A theorem allowing to derive deterministic evolution equations from stochastic evolution equations, II: The non-Markovian extension

Deterministic evolution equations of classical as well as quantum mechanical models are derived from a set of non-Markovian stochastic evolution equations after an average over realization using a theorem. Examples are given, show that deterministic differential equations that contain derivatives with respect to time higher than or equal to two can be derived after a Taylor series expansion of the dynamical variables. It is shown that the derivation of such deterministic differential equations can be done by solving a set of linear equations that increase in number after increasing the number of previous time steps in the updating rules that define a given model. Two explicit examples, the first containing updating rules that depend on two previous time steps and the second on three, are worked in some detail in order to show some features of the linear transformation that allow one to obtain the deterministic differential equations.     

A theorem allowing the derivation of deterministic evolution equations from stochastic evolution equations. III The Markovian-non-Markovian mix

The proof of a theorem that allows one to construct deterministic evolution equations from a set, with two subsets, containing two types of discrete stochastic evolution equation is developed. One subset evolves Markovianly and the other non-Markovianly. As an illustrative example, the deterministic evolution equations of quantum electrodynamics are derived from two sets of Markovian and non-Markovian stochastic evolution equations, of different type, after an average over realization, using the theorem. This example shows that deterministic differential equations that contain both first-order and second-order time derivatives can be derived after a Taylor series expansion of the dynamical variables. It is shown that the derivation of such deterministic differential equations can be done by solving a set of linear equations. Two explicit examples, the first containing updating rules that depend on one previous time step and the second containing updating rules that depend on two previous time steps, are given in detail in order to show step by step the linear transformations that allow one to obtain the deterministic differential equations.     

Discrete stochastic evolution rules and continuum evolution equations

The continuum evolution equations are derived from updating rules for three classes of stochastic models. The first class corresponds to models whose stochastic continuum equations are of the Langevin type obtained after carrying out a “local average” known as coarse-graining. The second class consists of a hierarchy of continuum equations for the correlations of the dynamical variables obtained after making an average over realizations. This average generates a hierarchy of deterministic partial differential equations except when the dynamical variables do not depend on the values of the neighboring dynamical variables, in which case a hierarchy of ordinary differential equations is obtained. The third class of evolution equations for the correlations of the dynamical variable constitutes another hierarchy after calculating an average over both realizations and all the sites of the lattice. This double average generates a hierarchy of deterministic ordinary differential equations. The second and third classes of equations are truncated using a mean field (m,n)-closure approximation in order to obtain a finite set of equations. Illustrative examples of every class are given.     

Calculating adiabatic evolution of the perturbed DNLS/MNLS solitons

Adiabatic evolution equations for parameters of the perturbed DNLS/MNLS solitons obtained by recently developed direct perturbation theory [Phys. Rev. E 65 (2002) 066608] are reformulated in simpler forms for practical use. A symbolic computation technique is developed to calculate them, based on the residue theorem. Effects of the third-order dispersion on the MNLS solitons are studied as an example.     

A modified quantum adiabatic evolution for the Deutsch–Jozsa problem

Deutsch–Jozsa algorithm has been implemented via a quantum adiabatic evolution by S. Das et al. [S. Das, R. Kobes, G. Kunstatter, Phys. Rev. A 65 (2002) 062310]. This adiabatic algorithm gives rise to a quadratic speed up over classical algorithms. We show that a modified version of the adiabatic evolution in that paper can improve the performance to constant time.     

Nonlinear evolution of surface gravity waves over highly variable depth

New nonlinear evolution equations are derived that generalize those presented in a Letter by Matsuno [Phys. Rev. Lett. 69, 609 (1992)]] and a terrain-following Boussinesq system recently deduced by Nachbin [SIAM J Appl. Math. 63, 905 (2003)]]. The regime considers finite-amplitude surface gravity waves on a two-dimensional incompressible and inviscid fluid of, highly variable, finite depth. A Fourier-type operator is expanded in a wave steepness parameter. The novelty is that the topography can vary on a broad range of scales. It can also have a complex profile including that of a multiply valued function. The resulting evolution equations are variable coefficient Boussinesq-type equations. The formulation is over a periodically extended domain so that, as an application, it produces efficient Fourier (fast-Fourier-transform algorithm) solvers.     

