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Journal of Modern Optics

Time-dependent Pulses in Quantum Optics Exact Results for Optical-double-resonance and Resonance-fluorescence Spectra

P.A. Rodgers, S. Swain

Published May 1, 1987• 12 citations
Publisher

Abstract

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.

DOI
10.1080/09500348714550661

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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 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 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.
Resonance fluorescence from the two-level atom in the field of a resonant, exponentially-decaying pulse is considered when the spontaneous linewidth may be neglected. The analytical expressions for the physical spectra are given and discussed for any field strength and any pulse decay time. The approximation schemes for large and small pulse areas are presented and illustrated by numerical results. In the latter case line narrowing is obtained, while the former corresponds to the near-adiabatic case.
We compute the time-dependent spectrum of fluorescence by two-level atoms after a strong resonant exciting laser pulse is turned off abruptly. The behavior of the "delayed" three-peaked spectrum is shown to be dependent on the natural lifetime of the upper state and on the bandwidth of the interferometer used in the measurement. A prompt increase in the central-peak intensity and an oscillating decay in the two side peaks are predicted to occur immediately after the laser is turned off. These are manifestations of the "undressing" of the atomic states. Our calculation permits consideration of the question of the speed of the undressing and its observation.
We study the transient fluorescence spectra of two-level atoms driven by a strong optical field when the atoms are initially prepared in pure dressed states of the atom-field Hamiltonian. Spectra obtained here differ qualitatively from those observed in the steady state. In the case of steady-state resonant excitation, a three-peaked spectrum symmetric about the laser frequency is obtained; with pure-dressed-state preparation, the symmetry of the spectrum about the driving field is broken---one side peak is initially enhanced and the other is initially suppressed. Furthermore, the temporal oscillations of fluorescence intensity observed in previous calculations of transient fluorescence spectra appropriate to atoms prepared in a superposition of dressed states [J. H. Eberly et al., J. Phys. B 13, 217 (1980)] are suppressed when pure-dressed-state preparation is considered.