The study of physics at the atomic scale led to the development of quantum mechanics in the early twentieth century. Since then, quantum mechanics has developed into one of the most successfully tested of all physical theories. Central to quantum mechanics is the concept of coherence. Keeping quantal coherence over large time scales or macroscopic length scales has proven to be a difficult, but fruitful endeavour both theoretically and experimentally. Two manifestations of this so-called macroscopic quantum coherence will be investigated in this thesis; the century-old field of superconductivity and the decade-old field of cold atomic gases. Yttrium barium copper oxide is a layered superconductor whose transition temperature can be changed by controlling the amount of oxygen found between the copper oxide planes. Motivated by recent experiments where the penetration depth along the direction perpendicular to the copper oxygen planes was measured on extremely underdoped samples, a theoretical model is constructed which phenomenologically explains the observed electrodynamic properties. The field of atomic physics underwent a revolution in 1994 when dilute atomic gases were cooled to nanokelvin temperatures, which resulted in the much sought after Bose-Einstein condensate. In the past decade, ultra-cold atomic experiments have been used to study fundamental quantum mechanics and more recently, with the application of optical lattices, many-body physics. The second project contained in this thesis investigates a method to engineer the emergence of Dirac Ferinions in an ultra-cold Fermionic gas with the application of an optical lattice. The Hamiltonian governing the low energy properties of this system is well known, and is shown to undergo a quantum phase transition where the low energy Fermionic quasiparticles acquire a mass, due to the appearance of "antiferromagnetic" ordering.
The study of physics at the atomic scale led to the development of quantum mechanics in the early twentieth century. Since then, quantum mechanics has developed into one of the most successfully tested of all physical theories. Central to quantum mechanics is the concept of coherence. Keeping quantal coherence over large time scales or macroscopic length scales has proven to be a difficult, but fruitful endeavour both theoretically and experimentally. Two manifestations of this so-called macroscopic quantum coherence will be investigated in this thesis; the century-old field of superconductivity and the decade-old field of cold atomic gases. Yttrium barium copper oxide is a layered superconductor whose transition temperature can be changed by controlling the amount of oxygen found between the copper oxide planes. Motivated by recent experiments where the penetration depth along the direction perpendicular to the copper oxygen planes was measured on extremely underdoped samples, a theoretical model is constructed which phenomenologically explains the observed electrodynamic properties. The field of atomic physics underwent a revolution in 1994 when dilute atomic gases were cooled to nanokelvin temperatures, which resulted in the much sought after Bose-Einstein condensate. In the past decade, ultra-cold atomic experiments have been used to study fundamental quantum mechanics and more recently, with the application of optical lattices, many-body physics. The second project contained in this thesis investigates a method to engineer the emergence of Dirac Ferinions in an ultra-cold Fermionic gas with the application of an optical lattice. The Hamiltonian governing the low energy properties of this system is well known, and is shown to undergo a quantum phase transition where the low energy Fermionic quasiparticles acquire a mass, due to the appearance of "antiferromagnetic" ordering.
Physics of ultra cold atoms and molecules is one of the most rapidly developing areas of the modern atomic, molecular, optical physics and quantum optics.This area covers several sub-areas of diverse importance, but definitely two trends can be easily identifies: i) fundamental research on properties of ultra cold atoms and molecules, and in particular atomic and molecular condensates trapped and lattice gases; here the fundamental questions of nature and form of phase transitions in degenerate systems are being posed and studied; ii) more applications oriented research on ultracold atoms and molecules, involving in particular applications in precision measurements, atomic clocks, frequency standards, atom and molecule interferometry, quantum information, quantum engineering, "super" chemistry etc.This thesis concerns both of these major trends.The applications, or more specifically quantum engineering aspect are discussed in the first four chapters of the thesis, which deal with optical manipulations of ultra cold gases, in particular optical generation of vortices in Bose condensates, quasi-solitons in Fermi gases, and retarded dipole-dipole interactions in Bose, or Fermi gases.This thesis consists of four main chapters.Chapter 2, "Optical generation of vortices in Bose Einstein Condensates".In this chapter we study theoretically the generation of vortices by phase imprinting in trapped Bose-Einstein condensates.Phase imprinting is achieved by passing a short off-resonance laser pulse through an appropriately designed absorption plate, and impinging it on a condensate.We answer the fundamental question of how the circulation and the vortex are introduced into the system, and discuss the creation of vortex arrays.We discuss vortex dynamics and interactions in such configurations.Various methods of vortex detection based on interference combined with Bragg scattering are also analyzed.In Chapter 3, "Fermionic 'Solitons'" we investigate in detail the concept of low energy excitations of ultra cold Fermi gases.This excitations are generated using the