In this thesis I present the work done during my PhD in the area of low dimensional quantum gases. The chapters of this thesis are self contained and represent individual projects which have been peer reviewed and accepted for publication in respected international journals. Various systems are considered, the first of which is a two particle model which possesses an exact analytical solution. I investigate the non-classical correlations that exist between the particles as a function of the tunable properties of the system. In the second work I consider the coherences and out of equilibrium dynamics of a one-dimensional Tonks-Girardeau gas. I show how the coherence of the gas can be inferred from various properties of the reduced state and how this may be observed in experiments. I then present a model which can be used to probe a one-dimensional Fermi gas by performing a measurement on an impurity which interacts with the gas. I show how this system can be used to observe the so-called orthogonality catastrophe using modern interferometry techniques. In the next chapter I present a simple scheme to create superposition states of particles with special emphasis on the NOON state. I explore the effect of inter-particle interactions in the process and then characterise the usefulness of these states for interferometry. Finally I present my contribution to a project on long distance entanglement generation in ion chains. I show how carefully tuning the environment can create decoherence-free subspaces which allows one to create and preserve entanglement.
In this thesis I present the work done during my PhD in the area of low dimensional quantum gases. The chapters of this thesis are self contained and represent individual projects which have been peer reviewed and accepted for publication in respected international journals. Various systems are considered, the first of which is a two particle model which possesses an exact analytical solution. I investigate the non-classical correlations that exist between the particles as a function of the tunable properties of the system. In the second work I consider the coherences and out of equilibrium dynamics of a one-dimensional Tonks-Girardeau gas. I show how the coherence of the gas can be inferred from various properties of the reduced state and how this may be observed in experiments. I then present a model which can be used to probe a one-dimensional Fermi gas by performing a measurement on an impurity which interacts with the gas. I show how this system can be used to observe the so-called orthogonality catastrophe using modern interferometry techniques. In the next chapter I present a simple scheme to create superposition states of particles with special emphasis on the NOON state. I explore the effect of inter-particle interactions in the process and then characterise the usefulness of these states for interferometry. Finally I present my contribution to a project on long distance entanglement generation in ion chains. I show how carefully tuning the environment can create decoherence-free subspaces which allows one to create and preserve entanglement.
In this thesis, I present my work during my Ph.D. in quantum Darwinism. It is a original project, which has been peer-reviewed and accepted for publication in a respected international journal. In the first chapters, I recalled the essential concepts to understand quantum Darwinism, and in the two last, I present my works. The quantum Darwinism concept was constructed to explain a possible quantum-to-classical transition. My main interests are to find the quantum Darwinism implications in the system’s dynamics and to find more realistic models where it applies. The first work was finished and published while the second is still in progress. In the first model I studied quantum Darwinism in a many-body system of quantum harmonic oscillators where the subenvironments neither interact nor correlate between themselves. Through analytical and numerical calculations, we observed quantum Darwinism from two different approaches. We show quantum Darwinism through the BPH approach in a model for the first time. In contrast with a recently published work, we also show that quantum Darwinism can be observed even in a system with a high degree of non-Markovianity, and proposed a more suitable way to quantify it. In the second work, we investigate quantum Darwinism in a more realistic model where the subenvironments can become strongly correlated. The model consists of a two-level atom interacting with a fermionic environment and was used to study orthogonality catastrophe. As both orthogonality catastrophe and quantum Darwinism comes with decoherence, we are investigating if these two concepts are correlated. Beyond that, we chose this model because it can be mapped in a bosonic model of the Tonks-Girardeau gas enabling an experimental implementation with cold atoms. We are performing numerical and analytical calculations in this work. In the sixth chapter, I present the first result and describe what is missing.
