A dynamical formulation of one-dimensional scattering theory and its applications in optics

We develop a dynamical formulation of one-dimensional scattering theory where the reflection and transmission amplitudes for a general, possibly complex and energy-dependent, scattering potential are given as solutions of a set of dynamical equations. By decoupling and partially integrating these equations, we reduce the scattering problem to a second order linear differential equation with universal initial conditions that is equivalent to an initial-value time-independent Schrödinger equation. We give explicit formulas for the reflection and transmission amplitudes in terms of the solution of either of these equations and employ them to outline an inverse-scattering method for constructing finite-range potentials with desirable scattering properties at any prescribed wavelength. In particular, we construct optical potentials displaying threshold lasing, antilasing, and unidirectional invisibility.     

Squeezed states and quantum chaos

We examine the dynamics of a wave packet that initially corresponds to a coherent state in the model of a quantum rotator excited by a periodic sequence of kicks. This model is the main model of quantum chaos and allows for a transition from regular behavior to chaotic in the classical limit. By doing a numerical experiment we study the generation of squeezed states in quasiclassical conditions and in a time interval when quantum-classical correspondence is well-defined. We find that the degree of squeezing depends on the degree of local instability in the system and increases with the Chirikov classical stochasticity parameter. We also discuss the dependence of the degree of squeezing on the initial width of the packet, the problem of stability and observability of squeezed states in the transition to quantum chaos, and the dynamics of disintegration of wave packets in quantum chaos. Zh. éksp. Teor. Fiz. 113, 111–127 (January 1998)     

Reggeon phases and the dual bootstrap program

Analyticity requirements lead to complex Regge vertices in multiparticle production amplitudes. Their effect is discussed within a dual, unitary and exchange-degenerate multi-Regge model. A resulting complex Regge vertex is incorporated in fixed-t model calculations of the reggeon bootstrap and the topological pomeron. The bootstrap condition for particle-particle scattering is unaffected but new j-plane singularities appear in Regge-Regge scattering amplitudes. The pomeron and reggeon intercepts determine the two parameters of our Regge vertex and admit two possible solutions: one is close to the naive phase rule (real vertex) and the other is complex and leads to a high lying non-leading pole in the pomeron channel.     

Reggeon field theory: Formulation and use

We formulate and discuss Reggeon field theory, which enables one to systematically analyze the exchange of Regge poles and associated branch points in high energy hardron scattering. The field theory is first motivated by a consideration of hybrid Feynman graphs, and then a more general derivation from crossed-channel multiparticle unitarity relations is given. Rules for Reggeon interaction and propagation are formulated. We treat in some detail the problem of the Pomeron or vacuum pole which has α(0) = 1 and is responsible for diffractive processes. In particular the renormalization group analysis of Reggeon field theory is presented and the structure of Pomeron partial wave amplitudes is elucidated. Also the question of Pomeron or absorptive corrections to secondary trajectories (both fermion and boson) is considered. We make some comments on important problems yet remaining in Reggeon field theory; in particular, we stress the study of its s-channel content.     

The complex Kohn variational method applied to N-d scattering

The three-nucleon ground state and the N-d scattering states are obtained using variational principles. The wave function of the system is decomposed into angular-spin-isospin channels and the corresponding two dimensional spatial amplitudes are expanded in a correlated polynomial basis. For the scattering states, the complex form of the Kohn variational principle is used to determine the S-matrix. Special attention is given to the convergence pattern of the phase-shift and mixing parameters. The calculations have been performed using realistic local NN potentials and three-nucleon forces. Important features of the method are anomaly-free solutions and the low dimensionality of the matrices involved allowing for the inclusion of a large number of states. Very precise and stable numerical results have been obtained.     

Inverse Scattering Transform for the Discrete Focusing Nonlinear Schrödinger Equation with Nonvanishing Boundary Conditions

In this article we develop the direct and inverse scattering theory of the Ablowitz-Ladik system with potentials having limits of equal positive modulus at infinity. In particular, we introduce fundamental eigensolutions, Jost solutions, and scattering coefficients, and study their properties.We also discuss the discrete eigenvalues and the corresponding norming constants. We then go on to derive the left Marchenko equations whose solutions solve the inverse scattering problem. We specify the time evolution of the scattering data to solve the initial-value problem of the corresponding integrable discrete nonlinear Schrödinger equation. The one-soliton solution is also discussed.     

Mesoscopic quantum coherence in an optical lattice

We observe the quantum coherent dynamics of atomic spinor wave packets in the double-well potentials of a far-off-resonance optical lattice. With appropriate initial conditions the system Rabi oscillates between the left and right localized states of the ground doublet, and at certain times the wave packet corresponds to a coherent superposition of these mesoscopically distinct quantum states. The atom/optical double-well potential is a flexible and powerful system for further study of quantum coherence, quantum control, and the quantum/classical transition.     

