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Physical Review A

Rabi resonances induced by an off-resonant, stochastic field

James C. Camparo, John G. Coffer, R. P. Frueholz

Published Nov 1, 1998• 18 citations
Publisher

Abstract

When an atom interacts with a phase-fluctuating field of fairly arbitrary spectral character, the Fourier spectrum of atomic population variations manifests a ``bright line'' at the atomic system's Rabi frequency. This bright line is termed a Rabi resonance. Here, we generalize our previous studies of this phenomenon by considering the characteristics of the Rabi resonance when it is excited by an off-resonant stochastic field. We find both experimentally and theoretically that (i) the Rabi resonance occurs at the Rabi nutational frequency, $\ensuremath{\Omega}=\sqrt{{\ensuremath{\Delta}}^{2}+{\ensuremath{\omega}}_{1}^{2}},$ where \ensuremath{\Delta} is the detuning and ${\ensuremath{\omega}}_{1}$ is the Rabi frequency, (ii) that the strength of the Rabi resonance is maximized when |\ensuremath{\Delta}| equals ${\ensuremath{\omega}}_{1},$ and (iii) that the strength of the Rabi resonance is an asymmetric function of detuning.

DOI
10.1103/physreva.58.3873

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Underlying simplicity of atomic population variations induced by a stochastic phase-fluctuating field

R. P. Frueholz, James C. Camparo

We have examined the temporal evolution of a two-level atom in response to stochastic phase-fluctuating fields by applying perturbation theory to the optical Bloch equations written in the instantaneous frame. Our results show that the general nature of the atomic population variations is relatively simple, being composed of adiabatic and nonadiabatic components. The adiabatic response results from Fourier components of the phase variation below the system's Rabi frequency and is proportional to the product of the relatively slow phase variation and its first derivative. The nonadiabatic response has its maximum amplitude at the system's Rabi frequency, for which the atom behaves like a simple harmonic oscillator whose resonance frequency is the Rabi frequency. In this sense the atom acts as a tunable narrow-band filter. \textcopyright{} 1996 The American Physical Society.

