rydberg constant hydrogen spectrum experiment

• For a given order (say m=3), is the angle of diffraction for red light larger or smaller than the angle for blue light? Explain.
• In part 2 of this experiment, how will you determine the spacing d of your diffraction grating?
• What is the relation between the wavelength l and the frequency f of a photon?
• What is the relation between the energy E and the frequency f of a photon?
• Sketch an energy level diagram of the hydrogen atom with the various levels labeled with the proper value of the quantum number n. Indicate on your diagram which transitions cause the four lines of the Balmer series.
• The colors of the four lines of the hydrogen spectrum are: red, blue-green, and two shades of violet. Which initial states n i = 3, 4, 5, or 6 correspond to these colors? (Hint: The colors of the visible spectrum, from longest to shortest wavelengths, are: red, orange, yellow, green, blue, violet.)
• Transitions to the n=1 (ground state) level from higher levels in the hydrogen atom never produce visible wavelength photons. Are the wavelengths produced by transitions to the n=1 level longer or shorter than visible wavelengths? Explain.
• In part 2 of this experiment, which quantities are given, which are measured directly, and which quantities are calculated?

Spectrum of Atomic Hydrogen

Adapted from "Spectrum of Atomic Hydrogen" in " Advanced Physics with Vernier – Beyond Mechanics " ©Vernier Software & Technology

Reading the Review of Bohr Theory is recommended before starting this experiment.

In this experiment, you will:

Observe light sources through a diffraction grating. Use a photodiode array spectrometer to make a qualitative investigation of the spectrum of several different kinds of light sources and categorize their spectrum as broadband continuum, narrowband continuum, line or composite. Perform a quantitative study of the visible spectrum of hydrogen. Determine the wavelengths of the emission lines in the visible spectrum of excited hydrogen gas. Determine the energies of the photons corresponding to each of these wavelengths. Use Bohr Theory to relate the photons’ energies to specific transitions between energy levels. Use your data and the values for the electron transitions to determine a value for Rydberg’s constant for hydrogen and determine the lowest energy Bohr state for your hydrogen spectrum.. Compare your experimental value for the Rydberg constant to the accepted value.

Preliminary INVESTIGATION

•  Plug in the straight-filament lamp. View the spectrum through the diffraction grating. When finished, remove power from the lamp. Describe in your notebook what you observed.
• If necessary, rotate the carousel to place the hydrogen discharge tube in. Turn on the power and observe the bright line spectrum of hydrogen through the diffraction grating. Turn the power off (the hydrogen tube has a very limited lifetime). Record your observations.
• Use your grating to look at the overhead fluorescent lights.
• Discuss the differences in the spectra from these two sources.

Warning: Never touch the ends of the optical fiber and never allow the fiber to kink tightly .   The optical cables are fragile and easily damaged.

• If it is not connected, connect the Vernier spectrometer to a USB port on the computer. Double Click ‘ Spectral Studies.cmbl’ in the PHY 173 or PHY 183 folder, as appropriate, to Start Logger Pro. If not already connected, carefully connect the optical fiber cable to your spectrometer by removing the plastic protection sheath, inserting the bundle into the screw fixture on the spectrometer and gently tightening the outer shell until it is just snug.
• Choose Change Units ►Spectrometer ►Intensity from the Experiment menu. The software will measure the intensity in relative units.
• Next, choose Set Up Sensors ►Spectrometer from the Experiment menu. Change the data-collection duration to 40 ms if it is not set already.
• If not, right click on the graph, go to Graph Options, then axes and click the ‘Log axis’ box.
• Set the maximum value to 5 and the minimum value to 5E-4.
• This setting will compress the tall features on the graph allowing smaller features to be identified as being present. The value around 0.001 represents no light detected by the spectrometer.

Qualitative classification of light sources

Be sure to record the name of your light source in a table – i.e. green LED, helium tube, etc – as you take your data. Otherwise you will have a lot of curves but may not know what they are.

• Point your fiber at the overhead fluorescent lights.
• Click ►Collect to start the program taking data and position the optical fiber until you have a plot that is above 0.5 at its highest point but less than one. Stop the data collection. (The next time you click Collect, you will be asked if you want to store or discard the previous data – always choose to store unless you are repeating on the same lamp. Be sure to record the data name beside the light source you recorded earlier.)
• If it is a sunny day, turn the room lights off and point your fiber at the windows and repeat step #4.
• You have a bulb and two batteries and some wires. Make the bulb light. And repeat step #4. Disconnect the lamp when finished.
• You have a box with three pushbuttons and a snap to a battery pack.  If not attached, place the snap on the connector of the battery holder that uses 3 "AA" batteries - Never connect the LED Box to a 9-volt battery! If you look on the side of the box there is a small hole that will show light when one of the buttons is pushed.  The hole is intentionally too small to be able to insert the optical fiber.  The LEDs are too bright for the spectrometer to measure them directly, so there is a 'process' involved:
• Click on 'Collect' in Logger Pro
• Hold the end of the fiber up to the hole and 'tilt' the fiber until a reasonably large spectra is observed.
• Click 'Stop' on Logger Pro to capture the spectrum.
• Label the curve in Logger Pro in the same manner you labeled the spectrum for the lamp.
• Set the LED box aside, beings sure that nothing is pressing on the buttons.
• Your discharge lamp fixture (the green cylinder with the black carousel) has a bracket that will hold the end of the optical fiber in position relative to the lamp. Place the optical fiber in the notch and lightly tighten the screw on the hexagonal metallic part of the end of the fiber optic cable. Do not over-tighten. The carousel can be rotated (with the power off) to change light sources. The lamp can be gently rocked from side to side while on to better position the lamp and optical fiber if need be. Notes: Hydrogen tubes have a limited lifetime, so turn the tube off when you are not taking data. Air tubes may take several minutes to start.
• For each of the lamps in the discharge carousel repeat step #4. You should have the intensity of the strongest line above 0.4 but no lines should be ‘clipped’ at the value 1.0 (or 10 0 on the graph). If necessary, rock the carousel gently to change the amount of light entering the fiber. Turn the lamp off as soon as you stop data collection.
• Save your experiment on the Desktop.
• Change to page ‘2: No Color’ in Logger Pro which is the same graph without the color spectrum background.
• Print a graph of your curves (without the data tables) and include in your lab report.

