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Journal of the Optical Society of America B

Atomic absorption in cross-correlated time-delayed stochastic laser fields

Klaus M. Gheri, Monika A. M. Marte, P. Zoller

Published Aug 1, 1991• 6 citations
Publisher

Abstract

A theory of atomic transitions in correlated stochastic laser fields is developed for the case in which one pulse is assumed to be arbitrarily strong (beyond the fast fluctuation expansion), while the second, time-delayed, pulse is treated in perturbation theory. The statistics of the driving laser field is assumed to be that of a Markov process. We illustrate the technique by calculating the absorption spectrum of a two-level atom in a pump field and cross-correlated probe laser where the field statistics is described by the phase diffusion model.

DOI
10.1364/josab.8.001559

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A theory of atomic transitions in correlated stochastic laser fields is developed for the case in which one pulse is assumed to be arbitrarily strong (beyond the fast fluctuation expansion), while the second, time-delayed, pulse is treated in perturbation theory. The statistics of the driving laser field is assumed to be that of a Markov process. We illustrate the technique by calculating the absorption spectrum of a two-level atom in a pump field and cross-correlated probe laser where the field statistics is described by the phase diffusion model.
We have measured the absorption of a weak, stochastic field in the presence of a strong, degenerate stochastic pump that saturates the 3S1/2(F = 2, m f =2) → 3P3/2(F = 3, m f = 3) transition in atomic sodium. The probe is a time delayed replica of the pump field, and the net field obeys non-Markovian phase diffusing statistics. The absorption (gain) of the probe field was measured as a function of the time delay. We use cw lasers and impose phase diffusing statistics (δ-correlated frequency fluctuations) with a system of electro-optics. The time delay is provided by lengths of optical fiber. Our results agree well with a recent theory for atomic response to non-Markovian fields, valid for arbitrary strengths and bandwidths of the pump field and a weak time delayed probe. For both pump and probe fields resonant with the transition, we observe maximum gain for a probe time delay of approximately 0.8 lifetimes of the upper state (γ−1).
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.
The dependence of the electronic absorption spectrum of a dye molecule on the time of delay between the excitation of the molecule by a pump-laser pulse and the probing of the absorption by a test-laser pulse is studied. The effect of dephasing of the molecular transition arising from the interaction with dissipative systems is described by a stochastic modulation of the electronic transition frequency. It is shown that in the case when the two laser pulses are short compared with the correlation time of the frequency modulation and when the spectral linewidth of the pump-laser light is smaller than the bandwidth of the equilibrium absorption spectrum, hole burning appears although the molecules have equal mean transition frequencies. This effect arises from non-Markovian correlations between the dephasings of the molecule during the actions of the pump- and the test-laser pulse. With increasing delay time the holes are smoothed out and disappear when the delay time exceeds the correlation time. Such effects were found in recent femtosecond absorption experiments by Shank et al. [in Ultrafast Phenomena V, dited by G. R. Fleming and A. E. Siegman (Springer, Berlin, 1986), p. 179].
The authors apply the recently developed theory of multiplicative stochastic correlation functions with Markovian random jumps to the problem of resonance fluorescence from a two-level atom in a strong laser field. The finite bandwidth of a single-mode laser or the appearance of distinct lines in a multimode laser is considered to be brought about by a random mode-switching process. The two-time dipole correlation functions, which determine the spectral distribution of the fluorescence and the temporal correlations between the emitted photons are averaged exactly over the stochastics of the driving field. The authors' explicit expressions only involve single-time averages, which can be easily evaluated for a given probability distribution. These results generalise and modify earlier results. The authors obtain closed expressions, rather than implicit differential equations, which can handle the continuous case as well, and take into account the initial correlations, which always arise when the laser coherence time is finite. They illustrate the dependence on the stochastics of the laser model by comparing the multimode description with the phase-diffusion process and the Lorentz wave, which all give rise to the same laser profile, but to a different fluorescence spectrum.
We have measured the absorption from a weak time-delayed probe field by a two-level atomic system [Na 3s1/2{\mathit{s}}_{1/2}s1/2​(F=2,mF{\mathit{m}}_{\mathit{F}}mF​=2)\ensuremath{\rightarrow}3p3/2{\mathit{p}}_{3/2}p3/2​(F=3,mF{\mathit{m}}_{\mathit{F}}mF​=3)] saturated by a phase-diffusing pump field. The pump and probe, derived from the same artificially-noise-modulated phase-diffusing laser beam, are frequency degenerate and resonant with the atomic transition. The probe is a time-delayed replica of the pump. Our results, carried out in a regime where the field bandwidth and the Rabi frequency are comparable, provide confirmation of recent theoretical predictions [K. Gheri, M. A. M. Marte, and P. Zoller, J. Opt. Soc. Am. B 5, 1559 (1991)] for the atomic response to the non-Markovian composite of a pump and time-delayed probe, valid for arbitrary bandwidths and intensities. Amplification of the probe field is observed for delays comparable to the lifetime of the upper level and again for very long delays, if the bandwidth is less than the decay rate of the upper state.
The dependence of the multiphoton absorption probability for an atomic system on the statistical properties of the incident light is investigated. It is shown, by means of nthn\mathrm{th}nth-order time-dependent perturbation theory, that the multiphoton absorption probability depends on the (n,n)th(n, n)\mathrm{th}(n,n)th normally ordered correlation functions of the electromagnetic field. For stationary fields, the transition probability per unit time is constant and is directly related to the reduced cross-spectral correlation function of the electromagnetic field. This expression for the transition probability is then analyzed under various conditions of coherence. Explicit calculations are carried out for thermal light and for laser light, and the transition rates are compared when each has a Lorentzian spectrum. In particular, when the width of the final atomic level is much larger than the bandwidth of the field, the transition rate for thermal light is nnn! times higher than with laser light of the same mean intensity. The other extreme case, i.e., when the width of the final atomic level is much smaller than the bandwidth of the field, is also considered in detail. An equation for the density operator of the field is derived; this equation governs the change in the statistical properties of the photon system in nnn-photon absorption processes. On the basis of this equation, the time-dependent properties of the photon system are then studied and some physical consequences of these equations are discussed. Some numerical solutions relating to the time dependence of the factorial moments in two-photon absorption processes are presented.