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Physical Review A

Energy transfer between slowly moving atoms—the case of no crossing point

M. G. Payne, Munir Hasan Nayfeh

Published Feb 1, 1976• 34 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.13.595

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Journal of the Optical Society of America B•Aug 1, 1992•5 citations
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 \ensuremathΔE\ensuremathτ\ensuremath≫\ensuremathℏ\ensuremath{\Delta}E\ensuremath{\tau}\ensuremath{\gg}\ensuremath{\hbar}\ensuremathΔE\ensuremathτ\ensuremath≫\ensuremathℏ, with \ensuremathΔE\ensuremath{\Delta}E\ensuremathΔE being the energy discrepancy and \ensuremathτ=bv\ensuremath{\tau}=\frac{b}{v}\ensuremathτ=vb​ 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 \ensuremathΔE=0\ensuremath{\Delta}E=0\ensuremathΔE=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.
A general approximate method for computing the cross sections for certain inelastic collision processes is developed. It is based on the possibility of obtaining, with the use of the impact-parameter method, an exact expression for the transition amplitude in a special case. The dependence of the cross section on the parameters of coupling constant, velocity, and energy difference between states is discussed for a two-level model. Detailed calculations are made of cross section for collisions of the type $A(^{2}P_{\frac{3}{2}})+B\ensuremath{\rightleftharpoons}A(^{2}P_{\frac{1}{2}})+B$, where AAA is an excited atom of an alkali metal and BBB is an atom of a noble gas. An Appendix contains a review of the effective long-range interaction of atoms.
A two-level model of an atomic system is investigated in order to study the cross section for certain inelastic collision processes. An impact-parameter method is employed. The time-dependent Schr\"odinger equation has been integrated numerically for some simple interaction potentials. Cross sections for excitation have been determined and are presented as functions of the parameters describing the model. The results are compared with those obtained by certain approximate methods. The effects of inclusion of diagonal matrix elements in the interaction Hamiltonian are found to be large. For a fixed form of the interaction potential, two cross sections, which may differ quite substantially, are obtained, depending on the algebraic sign of the coupling parameters or the energy difference between the states.
This thesis discusses different aspects of time-dependent two-level quantum systems and their coherent dynamics. In the introductory chapters, we present some basic models used to study them and some basic results related to the level-crossing problems generally. As these systems are only rarely analytically solvable, some emphasis is put on the existing approximative theories. In particular, the generalization of the method of Dykhne, Davis and Pechukas (DDP), originally developed for problems in near-adiabatic region, is studied further. In the DDP theory, the final populations of different states are determined by the complex zero points of the eigenenergies. Superparabolic level-crossing models are introduced in this thesis and they prove to be simple but versatile test models for these studies. It is shown that, by considering all of the zero points, one can obtain accurate approximations also for highly non-adiabatic regions. In Chapter 4, a general differential geometric framework for time-dependent level-crossing models is developed and its basic character is discussed. A natural way of associating a plane curve to a time-dependent two-level model is introduced. This association allows one to use all of the mathematical results concerning plane curves to study these models. As an example, we use the Four-vertex theorem to discuss the adiabatic limit of certain type of systems. We also consider the prospect of enhancing the population transfer by transforming the time-dependent coupling pulse into a zero-area coupling. This is done either smoothly or by an abrupt jump in the phase of the coupling. It has been shown earlier, that one can obtain a complete population inversion (CPI) in a robust way with strong, non-resonant zero-area pulses. We study this in the more general time-dependent setting in Chapter 5, where also the energy levels are driven in a time-dependent fashion. This allows to transport the CPI phenomenon from the strong coupling region towards more moderate couplings. As an example of the smooth case, we consider the SechTanh-model, and show that even the nearest-zero DDP approximation can be used, despite the highly non-adiabatic origin of the phenomenon. We develop an approximative scheme also for the phase-jump case, using the parabolic model as an example. As a limiting case, we derive an approximative formula for the transition probability, which has some universal character and which proves to be useful in the most interesting case where the robust CPI is obtained. It is shown that even the highly suppressed transitions in tunnelling cases can be strongly enhanced in the phase-jump scenario.
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.
Excitation transfer process between atoms is investigated theoretically in the framework of the two-state approximation. Two methods are proposed with regard to the treatment of the non-local potential which results from the projection-operator formalism. One is a local potential approximation and the other is an extension of the method of Chadan. A new expression for the cross section is found. This formula is more accurate than the usual distorted wave one. The matrix elements of a coupling potential with respect to the wave functions of the relative motion are estimated by making use of the JWKB approximation. The new cross section formula obtained will be useful for a wide class of collision processes in which initial and final states are mutually orthogonal.
An extension to a previous model [ J. Opt. Soc. Am. B8, 1594 ( 1991)], which described the collision between two atoms in a cooling laser field, is made, which allows diffusive effects to be analyzed. A method similar to that used in Monte Carlo studies is developed, except that the diffusion is now determined exactly at all times. This approach relies directly on making a local classical path approximation for the atoms’ trajectories. For the simple case of a single two-level atom in the cooling field, the energy of the atom reaches equilibrium at the Doppler limit, as one would expect. In the full collisional situation we make the assumption that each atom undergoes a series of binary collisions. For sufficiently high atomic densities these collisions provide a limit on the lowest temperature that may be achieved by the cooling process.