The success of quantum physics in description of various physical interaction phenomena relies primarily on the accuracy of analytical methods used. In quantum mechanics, many of such interactions such as those found in quantum optomechanics and quantum computing have a highly nonlinear nature, which makes their analysis extraordinarily difficult using classical schemes. Typically, modern quantum systems of interest nowadays come with four basic properties: (i) quantumness, (ii) openness, (iii) randomness, and (iv) nonlinearity. The newly introduced method of higher-order operators targets analytical solutions to such systems, and while providing at least mathematically approximate expressions with improved accuracy over the fully linearized schemes, some cases admit exact solutions. Many different applications of this method in quantum and classically nonlinear systems are demonstrated throughout. This review is purposed to provide the reader with ease of access to this recent and well-established operator algebra, while going over a moderate amount of literature review. The reader with basic knowledge of quantum mechanics and quantum noise theory should be able to start using this scheme to his or her own problem of interest.
The success of quantum physics in description of various physical interaction phenomena relies primarily on the accuracy of analytical methods used. In quantum mechanics, many of such interactions such as those found in quantum optomechanics and quantum computing have a highly nonlinear nature, which makes their analysis extraordinarily difficult using classical schemes. Typically, modern quantum systems of interest nowadays come with four basic properties: (i) quantumness, (ii) openness, (iii) randomness, and (iv) nonlinearity. The newly introduced method of higher-order operators targets analytical solutions to such systems, and while providing at least mathematically approximate expressions with improved accuracy over the fully linearized schemes, some cases admit exact solutions. Many different applications of this method in quantum and classically nonlinear systems are demonstrated throughout. This review is purposed to provide the reader with ease of access to this recent and well-established operator algebra, while going over a moderate amount of literature review. The reader with basic knowledge of quantum mechanics and quantum noise theory should be able to start using this scheme to his or her own problem of interest.
A central problem in modern mathematical physics is to study interacting systems beyond the reach of perturbation theory. This thesis studies interacting quantum systems using the mathematical theory of operator algebras. In the first part of this thesis, we consider how to construct quantum many-body systems from one-particle systems. This problem has a well-established solution - namely, the second quantization (or the Gaussian functor). However, it seems difficult for the second quantization to go beyond generalized free field constructions without the help of perturbation theory. The second quantization takes real Hilbert spaces as inputs. Using only information from one-particle Hilbert spaces, it is mathematically impossible for the second quantization to produce connected higher-point correlation functions. Moreover, from the perspectives of mathematical quantization and operator algebra theory, it is more natural to have a quantization procedure that takes an algebra along with a functional as the input. Such a pair of mathematical objects contains more information about the underlying one-particle system and includes the Hilbert space of states (via the GNS construction) as part of the data. Inspired by quantum probability theory, we introduce an alternative approach to construct quantum many-body systems. This approach is based on a noncommutative generalization of the classical Poisson random measure. We call this construction Poissonization. Mathematically, Poissonization is a functor from the category of von Neumann algebras with normal semi nite faithful weights to the category of von Neumann algebras with normal faithful states. It is a natural adaptation of the second quantization to the context of von Neumann algebras. This thesis discusses several properties of Poissonization, and uses Poissonization as a tool to construct various toy models of algebraic quantum field theories relevant to high energy physics. This collection of examples aims at demonstrating the versatility of Poissonization. In the second part of this thesis, we study the noisy interactions between an open quantum system and its environment. One of the challenges in quantum information science is to control open quantum systems with a large number of qubits. An important aspect of many-body systems is the emergence of collective phenomena. One collective noise model is an open atomic system in an electromagnetic environment. This model was considered by Dicke in the 50's. In this thesis, we study the entropic decay in Dicke's model and other related collective noise models. Specifically, we develop a general framework to estimate the spectral gap and modified logarithmic Sobolev constant of these collective noise models. In addition, we study the necessary mixing conditions a general Dicke's model must satisfy in order to have a unique equilibrium state.
Abstract Quantum mechanics was created mainly to account for microscopic phenomena that occur on a scale of order IO−10 m. It has turned out to be an astonishingly successful theory which applies on a scale down to at least 10−18 m. Quantum theory has become established as part of the essential core of physics. The purpose of this introductory chapter is twofold. Firstly, in Sections 1.1–1.7 we review the formalism of elementary nonrelativistic quantum mechanics. In this review we attempt to give enough of the terminology and basics of the subject so that a reader who is not already familiar with quantum mechanics will be able to follow the applications considered later in the book. For the reader who finds our discussion too brief, the literature on quantum mechanics is vast and we recommend in particular Refs 1–5. The second purpose of this chapter is to present the basic tools of operator methods as they are used in this book: thus in Sections 1.8–1.11 we discuss commutation relations, shift operators, and the virial, hypervirial, and Hellmann Feynman theorems.
