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Physical Review

Transition Probability due to Random Perturbations

Amnon Yariv, W. H. Louisell

Published Jan 15, 1962• 0 citations
Publisher

Abstract

The subject of transition probabilities induced by random perturbations is treated. The expression for average transition rates between two energy levels is derived for the general case when the frequency spectrum of the perturbation and the distribution of the energy levels have comparable widths. The general expression takes an especially simple form for the special cases where the frequency bandwidth of the perturbation Hamiltonian is either much larger or much smaller than that of the energy level distribution. The formulas are applied, by way of illustration, to a number of physical problems.

DOI
10.1103/physrev.125.558

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Physical Review•Nov 1, 1968•212 citations
The subject of transition probabilities induced by random perturbations is treated. The expression for average transition rates between two energy levels is derived for the general case when the frequency spectrum of the perturbation and the distribution of the energy levels have comparable widths. The general expression takes an especially simple form for the special cases where the frequency bandwidth of the perturbation Hamiltonian is either much larger or much smaller than that of the energy level distribution. The formulas are applied, by way of illustration, to a number of physical problems.
We have seen in Section 1.2 that a transition between two discrete levels | i 〉 and | k 〉 is possible when the system is submitted to a perturbation which has a non-zero Fourier component close to the Bohr frequency ω 0 = ( E k − E i )/ħ of the transition. This is the case when the perturbation Ĥ 1 ( t ) is sinusoidal, of the form Ĥ 1 ( t ) = Ŵ cos(ω t + φ), and oscillates at a frequency ω close to ω 0 . But in many real physical situations, the perturbation is not perfectly mastered: it is a pure sine function only for times shorter than a limit, called the coherence time . At longer times, the oscillating perturbation has a phase and an amplitude that vary randomly. This is, for example, the case when the perturbation is due to an incident electromagnetic wave produced by a thermal lamp, in which the thermal fluctuations induce random variations of the amplitude and phase of the wave. As the perturbation is no longer a pure sinusoidal wave, its Fourier spectrum is no longer a Dirac delta function. It has some finite width around a mean frequency. For this reason a random perturbation is also called a ‘broadband’ perturbation. We shall determine in this complement the temporal evolution of the quantum system submitted to such a random perturbation. We shall see that the transition probability between two discrete levels is proportional to T for small interaction time T , and has an exponential behaviour at long times, a result which is very similar to that obtained for a transition between a discrete level and a continuum.
Resonance transition probabilities for a system in the presence of oscillatory perturbations at two or more different frequencies are discussed. It is shown that if resonance transitions are induced by a perturbation at one frequency, then the presence of the other perturbations at nonresonant frequencies alters the resonance frequency for the first perturbation. Theoretical expressions for the alteration of the resonance frequency are derived. Various applications of the derived formulas are discussed, including resonance experiments with more than one oscillatory field, molecular beam experiments in a nonuniform constant field, evaluation of "collision-narrowing" effects in nuclear paramagnetic resonance, etc. The effects of the extraneous oscillatory perturbations are shown to be especially important if their frequencies are close to Bohr frequencies for the perturbation.
A general expression for stimulated transition rates between two levels of an atomic system is developed. This expression not only applies for an arbitrary radiation field, but it also includes the effects of line broadening on transition rates. The general expression is used to obtain the rate equation for the spectral photon number density per steradian for a given pair of atomic energy levels. The result is applied to two special cases by integrating over the spectral and angular ranges of interest.
Sudden-modulation theory is used for the calculation of stationary saturation and polarization of two-level systems. The randomly fluctuating frequency of the two-level system is assumed to be a noncorrelated, purely discontinuous Markovian process. The stationary probability of field-induced transitions is calculated exactly. It is shown that if the frequency modulation is fast (or if the field is strong enough) the transition probability coincides with that obtained in the framework of perturbation theory when the latter is developed in its integrodifferential non-Markovian version. This probability differs from the kinetic probability calculated by Markovian perturbation theory (with fluctuating frequency) in its corresponding spectral wing shape and in the field dependence of its corresponding resonance width. This field dependence is smoother than that in Markovian theory and stabilizes when the resonance width is not equal to the natural transition width. Nevertheless, in the Lorentzian approximation the free-induction decay rate in the framework of exact (or non-Markovian) theory increases with the field in nearly the same way as that in Markovian theory. Almost the same is true of the field dependence of the free-induction decay amplitude in either theory, provided that the signal correlation before and after the field is switched off is neglected. The conditions are found under which the Lorentzian approximation of the transition probability is valid. These conditions are shown to break down when the theory (in the fast-frequency-modulation limit) leads to agreement with experiments reported by De Voe and Brewer [ Phys. Rev. Lett.50, 1269– 1272 ( 1983)].
The probability of an atom undergoing a transition to a given final state by means of two-photon absorption is expressed for arbitrary field states in terms of the second-order normally-ordered field correlation function, evaluated at the position of the atom. In the case of stationary fields, an expression for the absorption rate is found which reduces to a particularly simple form for narrow-bandwidth fields near resonance. The effect of field statistics is illustrated by comparing the absorption rate w2(ch)w_{2}^{}{}_{}{}^{(\mathrm{ch})}w2​​(ch) for chaotic or Gaussian light to the rate w2(l)w_{2}^{}{}_{}{}^{(l)}w2​​(l) for laser light. The rate for laser light is calculated within the context of a particular model based on the assumption of fixed field amplitude and random frequency modulation, and the chaotic light to which it is compared is assumed to have the same (Lorentzian) power spectrum. When the width ${\ensuremath{\kappa}}_{f}$ of the final atomic level is much greater than the bandwidth bbb of the field, we obtain the previously derived result w2(ch)w2(l)=2\frac{w_{2}^{}{}_{}{}^{(\mathrm{ch})}}{w_{2}^{}{}_{}{}^{(l)}}=2w2​​(l)w2​​(ch)​=2. When ${\ensuremath{\kappa}}_{f}\ensuremath{\ll}b$, on the other hand, the ratio between the two rates depends on the (mean) frequency of the field, and assumes the (maximum) value 4 when the field is exactly on resonance.