We develop an explicit description of a time-dependent response of fermionic condensates to perturbations. The dynamics of Cooper pairs at times shorter than the energy relaxation time can be described by the BCS model. We obtain a general explicit solution for the dynamics of the BCS model. We also solve a closely related dynamical problem - the central spin model, which describes a localized spin coupled to a "spin bath". Here, we focus on presenting the solution and describing its general properties, but also mention some applications, e.g. to nonstationary pairing in cold Fermi gases and to the issue of electron spin decoherence in quantum dots. A typical dynamics of the BCS and central spin models is quasi-periodic with a large number of frequencies and stable under small perturbations. We show that for certain special initial conditions the number of frequencies decreases and the solution simplifies. In particular, periodic solutions correspond to the ground state and excitations of the BCS model.
The spin degree of freedom has become subject to one of the broadest and most intensively studied fields of research in physics. The present thesis is part of this field and deals with so-called central spin models.
The thesis is devided into two parts. In the first one we investigate very basic properties of central spin models on an abstract level. We introduce a new, or rather enhanced, notion of quantum integrability. From this we derive an explicit and, to a certain extent, analytical relation between integrability and decoherence. Then we specify a model of two coupled Gaudin models with all coupling constants being equal to each other (``homogeneous''). We numerically study the spectral properties of the model and find the spectrum to exhibit systematically degenerate multiplets. We briefly discuss possible applications in solid state quantum information processing.
The second part is devoted to concrete physical considerations in the context of solid state quantum information processing. Here we consider two exchange coupled electron spins confined in semiconductor quantum dots and evaluate the hyperfine induced spin dynamics for the case of a weak magnetic field applied to the electron spins by exact methods.
In a first step we consider the two exchange coupled electron spins to interact with the same bath of nuclear spins via homogeneous couplings. In particular we derive a scaling law for the decoherence time. In a second step we try to turn vice into virtue by investigating to what extent it is possible to swap and entangle two separate nuclear spin baths. Surprisingly, it turns out that a nuclear swap is possible for weakly coupled nuclear spin baths of realistic size, but not for smaller ones. We indicate that under the same circumstances it should also be possible to fully entangle the baths.
We close the thesis with a summary and an outlook on further important questions.
In this thesis, we describe methods for solving quantum impurity models out of equilibrium, and consider several applications of these methods. Quantum impurity models describe systems in which a small subsystem (the impurity) is coupled to a large external environment (the bath). These models are useful in describing a wide variety of physical systems, including quantum dots, absorbed adatoms, and molecular junctions. They are also important for the study of lattice models, such as the Hubbard model, where they appear as auxiliary problems within the framework of dynamical mean field theory (DMFT). Many interesting phenomena in these systems, such as transport through a quantum dot, or optical pumping of a Mott insulator, occur away from equilibrium. Theoretical description of these phenomena requires methods that are capable of handling both real time evolution and strong correlation effects presenting a formidable challenge. In this thesis, we describe several methods for the solution of non-equilibrium quantum impurity models which meet this challenge. These methods are based on an expansion in the coupling between the impurity and the bath known as the hybridization expansion. Chapter 1 gives an introduction to the problems addressed in this thesis. Chapter 2 provides a brief overview of Keldysh formalism for nonequilibrium problems. Chapter 3 describes the bare hybridization expansion for the Anderson impurity model as well as its bold extensions. In particular we derive the non-crossing approximation (NCA), the one-crossing approximation (OCA), the inchworm expansion, and a bold expansion for the impurity Green's function. Chapter 4 describes the implementation of a quantum Monte Carlo algorithm based on the inchworm expansion. The rest of the thesis describes applications of these algorithms to problems of physical interest. Chapters 5 and 6 apply the methods previously described to understanding the Kondo effect in quantum dot systems. The first chapter examines the behavior of the Kondo effect under an applied voltage bias, a classic non-equilibrium strongly-correlated problem. The second chapter examines different ways of measuring the cloud of electrons which screen the impurity spin in the Kondo regime. Chapters 7 and 8 apply the same methods to the Hubbard model using the Dynamical mean-field theory (DMFT) mapping. The first chapter studies the Hubbard model in equilibrium using the numerically exact inchworm quantum Monte Carlo method. The second chapter studies the possibility of dynamic switching between the metal and Mott-insulator states of the Hubbard model using the one-crossing approximation (OCA). Finally chapter 9 gives some overall conclusions and suggests future directions for the development of hybridization expansion methods.