Introduction This chapter gives the physical and mathematical basis of decoherence in quantum systems, with particular emphasis on its importance in quantum information theory and quantum computation. We assume that the reader is already familiar with the basics of quantum mechanics, such as wave functions, the Schrödinger equation, and Hilbert spaces, which are covered in standard textbooks. Nevertheless, all basic concepts will be at least briefly defined and described. A working knowledge of graduate-level linear algebra and calculus is also assumed. In quantum information processing, the term decoherence is often used loosely to describe any kind of noise that can affect a quantum system. This can include sources of noise that seem, qualitatively, to be quite different from each other. These noise sources include: (i) Random driving forces from the environment (e.g., Brownian motion). (ii) Interactions that produce entanglement between the system and the environment. (iii) Statistical imprecision in the experimental controls on the system (e.g., timing errors, frequency fluctuations, etc.). Remarkably, all of these sources of noise can be described by an almost identical mathematical framework, in which pure quantum states can evolve into mixed quantum states, and the quantum effects necessary for information processing – interference, entanglement, and reversible evolution – are disrupted. The chapter begins by examining a highly schematic model of decoherence, involving systems of qubits, but uses this model to present the mathematical formalism used in describing more realistic systems as well.
Open quantum systems have drawn attention over decades due to its applicability in the foundation of theoretical physics, e.g. statistical mechanics, quantum mechanics and condensed matter physics. The dynamics of open quantum systems has been described as separate entities from their surrounding environment that consists of a very large number of modes, somehow coupled to the mode of the system. Even though the exact solution of the dynamical behavior of the system is impossible to calculate, we obtain a tentative solution using the crucial Markov approximation. The input-output formalism of the quantum Langevin equation (QLE) has been considered as a useful tool which provides a semi-classical description of the dynamics of the system, whereas the master equation provides a complete picture of the dynamics of the system expressed in terms of density matrix. While studying the dynamics of nonlinear system/environment coupling using QLE, we see, for a small value of external field, that the steady state system field does not change much from the steady state field obtained in the absence of nonlinear dissipation. However, in a case where a stronger external field is applied, we see that the deviation becomes substantial from the solution of linear system. We also see that the nonlinear coupling introduces significant difference in the cavity fluctuation spectrum. The description, therefore, provides a potential explanation of parametric effects in terms of nonlinear dissipation phenomena associated with the nonlinear coupling. Even though the theories developed in the context of open quantum systems have proven to be powerful tools, they do not provide a satisfactory platform to be implemented on non-linear Hamiltonians. We often approximate it by linearizing over nonlinear steady state field amplitude, and therefore, the interesting effects are often overlooked. The limitation of the analytics provokes us to simulate open quantum dynamics numerically. The numerical method consists of transformation of the environmental degrees of freedom to a one-dimensional many-body chain, and the computational technique includes numerical diagonalization and renormalization process. The time-adaptive density matrix renormalisation group (t-DMRG) is known as one of the most powerful techniques for the simulation of strongly-correlated many-body quantum systems. In this thesis, along with the theoretical modeling, we implement DMRG numerical scheme for the simulation of canonical S/B model by mapping it to one-dimensional harmonic chain with nearest neighbor interactions, and use the method to investigate the dynamics of the free dissipative system. The thermalization of open quantum systems is also studied by generating minimally entangled typical thermal states (METTS) through imaginary time evolution, and real-time evolving an empty system in the presence of the thermal bath. Further, we simulate coherently driven free dissipative Kerr nonlinear system numerically using Euler's method by solving Heisenberg equation of motion and t-DMRG algorithm, and demonstrate how the numerical results are analogous to classical bistability. By comparing with analytics, we see that the DMRG numerics is analogous to the quantum-mechanical exact solution obtained by mapping the equation of motion of the density matrix of the system to a Fokker-Plank equation. The comparison between two different numerical techniques shows that the semi-classical Euler's method determines the dynamics of the system field of one among two coherent branches, whereas DMRG numerics gives the superposition of both of them. Hence, DMRG-determined time dynamics undergoes generating non-classical states. Our approach of dealing with nonlinearity represents an important contribution in the developments of technique to study the dynamical and steady-state behavior of open quantum systems, which is a fundamental aspect of quantum physics.
An operational description of quantum phenomena concerns developing models that describe experimentally observed behaviour. $\textit{Higher-order quantum operations}\unicode{x2014}$quantum operations that transform quantum operations$\unicode{x2014}$are fundamental to modern quantum theory, extending beyond basic state preparations, evolutions, and measurements described by the Born rule. These operations naturally emerge in quantum circuit architectures, correlated open dynamics, and investigations of quantum causality, to name but a few fields of application. This Review Article provides both a pedagogical introduction to the framework of higher-order quantum operations and a comprehensive survey of current literature, illustrated through physical examples. We conclude by identifying open problems and future research directions in this rapidly evolving field.
We review the methods and use of random quantum states with particular emphasis on recent theoretical developments and applications in various fields. The guiding principle of the review is the idea that random quantum states can be understood as classical probability distributions in the Hilbert space of the associated quantum system. We show how this central concept connects questions of physical interest that cover different fields such as quantum statistical physics, quantum chaos, mesoscopic systems of both non-interacting and interacting particles, including superconducting and spin–orbit phenomena, and stochastic Schrödinger equations describing open quantum systems.