This completes the qualitative component of this lab.  Be sure to reflect in you lab notebook.

Quantitative analysis of the hydrogen spectrum.

Hide all of your curves except for the hydrogen spectrum. You should be able to see 4 or 5 peaks representing the location and intensity of the ‘spectral lines’. The most intense line should have an intensity less than 1.00. If this is the case you can use the data you have already taken. If not, you will need to acquire a new spectrum that shows 4 or, preferably, 5 peaks without clipping the most intense peak at a value of 1.00 (clipping refers to cutting the top off a curve, much like clipping a blade of grass while mowing). If you take new hydrogen data be sure to save your experiment file and to turn off the hydrogen discharge tube as soon as you finish.

EVALUATION OF DATA

 At this point, you should be on page '2: No Color' the only graph on your screen should be your hydrogen spectrum. If this is not the case, hide all of the other curves except for hudrogen before proceeding.

• Use the Examine tool to find the center wavelength for the red line you observed in the H-spectrum. Record the wavelength in a table in your lab notebook like the one shown below. Repeat this step for the remaining peaks.  If you need to be reminded of color, refer to the first page.
• Change to page ‘3: Rydberg’ in Logger Pro and enter your peak wavelengths in the appropriate column. The first line is a placeholder to keep the formula to convert the wavelength into photon energy - type over the values in that line. The photon energy for that wavelength will be calculated automatically (in eV) in the appropriate column. Copy those values for wavelength and change in energy into your data table in your notebook.

 We can, however, make use of the fact that D E = hf = h(c/λ) to obtain a form of the Rydberg equation that relates the energy of the emitted light to the initial and final energy states in terms of photon energy in eV.

Using Bohr's equation above, complete the table below and include it in your notebook. These are the lowest energy photons emitted for a given value of n-final.  The range of visible light is approximately 2 to 4 eV.  Compare your value for Delta-E to determine the spectral range as either infrared, visible, or ultraviolet.

• In your conclusion be sure to reflect on both the qualitative and quantitative part of this laboratory experiment.

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Chemistry Faculty Publications

Determination of the rydberg constant for the hydrogen atom.

Kurt W. Kolasinski , West Chester University of Pennsylvania Follow

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This laboratory video explains an experiment that uses the acquisition of the spectrum of H atoms across the visible and into the ultraviolet regions of the electromagnetic spectrum to determine a value of the Rydberg constant for the H atom.

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From Kolasinski's Physical Chemistry YouTube Channel

Recommended Citation

Kolasinski, K. W. (2022). Determination of the Rydberg Constant for the Hydrogen Atom. Retrieved from https://digitalcommons.wcupa.edu/chem_facpub/44

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Precise Measurement of Hydrogen’s Energy Levels

• Department of Physics and Laboratory for Atomic, Molecular, and Optical Research, Missouri University of Science and Technology, Rolla, MO, US

Physicists used to think they had a good idea of the size of the proton. Values derived from measurements of hydrogen’s emission spectrum and from electron-scattering experiments agreed with a proton radius of around 0.88 femtometers (fm). Then, in 2010, confidence was shaken by a spectral measurement that indicated a proton radius of approximately 0.84 fm [ 1 ]. In the years since, this “proton radius puzzle” has become even more of a head-scratcher, with some experiments supporting the original estimate and others finding an even greater discrepancy. Simon Scheidegger and Frédéric Merkt at the Swiss Federal Institute of Technology (ETH), Zurich, have now made precise new measurements of the transition energies between one of hydrogen’s metastable low-energy states and several of its highly excited states [ 2 ] (Fig. 1 ). These measurements allow the researchers to derive some of the atom’s properties, such as its ionization energy, with greater confidence, which should help clear up some of the confusion.

The 2010 study that “shrank the proton” (as the title of the editorial summary in Nature jokingly stated) concerned the 2 S –2 P 1/2 Lamb shift [ 1 ]. According to Dirac’s predictions, the 2 S and 2 P 1/2 levels of atomic hydrogen should be degenerate. The Lamb shift refers to the lifting of this degeneracy by quantum electrodynamic (QED) effects, the largest contribution being the electron “self-energy” due to interactions with virtual photons. Once this and other QED effects are accounted for, a tiny shift of the bound-state energy levels remains, which can be attributed to the proton’s finite size. By measuring this residual energy shift, one can determine the proton radius directly. The authors of the 2010 study did so using hydrogen atoms in which the electron was replaced by its heavier cousin, the muon, since the finite-size effect is stronger in this system.

Ever since that surprise result, researchers have tried to pin down the proton radius both directly, via the finite-size effect, and indirectly, via the Rydberg constant. The Rydberg constant relates an atom’s energy levels to other physical constants and is one of the key inputs used in calculations of the proton radius. Determining its value requires painstaking measurements of the transition energies between hydrogen’s various states. Several groups have made monumental efforts in this regard, but the values they derive for the proton radius have been all over the place. A 2018 measurement of the 1 S –3 S transition by a group in France gave a value of about 0.88 fm [ 3 ], a 2019 measurement of the classic Lamb shift (this time in regular hydrogen) by a group in Canada came up with a value of about 0.833 fm [ 4 ], and a 2017 measurement of the 2 S –4 P transition by a group in Germany suggested a similarly low value of about 0.834 fm [ 5 ]. In 2020, the group in Germany arrived at a slightly higher value of 0.848 fm [ 6 ]. In 2022, finally, from measurements of the 2 S –8 D transition, a group at Colorado State University proposed a “compromise value” of about 0.86 fm [ 7 ].

