arXiv:2506.04057 · Physical Review A

Precision measurement of Cs ( n F J ) quantum defects and calculations of scalar and tensor polarizabilities of the n S 1 / 2 , n P J , n D J , and n F J series

Pinrui Shen, Mariusz Pawlak, Donald W. Booth, Kent A. Nickerson, Haddad Miladi, H. R. Sadeghpour, James P. Shaffer

Published Jul 25, 2025• 3 citations

Abstract

In this paper, we extend our recent work on cesium $S$ and $D$ states [P. Shen et al., Phys. Rev. Lett. 133, 233005 (2024)] to the $F$ states. We present absolute frequency measurements of the $|6{S}_{1/2},F=3\ensuremath{\rangle}\ensuremath{\rightarrow}n{F}_{5/2,7/2}(n=28--68)$ Rydberg series to measure the spectrum of $^{133}\mathrm{Cs}$. Atomic spectra are obtained using a three-photon excitation scheme referenced to an optical frequency comb in a sample of ultracold $^{133}\mathrm{Cs}$. By globally fitting the absolute-frequency measurements to the modified Ritz formula, we determine the quantum defects of the $n{F}_{5/2}$ and $n{F}_{7/2}$ series. The ionization potential extracted for both series from the modified Ritz formula agrees with our measurements based on the $S$ and $D$ series. Fine-structure intervals are calculated and parameterized. The wave functions computed for the energies from the quantum defects are used to calculate transition dipole moments. We compare the reduced electric-dipole matrix elements with available benchmarks and find agreement within the precision of those works. The scalar and tensor polarizabilities of the $n{S}_{1/2}$, $n{P}_{J}$, $n{D}_{J}$, and $n{F}_{J}$ series are calculated based on the now more accurate set of wave functions. Moreover, we report the polarizability as a series in powers of the effective principal quantum number and find the main coefficients of the expansion. The results will be useful for calculating properties of $^{133}\mathrm{Cs}$ such as collision and decay rates, polarizabilities, and magic wavelengths.

DOI: 10.1103/dkb4-kb5b

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arXiv:2506.04057 · Physical Review A

Precision measurement of Cs ( n F J ) quantum defects and calculations of scalar and tensor polarizabilities of the n S 1 / 2 , n P J , n D J , and n F J series

Pinrui Shen, Mariusz Pawlak, Donald W. Booth, Kent A. Nickerson, Haddad Miladi, H. R. Sadeghpour, James P. Shaffer

Published Jul 25, 2025• 3 citations

arXiv HTML PDF

Abstract

In this paper, we extend our recent work on cesium

S

and

D

states [P. Shen et al., Phys. Rev. Lett. 133, 233005 (2024)] to the

F

states. We present absolute frequency measurements of the

|6{S}_{1/2},F=3\ensuremath{\rangle}\ensuremath{\rightarrow}n{F}_{5/2,7/2}(n=28--68)

Rydberg series to measure the spectrum of

^{133}\mathrm{Cs}

. Atomic spectra are obtained using a three-photon excitation scheme referenced to an optical frequency comb in a sample of ultracold

^{133}\mathrm{Cs}

. By globally fitting the absolute-frequency measurements to the modified Ritz formula, we determine the quantum defects of the

n{F}_{5/2}

and

n{F}_{7/2}

series. The ionization potential extracted for both series from the modified Ritz formula agrees with our measurements based on the

S

and

D

series. Fine-structure intervals are calculated and parameterized. The wave functions computed for the energies from the quantum defects are used to calculate transition dipole moments. We compare the reduced electric-dipole matrix elements with available benchmarks and find agreement within the precision of those works. The scalar and tensor polarizabilities of the

n{S}_{1/2}

n{P}_{J}

n{D}_{J}

, and

n{F}_{J}

series are calculated based on the now more accurate set of wave functions. Moreover, we report the polarizability as a series in powers of the effective principal quantum number and find the main coefficients of the expansion. The results will be useful for calculating properties of

^{133}\mathrm{Cs}

such as collision and decay rates, polarizabilities, and magic wavelengths.

