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Physical Review A

Delayed spectrum of two-level resonance fluorescence

X. Y. Huang, R. Tanaś, Joseph H. Eberly

Published Aug 1, 1982• 39 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.26.892

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Physical Review A•Sep 1, 1983•20 citations
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.
The time-dependent spectrum of the light scattered by a collisionally damped two-level atom weakly excited off resonance by a partially coherent laser field is evaluated. Both the collisional and the laser noise are treated as classical Markovian, Gaussian processes while, in addition, the atom is assumed to experience damping and fluctuations that are due to its coupling to an unspecified exterior reservoir. The scattered-field spectra are determined both at steady state and at times following the abrupt turn off of the laser field (the delayed spectrum). The distinct contributions to these spectra arising from the various noise sources as well as the finite bandwidth of the detecting interferometer are investigated in each case. Where appropriate, comparisons are made with related work of Saari [ Light Scattering in Solids ( Plenum, New York, 1979)]. A general inequality is also found that must be satisfied by the longitudinal and the transverse decay rates induced by the exterior reservoir. Finally, the inverse detuning of the laser is found to provide a limit on the time in which the undressing of the atom could be observed to occur following the turn off of the laser.
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.
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.
Attention is focused on the dynamic evolution of the emission spectrum following abrupt excitation of a near-resonant two-level absorber by a strong laser field. The multiphoton character of the absorption-emission process is evident in the AC Stark splitting of the spectrum, which begins approximately half an atomic lifetime after excitation, if the laser field is strong enough. Analytic expressions for the spectra are given of arbitrary laser field strength, atomic lifetime, laser-atom detuning, and the bandwidth of the spectrometer used to analyse the spectrum. Selected graphs are included to illustrate various time-dependent features of the spectrum. The influence on the transient spectrum of three different states of atomic excitation prior to laser irradiation and the effect on the spectrum of irradiation by a phase-noisy laser are also considered. Comparison of the transient spectra from atoms initially prepared in ' pi /2' and '3 pi /2' states show clearly that phase relationships between the irradiating field and the emitting dipole are important even in fluorescence.
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.
A method of calculation of multiple-time correlation functions of the resonance fluorescence light from a two-level atom driven by a realistic laser field with phase and amplitude fluctuations is developed. Both kinds of fluctuations are assumed to be Gaussian. The phase fluctuations are treated in the phase-diffusion model. The averaging over the amplitude fluctuations can be performed in a closed way when the relative mean-square amplitude fluctuation is small compared with the ratio of the amplitude correlation decay rate to the characteristic atomic relaxation rate. There is no restriction to Markovian amplitude fluctuations. Arbitrary values for the mean Rabi frequency are allowed. Results are presented for the intensity, the intensity correlation function, and the spectrum of the fluorescent light. It is shown that even in the case when the relative strength of the amplitude fluctuations is small, significant effects can occur due to the finite correlation length of the fluctuations. The results for fast amplitude correlation decay correspond to the situation without amplitude fluctuations. When the correlation time of the amplitude fluctuations becomes comparable with the atomic relaxation times, the Rabi oscillations of the intensity and the intensity correlation function are nearly washed out, and non-Lorentzian line shapes for the side peaks in the spectrum are observed. It is found that the ratio of the heights of the central peak to the side peak increases with increasing correlation time.