Experimental implementation of local adiabatic evolution algorithms by an NMR quantum information processor

Quantum adiabatic algorithm is a method of solving computational problems by evolving the ground state of a slowly varying Hamiltonian. The technique uses evolution of the ground state of a slowly varying Hamiltonian to reach the required output state. In some cases, such as the adiabatic versions of Grover's search algorithm and Deutsch-Jozsa algorithm, applying the global adiabatic evolution yields a complexity similar to their classical algorithms. However, using the local adiabatic evolution, the algorithms given by J. Roland and N.J. Cerf for Grover's search [J. Roland, N.J. Cerf, Quantum search by local adiabatic evolution, Phys. Rev. A 65 (2002) 042308] and by Saurya Das, Randy Kobes, and Gabor Kunstatter for the Deutsch-Jozsa algorithm [S. Das, R. Kobes, G. Kunstatter, Adiabatic quantum computation and Deutsh's algorithm, Phys. Rev. A 65 (2002) 062301], yield a complexity of order N (where N=2(n) and n is the number of qubits). In this paper, we report the experimental implementation of these local adiabatic evolution algorithms on a 2-qubit quantum information processor, by Nuclear Magnetic Resonance.     

A theory of open systems based on stochastic differential equations

For a model of an open quantum system—a concentrated ensemble consisting of similar atoms and interacting with a one-dimensional quantum vacuum environment with a zero photon density—quantum stochastic differential equations of a non-Wiener type of the general form have been obtained; based on the equations, kinetic equations describing a wide class of physical systems are derived. The distinctive feature of such systems is effects of suppression of collective spontaneous emission and stabilization of the excited state. For the open classical system exposed to the action of noise in the form of a Levy process of the general non-Gaussian kind, kinetic equations of the Fokker-Planck type with fractional derivatives have been obtained based on classical non-Wiener stochastic differential equations. This emphasizes the common base of the developed theory for different types of open systems, which is expressed in using the mathematical formalism of stochastic differential equations of the general non-Wiener type.     

Critical Behaviors in a Stochastic Local Limited One-Dimensional Rice-Pile Model

A stochastic local limited one-dimensional rice-pile model is numerically investigated. The distributions for avalanche sizes have a clear power-law behavior and it displays a simple finite size scaling. We obtain the avalanche exponents τ_s=1.54±0.10, β_s=2.17±0.10 and τ_T=1.80±0.10, β_T=1.46±0.10. This self-organized critical model belongs to the same universality class with the Oslo rice-pile model studied by K. Christensen et al. [Phys. Rev. Lett. 77 (1996) 107], a rice-pile model studied by L.A.N. Amaral et al. [Phys. Rev. E 54 (1996) 4512], and a simple deterministic self-organized critical model studied by M.S. Vieira [Phys. Rev. E 61 (2000) 6056].     

Thermal plumes and convection in highly compressible fluids

We present simple hydrodynamic equations of supercritical fluids close to the gas-liquid critical point. We numerically solve them to examine plume generation and convection under gravity. These results are in good agreement with the experiment [A. B. Kogan and H. Meyer, Phys. Rev. E 63, 056310 (2001)]. This Letter is a first study of transient behavior of convection, which is unique in compressible fluids due to the piston effect.     

Geometric measure of quantum discord over two-sided projective measurements

The original definition of quantum discord of bipartite states was defined over one-sided projective measurements, it describes quantum correlations more extensively than entanglement. Dakic, Vedral, and Brukner [Phys. Rev. Lett. 105 (2010) 190502] introduced a geometric measure of quantum discord, Luo and Fu [Phys. Rev. A 82 (2010) 034302] simplified the variation expression of this geometric measure. In this Letter we introduce a geometric measure of quantum discord over two-sided projective measurements. A simplified expression and a lower bound of this two-sided geometric measure are derived and explicit expressions are obtained for some special cases.     

Are avalanches in sandpiles a chaotic phenomenon?

We show that deterministic systems with strong nonlinearities seem to be more appropriate to model sandpiles than stochastic systems or deterministic systems in which discontinuities are the only nonlinearity. In particular, we are able to reproduce the breakdown of self-organized criticality found in two well known experiments, that is, a centrally fueled pile [Phys. Rev. Lett. 65 (1990) 1120] and sand in a rotating tray [Phys. Rev. Lett. 69 (1992) 2431]. By varying the parameters of the model we recover self-organized criticality, in agreement with other experiments and other models. We show that chaos plays a fundamental role in the dynamics of the system.     

