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International Journal of Quantum Chemistry

Absorption line shape of two coupled oscillators decaying in two different mediums

Arturo Hardisson, F. Mauricio, Santiago Velasco

Published Jun 1, 1977• 1 citation
Publisher

Abstract

Abstract The linear response theory is used to study the line shape of two coupled oscillators, decaying in two different baths, when one of the oscillators is coupled to the radiation field. The baths are considered to be thermally excited. The analysis is based on the computation of the correlation functions 〈a (t)ai (0)〉 and 〈ai (t)a (0)〉 of each oscillator, taking the coefficients associated with the operators a and ai in the Heisenbemg representation as the starting point of the calculation. The line shape in different cases is also discussed.

DOI
10.1002/qua.560110610

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Abstract The linear response theory is used to study the line shape of two coupled oscillators, decaying in two different baths, when one of the oscillators is coupled to the radiation field. The baths are considered to be thermally excited. The analysis is based on the computation of the correlation functions 〈a (t)ai (0)〉 and 〈ai (t)a (0)〉 of each oscillator, taking the coefficients associated with the operators a and ai in the Heisenbemg representation as the starting point of the calculation. The line shape in different cases is also discussed.
Absorption spectra are calculated for four model systems using two different methods for the calculation: The first is the standard linear response technique, while the second involves direct calculation of the energy dissipated in the sample interacting with the applied field, thus corresponding exactly to what is in fact measured experimentally. This latter technique, which we call the direct method, agrees with a linear response (as it must) for weakly applied fields for which the first-order perturbation analysis underlying linear response is in fact valid. For cases in which the system/field interaction is stronger, the linear approximation becomes poor, and the direct method, which includes the field-induced dynamics and in itself contains no approximations, is far better. The four systems considered consist of either single or coupled pairs of harmonic oscillators or two-level systems (spins) each in a thermal bath and coupled to the field by a displacement coupling. For both linear-response and direct calculations the system/field dynamics is solved exactly and the system/bath dynamics treated using a semigroup technique. For the harmonic oscillators, for which the population in any single excited state can never become large, the nonlinear effects (those present in the direct but not in the linear response) are quite minor. For the two-level/spin/systems, however, substantial corrections to the linear-response result are found in the direct calculation. These include power (or saturation) broadening and the appearance, when the two spins are coupled strongly to each other and one is coupled strongly to the field, of a new line at the averaged frequency of the two spins. For even stronger coupling, the line shape becomes very complex (corresponding to the full 15D algebra of the two spins plus the field), exhibiting several peaks and Fano antiresonances. The direct method, which is based on work of Lebowitz, seems to offer substantial advantages for the analysis of systems in which the applied field interactions are strong: This situation, common in microwave and magnetic resonance spectroscopy, is also relevant to high-energy laser studies.
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.
The effect of a driving or pump field on the emission and absorption line-shape functions for an atom is discussed. The pump field is assumed to oscillate at a frequency near the resonance frequency for transitions between a single pair of atomic states, and transitions between one of these states and any other state of the atom are analyzed. The pump field is treated classically, and atomic relaxation is treated in general terms. The emission and absorption line-shape function (the latter is defined as the rate of absorption of energy from a weak signal field, applied in addition to the pump field, as a function of the signal-field frequency) are found by evaluating the relevant two-time atomic correlation functions in the Markoff approximation. In the limit of high pump-field intensity, both the absorption and the emission spectra are doubly peaked at frequencies which differ from the usual resonance frequency by $\ifmmode\pm\else\textpm\fi{}\frac{1}{2} \ensuremath{\Omega}$, where $\ensuremath{\Omega}$ is the frequency of the pump-field-induced oscillations in the populations of the two strongly coupled states. The absorption and the emission spectra are represented by essentially the same function in the limit of high pump-field intensity, as they are also in the limit of vanishing pump-field intensity. For intermediate pump-field intensities, however, the two functions are quite different in form, and no simple proportionality exists between them.
The frequency distribution of resonant radiation emitted by a system of two identical interacting atoms is calculated. The separation between the atoms is assumed to be much larger than the size of their charge distributions. Initially one atom is excited while the second one is in the ground state. The two atoms are coupled through the radiation field and the electrostatic interaction. For not too large internuclear separations the line shape of the emitted radiation exhibits two distinct peaks, and the general features are identical with the result obtained from resonance fluorescence scattering. In the decay process, however, there is equal probability of scattering through the symmetric and antisymmetric states, and thus the peak heights of the emitted radiation are different from the ones in the resonance fluorescent case. For large internuclear separations the two peaks merge into the one characteristic of the line shape of a single isolated atom.
The processes of absorption and emission of radiation by a system of two interacting atoms is investigated. Separation between the atoms is assumed to be much larger than the atomic diameter, and interactions considered are those between the individual atoms and the radiation field and the electrostatic interaction between the atoms. Calculations include those for the time-dependent probability amplitude of the intermediate and final states as well as that of the emission line shape. The line shape of the emitted radiation consists of two peaks whose amplitude, separation, and half-width are functions of both the internuclear separation R of the atoms and the energy difference of the excited states of the two atoms. For a large energy difference, the peaks are distinct; for small energy differences, the peaks are distinct at smaller separations, but merge toward the line shape characteristic of a single, isolated atom as R increases.
We develop an expansion for the absorption line shape of a two-level system in a strong radiation field being perturbed by dephasing collisions with a thermal bath of foreign atoms. We assume that the bath atoms do not interact with the radiation field and with each other. The microscopic information relevant for line broadening in a strong field is expressed in terms of a hierarchy of dipole correlation functions. The latter may be calculated with the use of the Franck-Condon factors for the nuclear wave functions. The present theory reduces in the weak-field limit to the well-known linear unified theory of line broadening and in the impact limit we recover the Karplus-Schwinger formula.