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Journal of Physics B Atomic and Molecular Physics

Field correlation effects in laser-assisted electron scattering: the phase diffusion model

Fabio Trombetta, G. Ferrante, K. Wódkiewicz, P. Zoller

Published Jul 28, 1985• 19 citations
Publisher

Abstract

The authors present a theory of the potential scattering in the presence of a laser field taken with a well stabilised amplitude and a randomly fluctuating phase (phase diffusion model, PDM), and investigate the structure of the scattering lineshapes in the weak- and strong-field limits. The field, with its statistics, enters the theoretical treatment in a non-perturbative way. This gives the possibility of obtaining new information on the role of the statistical properties of the radiation fields in highly non-linear domains of the matter-field interaction, which complements in a significant way what is presently known. The PDM results are compared with those relative to a chaotic field (multimode laser field with a large number of uncorrelated modes). The broadening of the scattering lines is found to be controlled in an essential way by several parameters (field bandwidth, strength of the particle-field coupling, field intensity and so on); among them, the multiplicity of the exchanged photons becomes the most important only in the limit of weak coupling. In the strong-coupling limit, a chaotic field (i.e. a field undergoing amplitude and phase fluctuations) is found to broaden the scattering lines in a more effective way than a PDM field.

DOI
10.1088/0022-3700/18/14/016

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Journal of Physics B Atomic and Molecular Physics•Jan 14, 1980•69 citations
The authors present a theory of the potential scattering in the presence of a laser field taken with a well stabilised amplitude and a randomly fluctuating phase (phase diffusion model, PDM), and investigate the structure of the scattering lineshapes in the weak- and strong-field limits. The field, with its statistics, enters the theoretical treatment in a non-perturbative way. This gives the possibility of obtaining new information on the role of the statistical properties of the radiation fields in highly non-linear domains of the matter-field interaction, which complements in a significant way what is presently known. The PDM results are compared with those relative to a chaotic field (multimode laser field with a large number of uncorrelated modes). The broadening of the scattering lines is found to be controlled in an essential way by several parameters (field bandwidth, strength of the particle-field coupling, field intensity and so on); among them, the multiplicity of the exchanged photons becomes the most important only in the limit of weak coupling. In the strong-coupling limit, a chaotic field (i.e. a field undergoing amplitude and phase fluctuations) is found to broaden the scattering lines in a more effective way than a PDM field.
A quantum statistical theory of resonant optical phenomena is developed for the case when the fluctuations of the laser, used for exciting transitions, are important. The general equations describing the dynamics of a relaxing two-level atom (TLA) are solved exactly for different types of mean values and the two-time amplitude and intensity correlations. The phase-diffusion model is adopted for laser fluctuations. The exact results are used to analyze the effect of laser fluctuations on a number of optical effects-optical free-induction decay, adiabatic following, the spectrum of the scattered light from a relaxing TLA, the energy-absorption spectrum from a weak field, Hanle resonances, etc. In each case, the laser fluctuations are found to affect in an important way the characteristics of the above optical-resonance phenomena. For example, the spectrum of scattered radiation from a TLA has the usual three-peak structure for fields at resonance and with strengths $\ensuremath{\alpha}$ above the threshold and for $\ensuremath{\alpha}\ensuremath{\gg}{\ensuremath{\gamma}}_{c}$; however, now the peak heights (widths) are in the ratio 3x3x3x (23x\frac{2}{3x}3x2​), $x=\frac{[{\ensuremath{\gamma}}_{c}+\frac{({T}_{1}^{\ensuremath{-}1}+{T}_{2}^{\ensuremath{-}1})}{3}]}{[2{\ensuremath{\gamma}}_{c}+{T}_{2}^{\ensuremath{-}1}]}$, where ${\ensuremath{\gamma}}_{c}^{\ensuremath{-}1}$ is the correlation time for laser phase fluctuations. For ${\ensuremath{\gamma}}_{c}\ensuremath{\gg}\ensuremath{\alpha}$, one gets a single-peak spectrum. The laser field is treated as a second-quantized field with excitation in either a coherent state or a Fock state. The results, obtained by straightforward perturbation theory, but valid for arbitrary values of the relaxation parameters, detuning, and the laser correlation time, are also presented.
