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Acta Physica Sinica

THEORETICAL TREATMENT OF RADIATION TRAPPING IN A MULTIPLE LINES COUPLED SYSTEM

LAI RE, Jian Wang, Wang Chu-Ji, MA XING-XIAO, 中国科学技术大学化学物理系

Published Jan 1, 1997• 0 citations
Publisher

Abstract

The purpose of this paper is to propose a theoretical treatment of radiation trapping in a multiple lines coupled system in which coupling includes collisional energy transfer and cross absorption between the excited states. First, to describe the coupled system, the Holstein equation of radiation trapping is extended to an integral differential master equation group in which the vector and matrix form is adopted. Then by use of operator manipulations, the asymptotic solution in the form of a series expansion in terms of operators is presented. Finally, the asymptotic solution is reduced, leading to a result which can not only give the system a complete description, but also has far more advantages than the simple cut off solution. A numeral illustration of a two excited states system is also presented.

DOI
10.7498/aps.46.42

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The purpose of this paper is to propose a theoretical treatment of radiation trapping in a multiple lines coupled system in which coupling includes collisional energy transfer and cross absorption between the excited states. First, to describe the coupled system, the Holstein equation of radiation trapping is extended to an integral differential master equation group in which the vector and matrix form is adopted. Then by use of operator manipulations, the asymptotic solution in the form of a series expansion in terms of operators is presented. Finally, the asymptotic solution is reduced, leading to a result which can not only give the system a complete description, but also has far more advantages than the simple cut off solution. A numeral illustration of a two excited states system is also presented.
Abstract A differential integral equation system of Holstein type is used to describe the kinetic process in V E energy transfer between molecules and atoms with radiation trapping.A reduced solution of a finite series has been presented,which allows to describe the whole temporal behavior of fluorescence.The further derivative is an analytical expression of the fluorescence decay rate as a function of rate constants and the maximum eigenvalue of the integral operator appearing in the rate equations.This will be especially useful and convenient for experimental determinations.Results are schematically illustrated and mutually compared by means of numerical calculations.
A differential-integral equation system of Holstein type is used to describe the kinetic process in V-E energy transfer between molecules and atoms with radiation trapping.A reduced solution of a finite series has been presented,which allows to describe the whole temporal behavior of fluorescence.The further derivative is an analytical expression of the fluorescence decay rate as a function of rate constants and the maximum eigenvalue of the integral operator appearing in the rate equations.This will be especially useful and convenient for experimental determinations.Results are schematically illustrated and mutually compared by means of numerical calculations.
Theoretical treatment of resonance radiation trapping under steady-state conditions, particularly of quenching experiments, is presented. Applying operator manipulations, Holstein's equation of radiation trapping [Phys. Rev. 72, 1212 (1947)] is solved by a perturbation expansion, which can be physically interpreted in terms of multiple scatterings. The resulting expression shows that in a steady-state quenching experiment radiation trapping invalidates the relation of the Stern-Volmer type except when the amount of the buffer gas is small. In the limit of low buffer-gas pressure, a simple expression is given for the effective lifetime in the Stern-Volmer relation: the average time of the detected resonant photons remaining within the enclosure. While it is true that this suggested approach is perturbative in nature, the asymptotic behavior of the high-order expansion coefficients, however, enables us to reduce the summation over infinitely many alternatives to some leading terms. Such a reduction procedure helps curtail the computation time and improve the precision. Finally, some numerical illustrations of our suggested ideas are presented, which shows this approach is a good tool to treat the steady-state trapping effect.
The key to the treatment of the resonance radiation imprisonment lies in the solution to Holstein's equation. A new method to solve Holstein's equation is proposed in this paper, which integrates the equation directly by means of operator calculation and results in a formula for the excited particles in an infinite series in an integral operator. Our study on the behavior of the high-ordercoefficients of this series leads to the success in overcoming the difficulty to obtain the sum of infinite terms, and a formula of finite terms under any given precision is acquired, which applies to the whole time scale.
The transferred energy between two (three) radiators localized in two (three) cavities in interaction through an evanescent field is investigated. In the good-cavity limit the exact solution of the Schrödinger equation for this system is obtained. An interesting problem appears when the located excitation in one atom from one distinguished cavity is transferred to a second atom from another cavity. We analyze analytically and numerically the dynamics of the transferred energy as a function of the coupling constant between the cavities. Another interesting problem is the trapping condition for two atoms flying through different coupled cavities. For subsystems consisting of two atomic levels, we know that, only a dynamic trapping condition can be found. In this paper, we show that due to the coupling of the cavities, the trapping condition for two atoms flying through different cavities becomes possible. This effect depends on the coupling constants between the cavities and the dipole interaction between the radiators and cavity modes. A similar problem is studied in the bad-cavity limit, when the lifetime of photons in the cavity is smaller than the decay time of excited atoms. In this case, we write the analytical solution of the master equation and show its entanglement. We obtain that entangled states arise from non-entangled initial states of the atoms. The exact description of such a system opens up new possibilities for the processing of quantum information with two optical cavities.
The classical path treatment of a collision between an excited atom and another atom with a nearly resonant energy level leads to coupled differential equations which have been discussed previously by many workers. Here we present an elementary procedure for solving this system of equations in the case of no crossing point and in the limit where $\ensuremath{\Delta}E\ensuremath{\tau}\ensuremath{\gg}\ensuremath{\hbar}$, with $\ensuremath{\Delta}E$ being the energy discrepancy and $\ensuremath{\tau}=\frac{b}{v}$ being a measure of the time of collision. Even though the method is designed to take advantage of the slowness of the collision, it is also shown to be exact if $\ensuremath{\Delta}E=0$. The form of the solution is very similar to that of Vainshtein et al., but the present derivation avoids several of their approximations which individually are not as accurate as their final result. It is also pointed out that the coupled equations which arise for the energy-transfer problem also arise in connection with the interaction of laser pulses with atoms. Hence, the method has application to a wide class of multistate problems. A simple laser-interaction problem is discussed as a second example of the method.