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Physical Review A

Holes in Spectral Lines

Peter R. Fontana, Rajendra P. Srivastava

Published Jun 1, 1973• 9 citations
Publisher

Abstract

The decay of an atom in the presence of a static perturbation is investigated. The perturbation couples a decaying state with a nondecaying state. A "hole" appears in the emission line at a frequency equal to the frequency difference between the nondecaying state and the ground state. The effect is explained by analyzing the phase relationships of the radiation emitted by the two coupled states.

DOI
10.1103/physreva.7.1866

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The decay of an atom in the presence of a static perturbation is investigated. The perturbation couples a decaying state with a nondecaying state. A "hole" appears in the emission line at a frequency equal to the frequency difference between the nondecaying state and the ground state. The effect is explained by analyzing the phase relationships of the radiation emitted by the two coupled states.
The radiative decay of an atom with two excited states, one decaying and the other non-decaying, coupled by a time dependent perturbation, is investigated. The hole in the emission line is found to be shifted by an amount equal to the frequency of the time dependent perturbation. The practical importance of this effect is discussed. It is also noticed that the two methods of solution, one by making the rotating-wave approximation and the other by a perturbation expansion, yield different results, especially in the case when the two excited states are strongly coupled.
The excitation and decay probabilities for resonance scattering of radiation from an atom with two coupled excited states in an external static field are calculated as a function of time and frequency. Various oscillatory terms are found in the probabilities. These oscillations depend on the frequency of radiation, the external coupling and the energy difference between the excited states. Two special cases are investigated in detail. In the first case where one of the excited states does not decay a 'hole' appears in the emission line at a frequency equal to the frequency difference between the ground state and the unperturbed non-decaying excited state. In the second case where the two excited states decay with the same rate, one of the two lines in the emission spectrum is suppressed and the other is enhanced.
Several calculations have been carried out using the transition operator technique. The radiative decay in a two-level quantum system with excited state coupled by an external perturbation has been investigated and an expression obtained for P(t), the probability of finding the atom in its excited state at time t. This is seen to possess a steady value for yt >> 1 and a third term which decays at nearly half the rate of its second. Next radiative decay of an atom with two close-lying excited states is considered and |bq| 2 , the spectral density for spontaneous emission from the uppermost level, calculated. This shows that the proximity of the upper two levels makes the line-shape non-Lorentxian. Heisenberg equations of motion were then derived for the transitions operators of an ℓ-level atom undergoing radiative decay, the possibility of overlapping pairs of levels being ignored. These master equations were then used to obtain the spectral profiles of certain atomic lines, employing the so-far avoided Markoff approximation. First the spectral density of the scattered radiation was found for transitions between levels 5P3/2 and 4S (ground state), and 6S and 5P 3/2, in the potassium atom when driving fields of arbitrary strength coupled levels 6S and 4P3/2, and 4P 3/2 and 4S. Triple-peaked profiles were obtained. Spectral profiles for emission were then derived for transitions between levels 3P and (ground state), and 3P and in the hydrogen atom when a driving field of arbitrary strength coupled levels The former profile was double the latter quadruple-peaked. Lastly master equations were derived for an atom, with two excited levels undergoing radiative decay, when allowance was made for thepossibility of their overlapping. A driving field of arbitrary magnitude coupled the uppermost and ground states. The spectrum for the scattered light resulting from transitions between the uppermost and ground state was found and graphs computed for various field strengths and separations of the excited states.
A two-level atom interacting simultaneously with an external static field is reinvestigated. Here, first the Hamiltonian of the atom and the external field is diagonalized and then the interaction of the radiation field is considered. This method yields, directly, the state populations for the atom, and the lineshape and linewidth of the spontaneously emitted radiation. The earlier result on the state populations by Bali and Higgins has been compared with the present work and found to be in good agreement. The line shape of the emitted radiation is Lorenzian and the halfwidth is found to be increasing with increasing external perturbation. For the case where the atom is initially in the excited state, the intensity of the emitted radiation decreases with increasing perturbation and finally reaches a saturation limit. However, when the atom is considered to be in the ground state, initially, the intensity increases with increasing perturbation and it again reaches a saturation limit at a large perturbation field.
The radiative decay of an atom with two excited states coupled by an external perturbation is investigated. The differential equations of motion are Fourier transformed and the probability amplitudes are obtained by contour integration. The real parts of the poles in the complex plane are the perturbed energies of the excited states, and the imaginary parts yield the decay characteristics. The decay probabilities of the excited states contain three different decay terms; two exponential decays and one modulated exponential decay. The probabilities of the final states give the frequency distribution of the emitted photons as a function of time. In an Appendix, the Heitler-Ma formalism is used to eliminate the final states of the system, and the resulting equations which contain damping terms are compared with the phenomenological method.