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Physical Review A

Observation of the Rabi-resonance spectrum

James C. Camparo, R. P. Frueholz

Published Dec 1, 1988• 20 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.38.6143

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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.
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 consider the interaction of a three-level system with phase-modulated resonant fields in the \ensuremath{\Lambda} excitation scheme. We treat theoretically the case of a sinusoidal phase modulation, a phase step perturbation, and a stochastic phase modulation. The appearance of a Rabi resonance both in the spectrum of the optical transmitted signal (electromagnetically induced transparency) and in the spectrum of the microwave emission (coherent population trapping maser) is considered in detail. All the theoretical results are compared with the analogous ones reported for the two-level system and with our experimental observations obtained for the case of rubidium in a buffer gas.
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.
The coupling between microwave fields and atoms (or atomlike systems) is inherently weaker than for optical fields, making microwave signal manipulation for applications like quantum information processing technically challenging. In order to better understand this coupling and to develop tools for measuring it, we explore the microwave coupling to atoms using the ``atomic candle'' technique, and push it beyond the bounds of the small-signal regime in order to deliver a larger signal. In familiar two-level systems, responses beyond the usual Rabi oscillations can arise when a single-tone drive is phase modulated, causing the steady-state populations to oscillate at integer harmonics of the modulation frequency. Resonant behavior of the first two harmonics for frequencies near the Rabi frequency, known as $\ensuremath{\alpha}$ and $\ensuremath{\beta}$ Rabi resonances, is widely used for microwave-field magnetometry and as a power standard known as the atomic candle. Here, we explore Rabi resonances beyond the small-signal approximation and report upon experimental observations of higher-order-harmonic population response for microwave hyperfine transitions in cold 87Rb^{87}\mathrm{Rb}87Rb atoms, which we compare to numerical simulations.
Quantum Rabi oscillation in cavity quantum electrodynamics is at the very heart of the interaction of an atomic system with a single electromagnetic field in a resonator. It has proved a powerful tool to entangle the atom with the field in the case of an empty cavity (vacuum Rabi oscillation). On the other side, as soon as the field is macroscopic, its state seems unaffected and one observes the well-known phenomenon of classical Rabi oscillation which only affects the atomic populations. We propose to experimentally investigate the frontier between those two behaviours, when one has to take into account the fluctations of the photon number. In order to do so, we let interact circular Rydberg atoms with a microwave resonator in which a mesocopic field is injected (a few tens of photons). We observe a strong correlation between specific atomic states and the phase of the injected field, which is measured by a homodyne technique. Those correlations explain in an easy way all the standard features of quantum Rabi oscillations (especially the collapses and revivals). Moreover they allow for the preparation of atom-field entangled state as well as non classical Schroedinger cat like states for the field.
We have studied the transient response of undamped two-level atoms to excitation by a classical bichromatic field consisting of a strong resonant component and a moderate-strength, tunable, ``perturber'' component. It is found that the strong field tends to stabilize the atom against the influence of the perturber field. Exceptions to this general result occur when the perturber is detuned from the strong field by the Rabi frequency of the latter or its subharmonics. We believe this stabilization phenomenon is a semiclassical analog of the effect that leads to the dynamical modification of spontaneous emission [M. Lewenstein and T. W. Mossberg, Phys. Rev. Lett. 59, 775 (1987)], and the presence of detunings at which the stabilization fails provides a simple motivation for the Rabi subharmonic resonances found in certain cw bichromatic spectra. Interestingly, when the perturber field is relatively weak, detuned from the atomic resonance by the strong-field Rabi frequency, and turned on with a specific initial phase relative to the strong field, one finds that the bichromatic field can be employed to dressed-state polarize initially ground-state atoms. In a related vein, we find that perturbations at the subharmonic detunings lead to interesting phase-dependent atomic dynamics. Results were obtained using both numerical and approximate analytical techniques entirely within the context of the rotating-wave approximation. The validity of the latter is ensured by demanding that all detunings and Rabi frequencies are vanishingly small compared to the atomic transition frequency.