In this Master's thesis, the behavior of ultracold, synthetically spin orbit coupled, weakly interacting, two-component Bose gases residing on Bravais lattices, is explored analytically. In particular, utilizing a mean field approach, general expressions for the chemical potentials are deduced for the superfluid phase, and non-diagonal expressions for the Hamiltonian are subsequently provided. An application of these expressions is presented for the case of a pure condensate residing on a square lattice, yielding an analytically derived phase diagram that coincides with numerical results found in the literature. Finally, future continuations of this work are suggested.
In this Master's thesis, the behavior of ultracold, synthetically spin orbit coupled, weakly interacting, two-component Bose gases residing on Bravais lattices, is explored analytically. In particular, utilizing a mean field approach, general expressions for the chemical potentials are deduced for the superfluid phase, and non-diagonal expressions for the Hamiltonian are subsequently provided. An application of these expressions is presented for the case of a pure condensate residing on a square lattice, yielding an analytically derived phase diagram that coincides with numerical results found in the literature. Finally, future continuations of this work are suggested.
A weakly interacting, spin-orbit coupled, two-component, ultracold Bose gas bound to a Bravais lattice is studied. Motivated by recent experimental advances in the field of synthetically spin-orbit coupled, ultracold, neutral atomic gases showing Bose-Einstein condensation, an analytic framework with which to describe such systems in the superfluid regime is presented. This is applied to a Rashba spin-orbit-coupled Bose gas in a two-dimensional optical lattice. The exotic nature of Bose-Einstein condensation in the presence of spin-orbit coupling is an interesting study by itself. Additionally, when the optical lattice is introduced, the system provides a highly controllable experimental testing ground for numerous condensed matter physics phenomena. Five phases of the system are considered, and their excitation spectra, critical superfluid velocities and free energies are found. In obtaining the free energy, the effects of terms in the Hamiltonian that are linear in excitation operators are included, and such terms have not been studied previously in this context. Minimization of the free energy at zero temperature is used to confirm the phase diagrams reported in the literature, where it has usually been obtained by neglecting the effect of excitations. The plane and stripe wave phases in the phase diagram are bosonic analogues of Fulde-Ferrell-Larkin-Ovchinnikov states in superconductors involving nonzero condensate momenta.
Motivated by previous work on drag effects in superfluid Bose-Einstein condensates, the Bose-Hubbard model on the square lattice is considered using mean-field theory, and a method for computing the superfluid drag density that does not rely on Galilean invariance is employed to re-derive the drag in the weakly interacting two-component Bose-Einstein condensate. The same method is then used on the Bose-Hubbard model generalized to three-components. An exact analytic expression is not possible because of the complexity of the resulting diagonalization problem. However, Rayleigh-Schrödinger perturbation theory is used to fourth order, which yields an analytic perturbative expression for the zero temperature drag, in addition to exact numerical diagonalization to find the exact behaviour at zero and finite temperatures within mean-field theory. It is found that the presence of a third component can alter the drag in a non-trivial manner: The drag between the two initial boson components can be completely mediated by the third, and it can be both enhanced and diminished depending on the inter-component interaction strengths and signs. An attempt is also made at finding the drag in a weakly interacting two-component Bose-Einstein condensate with spin-orbit coupling, but complications arise, which are discussed. Finally, a phase diagram for the two-component condensate with interactions and spin-orbit coupling is constructed, describing the spin-imbalance, degeneracy, and distribution of spin at zero temperature. A qualitative discussion of the finite temperature behaviour suggests that the inhomogeneous weakly interacting and spin-orbit coupled superfluid Bose-Einstein condensate may in some regions in parameter space favor spin-imbalance as the temperature is increased.
The excitation spectrum of a spin-orbit coupled Bose-Einstein condensate is investigated by the use of the Bose-Hubbard model and a mean-field approach. The undiagonalized Hamiltonian matrix for a general inversion-symmetric Bravais lattice is found. Analytical expressions for the excitation spectrum is then found in the limiting case of on-site interactions and nearest-neighbor hopping on a one-dimensional chain and a two-dimensional quadratic lattice. The densities of particles in the ground state acting as mean-field parameters are found self-consistently by minimization of the free energy, and they are seen to be in agreement with published literature.
Since the realization of Bose-Einstein condensation in 1995, experimentalists in ultracold atomic gas community have developed a set of entirely new methods to study quantum gases. In ultracold atomic gases, implementation of synthetic gauge fields, the spin-orbit coupling, optical lattices and Feshbach resonances, provides a clean and highly controllable platform to tune the parameters of the system, which are typically fixed for real materials. Motivated by the rapid developments of experimental progresses, I investigate the ultracold atomic gases with synthetic gauge fields. \n \nIn this thesis, a bosonic gas in optical lattices in the presence of an effective staggered magnetic flux will be studied, using a generalized Bose-Hubbard model. Phase transition from the Mott insulating phase to the superfluid phase will be discussed. Novel superfluid phases including the stripe phase and the plane wave phase will be discussed. Furthermore, the existence of a tricritical point in the phase diagram is predicted. \n \nTo study Fermi systems with synthetic gauge fields, a spin-1=2 Fermi gas with Rashba spin-orbit coupling is investigated by studying the structure of vortex line through the Bardeen-Cooper-Schrieffer to Bose-Einstein condensation (BCSBEC) crossover. Due to the spin-orbit coupling, the s-wave interacting Fermi gas behaves like an effective p-wave paired system. A discussion about the vortex line structure will be provided by calculating the spatial distribution of the gap function, the density, the particle current, and also the elementary excitations. It has been found that the most robust superfluid occurs around the unitary point, indicated by the smallest healing length and the largest critical current. \n \nFinally, motivated by the recent progress of p-wave resonantly Fermi gas in experiment, an extension of Nozières-Schmitt-Rink method to the p-wave case will be given. An investigation of the normal state properties will be given by calculating the equation of state and the p-wave contacts, which have been measured by the experiment. Furthermore, the superfluid transition temperature with realistic parameters appropriate for experiment is calculated and is found to be within experimental reach.
