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arXiv:1902.01971 · Physical Review A

Microwave Rabi resonances beyond the small-signal regime

А. А. Третьяков, Lindsay J. LeBlanc

Published Apr 2, 2019• 19 citations
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Abstract

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 $^{87}\mathrm{Rb}$ atoms, which we compare to numerical simulations.

DOI
10.1103/physreva.99.043402

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Physical Review A•Nov 25, 2019•4 citations
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 \ensuremathα\ensuremath{\alpha}\ensuremathα and \ensuremathβ\ensuremath{\beta}\ensuremathβ 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.
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.
Just as it is possible to stabilize the frequency of an electromagnetic field to an atomic resonance between energy eigenstates, so it is possible to stabilize the intensity or brightness of a field to an atomic Rabi-resonance. We term such a device an "atomic candle." We describe the operation of a microwave atomic candle based on the 0-0 ground state hyperfine transition in /sup 87/Rb, and our experiments examining its stability and sensitivity to various experimental parameters. Specifically, we have measured the microwave power stability of our candle, relative to the peak of the Rabi-resonance, and obtained /spl sigma//sub /spl Delta/P/P/(/spl tau/)=9/spl times/10/sup -7/+10/sup -7//spl radic//spl tau/. Additionally, we examined the Rabi-resonance shift as a function of various experimental parameters including microwave detuning from the 0-0 resonance and resonance-cell temperature. The paper concludes with a discussion of a novel application of the atomic candle: the precision measurement of a material's complex dielectric constant.
Just as it is possible to stabilize the frequency of an electromagnetic field to an atomic resonance between energy eigenstates, it is possible to stabilize the intensity or brightness of a field to an atomic Rabi resonance. For ease of reference, such a stabilized field is termed an ``atomic candle.'' Here, we describe a microwave atomic candle based on the (F=2,(F=2,(F=2, mF=0){m}_{F}=0)mF​=0)--(F=1,(F=1,(F=1, mF=0){m}_{F}=0)mF​=0) ground-state hyperfine transition of 87Rb{}^{87}\mathrm{Rb}87Rb (6834.7 MHz) as observed in an optically pumped vapor. We discuss the atomic candle's operation and the intensity stability of the field it produces (relative to the Rabi-resonance peak). We then discuss our use of the candle to investigate Rabi-resonance shifts. Specifically, we find that the Rabi resonance depends on the ac Stark shift of the hyperfine transition, the microwave field-atom detuning, and the alkali-metal vapor temperature.
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.
We have recently presented a technique for measuring unknown microwave (MW) fields based on Rabi resonances induced by the interaction of atoms with a phase-modulated MW field. The single-peak feature of the measurement model makes the technique a valuable tool for simple and fast field measurement. Here we demonstrated a detailed investigation of Rabi resonance of Cs atoms inside a vapour cell. For illustration, we used the Rabi resonance-based field detection technique to determine the field strength inside an X-band cavity for applied power in the range of −21 to 20 dBm. The difference between the practical measurement and the simulation was approximately about 3% for an applied power of 20 dBm, and a higher accuracy could be expected for a larger power input.
The coherent interaction of Rydberg atoms with microwave fields in the ultrastrong-driving regime, in which the Rabi frequency is of the same order of magnitude as the transition frequency, has been studied for states with principal quantum number n=105n=105n=105 in helium. Experiments were performed in pulsed supersonic beams, and the effects of the ultrastrong 1.280 GHz microwave driving field, tuned to near resonance with the $1s105s\phantom{\rule{0.16em}{0ex}}^{3}S_{1}\ensuremath{\rightarrow}1s105p\phantom{\rule{0.16em}{0ex}}^{3}P$ transition, were identified from Autler-Townes splittings of the $1s3p\phantom{\rule{0.16em}{0ex}}^{3}P_{2}\ensuremath{\rightarrow}1s105s\phantom{\rule{0.16em}{0ex}}^{3}S_{1}$ transition by cw laser spectroscopy. The microwave field strength was calibrated in situ in the apparatus from Autler-Townes splittings measured in the weak-driving regime in which the rotating-wave approximation holds. The results of the experiments were compared to the energy-level structure of the atoms in the presence of the microwave field calculated using Floquet methods. From this comparison, the microwave-field strengths for which the rotating-wave approximation and the two-level approximation break down have been identified. The feasibility of implementing microwave traps for Rydberg atoms and molecules, which operate in the ultrastrong-driving regime, has been evaluated.