Showing papers on "Superposition principle published in 1999"

Wave–particle duality of C 60 molecules

Abstract: Quantum superposition lies at the heart of quantum mechanics and gives rise to many of its paradoxes. Superposition of de Broglie matter waves1 has been observed for massive particles such as electrons2, atoms and dimers3, small van der Waals clusters4, and neutrons5. But matter wave interferometry with larger objects has remained experimentally challenging, despite the development of powerful atom interferometric techniques for experiments in fundamental quantum mechanics, metrology and lithography6. Here we report the observation of de Broglie wave interference of C60 molecules by diffraction at a material absorption grating. This molecule is the most massive and complex object in which wave behaviour has been observed. Of particular interest is the fact that C60 is almost a classical body, because of its many excited internal degrees of freedom and their possible couplings to the environment. Such couplings are essential for the appearance of decoherence7,8, suggesting that interference experiments with large molecules should facilitate detailed studies of this process.

Fast-sum method for the elastic field of three-dimensional dislocation ensembles

Abstract: The elastic field of complex shape ensembles of dislocation loops is developed as an essential ingredient in the dislocation dynamics method for computer simulation of mesoscopic plastic deformation. Dislocation ensembles are sorted into individual loops, which are then divided into segments represented as parametrized space curves. Numerical solutions are presented as fast numerical sums for relevant elastic field variables (i.e., displacement, strain, stress, force, self-energy, and interaction energy). Gaussian numerical quadratures are utilized to solve for field equations of linear elasticity in an infinite isotropic elastic medium. The accuracy of the method is verified by comparison of numerical results to analytical solutions for typical prismatic and slip dislocation loops. The method is shown to be highly accurate, computationally efficient, and numerically convergent as the number of segments and quadrature points are increased on each loop. Several examples of method applications to calculations of the elastic field of simple and complex loop geometries are given in infinite crystals. The effect of crystal surfaces on the redistribution of the elastic field is demonstrated by superposition of a finite-element {ital image force} field on the computed results. {copyright} {ital 1999} {ital The American Physical Society}

Creation of coherent atomic superpositions by fractional stimulated Raman adiabatic passage

Abstract: We discuss the properties of a simple and robust scheme for preparing atoms and molecules in an arbitrary preselected coherent superposition of quantum states - fractional stimulated Raman adiabatic passage (fractional STIRAP) - first proposed by Marte et al (1991 Phys. Rev. A 44 R4118). As in STIRAP, the Stokes pulse arrives before the pump pulse, but unlike STIRAP, here the two pulses terminate simultaneously, while maintaining a constant ratio of amplitudes. We extend earlier research and suggest a realization of this scheme with two smoothly varying delayed laser pulses (which can be derived from a single laser), the parameters of the created superposition being controlled by the polarization of the delayed pulse. Furthermore, we provide simple analytic estimates of the robustness of this process against variations in the laser intensity, laser frequency and pulse delay. Finally, we discuss an extension to multistate systems which provides a possibility for creating coherent superpositions of more than two states.

Do's and don'ts in Fourier analysis of steady-state potentials

Abstract: Fourier analysis is a powerful tool in signal analysis that can be very fruitfully applied to steady-state evoked potentials (flicker ERG, pattern ERG, VEP, etc) However, there are some inherent assumptions in the underlying discrete Fourier transform (DFT) that are not necessarily fulfilled in typical electrophysiological recording and analysis conditions Furthermore, engineering software-packages may be ill-suited and/or may not fully exploit the information of steady-state recordings Specifically: * In the case of steady-state stimulation we know more about the stimulus than in standard textbook situations (exact frequency, phase stability), so 'windowing' and calculation of the 'periodogram' are not necessary * It is mandatory to choose an integer relationship between sampling rate and frame rate when employing a raster-based CRT stimulator * The analysis interval must comprise an exact integer number (eg, 10) of stimulus periods * The choice of the number of stimulus periods per analysis interval needs a wise compromise: A high number increases the frequency resolution, but makes artifact removal difficult; a low number 'spills' noise into the response frequency * There is no need to feel tied to a power-of-two number of data points as required by standard FFT, 'resampling' is an easy and efficient alternative * Proper estimates of noise-corrected Fourier magnitude and statistical significance can be calculated that take into account the non-linear superposition of signal and noise These aspects are developed in an intuitive approach with examples using both simulations and recordings Proper use of Fourier analysis of our electrophysiological records will reduce recording time and/or increase the reliability of physiologic or pathologic interpretations

