Exact Resonance Research Articles

Exact steady states of the periodically forced and damped Duffing oscillator

We present an analytical methodology to compute the exact steady states of the strongly nonlinear Duffing oscillator with arbitrary viscous damping and under the action of a time-periodic excitation. In contrast to current analytical approaches that consider harmonic excitations to produce approximate steady-state solutions, the excitations considered in this work are multi-harmonic and are computed in closed form. We consider the exact fundamental resonance plot of this strongly nonlinear system and present an exact depiction of the dynamic balance between the internal and external forces leading to it in terms of rotating vectors in the complex plane with modulated amplitudes and modulated frequencies of rotation. Moreover, an infinity of exact steady-state solutions seems to be possible, each corresponding to a different time-periodic excitation. Generalization of the presented methodology to systems with other type of nonlinearities or with many degrees of freedom is presumably straightforward.

Steady state obliquity of a rigid body in the spin–orbit resonant problem: application to Mercury

We investigate the stable Cassini state 1 in the p : q spin–orbit resonant problem. Our study includes the effect of the gravitational potential up to degree and order 4 and p : q spin–orbit resonances with \(p,q\le 8\) and \(p\ge q\). We derive new formulae that link the gravitational field coefficients with its secular orbital elements and its rotational parameters. The formulae can be used to predict the orientation of the spin axis and necessary angular momentum at exact resonance. We also develop a simple pendulum model to approximate the dynamics close to resonance and make use of it to predict the libration periods and widths of the oscillatory regime of motions in phase space. Our analytical results are based on averaging theory that we also confirm by means of numerical simulations of the exact dynamical equations. Our results are applied to a possible rotational history of Mercury.

Non-exponential decoherence of radio-frequency resonance rotation of spin in storage rings

Precision experiments, such as the search for electric dipole moments of charged particles using radiofrequency spin rotators in storage rings, demand for maintaining the exact spin resonance condition for several thousand seconds. Synchrotron oscillations in the stored beam modulate the spin tune of off-central particles, moving it off the perfect resonance condition set for central particles on the reference orbit. Here we report an analytic description of how synchrotron oscillations lead to non-exponential decoherence of the radiofrequency resonance driven up-down spin rotations. This non-exponential decoherence is shown to be accompanied by a nontrivial walk of the spin phase. We also comment on sensitivity of the decoherence rate to the harmonics of the radiofreqency spin rotator and a possibility to check predictions of decoherence-free magic energies.

The effect of electromagnetically induced transparency in a potassium nanocell

The effect of electromagnetically induced transparency (EIT) has been experimentally implemented for the first time for the (4S 1/2–4P 1/2–4S 1/2) Λ-system of potassium atom levels in a nanocell with a 770-nm-thick column of atomic vapor. It is shown that, at such a small thickness of the vapor column, the EIT resonance can be observed only when the coupling-laser frequency is in exact resonance with the frequency of the corresponding atomic transition. The EIT resonance disappears even if the coupling-laser frequency differs slightly (by ~50 MHz) from that of the corresponding atomic transition, which is due to the high thermal velocity of K atoms. The EIT resonance and related velocity selective optical pumping resonances caused by optical pumping (formed by the coupling) can be simultaneously recorded because of the small (~462 MHz) hyperfine splitting of the lower 4S 1/2 level.

On-the-Fly Generation of Differential Resonance Scattering Probability Distribution Functions for Monte Carlo Codes

