupling between levels. Additionally, we examine strong field control of a four-level atomic interferometer and show how interference in a target state changes from resonant pathways in the frequency domain to time-domain interference via a singe path. Further, we use shaped femtosecond pulses to demonstrate a phenomenon in which a three-level atom becomes a modulator of an ultrafast pulse. The results are based on a pump-probe scheme that is very similar to Electromagnetically Induced Transparency EIT). Important dynamics associated with a time-dependent coupling field are examined. Lastly, we extend previous work on two-photon driven superfluorescence from a shaped ultrafast drive laser and show how stimulated emission near threshold can turn modest coherent control yields into essentially perfect discrimination between systems where a control factor of about 104 is achieved between atomic and molecular species.