Embedded random matrix ensembles for complexity and chaos in finite interacting particle systems

Universal properties of simple quantum systems whose classical counter parts are chaotic, are modeled by the classical random matrix ensembles and their interpolations/deformations. However for finite interacting many-particle systems such as atoms, molecules, nuclei and mesoscopic systems (atomic clusters, helium droplets, quantum dots, etc.) for wider range of phenomena, it is essential to include information such as particle number, number of single-particle orbits, lower particle rank of the interaction, etc. These considerations led to resurgence of interest in investigating in detail the so-called embedded random matrix ensembles and their various deformed versions. Besides giving a overview of the basic results of embedded ensembles for the smoothed state densities and transition matrix elements, recent progress in investigating these ensembles with various deformations, for deriving a statistical mechanics (with relationships between quantum chaos, thermalization, phase transitions and Fock space localization, etc.) for isolated finite systems with few particles is briefly discussed. These results constitute new progress in deriving a basis for statistical spectroscopy (introduced and applied in nuclear structure physics and more recently in atomic physics) and its domains of applicability.     

The different effect of electron-electron interaction on the spectrum of atoms and quantum dots

Though atoms and quantum dots typically contain a comparable number of electrons, the number of discrete levels resolved in spectroscopy experiments is very different for the two systems. In atoms, hundreds of levels are observed while in quantum dots that number is usually smaller than 10. In the present work, this difference is traced to the different confining potentials in these systems. In atoms, the soft confining potential leads to large spatial extent of the excited electron's wave function and hence to weak Coulomb interaction with the rest of the atomic electrons. The resulting level broadening is smaller than the single particle level spacing and decreases as the excitation energy is increased. In quantum dots, on the other hand, the sharp confining potential results in electron-electron scattering rates that grow rapidly with energy and fairly quickly exceed the approximately constant single particle level spacing. The number of discrete levels in quantum dots is hence limited by electron-electron interaction, whose effect is negligible in atoms. Received 3 April 2000 and Received in final form 7 August 2000     

基于量子并行粒子群优化算法的分数阶混沌系统参数估计

分数阶混沌系统参数估计的本质是多维参数优化问题, 其对于实现分数阶混沌控制与同步至关重要. 提出一种基于量子并行特性的粒子群优化新算法, 用于解决分数阶混沌的系统参数估计问题. 利用量子计算的并行特性, 设计出了一种新的量子编码, 使每代运算的可计算次数呈指数增加. 在此基础上, 构建了由量子当前旋转角、个体最优旋转角和全局最优旋转角共同组成的粒子演化方程, 以约束粒子在量子空间中的运动行为, 使算法的搜索能力得到了较大提高. 以分数阶Lorenz混沌系统和分数阶Chen混沌系统的参数估计为例, 进行了未知参数估计的数值仿真, 结果显示本算法具有良好的有效性、鲁棒性和通用性.     

Bohr correspondence principle for large quantum numbers

Periodic systems are considered whose increments in quantum energy grow with quantum number. In the limit of large quantum number, systems are found to give correspondence in form between classical and quantum frequency-energy dependences. Solely passing to large quantum numbers, however, does not guarantee the classical spectrum. For the examples cited, successive quantum frequencies remain separated by the incrementhI ^?1, whereI is independent of quantum number. Frequency correspondence follows in Planck's limit,h → 0. The first example is that of a particle in a cubical box with impenetrable walls. The quantum emission spectrum is found to be uniformly discrete over the whole frequency range. This quality holds in the limitn → ∞. The discrete spectrum due to transitions in the high-quantum-number bound states of a particle in a box with penetrable walls is shown to grow uniformly discrete in the limit that the well becomes infinitely deep. For the infinitely deep spherical well, on the other hand, correspondence is found to be obeyed both in emission and configuration. In all cases studied the classical ensemble gives a continuum of frequencies.     

Effective field theory,large number of particle species,and holography

An effective quantum field theory (QFT) with a manifest UV/IR connection, so as to be valid for arbitrarily large volumes, can successfully be applied to the cosmological dark energy problem as well as the cosmological constant (CC) problem. Motivated by recent approaches to the hierarchy problem, we develop such a framework with a large number of particle species. When applying to systems on the brink of experiencing a sudden collapse to a black hole, we find that the entropy, unlike the total energy, now becomes an increasing function of the number of field species. An internal consistency of the theory is then used to infer the upper bound on the number of particle species, showing consistency with the holographic Bekenstein–Hawking bound. This may thus serve to fill in a large gap in entropy of any non-black hole configuration of matter and the black holes. In addition, when the bound is saturated the entanglement entropy matches the black hole entropy, thus solving the multiplicity of species problem. In a cosmological setting, the maximum allowable number of species becomes a function of cosmological time, reaching its minimal value in a low-entropy post-reheating epoch.     

