Mueller Research Group

Department of Physics

Laboratory of Atomic and Solid State Physics

Cornell UniversityIthaca, NY • 14853 • (607)255-1568

Ultracold Atom Theory

Group Theses

  • Erich Mueller -- Theory of Quantum Degenerate Gases 2001

    Motivated by experiments on cold alkali atoms, I present a theoretical study of weakly interacting quantum degenerate particles. These experiments observe a wide range of phenomena and consequently this thesis has a broad scope, involving issues of coherence, stability, superfluidity, and kinetics. I explore six topics which exemplify the rich and exciting physics of cold alkali atoms: first, the effect of interactions on the transition temperature of a dilute Bose gas; second, the connection between broken non-gauge symmetries and the appearance of multiple condensates in a Bose gas; third, the role of thermally activated “phase slip” events in destroying superfluidity near the critical temperature of a Bose gas; fourth, mechanical instabilities in clouds of attractive bosons; fifth, the kinetics of a gas of partially condensed atoms; and sixth, the nonlinear optical properties of an atomic gases.


    Atomic clouds in a rotating optical lattice are at the intellectual intersection of several major paradigms of condensed matter physics. An optical lattice sim- ulates the periodic potential ubiquitous in solid state physics, while rotation probes the superfluid character of these cold atomic gases by driving the forma- tion of quantized vortices. Here we explore the theory of vortices in an optical lattice.

    We first provide a detailed introduction section aimed at providing the reader with the information necessary to understand and appreciate the re- search presented in later chapters.

    Next we study an infinite square lattice configuration of vortices in a rotat- ing optical lattice near the superfluid–Mott-insulator transition. We find a se- ries of abrupt structural phase transitions where vortices are pinned with their cores only on plaquettes or only on sites. We discuss connections between these vortex structures and the Hofstadter-butterfly spectrum of free particles on a rotating lattice.

    We then investigate vortex configurations within a harmonically trapped Bose-Einstein condensate in a rotating optical lattice. We find that proximity to the Mott insulating state dramatically affects the vortex structures. To illus- trate we give examples in which the vortices: (i) all sit at a fixed distance from the center of the trap, forming a ring, or (ii) coalesce at the center of the trap,

    forming a giant vortex. We model time-of-flight expansion to demonstrate the experimental observability of our predictions.

    Finally for a trapped gas far from the Mott regime, the competition between vortex-vortex interactions and pinning to the optical lattice results in a compli- cated energy landscape, which leads to hysteretic evolution. The qualitative structure of the vortex configurations depends on the commensurability be- tween the vortex density and the site density – with regular lattices when these are commensurate, and concentric rings when they are not. Again we model the imaging of these structures by calculating time-of-flight column densities. As in the absence of the optical lattice, the vortices are much more easily observed in a time-of-flight image than in-situ.


    This doctoral dissertation is concerned with the physics of strongly interacting cold al- kali atoms at low temperatures near a Feshbach resonance. In Chapter 1, we establish a connection between superfluid 4He and the BCS theory of superconductivity, and cold alkali atoms. We give the history of cold atoms, describing the significant achievements, pitfalls and challenges.

    In Chapter 2, we explore the thermodynamics of strongly interacting Bosonic atoms. We explore the stability of atomic Bosonic condensates near a Feshbach resonance. We show that the experimentally attained atomic condensate is a saddle point of the free en- ergy, but the kinetics of its decay is slow. We also show that there is a second, higher den- sity condensate branch which has an Ising-like phase transition to a molecular (paired) condensate when ramped across the Feshbach resonance. We argue that due to the high density, inelastic 3-body processes possibly render this transition unobservable.

    In Chapter 3, we explore the thermodynamics of Fermionic atoms near a Feshbach resonance. We determine the zero-temperature (T ≪ TF) pair propagator for a spin- imbalanced mixture of up and down spin Fermions, and use it to show that such a mixture becomes completely polarized at μ↓ = −0.9μ↑. We also determine the Thouless criterion for superfluidity in a spin-imbalanced Fermi mixture, and construct a phase diagram of such a system at zero temperature. We then compare our results with experiments per- formed by two different groups. We find that interaction modifications to the minority spin self-energy inferred from our analysis is roughly double those observed in experi- ments. This discrepancy is consistent with the expected accuracy of the theory.

    In Chapter 4, we extend our analysis of the preceding chapter to calculate the surface tension of an interface between spin-polarized Fermions in the normal and superfluid phases. We show that, as expected, this surface tension decreases with increasing tem- perature and vanishes at a tricritical temperature, above which the transition becomes continuous. We also calculate the thickness of the interface; at T = 0, this is a few in- terparticle spacings, but diverges at the tricritical temperature. To compare with a set of relevant experiments, we also develop a phenomenological model for surface tension, and conclude that experimental surface tensions are an order of magnitude higher than what our microscopic calculation yields. We hypothesize possible mechanisms.

