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Physical Review A

General atomic response to resonant, phase-fluctuating fields in the adiabatic limit

R. P. Frueholz, James C. Camparo

Published Jul 1, 1995• 3 citations
Publisher

Abstract

Though the interaction of a two-level atom with a field undergoing some arbitrary pattern of phase variation may be obtained by a numerical solution of the density-matrix equations, theory has proven to be problematic in providing a general, intuitive picture of the atom's response to a phase-varying field. The convolution picture of the fluctuating field-atom interaction, known to spectroscopists for decades, is only valid in the regime of weak fields. Here we discuss a complementary intuitive view that is valid in the regime of strong, resonant fields. Specifically, when an atom's response to a phase-varying field is considered in the instantaneous frame, the Bloch-vector trajectory always traces out a ``figure-eight''--type pattern, reminiscent of a Lissajous figure. This observation is demonstrated for a number of fields whose phase-varying patterns span a broad range (i.e., sinusoidal, two-frequency quasiperiodic, chaotic, stochastic), and analysis of the Bloch-vector equations then reveals the complete generality of this response. The analysis also reveals that the temporal variation of the atomic population to phase fluctuations in the strong, resonant field regime is nearly proportional to a simple product of the field's phase and its first derivative.

DOI
10.1103/physreva.52.472

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Though the interaction of a two-level atom with a field undergoing some arbitrary pattern of phase variation may be obtained by a numerical solution of the density-matrix equations, theory has proven to be problematic in providing a general, intuitive picture of the atom's response to a phase-varying field. The convolution picture of the fluctuating field-atom interaction, known to spectroscopists for decades, is only valid in the regime of weak fields. Here we discuss a complementary intuitive view that is valid in the regime of strong, resonant fields. Specifically, when an atom's response to a phase-varying field is considered in the instantaneous frame, the Bloch-vector trajectory always traces out a ``figure-eight''--type pattern, reminiscent of a Lissajous figure. This observation is demonstrated for a number of fields whose phase-varying patterns span a broad range (i.e., sinusoidal, two-frequency quasiperiodic, chaotic, stochastic), and analysis of the Bloch-vector equations then reveals the complete generality of this response. The analysis also reveals that the temporal variation of the atomic population to phase fluctuations in the strong, resonant field regime is nearly proportional to a simple product of the field's phase and its first derivative.
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.
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.
We have examined the population oscillation amplitudes, associated attractors, and the order-1 Renyi entropy of Bloch-vector trajectories for a two-level atom interacting with an electromagnetic field, whose phase varies in a two-frequency quasiperiodic manner. Distinct differences in these quantities are observed as the strength of the exciting field varies from weak (nonadiabatic dynamics) to strong (adiabatic dynamics). We find, though, an underlying homology. In all cases the response is two-frequency quasiperiodic with a resulting two-dimensional attractor for the quantum system. Additionally, correlated with observed Rabi resonances, there are abrupt increases in the entropy of the system. A key finding is that Rabi resonances are more than simple increases in the amplitude of a quantum system's population oscillations; the atomic dynamics abruptly becomes more disordered at these resonances.
The effect of field fluctuations on the motion of a two-level atom in resonant laser radiation is investigated theoretically. The internal motion of the atom is treated quantum mechanically (optical Bloch equations); the translational motion, which is coupled to the internal motion, is treated classically (Ehrenfest's theorem); and the fluctuating radiation is treated as a classical electromagnetic wave with stochastic phase (phase-diffusion model). An expression for the radiation force and equations of motion for the ensemble-averaged Bloch vector are derived and used to calculate (i) the radiation force in a plane running wave, (ii) the steady-state radiation force in a general standing wave, and (iii) the dissipative force in a plane standing wave in the limit of strong phase fluctuations. In case (iii) it is found that the atomic velocity is damped when the radiation is tuned above resonance, which is opposite to the case of a weak coherent standing wave, in which damping occurs for radiation tuned below resonance. It is shown that fluctuations have a substantial effect on the radiation force only when the spectral width of the radiation approaches or exceeds the spontaneous emission rate.
In dense media, the Lorentz local-field condition results in a generalization of the usual semiclassical Maxwell-Bloch formulation for two-level atoms in which the optical Bloch equations are recast in terms of macroscopic spatially averaged atomic variables. We derive an adiabaticlike, or stationary, approximation for the generalized Bloch equations in the ultratransient limit and establish validity criteria. The significance of this quasiadiabatic approximation is that a point response solution of the generalized Bloch equations is obtained in terms of the field strength, thereby allowing the atomic response to be expressed as a nonlinear susceptibility that can be used to predict and analyze propagation effects in dense media. The quasiadiabatic approximation is quite general, allowing for a combination of detuning, chirping, and a variety of local-field and mean-field effects. It spans the range from cases for which local-field effects can be neglected, as in the usual adiabatic approximation, to cases in which local-field effects cause a large inversion-dependent frequency renormalization. The approximation is used to interpret previous numerical results for the resonant interaction of ultrashort pulses with dense media and it is shown that dense media can exhibit behavior analogous to adiabatic following and adiabatic inversion. The nonlinear index of refraction in the quasiadiabatic limit is purely real for large detunings. However, at and near resonance, the nonlinear index is purely imaginary. The imaginary index relates to coherent reflectivity arising from the reaction field of the cooperating atoms, rather than absorption, and transmission and reflection coefficients for thin films are derived. In the intermediate range of detunings, the intensity-dependent index of refraction can undergo a phase transition due solely to the variation of the field intensity during excitation by an ultrashort pulse. Propagation effects are investigated using a finite-difference time-domain method to integrate the generalized Bloch-Maxwell equations and the results of the numerical simulations are analyzed in the context of the nonlinear index of refraction obtained using the quasiadiabatic approximation.
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.