This thesis mainly concerned with traffic and transport problems. Traffic flow is understood as a many-body system in which moving particles have asymmetric interactions. The Totally Asymmetric Simple Exclusion Process (TASEP) is a non-equilibrium lattice model which has been used widely in traffic modelling. In fact, it is the paradigmatic model for most of transport processes. In this model, objects move forward in one direction along a line segment and they are not allowed to pass each other. Traffic jam and Protein synthesis are two interesting examples which can be studied using this model. For traffic jam model, the above rules apply to cars travelling on a one-way road. This model will provide us with information on how much traffic the road can carry, where to put traffic lights, how wide should be the road and where to put the intersections. For the other example, protein synthesis, the TASEP could model ribosome moving on mRNA strands to produce proteins. Understanding the mechanism of this process not only provides us information about the regulation and control of protein synthesis, but also may be helpful to find therapeutic applications, as ribosome is the target of many antibiotics. For such non-equilibrium systems, which have interactions with the environment, there are no general framework such as the laws of thermodynamics and statistical mechanics for closed systems in equilibrium. This increases the importance of using numerical techniques. In the TASEP model, the road (in vehicular traffic model) or the mRNA strands (in Protein synthesis) is represented a lattice of sites which are either occupied by exactly one particle or are empty. The Matrix Product States (MPS) representation is a useful approach to describe the probability distribution of these stochastic systems. The main idea is to efficiently represent the probability distribution of the stochastic system by a tensor network ansatz, which truncating unimportant degrees of freedom, can be a solution to the problem of the exponential increase in parameters with the size of the system, and correspondingly, the high computational costs. We can formulate the stochastic process in terms of an evolution operator similar to the Hamiltonian for a quantum many-body system. We use the adapted version of the Time Evolving Block Decimation (TEBD) algorithm — which is used for simulation of one-dimensional quantum systems — to study the evolution of the system and extract useful properties of the model in the steady state, such as density profile, current, velocity, correlation functions, etc. Then, we introduce an approach for evaluating the behaviour of a tagged particle in a system with open boundary condition and compute the distance travelled by this particle, the number of particle entered to the system during an interval time and time-dependent correlation functions.
This thesis mainly concerned with traffic and transport problems. Traffic flow is understood as a many-body system in which moving particles have asymmetric interactions. The Totally Asymmetric Simple Exclusion Process (TASEP) is a non-equilibrium lattice model which has been used widely in traffic modelling. In fact, it is the paradigmatic model for most of transport processes. In this model, objects move forward in one direction along a line segment and they are not allowed to pass each other. Traffic jam and Protein synthesis are two interesting examples which can be studied using this model. For traffic jam model, the above rules apply to cars travelling on a one-way road. This model will provide us with information on how much traffic the road can carry, where to put traffic lights, how wide should be the road and where to put the intersections. For the other example, protein synthesis, the TASEP could model ribosome moving on mRNA strands to produce proteins. Understanding the mechanism of this process not only provides us information about the regulation and control of protein synthesis, but also may be helpful to find therapeutic applications, as ribosome is the target of many antibiotics. For such non-equilibrium systems, which have interactions with the environment, there are no general framework such as the laws of thermodynamics and statistical mechanics for closed systems in equilibrium. This increases the importance of using numerical techniques. In the TASEP model, the road (in vehicular traffic model) or the mRNA strands (in Protein synthesis) is represented a lattice of sites which are either occupied by exactly one particle or are empty. The Matrix Product States (MPS) representation is a useful approach to describe the probability distribution of these stochastic systems. The main idea is to efficiently represent the probability distribution of the stochastic system by a tensor network ansatz, which truncating unimportant degrees of freedom, can be a solution to the problem of the exponential increase in parameters with the size of the system, and correspondingly, the high computational costs. We can formulate the stochastic process in terms of an evolution operator similar to the Hamiltonian for a quantum many-body system. We use the adapted version of the Time Evolving Block Decimation (TEBD) algorithm — which is used for simulation of one-dimensional quantum systems — to study the evolution of the system and extract useful properties of the model in the steady state, such as density profile, current, velocity, correlation functions, etc. Then, we introduce an approach for evaluating the behaviour of a tagged particle in a system with open boundary condition and compute the distance travelled by this particle, the number of particle entered to the system during an interval time and time-dependent correlation functions.