Improved quantum “Ping-pong” protocol based on GHZ state and classical XOR operation

In order to transmit the secure message, a deterministic secure quantum direct communication protocol which was called Ping-pong protocol was proposed by Bostr m and Felbinger [Bostr m K, et al. Phys Rev Lett, 2002, 89: 187902]. But the protocol was proved very vulnerable, and can be attacked by an eavesdropper. An improved Ping-pong protocol is presented to overcome the problem. The GHZ state particles are used to detect eavesdroppers, and the classical XOR operation which serves as a one-time-pad is used ...     

Quantum contextuality for rational vectors

The Kochen-Specker theorem states that noncontextual hidden variable models are inconsistent with the quantum predictions for every yes-no question on a qutrit, corresponding to every projector in three dimensions. It has been suggested [D.A. Meyer, Phys. Rev. Lett. 83 (1999) 3751] that the inconsistency would disappear when restricting to projectors on unit vectors with rational components; that noncontextual hidden variables could reproduce the quantum predictions for rational vectors. Here we show that a qutrit state with rational components violates an inequality valid for noncontextual hidden-variable models [A.A. Klyachko et al., Phys. Rev. Lett. 101 (2008) 020403] using rational projectors. This shows that the inconsistency remains even when using only rational vectors.     

Dynamics of reorientation of a constrained nematic elastomer

A model is proposed for the reorientation dynamics of a confined nematic liquid crystal elastomer, where the effect of crosslinks is to couple the director to deformations of the elastic matrix. The model combines the (equilibrium) `neo-classical' theory of liquid crystal rubber elasticity with the simplest time evolution equations for a system described by two coupled, non-conserved order parameters. Relaxation from an orientation imposed by an electric field is studied as a function of elastic softness, starting angle, surface pretilt, and the relative mobilities of director and strain. Most importantly, the absence of a `semi-soft' elastic threshold changes the long-time behaviour of the effective refractive index of the medium from exponential to inverse power law decay. Predictions are compatible with recent experimental results by Chang, Chien and Meyer [Phys. Rev. E 56, 595 (1997)]. Received 22 June 1998     

The periodically kicked quantum spin

It is shown that the same kind of deterministic chaos that occurs in classical systems can occur in certain quantum mechanical, many-body systems. The example of the physical realization of the periodically kicked quantum spin (PKQS) is considered in detail. The quantum mechanical equations of motion for this system can be converted into the three-dimensional PKQS map, which exhibits deterministic chaos and Arnold diffusion. Although the case of quantum spin s= 1/2 is assumed, it is shown that the same map results for s=1 (but not for s>/=3/2), and for a suitably chosen classical particle with orbital angular momentum. A simple generalization of the PKQS model gives rise to stochastic webs on the surface of the unit sphere very similar to the Zaslavsky stochastic webs in a plane.     

Symmetry of quantum phase space in a degenerate Hamiltonian system

The structure of the global "quantum phase space" is analyzed for the harmonic oscillator perturbed by a monochromatic wave in the limit when the perturbation amplitude is small. Usually, the phenomenon of quantum resonance was studied in nondegenerate [G. M. Zaslavsky, Chaos in Dynamic Systems (Harwood Academic, Chur, 1985)] and degenerate [Demikhovskii, Kamenev, and Luna-Acosta, Phys. Rev. E 52, 3351 (1995)] classically chaotic systems only in the particular regions of the classical phase space, such as the center of the resonance or near the separatrix. The system under consideration is degenerate, and even an infinitely small perturbation generates in the classical phase space an infinite number of the resonant cells which are arranged in the pattern with the axial symmetry of the order 2&mgr; (where &mgr; is the resonance number). We show analytically that the Husimi functions of all Floquet states (the quantum phase space) have the same symmetry as the classical phase space. This correspondence is demonstrated numerically for the Husimi functions of the Floquet states corresponding to the motion near the elliptic stable points (centers of the classical resonance cells). The derived results are valid in the resonance approximation when the perturbation amplitude is small enough, and the stochastic layers in the classical phase space are exponentially thin. The developed approach can be used for studying a global symmetry of more complicated quantum systems with chaotic behavior. (c) 2000 American Institute of Physics.     

Probability representation of the quantum evolution and energy-level equations for optical tomograms

The von Neumann evolution equation for the density matrix and the Moyal equation for the Wigner function are mapped onto the evolution equation for the optical tomogram of the quantum state. The connection with the known evolution equation for the symplectic tomogram of the quantum state is clarified. The stationary states corresponding to quantum energy levels are associated with the probability representation of the von Neumann and Moyal equations written for optical tomograms. The classical Liouville equation for optical tomogram is obtained. An example of the parametric oscillator is considered in detail.