Scheidegger and Merkt, like some of their predecessors, start from hydrogen’s metastable 2 S level. This state has a natural lifetime of 0.122 seconds and provides a convenient “launchpad” for transitions to high principal quantum numbers. But the researchers go higher than most, accomplishing measurements of transitions from that 2 S launchpad into the realm of highly excited “Rydberg” states with principal quantum numbers n of 20–24. These Rydberg states have undefined angular momenta but, due to the application of an external electric field, defined Stark-state parabolic quantum numbers. This field lets the researchers control and distinguish between different fine-structure-resolved and hyperfine-structure-resolved hydrogen eigenstates that would otherwise overlap in energy. Scheidegger and Merkt find that all of the measured transitions can be fitted by a unified theoretical model that takes the quantum numbers of the Rydberg states into account. In fact, they use the observed splitting to calibrate the electric field. This in itself is no small achievement, as it requires the diagonalization of complicated hyperfine-resolved matrices of the Stark operator.

In order to understand the rationale of their experiment, one needs to know that, to a good approximation, S states, and only S states, have a nonvanishing probability density at the nucleus. Together with their spherical symmetry, this makes these states sensitive to the proton radius, with the nuclear-size correction—that is, the energy-level adjustment needed to account for the proton’s finite size—proportional to 1/ n 3 . In fact, it is the sensitivity of the 2 S state to the proton radius and the insensitivity of the 2 P states, which makes it possible to determine the proton radius using the 2 S –2 P 1/2 Lamb shift alone.

But wait. Scheidegger and Merkt’s goal is to determine a precise, accurate value of the Rydberg constant from measurements of transition energies. If the energy levels of S states—including their 2 S launchpad—depend on the nuclear size, how can they obtain a value that’s independent of the proton radius?

The answer lies in the additional input that they use—namely, the value of the Lamb shift transition 2 S –2 P 1/2 measured by the group in Canada in 2019 [4]. By adding that value to the frequency of the transition from the 2 S state to a highly excited Rydberg state, Scheidegger and Merkt effectively measure the transition from the 2 P 1/2 state to the highly excited Rydberg state. Now, both the upper and lower levels are largely independent of the proton radius. One can always find such combinations of frequencies that are independent of the proton radius and therefore give the Rydberg constant. Here, the lucky circumstance is that the prefactors that correct for the proton radius are unity. For example, if one wanted to determine the Rydberg constant on the basis of the 1 S –2 S and 2 S –4 P transition frequencies, the prefactors would be 1/7 and −1. In Scheidegger and Merkt’s work, by contrast, the simple sum of the 2 S –2 P 1/2 and the 2 S -to- n = 20 frequencies eliminates the proton radius.

Scheidegger and Merkt’s determination of the Rydberg constant comes with a caveat: other measurements of the classic Lamb shift transition 2 S –2 P 1/2 exist—notably, by Lundeen and Pipkin [ 8 ] and Hagley and Pipkin [ 9 ]. Those measurements, although carried out in the 1980s and 1990s, are not much less precise than the 2019 measurement by the group in Canada [4] but indicate a larger proton radius. The reliability of Scheidegger and Merkt’s determination depends on which group’s measurement of the 2 S –2 P 1/2 Lamb shift is most accurate. In order to put this dependence into perspective, note that classic Lamb shift measurements need to overcome an important obstacle—the extremely short lifetime of the hydrogen 2 P state. This state is one of the shortest-lived excited states in all neutral atoms, and measuring its transitions presents a major challenge. Thus, we can conclude that there remains work to be done before we can be confident about the true value of the Rydberg constant.

This caveat does not apply to Scheidegger and Merkt’s measurement of the value of the ionization energy of atomic hydrogen. The ionization energy can be determined very reliably, because the only additional input data needed are the hydrogen 1 S –2 S frequency and the 1 S hyperfine frequency. These frequencies are both known to have more than sufficient accuracy, as measurements are facilitated by the very small natural linewidths of both the 1 S –2 S frequency and of the 1 S hyperfine transition [ 6 – 10 ]. As a result, the value for the ionization energy of hydrogen obtained by Scheidegger and Merkt constitutes not only the most precisely known ionization energy yet known of any bound system but also one of the most precisely known of all physical constants.

• R. Pohl et al. , “The size of the proton,” Nature 466 , 213 (2010) .
• S. Scheidegger and F. Merkt, “Precision-spectroscopic determination of the binding energy of a two-body quantum system: The hydrogen atom and the proton-size puzzle,” Phys. Rev. Lett. 132 , 113001 (2024) .
• H. Fleurbaey et al. , “New measurement of the 1 S –3 S transition frequency of hydrogen: Contribution to the proton charge radius puzzle,” Phys. Rev. Lett. 120 , 183001 (2018) .
• N. Bezginov et al. , “A measurement of the atomic hydrogen Lamb shift and the proton charge radius,” Science 365 , 1007 (2019) .
• A. Beyer et al. , “The Rydberg constant and proton size from atomic hydrogen,” Science 358 , 79 (2017) .
• A. Grinin et al. , “Two-photon frequency comb spectroscopy of atomic hydrogen,” Science 370 , 1061 (2020) .
• A. D. Brandt et al. , “Measurement of the 2 S 1/2 –8 D 5/2 transition in hydrogen,” Phys. Rev. Lett. 128 , 023001 (2022) .
• S. R. Lundeen and F. M. Pipkin, “Measurement of the Lamb shift in hydrogen, n = 2,” Phys. Rev. Lett. 46 , 232 (1981) .
• E. W. Hagley and F. M. Pipkin, “Separated oscillatory field measurement of hydrogen 2 S 1/2– 2 P 3/2 fine structure interval,” Phys. Rev. Lett. 72 , 1172 (1994) .
• Th. Udem et al. , “Phase-coherent measurement of the hydrogen 1 S –2 S transition frequency with an optical frequency interval divider chain,” Phys. Rev. Lett. 79 , 2646 (1997) .

