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Physical Review A

Observation of the stochastic realization shift in the weak-field limit

Martin P. van Exter, D. Boersma, A. K. Jansen van Doorn, J. P. Woerdman

Published Apr 1, 1994• 3 citations
Publisher

Abstract

The amplitude-phase coupling of a stochastically varying field has been predicted to lead to a shift of the effective resonance frequency when this field interacts with a two-level system. Experimentally, we have generated such a field by passing a semiconductor laser beam through a Michelson interferometer, thereby converting part of the intrinsic phase flucutations into amplitude fluctuations. We find indeed a shifted resonance frequency when this beam interacts with the Rb ${\mathit{D}}_{2}$ transition. Depending on the path-length difference of the interferometer, shifts of more than 10 MHz have been measured. The experiments have been performed in the weak-field limit. This allows a convolution-type interpretation of the stochastic realization shift.

DOI
10.1103/physreva.49.2861

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Physical Review A•Jul 1, 1989•18 citations
The amplitude-phase coupling of a stochastically varying field has been predicted to lead to a shift of the effective resonance frequency when this field interacts with a two-level system. Experimentally, we have generated such a field by passing a semiconductor laser beam through a Michelson interferometer, thereby converting part of the intrinsic phase flucutations into amplitude fluctuations. We find indeed a shifted resonance frequency when this beam interacts with the Rb D2{\mathit{D}}_{2}D2​ transition. Depending on the path-length difference of the interferometer, shifts of more than 10 MHz have been measured. The experiments have been performed in the weak-field limit. This allows a convolution-type interpretation of the stochastic realization shift.
Several years ago, a resonance-shift phenomenon was predicted to exist, termed the stochastic realization shift (SRS), which arises as a consequence of correlated amplitude and frequency variations in a resonant field. An interesting aspect of the phenomenon is that the sign of the SRS in weak and strong fields has been shown to differ. Here, we demonstrate that the SRS may be conveniently studied over a broad range of field strengths via magnetic resonance in an optically pumped alkali-metal vapor. Specifically, we study the SRS in the ground state 0-0 hyperfine transition of Rb87{\mathrm{Rb}}^{87}Rb87 for a field undergoing stochastic (yet correlated) amplitude-frequency fluctuations. Under our experimental conditions we find no evidence for a change in sign of the SRS as the field strength undergoes a transition from weak to strong. Further, our results show that in strong fields the magnitude of the SRS may be influenced by the field's spatial variations. \textcopyright{} 1996 The American Physical Society.
We report an experiment on the observation of Hanle resonances in sinusoidally amplitude modulated, weak, phase diffusing optical fields. The J = 0 to J = 1 transition in an atomic beam of barium was probed with a frequency stabilized dye laser, whose bandwidth and bandshape were precisely controlled with an extra-cavity noise modulation system. Phase sensitive detection of the Hanle fluorescence signal at the amplitude modulation frequency revealed resonances in the in-phase and in-quadrature parts, whose position, shape, and strength were sensitive to the laser bandwidth. These resonances, which were Lorentzian or dispersion shaped or a combination of the two, occurred when the Zeeman splitting of the exited sublevels equaled the modulation frequency for monochromatic light and at splitting of half the modulation frequency for broadband light. The results are in good agreement with the theoretical predictions of Saxena and Agarwal.1
We have investigated the ac Stark shift of a two-photon transition induced by a model stochastic field by using a Monte Carlo technique. The model field has a stochastic single mode, which can possess both colored amplitude and colored frequency fluctuations. The stochastic field’s modification of the ac Stark shift is quantified by a parameter M, which is the ratio of the Stark shift in the model stochastic field to the Stark shift that would be obtained in a phase-diffusion field of equivalent linewidth (i.e., a field with frequency fluctuations but no amplitude fluctuations). In the regime of weak fields, below saturation of the bound-bound transition, we find that M is greater than unity: the Stark shift is enhanced by the stochastic field. Moreover, in this regime M is an increasing function of the field’s degree of photon bunching. In strong fields, where the bound–bound transition is nearly saturated, the enhancement of the Stark shift is diminished, so much so that in extremely strong fields M is actually less than unity. Our calculations indicate that the ac Stark shift’s modification occurs through two distinct processes, depending on the strength of the field. In weak fields enhancement of a multiphoton transition’s Stark shift is influenced primarily by the intrinsic correlation between ac-Stark-shift fluctuations and Rabi-frequency fluctuations. In strong fields the fluctuating Stark shifts give rise to an asymmetric resonance line shape in a fashion analogous to inhomogeneous broadening. The line shape’s peak position then has a sublinear dependence on the stochastic field’s intensity, and this yields a diminished value of M.
Summary form only given. We report on the investigation of the parametric phase shift of a fixed coupling laser field in a lambda type three level system. The parameter under variation is the frequency of the probing field acting on the second transition. In the vicinity of the two photon resonance in this coherent population trapping system the phase of both fixed coupling field and swept probe field undergo a very rapid change. Outside this relatively narrow structure the phase of the probe field follows the usual one photon dispersion line. A semi-classical density matrix model yields analytically expressions for the parametric dispersion and absorption. The system we used was a three level system in the D2 line of cesium obtained from a cesium atomic beam. The phase shift of the coupling beam was measured with a three beam heterodyne interferometer relative to an off-resonant reference laser field. This setup is less sensitive to acoustic and vibrational noise than a homodyne interferometer and provides a reasonable signal to noise ratio even at coupling powers of a few nW. The power of both coupling and probe field was varied over several orders of magnitude and the obtained phase shifts are in good agreement with the semi-classical model.
Recently, Narducci et al.1 studied V-configuration three-level atoms and predicted that quantum fluctuations associated with a "strong" optical transition driven by a coherent field can be stabilized through excitation of a coupled "weak" transition. This stabilization is manifest in a narrowing, to subnatural values, of the fluorescence linewidths associated with the strong transition as the weak transition is driven more and more strongly. We have experimentally investigated this problem and provide the first demonstration of the effect. In the experiment, a beam of Ba atoms was intercepted by two copropagating cw laser beams. One of the beams, of Rabi frequency Ω1, was tuned into resonance with the strong, 553.5 nm, 1S0→3P1 transition, while the other beam, of Rabi frequency Ω2, was tuned into resonance with the weak, 791.1 nm, 1S0→3P1 transition. The fluorescence spectrum of the strong transition was recorded for the case when Ω1≈35 MHz and Q2≈45 MHz and, it was found that the spectrum was three-peaked. Most interestingly, the measured linewidth (FWHM) for the central peak was 16 MHz, or 2 MHz less than the 18 MHz natural width of the strong transition even before the substantial instrumental broadening (≈8 MHz) of the peak was deconvolved.
A phase-diffusion stochastic field with frequency fluctuations obeying Gaussian Markovian statistics is produced at microwave frequency and used to drive a two-level spin system. The system is tuned to the two-photon (TP) resonance and its TP-induced second-order response is measured as a function of the characteristic parameters of the frequency noise. It is found that, when the statistics of the driving field are varied from the fast- to the slow-modulation limit, the power spectrum of the nonlinear response of the system varies in a continuous way from a Lorentzian form toward a Gaussian one, and its relative width (normalized to the input spectral width) varies from 4.0\ifmmode\pm\else\textpm\fi{}0.4 to 2.0\ifmmode\pm\else\textpm\fi{}0.2. The experimental results are interpreted on the basis of a semiclassical calculation which takes into account the stochastic nature of the field. The effect of the filter action of the cavity modes is discussed. The relationship with previous experiments and theoretical calculations of TP absorption is also examined.