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Interacting ultracold gases in optical lattices: non-equilibrium dynamics and effects of disorder
(2012)
This dissertation aims at giving a theoretical description of various applications of ultracold gases. A particular focus is cast upon the dynamical evolution of bosonic condensates in non-equilibrium by means of the time-dependent Gutzwiller method. Ground state properties of strongly interacting fermionic atoms in box and speckle disordered lattices are investigated via real-space dynamical mean-field theory. ...
In this thesis we have studied the physics of different ultracold Bose-Fermi mixtures in optical lattices, as well as spin 1=2 fermions in a harmonic trap. To study these systems we generalized dynamical mean-field theory for a mixture of fermions and bosons, as well as for an inhomogeneous environment. Generalized dynamical mean-field theory (GDMFT) is a method that describes a mixture of fermions and bosons. This method consists of Gutzwiller mean-field for the bosons, and dynamical mean-field theory for the fermions, which are coupled on-site by the Bose-Fermi density-density interaction and possibly a Feshbach term which converts a pair of up and down fermions into a molecule, i.e. a boson. We derived the self-consistency equations and showed that this method is well-controlled in the limit of high lattice coordination number z. We develop real-space dynamical mean-field theory for studying systems in an inhomogeneous environment, e.g. in a harmonic trap. The crucial difference compared to standard DMFT is that we are taking into account that different sites are not equivalent to each other and thus take into account the inhomogeneity of the system. Different sites are coupled by the real-space Dyson equation. ...
Seit Anbeginn der Festkörperphysik ist die Frage, warum manche Materialien metallisch sind, andere dagegen isolierend, von zentraler Bedeutung. Eine erste Erklärung wurde durch die Bändertheorie [23, 44] gegeben. Die Elektronen sind dem periodischen Potential der Rumpfatome ausgesetzt, wodurch ein Energiespektrum bestehend aus Bändern erzeugt wird und die Füllung dieser Bänder bestimmt die Leitungseigenschaften des Festkörpers. ...
The miniaturization of electronics is reaching its limits. Structures necessary to build integrated circuits from semiconductors are shrinking and could reach the size of only a few atoms within the next few years. It will be at the latest at this point in time that the physics of nanostructures gains importance in our every day life. This thesis deals with the physics of quantum impurity models. All models of this class exhibit an identical structure: the simple and small impurity only has few degrees of freedom. It can be built out of a small number of atoms or a single molecule, for example. In the simplest case it can be described by a single spin degree of freedom, in many quantum impurity models, it can be treated exactly. The complexity of the description arises from its coupling to a large number of fermionic or bosonic degrees of freedom (large meaning that we have to deal with particle numbers of the order of 10^{23}). An exact treatment thus remains impossible. At the same time, physical effects which arise in quantum impurity systems often cannot be described within a perturbative theory, since multiple energy scales may play an important role. One example for such an effect is the Kondo effect, where the free magnetic moment of the impurity is screened by a "cloud" of fermionic particles of the quantum bath.
The Kondo effect is only one example for the rich physics stemming from correlation effects in many body systems. Quantum impurity models, and the oftentimes related Kondo effect, have regained the attention of experimental and theoretical physicists since the advent of quantum dots, which are sometimes also referred to as as artificial atoms. Quantum dots offer a unprecedented control and tunability of many system parameters. Hence, they constitute a nice "playground" for fundamental research, while being promising candidates for building blocks of future technological devices as well.
Recently Loss' and DiVincenzo's p roposal of a quantum computing scheme based on spins in quantum dots, increased the efforts of experimentalists to coherently manipulate and read out the spins of quantum dots one by one. In this context two topics are of paramount importance for future quantum information processing: since decoherence times have to be large enough to allow for good error correction schemes, understanding the loss of phase coherence in quantum impurity systems is a prerequisite for quantum computation in these systems. Nonequilibrium phenomena in quantum impurity systems also have to be understood, before one may gain control of manipulating quantum bits.
As a first step towards more complicated nonequilibrium situations, the reaction of a system to a quantum quench, i.e. a sudden change of external fields or other parameters of the system can be investigated. We give an introduction to a powerful numerical method used in this field of research, the numerical renormalization group method, and apply this method and its recent enhancements to various quantum impurity systems.
The main part of this thesis may be structured in the following way:
- Ferromagnetic Kondo Model,
- Spin-Dynamics in the Anisotropic Kondo and the Spin-Boson Model,
- Two Ising-coupled Spins in a Bosonic Bath,
- Decoherence in an Aharanov-Bohm Interferometer.
In dieser Arbeit wurde das Verhalten von repulsiv gebundenen Teilchenpaaren (Dimeren) in eindimensionalen optischen Gittern untersucht. Repulsiv gebundene Teilchenpaare sind metastabile Zustände, die nicht im freien Raum, dafür aber in geordneten Potentialen, wie optische Gitter sie darstellen, vorkommen können. In einem analytischen Teil beschäftigten wir uns mit der Herleitung effektiver Hamiltonians für Dimersysteme. Diese wurden dann unter Verwendung des Time Evolving Block Decimation-Algorithmus (TEBD) numerisch untersucht...
