Ultracold matter opens a gate to the manipulation with a new quantum phase of matter realized by all the atoms populating the same quantum state and sharing coherently the same macroscopic wavefunction. It is one of the most challenging research areas in physics nowadays. Ultracold bosonic and fermionic quantum systems are playgrounds for probing fundamental condensed-matter physics problems, phenomena in quantum optics and quantum information processing, understanding atomic and molecular physics, simulating basic phenomena in nonlinear dynamics including nonlinear optics, hydrodynamics etc. The main reason for this is the possibility to control internal atom properties and interactions between atoms by externally changing the potential in which the cold atom system resides. We mainly theoretically study the dynamics of dipolar Bose-Einstein condensates in deep optical lattices in the mean field approximation to interpret observed and predict new phenomena in the experiments and to model different nonlinear phenomena in photonic systems.

Bose-Einstein condensation is a quantum phenomenon proposing occupation of the lowest quantum state by ensembles of identical bosons at very low temperature. Formed boson condensate can be represented as a fluid in the mean-field approximation accurately described by the Gross-Pitaevskii (GP), i.e. the nonlinear Schrodinger equation (NLSE). Due to the two-body interactions, the Bose-Einstein condensate (BEC) is intrinsically a nonlinear system with advantage over other nonlinear systems consisting in easy experimental manageability by means of the Feshbach resonance, variation of the effective atomic mass and the stability properties with help of optical lattice (OL), and preparation of suitable initial conditions by phase and density engineering. Sensitivity of the BEC system to external magnetic and electromagnetic fields implies easy control of its properties.

Our study is mainly developed to the investigation of the creation, dynamical and symmetry properties of diverse localized structures and periodic patterns in dipolar BECs in one-dimensional (1D) and two-dimensional (2D) very deep OLs. We theoretically model the corresponding cigar-shaped and pancake-shaped BEC as an array of dipolar droplets distributed inside the OL potential wells by the set of difference-differential 1D and 2D the GP equation with cubic nonlinearity, or the nonpolinomial nonlinear Schrodinger equation (NPSE), respectively. By numerically solving the model equations we found and analyzed properties of the localized solutions of the discrete solitary type: fundamental on-site and inter-site, unstaggered and staggered solitons in the 1D system; fundamental on-site, inter-site, hybrid modes and corresponding ones on the finite background, as well as vortex solitons with different topological charge in the 2D BEC. The BEC systems with contact and dipole-dipole (DD) interaction of both attractive and repulsive nature, as well as isotropic and anisotropic character of the DD interaction are investigated. Let mention the main findings: the nonlocal DD interaction can enlarge the localized mode existence and stability areas in some circumstances and suppress the collapse instability under certain combinations of the attractive contact and attractive isotropic DD interactions in the dipolar BECs. In general, we shown that the presence of the long-ranged DD interactions is crucial for interpretation of many previously observed experimental findings in BECs with Chromium, Dysprosium and Erbium atoms.

In addition we discovered the stable propagating double and triple periodic patterns with respect to the period of underlying lattice in dipolar cigar-shaped BEC owing to the DD interaction among the BEC droplets reminding to the charge-density waves in the solid state context. We found that the double- and triple-periodic patterns emerge via phase transitions of the second and first kind, respectively, and that they can be stable if the DD interactions are repulsive and sufficiently strong, in comparison with local repulsive nonlinearity. Involving the droplet dynamics inside the OL wells into the calculation we obtained two normal modes propagating on top of the stable uniform background: high-frequency (‘optical phonon’) mode and low-frequency (‘acoustic phonon’) mode, the first being created only in the presence of the repulsive DD interaction.

In collaboration with the group at LENS in Florence, Italy, we experimentally investigated the problems of the BEC management on atom chips by light and magnetic fields and realization of a multi-path interferometer that uses RF-coupling between the Zeeman states. The corresponding theoretical model for BEC on atom chip with integrated photonic and metal-wire components has been developed. These new perspectives in the coherent manipulation of matter waves could be ultimate goal of atomic optics, particularly for chip-based atomic interferometers with BECs.

Taking the mathematical model from the BEC studies as a base we modeled the light propagation through the 1D photonic lattice with local and nonlocal nonlinearity by the difference-differential Salerno equation and searched for events characterized by very high amplitude, so called extreme events (EE). The preferable conditions for their creation are shown to correspond to the weakly nonintegrable case, where the transport of the energy among the lattice sites was enhanced by the nonlocal nonlinearity. The next step in the field of the EEs was the study of the EE in the context of the light or matter wave propagation in the 2D disordered nonlinear OL. Transient and persistent EEs are statistically investigated and their formation and dynamical properties are correlated with the origins of the localization phenomena. We intensively continue studying the exotic EEs in the context of light and matter waves which would be bases for improving the predictability and control of the similar events in water-waves media, telecommunication channels and atomic-lasers systems.

The ultra-cold quantum gases are also used for emulation of fundamental effects in condensed-matter physics, such as the spin-orbit coupling (SOC). In cotranst to the complex situations in solids, the 'synthetic' SOC, induced by appropriate laser illumination of atomic gases in the combination with a magnetic field, can be precisely controlled in the experiment. Therefore, one of our research interests is focused on studying the SOC phenomena in the ultra-cold quantum gases.