Decoherence of odd compass states in the phase-sensitive amplifying/dissipating environment

We study the evolution of odd compass states (specific superpositions of four coherent states), governed by the standard master equation with phase-sensitive amplifying/attenuating terms, in the presence of a Hamiltonian describing a parametric degenerate linear amplifier. Explicit expressions for the time-dependent Wigner function are obtained. The time of disappearance of the so called “sub-Planck structures” is calculated using the negative value of the Wigner function at the origin of phase space. It is shown that this value rapidly decreases during a short “conventional interference degradation time” (CIDT), which is inversely proportional to the size of quantum superposition, provided the anti-Hermitian terms in the master equation are of the same order (or stronger) as the Hermitian ones (governing the parametric amplification). The CIDT is compared with the final positivization time (FPT), when the Wigner function becomes positive. It appears that the FPT does not depend on the size of superpositions, moreover, it can be much bigger in the amplifying media than in the attenuating ones. Paradoxically, strengthening the Hamiltonian part results in decreasing the CIDT, so that the CIDT almost does not depend on the size of superpositions in the asymptotical case of very weak reservoir coupling. We also analyze the evolution of the Mandel factor, showing that for some sets of parameters this factor remains significantly negative, even when the Wigner function becomes positive.     

Resolving the black-hole information paradox by treating time on an equal footing with space

Pure states in quantum field theory can be represented by many-fingered block-time wave functions, which treat time on an equal footing with space and make the notions of “time evolution” and “state at a given time” fundamentally irrelevant. Instead of information destruction resulting from an attempt to use a “state at a given time” to describe semi-classical black-hole evaporation, the full many-fingered block-time wave function of the universe conserves information by describing the correlations of outgoing Hawking particles in the future with ingoing Hawking particles in the past.     

An optical study of the correlation between growth kinetics and microstructure of μc-Si grown by SiH4-H2 PECVD

Fully microcrystalline silicon, μc-Si, thin films have been deposited on corning glass by plasma enhanced chemical vapor deposition (PECVD) using SiH₄-H₂. The effects of the surface treatment and of the deposition temperature on microstructure of μc-Si films are investigated by “in situ” laser reflectance interferometry (LRI), “ex situ” spectroscopic ellipsometry (SE) and Raman spectroscopy. LRI indicated the existence of a “crystalline seeding time”, which is indicative of the crystallite nucleation, and depends on substrate treatments. Longer “crystalline seeding time” results in a lower density of crystalline nuclei, which grow laterally, yielding to complete suppression of the amorphous incubation layer and to growth of very dense, fully crystalline layer at a growth temperature as low as 120 °C.     

A physical limitation of the Wigner “distribution” function in molecular transport

We present an example revealing that the sign of the “momentum” P   of the Wigner “distribution” function f(q,P)$f(q,P)$ is not necessarily associated with the direction of motion in the real world. This aspect, which is not related to the well-known limitation of the Wigner function that traces back to the Heisenberg?s uncertainty principle, is particularly relevant in transport studies, wherein it is helpful to distinguish between electrons flowing from electrodes into devices and vice versa.     

Number-phase entropic uncertainty relations and Wigner functions for solvable quantum systems with discrete spectra

In this Letter, the “number-phase entropic uncertainty relation” and the “number-phase Wigner function” of generalized coherent states associated to a few solvable quantum systems with non-degenerate spectra are studied. We also investigate time evolution of “number-phase entropic uncertainty” and “Wigner function” of the considered physical systems with the help of temporally stable Gazeau-Klauder coherent states.     

A last updating evolution model for online social networks

As information technology has advanced, people are turning to electronic media more frequently for communication, and social relationships are increasingly found on online channels. However, there is very limited knowledge about the actual evolution of the online social networks. In this paper, we propose and study a novel evolution network model with the new concept of “last updating time”, which exists in many real-life online social networks. The last updating evolution network model can maintain the robustness of scale-free networks and can improve the network reliance against intentional attacks. What is more, we also found that it has the “small-world effect”, which is the inherent property of most social networks. Simulation experiment based on this model show that the results and the real-life data are consistent, which means that our model is valid.     