About the Author

Ulrich D. Jentschura studied at the University of Munich and obtained his PhD and habilitation degree from Dresden University of Technology. He is a Fellow of the American Physical Society. He is a professor of physics at Missouri University of Science and Technology. His research focuses on quantum electrodynamics, lasers, dynamical processes involving atoms in dressed environments, and related areas. Recently, he and Gregory Adkins from Franklin and Marshall College, Pennsylvania, published a book entitled Quantum Electrodynamics: Atoms, Lasers and Gravity (World Scientific, Singapore, 2022).

Precision-Spectroscopic Determination of the Binding Energy of a Two-Body Quantum System: The Hydrogen Atom and the Proton-Size Puzzle

Simon Scheidegger and Frédéric Merkt

Phys. Rev. Lett. 132 , 113001 (2024)

Published March 11, 2024

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Rydberg electromagnetically induced transparency of 85 Rb vapor in Ar, Ne and N 2 gases

An experimental study on Rydberg electromagnetically induced transparency (EIT) in rubidium (Rb) vapor cells containing inert gases at pressures ≤ 5 absent 5 \leq 5 ≤ 5  Torr is reported. Using an inert-gas-free Rb vapor cell as a reference, we measure frequency shift and line broadening of the EIT spectra in Rb vapor cells with argon, neon or nitrogen gases at pressures ranging from a few mTorr to 5 Torr. The results qualitatively agree with a pseudo-potential model that includes s 𝑠 s italic_s -wave scattering between the Rydberg electron and the inert-gas atoms, and the effect of polarization of the inert-gas atoms by the Rydberg atoms. Our results are important for establishing Rydberg-EIT as an all-optical and non-intrusive spectroscopic probe for field diagnostics in low-pressure radio-frequency discharges.

I Introduction

Rydberg atoms are widely used for electric field sensing because of their large electric polarizabilities  [ 1 ] . Combining optical and microwave excitation of Rydberg states with electromagnetically induced transparency (EIT)  [ 2 ] allows spectroscopic measurement of Stark shifts of Rydberg states in response to local electric fields. Recently, Rydberg-EIT has emerged as a broadband, sensitive and SI-traceable field-sensing technique for microwave electrometry  [ 3 , 4 ] , radio-frequency (rf) reception   [ 5 ] , magnetic field  [ 6 ] and dc electric field sensing  [ 7 ] . Recently, Rydberg atoms have been utilized to measure the macroscopic and microscopic electric fields in cold ion clouds revealing signatures of Holtsmark field distribution and opening an avenue for forthcoming studies on plasma field diagnostics  [ 8 ] . Furthermore, the principles of dc electric field sensing using Rydberg states with high angular momentum ( ℓ ≤ 6 ℓ 6 \ell\leq 6 roman_ℓ ≤ 6 ) have been demonstrated in room temperature vapor cells  [ 9 ] .

Rydberg-EIT-based techniques are promising candidates for in-situ electric field measurement in delicate environments because EIT can enable all-optical signal acquisition. This eliminates or reduces the need for conducting components that must be introduced in the system under test. Such environments include the aforementioned ion clouds and sheath regions of plasma, both of which are characterized by complicated and potentially time-varying electric fields. There is a growing interest in all-optical non-invasive plasma diagnostics methods  [ 10 ] to avoid the limitations of traditional electrostatic probes caused by probe-induced perturbation  [ 11 ] .While Stark-effect-based field measurements have been conducted using laser-induced fluorescence spectroscopy of Rydberg states in plasma discharges  [ 12 , 13 , 14 ] , this method is limited to hydrogen and other types of low-pressure plasmas. Laser spectroscopy of Rydberg states has been used to measure electric fields in the range of 10 V/cm to 50 V/cm in Coulomb-expanding one-component Rb + micro-plasmas  [ 15 ] . Other non-invasive field diagnostic methods, such as emission spectroscopy  [ 16 , 17 ] or field-induced non-linear effects e.g. , four-wave mixing (E-CARS)  [ 18 , 19 ] and second-harmonic generation (E-FISH)  [ 20 , 21 ] are suitable for fields stronger than 100 ⁢ V/cm 100 V/cm 100\,\text{V/cm} 100 V/cm . On the other hand, EIT-based field sensing has been demonstrated in laser-generated alkali plasmas by optically interrogating locally embedded alkali Rydberg atoms at the sub-V/cm regime  [ 22 , 23 , 8 ] .

