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Physical Review A

Line shapes of atomic-candle-type Rabi resonances

John G. Coffer, Brett A. Sickmiller, A. Presser, James C. Camparo

Published Aug 8, 2002• 49 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.66.023806

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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\ensuremathωm.2{\ensuremath{\omega}}_{m}.2\ensuremathωm​. 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\ensuremathδ\ensuremathΩ1/2/\ensuremathΩ)2\ensuremath{\delta}{\ensuremath{\Omega}}_{1/2}/\ensuremath{\Omega})2\ensuremathδ\ensuremathΩ1/2​/\ensuremathΩ) 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.
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.
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.
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.
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, so too is it possible to stabilize the amplitude (or 'brightness') of a field to atomic parametric resonances, the so-called Rabi-resonances. For ease of reference, and by analogy to the atomic clock, we have coined the term 'atomic candle' for this quantum-mechanical, amplitude-stabilization system. Though the atomic candle was originally developed to stabilize microwave power in gas-cell atomic clocks, thereby eliminating a source of timekeeping instability in these devices, the atomic candle's applications extend well beyond the area of precise timekeeping. Basically, the atomic-candle provides a means for detecting and controlling subtle amplitude changes in electromagnetic fields at very low Fourier frequencies (i.e., f < 0.1 Hz). In the present work, we discuss a number of atomic candle applications: laser stabilization, absorption/refractive-index measurements, observations of cavity mode stability over very long time scales (i.e., 50 days), and measurements of low-frequency absorption/scattering fluctuations along a propagation path.
Large-amplitude oscillations are predicted in the free-induction decay (FID) of atomic or spin two-level quantum systems that are coherently prepared by a resonant electromagnetic field in the form of a square amplitude pulse of duration TTT. The effect should be detectable in nuclear magnetic resonance (NMR) or in optical resonance experiments when the transition line shape is inhomogeneously broadened (${T}_{2}^{*}\ensuremath{\ll}{T}_{2}$) and the applied field is intense enough that the pulse area $\ensuremath{\chi}T\ensuremath{\ge}2\ensuremath{\pi}$, where $\ensuremath{\chi}$ is the Rabi frequency. Physically, the effect is due to Rabi oscillations which are generated in the preparative stage and are then reproduced because of atomic memory in the radiative period which follows. The atomic polarization derived from the Bloch equations assumes the form of an integral over the inhomogeneous line shape, and can be evaluated numerically for arbitrary linewidth $\ensuremath{\sigma}$ or analytically when $\ensuremath{\sigma}\ensuremath{\rightarrow}\ensuremath{\infty}$. The characteristics of the oscillation are unusual since the oscillation frequency, which is of order $\ensuremath{\chi}$, increases with time while the oscillation envelope vanishes identically for times $t\ensuremath{\ge}2T$. Coherent emission is therefore confined to one pulse width TTT immediately following the pulse. Previous NMR and infrared FID experiments with small-area pulses ($\ensuremath{\chi}T\ensuremath{\le}4\ensuremath{\pi}$), interpreted erroneously in terms of an edge echo effect, support the oscillatory FID behavior described here. For the case $\ensuremath{\chi}T>4\ensuremath{\pi}$, an experimental test would be even more decisive, as the oscillations are more pronounced.