Exact statistical properties of the zeros of complex random polynomials

Abstract: The zeros of complex Gaussian random polynomials, with coefficients such that the density in the underlying complex space is uniform, are known to have the same statistical properties as the zeros of the coherent state representation of one-dimensional chaotic quantum systems. We extend the interpretation of these polynomials by showing that they also arise as the wavefunction for a quantum particle in a magnetic field constructed from a random superposition of states in the lowest Landau level. A study of the statistical properties of the zeros is undertaken using exact formulae for the one- and two-point distribution functions. Attention is focused on the moments of the two-point correlation in the bulk, the variance of a linear statistic, and the asymptotic form of the two-point correlation at the boundary. A comparison is made with the same quantities for the eigenvalues of complex Gaussian random matrices.

Phase-dependent nonlinear optics with double- Λ atoms

Abstract: We present a theory of a continuous-wave light propagation in a medium of atoms with a double-$\ensuremath{\Lambda}$ configuration of levels. This is a configuration with a closed cycle of radiation-induced transitions. An interference of excitation channels in such closed-loop systems leads to a strong dependence of the atomic state on the relative phase and the relative amplitudes of applied electromagnetic waves. Therefore, the medium response may be controlled by the phases. On the other hand, the phases themselves change during the propagation. Thus the state of the medium and all the field parameters are tightly coupled to each other in the present problem. We consider the propagation of four-frequency laser radiation through the double-$\ensuremath{\Lambda}$ medium for two situations. At resonant or near-resonant excitation of atoms, both the medium and the field evolve into a nonabsorbing state. This state implies specific coherent superposition for atoms (``dark state''), and particular relations for the field phases, amplitudes, and frequencies. In this way, the propagation results in the phase, amplitude, and frequency matching of the laser waves. In the second case, one $\ensuremath{\Lambda}$ system in double-$\ensuremath{\Lambda}$ atoms is excited resonantly, while the second $\ensuremath{\Lambda}$ system is far off resonance. Such an excitation scheme ensures the preparation of atoms in the nearly dark state throughout the medium. Therefore, the total light energy is dissipated very weakly, whereas each individual laser wave can vary considerably along the propagation path. We have found that the resonant fields change as much as the far-detuned ones. The intensities oscillate sinusoidally with the optical length, with the energy being transferred back and forth between two waves in each frequency pair, resonant and far detuned. This gives the possibility for an almost lossless amplification of two of the laser waves, or an even generation of one of them.

On the nature of Thermal Diffusion in binary Lennard-Jones liquids

Abstract: The aim of this study is to understand deeper the thermal diffusion transport process (Ludwig-Soret effect) at the microscopic level. For that purpose, the recently developed reverse nonequilibrium molecular dynamics method was used to calculate Soret coefficients of various systems in a systematic fashion. We studied binary Lennard-Jones (LJ) fluids near the triple point (of one of the components) in which we separately changed the ratio of one of the LJ parameters mass, atomic diameter and interaction strength while keeping all other parameters fixed and identical. We observed that the magnitude of the Soret coefficient depends on all three ratios. Concerning its sign we found that heavier species, smaller species and species with higher interaction strengths tend to accumulate in the cold region whereas the other ones (lighter, bigger or weaker bound) migrate to the hot region of our simulation cell. Additionally, the superposition of the influence of the various parameters was investigated as well as more realistic mixtures. We found that in the experimentally relevant parameter range the contributions are nearly additive and that the mass ratio often is the dominating factor.