In this thesis, we have investigated strongly correlated bosonic gases in an optical lattice, mostly based on a bosonic version of dynamical mean field theory and its real-space extension. Emphasis is put on possible novel quantum phenomena of these many-body systems and their corresponding underlying physics, including quantum magnetism, pair-superfluidity, thermodynamics, many-body cooling, new quantum phases in the presence of long-range interactions, and excitational properties. Our motivation is to simulate manybody phenomena relevant to strongly correlated materials with ultracold lattice gases, which provide an excellent playground for investigating quantum systems with an unprecedented level of precision and controllability. Due to their high controllability, ultracold gases can be regarded as a quantum simulator of many-body systems in solid-state physics, high energy astrophysics, and quantum optics. In this thesis, specifically, we have explored possible novel quantum phases, thermodynamic properties, many-body cooling schemes, and the spectroscopy of strongly correlated many-body quantum systems. The results presented in this thesis provide theoretical benchmarks for exploring quantum magnetism in upcoming experiments, and an important step towards studying quantum phenomena of ultracold gases in the presence of long-range interactions.
Hofstadter-Hubbard physics
(2020)
The Hofstadter model, besides the Haldane and Kane-Mele models, is the most common tight-binding model which hosts topologically nontrivial states of matter. In its time-reversal-symmetric formulation the model can even describe topological insulators. Experimentally, the Hofstadter model was realized with ultracold quantum gases in optical lattices which is a wellcontrolled way to engineer quantum states of tight-binding Hamiltonians. Another established control parameter in ultracold quantum gases are twoparticle, on-site interactions, also known as Hubbard interactions. This work aims at introducing the reader to the concepts of topological states of matter, a collection of corresponding tight-binding models, and the methodology to treat interacting topological states with dynamical mean-field theory.We present recent results for inhomogeneous, interacting systems, spinimbalanced magnetic systems, propose experimental detection methods, and extensions to three-dimensional topological states.
In the course of this thesis we discuss a certain kind of supersolid, the lattice-supersolid, which can be realized using quantum gases in an optical lattice trap. The lattice-supersolid, which simultaneously possesses off-diagonal and diagonal long-range order in its density matrix and also breaks the discrete translational symmetry of an underlying lattice, is induced by self-ordering of the gas due to strong long-range van der Waals interactions. In the considered scenario, the interactions are facilitated by the excitation of atomic Rydberg states, which exhibit enhanced van der Waals forces.
In the first part of this thesis (chapters 1-3), we review the relevant basics of quantum gases, Rydberg physics and introduce the extended Bose-Hubbard model. We start with the relevant methods and devices of the vast toolbox available in common quantum gas experiments, as well as consider the main concepts behind superfluidity and supersolidity. This is followed by an introduction of some basic concepts of Rydberg atoms in quantum many-body systems, with a focus on the facilitation of long-range interactions and the implementation in a theoretical model. Thereafter a brief introduction is given, on the realization of the Bose-Hubbard model in optical lattice systems and its extension to include Rydberg states, which concludes the introductory part of this thesis.
In the following part (chapters 4-6), we introduce the theoretical tools used to derive the results presented in the final part. First, an introduction to a real-space extension of bosonic dynamical mean-field theory (RB-DMFT) for bosonic systems with long-range interactions in the Hartree approximation is given. This method is based on the non-perturbative self-consistent evaluation of the lattice Green’s function, which also incorporates the effect of nearest neighbor correlations due to the non-condensed particles. Then we focus on a quasiparticle expansion of the Bose-Hubbard model, which has its foundation in linearized fluctuations of a static mean-field ground-state, allowing for the prediction of a vast range of experimentally relevant observables. Lastly, we introduce an efficient truncation scheme for the local bosonic Fock-basis, which allows for the simulation of phases with high condensate density at a vastly reduced computational effort.
In the final part (chapters 7 and 8), we discuss the application of both methods to itinerant bosonic gases in two-dimensional optical lattices, in order to predict the equilibrium ground-state phases, as well as the signatures of supersolidity and its formation in spectral functions and the dynamic and static structure factor. Specifically, we focus on two limiting cases. Firstly, we consider a two-component gas, as realized by two hyperfine ground states, for example, of rubidium-87, where one component is off-resonantly excited to a Rydberg state, which generates a soft-core shaped interaction potential. Secondly, we discuss the opposing limit, using near-resonant excitations
of Rydberg states, where the interacting component now directly corresponds to the Rydberg state, which interacts via a van der Waals potential. In both cases we discuss the rich variety of supersolid phases, which are found for a wide range of parameters. We also discuss how some of these phases can be realized in experiment.
In the subsequent appendices (A to D) we discuss some methodological details. Most notably, we consider the possible Fock-extension of the Hartree approximation (appendix A), introduced in the RB-DMFT treatment of the extended Bose-Hubbard model.