DOI: 10.1103/dkb4-kb5b

Related papers

Precision measurement of the ionization energy and quantum defects of K39 i

Michael Peper, Felix Helmrich, Jonas Butscher +4 more

We present absolute-frequency measurements in ultracold $^{39}\mathrm{K}$ samples of the transitions from the $4{s}_{1/2}$ ground state to $n{p}_{1/2}$ and $n{p}_{3/2}$ Rydberg states. A global nonlinear regression of the $n{p}_{1/2}$ and $n{p}_{3/2}$ term values yields an improved wave number of 35009.8139710(22)${}_{\mathrm{sys}}{(3)}_{\mathrm{stat}}\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1}$ for the first ionization threshold of $^{39}\mathrm{K}$ and the quantum defects of the $n{p}_{1/2}$ and $n{p}_{3/2}$ series. In addition, we report the frequencies of selected one-photon transitions ${n}^{\ensuremath{'}}{s}_{1/2}\ensuremath{\leftarrow}n{p}_{3/2}, {n}^{\ensuremath{'}}{d}_{j}\ensuremath{\leftarrow}n{p}_{3/2}, {n}^{\ensuremath{'}}{f}_{{j}^{\ensuremath{'}}}\ensuremath{\leftarrow}n{d}_{j}$, and ${n}^{\ensuremath{'}}{g}_{{j}^{\ensuremath{'}}}\ensuremath{\leftarrow}n{f}_{j}$ and two-photon transitions $n{f}_{{j}^{\ensuremath{'}}}\ensuremath{\leftarrow}n{p}_{j}$ determined by millimeter-wave spectroscopy, where $j$ is the total angular-momentum quantum number. By combining the results from the laser and millimeter-wave spectroscopic experiments, we obtain improved values for the quantum defects of the ${s}_{1/2}, {d}_{3/2}, {d}_{5/2}, {f}_{j}$ , and ${g}_{j}$ states. For the ${d}_{j}$ series, the inverted fine structure was confirmed for $n\ensuremath{\ge}32$. The fine-structure splitting of the $f$ series is less than 100 kHz at $n=31$ , significantly smaller than the hydrogenic splitting, and the fine structure of the $g$ series is regular for $n\ensuremath{\ge}30$, with a fine-structure splitting compatible with the hydrogenic prediction. From the measured quantum defects of the $f$ and $g$ series we derive an estimate for the static dipole ${\ensuremath{\alpha}}_{\mathrm{d}}$ and quadrupole ${\ensuremath{\alpha}}_{\mathrm{q}}$ polarizabilities of the ${\mathrm{K}}^{+}$ ion core. Additionally, the hyperfine splitting of the $4{s}_{1/2}$ ground state of $^{39}\mathrm{K}$ was determined to be 461.719700(5) MHz using radio-frequency spectroscopy and Ramsey-type interferometry.We present absolute-frequency measurements in ultracold $^{39}\mathrm{K}$ samples of the transitions from the $4{s}_{1/2}$ ground state to $n{p}_{1/2}$ and $n{p}_{3/2}$ Rydberg states. A global nonlinear regression of the $n{p}_{1/2}$ and $n{p}_{3/2}$ term values yields an improved wave number of 35009.8139710(22)${}_{\mathrm{sys}}{(3)}_{\mathrm{stat}}\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1}$ for the first ionization threshold of $^{39}\mathrm{K}$ and the quantum defects of the $n{p}_{1/2}$ and $n{p}_{3/2}$ series. In addition, we report the frequencies of selected one-photon transitions ${n}^{\ensuremath{'}}{s}_{1/2}\ensuremath{\leftarrow}n{p}_{3/2}, {n}^{\ensuremath{'}}{d}_{j}\ensuremath{\leftarrow}n{p}_{3/2}, {n}^{\ensuremath{'}}{f}_{{j}^{\ensuremath{'}}}\ensuremath{\leftarrow}n{d}_{j}$, and ${n}^{\ensuremath{'}}{g}_{{j}^{\ensuremath{'}}}\ensuremath{\leftarrow}n{f}_{j}$ and two-photon transitions $n{f}_{{j}^{\ensuremath{'}}}\ensuremath{\leftarrow}n{p}_{j}$ determined by millimeter-wave spectroscopy, where $j$ is the total angular-momentum quantum number. By combining the results from the laser and millimeter-wave spectroscopic experiments, we obtain improved values for the quantum defects of the ${s}_{1/2}, {d}_{3/2}, {d}_{5/2}, {f}_{j}$, and ${g}_{j}$ states. For the ${d}_{j}$ series, the inverted fine structure was confirmed for $n\ensuremath{\ge}32$. The fine-structure splitting of the $f$ series is less than 100 kHz at $n=31$, significantly smaller than the hydrogenic splitting, and the fine structure of the $g$ series is regular for $n\ensuremath{\ge}30$, with a fine-structure splitting compatible with the hydrogenic prediction. From the measured quantum defects of the $f$ and $g$ series we derive an estimate for the static dipole ${\ensuremath{\alpha}}_{\mathrm{d}}$ and quadrupole ${\ensuremath{\alpha}}_{\mathrm{q}}$ polarizabilities of the ${\mathrm{K}}^{+}$ ion core. Additionally, the hyperfine splitting of the $4{s}_{1/2}$ ground state of $^{39}\mathrm{K}$ was determined to be 461.719700(5) MHz using radio-frequency spectroscopy and Ramsey-type interferometry.