Intrinsic Entropy of Squeezed Quantum Fields and Nonequilibrium Quantum Dynamics of Cosmological Perturbations

Density contrasts in the universe are governed by scalar cosmological perturbations which, when expressed in terms of gauge-invariant variables, contain a classical component from scalar metric perturbations and a quantum component from inflaton field fluctuations. It has long been known that the effect of cosmological expansion on a quantum field amounts to squeezing. Thus, the entropy of cosmological perturbations can be studied by treating them in the framework of squeezed quantum systems. Entropy of a free quantum field is a seemingly simple yet subtle issue. In this paper, different from previous treatments, we tackle this issue with a fully developed nonequilibrium quantum field theory formalism for such systems. We compute the covariance matrix elements of the parametric quantum field and solve for the evolution of the density matrix elements and the Wigner functions, and, from them, derive the von Neumann entropy. We then show explicitly why the entropy for the squeezed yet closed system is zero, but is proportional to the particle number produced upon coarse-graining out the correlation between the particle pairs. We also construct the bridge between our quantum field-theoretic results and those using the probability distribution of classical stochastic fields by earlier authors, preserving some important quantum properties, such as entanglement and coherence, of the quantum field.     

On equations for the evolution of collective phenomena in fermion systems

Based on a Hamiltonian of a charged particle system with an intrinsic magnetic moment in an external electromagnetic field with the field of magnetic moments, quantum hydrodynamic equations are derived, including the equations for densities of particle number, momentum, magnetic moment, and energy. In the self-consistent field approximation, a closed system of equations is obtained, which provides the basis for investigation of collective physical phenomena in distributed quantum systems. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 74–80, December, 2007.     

On the finiteness of discrete spectrum in the n-particle problem

In spaces of functions of any given symmetry sufficient conditions have been obtained for the discrete spectrum finiteness of the energy operators of many particle quantum systems in the absence of external fields.The most general case is considered when the beginning of the essential spectrum of the Hamiltonian is defined by breaking the initial system into any number of stable subsystems. Results obtained are applicable, in particular, to systems with a short-range and to systems with long-range action (in the case of interaction compensation).     

Product structure of heat phase space and branching Brownian motion

A generical formalism for the discussion of Brownian processes with non-constant particle number is developed, based on the observation that the phase space of heat possesses a product structure that can be encoded in a commutative unit ring. A single Brownian particle is discussed in a Hilbert module theory, with the underlying ring structure seen to be intimately linked to the non-differentiability of Brownian paths. Multi-particle systems with interactions are explicitly constructed using a Fock space approach. The resulting ring-valued quantum field theory is applied to binary branching Brownian motion, whose Dyson-Schwinger equations can be exactly solved. The presented formalism permits the application of the full machinery of quantum field theory to Brownian processes.     

Semiclassical elementary particle models

Some simple models of elementary particles are discussed; they may be described as semiclassical, quark, shell models. Particles are assumed to be composed of spherical concentric charged shells. Three basic types of shell are allowed, quantum numbers are associated with each type such as to establish a quantum number correspondence between the shell types and the (p, n, ) quarks. Particles are identified through the quantum numbers of their constituent shells (quarks).The basic assumptions underlying the models considered are relationships between the electromagnetic energy associated with elementary particles (quark systems) and particle masses. The electromagnetic energy is represented classically; the models are semiclassical in that the shell radii are related to particle Compton wavelengths.Particle mass and magnetic moment formulas are derived, possible values for quark masses are suggested, and possible connections of the models considered with particle symmetry schemes are discussed.     

Classical Systems, Standard Quantum Systems, and Mixed Quantum Systems in Hilbert Space

Traditionally, there has been a clear distinction between classical systems and quantum systems, particularly in the mathematical theories used to describe them. In our recent work on macroscopic quantum systems, this distinction has become blurred, making a unified mathematical formulation desirable, so as to show up both the similarities and the fundamental differences between quantum and classical systems. This paper serves this purpose, with explicit formulations and a number of examples in the form of superconducting circuit systems. We introduce three classes of physical systems with finite degrees of freedom: classical, standard quantum, and mixed quantum, and present a unified Hilbert space treatment of all three types of system. We consider the classical/quantum divide and the relationship between standard quantum and mixed quantum systems, illustrating the latter with a derivation of a superselection rule in superconducting systems.     

Quantum Potential in Relativistic Dynamics

The experimental confirmation of nonlocality has renewed interest in Bohm's quantum potential. The construction of quantum potentials for relativistic systems has encountered difficulties which do not arise in a parametrized formulation of relativistic quantum mechanics known as Relativistic Dynamics. The purpose of this paper is to show how to construct a quantum potential in the relativistic domain by deriving a relativistically invariant quantum potential using Relativistic Dynamics. The formalism is applied to three relativistic scalar particle models: a single particle interacting with a scalar potential; N particles interacting with a scalar potential; and a single particle interacting with an electromagnetic 4-vector potential.     

Power-law scalings in weakly-interacting Bose gases at quantum criticality

Weakly interacting quantum systems in low dimensions have been investigated for a long time, but there still remain a number of open questions and a lack of explicit expressions of physical properties of such systems. In this work, we find power-law scalings of thermodynamic observables in low-dimensional interacting Bose gases at quantum criticality. We present a physical picture for these systems with the repulsive interaction strength approaching zero; namely, the competition between the kinetic and interaction energy scales gives rise to power-law scalings with respect to the interaction strength in characteristic thermodynamic observables. This prediction is supported by exact Bethe ansatz solutions in one dimension, demonstrating a simple 1/3-power-law scaling of the critical entropy per particle. Our method also yields results in agreement with a non-perturbative renormalization-group computation in two dimensions. These results provide a new perspective for understanding many-body phenomena induced by weak interactions in quantum gases.     

