Rydberg-EIT of ⁸⁵Rb vapor in a cell with Ne buffer gas

Atoms in highly-excited Rydberg states exhibit sensitivity to their environment. When the Rydberg atoms are perturbed by dense ground-state perturbers, their interaction with the surrounding medium gives rise to a frequency shift, which can be attributed to two main effects Alekseev and Sobel’man (1966); Omont (1977); Asaf et al. (1993). The dominant effect arises from the scattering of the Rydberg electron by the perturbers within a Rydberg-atom volume of $\sim\frac{4}{3}\pi(2n^{2}a_{0})^{3}$. This scattering effect can be explained by a Fermi interaction Fermi (1934), and the resulting angular frequency shift in units of rad/s is given by

where $\hbar$ is the reduced Planck’s constant, $e$ is the elementary charge in C, $a_{s}$ is the low-energy s-wave scattering length in meters, and $N$ is the volume density of the buffer gas atoms.

The second effect originates from the interaction between the ion core of the Rydberg atom and the perturbers. When a Rydberg atom is immersed in a medium containing ground-state atoms or molecules, the atomic electric field induces a polarization in the perturbers, leading to a frequency shift. The frequency shift due to the polarization effect can be obtained from the impact approximation Alekseev and Sobel’man (1966); Omont (1977), and is given by (in units of rad/s)

where $\alpha$ represents the polarizability of the perturber in Cm²/V, and $v$ is the mean relative velocity between the Rydberg atoms and the perturbers in m/s. The total energy shift experienced by the Rydberg atom is the sum of both effects, $\Delta\omega_{\text{total}}=\Delta\omega_{\text{sc}}+\Delta\omega_{\text{p}}.$

In addition to frequency shifts, polarization and electron scattering can also lead to level decays, denoted $\gamma_{\text{p}}$ and $\gamma_{\text{sc}}$, respectively. It was found that the level decay mainly comes from the polarization of the perturbing atoms Omont (1977), i.e., $\gamma_{\text{p}}>>\gamma_{\text{sc}}$, and that the decay rate

The value of $\gamma_{\text{p}}$ is equivalent with a full-width at half-maximum (FWHM) line broadening in units of rad/s.

Through two-photon, Doppler-free spectroscopy, the broadening and shifts of Rb $nS$ and $nD$ Rydberg levels in the presence of inert perturbers has been experimentally observed in Brillet and Gallagher (1980) for He, Ar, Ne, Kr, and Xe, in Weber and Niemax (1982) for He, Ar, and Xe, in Bruce et al. (1982) for He and Ar, and in Thompson et al. (1987) for Ne, Kr, and H₂. For these experiments, effects of buffer-gas pressure broadening by the intermediate $5P_{1/2}$ and $5P_{3/2}$ levels could be ignored because they were very far off-resonance. This is, however, not the case in our work. Pressure broadening of the Rb $D_{2}$ line due to binary interactions with noble gases has been observed in Ottinger et al. (1975) for pressures of up to 1.1 kTorr. Recently, ultra-high pressures of He and Ar on the order of $10^{5}~{}$Torr interacting with a Rb vapor were spectroscopically studied in Ockenfels et al. (2022). In the present work, the Ne pressure is 5 Torr, which leads to a $D_{2}$ line broadening on the order of $\gamma_{D_{2}}\sim 2\pi\times 50~{}$MHz Ottinger et al. (1975). This is consistent with our Doppler-free saturated absorption spectra, which only show marginal remnants of the buffer-gas-free $5P_{3/2}$ hyperfine features. Since we observe Rydberg-EIT linewidths that are considerably larger, at the level of precision of our current study we neglect the effect of $\gamma_{D_{2}}$ on the Rydberg-EIT linewidth. As such, based on $\gamma_{p}>>\gamma_{sc}$ and $\gamma_{p}>\gamma_{D_{2}}$, we compare our measured Rydberg-EIT linewidths only with estimates for $\gamma_{p}$.

We perform the EIT experiment using ⁸⁵Rb atoms with an energy-level diagram shown in Fig. 1 (a). The probe laser ($\lambda_{\text{p}}$) with a wavelength of $\lambda_{p}=780$ nm is close to resonance with the $F=3$ to $F^{\prime}=4$ hyperfine component of the 5$S_{1/2}$ $\leftrightarrow$ 5$P_{3/2}$ transition. The EIT signal is measured as a function of the coupling laser (wavelength $\lambda_{\text{c}}\approx 480$ nm), which is scanned over the 5$P_{3/2}$ $\leftrightarrow$ $nD_{5/2}$ transition. The coupler detuning is denoted by $\Delta_{c}$.