To extend EIT-based field diagnostics to dc and rf discharges, it is crucial to characterize the effect of gases commonly used to sustain such plasmas on Rydberg-EIT. Rydberg-EIT proceeds through a near-resonant intermediate state and it involves a coherent quantum-interference effect  [ 24 ] . This situation substantially differs from past experiments in which gas-induced Rydberg-line shifts and pressure broadening have been explored using multi-photon spectroscopy methods without resonant intermediate states  [ 25 , 26 , 27 , 28 ] . Recent EIT experiments have demonstrated ladder-type EIT in the 5 ⁢ S 1 / 2 ↔ 5 ⁢ P 1 / 2 ↔ 5 ⁢ D 3 / 2 ↔ 5 subscript 𝑆 1 2 5 subscript 𝑃 1 2 ↔ 5 subscript 𝐷 3 2 5S_{1/2}\leftrightarrow 5P_{1/2}\leftrightarrow 5D_{3/2} 5 italic_S start_POSTSUBSCRIPT 1 / 2 end_POSTSUBSCRIPT ↔ 5 italic_P start_POSTSUBSCRIPT 1 / 2 end_POSTSUBSCRIPT ↔ 5 italic_D start_POSTSUBSCRIPT 3 / 2 end_POSTSUBSCRIPT cascade of Rb  [ 29 ] , as well as Rydberg-EIT in room-temperature vapor cells containing 5 ⁢ Torr 5 Torr 5\,\text{Torr} 5 Torr of neon  [ 30 ] . In the present paper, we extend earlier work by exploring the effect of the inert gases argon, neon and nitrogen on rubidium Rydberg-EIT over a pressure range from 15 ⁢ mTorr 15 mTorr 15\,\text{mTorr} 15 mTorr to 5 ⁢ Torr 5 Torr 5\,\text{Torr} 5 Torr . In the analysis of the experimental data we find that the semi-classical pseudo-potential model reliably predicts the sign and magnitude of corresponding pressure- and gas-species-dependent line shifts and broadening. Our study indicates that Rydberg EIT could be suitable to analyze electric fields in Ar plasmas in the range of tens of mTorr, including inductively coupled plasmas.

The paper is organized as follows. We describe details of the experimental setup and the Rydberg-EIT scheme in Sec.  II . EIT spectra obtained in inert-gas cells and the deduced line shifts and broadening are presented in Sec.  III . In Sec.  IV , the pseudo-potential model for the interactions between a Rydberg atom and an inert-gas atom is presented. We estimate the semi-classical predictions for line shifts and broadening based on s 𝑠 s italic_s -wave scattering length and polarizability values of common inert gases found in the literature. We discuss the validity of the semi-classical model and implications for plasma field diagnostics in Sec.  V . The paper is concluded in Sec.  VI .

II EXPERIMENTAL SETUP

The EIT signals from the signal line in Fig.  1 are recorded in the presence of Ar, Ne or N 2 . The probe and coupler beams are focused to respective beam waists of 150 ⁢ μ m 150 μ m 150\,\text{$\mu$m} 150 italic_μ m and 170 ⁢ μ m 170 μ m 170\,\text{$\mu$m} 170 italic_μ m and counter-propagated through a 5 ⁢ cm 5 cm 5\,\text{cm} 5 cm vapor cell containing Rb and one of the specified inert gases. The optical powers in the signal line are ≈ \approx ≈ 22 ⁢ mW 22 mW 22\,\text{mW} 22 mW for the coupler beam and ≈ \approx ≈ 1.7 ⁢ μ W 1.7 μ W 1.7\,\text{$\mu$W} 1.7 italic_μ W for the probe beam; the latter corresponds to a Rabi frequency of 30 ⁢ MHz 30 MHz 30\,\text{MHz} 30 MHz . The coupler beam in the signal line is pulsed at 15 ⁢ kHz 15 kHz 15\,\text{kHz} 15 kHz at a 50 % duty cycle using an acousto-optic modulator (AOM) for lock-in detection of the EIT signal, which is required to provide sufficient sensitivity to observe weak EIT lines in cells with higher inert-gas pressures. The EIT-probe power in the signal line is detected using the same types of photo-diode and TIA as in the reference line. The TIA output is sent to a dual-channel lock-in amplifier (MFLI-500, Zurich Instruments), which is referenced to the coupler-beam pulse sequence. The phase of the lock-in amplifier is set by minimizing the y 𝑦 y italic_y -output of the lock-in amplifier on a strong EIT line. The recorded signal is the x 𝑥 x italic_x -output of the lock-in amplifier.

III RESULTS

EIT measurements were made at pressures ranging from 15 ⁢ mTorr 15 mTorr 15\,\text{mTorr} 15 mTorr to 5 ⁢ Torr 5 Torr 5\,\text{Torr} 5 Torr of Ar and 50 ⁢ mTorr 50 mTorr 50\,\text{mTorr} 50 mTorr to 5 ⁢ Torr 5 Torr 5\,\text{Torr} 5 Torr of Ne and N 2 . Figs.  3 and 4 show the shift and the broadening of the Rydberg-EIT lines versus pressure, respectively. As discussed in the next Section, the estimated inert-gas-induced broadening is ≈ \approx ≈ 1 ⁢ MHz 1 MHz 1\,\text{MHz} 1 MHz in low pressure cells ( i.e. 50 ⁢ mTorr 50 mTorr 50\,\text{mTorr} 50 mTorr or smaller), which is about 5 times smaller than the inert-gas-free EIT linewidth and not discernible within our experimental precision. Therefore, we infer the quantitative effect of the inert gases from observations in the higher-pressure cells ( 0.5 ⁢ Torr 0.5 Torr 0.5\,\text{Torr} 0.5 Torr and 5 ⁢ Torr 5 Torr 5\,\text{Torr} 5 Torr ), where shifts and broadening become more significant.

