In the first part, we investigate the effect of long range particle exchange in ideal bosonic chains. We establish that by using the Heisenberg formalism along with matrix product state representation we can study the evolution as well as the ground state of bosonic arrangements while including terms beyond next-neighbour hopping. The method is then applied to analyse the quench dynamics of condensates in a trapping potential and also to study the emergence of entanglement as a result of collision in boson chains. In the second part, we study the ground state as well as the dynamics of 1D boson arrangements with local repulsive interactions and nearest-neighbour exchange using numerical techniques based on time evolving block decimation (TEBD). We focus on the development of quantum correlations between the terminal places of these arrangements. We find that long-range entanglement in the ground state arises as a result of intense boson tunnelling taking place across the whole chain in systems with appropriate hopping coefficients. Additionally, we identify the perturbations necessary to increase the entanglement between the end sites above their ground state values. In the final part, we study the wave function of a kicked condensate using a perturbative approach and compare the results obtained in this way with numerical simulations.
In the first part, we investigate the effect of long range particle exchange in ideal bosonic chains. We establish that by using the Heisenberg formalism along with matrix product state representation we can study the evolution as well as the ground state of bosonic arrangements while including terms beyond next-neighbour hopping. The method is then applied to analyse the quench dynamics of condensates in a trapping potential and also to study the emergence of entanglement as a result of collision in boson chains. In the second part, we study the ground state as well as the dynamics of 1D boson arrangements with local repulsive interactions and nearest-neighbour exchange using numerical techniques based on time evolving block decimation (TEBD). We focus on the development of quantum correlations between the terminal places of these arrangements. We find that long-range entanglement in the ground state arises as a result of intense boson tunnelling taking place across the whole chain in systems with appropriate hopping coefficients. Additionally, we identify the perturbations necessary to increase the entanglement between the end sites above their ground state values. In the final part, we study the wave function of a kicked condensate using a perturbative approach and compare the results obtained in this way with numerical simulations.
We study the ground state as well as the dynamics of chains of bosons with local repulsive interactions and nearest-neighbor exchange using numerical techniques based on density-matrix decimation. We explore the development of entanglement between the terminal sites of such chains as mechanisms are invoked to concentrate population in these sites. We find that long-range entanglement in the ground state emerges as a result of transfer taking place throughout the chain in systems with appropriate hopping coefficients. Additionally, we find appropriate perturbations to increase the entanglement between the end sites above their ground-state values.
We study the effect of long range particle exchange in bosonic arrangements. We show that by combining the solution of the Heisenberg equations of motion with matrix product state representation it is possible to investigate the dynamics as well as the ground state while including particle exchange beyond next-neighbours sites. These ideas are then applied to study the emergence of entanglement as a result of scattering in boson chains. We propose a scheme to generate highly entangled multi-particle states that exploits collision as a powerful entangling mechanism.
We study non-equilibrium and strongly-correlated dynamics in two contexts. We begin by analysing quantum many-body systems out of equilibrium through the lens of cold atomic impurities in Bose gases. Such highly-imbalanced mixtures provide a controlled arena for the study of interactions, dissipation, decoherence and transport in a many-body quantum environment. Specifically we investigate the oscillatory dynamics of a trapped and initially highly-localised impurity interacting with a weakly-interacting trapped quasi low-dimensional Bose gas. This relates to and goes beyond a recent experiment by the Inguscio group in Florence. We witness a delicate interplay between the self-trapping of the impurity and the inhomogeneity of the Bose gas, and describe the dissipation of the energy of the impurity through phononic excitations of the Bose gas. We then study the transport of a driven, periodically-trapped impurity through a quasi one-dimensional Bose gas. We show that placing the weakly-interacting Bose gas in a separate periodic potential leads to a phononic excitation spectrum that closely mimics those in solid state systems. As a result we show that the impurity-Bose gas system exhibits phonon-induced resonances in the impurity current that were predicted to occur in solids decades ago but never clearly observed. Following this, allowing the bosons to interact strongly, we predict the effect of different strongly-correlated phases of the Bose gas on the motion of the impurity. We show that, by observing the impurity, properties of the excitation spectrum of the Bose gas, e.g., gap and bandwidth, may be inferred along with the filling of the bosonic lattice. In other words the impurity acts as a probe of its environment. To describe the dynamics of such a strongly-correlated system we use the powerful and near-exact time-evolving block decimation (TEBD) method, which we describe in detail. The second part of this thesis then analyses, for the first time, the performance of this method when applied to simulate non-equilibrium classical stochastic processes. We study its efficacy for a well-understood model of transport, the totally-asymmetric exclusion process, and find it to be accurate. Next, motivated by the inefficiency of sampling-based numerical methods for high variance observables we adapt and apply TEBD to simulate a path-dependent observable whose variance increases exponentially with system size. Specifically we calculate the expected value of the exponential of the work done by a varying magnetic field on a one-dimensional Ising model undergoing Glauber dynamics. We confirm using Jarzynski's equality that the TEBD method remains accurate and efficient. Therefore TEBD and related methods complement and challenge the usual Monte Carlo-based simulators of non-equilibrium stochastic processes.
