Excitation spectra for the V configuration of a three-level atom in resonance with two laser fields

1985 ◽  
Vol 63 (2) ◽  
pp. 144-150
Author(s):  
D. A. Hutchinson
Keyword(s):  
Excited States ◽  
Excitation Spectrum ◽  
Ground State ◽  
Central Peak ◽  
Decay Rates ◽  
Closed Form Expression ◽  
Form Expression ◽  
Excitation Spectra ◽  
Level Atom ◽  
Special Cases

The excitation spectrum is calculated for a three-level atom interacting with two strong electromagnetic fields. The two fields are in resonance with the atomic transition frequencies from the ground state to the two excited states. The excitation spectrum consists of a central peak and two pairs of side bands for each of the two transitions. If the decay rates of the two excited states are equal a relatively simple closed form expression is derived for the excitation spectrum. For unequal decay rates numerical methods are used to determine the excitation spectrum for selected special cases.

scholarly journals A Note on a Triple Integral

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2056
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Closed Form ◽  
Closed Form Expression ◽  
Polynomial Functions ◽  
Fundamental Constants ◽  
Form Expression ◽  
Zero Distribution ◽  
Special Cases ◽  
Almost All

A closed form expression for a triple integral not previously considered is derived, in terms of the Lerch function. Almost all Lerch functions have an asymmetrical zero-distribution. The kernel of the integral involves the product of the logarithmic, exponential, quotient radical, and polynomial functions. Special cases are derived in terms of fundamental constants; results are summarized in a table. All results in this work are new.

Evaluation of Luminescence Decay Measurements Probed on Pure and Doped Pt(IV) Hexahalogeno Complexes. II. Molecular Properties Obtained from Temperature Dependent Lifetime Curves

1997 ◽  
Vol 52 (5) ◽  
pp. 447-456
Author(s):  
Ingo Biertümpel ◽  
Hans-Herbert Schmidtke
Keyword(s):  
Excited States ◽  
Ground State ◽  
Energy Levels ◽  
Decay Rates ◽  
Rate Equations ◽  
Temperature Dependent ◽  
Lifetime Measurements ◽  
Thermally Activated ◽  
Ground State Energies ◽  
Higher Excited States

Abstract Lifetime measurements down to nearly liquid helium temperatures are used for determining energy levels and transition rates between excited levels and relaxations into the ground state. Energies are obtained from temperature dependent lifetimes by fitting experimental curves to model functions pertinent for thermally activated processes. Rates are calculated from solutions of rate equations. Similar parameters for pure and doped Pt(IV) hexahalogeno complexes indicate that excited levels largely belong to molecular units. Some of the rates between excited states are only somewhat larger than decay rates into the ground state, which is a consequence of the polyexponential decay measured also at low temperature (2 K). In the series of halogen complexes, the rates between spinorbit levels resulting from 3T1g increase from fluorine to bromine, although energy splittings become larger. Due to the decreasing population of higher excited states in this series, K^PtFö shows a tri-exponential, K2PtCl6 a bi-exponential and FoPtBr6 a mono-exponential decay. In the latter case the population density of higher excited states relaxes so fast that emission occurs primarily from the lowest excited Γ3(3T1g) level. Phase transitions and emission from chromophores on different sites can also be observed.

Shear compliance of two-dimensional pores possessing N -fold axis of rotational symmetry

2008 ◽  
Vol 464 (2091) ◽  
pp. 759-775 ◽  
Author(s):  
Thushan C Ekneligoda ◽  
Robert W Zimmerman
Keyword(s):  
Rotational Symmetry ◽  
Scaling Law ◽  
Mapping Function ◽  
Closed Form Expression ◽  
Fold Axis ◽  
Two Dimensional ◽  
Form Expression ◽  
Special Cases ◽  
Fourfold Symmetry ◽  
Shear Compliance

We use the complex variable method and conformal mapping to derive a closed-form expression for the shear compliance parameters of some two-dimensional pores in an elastic material. The pores have an N -fold axis of rotational symmetry and can be represented by at most three terms in the mapping function that conformally maps the exterior of the pore into the interior of the unit circle. We validate our results against the solutions of some special cases available in the literature, and against boundary-element calculations. By extrapolation of the results for pores obtained from two and three terms of the Schwarz–Christoffel mapping function for regular polygons, we find the shear compliance of a triangle, square, pentagon and hexagon. We explicitly verify the fact that the shear compliance of a symmetric pore is independent of the orientation of the pore relative to the applied shear, for all cases except pores of fourfold symmetry. We also show that pores having fourfold symmetry, or no symmetry, will have shear compliances that vary with cos 4 θ . An approximate scaling law for the shear compliance parameter, in terms of the ratio of perimeter squared to area, is proposed and tested.