We find that Ar induces a large negative frequency shift, unlike other inert gases considered here. The Rydberg-EIT line is shifted by − 1485 ± 25 ⁢ MHz plus-or-minus 1485 25 MHz -1485\pm 25\,\text{MHz} - 1485 ± 25 MHz and broadened by 610 ± 30 ⁢ MHz plus-or-minus 610 30 MHz 610\pm 30\,\text{MHz} 610 ± 30 MHz in the 5 ⁢ Torr 5 Torr 5\,\text{Torr} 5 Torr Ar cell. It should be noted that the log-log plot in Fig.  3 shows the absolute value of the frequency shift in Ar. In the 5 ⁢ Torr 5 Torr 5\,\text{Torr} 5 Torr Ne cell, we observe a moderate positive frequency shift of 105 ± 5 ⁢ MHz plus-or-minus 105 5 MHz 105\pm 5\,\text{MHz} 105 ± 5 MHz and a line broadening of 140 ± 5 ⁢ MHz plus-or-minus 140 5 MHz 140\pm 5\,\text{MHz} 140 ± 5 MHz . The results in the 5 ⁢ Torr 5 Torr 5\,\text{Torr} 5 Torr Ne cell agree well with earlier findings  [ 30 ] (which were obtained without using lock-in detection). At 5 Torr of N 2 we observe a large line broadening of 600 ± 85 ⁢ MHz plus-or-minus 600 85 MHz 600\pm 85\,\text{MHz} 600 ± 85 MHz , which contrasts with a rather small positive frequency shift of only 68 ± 20 ⁢ MHz plus-or-minus 68 20 MHz 68\pm 20\,\text{MHz} 68 ± 20 MHz .

The experimental uncertainties in the line shifts and broadenings in the cells with pressures below 0.5 ⁢ Torr 0.5 Torr 0.5\,\text{Torr} 0.5 Torr are  ≈ \approx ≈ 1 ⁢ MHz 1 MHz 1\,\text{MHz} 1 MHz and are attributed to coupler-laser frequency calibration uncertainty. In the high-pressure cells ( 5 ⁢ Torr 5 Torr 5\,\text{Torr} 5 Torr ), the EIT signals are two to three orders of magnitude weaker than in the low-pressure cells, which is due to large inert-gas-induced EIT line broadening. The large linewidths in the high-pressure cells result in larger experimental uncertainties, which are determined by visual inspection of the spectra. An important systematic uncertainty in our experiments arises from the uncertainty of the inert-gas pressures within the cells. The reported pressures are nominal manufacturer values, which have up to 5 ⁢ % 5 % 5\,\text{\%} 5 % error in the high-pressure cells and 10 ⁢ % 10 % 10\,\text{\%} 10 % for the cells with pressures below 50 ⁢ mTorr 50 mTorr 50\,\text{mTorr} 50 mTorr   [ 32 ] .

IV Analysis

Pressure broadening and line shifts of Rydberg levels in low-pressure background gases have been studied extensively since the 1960s using single or two-photon spectroscopy. The collisional shifts of Rydberg transitions were first explained by Fermi using a so-called pseudo-potential  [ 33 ] that treats the ionic core and the Rydberg electron separately in their corresponding interactions with perturbers that can be inert-gas atoms or molecules. The model includes two components: scattering between the perturbers and the Rydberg electron, and a long-range attractive potential experienced by the perturber atoms/molecules due to polarization by the ionic core of the Rydberg atom. A semi-classical treatment is afforded by the impact approximation, in which radiative coupling or change in the Rydberg-electron wave function during the collision are ignored. The scattering is well-approximated by the s 𝑠 s italic_s -wave scattering due to the low kinetic energy of the bound Rydberg electron.

Refs.  [ 34 ] and  [ 35 ] have expanded Fermi’s treatment by including phase shifts from scattering. In this picture, the frequency shift ( Δ r subscript Δ 𝑟 \Delta_{r} roman_Δ start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) and broadening ( γ r subscript 𝛾 𝑟 \gamma_{r} italic_γ start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) of a Rydberg line are given by the sum of respective contributions from the scattering and polarization effect; i.e. ,

Subscripts s ⁢ c 𝑠 𝑐 sc italic_s italic_c and p 𝑝 p italic_p denote the scattering and polarization contributions, respectively, and are expressed as

where a s subscript 𝑎 𝑠 a_{s} italic_a start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT and α 𝛼 \alpha italic_α denote the s 𝑠 s italic_s -wave electron scattering length and the dipole polarizability of the perturber, respectively, v 𝑣 v italic_v is the root-mean-square velocity of the perturbers, n 𝑛 n italic_n is the effective quantum number of the Rydberg state, and N 𝑁 N italic_N is the volume number density of perturbers. All physical quantities in Eq.  2 are in SI units, including the line shift and the broadening being in units of Hz. The negative sign of Δ p subscript Δ 𝑝 \Delta_{p} roman_Δ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT is due to the attractive polarization potential produced by the ionic core. However, the Rydberg-electron scattering can induce a positive or negative shift, depending on the sign of the scattering length.

We have surveyed the electric-dipole polarizability, α 𝛼 \alpha italic_α , and the s 𝑠 s italic_s -wave scattering length, a s subscript 𝑎 𝑠 a_{s} italic_a start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT , of the relevant inert gases in the existing literature. Reported polarizability values from theoretical predictions and several experiments demonstrate excellent agreement with each other. Techniques such as dielectric constant gas thermometry  [ 36 ] , optical spectroscopy  [ 37 ] and capacitance measurement  [ 38 ] have been used to measure α 𝛼 \alpha italic_α for common inert gases. The vast majority of results reported in literature for Ar, Ne and N 2 agree with each other to within 1 % percent 1 1\% 1 % . The variation range of the reported α 𝛼 \alpha italic_α -values has a negligible effect on the uncertainties of line shifts and broadenings relevant in our work.