Current Monte Carlo codes use one of three models: (1) the asymptotic scattering model, (2) the free gas scattering model, or (3) the S(α,β) model, depending on the neutron energy and the specific Monte Carlo code. This paper addresses the consequences of using the free gas scattering model, which assumes that the neutron interacts with atoms in thermal motion in a monatomic gas in thermal equilibrium at material temperature T. Most importantly, the free gas model assumes the scattering cross section is constant over the neutron energy range, which is usually a good approximation for light nuclei, but not for heavy nuclei, where the scattering cross section may have several resonances in the epithermal region. Several researchers in the field have shown that the exact resonance scattering model is temperature dependent, and neglecting the resonances in the lower epithermal range can underpredict resonance absorption due to the upscattering phenomenon mentioned above, leading to an overprediction of keff by several hundred pcm. Existing methods to address this issue involve changing the neutron weights or implementing an extra rejection scheme in the free gas sampling scheme, and these all involve performing the collision analysis in the center-of-mass (CM) frame, followed by a conversion back to the laboratory frame to continue the random walk of the neutron.The goal of this paper was to develop a sampling methodology that (1) accounted for the energy-dependent scattering cross sections in the collision analysis and (2) was performed in the laboratory frame, avoiding the conversion to the CM frame. The energy dependence of the scattering cross section was modeled with even-ordered polynomials (second and fourth order) to approximate the scattering cross section in Blackshaw’s equations for the moments of the differential scattering probability distribution functions. These moments were used to sample the outgoing neutron speed and angle in the laboratory frame on the fly during the random walk of the neutron. Results for criticality studies on fuel pin and fuel assembly calculations using methods developed in this paper showed very close comparison to results using the reference Doppler-broadened rejection correction scheme.

Resonant structure, formation and stability of the planetary system HD155358

Two Jovian-sized planets are orbiting the star HD155358 near exact mean motion resonance (MMR) commensurability. In this work we re-analyze the radial velocity (RV) data previously collected by Robertson et al. (2012). Using a Bayesian framework we construct two models - one that includes and one that excludes gravitational planet-planet interactions (PPI). We find that the orbital parameters from our PPI and noPPI models differ by up to $2\sigma$, with our noPPI model being statistically consistent with previous results. In addition, our new PPI model strongly favours the planets being in MMR while our noPPI model strongly disfavours MMR. We conduct a stability analysis by drawing samples from our PPI model's posterior distribution and simulating them for $10^9$ years, finding that our best-fit values land firmly in a stable region of parameter space. We explore a series of formation models that migrate the planets into their observed MMR. We then use these models to directly fit to the observed RV data, where each model is uniquely parameterized by only three constants describing its migration history. Using a Bayesian framework we find that a number of migration models fit the RV data surprisingly well, with some migration parameters being ruled out. Our analysis shows that planet-planet interactions are important to take into account when modelling observations of multi-planetary systems. The additional information that one can gain from interacting models can help constrain planet migration parameters.

Influence of multiphoton detunings from resonance on adiabatic processes in a five-level system

The Schrodinger equation for a five-level system has been analyzed numerically for different values of multiphoton detuning from exact resonances. The range of admissible deviations for which the effect of multiphoton detunings on the eigenstate of the interaction Hamiltonian imitating a three-level system is insignificant has been determined. The effective population transfer and the possibility of double storage of optical information have been demonstrated.

CLASSIFICATION OF SATELLITE RESONANCES IN THE SOLAR SYSTEM

ABSTRACT Several pairs of solar system satellites occupy mean motion resonances (MMRs). We divide these into two groups according to their proximity to exact resonance. Proximity is measured by the existence of a separatrix in phase space. MMRs between Io–Europa, Europa–Ganymede, and Enceladus–Dione are too distant from exact resonance for a separatrix to appear. A separatrix is present only in the phase spaces of the Mimas–Tethys and Titan–Hyperion MMRs, and their resonant arguments are the only ones to exhibit substantial librations. Could there be a causal connection between the libration amplitude and the presence of a separatrix? Our suspicions were aroused by Goldreich & Schlichting, who demonstrate that sufficiently deep in a MMR, eccentricity damping could destabilize librations. However, our investigation reveals that libration amplitudes in both the Mimas–Tethys and Titan–Hyperion MMRs are fossils. Although the Mimas–Tethys MMR is overstable, its libration amplitude grows on the tidal damping timescale of Mimas’s inclination, which is considerably longer than a Hubble time. On the other hand, the Titan–Hyperion MMR is stable, but tidal damping of Hyperion’s eccentricity is too weak to have affected the amplitude of its libration.

Gain and index guiding in a Raman generator with diffraction-free pump beams

Gain spectrum for a diffraction-free solution of the steady-state paraxial wave equation for the Stokes amplitude is shown to be frequency-asymmetric relative to an exact Raman resonance. The asymmetry is directly related to a light guiding originated due to Raman-induced change of the refractive index and imprinted in the Stokes far field. The index guiding with a zero-order Bessel beam pump is experimentally observed at the Stokes generation in high-pressure hydrogen and barium nitrate, and reasonable agreement with numerical data is demonstrated.