Two-component Fermi gases model the behavior of many systems in different fields of physics, and one of their interesting features is that they condense into superfluids at low temperatures. Ultracold atoms experiments represent one realization of such Fermi gases, and their great flexibility sparked active research into their phase structure. In the literature, there are studies of the phase structure using functional methods, mean-field approximations, and other approaches. In the present work, we aim to perform ab-initio Monte-Carlo simulations of the system to probe their phase structure for inhomogeneous phases in the presence of spin imbalance. Such simulations generally require a bosonization of the theoretical description of the system that rewrites the theory in terms of an auxiliary bosonic field. For this auxiliary field, many possible choices achieve this, and previous studies often use a field that corresponds to a density of fermions. In the present work, we develop a novel approach to this problem by bosonizing the system in terms of the so-called pairing field which corresponds to the superfluid order parameter. Even in the absence of mass- or spin-imbalance, this approach exhibits a sign problem that presents a problem to many Monte-Carlo methods. To circumvent this sign problem, we base our simulation on the Complex Langevin method. The central question of this thesis is if a simulation of a pairing-field-based formalism using the Complex-Langevin method is suitable to study the phase structure of two-component Fermi gases in the presence of spin imbalance. To this end, we develop a lattice theory based on the pairing field by discretizing the continuous Hamiltonian and performing a rigorous derivation of the path integral from there. We pay special attention to the derivation of lattice derivative operators and rescale the theory to be dimensionless. Beyond that, we develop an efficient notation for lattice theories that makes their handling arguably easier than that of continuum theories. With the obtained theory, we derive the Langevin equation for the system together with the expressions we need to sample observables and develop a numerical simulation on that basis. We use the simulation to study 0+1-dimensional systems as a proof-of-concept. Specifically, we calculate density equations of state and two-point functions for a range of dimensionless couplings and compare them to exact solutions. Additionally, we compare our obtained results for the two-point functions to general discussions of the analytic properties of correlation functions to gain additional insight into the qualitative and numerical behavior of our simulation. We found that our results for both density equations of state and two-point functions are in excellent agreement with the exact solutions. Going forward, we plan to study systems in d>0 spatial dimensions, beginning with simulations of 1+1-dimensional systems. In d=3 dimensions the pairing field's correspondence to the superfluid order parameter may allow us to efficiently study the spontaneous breaking of the U(1) symmetry of the system. On that basis, we can probe the phase diagram of the system for inhomogeneous phases at finite spin imbalance in future studies.
The dissertation explores the effects of synthetic spin-orbit coupling on the behaviour of quantum gases in several different contexts. We first study realistic methods to create vortices in spin-orbit-coupled (SOC) Bose-Einstein condensates (BEC). We propose two different methods to induce thermodynamically stable static vortex configurations: (1) to rotate both the Raman lasers and the anisotropic trap; and (2) to impose a synthetic Abelian field on top of synthetic spin-orbit interactions. We solve the Gross-Pitaevskii equation for several experimentally relevant regimes and find new interesting effects such as spatial separation of left- and right-moving spin-orbit-coupled condensates, and the appearance of unusual vortex arrangements. Next we consider cold atoms in an optical lattice with synthetic SOC in the Mott-insulator regime. We calculate the parameters of the corresponding tight-binding model and derive the low-energy spin Hamiltonian which is a combination of Heisenberg model, quantum compass model and Dzyaloshinskii-Moriya interaction. We find that the Hamiltonian supports a rich classical phase diagram with collinear, spiral and vortex phases. Next we study the time evolution of the magnetization in a Rashba spin-orbit-coupled Fermi gas, starting from a fully-polarized initial state. We model the dynamics using a Boltzmann equation, which we solve in the Hartree-Fock approximation. The resulting non-linear system of equations gives rise to three distinct dynamical regimes controlled by the ratio of interaction and spin-orbit-coupling strength λ: for small λ, the magnetization decays to zero. For intermediate λ, it displays undamped oscillations about zero and for large λ, a partially magnetized state is dynamically stabilized. Motivated by an interesting stripe phase which appears in BEC with SOC [Li et al., Phys. Rev. Lett. 108, 225301 (2011)], we study the finite-temperature phase diagram of a pseudospin-1/2 Bose gas with contact interactions. We show that strong inter-spin interactions can lead to the appearance of magnetically ordered phases at temperatures above the superfluid transition. For the case of inter-spin attraction, we also discuss the possibility of a bosonic analogue of the Cooper-paired phase, however this state is not energetically favourable. We extend our calculations to a spin-orbit-coupled Bose gas to investigate the possibility of stripe ordering in the normal phase. However, within our approximations, we do not find an instability towards stripe formation. Finally, we consider a two-dimensional Bose gas at zero temperature with an underlying quartic single-particle dispersion in one spatial direction. This Hamiltonian can be realized using the NIST scheme of spin-orbit coupling [Y.-J. Lin, K. Jimenez-Garcia, and I. B. Spielman, Nature 471, 83 (2011)], or using the shaken lattice scheme of Parker et al. [C. V. Parker, L.-C. Ha and C. Chin, Nature Physics 9, 769 (2013)]. By numerically comparing energies of various trial wave-functions, we show that, at low densities, the ground state is strongly correlated, in contrast to a typical mean-field BEC. The trial wave-function with the lowest energy is of Jastrow-type and it describes a state with finite, but strongly reduced, condensate fraction.