    In Chapter 5, we calculate the finite temperature phase diagram of a Bose-Fermi mixture produced from a spin-imbalanced two-component Fermi gas deep in the BEC phase. We show that there is a discontinuous transition between the superfluid and nor- mal phase, with an entropy of mixing sufficient to cool the system down. We detail the construction of such a cooling scheme to cool a Fermi system below what is possible evaporatively, and find that the cooling efficiency is comparable to typical evaporative schemes.

    In Chapter 6, we shift our focus from thermodynamics to dynamics. We calculate shifts in the energy spectrum of a spin-balanced Fermionic superfluid of Cooper pairs due to the presence of energetically close states coupled by a Feshbach resonance. These shifts manifest themselves as clock-shifts in the radio-frequency spectrum of the super- fluid. In addition to a broad asymmetric peak coming from the break-up of Cooper pairs, we find (for certain parameter ranges) a sharp, symmetric “bound-bound” spectral line coming from the conversion of Cooper pairs in one channel to pairs or molecules in an- other channel. Our theory shows remarkable quantitative agreement with experiments performed by an experimental group.


    This thesis describes how to create and probe novel phases of matter and exotic (non-quasiparticle) behavior in cold atomic gases. It focuses on situations whose physics is relevant to condensed matter systems, and where open questions about these latter systems can be addressed. It also attempts to better understand several experimental anomalies in condensed matter systems.

    The thesis is divided into five parts. The first section or chapter of each part gives an introduction to the motivation and background for the physics of that part; the last section or chapter gives an outlook for future studies. Parts 1-4 (Chapters 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, and 15) intro- duce and show different facets of how to learn about novel physics relevant to condensed matter using cold atomic systems. Part 5 ( Chapters 16, 17, 18, and 19) constrains explanations of several ill-understood phenomena occurring in low-temperature quantum solids and condensed matter systems and attempts to construct mechanisms for their behavior.

    Part 1 (Chapters 1 and 2) generally motivates and introduces condensed matter, cold atoms, and many-body physics. Part 2 (Chapters 3, 4, 5, 6, and 7) introduces optical lattice physics and describes ways of spectroscopically prob- ing many-body physics, especially dynamics, near quantum phase transitions in these systems. Part 3 (Chapters 8, 9, 10, 11, and 12) discusses the effects of rotation and how this can create exotic states, as well as alternative methods of creating exotic states. This leads us to study optical lattices where particles pos- sess non-trivial correlations between particles even within a site in Chapters 10 and 11. Part 4 (Chapters 13, 14, and 15) introduces another route to studying exotic physics in cold atoms: examining finite temperature behavior near second order quantum phase transitions. In the “quantum critical regime” occurring near these transitions, non-quasiparticle behavior generically manifests. This behavior has been hidden in previous analyses of data, but Chapter 14 introduces a set of tools required to extract universal quantum critical behavior from standard observ- ables in cold atoms experiments. Chapter 15 discusses near-term opportunities to use these tools to impact fundamental, open questions in condensed matter physics. Part 5 (Chapters 16, 17, 18, and 19) introduces several anomalous or interest- ing experimental results from low-temperature and solid state physics, constrains possible explanations, and attempts to construct mechanisms for their behavior. Chapter 17 shows that collisional properties between quasi-two-dimensional spin- polarized hydrogen atoms are dramatically modified by the presence of a helium film, on which they are invariably adsorbed in present experiments. Chapter 18 constrains theories regarding recent observations on atomic hydrogen defects in molecular hydrogen quantum solids. Finally, Chapter 19 proposes a mechanism for supersolidity that could account for the experimental observations at that time. Since then, it probably has been falsified, but still contains an intriguing mecha- nism for coexistence of superfluidity and solidity that involves disorder.


    Rapid progress in the field of ultracold atoms allows the study of many new and old models of quantum many-body physics. In this doctoral dissertation we theoretically explore exotic phases of ultracold quantum gases, with a special focus spin-imbalanced attractive Fermi gases in lower dimensional situations.

    Chapter 2 reviews the mean-field theory approach to pairing in two- component Fermi gases. Applications of this theory are illustrated in Chapter 3, where we discuss mostly well-known results of mean-field theory applied to imbalanced Fermi gases. Adapted from the author’s prior publications, Chap- ters 4, 5 use the theory developed in Chapters 2, 3.

    In Chapter 6 we discuss the physics of Fermi gases, squeezed into one spatial dimension. In this and Chapter 7, we go beyond mean-field theory, approach- ing the problem through the Bethe ansatz, exact solutions to few-body problems and Fermi-Bose mappings (“fermionization”). We also show results from a joint effort with the experimental group of Randy Hulet at Rice University to experi- mentally realize and probe a strongly interacting one dimensional paired Fermi gas.

