Quantum chaos and fractals with atoms in cavities

We study the coupled translational, electronic, and field dynamics of the combined system “a two-level atom + a single-mode quantized field + a standing-wave ideal cavity”. In the semiclassical approximation with a point-like atom, interacting with the classical field, the dynamics is described by the Heisenberg equations for the atomic and field expectation values which are known to produce semiclassical chaos under appropriate conditions. We derive Hamilton–Schrödinger equations for probability amplitudes and averaged position and momentum of a point-like atom interacting with the quantized field in a standing-wave cavity. They constitute, in general, an infinite-dimensional set of equations with an infinite number of integrals of motion which may be reduced to a dynamical system with four degrees of freedom if the quantized field is supposed to be initially prepared in a Fock state. This system is found to produce semiquantum chaos with positive values of the maximal Lyapunov exponent. At exact resonance, the semiquantum dynamics is regular. At large values of detuning |δ|1, the Rabi atomic oscillations are usually shallow, and the dynamics is found to be almost regular. The Doppler–Rabi resonance, deep Rabi oscillations that may occur at any large value of |δ| to be equal to |αp₀|, is found numerically and described analytically (with α to be the normalized recoil frequency and p₀ the initial atomic momentum). Two gedanken experiments are proposed to detect manifestations of semiquantum chaos in real experiments. It is shown that in the chaotic regime values of the population inversion z_out, measured with atoms after transversing a cavity, are so sensitive to small changes in the initial inversion z_in that the probability of detecting any value of z_out in the admissible interval [−1,1] becomes almost unity in a short time. Chaotic wandering of a two-level atom in a quantized Fock field is shown to be fractal. Fractal-like structures, typical with chaotic scattering, are numerically found in the dependence of the time of exit of atoms from the cavity on their initial momenta.     

Beyond the Mean-Field Model of the Ring Cavity

The ring cavity device with large diffraction path in the free-space of the cavity cannotbe described within the mean-field model. It is shown to generate a large variety of monoconicaland multiconical patterns with wave and/or Türing modes, for anonlinear medium either made of two-level atoms or with a χ ⁽²⁾ crystal.Even in the limit of a single-longitudinal mode operation, monoconical structures can be differentfrom those predicted by the mean-field model. For instance, chaotic localized structures with anatomic medium and square patterns with a DOPO are presented.     

Optical bistability outside the rotating wave approximation in mixed species

Optical bistable behavior for a system of inhomogeneously broadened two sorts of two-level atoms in a ring cavity is investigated outside the rotating wave approximation (RWA). The model Maxwell–Bloch equations are treated with Fourier decomposition up to first harmonic. The first harmonic output field component exhibits reversed or mushroom bistability simultaneously with bi- and double-bistability in the fundamental field component. Inhomogeneous broadening and transverse field effects are also considered.     

Non-Equilibrium States of a Photon Cavity Pumped by an Atomic Beam

We consider a beam of two-level randomly excited atoms that pass one-by-one through a one-mode cavity. We show that in the case of an ideal cavity, i.e. no leaking of photons from the cavity, the pumping by the beam leads to an unlimited increase in the photon number in the cavity. We derive an expression for the mean photon number for all times. Taking into account leaking of the cavity, we prove that the mean photon number in the cavity stabilizes in time. The limiting state of the cavity in this case exists and it is independent of the initial state. We calculate the characteristic functional of this non-quasi-free non-equilibrium state. We also calculate the total energy variation in both the ideal and the open cavities as well as the entropy production in the ideal cavity.     

Application of the Jacobi functional equation and the ATS theorem in a quantum optical model

A new method is devised to study the atomic inversion in the model of a two-level atom interacting with a single quantized mode of the (initially coherent) electromagnetic field in an ideal resonant cavity. The method is based on number-theoretic results applied to the approximation of special series, specifically, on the functional equation for Jacobi theta functions and the ATS theorem. New asymptotic formulas are derived, with the help of which the behavior of the atomic inversion function on various time intervals can be determined in detail depending on the parameters of the system.     

Dynamical echo in two-state quantum systems

Evolution of quantum fidelity for two-level systems is studied in the context of periodic echo. For time independent case, we obtain a formal condition on the governing Hamiltonians under which the systems display periodic quantum echo. In addition, a revisit of single spin-1/2 system exposed to uniform rotating magnetic field elucidates that the well known Rabi oscillation is a simple mechanism to generate such echoes.     

On the Dimension of the Space of Dark States in the Tavis–Cummings Model

Mathematical Notes - The space of minimal energy of a qubit system is the dark subspace of quantum states of a system of two-level atoms in the finite-dimensional Tavis–Cummings (TC) model of...     