same idea of phase imprinting as that employed in the generation of vortices in BEC.We address in this chapter some questions concerning the generation (via phase imprinting) of soliton-like solutions in a single-component Fermi gas of neutral atoms at zero and finite temperatures.By using both numerical and analytical calculations, we find the conditions under which it is possible to find in non-interacting Fermi gases, quasisolitons which resemble the properties of solitons in non-linear integrable equations.We present the results for both spatially homogeneous and trapped cases, and emphasize the importance of Fermi statistics and the absence of interactions for the existence of such solutions.In Chapter 4, "Generating Dipole Condensate" we keep studying the possibilities of optical manipulations of ultracold gases.In particular, we study two different ways to laser-induce dipole-dipole interactions in trapped Bose-Einstein condensates.The first one employs the fact that those atoms in high Rydberg states posses a large dipolar moment.We first analyze the idea of imposing the dipole moment as a time average over short stroboscopic pulses.These pulses are coherently transporting the atoms from the ground atomic state to some high Rydberg state and back.The second idea is based on the observation that in strong static or oscillatory electric fields, atoms polarize and acquire dipole moment.We analyze the possibilities and limitations of inducing such a high electric field using lasers of low frequency.In Chapter 5, "Dipolar BCS", we continue our study of trapped fermionic gases, with the analysis of the BCS transition in trapped dipolar fermionic gases.We find that the critical strength of the dipole-dipole interaction to achieve BCS pairing, depends on the trap geometry.We find the particular form of this dependence, and present the solutions for the order parameter.This first chapter is meant as a general introduction to the thesis.This introduction is by no means exhaustive, and should be rather treated as a collection of basic facts concerning ultra cold gases, without rigorous derivations.The last subsection of the introductory chapter, discusses very briefly the most important achievements and trends in the field and the current state of the art.
Remarkable experimental advances in the past few decades have paved the way for a new era of ultra-cold physics. Experimentalists can now cool large samples of atoms down to temperatures where quantum mechanical effects dominate, and control their bulk geometrical structure and microscopic interactions with ease. The result is a wide open testing ground for quantum many-body physics. In this talk, I will provide an overview of the important concepts and methods that form the foundations of this field, before steering the discussion to Bose gases, or gases with constituents that obey Bose-Einstein statistics. At ultra-cold temperatures, gases of bosons condense into a single macroscopicquantum mechanical state, forming a Bose-Einstein condensate. One of the more interesting properties of Bose-Einstein condensates is that they are superfluid, and can exhibit dissipationless flow. If the bosons possess large dipole moments, their interaction character changes signifi cantly, as does their collective superfluid behavior. I will discuss some of the theoret ical and computational tools used to model Bose-Einstein condensates, demonstrate how dipolar interactions play a profound role in the basic properties of these novel superfluids, and provide an outlook for the future of this exciting field.
Advances in experiments on dilute ultracold atomic gases have given us access to highly tunable quantum systems. In particular, there have been substantial improvements in achieving different kinds of interaction between atoms. As a result, utracold atomic gases oer an ideal platform to simulate many-body phenomena in condensed matter physics, and engineer other novel phenomena that are a result of the exotic interactions produced between atoms. In this dissertation, I present a series of studies that explore the physics of dilute ultracold atomic gases in different settings. In each setting, I explore a different form of the inter-particle interaction. Motivated by experiments which induce artificial spin-orbit coupling for cold fermions, I explore this system in my first project. In this project, I propose a method to perform universal quantum computation using the excitations of interacting spin-orbit coupled fermions, in which effective p-wave interactions lead to the formation of a topological superfluid. Motivated by experiments which explore the physics of exotic interactions between atoms trapped inside optical cavities, I explore this system in a second project. I calculate the phase diagram of lattice bosons trapped in an optical cavity, where the cavity modes mediates effective global range checkerboard interactions between the atoms. I compare this phase diagram with one that was recently measured experimentally. In two other projects, I explore quantum simulation of condensed matter phenomena due to spin-dependent interactions between particles. I propose a method to produce tunable spin-dependent interactions between atoms, using an optical Feshbach resonance. In one project, I use these spin-dependent interactions in an ultracold Bose-Fermi system, and propose a method to produce the Kondo model. I propose an experiment to directly observe the Kondo effect in this system. In another project, I propose using lattice bosons with a large hyperfine spin, which have Feshbach-induced spin-dependent interactions, to produce a quantum dimer model. I propose an experiment to detect the ground state in this system. In a final project, I develop tools to simulate the dynamics of fermionic superfluids in which fermions interact via a short-range interaction.
