In this dissertation we proposed and modeled five ultra cold atoms experiments that make use of quantum effects to produce novel macroscopic phenomena. The first of these experiments is a proposal to produce a p-wave superfluid using two different atomic species.The pairing species is trapped in two-dimensions and the other species is allowed to move in three-dimensions and mediates the p-wave pairing interaction. We confirm our predictions using the functional Renormalization Group method.The next experiment studies how macroscopic cat states can be produced and detected in bosonic interferometers. A bosonic interferometer is composed of ultra-cold bosons that can condense in one of two modes. Tunneling and interactions are the primary contributions to the dynamics and in cooperation can produce highly entangled states such as Schrodinger cat states. We show how the quantum Fisher information, a tool in quantum metrology, can be used to perform a partial state tomography to determine the nature of the superposition in experimental cat states.In the last three proposals, we study the long time equilibration dynamics of three different proposed experiments. The first proposed experiment takes place in the same bosonic interferometer as discussed in the Schr{\\"o}dinger cat work, and we identify a novel mechanism for the breakdown of thermalization with the slow dynamics of an unstable fixed point in a semi-classical approximation. The second proposed experiment also takes place in the same bosonic model, but involves a periodic kick to the interaction strength. We find that these dynamics can produce a time-crystal-like state where discrete time-translation symmetry is broken in the long time dynamics. Finally, we study the long time dynamics of a clean fermionic chain and a disordered fermionic chain that are coupled by density density interactions. In this proposal, we find that the effects of disorder can be transferred to the clean chain and prevent thermalization.
In this dissertation we proposed and modeled five ultra cold atoms experiments that make use of quantum effects to produce novel macroscopic phenomena. The first of these experiments is a proposal to produce a p-wave superfluid using two different atomic species.The pairing species is trapped in two-dimensions and the other species is allowed to move in three-dimensions and mediates the p-wave pairing interaction. We confirm our predictions using the functional Renormalization Group method.The next experiment studies how macroscopic cat states can be produced and detected in bosonic interferometers. A bosonic interferometer is composed of ultra-cold bosons that can condense in one of two modes. Tunneling and interactions are the primary contributions to the dynamics and in cooperation can produce highly entangled states such as Schrodinger cat states. We show how the quantum Fisher information, a tool in quantum metrology, can be used to perform a partial state tomography to determine the nature of the superposition in experimental cat states.In the last three proposals, we study the long time equilibration dynamics of three different proposed experiments. The first proposed experiment takes place in the same bosonic interferometer as discussed in the Schr{\"o}dinger cat work, and we identify a novel mechanism for the breakdown of thermalization with the slow dynamics of an unstable fixed point in a semi-classical approximation. The second proposed experiment also takes place in the same bosonic model, but involves a periodic kick to the interaction strength. We find that these dynamics can produce a time-crystal-like state where discrete time-translation symmetry is broken in the long time dynamics. Finally, we study the long time dynamics of a clean fermionic chain and a disordered fermionic chain that are coupled by density density interactions. In this proposal, we find that the effects of disorder can be transferred to the clean chain and prevent thermalization.
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.
We study quantum phases and transitions in ultra-cold quantum gases, beyond the scope of conventional condensed matter systems. The study focuses on three major thrusts. The first thrust is the study of cold ensembles of bosonic or fermionic atoms in the higher orbital bands of optical lattices. Orbital and topological quantum physics is systematically explored for various lattice geometries, to be outlined below. The second is the low dimensional spin-imbalanced fermions in the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase. We analyze its thermal instability towards Berezinsky-Kosterlitz-Thouless (BKT) phase transitions driven by different topological defects at finite temperature. The third is the theoretical search for a mechanism to achieve a Bose-Einstein crystal phase for bosons in the ground state. A new class of two-body interactions characterized by strong momentum-dependence is found to give rise to such a quantum state. It is different than but related to the long-debated concept of supersolidity. The first thrust forms the main part of the dissertation. Meta-stable phases of high orbital bosons and fermions have been explored for different lattice geometries such as square and bipartite lattices. Exotic Mott and superfluid phases are studied in both one and two dimensions. Quantum and thermal phase transitions, associated with time reversal symmetry breaking and their experimental signatures are discussed. In particular a quantum ``disordered" superfluid phase, beyond the mean field description, is found in one dimension. The challenging problem of probing time reversal symmetry breaking is solved. Besides, a one-dimensional fermionic optical ladder with coupled s- and p-orbitals is shown to mimic spin-orbit interactions and exhibit topological insulator and topological superconducting phases. Charge fractionalization is shown to be realizable in such an sp-orbital ladder.
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.