Global well‐posedness of classical solutions with large oscillations and vacuum to the three‐dimensional isentropic compressible Navier‐Stokes equations

We establish the global existence and uniqueness of classical solutions to the Cauchy problem for the isentropic compressible Navier‐Stokes equations in three spatial dimensions with smooth initial data that are of small energy but possibly large oscillations with constant state as far field, which could be either vacuum or nonvacuum. The initial density is allowed to vanish, and the spatial measure of the set of vacuum can be arbitrarily large; in particular, the initial density can even have compact support. These results generalize previous results on classical solutions for initial densities being strictly away from vacuum and are the first for global classical solutions that may have large oscillations and can contain vacuum states. © 2012 Wiley Periodicals, Inc.     

Dynamics of an n level system of atoms interacting with laser fields

This paper is an extension of Fujii et al. (quant—ph/0307066), and in this one we again treat a model of an atom with n energy levels interacting with n(n ? 1)/2 external laser fields, which is a natural extension of the usual two-level system. Then the rotating wave approximation (RWA) is assumed from the beginning. To solve the Schrödinger equation we set the consistency condition in our terminology and reduce it to a matrix equation with symmetric matrix Q consisting of coupling constants. However, to calculate exp(?itQ) explicitly is not easy. In the case of three-and four-level systems we determine it in a complete manner, so our model in these levels becomes realistic. Lastly, we make a comment on cavity QED quantum computation based on three energy levels of atoms as a forthcoming target.     

Statistical approach to the Jaynes-Cummings model

We study the dynamics of systems consisting of interacting two-level atoms and a field (microcavities). Such systems include the Jaynes-Cummings model. We formulate the problem and present a short history of it, derive a generalized kinetic equation for the system, find its solution, and show that this model allows describing photon emission and absorption.     

Damping of the free oscillations of elastic systems using gyroscopes

The damping of the free oscillations of an elastic system of general form to which a two-stage gyroscope is connected is considered. The coupling between the elastic system and the outer frame of the gyroscope is rigid, while the coupling between the frames is elasto-viscous. The parameters of this coupling must be chosen so that the free oscillations corresponding to the first mode of oscillations are damped as rapidly as possible. The solution of the problem is given in explicit form for an elastic system with one degree of freedom.     

Vertical oscillations of multimass system on elastic half space with a cylindrical cavity

A problem on oscillations of a multimass system (MS) is considered on an elastic half space with a cylindrical cavity. Equations of motions of an MS are given, which are modeled by masses that are connected by springs and dampers. A motion of the half space with a cavity is characterized by a transmitting function,which is known from a solution of a contact problem with vertical oscillations of a die on the half space given. The conditions of interrelation of the MS with the base close the system of algebraic linear equations for determining amplitudes of oscillations of each element of the MS. Translated from Dinamicheskie Sistemy, No. 7, pp. 13–18, 1988.     

A qualitative investigation of the oscillations of a pendulum with a periodically varying length and a mathematical model of a swing

The behaviour of the amplitude-frequency characteristics of families of periodic solutions, produced from the equilibrium position of a system, is established by a qualitative investigation of the equation of the oscillations of a pendulum, the length of which is an arbitrary periodic function of time. The non-local conditions for their stability and instability, expressed in terms of the amplitude and frequency of the oscillations, are obtained. The results are used when discussing the parametric and self-excited oscillatory model of a swing. In the parametric model the length of a swing is a specified periodic function of time, and in the self-excited oscillatory model it is a function of the phase coordinates of the system. For an appropriate choice of these functions, both systems have a common periodic solution. It is shown that the parametric model leads to an erroneous conclusion regarding the instability of the periodic mode, which is in fact realized in the oscillations of a swing, whereas the self-excited oscillatory model indicates its stability.     

Dynamical symmetries and temporal segmentation

Summary Segmentation of a mixed input into recognizable patterns is a task that is common to many perceptual functions. It can be realized in neural models through temporal segmentation: formation of staggered oscillations such that within each period every nonlinear oscillator peaks once and is dominant for a short while. We investigate such behavior in a symmetric dynamical system. The fully segmented mode is one type of limit cycle that this system can exhibit. We discuss its symmetry classification and its dynamical characterization. We observe that it can be sustained for only a small number of segments and relate this fact to a limitation on the appearance of narrow subharmonic oscillations in our system.     