Wavefield extrapolation by nonstationary phase shift

Abstract: The phase‐shift method of wavefield extrapolation applies a phase shift in the Fourier domain to deduce a scalar wavefield at one depth level given its value at another. The phase‐shift operator varies with frequency and wavenumber, and assumes constant velocity across the extrapolation step. We use nonstationary filter theory to generalize this method to nonstationary phase shift (NSPS), which allows the phase shift to vary laterally depending upon the local propagation velocity. For comparison, we derive an analytic form for the popular phase shift plus interpolation (PSPI) method in the limit of an exhaustive set of reference velocities. NSPS and this limiting form of PSPI can be written as generalized Fourier integrals which reduce to ordinary phase shift in the constant velocity limit. In the (x, ω) domain, these processes are the transpose of each other; however, only NSPS has the physical interpretation of forming the scaled, linear superposition of laterally‐variable impulse responses (i.e.,Huygen...

Coherent Control of Chemical Reactions

Abstract: The traditional goals of chemical kinetics are to measure the rates of chemical reactions and to understand their mechanisms on a molecular level. With the advent of lasers and molecular beams, it has become possible to study the reactions of molecules in individual quantum states and to explain their behavior in terms of single collisions governed by well-defined potential energy surfaces. Likewise, numerical methods have been refined to the point that it is possible to predict diverse properties of elementary reactions from first principles. In recent years, the challenge has shifted from measuring and calculating rates of reactions to devising methods of controlling their outcome. The idea of controlling the yield and product distribution of a reaction is, of course, a very old one. The field of catalysis, for example, is devoted to finding means of enhancing the natural yield of a reaction. Similarly, temperature and pressure have been used for decades in the chemical industry to alter reaction rates. With the development of narrow-band lasers, it has become possible to excite selectively a single molecular mode. Provided energy transfer is slow as compared to reaction, such mode-selective preparation of a molecule can be used to alter the outcome of its reaction.1-3 For example, if the OH bond in HOD is vibrationally excited, that bond becomes more reactive, and collisions with Cl atoms preferentially yield HCl rather than DCl.4 A more general, photochemical method for altering reaction pathways exploits the phase property of lasers and has become known as coherent control, or phase control.5 Two main schemes of coherently controlling physical and chemical processes have been developed over the past decade, based on similar concepts but differing in the properties of the electromagnetic fields that are used. The first, introduced by Tannor and Rice,6 uses an ultrashort pulse of laser light to create a coherent superposition of energy-resolved eigenstates, namely, a wave packet. Such carefully phased superposition states are, in general, nonstationary. By altering the amplitudes and phases of the light pulses, it is possible to modify the content of the superposition state and thus control the motion of the wave packet. Typically, a vibrational or electronic wave packet is generated at time t ) 0 by a short laser pulse. At some later time, when the wave packet has evolved to a desired configuration, a second pulse triggers the reaction. An experimental example of this method is control of the electronic branching ratio of Na atoms in the photodissociation of NaI, which was achieved by varying the timing between two transform-limited ultrashort pulses.7 Optimal control theory, introduced by Rabitz and co-workers8 and subsequently developed by several groups,5 employs a feedback loop to optimize the spectral content and temporal shape of a pulse in order to maximize the yield of a given product. Experimental demonstrations of the control of branching ratios by means of optimally tailored laser pulses were reported by Bardeen et al.9 and by Assion et al.10 The use of the intensity property of short pulses to control reactions was also reported.11 A second method was introduced by Brumer and Shapiro12 and is the subject of the present paper. In this approach, two long laser pulses (in principle, they could be continuous beams) excite an atom or a molecule from an initial state to a final state. The frequencies of the lasers are chosen so that the target absorbs either m photons from the first laser or n photons from the second laser to reach the same final state.13 That is, the laser frequencies satisfy the relation mωm ) nωn. An important property of the laser beams is that they have a well-defined phase relation; that is, the phases of the two electromagnetic fields differ by a controllable amount, φ. As we will show, by varying the relative phase of the two beams, it is possible to control the branching ratio of the reaction. It is useful to think of the latter method as an analog of Young’s two-slit experiment.14 In that experiment, particles emerging from two slits create a pattern on a screen. If only one slit is open, the result is a diffraction pattern produced by that slit. If both slits are open, Robert J. Gordon obtained his doctorate from Dudley Herschbach at Harvard University in 1970. After postdoctoral studies at Caltech and the Naval Research Laboratory, he came to the University of Illinois at Chicago, where he is a professor of chemistry. Prof. Gordon’s research interests include experimental studies of the spectroscopy and reaction dynamics of small molecules.