Wavepacket dynamics, quantum reversibility, and random matrix theory

We introduce and analyze the physics of “driving reversal” experiments. These are prototype wavepacket dynamics scenarios probing quantum irreversibility. Unlike the mostly hypothetical “time reversal” concept, a “driving reversal” scenario can be realized in a laboratory experiment, and is relevant to the theory of quantum dissipation. We study both the energy spreading and the survival probability in such experiments. We also introduce and study the “compensation time” (time of maximum return) in such a scenario. Extensive effort is devoted to figuring out the capability of either linear response theory or random matrix theory (RMT) to describe specific features of the time evolution. We explain that RMT modeling leads to a strong non-perturbative response effect that differs from the semiclassical behavior.     

On infinite walls in deformation quantization

We examine the deformation quantization of a single particle moving in one dimension (i) in the presence of an infinite potential wall, (ii) confined by an infinite square well, and (iii) bound by a delta function potential energy. In deformation quantization, considered as an autonomous formulation of quantum mechanics, the Wigner function of stationary states must be found by solving the so-called ?-genvalue (“stargenvalue”) equation for the Hamiltonian. For the cases considered here, this pseudo-differential equation is difficult to solve directly, without an ad hoc modification of the potential. Here we treat the infinite wall as the limit of a solvable exponential potential. Before the limit is taken, the corresponding ?-genvalue equation involves the Wigner function at momenta translated by imaginary amounts. We show that it can be converted to a partial differential equation, however, with a well-defined limit. We demonstrate that the Wigner functions calculated from the standard Schrödinger wave functions satisfy the resulting new equation. Finally, we show how our results may be adapted to allow for the presence of another, non-singular part in the potential.     

A comment on the paper “Stochastic feedback, nonlinear families of Markov processes, and nonlinear Fokker-Planck equations” by T.D. Frank

The purpose of this comment is to correct mistaken assumptions and claims made in the paper “Stochastic feedback, nonlinear families of Markov processes, and nonlinear Fokker-Planck equations” by T. D. Frank [T.D. Frank, Stochastic feedback, non-linear families of Markov processes, and nonlinear Fokker-Planck equations, Physica A 331 (2004) 391]. Our comment centers on the claims of a “non-linear Markov process” and a “non-linear Fokker-Planck equation.” First, memory in transition densities is misidentified as a Markov process. Second, the paper assumes that one can derive a Fokker-Planck equation from a Chapman-Kolmogorov equation, but no proof was offered that a Chapman-Kolmogorov equation exists for the memory-dependent processes considered. A “non-linear Markov process” is claimed on the basis of a non-linear diffusion pde for a 1-point probability density. We show that, regardless of which initial value problem one may solve for the 1-point density, the resulting stochastic process, defined necessarily by the conditional probabilities (the transition probabilities), is either an ordinary linearly generated Markovian one, or else is a linearly generated non-Markovian process with memory. We provide explicit examples of diffusion coefficients that reflect both the Markovian and the memory-dependent cases. So there is neither a “non-linear Markov process”, nor a “non-linear Fokker-Planck equation” for a conditional probability density. The confusion rampant in the literature arises in part from labeling a non-linear diffusion equation for a 1-point probability density as “non-linear Fokker-Planck,” whereas neither a 1-point density nor an equation of motion for a 1-point density can define a stochastic process. In a closely related context, we point out that Borland misidentified a translation invariant 1-point probability density derived from a non-linear diffusion equation as a conditional probability density. Finally, in the Appendix A we present the theory of Fokker-Planck pdes and Chapman-Kolmogorov equations for stochastic processes with finite memory.     

Dixon's extended bodies and impulsive gravitational waves

The “reaction” of an extended body to the passage of an exact plane gravitational wave is discussed following Dixon's model. The analysis performed shows several general features, e.g. even if initially absent, the body acquires a spin induced by the quadrupole structure, the center of mass moves from its initial position, as well as certain “spin-flip” or “spin-glitch” effects which are being observed.     