Different indicators for Markovian and non-Markovian dynamics

The Markovianity/non-Markovianity of two different systems are discussed by means of the quantum speed limit time and quantum Fisher information. The first system is described by a central mass particle interacts locally with its surrounding particles, while the second and third models consist of a single qubit interacts with a non-detuning Lorentzian cavity and with a thermal reservoir, respectively. For the first model, the large distance between the central particle and the surrounding particles is guaranty for a fixed quantum speed limit, while the driving time plays the central role on the fixed behavior of the quantum speed limit time. Due to the stable behavior of the quantum speed limit time and the quantum Fisher information, the exchange information between the systems and their surrounding is limited. The distance between the central mass particle and its surrounding particle plays the main role on predicating the Markovianity/non-Markovianity. For the second system the driving time is an important parameter that control on the Markovianity/non-Markovianity behavior. Finally the third model proves that non-Markovianity dynamic may increase the speed and the sensitivity of the open system.     

Projection operator approach to transport in complex single-particle quantum systems

We discuss the time-convolutionless (TCL) projection operator approach to transport in closed quantum systems. The projection onto local densities of quantities such as energy, magnetization, particle number, etc. yields the reduced dynamics of the respective quantities in terms of a systematic perturbation expansion. In particular, the lowest order contribution of this expansion is used as a strategy for the analysis of transport in “modular” quantum systems corresponding to quasi one-dimensional structures which consist of identical or similar many-level subunits. Such modular quantum systems are demonstrated to represent many physical situations and several examples of complex single-particle models are analyzed in detail. For these quantum systems lowest order TCL is shown to represent an efficient tool which also allows to investigate the dependence of transport on the considered length scale. To estimate the range of validity of the obtained equations of motion we extend the standard projection to include additional degrees of freedom which model non-Markovian effects of higher orders.     

Pearson Global Indicator and the Quantum Mechanics of p-Dimensional Systems

We extend the results of Guiasu (1992a) top-dimensional systems. We use quantum mechanics in orderto extend the basic mathematical model from Guiasu(1992a) to systems with more dimensions. Twomultidimensional quantum systems are presented as applicationsof the mathematical results (the p-dimensional isotropicharmonic oscillator and the free particle in ap-dimensional box).     

Infinite-volume limit of continuous n-particle quantum systems

The Feynman–Kac theorem is applied in order to establish the infinite-volume limit behaviour of the free energy per particle of continuous n-particle quantum systems with bounded separable 2-body interactions defined in the configuration space of particle positions. The mean-field character of such systems is demonstrated.A similar technique is applied to n-particle quantum systems with separable interactions defined in the space of particle momenta and spins. Three examples of systems with separable interactions are given and solved, one of which deals with an electron gas interacting with localized impurity spins in a dilute magnetic alloy (DMA) and extension of Kondo’s resistivity formula for DMA to temperatures close to 0 K.Most of the results are generalizations or more detailed presentations of those published earlier.     

Study of Certain Aspects of Anharmonic,Time-Dependent and Damped Harmonic Oscillator Systems

The present review is devoted to the study of certain aspects of anharmonic, time-dependent and damped oscillator(s) system using different theoretical techniques. A theoretical understanding of these systems is important for application in many problems in physics, mechanics and other fields. We discuss in detail the difficulties in the theoretical analysis of the problem. In particular we discuss here the regular, well-behaved perturbative solution, the large quantum number behaviour of anharmonic oscillator(s) using the technique of coherent states, exact solution of quantum anharmonic oscillators, the electromagnetic radiation emitted by a charged particle executing damped anharmonic oscillator motion using Krylov-Bogoliubov approximation method, use of invariants to obtain solution and coherent states of time-dependent oscillator(s), the derivation of perturbative frequencies of a damped coupled anharmonic oscillators system using suitable canonical transformation in the framework of Hamilton-Jacobi formalism and the quantisation and construction of coherent states of a damped oscillator using time-dependent operators.     

Monte Carlo Study of Localization on a One-Dimensional Lattice

We study the equilibrium properties of a single quantum particle interacting with a classical lattice gas. We develop a path-integral formalism in which the quantum particle is represented by a closed, variable-step random walk on the lattice. After demonstrating that a Metropolis algorithm correctly predicts the properties of a free particle, we extend it to investigate the behavior of the quantum particle interacting with the lattice gas. Evidence of weak localization is observed under conditions of quenched disorder, while self-trapping clearly occurs for the fully annealed system. Compared with continuous space systems, convergence of Monte Carlo simulations in this minimum model is orders of magnitude faster in cpu time. Therefore the system behavior can be investigated for a much larger domain of thermodynamic parameters (e.g., density and temperature) in a reasonable time.