Application of the low-finesse γ -ray frequency comb for high-resolution spectroscopy

High finesse frequency combs (HFC) with large ratio of the frequency spacing to the width of the spectral components have demonstrated remarkable results in many applications such as precision spectroscopy and metrology. We found that low finesse frequency combs having very small ratio of the frequency spacing to the width of the spectral components are more sensitive to the exact resonance with absorber than HFC. Our method is based on time domain measurements reviling oscillations of the radiation intensity after passing through an optically thick absorber. Fourier analysis of the oscillations allows to reconstruct the spectral content of the comb. If the central component of the incident comb is in exact resonance with the single line absorber, the contribution of the first sideband frequency into oscillations is exactly zero. We demonstrate this technique with gamma-photon absorbtion by M\"{o}ssbauer nuclei providing the spectral resolution beyond the natural broadening.

The influence of deuteration of complex organic molecules on their fluorescence quantum yield (a review)

Data on the influence of deuteration of organic molecules and/or solvent on their fluorescence quantum yield are summarized. It is demonstrated that the effect of deuteration is normal, i.e., deuteration increases the fluorescence quantum yield due to decrease in the internal conversion probability, for organic molecules exhibiting fluorescence in the red region of spectrum, in which the internal conversion competes with the emission of fluorescence. In the case in which the probability of internal conversion is low, i.e., energy of level S 1 ≥ 16000–20000 cm–1, the intersystem crossing probability from the S 1 state to the set of triplet levels depends on the relative position of S 1 and the nearest triplet level. In so doing, the effect of deuteration can be either normal or anomalous, depending on whether the resonance between the S 1 level and the level nearest to it, the T n level, improves or deteriorates when these levels shift as a result of deuteration. In dilute vapors, cooled supersonic jets, and crystalline matrices, including Shpolskii matrices, the effect of deuteration at helium temperatures depends on exact resonance between the interacting levels. In the case of dilute vapors and cooled jets, the effect also depends on the vibronic level being excited. Deuteration of OH and NH groups, as a rule, slows proton transfer in the S 1 state of the molecule that occurs with their involvement, leading to normal effect of deuteration.

Laboratory Experiments on the Effects of a Variable Current Field on the Spectral Geometry of Water Waves

AbstractLaboratory experiments were performed to investigate the effects of a coflowing current field on the spectral shape of water waves. The results indicate that refraction is the main factor in modulating wave height and overall wave energy. Although the structure of the current field varies considerably, some current-induced patterns in the wave spectrum are observed. In high frequencies, the energy cascading generated by nonlinear interactions is suppressed, and the development of a spectral tail is disturbed, as a consequence of the detuning of the four-wave resonance conditions. Furthermore, the presence of currents slows the downshifting of the spectral peak. The suppression of the high-frequency energy under the influence of currents is more prominent as the spectral steepness increases. The energy suppression is also more accentuated and long-standing along the fetch when the directional spreading of waves is sufficiently broad. This result indicates that the current-induced detuning of resonant conditions is more effective when exact resonances are the primary mechanism of nonlinear interactions than when quasi resonances prevail (directionally narrow cases). Additionally, the directional analysis shows that the highly variable currents broaden the directional spreading of waves. The broadening is suggested to be related to random refraction and scattering of wave rays. The random disturbance of wavenumbers alters the nonlinear interaction conditions and weakens the energy exchanges among wave components, which is expressed in the suppression of the high-frequency energy.

Nonresonant two-photon transitions in length and velocity gauges

We reexamine the invariance of two-photon transition matrix elements and corresponding two-photon Rabi frequencies under the "gauge" transformation from the length to the velocity gauge. It is shown that gauge invariance, in the most general sense, only holds at exact resonance, for both one-color as well as two-color absorption. The arguments leading to this conclusion are supported by analytic calculations which express the matrix elements in terms of hypergeometric functions, and ramified by a "master identity" which is fulfilled by off-diagonal matrix elements of the Schroedinger propagator under a the transformation from the velocity to the length gauge. The study of the gauge dependence of atomic processes highlights subtle connections between the concept of asymptotic states, the gauge transformation of the wave function, and infinitesimal damping parameters for perturbations and interaction Hamiltonians that switch off the terms in the infinite past and future [of the form exp(-epsilon * | t |)]. We include a pertinent discussion.