Area in phase space as determiner of transition probability: Bohr-Sommerfeld bands,Wigner ripples,and Fresnel zones

We consider an oscillator subjected to a sudden change in equilibrium position or in effective spring constant, or both—to a squeeze in the language of quantum optics. We analyze the probability of transition from a given initial state to a final state, in its dependence on final-state quantum number. We make use of five sources of insight: Bohr-Sommerfeld quantization via bands in phase space, area of overlap between before-squeeze band and after-squeeze band, interference in phase space, Wigner function as quantum update of B-S band and near-zone Fresnel diffraction as mockup Wigner function.     

The dense α-√3 × √3Pb/Si(1 1 1) phase: A comprehensive STM and SPA-LEED study of ordering, phase transitions and interactions

The T-θ phase diagram for the system Pb/Si(1 1 1) was determined in the coverage range 6/5 ML < θ < 4/3 ML from complementary STM and SPA-LEED experiments. This coverage is within the range where a “Devil’s Staircase” (DS) has been realized. The numerous DS phases answer conflicting information in the Pb/Si(1 1 1) literature and update the previously published phase diagram. The measurements reveal the thermal stability of the different linear DS phases with the transition temperature found to be a function of phase period. Because of additional complexity in the experimental system (i.e. two-dimensionality and 3-fold symmetry) the linear DS phases transform at higher temperature into commensurate phases of 3-fold symmetry HIC (historically named “hexagonal incommensurate phase”). Different types of HIC phases have been discovered differing in the size of the supercell built out of √3 × √3 domains separated by domain walls of the √7 × √3 phase. The detailed structures of these HIC phases (coverage, binding site, twist angle, etc.) have been deduced from the comparison of STM images and diffraction patterns. After heating the system to even higher temperature the HIC phase transforms into the disordered phase. For sufficiently high coverage a SIC (“striped incommensurate phase” which is also built from √3 × √3 domains but meandering √7 × √3 domain walls) is observed which also disorders at high temperatures.     

Exact integral operator form of the Wigner distribution-function equation in many-body quantum transport theory

A formal derivation of a generalized equation of a Wigner distribution function including all many-body effects and all scattering mechanisms is given. The result is given in integral operator form suitable for application to the numerical modeling of quantum tunneling and quantum interference solid state devices. In the absence of scattering and many-body effects, the result reduces to the noninteracting-particle Wigner distribution function equation, often used to simulate resonant tunneling devices. The derivation uses a Weyl transform technique which can easily incorporate Bloch electrons. Weyl transforms of self-energies are derived. Various simplifications of a general quantum transport equation for semiconductor device analysis and self-consistent numerical simulation of a quantum distribution function in the phase-space/frequency-time domain are discussed. Recent attempts to include collisions in the Wigner distribution-function approach to the numerical simulation of tunneling devices are clearly shown to be non-self-consistent and inaccurate; more accurate numerical simulation is needed for a deeper understanding of the effects of collision and scattering.     

Correspondence between the classical and quantum canonical transformation groups from an operator formulation of the wigner function

An explicit expression of the Wigner operator is derived, such that the Wigner function of a quantum state is equal to the expectation value of this operator with respect to the same state. This Wigner operator leads to a representation-independent procedure for establishing the correspondence between the inhomogeneous symplectic group applicable to linear canonical transformations in classical mechanics and the Weyl-metaplectic group governing the symmetry of unitary transformations in quantum mechanics.     

Poincaré covariance and κ-Minkowski spacetime

A fully Poincaré covariant model is constructed as an extension of the κ-Minkowski spacetime. Covariance is implemented by a unitary representation of the Poincaré group, and thus complies with the original Wigner approach to quantum symmetries. This provides yet another example (besides the DFR model), where Poincaré covariance is realised à la Wigner in the presence of two characteristic dimensionful parameters: the light speed and the Planck length. In other words, a Doubly Special Relativity (DSR) framework may well be realised without deforming the meaning of “Poincaré covariance”.     