We simultaneously extract EIT signals from two EIT beam lines, as shown in Fig. 1 (b). The upper beam line, which serves to produce a reference spectrum, utilizes a buffer-gas-free Rb vapor cell. The reference EIT signal allows us to calibrate the frequency axis for $\Delta_{c}$, as well as to mark the coupler-laser frequency of the shift-free EIT line. In the lower (signal) beam line, the EIT signal is acquired using a cell that contains Rb vapor and a Ne buffer gas of nominally 5 Torr pressure. The probe and coupler lasers are split between the reference and signal beam lines using polarizing beam-splitter cubes (PBS), and are then counter-propagated through the respective cells. In each cell, both beams have parallel linear polarizations. After passage through the cells, probe and coupler beams are separated using dichroic optics. The reference and signal probe beams are simultaneously detected using a pair of identical silicon photo-diodes and low-noise transimpedance amplifiers (TIAs), and the respective data traces are recorded. We perform 100 scans per data set, and present averages over the 100 scans.

The Rb vapor cell in the reference (upper) beam line in Fig. 1 (c) is held at room temperature (291 K). The probe and coupling beams in the reference line are approximately Gaussian and have $1/e^{2}$ drop-off radii of the intensity distribution of $w_{0}=300~{}\mu$m and 500 $\mu$m, respectively. In the signal (lower) beam line, the cell that contains the buffer gas is heated to 303 K, which results in about 70% peak absorption on the 5$S_{1/2}$, F$=3$ $\leftrightarrow$ 5$P_{3/2}$, F^′ transition. The probe and coupling beams in the signal line have Gaussian beam parameters of $w_{0}\approx 150~{}\mu$m.

The reference EIT signal is shown in Fig. 1 (c). The strongest EIT peak is from the 5$S_{1/2}$, $F=3$ $\leftrightarrow$ $5P_{3/2}$, $F^{\prime}=4$ $\leftrightarrow$ 34$D_{5/2}$ cascade, which has the largest electric dipole moment and is the least diminished by optical pumping into the uncoupled 5$S_{1/2}$, $F=2$ level. The two small peaks that bracket the -100 MHz mark are from the intermediate hyperfine states 5$P_{3/2}$, $F^{\prime}=3$ and $F^{\prime}=2$. These are small in size mostly due to optical pumping during the atom-field interaction time (which is a few $\mu$s in the buffer-gas-free cell). All observed frequency splittings between the $5P_{3/2}$ hyperfine peaks carry a Doppler scaling factor of $(\lambda_{p}/\lambda_{c}-1)=0.63$. The leftmost peak in Fig. 1 (c) is attributed to the 5$S_{1/2}$, $F=3$ $\leftrightarrow$ $5P_{3/2}$, $F^{\prime}=4$ $\leftrightarrow$ 34$D_{3/2}$ cascade. Noting that the Doppler scaling factor for Rydberg lines is unity, the splitting between the largest 34$D_{5/2}$ peak and the 34$D_{3/2}$ peak in Fig. 1 (c) equals the 34$D_{J}$ fine-structure splitting, which is 306.057 MHz. This splitting is used for calibration of the frequency axis of the reference and signal spectra, which are simultaneously acquired.

A typical Rydberg-EIT signal at a low probe intensity, obtained from the cell with buffer gas, is shown in Fig. 1 (d). The EIT signal from that cell has an asymmetric shape, and in the case shown the peak is shifted positively from the reference EIT line by 68 $\pm$ 1 MHz. In the following we study the dependence of shift and line width on principal quantum number $n$. We will then explore effects observed at high probe intensity.