    In Chapter 8, after a brief introduction to rapidly rotating two dimensional Bose gases, we introduce a new protocol to create few atom fractional quantum Hall states.

    Finally, in Chapter 9 we study the effects of two-body losses on lattice Bose gases with hardcore interactions in one and two spatial dimensions.


    This thesis represents a body of work investigating the physics of strongly inter- acting quantum particles, confined to a two-dimensional plane in a nontrivial vector potential (such as a transverse magnetic field), commonly referred to as quantum Hall physics. The result which anchors these studies is my discovery, reported in 2010 (PRL 105, 215303), of a lattice model in which there is a de- generate manifold of single-particle states, with wavefunctions matching those of the lowest Landau level of continuum particles, providing a bridge between continuum and lattice physics. Within this model anyonic states are robust and thus amenable to observation. Building on this result, I numerically demon- strate the braiding of anyons in many-body quantum Hall states of bosons, con- firming their anyonic statistics. I also study the equation of state of quantum Hall bosons for various flux densities and choices of hopping parameters, with the goal of quantifying the effects of finite temperature and examining the feasi- bility of observing these states in a system of cold atoms. I then propose a new architecture for superconducting qubits, where flux states of circulating current combined with superconducting transformers and tuned voltage offsets mimic the physics of charged particles in a magnetic field, allowing boson quantum Hall physics to be studied in an environment free of charge noise. Finally, I review the work presented here and speculate on its possible applications to topological quantum computing.


    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 control- ling 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 experi- mentalist 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 par- ticular, 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 Bo- goliubov 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 de- rive 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 underly- ing excitation spectrum influences the long and short time behavior of the cor- relation 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, ap- plicable 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 de- rive 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 dynam- ics 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 experimen- tal 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.


    Advances in experimental atomic systems have given us access to highly tunable quantum systems, and to an unprecedented range of observables of these systems. One fundamental system that has been made accessible in this way is a gas of bosons trapped in a periodic potential.

    We present here a several studies of the many-body physics of bosons in optical lattices. We discuss the ferromagnetic effects of a single vacancy in a two-species gas on a lattice. We present a derivation of the superfluid density, the order param- eter for the superfluid state, and discuss the mathematical subtleties of calculating it. Finally, we calculate the dynamics of bosonic lattice systems, in the presence of inelastic light scattering used as a density measurement, and during a ramp of the interaction strength from the Mott to the superfluid phase.


    Motivated by experiments on cold alkali atoms, I present a theoretical study of weakly interacting quantum degenerate particles. These experiments observe a wide range of phenomena and consequently this thesis has a broad scope, involving issues of coherence, stability, superfluidity, and kinetics. I explore six topics which exemplify the rich and exciting physics of cold alkali atoms: first, the effect of interactions on the transition temperature of a dilute Bose gas; second, the connection between broken non-gauge symmetries and the appearance of multiple condensates in a Bose gas; third, the role of thermally activated “phase slip” events in destroying superfluidity near the critical temperature of a Bose gas; fourth, mechanical instabilities in clouds of attractive bosons; fifth, the kinetics of a gas of partially condensed atoms; and sixth, the nonlinear optical properties of an atomic gases.

  • Bhuvanesh Sundar -- MANY-BODY PHYSICS USING COLD ATOMS 2016

    Advances in experiments on dilute ultracold atomic gases have given us access to highly tunable quantum systems. In particular, there have been substantial im- provements in achieving different kinds of interaction between atoms. As a result, utracold atomic gases offer 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.


    This thesis presents a series of theoretical studies of ultra cold atomic systems which model and propose experiments, and develop new computa- tional techniques in order to elucidate aspects of many-body physics and non- equilibrium dynamics. In the first two studies I model the dynamics of non- linear solitonic excitations in ultracold fermionic superfluids: the first simu- lates recent experiments and supports the hypothesis that the solitons generated in those experiments are unstable to the formation of vortex rings; the second demonstrates how population imbalance between up and down spin fermions can be used to prevent this instability. In the next study I discuss a method for generating and probing topologically protected edge states using periodically driven optical lattices potentials. Next I use a perturbative approach to study the spectral density of fermions with strong attractive interactions in the nor- mal phase. After that I develop a novel cluster expansion technique to model the dynamics of interacting fermions in a disordered optical lattice. Finally I ap- ply a Ginzurg-Landau theory to model experimental studies of superfluid 3He embedded in nematically ordered aerogel, finding evidence for a new phase of matter –the “polar phase”– which is not seen in bulk 3He.