Hydrodynamics and phases of flocks

We review the past decade’s theoretical and experimental studies of flocking: the collective, coherent motion of large numbers of self-propelled “particles” (usually, but not always, living organisms). Like equilibrium condensed matter systems, flocks exhibit distinct “phases” which can be classified by their symmetries. Indeed, the phases that have been theoretically studied to date each have exactly the same symmetry as some equilibrium phase (e.g., ferromagnets, liquid crystals). This analogy with equilibrium phases of matter continues in that all flocks in the same phase, regardless of their constituents, have the same “hydrodynamic”—that is, long-length scale and long-time behavior, just as, e.g., all equilibrium fluids are described by the Navier-Stokes equations. Flocks are nonetheless very different from equilibrium systems, due to the intrinsically nonequilibrium self-propulsion of the constituent “organisms.” This difference between flocks and equilibrium systems is most dramatically manifested in the ability of the simplest phase of a flock, in which all the organisms are, on average moving in the same direction (we call this a “ferromagnetic” flock; we also use the terms “vector-ordered” and “polar-ordered” for this situation) to exist even in two dimensions (i.e., creatures moving on a plane), in defiance of the well-known Mermin-Wagner theorem of equilibrium statistical mechanics, which states that a continuous symmetry (in this case, rotation invariance, or the ability of the flock to fly in any direction) can not be spontaneously broken in a two-dimensional system with only short-ranged interactions. The “nematic” phase of flocks, in which all the creatures move preferentially, or are simply oriented preferentially, along the same axis, but with equal probability of moving in either direction, also differs dramatically from its equilibrium counterpart (in this case, nematic liquid crystals). Specifically, it shows enormous number fluctuations, which actually grow with the number of organisms faster than the “law of large numbers” obeyed by virtually all other known systems. As for equilibrium systems, the hydrodynamic behavior of any phase of flocks is radically modified by additional conservation laws. One such law is conservation of momentum of the background fluid through which many flocks move, which gives rise to the “hydrodynamic backflow” induced by the motion of a large flock through a fluid. We review the theoretical work on the effect of such background hydrodynamics on three phases of flocks—the ferromagnetic and nematic phases described above, and the disordered phase in which there is no order in the motion of the organisms. The most surprising prediction in this case is that “ferromagnetic” motion is always unstable for low Reynolds-number suspensions. Experiments appear to have seen this instability, but a quantitative comparison is awaited. We conclude by suggesting further theoretical and experimental work to be done.     

Learning in synergetic systems for pattern recognition and associative action

Synergetic systems are in particular physical systems which can produce spatial or temporal patterns by means of the interaction of their individual parts. We show how such a system can be devised or even learn by itself to reproduce given patterns described by their probability distribution function. If an initial state close to one of the learned patterns is presented to such a system, it will pull the initial state into an attractor belonging to the learned state (pattern recognition via associative memory). Furthermore we show how such a system can be devised or can learn to perform any prescribed stationary continous Markov process. If a set of incomplete or partly incorrect initial data is offered to such a system, it may correct it and perform associative action.     

Measurement of quantum states and the Wigner function

In quantum mechanics, the state of an individual particle (or system) is unobservable, i.e., it cannot be determined experimentally, even in principle. However, the notion of measuring a state is meaningful if it refers to anensemble of similarly prepared particles, i.e., the question may be addressed: Is it possible to determine experimentally the state operator (density matrix) into which a given preparation procedure puts particles. After reviewing the previous work on this problem, we give simple procedures, in the line of Lamb's operational interpretation of quantum mechanics, for measuring a translational state operator (whether pure or mixed), via its Wigner function. These procedures closely parallel methods that might be used in classical mechanics to determine a true phase space probability distribution; thus, the Wigner function simulates such a distribution not only formally, but operationally also.

E. P. Wigner⁽¹⁾