Since the achievement of Bose-Einstein condensation (BEC), the progress in matterwave \nphysics has been immense. Among the many recent achievements there is \nthe miniaturization of atom traps, demonstration of the superfluid-Mott insulator \nquantum-phase transition in optical lattices and the experimental demonstration of \nthe BEC-BCS crossover in ultra-cold gases. \nMiniaturization of atom traps using micro-structured wires on a chip is one important \nstep towards an on-chip cold-atom device. These so-called \\atom chips" provide \nhigh control and versatility for trapping and guiding the ultra-cold atomic clouds. \nParticularly interesting is the use of these microchips to build mesoscopic devices \nfor cold atomic clouds as, for instance, in the case of an atom-cloud interferometer. \nHowever, these mesoscopic devices require coherent transport of the atom cloud. A \ngeneral method to treat decoherence due to current fluctuations in multi-wire atomchip \ntraps is presented in the rst part of this thesis. The decoherence rate Γ shows a \nstrong dependence on the distance between the wire and the atom cloud, r0, scaling \nas Γ ~ r-4 0 for a single atom waveguide. Considering an interferometer device, a \nstrong dependence of the decoherence rate on the trap geometry is found. \n \nStudying many-body eects in ultra-cold quantum gases is another important \nresearch eld. Experiments using ultra-cold quantum gases in optical lattices have \ndemonstrated the superuid-Mott insulator quantum phase transition and manybody \nentanglement. Optical lattices are based on a periodic modulation of the light \nintensity, generated by retro-reected laser beams. Correlations of the atomic cloud \nbetween dierent lattice sites of the optical lattice play a central role in these manybody \nexperiments. The dierent phases of the superuid-Mott insulator system can \nbe characterized by the dierent behavior of the inter-lattice site correlations. There \nare several numerical methods such as Quantum Monte Carlo (QMC) simulations, \nDensity Matrix Renormalization Group (DMRG) simulations, exact-diagonalization, \nor the Gutzwiller ansatz, to investigate the dynamics of an ultra-cold gas in an optical \nlattice theoretically. The Gutzwiller method, corresponding to the mean-eld solution, \nallows for the treatment of large lattice sizes. Mean-eld approaches have proven \nto be very useful to describe many-body physics. However, diculties arise in the correct \ndescription of the behavior of the decay of inter-lattice site correlations. Based on \nthe Gutzwiller approach, we have developed a method which allows the successive inclusion \nof inter-lattice site correlations. Comparing the results for the particle-number \nuctuations and the correlation function obtained from pure Gutzwiller calculations, \nto calculations which perturbatively include short-range correlations and calculations \n \nusing \\quasi-exact methods", showed a considerable improvement relative to the pure \nGutzwiller results due to the inclusion of short-range correlations. \nMany-body eects do not only arise in periodic potentials, but become increasingly \nimportant at ultra-low temperatures. The formation of Bose-Einstein condensates \nrequires an overlap of the atom wavefunctions and, hence, the formation of \na single condensate wavefunction. Another example of a many-body state is the \nsuperuid-BCS state, commonly used as a description of superconductivity. Here, \nfermions in dierent hyperne states form Cooper pairs. Experiments with ultracold \nquantum gases enable a variation of the interparticle interaction, e.g. , by using \na Feshbach resonance. Using Feshbach resonances to tune the interaction strength \nhas enabled the experimental observation of the crossover from a superuid-BCS \nstate to a Bose-Einstein condensate of molecules. A useful way to characterize the \ndierent states of ultra-cold quantum gases is to investigate the particle-number uctuations. \nIn this thesis we suggest to divide the atomic cloud into bins and consider \nthe atom-number uctuations in these bins. We calculate the full counting statistics \nfor dierent physical systems of ultra-cold gases (e.g. bosonic gases, fermionic gases, \nand spin mixtures). In particular, we consider the BCS-state as a rst trial example \nto show that there is a strong variation in the particle-number statistics at the \ncrossover from a superuid-BCS state to a Bose-Einstein condensate of molecules.