Wigner functions, squeezing properties, and slow decoherence of a mesoscopic superposition of two-level atoms

Abstract: Department of Quantum Physics, University of Ulm, D-89069 Ulm, Germany(Submitted to Physical Review A: March 26, 1999.)We consider a class of states in an ensemble of two-level atoms: a superposition of two distinctatomic coherent states, which can be regarded as atomic analogues of the states usually calledSchr¨odinger cat states in quantum optics. According to the relation of the constituents we definepolar and nonpolar cat states. The properties of these are investigated by the aid of the sphericalWigner function. We show that nonpolar cat states generally exhibit squeezing, the measure of whichdepends on the separation of the components of the cat, and also on the number of the constituentatoms. By solving the master equation for the polar cat state embedded in an external environment,we determine the characteristic times of decoherence, dissipation and also the characteristic time ofa new parameter, the non-classicality of the state. This latter one is introduced by the help of theWigner function, which is used also to visualize the process. The dependence of the characteristictimes on the number of atoms of the cat and on the temperature of the environment shows that thedecoherence of polar cat states is surprisingly slow.PACS number(s): 42.50.-p, 42.50.Fx, 03.65.BzI. INTRODUCTION

A new approach to bessel beams

Abstract: A new interpretation of the non-diffracting Bessel beams is given as a superposition of two fields described by Hankel functions. Within our picture, supported by the Sommerfeld radiation condition, we find that Bessel beams must be of finite transverse extension. Our approach also easily explains propagation characteristics of Bessel-Gauss beams and, in general, the propagation of amplitude modulated Bessel beams. Consequences are discussed and contrasted with the classic theory of Bessel beams.

Problem of plane EM wave self-action in multilayer structure: an exact solution

Abstract: To solve exactly the problem of plane EM wave interaction with multilayer nonlinear structures a non-traditional theoretical method is applied. Wave equation's solution in each layer of a structure is presented in the special form of a single expression. This form of field presentation does not require the fulfillment of superposition principle, contrary to the traditional approach. Multilayer structures of threefold quarter-wavelength bilayers with alternate linear and nonlinear indices are investigated. Multistable dependences of reflection coefficients on incident electric field amplitude are obtained for negative and positive nonlinearity. The results of investigation not only permits to deeply understand the phenomena of intense electromagnetic wave interaction with multilayer structures but also will be very useful for elaboration of nonlinear devices.

Quantum Superposition of Multiple Clones and the Novel Cloning Machine

Abstract: we envisage a novel quantum cloning machine, which takes an input state and produces an output state whose success branch can exist in a linear superposition of multiple copies of the input state and the failure branch exist in a superposition of composite state independent of the input state. We prove that unknown non-orthogonal states chosen from a set $\cal S$ can evolve into a linear superposition of multiple clones by a unitary process if and only if the states are linearly independent. We derive a bound on the success probability of the novel cloning machine. We argue that the deterministic and probabilistic clonings are special cases of our novel cloning machine.