The nonequilibrium effects of dissipation and drive play a key role in an immense variety of nanoscale and mesoscale quantum systems. To understand the behaviour of open quantum systems, we need accurate methods that capture the influence of the environment on the system, while managing the exponentially large Hilbert space required to describe the system. Tensor network algorithms offer an efficient way to approach this challenge. In this thesis, we develop and apply tensor network techniques to study the dynamics and steady states of various open quantum systems. The first part of the thesis focuses on the driven dissipative many body physics in coupled cavity arrays described by Born-Markov master equations. We extend transfer matrix product operator methods to Liouvillian dynamics, and utilize them to compute dynamical correlation functions and fluorescence spectrum of an infinite coupled cavity array in 1D. We also investigate thermalization, and observe the emergence of a quasi-thermal steady state with a negative effective temperature. In another study, we use infinite projected entangled pair state (iPEPS) methods to compute steady states of coupled cavity lattices in 2D. We find that a straightforward adaptation of iPEPS to Liouvillian dynamics is unstable, contradicting a recent publication in the field. The second part investigates more general systems involving strong couplings and structured environments that induce non-Markovian dynamics. We develop a powerful time-evolving matrix product operator (TEMPO) algorithm that builds on Feynman-Vernon influence functional formalism, and uses matrix product states (MPS) to represent the temporal non-Markovian correlations efficiently. We apply TEMPO to study the localization phase transition of the spin-boson model and the dynamics of two spatially separated two-level systems coupled to a common environment. Finally, we propose the Toblerone TEMPO algorithm, which extends TEMPO to many-body systems interacting with general bosonic environments.
We consider the general problem of determining the steady state of stochastic nonequilibrium systems such as those that have been used to model (among other things) biological transport and traffic flow. We begin with a broad overview of this class of driven diffusive systems - which includes exclusion processes - focusing on interesting physical properties, such as shocks and phase transitions. We then turn our attention specifically to those models for which the exact distribution of microstates in the steady state can be expressed in a matrix product form. In addition to a gentle introduction to this matrix product approach, how it works and how it relates to similar constructions that arise in other physical contexts, we present a unified, pedagogical account of the various means by which the statistical mechanical calculations of macroscopic physical quantities are actually performed. We also review a number of more advanced topics, including nonequilibrium free energy functionals, the classification of exclusion processes involving multiple particle species, existence proofs of a matrix product state for a given model and more complicated variants of the matrix product state that allow various types of parallel dynamics to be handled. We conclude with a brief discussion of open problems for future research.
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.
[eng] Quantum many-body dynamics is a vast field of physics at the intersection of statistical mechanics, condensed matter, quantum information, and computational physics. Understanding the time evolution of observable correlations—such as those involved in material responses—is both a milestone for technological advancement and a corner-stone for generating fundamental knowledge. At the intersection of emergent collective behavior in many-body systems, quantum physics, and information theory lies the classical simulation of out-of-equilibrium phenomena. Without numerical tools, progress in disciplines that require heavy computations with high-dimensional vectors and matrices would be limited to purely analytical predictions and constrained by the available expertise and resources needed to build benchmarking experiments. The goal of this thesis is to advance the classical simulation of quantum many-body systems using Tensor Networks. This technique has proven fruitful for predicting equilibrium ground states and thermal properties of one-dimensional systems, yet it still faces challenges when applied to out-of-equilibrium scenarios. In brief, the dynamics of experimentally accessible expectation values can be represented as a 2D Tensor Network, which must be contracted using iterative methods such as Time-Evolving Block Decimation. However, the memory and computational time required for these calculations—which reflect the methods’ complexity—are known to grow exponentially with the simulated physical time in generic systems. This limitation is often referred to in the literature as the entanglement barrier. This manuscript tackles the problem through the three inequivalent ways in which the 2D Tensor Network can be contracted. In Chapter 3, we study how Schrödinger-picture contraction from the state edge of the network can be modified to improve the convergence of iterative contraction and reduce truncation errors in small systems. To this end, we explore physical decoherence in optimized bases and artificial generalized decoherence, finding that both alternatives show promise for new protocols that overcome the entanglement barrier. In Chapter 4, we explore Heisenberg-picture contraction from the observable edge, combining the concept of generalized decoherence with Pauli Weight truncation. By thoroughly characterizing the Pauli weight requirements to accurately describe the time evolution of local expectation values from initial product states, we conclude that truncating off-diagonal Pauli Weights is an effective scheme for simulating long-time dynamics—and we successfully implement it. We conclude with a spatial characterization of Pauli Weight spreading, bridging Tensor Network simulations and the field of operator hydrodynamics. Finally, in Chapter 5, we focus on the transverse contraction of the 2D Tensor Net-work using the light-cone algorithm, analyzing its variants in terms of truncation strategies and convergence across different models. This leads to a systematic characterization of the method based on the integrability of the underlying Hamiltonians and single-spin noise models.