This thesis describes aspects of dynamics of solid state systems which are relevant to quantum metrology and computation. It may be divided into three research directions (parts). For the first part, a new method to enhance precision measurements that makes use of a sensor’s environment to amplify its response to weak external perturbations is described. In this method a “central” spin is used to sense the dynamics of surrounding spins, which are affected by the external perturbations that are being measured. The enhancement in precision is determined by the number of spins that are coupled strongly to the central spin and is resilient to various forms of decoherence. For polarized environments, nearly Heisenberg-limited precision measurements can be achieved. The second part of the thesis focuses on the decoherence of Majorana fermions. Specializing to the experimentally relevant case where each mode interacts with its own bath we present a method to study the effect of external perturbations on these modes. We analyze a generic gapped fermionic environment (bath) interacting via tunneling with individual Majorana modes - components of a qubit. We present examples with both static and dynamic perturbations (noise), and derive a rate of information loss for Majorana memories, that depends on the spectral density of both the noise and the fermionic bath. For the third part of the thesis we discuss vortices in topological superconductors which we model as closed finite systems, each with an odd number of real fermionic modes. We show that even in the presence of many-body interactions, there are always at least two fermionic operators that commute with the Hamiltonian. There is a zero mode corresponding to the total Majorana operator [1] as well as additional linearly independent zero modes, one of which is continuously connected to the Majorana mode in the non-interacting limit. We also show that in the situation where there are two or more well separated vortices their zero modes have non-Abelian Ising statistics under braiding.
This PhD thesis is concerned with the investigation of Extended Hubbard Models on one-dimensional lattices (chains); the approach used to such investigation is that of determining exact results. The extended Hubbard models are models of correlated electrons on lattices; they are the extensions of a model proposed by Hubbard in \\cite{HUB} to explain the metal-insulator transition in
d-transition compounds as the effect of electronic correlations, usually neglected in band theories. Later on, a need for extensions of such model was first felt within the context of high-temperature superconductivity; indeed, the different behaviour of doped cuprates with respect to conventional superconductors suggested that a microscopic theory of this phenomenon could be formulated in terms of electron-electron interaction, without explicitly invoking the phonon coupling, like in BCS theory. Although an exhaustive microscopic formulation for cuprate superconductivity has not been achieved yet, the Extended Hubbard Models have been in the meanwhile applied to other problems in condensed matter; they are actually considered the reference paradigm for treating all those systems in which electronic correlations cannot be neglected. \\\\As observed above, this thesis focuses in particular on 1-D chains: indeed in last years the progress in technological synthesization and small-scale observations has shown that there are many classes of materials exhibiting a strong one-dimensional character, such as organic (super)conductors, linear-chain polymers, quantum wires, quantum-dot chains, one-dimensional cuprates. Many experimental results confirm that such materials exhibit a quite rich behaviour, with different phases, charge-spin separation, as well as a strong electronic correlations. However, a sufficient understanding of these phenomena on a theoretical point of view has not been reached yet. This is due to the fact the usual theoretical methods can be shown to fail in D=1, and theoreticians are thus forced to develop new schemes for approaching the problem of many interacting fermions in such dimension. The latter therefore represents a fascinating challenge. Some progress has been made and many aspects are nowadays more clear; nevertheless, due to the lack of traditional techniques, the need for a comparison with exact results is perhaps more urgent in D=1 than in higher dimensions. The aim of this thesis is to provide some results to the research in this field. The present work is organized as follows: the two first chapters are introductory, while chapters 3 and 4 mostly contain the author's original contribution. More explicitly, {\\bf chapter 1} is an introduction to the Hubbard models (both the ordinary and the Extended) and is meant for the reader who is non-familiar with these models. We tried to emphasize the motivation that historically led to their introduction and the physical contexts in which they are applied, not necessarily in one-dimension. \\\\{\\bf Chapter 2} is instead devoted to general problematics of one dimensional physics. We have tried to stress that there is a number of physically interesting materials that can be regarded as actual 1-D systems; we give a brief account of the difficulties leading the usual theoretical approaches, such as mean-field and Fermi Liquid, to fail in D=1. While briefly describing the important results of the proposed alternative formulations (Luttinger Liquid, Renormalization Group), we also point out the points that are still unsatisfactory, and emphasize the importance that exactly integrable models have in this field. \\\\Matching the subjects of chapter 1 and 2, the subsequent ones contain the results of our investigation of exactly solvable Extended Hubbard model on 1-D chains. \\\\We wish to emphasize to the reader non familiar with integrable systems that the literature on this subject is often strongly concerned with the mathematical aspects of the involved methods, leaving the determination of the physical features of the derived integrable models as a secondary issue. On the contrary, we believe that both aspects of the research deserve a relevant role in the study. For these reasons, while chapter 3 contains new results about the general formulation of integrability for fermionic systems, chapter 4 is devoted to the investigation of the physical features for some models whose integrability has been proved in the previous chapter. \\\\More explicitly, in {\\bf chapter 3} (apart from a preliminary introduction of sec.3.1) the theory of integrable {\\it fermionic} systems has been developed within the frame of the Quantum Inverse Scattering Method (QISM). The latter is a technique for obtaining integrable models, traditionally formulated with success for spin systems; its more recent application to electron systems was leaving many unsatisfactory aspects. To overcome this weaknesses, in sec.3.2 we both clarify the theory of matrix representation for fermionic operators, and describe a method for obtaining integrable models of fermions (straightforwardly readable in terms of fermion operators), starting from the the solutions of a