The classical field limit of scattering theory for non-relativistic many-boson systems. I

We study the classical field limit of non-relativistic many-boson theories in space dimensionn≧3. When ?→0, the correlation functions, which are the averages of products of bounded functions of field operators at different times taken in suitable states, converge to the corresponding functions of the appropriate solutions of the classical field equation, and the quantum fluctuations are described by the equation obtained by linearizing the field equation around the classical solution. These properties were proved by Hepp [6] for suitably regular potentials and in finite time intervals. Using a general theory of existence of global solutions and a general scattering theory for the classical equation, we extend these results in two directions: (1) we consider more singular potentials, (2) more important, we prove that for dispersive classical solutions, the ?→0 limit is uniform in time in an appropriate representation of the field operators. As a consequence we obtain the convergence of suitable matrix elements of the wave operators and, if asymptotic completeness holds, of theS-matrix.     

Conditional Generation of Multiparticle Entanglement via Cavity QED

We propose a simple scheme to not only generate GHZ states and W states of the multiparticle but also form a new category of multiparticle entangled states by letting the λ-type three-level atoms simultaneously interacting with a coherent cavity field followed by the selective measurements on the cavity mode. We investigate the influence of the cavity dissipation on the generated entangled state and discuss the experimental feasibility of our scheme. It is shown that the intensity of the coherent cavity field plays an instructive role in contribution to state preparation process while the cavity decay and the detuning between the atoms and cavity mode result in the deterioration of the generated entangled state.     

Coherent states of the inverted Caldirola-Kanai oscillator with time-dependent singularities

The coherent states for a system of time-dependent singular potentials coupled to inverted CK (Caldirola-Kanai) oscillator are investigated by employing invariant operator method and Lie algebraic approach. We considered Coulomb potential and inverse quadratic potential as singularities of the system. The spectrum of quantum states is discrete for λ < 0 while continuous for λ ? 0. The probability densities for both Fock state and coherent state are converged to the center as time goes by according to the dissipation of energy. We confirmed that the probability density in the coherent state oscillates back and forth like a classical wave packet.     

Network models: Action formulation

We develop a technique to formulate quantum field theory on an arbitrary network, based on different randomly disposed sets of scattering points. We define the R-matrix of the whole network as a product of R-matrices attached to each scattering node. Then an action is formulated for a network in terms of fermionic fields, which allows to calculate the transition amplitudes as Green functions. On so-called bubble and triangle diagrams it is shown that the method produces the same results as the one which uses the generalized star product. The approach allows to extend network models by including multiparticle interactions at the scattering nodes.     

Mechanism of elastic and inelastic proton scattering on a 15C nucleus in diffraction theory

The amplitudes for elastic and inelastic proton scattering on the neutron-rich nucleus ¹⁵C (to its J ^?? = 5/2⁺ level in the latter case) in inverse kinematics were calculated within Glauber diffraction theory. First- and second-order terms were taken into account in the multiple-scattering operator. The ¹⁵C wave function in the multiparticle shell model was used. This made it possible to calculate not only respective differential cross sections but also the contribution of proton scattering on nucleons occurring in different shells. The differential cross sections for elastic and inelastic scattering were calculated at the energies of 0.2, 0.6, and 1 GeV per nucleon.     

Sommerfeld-watson representation for double-spectral functions

We discuss, for the case of pion-pion scattering, a closed system of equations which may be used for a self-consistent calculation of partial-wave amplitudes. It is shown that, for a given sufficiently small input function, the equations have a locally unique solution in a particular Banach space of doubly Hölder continuous partial wave amplitudes. At a fixed point, the scattering amplitude is shown to satisfy both a crossing symmetric unsubtracted Mandelstam representation and the elastic unitarity condition. In this initial study the partial-wave amplitudes are holomorphic in the right half complex angular-momentum plane.     

Scattering of electromagnetic waves from two-dimensional perfectly conducting random rough surfaces – study with the curvilinear coordinate method

We present a method giving the bi-static scattering coefficient of two-dimensional (2-D) perfectly conducting random rough surface illuminated by a plane wave. The theory is based on Maxwell's equations written in a nonorthogonal coordinate system. This method leads to an eigenvalue system. The scattered field is expanded as a linear combination of eigensolutions satisfying the outgoing wave condition. The boundary conditions allow the scattering amplitudes to be determined. The Monte Carlo technique is applied and the bi-static scattering coefficient is estimated by averaging the scattering amplitudes over several realizations. The random surface is represented by a Gaussian stochastic process. Results are compared to published numerical and experimental data. Comparisons are conclusive.     