Resonance fluorescence spectra from a three level atom interacting with a strong bichromatic field: induced two photon transitions

1983 ◽  
Vol 61 (1) ◽  
pp. 15-29 ◽  
Author(s):  
Douglas A. Hutchinson ◽  
Christine Downie ◽  
Constantine Mavroyannis
Keyword(s):  
Ground State ◽  
Rabi Frequency ◽  
Resonance Fluorescence ◽  
Excitation Spectra ◽  
Laser Fields ◽  
Two Photon ◽  
Photon Transition ◽  
Level Atom ◽  
Photon Excitation ◽  
The One

This investigation describes the interaction of a three level atom with two laser fields. One of the transitions from the ground state is in resonance with twice the frequency of the first laser and the other transition from the ground state is in resonance with the second laser. The Green's function formalism is used to derive expressions from which the induced two photon and one photon excitation spectra are computed. Also, approximate expressions are derived for the excitation spectra in the appropriate frequency regions. These results agree well with the numerical computations based upon the precise expressions. The interference between the two transitions produce some splittings; these splittings depend upon the Rabi frequency of the one photon transition. The intensities of the weak peaks depend upon the ratio of the Rabi frequency of the two photon transition to the frequency of the first laser. Some features of the excitation spectra are interpreted in terms of previous knowledge about the behavior of two level atoms in strong laser fields.

Resonance fluorescence spectra from a three-level atom interacting with a strong bichromatic field

1980 ◽  
Vol 58 (11) ◽  
pp. 1570-1579 ◽  
Author(s):  
M. P. Sharma ◽  
A. Balbin Villaverde ◽  
Constantine Mavroyannis
Keyword(s):  
Closed Form ◽  
Spectral Function ◽  
Excitation Spectrum ◽  
Electromagnetic Fields ◽  
Fluorescence Spectra ◽  
Central Peak ◽  
Relative Strength ◽  
Resonance Fluorescence ◽  
Level Atom ◽  
Bichromatic Field

We have studied the fluorescence spectra arising from the interaction of a three-level atom with two strong electromagnetic fields whose initially populated modes are equal to the two atomic transition frequencies, respectively. The Green's function formalism has been used to calculate the excitation spectrum of the system. An expression for the spectral function describing the excitation spectrum of the system has been derived in a closed form in the limit of high photon densities. Numerical computation of the expression for the spectral function indicates that at each transition frequency there may exist either one pair or two pairs or three pairs of sidebands, in addition to the central peak, depending upon the relative strength of the Rabi frequencies involved.

The Gerber-Shiu Discounted Penalty Function for Risk Process with Double Markovian Environment

2010 ◽  
Vol 108-111 ◽  
pp. 1103-1108
Author(s):  
Wen Guang Yu
Keyword(s):  
Penalty Function ◽  
Risk Process ◽  
Closed Form Expression ◽  
Standard Wiener Process ◽  
Form Expression ◽  
Expected Discounted Penalty Function ◽  
Premium Rate ◽  
Special Cases ◽  
Discounted Penalty Function ◽  
Expected Discounted Penalty

In this paper, we study the Gerber-Shiu discounted penalty function. We shall consider the case where the discount interest process and the occurrence of the claims are driven by two distinguished Markov process, respectively. Moreover, in this model we also consider the influence of a premium rate which varies with the level of free reserves. Using backward differential argument, we derive the integral equation satisfied by the expected discounted penalty function via differential argument when interest process in every state is perturbed by standard Wiener process and Poisson process. In some special cases, closed form expression for these quantities are obtained.

scholarly journals Exploring Novel Surface Representations via an Experimental Ray-Tracer in CGA

2021 ◽  
Vol 31 (2) ◽  
Author(s):  
Hugo Hadfield ◽  
Sushant Achawal ◽  
Joan Lasenby ◽  
Anthony Lasenby ◽  
Benjamin Young
Keyword(s):  
Closed Form Expression ◽  
Conformal Transformations ◽  
Square Root ◽  
Form Expression ◽  
Point Pair ◽  
Special Cases ◽  
Pure Grade ◽  
Surface Representations ◽  
Geometric Primitives ◽  
Vector Element

AbstractConformal Geometric Algebra (CGA) provides a unified representation of both geometric primitives and conformal transformations, and as such holds significant promise in the field of computer graphics. In this paper we implement a simple ray tracer in CGA with a Blinn–Phong lighting model, before putting it to use to examine ray intersections with surfaces generated from the direct interpolation of geometric primitives. General surfaces formed from these interpolations are rendered using analytic normals. In addition, special cases of point-pair interpolation, which might find use in graphics applications, are described and rendered. A closed form expression is found for the derivative of the square root of a scalar plus 4-vector element with respect to a scalar parameter. This square root derivative is used to construct an expression for the derivative of a pure-grade multivector projected to the blade manifold. The blade manifold projection provides an analytical method for finding the normal line to the interpolated surfaces and its use is shown in lighting calculations for the ray tracer and in generating vertex normals for exporting the evolved surfaces as polygonal meshes.