In this PhD thesis we investigate some properties of one-dimensional quantum systems, focusing on two important aspects of integrable models: Their entanglement properties at equilibrium and their dynamical correlators after a quantum quench. The first part of the thesis will therefore be devoted to the study of the entanglement entropy in one-dimensional integrable systems, with a special focus on the XYZ spin-1/2 chain, which, in addition to being integrable, is also an interacting theory with non perturbative solutions. We derive its bipartite Renyi entropies in the thermodynamic limit and its behaviour in different phases and for different values of the mass-gap is analysed, both analytically and numerically. In particular it is worth mentioning that the numerical analysis of the entropies of the XYZ model presented in this dissertation has never been published in literature. In the second part of the thesis we study the dynamics of correlators after a quantum quench, preparing the system in a squeezed coherent initial state. The emphasis will be on the Transverse Field Ising Chain and the O(3) non-linear sigma model, the latter studied by means of the semi-classical theory, the former by a form-factor approach. Moreover in the last chapter we outline a general result about the dynamics of correlation functions of local observables after a quantum quench. In particular we show that if there are not long-range interactions in the final Hamiltonian, then the dynamics of the model at long times (non equal- time correlations) is described by the same statistical ensemble that describes its statical properties (equal-time correlations). For the Transverse Field Ising Chain this result means that its dynamics is described by a Generalized Gibbs Ensemble.
This thesis studies four related projects: tuning entanglement for spin systems, de-coherence and dynamics of entanglement, entanglement as measure of electron correlation in chemical systems and quantum algorithms to search for global minima. Chapter 1 presents a general introduction and the motivation for these research areas. It is composed of a brief review of the history and the future of modern computers and the state-of-art areas in quantum computation and information. Chapter 2 will be dedicated to introducing the concept of entanglement, its importance, quantification and applications. Then in Chapter 3, the properties of a set of localized spins, which are coupled to an external magnetic field through exchange interaction, are investigated. It is shown that entanglement can be controlled and tuned by varying the anisotropy parameter in the Hamiltonian, and by introducing impurities into the systems. Furthermore, the scheme for amplifying the internal entanglement by external interaction is also studied. In Chapter 4, the applications of the entanglement concept for de-coherence and dynamics is considered. In particular, the dynamics of entanglement for one-dimensional spin systems, which are coupled through nearest neighbor exchange interaction and subject to an external time-dependent magnetic field, is investigated. Using the two-site density matrix, the time-dependent entanglement of formation between nearest neighbor qubits is calculated. It is found that the entanglement can be localized between nearest neighbor qubits for certain values of the external time-dependent magnetic field. In quantum chemistry calculations, the correlation energy is defined as the difference between the Hartree-Fock limit energy and the exact solution of the non-relativistic Schrödinger equation. In Chapter 5, the application of entanglement as an alternative measure of the electron correlation in quantum chemistry calculations is proposed. Entanglement is directly observable and it is one of the most striking properties of quantum mechanics. Finally, Chapter 6 presents the latest research results in developing quantum algorithms for searching the global minima. In this Chapter, the modified Grover’s quantum algorithm using modest numbers of quantum bits to find a global minimum for real problems is demonstrated.
In this thesis we study the physics of quantum many-body systems confined to one-dimensional geometries. The work was motivated by the recent success of experimentalists in developing atom traps, capable of restricting the motion of the individual atoms to a single spatial dimension. Specifically, we look at aspects of the one-dimensional Bose gas including; excitation spectrum, correlation functions, and dynamical behaviour. In Chapter \ref{ch:excitation1D} we consider the Lieb-Liniger model of interacting bosons in one-dimension. We numerically solve the equations arising from the Bethe ansatz solution for the exact many-body wave function in a finite-size system of up to twenty particles for attractive interactions. We discuss novel features of the solutions, including deviations from the well-known string solutions due to finite size effects. We present excited state string solutions in the limit of strong interactions and discuss their physical interpretation, as well as the characteristics of the quantum phase transition that occurs as a function of interaction strength in the mean-field limit. Our results are compared to those obtained via exact diagonalization of the Hamiltonian in a truncated basis. In Chapter \ref{ch:g2} we analytically calculate the spatial nonlocal pair correlation function for an interacting uniform one dimensional Bose gas at finite temperature and propose an experimental method to measure nonlocal correlations. Our results span six different physical realms, including the weakly and strongly interacting regimes. We show explicitly that the characteristic correlation lengths are given by one of four length scales: the thermal de Broglie wavelength, the mean interparticle separation, the healing length, or the phase coherence length. In all regimes, we identify the profound role of interactions and find that under certain conditions the pair correlation may develop a global maximum at a finite interparticle separation due to the competition between repulsive interactions and thermal effects. In Chapter \ref{ch:casimirdrag} we study the drag force below the critical velocity for obstacles moving in a superfluid. The absence of drag is well established in the context of the mean-field Gross-Pitaevskii theory. We calculate the next order correction due to quantum and thermal fluctuations and find a non-zero force acting on a delta-function impurity moving through a quasi-one-dimensional Bose-Einstein condensate at all subcritical velocities and at all temperatures. The force occurs due to an imbalance in the Doppler shifts of reflected quantum fluctuations from either side of the impurity. Our calculation is based on a consistent extension of Bogoliubov theory to second order in the interaction strength, and finds new analytic solutions to the Bogoliubov-de Gennes equations for a gray soliton. In Chapter \ref{ch:solitons} we study the effect of quantum noise on the stability of a soliton. We find the soliton solutions exactly define the reflectionless potentials of the Bogoliubov-de Gennes equations. This results in complete stability of the solitons in a purely one dimensional system. We look at the modifications to the density profile of a black soliton due to quantum fluctuations.