To study the effect of the buffer gas on the EIT signals for a range of different Rydberg states, we take Rydberg-EIT data over an $n$-range of 34 to 46. We extract the relative frequency of the EIT peak in the signal beam, which equals the frequency shift of the EIT caused by the buffer gas. The frequency-shift results are shown in Fig. 2 (a) for probe powers of 0.06, 0.13, and 0.22 $\mu$W, corresponding to the probe Rabi frequencies listed. The observed frequency shifts are in the range of 60-73 MHz and averages to about 67 MHz, with the low-power data clustering around 70 MHz. This result agrees well with the observations in Thompson et al. (1987) where the shift rate was measured to be 12$\pm$1 MHz/Torr for high $n$ (which would lead to a shift of 60$\pm$5 MHz for 5 Torr of Ne buffer gas). Further, the shift decreases by up to about 10 MHz when the probe power increases, and it appears to overall decrease by a few MHz when $n$ increases. The weak $n$-dependence agrees with a semi-classical calculation in Henry and Herman (2002), which shows that the frequency shift of the Rydberg $nD$ state slightly decreases as $n$ increases.

For a quantitative comparison with theory, we first calculate the electron-scattering shifts using Eq. 1. For the low-energy s-wave scattering length, $a_{s}$, several of the previously computed values include, from low to high, $a_{s}=0.2a_{0}$  Hoffmann and Skarsgard (1969), $0.227a_{0}$ Fedus (2014), and $0.24a_{0}$ Omont (1977). A table listing both theoretical and experimental values for $a_{s}$ is provided in Cheng et al. (2014). Over the range $0.2a_{0}<a_{s}<0.24a_{0}$, $\Delta\omega_{\text{sc}}/(2\pi)$ varies from 195 MHz to 234 MHz. The shift from the Ne polarization due to the Rydberg atom, obtained from Eq. 2 and using $\alpha=2.66$ in atomic units Omont (1977), is $\Delta\omega_{\text{p}}/(2\pi)=-122$ MHz. The net shift from these two effects, $\Delta\omega_{\text{total}}/(2\pi)$, then ranges between 73 MHz and 112 MHz. It is seen that the shift from the calculation has the same sign in theory and experiment, confirming that the low-energy s-wave scattering length $a_{s}$ is positive (which is not the case for some other buffer gases) and dominant. Furthermore, depending on what exact value for $a_{s}$ is adopted, the calculated net shift is about 10% to 60% larger than the average experimental value of $\approx 67$ MHz from the previous paragraph. Hence, we claim good qualitative agreement.

To discuss these findings, we first note that the overall spread of data previously reported for $a_{s}$ has the largest effect on the net frequency shift, $\Delta\omega_{\text{total}}$, with our result being more consistent with the lower end of previously reported $a_{s}$-values ($a_{s}\approx 0.2a_{0}$). In fact, the ambiguity of estimates for $a_{s}$ alone may suffice to explain our observed deviation (if any) between measured line shifts and corresponding theoretical estimates. As to additional effects that might matter, we note that the Rydberg-EIT line may be slightly pulled to lower frequencies by the weak EIT peaks that are visible in Fig. 1 (c) but that are hidden in Fig. 1 (d). Furthermore, the Ne buffer gas pressure of 5 Torr has an uncertainty of 5% according to manufacturer information. Along the same line, one may speculate that differences in cell temperature during cell fabrication and the cell’s eventual operating temperature could in principle cause a mild buffer-gas density drop.

Further progress on comparing experimental buffer-gas-induced Rydberg-EIT shifts and theoretical estimates would require a refinement of theoretical models for $a_{s}$ as well as a full model for Rydberg-EIT that covers the effects of the buffer gas on all involved atomic levels. In experimental work, one may consider a determination of the absolute Ne density with an independent, quantitative method. These research directions are, however, outside the scope of our present work.

We next extract the FWHM of the EIT signal from the cell with buffer gas for several probe powers and $n$-values. As shown in Fig. 2 (b), the observed FWHM of the EIT signal from the cell with the buffer gas is between 110 and 125 MHz, which greatly exceeds the width of the reference EIT lines. The FWHM of the buffer-gas EIT slightly increases with an increase in $n$, but it is not significantly dependent on the probe power. This agrees well with the calculation in Henry and Herman (2002), which shows that, for Rydberg $nD$ states, the FWHM of the Rydberg line should slightly increase with $n$. A calculation of the broadening $\gamma_{p}$ from the polarization effect (Eq. 3) yields a FWHM of 144 MHz, which is $\approx 20\%$ larger than the experimentally observed width. We note that the effects that we neglect here could contribute somewhat to the experimentally observed line broadening, including broadening from the weak EIT peaks that are visible in Fig. 1 (c) but that are hidden in Fig. 1 (d), as well as the line broadening $\gamma_{D_{2}}$ [which is small compared to $\gamma_{p}$ but still substantially $>0$ (see Sec. I)]. The results on the EIT linewidth may also indicate that the exact density of Ne atoms in the cell could be slightly lower than the density one would have at 5 Torr and at room temperature.