The s 𝑠 s italic_s -wave scattering length is typically estimated by extrapolating experimentally measured electron collision cross-sections to zero energy electrons using modified effective range theory (MERT)  [ 39 , 40 ] . Experimentally reported values of a s subscript 𝑎 𝑠 a_{s} italic_a start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT have up to 20 ⁢ % 20 % 20\,\text{\%} 20 % of variation depending on the collision experiments, e.g. , crossed-beam, swarm and time-of-flight measurements, among others (see topical reviews in   [ 41 , 42 , 43 ] ), as well as specifics of the MERT model. As an example, in Table  1 we show the variation range of experimental values reported for a s subscript 𝑎 𝑠 a_{s} italic_a start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT of Ar. In Table  2 we provide the upper and lower bounds of a s subscript 𝑎 𝑠 a_{s} italic_a start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT -values reported for Ar, Ne and N 2 that we have found. The latter are used for the uncertainty ranges of the calculated shaded lines in Figs. 3 and 4.

After accounting for shifts of the ground state and the intermediate state in the utilized EIT energy-level cascade and assuming that the probe laser in on resonance, the final shift of the EIT resonance is given by  [ 6 ]

subscript Δ 𝑠 𝑐 subscript Δ 𝑝 \Delta_{r}=\Delta_{sc}+\Delta_{p} roman_Δ start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT = roman_Δ start_POSTSUBSCRIPT italic_s italic_c end_POSTSUBSCRIPT + roman_Δ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT , with different scaling factors. Notably, collisional shifts of the D 2 subscript 𝐷 2 D_{2} italic_D start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT line, as commonly reported in literature  [ 44 , 29 , 45 ] , only provide Δ e − Δ g subscript Δ 𝑒 subscript Δ 𝑔 \Delta_{e}-\Delta_{g} roman_Δ start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT - roman_Δ start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT , which is not sufficient to evaluate Eq.  3 . Under the assumption that the collisional shift of the ground state 5 ⁢ S 1 / 2 5 subscript 𝑆 1 2 5S_{1/2} 5 italic_S start_POSTSUBSCRIPT 1 / 2 end_POSTSUBSCRIPT is smaller than that of the excited electronic state 5 ⁢ P 3 / 2 5 subscript 𝑃 3 2 5P_{3/2} 5 italic_P start_POSTSUBSCRIPT 3 / 2 end_POSTSUBSCRIPT , one may conclude that Δ e ≈ Δ D 2 subscript Δ 𝑒 subscript Δ subscript 𝐷 2 \Delta_{e}\approx\Delta_{D_{2}} roman_Δ start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ≈ roman_Δ start_POSTSUBSCRIPT italic_D start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , the shift of the D 2 subscript 𝐷 2 D_{2} italic_D start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT line. The inert-gas-induced shift of the D 2 subscript 𝐷 2 D_{2} italic_D start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT line is ≈ \approx ≈ 2 ⁢ MHz/Torr 2 MHz/Torr 2\,\text{MHz/Torr} 2 MHz/Torr for Ne  [ 45 ] and ≈ \approx ≈ 6 ⁢ MHz/Torr 6 MHz/Torr 6\,\text{MHz/Torr} 6 MHz/Torr for Ar and N 2   [ 44 ] . The line broadening rates are ≈ \approx ≈ 10 ⁢ MHz/Torr 10 MHz/Torr 10\,\text{MHz/Torr} 10 MHz/Torr for Ne and ≈ \approx ≈ 20 ⁢ MHz/Torr 20 MHz/Torr 20\,\text{MHz/Torr} 20 MHz/Torr for Ar and N 2   [ 29 , 44 , 45 ] .These shifts and broadenings are a factor of three or more smaller than the respective Rydberg energy shifts and broadenings calculated from Eq.  2 . Therefore, we neglect the inert-gas effects on the D 2 subscript 𝐷 2 D_{2} italic_D start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT line at our current level of experimental precision. With this assumption, we attribute the Rydberg-EIT line shift and broadening solely to the Rydberg state, i.e. , Δ ≈ Δ r Δ subscript Δ 𝑟 \Delta\approx\Delta_{r} roman_Δ ≈ roman_Δ start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT and γ ≈ γ r 𝛾 subscript 𝛾 𝑟 \gamma\approx\gamma_{r} italic_γ ≈ italic_γ start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT .

subscript Δ 𝑝 subscript Δ 𝑠 𝑐 \Delta_{r}=\Delta_{p}+\Delta_{sc} roman_Δ start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT = roman_Δ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT + roman_Δ start_POSTSUBSCRIPT italic_s italic_c end_POSTSUBSCRIPT , as well as corresponding broadenings, for the inert gases studied in our work, in units MHz/Torr at 303 K. Eq.  2 reveals the linear dependence of line shift and broadening on inert gas pressure, which is an important result of the semi-classical model. We find in Fig.  3 that frequency shift scales linearly with inert-gas pressures and agrees with estimates from Eq.  2 within the uncertainty of our measurement.

V Discussion

The presented semi-classical model rests on the validity of impact approximation and the assumption that the Rydberg electron and the Rydberg-atom ionic core are decoupled in their respective interaction with the perturbers. We emphasize that both of these assumptions are applicable in our experimental regime of highly excited Rydberg levels and low inert-gas pressure, as discussed below.