Motivated by rapid experimental progress in the fields of ultracold atoms and quantum optics, I present a series of theoretical studies which explore collective phenomena in quantum gases of atoms and photons. In Chapter 1, I highlight the major developments in the research field and identify the overarching themes and motivations. I also provide a roadmap for the rest of the thesis and summarize the main results. The remaining eight chapters contain original studies, organized along three broad motifs. In Chapters 2 through 5, I investigate how the nature of collective excitations and quasiparticles can be explored in modern experiments. More specifically, I model the dynamics of a spin impurity in a Bose lattice gas, develop a protocol for observing fractionalized excitations or anyons in an optical cavity, and characterize the collective dynamics of Bogoliubov quasiparticles and domain walls in a Fermi superfluid. In Chapters 6 and 7, I examine unconventional superfluid phases in spin-imbalanced Fermi gases. In particular, I propose a novel technique for engineering the long-sought-after Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase and study the relative stability of exotic phases across a dimensional crossover. Finally, Chapters 8 and 9 are devoted to studies of kinetics in out-of-equilibrium systems. I model the formation of a Bose-Einstein condensate in a dimple trap and characterize the approach to thermal equilibrium in quasi-one-dimensional geometries.
Strongly correlated quantum systems display a myriad of complex, fascinating behaviour. The competition between charge, spin, orbital, or lattice degrees of freedom and their respective energy scales gives rise to rich and complex emergent phases. More recently, the non-equilibrium properties of correlated quantum many-body systems have come into the focus of active research, with remarkable advances. While the advent of tensor network based simulations on the theoretical side has allowed access to the full time-evolution of the quantum state as it traverses the Hilbert space, the experimental progress in the ultracold atom community has allowed physicists to enter the paradigm of analog quantum simulation. Their tunability and isolation from the environment makes them ideal platforms to emulate and study open problems in condensed matter physics and go beyond the simulational capabilities of state of the art classical simulations. In this thesis, we explore the dynamics of correlated Fermi gases. We develop and present different facets of applicability of radiofrequency modulation techniques to drive the system out of equilibrium. We begin by considering a homogeneous, three-dimensional Fermi gas in the BCS-BEC crossover and develop an off-resonant radiofrequency transfer scheme to excite the Higgs mode of the superfluid. In a subsequent study we investigate the tunability of the coherence between Cooper pairs by tuning the duration of interaction ramps of the internal interaction strength. These works highlight the possibility to externally tune and stabilise complex quantum many-body states away from equilibrium. Using quasi-exact matrix product state simulations, we study the response of an attractively interacting, one-dimensional Fermi-Hubbard model to weak radiofrequency perturbations. We reveal the emergence of two distinct dynamical regimes in the time evolution and are able to relate it back to the underlying excitation spectrum of the system. Finally, we explore the possibility to perform quantum quenches through radiofrequency π-pulses, achieving nearly complete population transfer between different interacting fermionic states. We reveal non-trivial dynamical effects in the pair correlation of the final state, as well as the excitation of a collective trap mode of the system.