Calculations on the Hanle effect in a J - J:i atomic transition are performed for laser fields with a) Brownian motion phase diffusion b) Gaussian amplitude fluctuations and c) fluctuating real and imaginary parts of the laser electric field amplitude (chaotic field model). We use the Fokker-Planck operator approach to stochastic differential equations. The zero-field level-crossing dip broadens at high laser intensity significantly less so as the laser bandwidth increases or if the amp]itu(le fluctuates.
We report results of calculations on the Hanle effect in a J=0\ensuremath{\leftrightarrows}J=1 atomic transition with three types of model fluctuating light fields: (a) the Brownian-motion phase-diffusion field, as produced in recent experiments by Arnett et al. [Phys. Rev. A 41, 2580 (1990)]; (b) Gaussian amplitude fluctuations; and (c) the chaotic field model, in which real and imaginary parts of the electric-field amplitude fluctuate. For the stochastic density-matrix equations, we use methods developed by Zoller and co-workers [e.g., Dixit, Zoller, and Lambropoulos, Phys. Rev. A 21, 1289 (1980)] employing the Fokker-Planck operator and leading to matrix continued-fraction expansions. The Hanle effect is of interest as a prototype for multisublevel atomic transitions. The width of the Hanle dip at zero magnetic field reflects the tendency of the light field to preserve the coherence between excited-state sublevels. For monochromatic light, the Hanle dip width increases as the square root of light intensity. When the laser bandwidth increases, power broadening of the coherence dip normally decreases. However, with the Brownian-motion phase-diffusion model, if the laser spectral profile is nearly Gaussian, broadening the laser up to several times the natural width of the atomic line does not diminish the Hanle dip width. With amplitude fluctuations, even in the limit of monochromatic light, power broadening of the Hanle dip with intensity is reduced by one-third to one-half depending on the particular model.
The authors investigate collisional redistribution in a laser field with arbitrary lineshape. The phase diffusion model cannot be correct in general due to its inherent assumption of zero correlation times. The authors give general expressions for the state of the atom, the dipole correlation function and the steady-state fluorescence spectrum in the presence of collisions and phase fluctuations with finite correlation times. It turns out that collisional effects decouple from the free-atom stochastic problem to a good approximation and that modifications due to the laser bandwidth can be quite different from collisional effects, although they are additive on a microscopic level. They evaluate the fluorescence spectrum explicitly in the limit where the three lines are well separated and show that the spectrum is determined by simple field correlation functions. The arbitrary phase fluctuations do not affect the line strengths but only the lineshapes. Collisions, on the other hand, can transfer intensity between the lines, due to the very small collision time. For free atoms and strong driving fields, the fluorescence spectrum becomes a convolution of the normalised laser spectrum with the fluorescence spectrum at monochromatic irradiation, but this does not hold any more when collisions are present.
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.
Presents a theoretical analysis of the effect of laser light statistics on two-photon resonant three-photon ionisation of sodium. The ionisation probability and the population of the atom in the resonant state are compared in the presence of an intensity stabilised laser with a diffusing phase (phase diffusion model) and light undergoing Gaussian amplitude fluctuations (chaotic field). In the limit of large bandwidth fields the resonant two-photon excitation is statistically decoupled from the following ionisation step. A decrease of the bandwidth results in a partial correlation between these two steps manifesting itself in an enhancement of the ionisation probability in the chaotic field. With increasing intensity and the onset of saturation of the bound-bound transition, the ionisation probability and resonant excited-state population in the chaotic field may become smaller than the one in the phase diffusing light. In the large-bandwidth limit the Stark shift in the chaotic field is enhanced, which has a pronounced effect on the effective power-law dependence of the ionisation probability.