Physical Review A•Oct 1, 1996•14 citations
When an atom interacts with a phase-fluctuating field of fairly arbitrary spectral character, the Fourier spectrum of atomic population variations manifests a ``bright line'' at the atomic system's Rabi frequency. This bright line is termed a Rabi resonance. Here, we generalize our previous studies of this phenomenon by considering the characteristics of the Rabi resonance when it is excited by an off-resonant stochastic field. We find both experimentally and theoretically that (i) the Rabi resonance occurs at the Rabi nutational frequency, \ensuremathΩ=\ensuremathΔ2+\ensuremathω12,\ensuremath{\Omega}=\sqrt{{\ensuremath{\Delta}}^{2}+{\ensuremath{\omega}}_{1}^{2}},\ensuremathΩ=\ensuremathΔ2+\ensuremathω12​​, where \ensuremath{\Delta} is the detuning and \ensuremathω1{\ensuremath{\omega}}_{1}\ensuremathω1​ is the Rabi frequency, (ii) that the strength of the Rabi resonance is maximized when |\ensuremath{\Delta}| equals \ensuremathω1,{\ensuremath{\omega}}_{1},\ensuremathω1​, and (iii) that the strength of the Rabi resonance is an asymmetric function of detuning.
Recent theoretical investigations have shown that when atoms resonantly interact with strong, phase-fluctuating fields of fairly arbitrary character there is an underlying simplicity in their temporal response to the field. Specifically, theory predicts that the atom's response can be divided into two distinct temporal components: an adiabatic component manifesting itself in the instantaneous frame of field-atom interactions as a Bloch-vector trajectory figure-eight pattern, and a nonadiabatic component manifesting itself as population variations oscillating at the Rabi frequency. Here, we experimentally verify theoretical expectations regarding the nonadiabatic component of the field's temporal response. In the presence of a resonant field suffering broadband phase noise we show that atomic population fluctuations display a resonant enhancement at Fourier frequencies corresponding to the Rabi frequency, and that the characteristics of this resonance are similar to those of a damped, driven harmonic oscillator.
When atoms interact with a phase-modulated field, the probability of finding the atom in the excited-state oscillates at the second harmonic of the modulation frequency, $2{\ensuremath{\omega}}_{m}.$ The amplitude of this oscillating probability is a resonant function of the Rabi frequency \ensuremath{\Omega}, and this is termed a \ensuremath{\beta} Rabi resonance. In this work, we examine the line shape of the \ensuremath{\beta} Rabi resonance both theoretically and experimentally. We find that a small-signal theory of the \ensuremath{\beta}-Rabi-resonance condition captures much of the line shape's character, and, in particular, that the resonance's ``line Q'' (i.e., $2\ensuremath{\delta}{\ensuremath{\Omega}}_{1/2}/\ensuremath{\Omega})$ is proportional to the modulation frequency. This result can be applied to the atomic candle, where \ensuremath{\beta} Rabi resonances are employed to stabilize field strength. Considering our results in the context of developing an optical atomic candle, we find that a free-running diode laser's intensity noise could be improved by orders of magnitude using the atomic candle concept.
A systematic method is presented for averaging the integral equations of motion for the atomic density matrix operator and its quantum correlation functions over the fluctuations of Markovian exciting fields with arbitrary bandwidth. The theory takes into account the statistics of the field to all orders. The method is applied to the investigation of resonance fluorescence in the presence of three different Markovian fields: (i) a phase-diffusion field, (ii) a chaotic field, and (iii) a Gaussian-amplitude field. It is shown that in the case of a resonant, intense phase-diffusion field a center line dip develops in the spectrum of resonance fluorescence when the Rabi frequency is approximately equal to the bandwidth of the field. In the case of amplitude fluctuations the sidebands of the on-resonance spectrum tend to reproduce the probability distribution for the amplitude of the exciting field.
The steady-state populations, the fluorescent intensity-intensity correlation function, and the resonance fluorescence and absorption spectra of a two-level atom bichromatically driven by a strong, coherent field and a weak, stochastic field with tunable carrier frequency and wide bandwidth are investigated analytically. Surprisingly, the population is inverted if the stochastic field is appropriately detuned from the coherent field, which is assumed resonant with the atomic transition. The resonance fluorescence spectrum still displays a three-peaked structure, but with linewidths and heights that depend upon the frequency and bandwidth of the stochastic field as well as the intensities of both fields. Asymmetric fluorescence spectra are found when the coherent field is detuned from the atomic resonance frequency. The intensity-intensity correlation function varies dramatically with the carrier frequency of the stochastic field. For appropriate choices of the parameters, the fluorescent emission of the atom is antibunched for all time. The absorption spectral features also deviate significantly from those of the monochromatically driven atom. By appropriately tuning the stochastic field, one finds that a sharp peak (absorption) or hole (gain) can occur at line center and the probe beam can be amplified at either the lower-frequency Rabi sideband or the higher-frequency sideband and the inversionless gain can be enhanced.
We examine the radiative properties of a two-level atom driven by a strong bichromatic field with frequencies ${\mathrm{\ensuremath{\omega}}}_{1}$=${\mathrm{\ensuremath{\omega}}}_{0}$-${\mathrm{\ensuremath{\delta}}}_{1}$ and ${\mathrm{\ensuremath{\omega}}}_{2}$=${\mathrm{\ensuremath{\omega}}}_{0}$+${\mathrm{\ensuremath{\delta}}}_{2}$, which can be asymmetrically placed about the atomic transition frequency ${\mathrm{\ensuremath{\omega}}}_{0}$ and can have different Rabi frequencies ${\mathrm{\ensuremath{\Omega}}}_{1}$ and ${\mathrm{\ensuremath{\Omega}}}_{2}$. Applying the optical Bloch equations for the bichromatic excitation, we derive an infinite set of equations of motion for the time evolution of the atomic variables. The equations are solved numerically by matrix inversion and Laplace transforms. Using the quantum regression theorem, we then solve for the steady-state total fluorescence intensity, and the resonance-fluorescence and absorption spectra. The spectra are found to depend on the frequency difference 2\ensuremath{\delta}=${\mathrm{\ensuremath{\delta}}}_{1}$+${\mathrm{\ensuremath{\delta}}}_{2}$, the average detuning \ensuremath{\Delta}=${\mathrm{\ensuremath{\omega}}}_{0}$-1/2(${\mathrm{\ensuremath{\omega}}}_{1}$+${\mathrm{\ensuremath{\omega}}}_{2}$), and the Rabi frequencies of the driving fields. For ${\mathrm{\ensuremath{\Omega}}}_{1}$=${\mathrm{\ensuremath{\Omega}}}_{2}$=\ensuremath{\Omega}, the total fluorescent intensity displays a series of maxima for \ensuremath{\Delta}=\ifmmode\pm\else\textpm\fi{}n\ensuremath{\delta}\ensuremath{\mp}${\mathrm{\ensuremath{\Omega}}}^{2}$/4\ensuremath{\delta}, where the n are the odd integers. The intensity-dependent shift from the resonances n\ensuremath{\delta} is explained as an analog of the generalized Bloch-Siegert shift. The resonance-fluorescence spectrum for \ensuremath{\Delta}\ensuremath{\ne}0 appears to contain more peaks than that for \ensuremath{\Delta}=0.This is due to the splitting of the central peak and the even sidebands into doublets. The absorption spectrum of a weak-probe beam for \ensuremath{\Delta}=0 and equal Rabi frequencies consists of a symmetric series of dispersionlike sidebands separated from ${\mathrm{\ensuremath{\omega}}}_{0}$ by integer multiples of \ensuremath{\delta}, together with a central absorption peak at ${\mathrm{\ensuremath{\omega}}}_{0}$, whose amplitude oscillates with \ensuremath{\Omega}. For \ensuremath{\Delta}\ensuremath{\ne}0 and/or ${\mathrm{\ensuremath{\Omega}}}_{1}$\ensuremath{\ne}${\mathrm{\ensuremath{\Omega}}}_{2}$, the odd sidebands remain dispersionlike, while the central peak and the even sidebands split into absorption-emission doublets. A simple physical interpretation of the spectral features is given in terms of the dressed-atom model.
We have examined the temporal evolution of a two-level atom in response to stochastic phase-fluctuating fields by applying perturbation theory to the optical Bloch equations written in the instantaneous frame. Our results show that the general nature of the atomic population variations is relatively simple, being composed of adiabatic and nonadiabatic components. The adiabatic response results from Fourier components of the phase variation below the system's Rabi frequency and is proportional to the product of the relatively slow phase variation and its first derivative. The nonadiabatic response has its maximum amplitude at the system's Rabi frequency, for which the atom behaves like a simple harmonic oscillator whose resonance frequency is the Rabi frequency. In this sense the atom acts as a tunable narrow-band filter. \textcopyright{} 1996 The American Physical Society.