Finally, we study the effects of probe power on the Rydberg-EIT signal in the signal beam line, which harbors the rubidium cell with the buffer gas, by varying the probe power from $P_{p}=0.06~{}\mu$W to 1.79 $\mu$W. The central probe-laser electric field is $E_{p}=\sqrt{2I_{p}/(c\epsilon_{0})}$, with central intensity $I_{p}=2P_{p}/(\pi w_{0}^{2})$ and $w_{0}\approx 150~{}\mu$m for the signal beam line. The probe Rabi frequency $\Omega_{p}/(2\pi)=\mu_{12}E_{p}/h$, where $\mu_{12}=1.892\,e\,a_{0}$ Steck (2021) is the probe-transition dipole moment. The EIT signals from the 34$D_{5/2}$ state are shown in Fig. 3 for five values of the probe Rabi frequency. The noise increase at high probe power is attributed to an increase in shot noise as well as an increase in TIA bandwidth at lower gain. Over the gain values used, the bandwidth of the TIA (model SRS SR570) increases from 200 Hz to 2 kHz with increasing probe power. Hence, the signals at higher probe powers have considerably larger noise on the utilized absolute-power scale. In Fig. 3, we include smoothed curves that allow for an easier comparison of the signal behavior over the entire probe-power range.

It can be seen in Fig. 3 that at low probe powers (Rabi frequency $\lesssim 10$ MHz, as used in Sec. III) the EIT line shape is invariant and the EIT signal strength is in proportion with probe power. In the limit of vanishing probe power such a linear behavior is expected. Also, the shape of the EIT signal is asymmetric, with a longer tail on the negative side. We speculate that this behavior may come in part from the blending of intermediate-state hyperfine structure, which adds to the $nD$-line broadening Ottinger et al. (1975). At Rabi frequencies above $\sim$10 MHz, the EIT peak position begins to shift negative, while shape and width of the peak still largely remain the same. At probe Rabi frequencies exceeding $\sim$20 MHz, the signals invert in shape and turn into electromagnetically-induced absorption (EIA), with the center of the EIA dip located $\sim 50$ MHz above the low-power EIT peak. The transition from EIT to EIA at high power may come from factors that involve optical pumping and velocity-changing collisions, the study of which could be the subject of future work.

We have observed Rydberg-EIT in a vapor cell containing 5 Torr of Ne buffer gas. Results obtained at low probe power have revealed frequency shifts of the EIT signals by about 70 MHz, as well as an increased FWHM EIT linewidth of about 120 MHz. These observations are largely unaffected by variations in the principal quantum number of the Rydberg states and in the probe power, as long as the probe Rabi frequency remains below about 10 MHz. The frequency shift is attributed to low-energy s-wave scattering between the Rydberg electron and the Ne atoms and polarization of there Ne atoms by the atomic electric field. The width of the signal is dominated by polarization of the Ne atoms. At high probe power, we observe a transition from EIT to EIA; this phenomenon awaits a future explanation.

Utilizing the Stark effect of Rydberg levels, potential applications of our research include non-invasive and spatially-resolved measurement of electric fields in low-pressure discharge plasmas in neon. Further, the electric fields of highly-charged dust particles in low-density plasma can potentially be mapped via Rydberg-EIT. For pressures below about 5 Torr, we also see applications in using Rydberg-EIT as a real-time, in-situ and non-invasive readout for buffer-gas density at a location of interest, which can have advantages over reading the buffer-gas pressure with a remote pressure gauge.

We acknowledge fruitful discussions with Prof. Eric Paradis (Eastern Michigan University), Dr. David A. Anderson (Rydberg Technologies Inc.), and Bineet Dash (University of Michigan). This project was supported by the U.S. Department of Energy, Office of Science, Office of Fusion Energy Sciences under award number DE-SC0023090. N.T. acknowledges funding from the NSRF via the Program Management Unit for Human Resources and Institutional Development, Research and Innovation (grant number B05F650024), and from the Office of the Permanent Secretary, Ministry of Higher Education, Science, Research and Innovation (Grant No. RGNS.64-067). R.C. acknowledges support from a Rackham Predoctoral Fellowship of the University of Michigan.