Evolution Of Quantum System Research Articles

Wave Packets Can Factorize Numbers

We draw attention to various aspects of number theory emerging in the time evolution of elementary quantum systems with quadratic phases. Such model systems can be realized in actual experiments. Our analysis paves the way to a new, promising and effective method to factorize numbers.

Noncommutative tori and universal sets of nonbinary quantum gates

We address the problem of universality in simulation of evolution of quantum system and in theory of quantum computations related with the possibility of expression or approximation of arbitrary unitary transformation by composition of specific unitary transformations (quantum gates) from given set. In an earlier paper application of Clifford algebras to constructions of universal sets of binary quantum gates Uk∈U(2n) was shown. For application of a similar approach to nonbinary quantum gates Uk∈U(ln), in present work we used rational noncommutative torus T1/l2n. A set of universal nonbinary two-gates is presented here as one example.

Quantum dissipation in unbounded systems.

In recent years trajectory based methodologies have become increasingly popular for evaluating the time evolution of quantum systems. A revival of the de Broglie--Bohm interpretation of quantum mechanics has spawned several such techniques for examining quantum dynamics from a hydrodynamic perspective. Using techniques similar to those found in computational fluid dynamics one can construct the wave function of a quantum system at any time from the trajectories of a discrete ensemble of hydrodynamic fluid elements (Bohm particles) which evolve according to nonclassical equations of motion. Until very recently these schemes have been limited to conservative systems. In this paper, we present our methodology for including the effects of a thermal environment into the hydrodynamic formulation of quantum dynamics. We derive hydrodynamic equations of motion from the Caldeira-Leggett master equation for the reduced density matrix and give a brief overview of our computational scheme that incorporates an adaptive Lagrangian mesh. Our applications focus upon the dissipative dynamics of open unbounded quantum systems. Using both the Wigner phase space representation and the linear entropy, we probe the breakdown of the Markov approximation of the bath dynamics at low temperatures. We suggest a criteria for rationalizing the validity of the Markov approximation in open unbound systems and discuss decoherence, energy relaxation, and quantum/classical correspondence in the context of the Bohmian paths.

Quantum Evolution in Fluctuating Backgrounds: Nonideal Clocks and Foam-like Spacetimes

I characterize good clocks, which are naturally subject to fluctuations, in statistical terms, obtain the master equation that governs the evolution of quantum systems according to these clocks, and find its general solution. This master equation is diffusive and produces loss of coherence. Moreover, real clocks can be described in terms of effective interactions that are nonlocal in time. Alternatively, they can be modeled by an effective thermal bath coupled to the system. I also study some aspects concerning the evolution of quantum low-energy fields in a foamlike spacetime, with involved topology at the Planck scale but with a smooth metric structure at large length scales. This foamlike structure of spacetime may show up in low-energy physics through loss of quantum coherence and mode-dependent energy shifts, for instance, which might be observable. Spacetime foam introduces nonlocal interactions that can be modeled by a quantum bath, and low-energy fields evolve according to a master equation that displays such effects. These evolution laws are similar to those for quantum mechanical systems evolving according to good nonideal clocks, although the underlying Hamiltonian structure in this case establishes some differences among both scenarios.

QSES's and the Quantum Jump

Abstract The stochastic methods in Hilbert space have been used both from a fundamental and a practical point of view. The result reported here concerns the application of these methods to model the evolution of quantum systems and does not enter into the question of their fundamental or practical status. It can easily be stated as follows: Once a quantum stochastic evolution scheme is assumed, the incompatibility of the Markov property with the notion of quantum jumps is easily established

Random-matrix model for quantum Brownian motion

We use a random band-matrix model for the system–bath interaction to derive a Markovian master equation for the time evolution of one-dimensional quantum systems weakly coupled to a heat bath. We study in detail the damped harmonic oscillator and discuss the fluctuation-dissipation relation. In the large-bandwidth, high-temperature limit, our master equation coincides with the equations derived by Agarwal and Caldeira–Leggett. This shows that the Markovian master equations for quantum Brownian motion are independent of model assumptions used in their derivation and, thus, universal.