Semi-classical treatments of the pseudo-potential model   [ 69 , 70 ] have established that the broadening cross-section reaches an asymptotic value for effective quantum numbers exceeding n max ≈ | a s | 1 / 3 / ( α ⁢ v 5 ) 1 / 18 subscript 𝑛 max superscript subscript 𝑎 𝑠 1 3 superscript 𝛼 superscript 𝑣 5 1 18 n_{\text{max}}\approx|a_{s}|^{1/3}/(\alpha v^{5})^{1/18} italic_n start_POSTSUBSCRIPT max end_POSTSUBSCRIPT ≈ | italic_a start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT 1 / 3 end_POSTSUPERSCRIPT / ( italic_α italic_v start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 1 / 18 end_POSTSUPERSCRIPT ( a s subscript 𝑎 𝑠 a_{s} italic_a start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT , α 𝛼 \alpha italic_α and v 𝑣 v italic_v in atomic units). The broadening cross-section at higher Rydberg levels arises from the polarization potential of the ionic core, whereas the effect of the Rydberg electron is limited to the scattering described by the Fermi interaction. For the inert gases studied in our work, it is n max ≈ 10 subscript 𝑛 max 10 n_{\text{max}}\approx 10 italic_n start_POSTSUBSCRIPT max end_POSTSUBSCRIPT ≈ 10 , whereas our experiment is performed with 36 ⁢ D J 36 subscript 𝐷 𝐽 36D_{J} 36 italic_D start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT states. Therefore, the use of Eq.  1 is well-justified, i.e. the components from long-range polarization potential and electron scattering can be treated separately.

Next, we consider the validity of the impact approximation, which holds when the timescale of collisions is much faster than the mean-free time, or, equivalently, when the mean number of perturbers occupying the interaction volume is much smaller than unity. This limits the perturber density to N ≪ 3 ⁢ v / ( π 2 ⁢ α ) much-less-than 𝑁 3 𝑣 superscript 𝜋 2 𝛼 N\ll 3v/(\pi^{2}\alpha) italic_N ≪ 3 italic_v / ( italic_π start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_α ) (in atomic units)   [ 69 ] , corresponding to ≈ \approx ≈ 100 ⁢ Torr 100 Torr 100\,\text{Torr} 100 Torr of inert-gas pressure. This is considerably larger than the highest pressures utilized in our experiments.

The validity of the semi-classical treatment is highlighted by the observed linear scaling of the line shift and broadening with inert-gas pressure at sufficiently high inert-gas pressure. Other approximations that are implicit in Eq.  2 include the s 𝑠 s italic_s -wave treatment of the Fermi interaction and the assumption of negligible inelastic scattering. While the former is generally justified for low-energy Rydberg electrons, the latter is more suitable for the interaction of atomic perturbers like Ne and Ar with Rydberg atoms, but less so for molecules. Non-polar molecular perturbers like N 2 possess a quadrupole moment and a rich rovibrational structure, which can give rise to several inelastic scattering processes  [ 71 ] . The study of such additional cross-sections in the context of Rydberg-EIT is beyond the scope of our present paper.

Our results have important implications for future applications of Rydberg-EIT to plasma field diagnostics. The gases studied in our experiment are common choices for low-pressure plasma discharges, including inductively coupled and dc plasmas. Rf discharges in gases at pressures above 1 ⁢ mTorr 1 mTorr 1\,\text{mTorr} 1 mTorr rely on power absorption through collisional heating. This mechanism is most efficient at gas pressures at which the effective rate of collisions between electrons and neutral atoms is close to the angular frequency of the rf power applied, i.e. , ν e ⁢ n ≈ 2 ⁢ π ⁢ f rf subscript 𝜈 𝑒 𝑛 2 𝜋 subscript 𝑓 rf \nu_{en}\approx 2\pi f_{\text{rf}} italic_ν start_POSTSUBSCRIPT italic_e italic_n end_POSTSUBSCRIPT ≈ 2 italic_π italic_f start_POSTSUBSCRIPT rf end_POSTSUBSCRIPT . For instance, Ar offers efficient rf power absorption at the standard frequency of 13.56 ⁢ MHz 13.56 MHz 13.56\,\text{MHz} 13.56 MHz over a range of pressures from 10 ⁢ mTorr 10 mTorr 10\,\text{mTorr} 10 mTorr to 100 ⁢ mTorr 100 mTorr 100\,\text{mTorr} 100 mTorr due to favorable values of ν e ⁢ n subscript 𝜈 𝑒 𝑛 \nu_{en} italic_ν start_POSTSUBSCRIPT italic_e italic_n end_POSTSUBSCRIPT   [ 72 , 73 ] . In this regime, the gas-induced EIT broadening is negligible in comparison to typical EIT linewidths, as shown in Fig.  3 , and the estimated line shift is less than 20 ⁢ MHz 20 MHz 20\,\text{MHz} 20 MHz for Ar and 2 ⁢ MHz 2 MHz 2\,\text{MHz} 2 MHz for Ne and N 2 . This indicates the viability of Rydberg-EIT as a diagnostic tool for electric fields in low-temperature rf plasma generated using the gases utilized in our study.

VI Conclusion

We have demonstrated that Rydberg-EIT presents a viable spectroscopic method in the presence of the commonly used inert gases Ar, Ne and N 2 at pressures up to about 5 ⁢ Torr 5 Torr 5\,\text{Torr} 5 Torr . Estimates based on the semi-classical pseudo-potential model of interactions between a Rydberg atom and an inert-gas atom agree well with our experimentally observed EIT line shifts and broadenings, within the achieved uncertainty. In the sub- 100 ⁢ mTorr 100 mTorr 100\,\text{mTorr} 100 mTorr pressure regime, inert-gas-induced broadening is negligible compared to the typical linewidth of our Rydberg-EIT signals. This pressure range is frequently used for low-temperature rf plasma discharges. Considering EIT shift, broadening and signal strength, we believe that Ar will be a particularly good choice in future plasma-physics applications of Rydberg-EIT.

Acknowledgments

We thank Dr. Ryan Cardman for early work on this experiment and Dr. David A. Anderson from Rydberg Technologies Inc. for valuable discussions. 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 Fundamental Fund 2023, Chiang Mai University. A.D. acknowledges support from the Rackham Predoctoral Fellowship at the University of Michigan.

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