Response and receptivity of the hypersonic boundary layer past a wedge to free-stream acoustic, vortical and entropy disturbances

This paper analyses the response and receptivity of the hypersonic boundary layer over a wedge to free-stream disturbances including acoustic, vortical and entropy fluctuations. Due to the presence of an attached oblique shock, the boundary layer is known to support viscous instability modes whose eigenfunctions are oscillatory in the far field. These modes acquire a triple-deck structure. Any of three elementary types of disturbance with frequency and wavelength on the triple-deck scales interacts with the shock to generate a slow acoustic perturbation, which is reflected between the shock and the wall. Through this induced acoustic perturbation, vortical and entropy free-stream disturbances drive significant velocity and temperature fluctuations within the boundary layer, which is impossible when the shock is absent. A quasi-resonance was identified, due to which the boundary layer exhibits a strong response to a continuum of high-frequency disturbances within a narrow band of streamwise wavenumbers. Most importantly, in the vicinity of the lower-branch neutral curve the slow acoustic perturbation induced by a disturbance of suitable frequency and wavenumbers is in exact resonance with a neutral eigenmode. As a result, the latter can be generated directly by each of three types of free-stream disturbance without involving any surface roughness element. The amplitude of the instability mode is determined by analysing the disturbance evolution through the resonant region. The fluctuation associated with the eigenmode turns out to be much stronger than the free-stream disturbances due to the resonant nature of excitation, and in the case of acoustic disturbances, to the well-known amplification effect of a strong shock. Moreover, excitation at the neutral position means that the instability mode grows immediately without undergoing any decay, or missing any portion of the unstable region. All these indicate that this new mechanism is particularly efficient. The boundary-layer response and coupling coefficients are calculated for typical values of parameters.

Pulse propagation and optically controllable switch in coupled semiconductor-double-quantum-dot nanostructures

The problem of pulse propagation is theoretically investigated through a coupled semiconductor-double-quantum-dot (SDQD) nanostructure. Solving the coupled Maxwell–Bloch equations for the SDQD and field simultaneously, the dynamic control of pulse propagation through the medium is numerically explored. It is found that when all the control fields are in exact resonance with their corresponding transitions, a weak Gaussian-shaped probe pulse is transmitted through the medium nearly without any significant absorption and losses so that it can preserve its shape for quite a long propagation distance. In contrast, when one of the control fields is not in resonance with its corresponding transition, the probe pulse will be absorbed by the QD medium after a short distance. Then we consider the probe pulses with higher intensities. It is realized that an intense probe pulse experiences remarkable absorption and broadening during propagation. Finally, we demonstrate that this SDQD system can be employed as an optically controllable switch for the wave propagation to transit from an absorbing phase to a perfect transparency for the probe field. The required time for such switch is also estimated through realistic values.

Consequences of tidal interaction between disks and orbiting protoplanets for the evolution of multi-planet systems with architecture resembling that of Kepler 444

We study orbital evolution of multi-planet systems with masses in the terrestrial planet regime induced through tidal interaction with a protoplanetary disk assuming that this is the dominant mechanism for producing orbital migration and circularization. We develop a simple analytic model for a system that maintains consecutive pairs in resonance while undergoing orbital circularization and migration. Migration times for each planet may be estimated once planet masses, circularization times and the migration time for the innermost planet are given. We applied it to a model system with the current architecture of Kepler 444 interacting with a protoplanetary disk, the evolution time for the system as a whole being comparable to current protoplanetary disk lifetimes. In addition we performed numerical simulations with input data obtained from this model. These indicate that although the analytic model is inexact, relatively small corrections to estimated migration rates yield systems for which period ratios vary by a minimal extent. Because of relatively large deviations from exact resonance in the observed system of up to $2\%,$ the migration times obtained in this way indicate only weak convergent migration such that a system for which the planets did not interact would contract by only $\sim 1\%$ although undergoing significant inward migration as a whole. We performed additional simulations to investigate how the system could undergo significant convergent migration before reaching its final state. These indicate migration times have to be significantly shorter and resonances significantly closer. Relative migration rates would then have to decrease allowing period ratios to increase to become more distant from resonances as the system approached its final state in the inner regions of the protoplanetary disk (abridged).