mathbbC-number equation (known as Yang-Baxter Equation). In sec.3.3 we first work out a general method of search for solutions of such equation; secondly we explicitly find 96 solutions, corresponding to 96 integrable Extended Hubbard models (most of which are new) for which a set of mutually commuting operators is proved to exist. In sec.3.4 we formulate a general theorem concerning the symmetries of the integrable models in QISM, and widely discuss the symmetries and supersymmetries for the class of extended Hubbard Models.\\\\ {\\bf Chapter 4} is devoted to the investigation of the physical properties of the integrable Extended Hubbard models found in chapter 3. We introduce a technique, called Sutherland's species technique, that, when applicable, yields an effective reduction of the Hilbert space and considerably simplifies the problem of diagonalizing the Hamiltonian. In fact, such technique is proved to be applicable to all the 96 models found above, and is used to determine the exact ground state phase diagram and the thermodynamics of the models. In particular we both recover in a quite straightforward way the phase diagram of a known model, and that of two new models (Model 2 and Model 3) that were not known before. \\\\The phase diagram exhibit structured forms with respect to the filling (density of electrons) and the on-site Coulomb repulsion (
U); the pressure and the compressibility in the ground state are also calculated. Model 2 exhibits a superconducting phase particularly enhanced towards positive values of
U, as well as a band-controlled insulator-superconductor transition at half filling; model 3 is the first example of exactly solved model with an asymmetric shape with respect to half-filling, due to the absence of particle-hole invariance of the Hamiltonian; a filling controlled metal-insulator transition is found to occur for this model. Generalizations to the other cases are also discussed. \\\\As to the thermodynamics, we compute the exact partition function of Model 3, which we believe to be particular worth of interest due to the above asymmetry. We then derive the exact specific heat of the model as a function of the temperature, which exhibits an interesting two-peak behaviour as well as a nearly universal crossing point. These results are discussed and compared with similar ones present in the literature. The chapters are of course linked each to another according to the above scheme; nevertheless, it has been a purpose of the author to make them as independently readable as possible. In particular, the reader who is more interested in the physical aspects of the models rather than in the general formalism of fermionic QISM can skip chapter 3 (at least in a first-sight reading) and pass directly to chapter 4; apart from some references to previous formulas the understanding should not be affected.\\\\{\\it \\noindent \\underline{Notational Remark:}} The customary symbol to denote the electron filling is
n; this has in fact been used throughout the two first introductory chapters. Nevertheless, in order to avoid possible misunderstanding with other extensive quantities, in chapter 3~and~4~the~symbol~has been changed into
rhoThe characterization of the dynamical structure of quantum many-body systems is an important and very active endeavor both experimentally and theoretically. Several tools, such as quantum-information theoretic observables and spectral measures, provide a powerful lens for characterizing correlations and their dynamics in quantum many-body systems. These quantities can reveal important features of experimentally realizable systems and provide insight into their control and dynamical behaviour. This thesis investigates nonequilibrium settings in which the spreading and redistribution of quantum correlations exhibit unconventional behaviour, with particular emphasis on the role of extended-range correlations. We consider two main classes of systems: lattice models undergoing quantum quenches and central-spin models motivated by experimental applications. For lattice systems, we study the dynamics of entanglement and local symmetry breaking in free fermionic and bosonic models, interacting integrable systems, and quantum circuits. We first investigate symmetry restoration following quenches from ground states of long-range fermionic Hamiltonians. We show that two states undergoing symmetry restoration can exhibit multiple inversions, manifested through multiple crossings of the distance measure, implying either the evasion of the quantum Mpemba effect or a more intricate route to its occurrence. This demonstrates that anomalous relaxation is not restricted to short times and can be understood through the momentum-dependent distribution and velocities of quasiparticle excitations. We then study quenches from initial states possessing extensive, spatially organized entanglement, moving beyond the conventional setting of weakly entangled initial states. For crosscap states, characterized by antipodal correlations and volume-law entanglement entropy, we uncover unconventional entanglement dynamics and formulate an antipodal quasiparticle picture in which initially entangled degrees of freedom originate from macroscopically separated points. We subsequently generalize this construction to entangled multipodal states, whose polygon-like correlation structure produces richer equilibrium and dynamical entanglement patterns. We further investigate charge-deformed crosscap states and show that their local symmetry-breaking dynamics can display the unusual phenomenon of dynamical symmetry breaking under symmetry-preserving evolution. Finally, we extend the crosscap construction to bosonic systems and describe the dynamics of the entanglement entropy, mutual information, and logarithmic negativity within a corresponding quasiparticle framework. The second class of systems considered concerns central-spin models and their application to quantum memories in semiconductor quantum dots. We investigate how disorder in couplings and frequencies modifies collective many-body dynamics and affects the storage and retrieval of quantum information. Using spectral, dynamical, and information-theoretic diagnostics, we characterize the resulting dynamical regimes and relate the loss of collectivity and the spreading of information through the spin bath to the performance of quantum-memory protocols. Overall, the results demonstrate how extended-range correlations, whether encoded in an initial state or generated by the interaction structure, can qualitatively reshape entanglement spreading, symmetry dynamics, and information retention in nonequilibrium quantum many-body systems.