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Physical Review A

Temporal response of an atom to a stochastic field: Resonant enhancement of population fluctuations at the Rabi frequency

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

Published Jul 1, 1997• 11 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.56.1007

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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.
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.
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.
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.
By use of a double-resonance technique, a spectrum of Rabi resonances has been observed. These resonances are enhancements in the dynamic response of a quantum system's population to rapid changes in the phase of a perturbing electromagnetic field. Theoretically, it is shown that the appearance of a spectrum, as opposed to a single Rabi resonance, is due to the fact that the phase of the field changes discretely by \ensuremath{\pi} radians, rather than sinusoidally. In the presence of \ensuremath{\pi}-radian phase changes, the quantum system's dynamic response is found to be similar to a damped harmonic oscillator driven by a \ensuremath{\delta}-function force term. In this harmonic-oscillator approximation the Rabi-resonance spectrum arises naturally, resulting from a resonant enhancement of the atomic coherence.
The nutational regime induced by a stochastic field with phase telegraph noise is experimentally investigated in a two-level system at microwave frequency. Regular oscillations are observed in the average transient response of the system with an effective Rabi frequency whose value depends both on the jump amplitude and on the mean dwell time of the phase noise. Motional-narrowing, weak-damping, and strong-damping regimes are observed on varying the dwell time and the jump amplitude. The experimental results are in fair agreement with theoretical predictions.