Locations of stationary/periodic solutions in mean motion resonances according to the properties of dust grains

The equations of secular evolution for dust grains in mean motion resonances with a planet are solved for stationary points. This is done including both Poynting-Robertson effect and stellar wind. The solutions are stationary in semimajor axis, eccentricity, and resonant angle, but allow the pericentre to advance. The semimajor axis of stationary solutions can be slightly shifted from the exact resonant value. The periodicity of the stationary solutions in a reference frame orbiting with the planet is analytically proved. The existence of periodic solutions in mean motion resonances means that analytical theory enables for dust particles also infinitely long capture times. The stationary solutions are periodic motions to which the eccentricity asymptotically approaches and around which the libration occurs. Using numerical integration of equation of motion are successfully found initial conditions corresponding to the stationary solutions. Numerically and analytically determined shifts of the semimajor axis form the exact resonance for the stationary solutions are in excellent agreement. The stationary solutions can be plotted by locations of pericenters in the reference frame orbiting with the planet. The pericenters are distributed in the space according to properties of dust particles.

Formation of metastable molecular states at the resonance scattering of two atoms in a laser radiation field

Using the Dirac’s method, the formation of metastable molecular states at the resonance scattering of two atoms in the laser radiation field is considered. Expressions for the metastable level populations and the resonance scattering cross sections are obtained. In the case of an exact resonance with the laser radiation, the graphs for populations and resonance scattering cross sections, which have two peaks due to the Autler–Townes effect, are obtained. These results play an important role in the study of the controlled chemical reaction and for the understanding of the processes in the quantum systems of the Bose–Einstein condensate at low temperatures, as well as in the various optical processes in atomic gases.

The theory of secondary resonances in the spin–orbit problem

We study the resonant dynamics in a simple one degree of freedom, time dependent Hamiltonian model describing spin-orbit interactions. The equations of motion admit periodic solutions associated with resonant motions, the most important being the synchronous one in which most evolved satellites of the Solar system, including the Moon, are observed. Such primary resonances can be surrounded by a chain of smaller islands which one refers to as secondary resonances. Here, we propose a novel canonical normalization procedure allowing to obtain a higher order normal form, by which we obtain analytical results on the stability of the primary resonances as well as on the bifurcation thresholds of the secondary resonances. The procedure makes use of the expansion in a parameter, called the detuning, measuring the shift from the exact secondary resonance. Also, we implement the so-called `book-keeping' method, i.e., the introduction of a suitable separation of the terms in orders of smallness in the normal form construction, which deals simultaneously with all the small parameters of the problem. Our analytical computation of the bifurcation curves is in excellent agreement with the results obtained by a numerical integration of the equations of motion, thus providing relevant information on the parameter regions where satellites can be found in a stable configuration.

Analysis of fuel temperature effects on reactivity of light water reactor fuel assemblies by using MVP-2 adopting an exact resonance elastic scattering model

ABSTRACTUsing the continuous-energy Monte Carlo code MVP-2 adopting a resonance elastic scattering model considering the thermal motion of a target nucleus (the exact model) for major heavy nuclides, analysis of fuel temperature effects on reactivity of mockup UO2 and MOX fuel assemblies for light water reactors was performed, and the results were compared with those of the conventional asymptotic model. A base condition was a hot operating condition with an in-channel void fraction of 40% and fuel temperature of 520 ℃ for the BWR fuel assemblies and a hot zero-power condition with fuel temperature of 284 ℃ for the PWR fuel assemblies. The fuel temperature of a high-temperature condition was 1500 ℃ for both types of assemblies. The calculated results showed that the exact model made the neutron multiplication factors at the high-temperature condition lower by −220 to −440 pcm (10−5 Δk) and the Doppler reactivity between the base- and high-temperature conditions more negative by 7% to 10% compared with those obtained by the asymptotic model. The energy-dependent reaction rates of capture and ν-fission were also analyzed to study the detail mechanism in the effect of the exact model on the assembly reactivity.