The study of out-of-equilibrium dynamics in ultra-cold gases is a new and exciting field, driven largely by the recent experimental advances in controlling and imaging cold clouds. The experimental and theoretical work thus far has been somewhat exploratory and largely numerical in nature, as the very paradigms for thinking about these systems are not well established. In this thesis I consider several different scenarios of ultra-cold bosonic and fermionic gases driven out of equilibrium and study their properties. In Chapter 1, I provide an overview of the phenomenology of ultra-cold gases, highlighting the timescales governing these systems and how the experimentalist can tune them. I discuss how cold gases can be cooled and trapped and discuss the basic physics behind optical lattices. I also discuss experimental probes of these gases, in particular the new high resolution imaging techniques developed recently at Chicago, Munich and Harvard. In Chapter 2, I discuss an early experiment (circa. 2008) which observed long lived spin dynamics in a thermal spin-1/2 Fermi gas. This experiment is an nice illustration of interesting physics resulting from the separation of timescales between spin and collisional dynamics. In my opinion, it is an excellent example of why cold gases are naturally suited to studying non-equilibrium dynamics. I simulate the experiment numerically using a collisionless Boltzmann equation and explain the observed spin dynamics both qualitatively and quantitively. In Chapter 3, I continue the discussion of spin waves in thermal gases by extending previous works on spin-1/2 gases to spin-1 Bose gases. In contrast to Chapter 2, the bulk of the work in this Chapter is analytic in nature. In particular, I find a spin wave instability in the thermal spin-1 Bose gas, which is the high temperature analog of the polar to ferromagnetic transition in a spin-1 Bose Einstein condensate. In Chapter 3, I turn my attention to bosonic systems and briefly review the the Bogoliubov mean-field theory. I calculate the momentum distribution and density-density correlation function of an interacting Bose gas within the Bogoliubov framework. Then I consider bosons in an optical lattice, and introduce the Bose Hubbard model. I calculate the mean-field phase diagram of the Bose Hubbard model and then consider fluctuations about the mean field, and derive the excitation spectrum of the lattice gas in the superfluid and insulating regimes. In Chapter 4, I ask what we learn by studying the dynamics of correlation functions following a sudden change in the interactions in a superfluid. Using the Bogoliubov theory developed in Chapter 3, I will show how the underlying excitation spectrum influences the long and short time behavior of the correlation functions. By considering a lattice dispersion, I study the analogous problem in a weak optical lattice and discuss how the lattice dispersion leads to additional features in the correlation functions. I will also discuss the timescale governing the revival of the condensate fraction in a quantum depleted gas. In Chapter 5, I derive equations of motion governing the dynamics of one and two body correlation functions in the single-band Bose Hubbard model, applicable to bosons in deep lattices. I then consider a simple quench from a Mott insulating initial state to a weakly interacting final state and produce analytic expressions describing the dynamics of correlations following such a quench. I discuss the timescale for the development of long range order following such a quench. I study the problem of chapter 4 using an equations of motion approach. This approach complements the Bogoliubov approach of Chapter 4. First, I derive exact expressions for a quench to a non-interacting state. I then consider how interactions redistribute quasi-momentum to first order in perturbation theory in different dimensions. In Chapter 6, I calculate the relevant timescales for local and global dynamics in trapped lattice Bose gases, a work done in collaboration with Dr. Kaden R. A Hazzard. Using a time-dependent Gutzwiller mean-field theory, I show that the timescale for local equilibration in these systems is fast in experimental terms. I then show that due to the spatial inhomogeneities inherent to cold gases, achieving global equilibrium can be quite complicated, sometimes taking longer than the lifetime of the experiment, an issue of practical importance to current day experiments. I continue this discussion in Chapter 7 which is a collaborative work with experimentalists David McKay and Prof. Brian DeMarco from the University of Toronto and the University of Illinois, Urbana Champaign. Using experimental and numerical methods, we show that the rapid timescales for local dynamics in interacting systems invalidates a frequently used cold atom technique for mapping out the momentum distribution of atoms in an optical lattice.