Quantum interference by a nonlocal double slit

Abstract: We report an interference experiment in which photon pairs generated by spontaneous parametric down-conversion produce a Young-type fourth-order interference pattern after being scattered by two different and spatially separated apertures, whose superposition defines a double slit. The experiment is compared with previous ones based on parametric down-conversion, and its nonlocal nature is discussed. A theoretical explanation is also provided.

Developing rigorous boundary conditions to simulations of discrete dislocation dynamics

Abstract: Mesoscale simulations have recently been developed in order to better understand the collective behaviour of dislocations and their effects on the mechanical response. Those simulations deal with dislocations discretized into segments which are allowed to move in a three-dimensional (3D) discrete network. This network is a sublattice of the original crystalline lattice network. The minimum distance between two points is defined by the annihilation distance for two edge dislocations, i.e. the minimum distance for which two edge dislocations can coexist without instantaneous collapse. The elastic theory can still be applied in the simulated volume, since the minimum distance is large compared to the dislocation core radius within which nonlinear expressions should be taken into account in the dislocation-dislocation interaction. This property allows us to use the superposition principle to enforce boundary conditions on the simulation box. This paper details the rigorous boundary conditions applied when the simulation box is supposed to be either a bulk crystal, a free standing film or a finite crystal submitted to a complex loading.

Spectral-Element Method for Levy-Type Plates Subject to Dynamic Loads

Abstract: Spectral-element method (SEM) is applied for the vibration analysis of the Levy-type rectangular plates subject to distributed dynamic loads In the solution procedure, both FFT and inverse FFT computer algorithms are efficiently used to obtain the dynamic responses in time domain Distributed dynamic load is approximated, as the superposition of equivalent dynamic line loads and the spatial coordinate dependence of equivalent line loads is removed by (spatial) FFT This procedure transforms the original plate (two-dimensional) problem into an effective beam (one-dimensional) problem so that the solution procedure for one-dimensional structures can be used Numerical tests show that SEM provides very accurate solutions when compared with conventional finite element solutions

Transducer field modeling in anisotropic media by superposition of Gaussian base functions

Abstract: Anisotropic structural materials like fiber composites, but also columnar-grained stainless steels, raise considerable problems for ultrasonic inspection due to the well-known wave propagation phenomena of skewing, splitting and distortion. In this respect, simulation and optimization in ultrasonic nondestructive testing have gained a considerable importance. Among a variety of methods for transducer field calculation, beam superposition has proven to be highly efficient. In this article, a Gaussian beam approach for anisotropic media is presented. The Gaussian base functions are obtained from relationships previously derived for Gaussian wave packets. Each function is furnished with coefficients fixing the beam waists and their position. To test the approach, the case of a piston radiator is addressed for general transversely isotropic media. Using Gaussian beam superposition instead of—as a reference—applying a point source superposition technique leads to an enormous reduction in computer run time.

A Quantum Algorithm for finding the Maximum

Abstract: This paper describes a quantum algorithm for finding the maximum among N items The classical method for the same problem takes O(N) steps because we need to compare two numbers in one step This algorithm takes O(sqrt(N)) steps by exploiting the property of quantum states to exist in a superposition of states and hence performing an operation on a number of elements in one go A tight upper bound of 68(sqrt(N)) for the number of steps needed using this algorithm was found These steps are the number of queries made to the oracle

The scattering of plane elastic waves by a one-dimensional periodic surface

Abstract: The two-dimensional scattering problem for time-harmonic plane waves in an isotropic elastic medium and an effectively infinite periodic surface is considered. A radiation condition for quasi-periodic solutions similar to the condition utilized in the scattering of acoustic waves by one-dimensional diffraction gratings is proposed. Under this condition, uniqueness of solution to the first and third boundary-value problems is established. We then proceed by introducing a quasi-periodic free field matrix of fundamental solutions for the Navier equation. The solution to the first boundary-value problem is sought as a superposition of single- and double-layer potentials defined utilizing this quasi-periodic matrix. Existence of solution is established by showing the equivalence of the problem to a uniquely solvable second kind Fredholm integral equation.

