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Journal of Modern Optics

Optical Double-resonance Spectra with Time-dependent Pulses

Peter Rodgers, S. Swain

Published Jul 1, 1989• 2 citations
Publisher

Abstract

The optical double-resonance spectrum for a three-level atom interacting with two monochromatic laser pulses with arbitrary temporal profiles is calculated by use of Light's perturbation theory. Unlike some earlier work, the theory is not restricted to exact resonance of the pump laser, or to pulse areas which are exact multiples of 2π. A variety of pulse profiles are investigated, and the spectra are found to depend strongly on the detuning of the pump laser.

DOI
10.1080/09500348914551001

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Physical Review A•Mar 1, 1988•17 citations
The optical double-resonance spectrum for a three-level atom interacting with two monochromatic laser pulses with arbitrary temporal profiles is calculated by use of Light's perturbation theory. Unlike some earlier work, the theory is not restricted to exact resonance of the pump laser, or to pulse areas which are exact multiples of 2π. A variety of pulse profiles are investigated, and the spectra are found to depend strongly on the detuning of the pump laser.
Exact solutions to the problem of a two-level atom interacting with an external field with a sech time profile are used to calculate the optical-double-resonance (ODR) and time-dependent resonance-fluorescence (RF) spectra for various values of the detuning. The probe in the ODR case is assumed to have either a sech or a flat time profile. For small pulse areas the exact ODR expressions obtained are very simple. For large pulse areas the ODR spectra for a flat probe differs markedly from the equivalent spectra for a sech probe, and for any pulse area they both depend strongly on the detuning. As the detuning increases, the extreme peaks on one side increase significantly before eventually decreasing whilst the peaks on the other side decrease rapidly and monotonically. Similar features are found for the RF spectrum, but the dependence on detuning is weaker.
The problem of a two-level atom interacting with a non-monochromatic radiation pulse is studied in the resonant approximation for arbitrary detunings, and without the resonant approximation. By using the time-dependent theory of perturbation the analytical expressions for the transition probability and induced dipole moment of the atom are obtained. It is shown that the upper level population of the atom exhibits oscillations that are absent in the resonant approximation. The temporal shape of the pulse is shown to affect significantly the atomic transition probability. The behaviour of the transition probability and dipole moment spectrum for arbitrary pulse shapes, phases and detunings is investigated. The interference between the spectral components arising in nonlinear processes is shown to play a considerable role in the formation of a dipole moment spectrum.
The responses of the time-dependent resonance fluorescence spectra of two-level atoms in the presence of (a) strong cw laser fields and (b) short pulsed laser fields to functional disturbances in these fields have been studied by the use of functional-sensitivity-analysis techniques. For the cw case, the transient and steady-state response regions are explored separately. It is found that the response to the field disturbance occurring during the earlier transient period is most prominent at the positions of the well-known three peaks of the corresponding spectrum. In contrast, the response to a disturbance taking place in or near the steady-state region tends to increase the separation between the peaks and lower the intensity of the line center, in exact agreement with the direct calculation of the perturbed spectrum. For the pulsed case, the response to the disturbance is most visible at the line center and at the accompanying multiple peaks caused by the population oscillation of the upper atomic level. The response strongly depends on the magnitude of the pulse area, thus giving rise to distinct asymptotic behavior for (2n+1)\ensuremath{\pi} and 2n\ensuremath{\pi} pulses. As the pulse area becomes large enough to cause significant population oscillation of the upper atomic level, the response of the spectrum shows a resemblance to that of the cw case.
A uniform asymptotic expression for the double optical resonance signal is obtained for arbitrary pulse shapes. The theory, which is applicable for resonant pump pulses with a total envelope area of greater than about 10 pi , predicts a multipeak structure for the double optical resonance spectrum. This is interpreted as arising from interference between probe excitation at two different parts of the pump pulse. Related phenomena have already been observed in similar, exact, calculations for specific pulse shapes.
A study is presented for the time-dependent resonance-fluorescence spectrum of two-level atoms irradiated by hyperbolic-secant pulsed lasers, allowing for an arbitrary spontaneous-emission rate from the excited level to the ground level. Novel features of the temporally transient spectrum have been observed. The spectrum is dominated by a central peak which oscillates in time and is accompanied by multiple side peaks. Asymptotically (i.e., after the pulse has passed) the side peaks disappear, leaving a tail at the line center which displays a Lorentzian shape of width equal to the sum of the spontaneous-emission rate and the width of the spectrometer considered and of magnitude proportional to the instantaneous intensity of the pulse. The exact behavior of the spectrum is dictated by the area of the engaged pulse and by how the atom is prepared initially. In particular, it is found that the spectrum can be asymmetric even when the laser is tuned to exact resonance if the atoms are initially prepared in a certain class of proper coherent states, irrespective of how the field amplitude may vary in time. An analysis of the spectral symmetry without resorting to any explicit evaluation of the spectrum itself is also presented. Furthermore, the associated optical Bloch equations are solved analytically in terms of hypergeometric functions in a similar manner to that done previously for the undamped two-level Schr\"odinger equation. Specific numerical results are presented for pulses of area equal to \ensuremath{\pi}, 2\ensuremath{\pi}, 3\ensuremath{\pi}, 4\ensuremath{\pi}, and 5\ensuremath{\pi} for both symmetric and asymmetric cases.