The study of interacting quantum many-body systems is one of the central challenges in modern physics. Interestingly, by understanding the physical properties of such systems from a microscopic description, one can explain many complex quantum phenomena such as the high-temperature superconductivity, fractional quantum Hall effect, etc. However, due to high complexity of these systems, theoretical investigations such as analytical and perturbation theories are limited to studying only a small subset of problems. In the past few decades, numerical simulation has emerged as an indispensable method for studying many phenomena, and has led to immense success and progress in recent years. The main obstacle in addressing quantum many-body systems is that the number of parameters required to represent a state of the system scales exponentially with the system size. However, one hardly needs the full Hilbert space but only a small subspace to describe physical properties of the system. From the point of view in quantum information theory, quantum entanglement is one of the main factors that is responsible for the complexity of studying quantum many-body systems. In particular, the amount of quantum entanglement in such system quantifies the correlation between constituents and therefore it determines how big the subspace is required to represent the system. In this thesis, we are interested in using tensor network states, a new emerging numerical technique to study dynamical and ground state properties of the system. The key ingredient is that a tensor network state which consists of a fairly small number of parameters can represent very well the complete wave function of a system. In one spatial dimension, Matrix Product State (MPS) has become a standard tool, combined with related algorithms such as Time Evolving Block Decimation algorithm (TEBD) to study dynamical properties of a locally-perturbed quantum lattice system. In this thesis, we introduce a technique called ``infinite boundary conditions'' that allows us to use a finite-size MPS to represent the state of an infinite-size system. As a result, one can simulate dynamics of a system for a longer time with a significant computational cost reduction compared to traditional methods. In two spatial dimensions, the tensor network known as Projected Entangled Pair States (PEPS) is emerging as a useful approach. However algorithms for optimizing the matrix elements of these tensors are complex and not very efficient. In particular, the efficiency of the optimization depends critically on the quality of the so-called ``environment'', which is expensive to compute. We have introduced a new method to accelerate the convergence of calculating the PEPS ground state by recycling the environment tensors for many optimization steps, vastly reducing the number of times that the full environment needs to be calculated.
In this dissertation a newly developed numerical method is presented, which is optimized for the time evolution of one-dimensional lattice systems with large local Hilbert spaces. This method extends the time-evolving block-decimation (TEBD) to include a local basis optimization (LBO), which has already been successfully combined with ground state methods. The algorithm is based on matrix product states (MPS), which can represent the quantum state of a one-dimensional chain in most cases with a number of parameters, that is not exponentially increasing with the chain length. The LBO causes a reduction of the simulation times that is linear in the bond dimension of the MPS. To demonstrate the advantages of this method, we apply the TEBD-LBO to electron-phonon (e-p) systems. In this thesis these are described by the Holstein model, which goes beyond semi-classical approximations and covers the full quantum statistics of the phonons. The understanding of the nonequilibrium dynamics of charge carriers coupled to lattice vibrations is of great importance for research areas like transport through quasi one-dimensional conductors, photo-generated phase transitions and time-resolved spectroscopy. First, the energy transfer from a highly excited electron to the phononic degrees of freedom on a small chain is studied. In the various parameter regimes different types of relaxation occur. In any case, after a certain time the system reaches a state, where on average no energy is exchanged between the electron and phonons. This can either mean a constant kinetic energy or oscillations with a constant amplitude and frequency. Next, long perfectly conducting leads without coupling to phonons were attached on both sides of the small chain. In the left lead an electron with density distribution in the shape of a Gaussian wave packet is injected with momentum towards the e-p coupled structure in the middle. This structure acts as an impurity in an otherwise perfectly conducting chain. The investigation shows resonance effects in the transmission and reflection at this impurity. Further, the electron can transfer a part of its energy permanently to the phonons, which results in a reduction of the velocity. Finally, two mechanisms are presented that lead to self-trapping of the electron on the e-p coupled structure.
In recent years, the systems comprising of bosonic atoms confined to optical lattices at ultra-cold temperatures have demonstrated tremendous potential to unveil novel quantum mechanical effects appearing in lattice boson models with various kinds of interactions. In this progress report, we aim to provide an exposition to recent advancements in quantum simulations of such systems, modeled by different 'non-standard' Bose-Hubbard models, focusing primarily on long-range systems with dipole-dipole or cavity-mediated interactions. Through a carefully curated selection of topics, which includes the emergence of quantum criticality beyond Landau paradigm, bond-order wave insulators, the role of interaction-induced tunneling, the influence of transverse confinement on observed phases, or the effect of cavity-mediated all-to-all interactions, we report both theoretical and experimental developments from the last few years. Additionally, we discuss the real-time evolution of systems with long-range interactions, where sufficiently strong interactions render the dynamics non-ergodic. And finally to cap our discussions off, we survey recent experimental achievements in this rapidly evolving field, underscoring its interdisciplinary significance and potential for groundbreaking discoveries.