Exotic stochastic processes from quantum chaotic environments

Stochastic processes are shown to emerge from the time evolution of complex quantum systems. Using parametric, banded random matrix ensembles to describe a quantum chaotic environment, we show that the dynamical evolution of a particle coupled to such environments displays a variety of stochastic behaviors, ranging from turbulent diffusion to Lévy processes and Brownian motion. Dissipation and diffusion emerge naturally in the stochastic interpretation of the dynamics. This approach provides a derivation of a fractional kinetic theory in the classical limit and leads to classical Lévy dynamics.

Time evolution for quantum systems at finite temperature

This paper investigates a new formalism to describe real time evolution of quantum systems at finite temperature. A time correlation function among subsystems will be derived which allows for a probabilistic interpretation. Our derivation is non-perturbative and fully quantized. Various numerical methods used to compute the needed path integrals in complex time were tested and their effectiveness was compared. For checking the formalism we used the harmonic oscillator where the numerical results could be compared with exact solutions. Interesting results were also obtained for a system that presents tunneling. A ring of coupled oscillators was treated in order to try to check self-consistency in the thermodynamic limit. The short time distribution seems to propagate causally in the relativistic case. Our formalism can be extended easily to field theories where it remains to be seen if relevant models will be computable.

Quantum dynamics via adiabatic ab initio centroid molecular dynamics

The ab initio path integral simulation method is combined with centroid molecular dynamics. This unification, and thus extension of these basic techniques, allows for the investigation of the real-time quantum dynamics in chemically complex many-body systems. The theory underlying the proposed ab initio centroid molecular dynamics (AICMD) technique is presented in detail. The real-time propagation of the nuclei is obtained in the quasiclassical approximation within the framework of centroid path integrals. Concurrently, the forces acting on the nuclei are computed from “on the fly” electronic structure calculations based on first-principle techniques such as, e.g., Hohenberg—Kohn—Sham density functional theory. AICMD can be considered as a quasiclassical generalization of standard Car—Parrinello ab initio molecular dynamics. At the same time, AICMD preserves the virtues of the ab initio path integral technique to generate exact time-independent quantum equilibrium averages. AICMD is well suited to investigate, in a quasiclassical sense, the real-time evolution of molecular quantum systems with complex interactions which cannot be satisfactorily represented by simple model potentials. In particular, the method permits the simulation of the dynamics of chemical reactions including quantum effects. AICMD is applicable to isolated systems in the gas phase such as molecules, clusters or complexes as well as to condensed matter, i.e. molecular liquids or solids.

Universality of quantum Brownian motion

Are Markovian master equations for quantum Brownian motion independent of model assumptions used in the derivation and, thus, universal? With the aim of answering this question, we use a random band-matrix model for the system-bath interaction to derive Markovian master equations for the time evolution of one-dimensional quantum systems weakly coupled to a heat bath. We study in detail two simple systems, the harmonic oscillator and the two-level system. Our results are in complete agreement with those of earlier models, like the Caldeira–Leggett model and, in the large-band limit, with the Agarwal equations (both with and without rotating-wave approximation). This proves the universality of these master equations.

Quantum evolution according to real clocks

We characterize good clocks, which are naturally subject to fluctuations, in statistical terms. We also obtain the master equation that governs the evolution of quantum systems according to these clocks and find its general solution. This master equation is diffusive and produces loss of coherence. Moreover, real clocks can be described in terms of effective interactions that are nonlocal in time. Alternatively, they can be modeled by an effective thermal bath coupled to the system.

Real time correlations at finite temperature for the ising model.