Classical breathers and quantum coherent states in discrete NLS systems

We present a comparison of quantum and “semiclassical” trajectories of coherent states that correspond to classical breather solutions of finite discrete nonlinear Schrödinger (DNLS) lattices. The main goal is to explain earlier numerical observations of recurrent return to the vicinity of initial coherent states corresponding to stable breathers that are also spatially localized. This effect can be considered as a quantum manifestation of classical spatial localization. We show that these phenomena are encoded in a simple expression for the distance between the quantum and semiclassical states that involves the basic frequencies of the classical and quantum systems, as well as the breather amplitude and quantum spectral decomposition of the system. A corollary is that recurrence phenomena are robust under perturbation of the initial conditions for stable breathers.     

Quantum treatment for two two-level atoms in interaction with an SU(1,1) quantum system

We consider the interaction between two identical two-level atoms prepared in superposition states and an SU(1, 1) quantum system prepared in the Perelemov coherent state. We determine the timedependent wave function through the Schrödinger equation for the resonance case, and, consequently, we obtain the density matrix. We consider the phenomenon of collapses and revivals of the atomic population inversion for different values of the parameters and show the coherent trapping. We investigate the entanglement in the system where we discuss the linear entropy for different values of the involved parameters and for some states. Finally we examine the second-order correlation function to distinguish between the classical and nonclassical behaviors. We show that the system is sensitive to the variation in both the Bargmann index k and the Perelomov coherent parameter μ, as well as the atomic phase parameters.     

De Sitter spacetime as a momentum measuring apparatus

We discuss the evolution of a quantum wave packet in the expanding de Sitter spacetime using the plane wave solutions of the Dirac equation. We concentrate on the case of large negative times when the packet approaches the event horizon and confirm that the evolution accords with that expected from the classical trajectories. We point out that in certain conditions the packet can split into two components that become localized at different parts of the horizon and that this effect can be seen, in an idealized sense, as a measuring process for the momentum of the particle, in direct analogy with the measurement of spin in a Stern-Gerlach experiment.     

Semiclassical scattering theory with complex trajectories. I. Elastic waves

The WKB approximation to the one-particle Schrödinger equation is used to obtain the wave function at a given point as a sum of semiclassical terms, each of them corresponding to a different classical trajectory ending up at the same point. Besides the usual, real trajectories, also possible complex solutions of the classical equations of motion are considered. The simplicity of the method makes its use easy in practical cases and allows realistic calculations. The general solution of the one-dimensional WKB equations for an arbitrary number of complex turning points is given, and the solution is applied to calculate the position of the Regge poles of the scattering amplitude. The solution of the WKB equations in three dimensions for a central analytical potential is also obtained in a way that can be easily generalized to N-dimensions, provided the problem is separable. A multiple reflection series is derived, leading to a separation of the scattering amplitude into a smooth “background” term (single reflection approximation) that can be treated using classical but complex trajectories and a second resonating term that can be treated using the Sommerfeld-Watson transformation. The physical interpretation of the complex solutions of the classical equations of motion is given: they describe diffractive effects such as Fresnel, Fraunhofer diffraction, or the penetration of the quantal wave into shadow regions of caustics. They arise also in the scattering by a complex potential in an absorptive medium. The comparison with exact quantal calculations shows an astonishingly good agreement, and establishes the complex semiclassical approximation as a quantitative tool even in cases where the potential varies rapidly within a fraction of a wavelength. An approximate property of classical paths is discussed. The general pattern of the trajectories depends only on the product ? = EΘ, and not on energy and angle separately. This property is confirmed by experiments and besides the signature it gives for the semiclassical behavior, it simplifies considerably the search for all trajectories scattering through the same angle. Finally, a general classification of the different types of elastic heavy ion cross sections is given.     

Coherent states of general time-dependent harmonic oscillator

By introducing an invariant operator, we obtain exact wave functions for a general time-dependent quadratic harmonic oscillator. The coherent states, both inx- andp-spaces, are calculated. We confirm that the uncertainty product in coherent state is always larger thankh/2 and is equal to the minimum of the uncertainty product of the number states. The displaced wave packet for Caldirola-Kanai oscillator in coherent state oscillates back and forth with time about the center as for a classical oscillator. The amplitude of oscillation with no driving force decreases due to the dissipation in the system. However, the oscillation with resonant frequency oscillates with a large amplitude, even after a sufficient time elapse.     

Efficient Entanglement Concentration Schemes for Separated Nitrogen-Vacancy Centers Coupled to Low-Q Microresonators

We present several efficient entanglement concentration protocols (ECPs) with the nitrogen-vacancy (N-V) centers coupled to low-Q microresonators. Based on the input-output process of ancillary coherent light pulse in low-Q microresonators, we can obtain the maximally entangled states among remote participants via local operations and classical communication. Our protocols use a conventional photon detector to discriminate the two coherent states |α〉 and |?α〉, which is more convenient than homodyne measurement. We discuss the feasibility of our protocols, and they may be beneficial for quantum repeaters and quantum information processing.