The Art of Modeling in Science and Engineering with Mathematica

Abstract: A First Look at Modeling The Physical Laws The Rate of the Variables: Dependent and Independent Variables The Role of Balance Space: Differential and Integral Balances The Role of Time: Unsteady State and Steady State Balances Information Derived from Model Solutions Choosing a Model Kick-Starting the Modeling Process Solution Analysis Practice Problems Analytical Tools: The Solution of Ordinary Differential Equations Definitions and Classifications Boundary and Initial Conditions Analytical Solutions of ODEs Nonlinear Analysis Laplace Transformation Practice Problems The Use of Mathematica in Modeling Physical Systems Handling Algebraic Expressions Algebraic Equations Integration Ordinary Differential Equations Partial Differential Equations Practice Problems Elementary Applications of the Conservation Laws Application of Force Balances Applications of Mass Balance Simultaneous Applications of the Conservation Laws Practice Problems Partial Differential Equations: Classification, Types, and Properties - Some Simple Transformations Properties and Classes of PDE PDEs of Major Importance Useful Simplifications and Transformations PDEs PDQ: Locating Solutions in the Literature Practice Problems Solution of Linear Systems by Superposition Methods Superposition by Addition of Simple Flows: Solutions in Search of a Problem Superposition by Multiplication: Product Solutions Solution of Source Problems: Superposition by Integration More Superposition by Integration: Duhamel's Integral and the Superposition of Danckwerts Practice Problems Vector Calculus: Generalized Transport Equations Vector Notation and Vector Calculus Superposition Revisited: Green's Functions and the Solution of PDEs by Green's Functions Transport of Mass Transport of Energy Transport of Momentum Practice Problems Analytical Solutions of Partial Differential Equations Separation of Variables Laplace Transformation and Other Integral Transforms The Method of Characteristics Practice Problems

Chemical Bonding as a Superposition Phenomenon

Abstract: The most profoundly "weird" aspects of Schrodinger's wave equation are known to originate in the quantum superposition principle and associated wavelike interference phenomena. We show that the characteristic phenomena of chemistry-covalent and coordinate bonding, resonance delocalization, aromaticity, H-bonding, hyperconjugation-can also be seen as special cases of a central superposition paradigm: the "donor-acceptor" interaction between filled and unfilled orbitals. Elementary consequences of this picture can be worked out with no more difficult mathematical steps than solution of the quadratic equation. A remarkable result of this treatment is that donor-acceptor superposition must lead to energy lowering ("bonding" or "stabilization") in a way that is virtually independent of details of the kinetic or potential energy terms in the Schrodinger equation. Thus, the deepest "explanation" of chemical bonding appears to lie in the superposition phenomenon itself, rather than in the comforting quasi-classic...

Beam-propagation factor and mode-coherence coefficients of hyperbolic-cosine Gaussian beams.

Abstract: Analytical expressions for the M2 factor and the mode-coherence coefficients of hyperbolic-cosine– (cosh-) Gaussian beams have been derived, and a simple method of producing cosh-Gaussian beams in an experiment, superposition of two decentered Gaussian beams, has been proposed. Our results will be useful for studying more-general Hermite–sinusoidal-Gaussian beams.

Evaluation of the modal structure of light beams composed of incoherent mixtures of Hermite-Gaussian modes.

Abstract: A new, to our knowledge, technique for determining the modal content of partially coherent beams that are made up of an incoherent superposition of Hermite-Gaussian modes is studied. The algorithm makes use of the intensity profile of the beam at an arbitrarily chosen transverse plane. Analytical derivations are presented for a Gaussian Schell-model source and flat-topped beams, as well as an analysis of their performances in the presence of experimental errors and noise. Numerical simulations are performed to test the accuracy and the stability of the recovery algorithm.