After having developed a method that measures real time evolution of quantum systems at a finite temperature, we present here the simplest field theory where this scheme can be applied to, namely the 1 + 1 Ising model. We will compute the probability that if a given spin is up, some other spin will be up after a time t, the whole system being at temperature T. We can thus study spatial correlations and relaxation times at finite T. The fixed points that enable the continuum real time limit can be easily found for this model. The ultimate aim is to get to understand real time evolution in more complicated field theories, with quantum effects such as tunneling at finite temperature.

Complete Control of Hamiltonian Quantum Systems: Engineering of Floquet Evolution

We propose a method for controlling the unitary evolution of quantum systems by switching on and off alternatively two distinct constant perturbations. We show how to find appropriate switching times in order to attain any desired evolution. We suggest an experimental realization of our method in controlling the translational motion of cold atoms.

“Classical” propagator and path integral in the probability representation of quantum mechanics

In the probability representation of the standard quantum mechanics, the explicit expression (and its quasiclassical van-Fleck approximation) for the “classical” propagator (transition probability distribution), which completely describes the quantum system's evolution, is found in terms of the quantum propagator. An expression for the “classical” propagator in terms of path integral is derived. Examples of free motion and harmonic oscillator are considered. The evolution equation in the Bargmann representation of the optical tomography approach is obtained.

Dynamics of a simple quantum system in a complex environment

We present a theory for the dynamical evolution of a quantum system coupled to a complex many-body intrinsic system/environment. By modelling the intrinsic many-body system with parametric random matrices, we study the types of effective stochastic models which emerge from random matrix theory. Using the Feynman-Vernon path integral formalism, we derive the influence functional and obtain either analytical or numerical solutions for the time evolution of the entire quantum system. We discuss thoroughly the structure of the solutions for some representative cases and make connections to well known limiting results, particularly to Brownian motion, Kramers classical limit and the Caldeira-Leggett approach.

Quasiadiabatic time evolution, avoided level crossings, and Berry’s phase

A nonperturbative theory for quasiadiabatic time evolution in quantum systems is presented. The dynamics of the passage through an avoided level crossing is given by a nonperturbative alternative to the Landau-Zener theory used for diatomic predissociation. The theory provides the theoretical framework for a quantitative understanding of the quantum steps in the hyteresis loops observed very recently in manganese acetate crystals. These steps correspond to avoided level crossings for the associated, fully adiabatic problem. A natural extention of the theory yields quasiadiabatic corrections to Berry's phase.

Nonperturbative real time propagation at finite temperature

A new formalism will be presented in order to study real time evolution of quantum systems at finite temperature. Probability distributions for time-correlated observables will be studied non-perturbatively and fully quantized. This works for various systems, including ones with tunneling phenomena. We have obtained good results with some computational methods which can be used on models with several degrees of freedom. Thus it looks feasible to study vacuum tunneling in real time for relevant field theories at finite T.

Evolution of quantum system in order domain of Chebyshev operator

A cosine transform between the order and angle of the Chebyshev operator is identified. Because the order and angle form a conjugate pair similar to energy and time, the Chebyshev state can be considered as a cosine-type evolution state in the order domain, analogous to a time-dependent wave packet. The order/angle formulation is analytically equivalent to the time/energy formulation, but the former may have some numerical advantages in certain applications. This is illustrated by examining the spectral method and the filter-diagonalization method in both formulations.

The collapses of wave functions

The collapse of the wave function has recently reemerged as a subject of extensive discussion in the quantum mechanical literature. In the present paper, wave function collapses occuring during the irreversible evolution of complex quantum systems, including those involved in measurement procedures, are described.

Time-dependent quantum systems and the invariant Hermitian operator.

We study the time evolution of quantum systems with a time-dependent Hamiltonian given by a linear combination of SU(1,1) and SU(2) generators. The invariant Hermitian operator is constructed in the same manner as for both the SU(1,1) and SU(2) systems. With the help of the invariant Hermitian operator we obtain not only the exact solutions of the Schr\"odinger equation but also the time-evolution operator. The adiabatic and nonadiabatic Berry phases are calculated with the exact solutions. \textcopyright{} 1996 The American Physical Society.