Abstract
This thesis includes two parts containing theoretical investigations about quantum transport problems in some bosonic, and fermionic systems.
The focus of the first part is to study Anderson localization in systems described by Hamiltonians with symplectic symmetry. Such systems are realized if spin-orbit coupling is present. For classical particles, the motion of a particle in a disordered medium is known to be diffusive. However, in quantum mechanics, due to the wave nature of the particles, interference effects play a major role, which manifest themselves in weak localization and Anderson localization phenomena.
In weak localization, interference phenomena give rise to corrections in the classical Drude-Boltzmann theory, nonetheless the system is still diffusive. On the other hand, under certain circumstances, the interference may totally suppress the diffusion and cause the energy eigen-functions to become localized, which is known as Anderson localization or strong localization. The transition from the diffusive (metallic) state to the localized (insulating) state is called the Anderson metal-insulating transition (MIT).
Recent experimental developments in cold atoms have provided the opportunity to access direct observables, like the distribution of the position and velocity of the particles. According to a series of works by Ghosh et al., the signatures of the Anderson transition are evident through the momentum distribution of plane waves scattered in a disordered medium. In momentum space, after propagation of a plane wave, two extrema are found. One is related to coherent back-scattering process (CBS), which is a signature of both weak and strong localization. The long time dynamics of the width of CBS profile characterizes the bulk phase of the system, whether it is in the insulating or the metallic regime. On the other hand, coherent forward-scattering (CFS) exists only in the insulating regime, where strong localization happens.
In the first section, we study these signatures for symplectic symmetry class in two dimensions. In order to prove the universality of the momentum signatures, an ensemble of random Hamiltonians are selected which only preserve the necessary symmetry, i.e., time reversal with broken spin rotation symmetry.
In the following section, I intend to answer a question arising from the symmetry dependence of the scaling function. As is confirmed through the literature, the scaling function of the conductance of a disordered hyper-cubic channel only depends on the dimension and the symmetry class of the Hamiltonian. Assuming the symmetry class changes by a continuously adjustable parameter. How does the scaling function suddenly jump from one type to another one even for an infinitesimal symmetry changing parameter? I answer this question by investigation of a system in which the symmetry changes from orthogonal to symplectic by adding a spin-orbit coupling interaction whose strength can be adjusted through a parameter called the spin-orbit coupling strength S. I will show that the symmetry dependence is violated in a cross-over regime for finite size systems, but it will be recovered for large enough sizes even for infinitesimal values of S.
The second part of the thesis is dedicated to vortices, their dynamics in Bose-Einstein condensates, and their application in atomtronic devices. Atomtronics is an emerging field that exploits coherent matter waves instead of electrons in an electronic circuit. Vortices and their dynamics have been extensively used in atomtronic devices, such as rotation sensors. In such systems, vortices are created by stirring a BEC trapped in a ring-shaped potential by a weak barrier. There is a critical rotation velocity of the stirrer at which the vortices are created, and this phenomena can be exploited as a criteria for measuring the rotation velocity. In an ideal scenario where the temperature is low enough that the effect of the non-condensed atoms is negligible, the system can be modeled by the Gross-Pitaevskii equation (GPE).
We present an algebraic equation to estimate this critical velocity instead of solving the partial differential equation GPE.
Later, finite temperature effects are studied in the framework of Hartree-Fock Bogoliubov approximation, and we conclude that the critical rotation velocity is robust against temperature fluctuations, due to the stabilising effect of the thermal cloud.
Throughout this thesis, we solely focus on the achievements obtained by the author, and in the case of necessity, the background theories are introduced briefly. The interested readers can consult the references cited in different sections. Furthermore, the cumbersome calculations, that may create inconvenience while reading the thesis, will be explained in the Appendix.