Compressibility of two-dimensional pores having n -fold axes of symmetry

2006 ◽  
Vol 462 (2071) ◽  
pp. 1933-1947 ◽  
Author(s):  
Thushan C Ekneligoda ◽  
Robert W Zimmerman
Keyword(s):  
Scaling Law ◽  
Mapping Function ◽  
Closed Form Expression ◽  
Fold Axis ◽  
Two Dimensional ◽  
Form Expression ◽  
Pore Compressibility ◽  
Special Cases ◽  
Axis Of Symmetry ◽  
Axes Of Symmetry

The complex variable method and conformal mapping are used to derive a closed-form expression for the compressibility of an isolated pore in an infinite two-dimensional, isotropic elastic body. The pore is assumed to have an n -fold axis of symmetry, and be represented by at most four terms in the mapping function that conformally maps the exterior of the pore into the interior of the unit circle. The results are validated against some special cases available in the literature, and against boundary-element calculations. By extrapolation of the results for pores obtained from three and four terms of the Schwarz–Christoffel mapping function for regular polygons, the compressibilities of a triangle, square, pentagon and hexagon are found (to at least three digits). Specific results for some other pore shapes, more general than the quasi-polygons obtained from the Schwarz–Christoffel mapping, are also presented. An approximate scaling law for the compressibility, in terms of the ratio of perimeter-squared to area, is also tested. This expression gives a reasonable approximation to the pore compressibility, but may overestimate it by as much as 20%.

scholarly journals Nuclear Weak Processes in Presupernova Stars

1996 ◽  
Vol 145 ◽  
pp. 165-172
Author(s):  
A. Ray ◽  
T. Kar ◽  
S. Sarkar ◽  
S. Chakravarti
Keyword(s):  
Excited States ◽  
Beta Decay ◽  
Ground State ◽  
Strength Function ◽  
Supernova Explosion ◽  
Decay Rates ◽  
Distribution Method ◽  
The Core ◽  
Weak Transition ◽  
Late Stages

The structure and the size of the core of massive presupernova stars are determined by the electron fraction and entropy of the core during its late stages of evolution; these in turn affect the subsequent evolution during gravitational collapse and supernova explosion phases. Beta decay and electron capture on a number of neutron rich nuclei can contribute substantially towards the reduction of the entropy and possibly the electron fraction in the core. Methods for calculating the weak transition rates for a number of nuclei for which no reliable rates exist (particularly for A > 60) are outlined. The calculations are particularly suited for presupernova matter density (p = 107 - 109 g/cc) and temperature (T = 2 - 6 × 109 °K). We include besides the contributions from the ground state and the known excited states, the Gamow-Teller (GT) resonance states (e.g. for beta decay rates, the GT+ states) in the mother nucleus which are populated thermally. For the GT strength function for transitions from the ground state (as well as excited states) we use a sum rule calculated by the spectral distribution method where the centroid of the distribution is obtained from experimental data on (p,n) reactions. The contribution of the excited levels and GT+ resonances turn out to be important at high temperatures which may prevail in presupernova stellar cores.

scholarly journals A Lévy Risk Model with Ratcheting Dividend Strategy and Historic High-Related Stopping

2020 ◽  
Vol 2020 ◽  
pp. 1-12 ◽  
Author(s):  
Aili Zhang ◽  
Zhang Liu
Keyword(s):  
Risk Model ◽  
Poisson Model ◽  
Risk Process ◽  
Closed Form Expression ◽  
Form Expression ◽  
Ruin Time ◽  
Scale Functions ◽  
Surplus Process ◽  
Dividend Payments ◽  
Special Cases

This paper focuses on the De Finetti’s dividend problem for the spectrally negative Lévy risk process, where the dividend is deducted from the surplus process according to the racheting dividend strategy which was firstly introduced in Albrecher et al. (2018). A major feature of the racheting strategy lies in which the dividend rate never decreases. Unlike the conventional studies, the closed form expression for the expected, accumulated, and discounted dividend payments until the draw-down time (rather than the ruin time) is obtained in terms of the scale functions corresponding to the underlying Lévy process. The optimal barrier for the ratcheting strategy is also studied, where the dividend rate can be increased. Finally, two special cases, where the scale functions are explicitly known, i.e., the Brownian motion with drift and the compound Poisson model, are considered to illustrate the main result.

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