The study of open quantum systems forms one of the main problems of modern physics. This thesis aims at contributing to this field, and focuses on the investigation of non-Markovian dynamics of open quantum systems. Here, as a special case, quantum many-body systems provide for a rich field of research. With the ability to control them being of essential importance for the development of quantum technologies, the characterization of their dynamical properties has also drawn a lot of interest lately. One of the main methodological tasks in the study of open quantum systems is to reduce the infinite degrees of freedom of the total system. In order to achieve this, the thesis makes use of the tensor network method matrix product states and combines it with the picture of the quantum stochastic Schrödinger equation. However, when computing the dynamics of many-body systems of system size N, even the reduced system dynamics scale with 2N, making larger system sizes difficult to access. Therefore, especially when computing non-Markovian time-dynamics where the information backflow from the reservoir has to be taken into account, efficiently constructed algorithms are required. This thesis proposes these type of algorithms and demonstrates their application. They enable an access to larger system sizes, thus making it possible to investigate non-equilibrium steady-states of quantum many-body systems in a non-Markovian interaction for system sizes up to N=30. This thesis consists of three parts. The first part provides an introduction to the theory of open quantum systems and to the methodology for modeling them. The second part examines many-body system-reservoir interaction in the non-Markovian regime by investigating the case of fully coherent self-feedback. It uses the one-dimensional Heisenberg spin-1/2 chain as a paradigmatic and recently intensely studied many-body system. It is demonstrated that this gives rise to new conditions for the feedback phase and to highly non-trivial states within the system as well as to the possibility of partially characterizing the system non-invasively and storing excitation within it. Next, a novel approach to the boundary-driven Heisenberg chain is presented, where the chain is exposed to a coherent driving field, thus enabling a comparison to the well-studied incoherently driven model. The third part demonstrates the use of an external pump to control the feedback phase of a Lambda-type three-level system, where the external pump pulse is shown to be a new control parameter. The thesis concludes with the proposition to enhance the indistinguishability of two photons by feedback control.
Recently, the possibility of investigating single molecule, or single spin observables, as well as the necessity of a better understanding of the mechanisms underlying quantum dynamics in order to obtain nanoscale devices and nanostructered materials suitable for quantum computing tasks, have revived the interest in foundational aspects of quantum statistical mechanics. This thesis aims to give a contribution to this field by re-considering the statistical characterization of a quantum system at the light of some paradigmatic changes in our understanding of quantum theory which have taken place in the last two decades. In particular the impressive development of quantum information theory has changed the perceptions of quantum entanglement: for a long time it has been considered a somewhat paradoxical property of the matter at the atomic scale, but now it is regarded as an essential and ubiquitous phenomenon whose consequences are affecting the very macroscopic world that we experience. Still the decoherence program has brought out the importance of considering a quantum system together with its environment in order to clarify some key aspects of quantum dynamics. Thus, we start from the idea that quantum correlations are ubiquitous and somewhat uncontrollable in systems with many degrees of freedom which are typically considered in statistical mechanics. As a consequence we assume the standpoint that quantum statistical mechanics has not to be based on the underlying idea of a collection of many, independent quantum systems but rather it has to emerge at the level of a global wavefunction (pure state) which describes the system as well as its environment as a whole. In order to investigate the consequences of these assumptions we study the equilibrium distribution of an isolated quantum system. This is defined, in analogy with the ergodic foundations of classical statistical mechanics, on the basis of the time evolution of the quantum state. Then, we study the emergence of thermodynamic properties in a quantum system by studying the probability distribution of some function of interests, as the entropy and the equilibrium state of a subsystem, on Ensembles of Pure States. Such a probability distribution is derived from the geometry of the Hilbert space, and the theoretical tools suitable for its characterization are developed. On the one hand we perform a numerical sampling of the ensemble distributions by employing Monte Carlo techniques, on the other hand simpler analytical approximation of the geometrical distributions are derived by means of a maximum entropy principle. Model systems composed of an ensemble of spins are chosen to illustrate the salient features which emerge from the developed theoretical framework: the main point is that the Ensemble Distributions of “thermodynamic observables” (entropy or equilibrium state of a subsystem) are sharply peaked around a typical value. From the analysis it emerges that each of the overwhelming majority of the wavefunctions which has appreciable weight in the considered ensemble, is characterized by the same value of the “macroscopic” functions. This is a striking evidence of the “typicality” of these properties. In the essence, our impossibility to know the state of the system in detail does not matter, just for the remarkable fact that almost all quantum states behave essentially in the same way. By virtue of this typicality the study of the behaviour of the typical values of the thermodynamic function become meaningful. Notably, under certain conditions, one recovers the results of standard statistical mechanics, that is, the equilibrium average of the state of a subsystem can be cast in the Boltzmann canonical form at the temperature given by the usual thermodynamical relation . In the second part of the thesis we consider the dynamical aspects of the equilibrium state of a subsystem interacting with its environment. The fluctuations around the equilibrium average critically depends on the entanglement between the system and the environment and on the form of the interaction Hamiltonian. The connection between the dynamics of the fluctuations of an observable at the equilibrium and the relaxation toward the equilibrium from a “non typical” initial value is also investigated with the aid of simple model systems. The study presented in this thesis was partly motivated by a critical analysis of the statistical methods available for the theoretical modelling of magnetic resonance experiments. One of these, the Stochastic Liouville Equation, has been employed in a work completed during the first year of my Ph.D. program in order to interpret some feature of a two dimensional electron spin resonance experiment, [Fresch B., Frezzato D., Moro G. J., Kothe G., Freed J. H.; J. Phys. Chem. B., 110, 24238, (2006)].