This thesis investigates thermalisation, correlations and entanglement in Bose-Einstein condensates. Bose-Einstein condensates are ultra-cold collections of identical bosonic atoms which accumulate in a single quantum state, forming a mesoscopic quantum object. They are clean and controllable quantum many-body systems that permit an unprecedented degree of experimental flexibility compared to other physical systems. Further, a tractable microscopic theory exists which allows a direct and powerful comparison between theory and experiment, propelling the field of quantum atom optics forward at an incredible pace. Here we explore some of the fundamental frontiers of the field, examining how non-classical correlations and entanglement can be created and measured, as well as how non-classical effects can lead to the rapid heating of atom clouds. We first investigate correlations between two weakly coupled condensates, a system analogous to a superconducting Josephson junction. The ground state of this system contains non-classical number correlation arising from the repulsion between the atoms. Such states are of interest because they may lead to more precise measurement devices such as atomic gyroscopes. Unfortunately thermal fluctuations can destroy these correlations, and great care is needed to experimentally observe non-classical effects. We show that adiabatic evolution can drive the isolated quantum system out of thermal equilibrium and decrease thermal noise, in agreement with a recent experiment [Esteve et al. Nature 455, 1216 (2008)]. This technique may be valuable for observing and using quantum correlated states in the future. Next, we analyse the rapid heating that occurs when a condensate is placed in a moving periodic potential. The dynamical instability responsible for the heating was the subject of much uncertainty, which we suggest was due to the inability of the mean-field approximation to account for important spontaneous scattering processes. We show that a model including non-classical spontaneous scattering can describe dynamical instabilities correctly in each of the regimes where they have been observed, and in particular we compare our simulations to an experiment performed at the University of Otago deep inside the spontaneous scattering regime. Finally, we proposed a method to create and detect entangled atomic wave-packets. Entangled atoms are interesting from a fundamental perspective, and may prove useful in future quantum information and precision measurement technologies. Entanglement is generated by interactions, such as atomic collisions in Bose-Einstein condensates. We analyse the type of entanglement generated via atomic collisions and introduce an abstract scheme for detecting entanglement and demonstrating the Einstein-Podolsky-Rosen paradox with ultra-cold atoms. We further this result by proposing an experiment where entangled wave-packets are created and detected. The entanglement is generated by the pairwise scattering that causes the instabilities in moving periodic potentials mentioned above. By careful arrangement, the instability process can be controlled to to produce two well-defined atomic wave-packets. The presence of entanglement can be proven by applying a series of laser pulses to interfere the wave-packets and then measuring the output populations. Realising this experiment is feasible with current technology.
Two models are discussed here to illustrate the quantum many-body phenomena in mixtures of ultra-cold atoms in optical lattices. The first model describes a mixture of two species of bosonic atoms of equal masses in optical lattices and the second describes a mixture of heavy bosonic atoms and light fermionic atoms in optical lattices. For both models, we assume the trap is present and use parameters typical in experiment. For the first model, the discussion is aimed at providing a thorough description of the collective behavior of the binary mixture in various interaction regions, with emphasis on two many-body phenomena, pairing and anti-pairing, as a result of the inter-species interaction. The pairing leads to a new type of superfluid order, called the paired superfluid (PSF) and the anti-pairing leads to another type of superfluid order, called the counter-flow superfluid (CFSF). In addition, we discuss the coexistence of charge density wave order with the three superfluid orders in the strong interaction region. We use both Luttinger liquid theory and the time evolving block decimation (TEBD) method to study this model in one dimension. The discussion is organized in three parts: the phase diagram and the correlation functions; the noise correlation functions; and the transport properties. Two phase diagrams are constructed to map the different orders in the parameter space. The correlation functions, include noise correlations, are carefully examined for the determination of the orders and for possible detection methods. In the end, the transport properties of the PSF and CFSF orders are studied through the dipole oscillation induced by trap displacement. For the second model, examining a mixture of heavy bosons and light fermions, the discussion is oriented toward determining the thermal properties of the mixture for attractive inter-species interactions. This work is motivated by experiments creating artificial molecules through optical and magnetic control of ultra-cold atoms. We use the strong coupling (SC) expansion method to evaluate the density profile, the onsite inter-species correlations, the density fluctuations and the entropy per particle. Analytical expressions are derived for all the quantities above as well as the partition function. To benchmark the accuracy, the SC calculations are compared with inhomogeneous dynamical mean field theory (IDMFT) and Monte Carlo (MC) simulation. From the calculations, we find that 1) the efficiency of creating pre-formed molecules is significantly increased by confining the mixtures onto optical lattices; 2) the temperature of the mixtures in optical lattices can be reliably estimated through the density gradient and the density fluctuations.
We explored interaction cold atoms in new quantum regimes that have no prior analogue in condensed matter materials in three related main topics. First, motivated by recent advance in orbitally tuned Feshbach resonance experiments, we analyze the ground-state phase diagram and related low-energy excitation spectra of a high partial wave interacting Bose gas. Remarkably different from what was previously known in the p-wave case, the atomic superfluid is found to be momentum-independent in the present d-wave case. What is more, we study the quantum fluctuations in the condensates of a mixture of bosonic atoms and molecules with interspecies p-wave interaction. Our analysis shows that the quantum phase of coexisting atomic and molecular condensates is unstable at the mean-field level. The quantum Lee-Huang-Yang correction to the mean-field energy provides a remarkable mechanism to self-stabilize the phase. The correlated order spontaneously breaks a rich set of global U(1) gauge, atomic spin, spatial rotation and translation, and time-reversal symmetries. Second, we study the dynamics of a non-integrable spin chain model composed of two ingredients - a nearest neighbor Ising coupling, and an infinite range XX interaction. Unlike other fast scrambling many-body systems, this model is not known to be dual to a black hole. We demonstrate that our model exhibits fast scrambling for a wide parameter regime, accompanied by a fast growth of the entanglement entropy, as well as a swift change in the magnetization. Third, to extend the study of the highly chaotic many-body model, we analyze the ground state phases of the model with a nearest neighbor XXZ interaction and an infinite range XX interaction. By employing spin-wave theory, we find that there is a large parameter regime where the continuous U(1) symmetry of this model is spontaneously broken, which is not possible in the absence of the infinite range interaction by the Mermin-Wagner theorem. Furthermore, we demonstrate that in the U(1) symmetry broken phase, the half chain entanglement entropy violates the area law logarithmically. Our work demonstrates that the interplay of short and long range interactions can lead to novel quantum phases of matter.