A model for describing near-resonance oscillations in an elastic layer

One-dimensional transverse oscillations in a layer of a non-linear elastic medium are considered, when one of the boundaries is subjected to external actions, causing periodic changes in both tangential components of the velocity. In a mode close to resonance, the non-linear properties of the medium may lead to a slow change in the form of the oscillations as the number of the reflections from the layer boundaries increases. Differential equations describing this process were previously derived. The equations obtained are hyperbolic and the change in the solution may both keep the functions continuous and lead to the formation of jumps. In this paper a model of the evolution of the wave patterns is constructed as integral equations having the form of conservation laws, which determine the change in the functions describing the oscillations of the layer as “slow” time increases. The system of hyperbolic differential equations previously obtained follows from these conservation laws for continuous motions, in which one of the variables is slow time, for which one period of the actual time serves as an infinitesimal quantity, while the second variable is the real time. For the discontinuous solutions of the same integral equations, conditions on the discontinuity are obtained. An analogy is established between the solutions of the equations obtained and non-linear waves propagating in an unbounded uniform elastic medium with a certain chosen elastic potential. This analogy enable discontinuities which may be physically realised to be distinguished. The problem of steady oscillations of an elastic layer is discussed.     

The dependence of the natural frequencies of a one-dimensional elastic system on its length

The dependence of the natural frequencies and modes of the oscillations of distributed elastic system with characteristics of the stiffness and density that are variable along a coordinate of the cross section for arbitrary boundary conditions is investigated. It is proved that the presence of an external elastic medium, described by the Winkler model, may lead to an increase in the natural frequencies of the lower oscillation modes when the length of a one-dimensional elastic system is increased. The fine properties of the change in the natural frequencies as a function of the length of the system and the number of the oscillation mode are also established. A numerical-analytical investigation of examples which illustrate the characteristic anomalous behaviour of the lowest natural frequencies is presented.     

Coherent Swing Instability of Power Grids

We interpret and explain a phenomenon in short-term swing dynamics of multi-machine power grids that we term the Coherent Swing Instability (CSI). This is an undesirable and emergent phenomenon of synchronous machines in a power grid, in which most of the machines in a sub-grid coherently lose synchronism with the rest of the grid after being subjected to a finite disturbance. We develop a minimal mathematical model of CSI for synchronous machines that are strongly coupled in a loop transmission network and weakly connected to the infinite bus. This model provides a dynamical origin of CSI: it is related to the escape from a potential well, or, more precisely, to exit across a separatrix in the dynamical system for the amplitude of the weak nonlinear mode that governs the collective motion of the machines. The linear oscillations between strongly coupled machines then act as perturbations on the nonlinear mode. Thus we reveal how the three different mode oscillations??local plant, inter-machine, and inter-area modes??interact to destabilize a power grid. Furthermore, we present a phenomenon of short-term swing dynamics in the New England (NE) 39-bus test system, which is a well-known benchmark model for power grid stability studies. Using a partial linearization of the nonlinear swing equations and the proper orthonormal decomposition, we show that CSI occurs in the NE test system, because it is a dynamical system with a nonlinear mode that is weak relative to the linear oscillatory modes.     

Optimal control of the motion of a multilink system in a resistive medium

The optimal control of the motion of a system consisting of a main body and one or two links joined to it by cylindrical joints in a resistive medium is investigated. The resistance force of the medium acting on the moving body is assumed to depend on their velocity. The control is accomplished through high-frequency angular oscillations of the links. The equations of motion are analysed, and the mean velocity of translational motion of the system is estimated under certain assumptions. Optimal control problems are formulated and solved, and the laws of control of the oscillations of the links for which the maximum mean velocity of motion is obtained are found as a result. The data obtained are in qualitative agreement with observations of the swimming of fish and animals. The results of this study can be used in developing mobile robots that move in a liquid.     

Large-Eddy Simulations of cavitation in a square surface cavity

We report on the development and application of a multiphase approach to the prediction of cavitation induced by high-speed flow over and within a square surface cavity. The approach entails employing a full cavitation model in conjunction with Large-Eddy Simulations in order to capture the initiation and development of bubble formations in turbulent-flow conditions. The incipient formation of the bubble cloud, and the flow processes of vortex shedding and shear-layer oscillations are tracked using the Volume of Fluid method. The validity of the computational approach was assessed by comparisons with experiments on cavitating flow over a hydrofoil. Application to the case of flow over and within a two-dimensional square cavity with cavitation clearly reveal the presence of traveling cavitation at the corner of the cavity trailing edge, and vortex cavitation within the cavity. It is shown that the collapse of cavitation bubbles results in an impact frequency that is higher than the frequency of the shear-layer oscillations. This implies that structural damage due to cavitation is likely to be most severe at the corner formed at the intersection of the cavity’s trailing edge and the flat surface upstream of it.     

Quantum integrability and quantum chaos in the micromaser

The time-dependent quantum Hamiltonians