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Physical Review A

Frequency-modulated excitation of a two-level atom

Michael W. Noel, W. M. Griffith, Thomas F. Gallagher

Published Sep 1, 1998• 62 citations
Publisher

Abstract

We present a detailed experimental study of the frequency-modulated excitation of a two-level atom, using a microwave field to drive transitions between two Rydberg Stark states of potassium. In the absence of a modulation the interaction is the standard model of the Rabi problem, producing sinusoidal oscillations of the population between the two states. In the presence of a frequency modulation of the interacting field, however, the time evolution of the system is significantly modified, producing square wave oscillations of the population, sinusoidal oscillations at a different frequency, or even sinusoidal oscillations built up in a series of stair steps. The three responses described above are each found in a different regime for the frequency of the modulation with respect to the unmodulated Rabi frequency: the low-, high-, and intermediate-frequency regimes, respectively.

DOI
10.1103/physreva.58.2265

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We present a detailed experimental study of the frequency-modulated excitation of a two-level atom, using a microwave field to drive transitions between two Rydberg Stark states of potassium. In the absence of a modulation the interaction is the standard model of the Rabi problem, producing sinusoidal oscillations of the population between the two states. In the presence of a frequency modulation of the interacting field, however, the time evolution of the system is significantly modified, producing square wave oscillations of the population, sinusoidal oscillations at a different frequency, or even sinusoidal oscillations built up in a series of stair steps. The three responses described above are each found in a different regime for the frequency of the modulation with respect to the unmodulated Rabi frequency: the low-, high-, and intermediate-frequency regimes, respectively.
By using the B -spline expansion technique and a model potential of the alkali atoms, the properties of frequency-modulated excitation of Rydberg potassium atoms in a static electric field and a microwave field are investigated by using the time-dependent two-level approach. We successfully reproduce the square wave oscillations in the low frequency, the stair step population oscillations in the intermediate frequency, and the multiphoton transitions in the high frequency with respect to the unmodulated Rabi frequency, which have been observed experimentally by Noel et al. [ Phys. Rev. A 58 2265 (1998) ]. Furthermore, we also numerically obtain the discretized Rabi oscillations predicted in the Landau—Zener accumulation model.
By using the B-spline expansion technique and a model potential of the alkali atoms, the properties of frequency- modulated excitation of Rydberg potassium atoms in a static electric field and a microwave field are investigated by using the time-dependent two-level approach. We successfully reproduce the square wave oscillations in the low frequency, the stair step population oscillations in the intermediate frequency, and the multiphoton transitions in the high frequency with respect to the unmodulated Rabi frequency, which have been observed experimentally by Noel et al. [Phys. Rev. A 58 2265 (1998)]. Furthermore, we also numerically obtain the discretized Rabi oscillations predicted in the Landau-Zener accumulation model.
A time-dependent perturbation method is proposed to study the properties of the frequency-modulated excitation of a potassium atom. Rabi oscillations in the absence of a modulation, square-wave oscillations in the presence of slow modulation, and radio-frequency multiphoton resonances in the presence of fast modulation have been calculated using this method. The numerical results are in excellent agreement with those of the experiment; novel explanations have been given to understand some of the experimental results.
We have studied the effect of phase shifted frequency modulation on two level atom with field interaction using Jaynes—Cummings model. Here the frequency of the interacting field is sinusoidally varying with time with a constant phase. Due to the presence of phase in the frequency modulation, the variation of population inversion with time is different from the standard case. There are no exact collapses and revivals in the variation of population inversion but it oscillates sinusoidally with time. In coherent field atom interaction the population inversion behaves as in the case of Fock state atom interaction, when frequency modulation with a non zero phase is applied. The study done with squeezed field has shown the same behavior of the population inversion.
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.