Electromagnetically-induced-transparency spectra of Rydberg atoms dressed with dual-tone radio-frequency fields

We examine spectral signatures of Rydberg atoms driven with near-resonant dual-tone radio-frequency (rf) fields in the regime of strong driving. We experimentally demonstrate and theoretically model a variety of nonlinear and multiphoton phenomena in the atomic Rydberg response that manifest in the electromagnetically-induced-transparency spectra. Our results echo previous studies of two-level atoms driven with bichromatic optical fields. In comparison to optical studies, the rf-driven Rydberg system utilizes a more complex excitation pathway and electromagnetic fields from two different spectral regimes: a two-photon optical excitation continuously creates highly excited Rydberg atoms, while rf fields drive resonant coupling between the Rydberg levels and generate strong mixing. However, our spectra reflect nearly identical effects of the dual-tone rf fields on the atomic Rydberg observables, showing detuning-dependent splittings and Rabi-frequency-dependent peak numbers and relative strengths, and avoided crossings at subharmonic resonances. We thus validate previous two-state models in this more complex physical system. In the context of Rydberg electrometry, we use these investigations to explore a technique in which we tune a known rf field to observe spectra which give the frequency and power of an unknown rf field using the complex dual-tone spectra.

Figure 1

Rydberg atom with applied dual-tone rf fields.

Figure 2

(a1)–(c1) Experimental and (a2)–(c2) theoretical waterfall plots showing Rydberg EIT spectra obtained through a simultaneous scan of $|{\mathrm{\Omega }}_1|=|{\mathrm{\Omega }}_2|=|\mathrm{\Omega }|$ with detunings ${\delta }_1=-{\delta }_2=\delta$ kept constant. The spectral features appear at coupling laser detunings spaced by the symmetric detuning $\delta$. The columns correspond to $\delta =35$, 55, and 85 MHz, from left to right. These locations are marked on the $x$ axis to emphasize the appearance of the spectral peaks at these locations. The features are reproduced in the theoretical Floquet quasienergy spectra, with waterfall plots over $\mathrm{\Omega }$ showing the mode occupations of Floquet modes against Floquet quasienergy $\epsilon$. The Floquet modes appear at quasienergies spaced by $\delta$, and this spacing remains constant as we scan $\mathrm{\Omega }$. The mode occupation has a sensitive dependence on $\mathrm{\Omega }$.

Figure 3

Bessel-function form of the mode occupation and corresponding EIT peak height for symmetrically detuned fields as the power-balanced Rabi frequencies are simultaneously scanned. (a) shows the line-outs (profile along the line) of the first three Floquet modes for $\delta =55\mathrm{MHz}$. We show the numerically computed mode occupations at intervals (circles) superposed on analytic Bessel functions (dash-dotted lines). In the same plot we show the experimentally obtained EIT peak heights (triangles), which are connected by lines to guide the eye. Blue, green, and purple elements in the plots depict the zero-, first-, and second-order Floquet modes. (b) shows the same line-outs for $\delta =85\mathrm{MHz}$.

Figure 4

(a1)–(c1) Experimental and (a2)–(c2) theoretical waterfall plots showing a simultaneous scan of ${\delta }_1=-{\delta }_2=\delta$ while the two Rabi frequencies $|{\mathrm{\Omega }}_1|=|{\mathrm{\Omega }}_2|=\mathrm{\Omega }$ are kept constant. The mode quasienergies are shown to increase with $\delta$ in each waterfall plot, and the splitting between the Floquet modes is linear in $\delta$ for the range of $\mathrm{\Omega }$ shown. Increasing the Rabi frequency has the effect of populating Floquet modes with a higher Floquet photon index. At higher driving Rabi frequencies we see an overall shift of the spectra toward lower energy [as seen in (c1)]. We attribute this shift to imperfect shielding of our reference cell from the applied rf fields.

Figure 5

(a1)–(e1) Experimental and (a2)–(e2) theoretical waterfall plots show a simultaneous scan of ${\delta }_1=-{\delta }_2=\delta$ while the two Rabi frequencies $|{\mathrm{\Omega }}_1|$, as labeled, and $|{\mathrm{\Omega }}_2|=1\sqrt{\text{mW}}$ are kept constant. The imbalance in Rabi frequencies is apparent from the asymmetric spectra, with the shift indicating the relative Rabi strengths as the Rabi frequency $|{\mathrm{\Omega }}_2|$ is larger or smaller than $|{\mathrm{\Omega }}_1|$.

Figure 6

(a1)–(g1) Experimental waterfall plots showing the obtained spectra against coupling laser frequency ${\delta }_c$ as the detuning ${\delta }_1$ is scanned for constant ${\delta }_2$ and Rabi frequencies $|{\mathrm{\Omega }}_1|=|{\mathrm{\Omega }}_2|=105$ MHz. (a2)–(g2) Numerically obtained Floquet quasienergy spectra showing Floquet-mode occupations against quasienergies $\epsilon$ as ${\delta }_1$ is scanned. The overlaid solid red lines indicate equal detunings of the two fields (${\delta }_1={\delta }_2$). The arrows and overlaid dashed blue horizontal lines in (d2) indicate the first and second subharmonic resonances marked at ${\delta }_1=\left|{\mathrm{\Omega }}_2\right|$ and ${\delta }_1=\left|{\mathrm{\Omega }}_2\right|/2$. The overlaid dashed blue vertical lines in (d1) and (d2) mark ${\delta }_c=\pm \left|{\mathrm{\Omega }}_2\right|/2$ and $\epsilon =0,\pm |{\mathrm{\Omega }}_2|/2$, respectively. The overlaid dashed red vertical lines show $\epsilon =|{\mathrm{\Omega }}_2|/2\pm |{\mathrm{\Omega }}_1|/4$.

Figure 7

The four-state model: (a) The data from Fig. 4 (symmetric, power-balanced fields as the detuning is scanned) show a structure that is not accounted for in the two-level model. (b) Including the fine states in the model reproduces these features. Here, for visibility we represent the Floquet-mode occupations as the widths of the spectral lines. The overlap of the Floquet quasienergies with the zero Floquet mode of $61D_{5/2}$ is shown in blue, while the overlap with the zero Floquet mode of $61D_{3/2}$ is shown in green. We scale the initial state occupations to reflect the relative peak heights of the $61D_{3/2}$ and $61D_{5/2}$ states in order to facilitate a comparison.

Figure 8

The 16-state model: (a) The data from Fig. 6 (a1), with asymmetric detunings and power-balanced Rabi frequencies, as the detuning ${\delta }_1$ is scanned. (b) Including the magnetic sublevels $m_J$ of the fine states in the model reproduces more features from the data. Here, for visibility we represent the Floquet-mode occupations as the widths of the spectral lines. The overlap of the Floquet quasienergies with the $m_J=\pm 1/2$ sublevels of the zero Floquet mode of $61D_{5/2}$ is shown in navy, the overlap with the $m_J=\pm 1/2$ sublevels of the zero Floquet mode of $61D_{3/2}$ is shown in green, and other $m_J$ components ($|m_J|>1/2$) of both lines are shown in red. We scale the initial-state occupations of the $m_J=\pm 1/2$ components of $61D_{3/2}$ and $61D_{5/2}$ to reflect their relative peak heights from a simple EIT scan and scale the initial-state occupations of all the other $m_J$ sublevels to be $10\%$ of the $m_J=\pm 1/2$ occupation of $61D_{5/2}$; in general, they would depend on the polarization anisotropy and mixing of the magnetic sublevels.