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The Journal of Chemical Physics

Transition Rates in Multilevel Systems

John H. Williamson

Published Mar 15, 1969• 1 citation
Publisher

Abstract

The transition rates between levels in a multilevel system can be measured by perturbing the system and observing the response. Previous work has shown that, in general, many different sets of transition rates could account for the same response. In consequence, statistical errors in the computed rates are much larger (e.g., 10 000 times larger) than those in the original data. It is shown here that it is possible to design the experiment so as to avoid this instability. The ideal experiment uses fairly high-frequency sinusoidal perturbations in place of the impulsive or steady-state perturbations usually employed. The application to the measurement of atomic lifetimes and collisional transition rates among excited states is discussed. The method is tested with data obtained from two 6-level electronic circuits which simulate a hydroxyl molecule and a hydrogen plasma, respectively.

DOI
10.1063/1.1671434

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Physical Review A•Feb 1, 1992•23 citations
The transition rates between levels in a multilevel system can be measured by perturbing the system and observing the response. Previous work has shown that, in general, many different sets of transition rates could account for the same response. In consequence, statistical errors in the computed rates are much larger (e.g., 10 000 times larger) than those in the original data. It is shown here that it is possible to design the experiment so as to avoid this instability. The ideal experiment uses fairly high-frequency sinusoidal perturbations in place of the impulsive or steady-state perturbations usually employed. The application to the measurement of atomic lifetimes and collisional transition rates among excited states is discussed. The method is tested with data obtained from two 6-level electronic circuits which simulate a hydroxyl molecule and a hydrogen plasma, respectively.
In time-domain measurements of a Poisson two-level system, the observed transition rates are always smaller than those of the actual system, a general consequence of a finite measurement bandwidth in an experiment. This underestimation of the rates is significant even when the measurement and detection apparatus are 10 times faster than the process under study. We derive here a quantitative form for this correction, using a straightforward state transition model that includes the detection apparatus, and provide a method for determining a system's actual transition rates from bandwidth-limited measurements. We support our results with computer simulations and experimental data from time-domain measurements of quasiparticle tunneling in a single-Cooper-pair transistor.
In an externally driven multilevel quantum system observation that the NEXT jump has not yet happened affects its future development. In previous work [Phys. Rev. A36, 929 (1987)] it was shown that this class of measurement makes it possible to observe remarkably long dark intervals -- or intermittency -- in the atomic fluorescence of an atom with 3 or more levels. Those calculations were carried out when the driven oscillations or Rabi flopping between the ground state and a strongly fluorescing state were fast compared to its lifetime. In systems with solid state Qubits the accessible parameter space is generally limited to the regime where oscillations are slower than the lifetime. In this paper we evaluate intermittency in atomic transitions, due to measurements with a null result, in this limit. During the dark periods the wave function of the continuously measured multilevel system is coherent.
There are many systems of physical interest for which a set of rate equations for level populations can provide insight. If the system has two (or more) different mechanisms for effecting transition between levels, total rates of transfer are usually taken as the sum of rates that the individual mechanisms would cause acting alone. Using the example of a hydrogen atom subjected to (ionic and electronic) collisions and external radiation, it is shown that when these individual mechanisms overlap, the total transfer rates must be modified to account for correlations between collisional and radiative processes. For a broad-band radiation field the modified rates have a simple physical interpretation. In the case of a narrow-band field the overlapping events may cause new coherences to appear and interpretation of the modified 'rates' is more complicated.
Strong driving of quantum systems opens opportunities for both controlling and characterizing their states. For theoretical studying of these systems properties we use the rate-equation formalism. The advantage of such approach is its relative simplicity. We used the formalism for description of a two-level system with further expanding it on a case of a multi-level system. Obtained theoretical results have good agreement with experiments. The presented approach can also be considered as one more way to explore properties of quantum systems and underlying physical processes such as, for instance, Landau–Zener–Stückelberg–Majorana transitions and interference.
An efficient computational algorithm for evaluating the evolution operator, and related induced transition probabilities, for a multilevel system interacting with an amplitude-modulated sinusoidal field of arbitrary strength, modulating envelope, frequency and phase, is developed. The algorithm is based on the Riemann product integral representation of the evolution operator in conjunction with Frazer's method of mean coefficients and avoids the introduction of commonly used perturbative or rotating wave approximations. Model calculations for the interaction of a two-level system with a Gaussian pulse are presented and used to discuss the dependence of time-resolved and steady-state frequency sweep spectra on some of the pulse characteristics.
We report the results of a series of pulsed and cw laser experiments which measure spontaneous and effective lifetimes of the Cs(5DJ) levels and investigate excitation-transfer collisions involving Cs(5DJ) atoms and ground-state-cesium perturbers. With cw excitation of the dipole-forbidden but electric-quadrupole-allowed 6S1/2\ensuremath{\rightarrow}5D5/2 transition, we monitor the ratio of sensitized to direct fluorescence, i.e., I5D3/2\ensuremath{\rightarrow}6P1/2/I5D5/2\ensuremath{\rightarrow}6P3/2. A rate-equation analysis of these data yields values for the Cs(5D5/2)+Cs(6S)\ensuremath{\rightarrow}Cs(5D3/2)+Cs(6S) excitation-transfer rate, and for the rate of quenching of Cs(5D) by ground-state perturbers. The role of out-of-multiplet quenching is discussed at length, and we have demonstrated that quenching by ground-state atoms dominates over that by cesium dimers under these conditions. In the pulsed-laser experiments, the temporal evolution of the 5DJ\ensuremath{\rightarrow}6S and cascade 6PJ\ensuremath{'}\ensuremath{\rightarrow}6S fluorescence was observed following either direct forbidden-line pumping of one fine-structure level, or indirect excitation of both fine-structure levels (i.e., molecular excitation followed by predissociation). Analysis of the buildup and decay rates of the various levels as a function of cesium density yields values for the natural lifetimes of the Cs(5DJ) levels, as well as for the excitation transfer and quenching rates. Best values for the lifetimes and cross sections for Cs-Cs collisions obtained from these combined experiments are \ensuremath{\tau}5D=1250\ifmmode\pm\else\textpm\fi{}115 ns, \ensuremath{\sigma}5/2\ensuremath{\rightarrow}3/2=36\ifmmode\pm\else\textpm\fi{}8 \AA{}2, and \ensuremath{\sigma}5D=30\ifmmode\pm\else\textpm\fi{}3 \AA{}2. In addition, numbers for the 6P level-quenching cross sections due to collisions with Cs atoms and Cs2 molecules, respectively, were obtained: \ensuremath{\sigma}6P(Cs)=6.6\ifmmode\pm\else\textpm\fi{}3.0 \AA{}2 and \ensuremath{\sigma}6P(Cs2)=863\ifmmode\pm\else\textpm\fi{}260 \AA{}2. In the discussion, we show how these new values can be used to reconcile the seemingly discrepant results of several previous studies.