arXiv:1907.02776Physical Review AJul 2, 201927 citations

Quantum defects of n F J levels of Cs Rydberg atoms

Jingxu Bai, Rong Song, Jiabei Fan +4 more

We present precise measurements of the quantum defects of cesium $n{F}_{J}$ Rydberg levels. We employ high-precision microwave spectroscopy of $(n+2){D}_{5/2}\ensuremath{\rightarrow}n{F}_{5/2,7/2}$ transitions for $n=45$ to 50 in a cold-atom setup. Cold cesium $(n+2){D}_{5/2}$ atoms, prepared via two-photon laser excitation, are probed by scanning weak microwave fields interacting with the atoms across the $n{F}_{5/2,7/2}$ resonances. Transition spectra are acquired using state-selective electric-field ionization and time-gated ion detection. Transition-frequency intervals are obtained by Lorentzian fits to the measured spectral lines, which have linewidths ranging between 70 and 190 kHz, corresponding to about one to three times the Fourier limit. A comprehensive analysis of relevant line-shift uncertainties and line-broadening effects is conducted. We find quantum defect parameters ${\ensuremath{\delta}}_{0}({F}_{5/2})=0.033\phantom{\rule{0.16em}{0ex}}415\phantom{\rule{0.16em}{0ex}}37(70)$ and ${\ensuremath{\delta}}_{2}({F}_{5/2})=\ensuremath{-}0.2014(16)$, as well as ${\ensuremath{\delta}}_{0}({F}_{7/2})=0.033\phantom{\rule{0.16em}{0ex}}564\phantom{\rule{0.16em}{0ex}}6(13)$ and ${\ensuremath{\delta}}_{2}({F}_{7/2})=\ensuremath{-}0.2052(29)$, for $J=5/2$ and $7/2$ , respectively. Fine-structure parameters ${A}_{\mathrm{FS}}$ and ${B}_{\mathrm{FS}}$ for Cs $n{F}_{J}$ are also obtained. Results are discussed in context with previous works, and the significance of the results is discussed.We present precise measurements of the quantum defects of cesium $n{F}_{J}$ Rydberg levels. We employ high-precision microwave spectroscopy of $(n+2){D}_{5/2}\ensuremath{\rightarrow}n{F}_{5/2,7/2}$ transitions for $n=45$ to 50 in a cold-atom setup. Cold cesium $(n+2){D}_{5/2}$ atoms, prepared via two-photon laser excitation, are probed by scanning weak microwave fields interacting with the atoms across the $n{F}_{5/2,7/2}$ resonances. Transition spectra are acquired using state-selective electric-field ionization and time-gated ion detection. Transition-frequency intervals are obtained by Lorentzian fits to the measured spectral lines, which have linewidths ranging between 70 and 190 kHz, corresponding to about one to three times the Fourier limit. A comprehensive analysis of relevant line-shift uncertainties and line-broadening effects is conducted. We find quantum defect parameters ${\ensuremath{\delta}}_{0}({F}_{5/2})=0.033\phantom{\rule{0.16em}{0ex}}415\phantom{\rule{0.16em}{0ex}}37(70)$ and ${\ensuremath{\delta}}_{2}({F}_{5/2})=\ensuremath{-}0.2014(16)$, as well as ${\ensuremath{\delta}}_{0}({F}_{7/2})=0.033\phantom{\rule{0.16em}{0ex}}564\phantom{\rule{0.16em}{0ex}}6(13)$ and ${\ensuremath{\delta}}_{2}({F}_{7/2})=\ensuremath{-}0.2052(29)$, for $J=5/2$ and $7/2$, respectively. Fine-structure parameters ${A}_{\mathrm{FS}}$ and ${B}_{\mathrm{FS}}$ for Cs $n{F}_{J}$ are also obtained. Results are discussed in context with previous works, and the significance of the results is discussed.

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Magic wavelengths for the State-insensitive Optical Trapping of Cs Atoms: case studies using the linearly and circularly polarized lights

Sukhjit Singh Sehra, Kiranpreet Kaur, B. K. Sahoo +1 more

We demonstrate magic wavelengths for the $6s ~ {^2}S_{1/2}-6p ~ {^2}P_{3/2,1/2}$ transitions in the Cs atom due to the circularly polarized light. In addition, we also obtain magic wavelengths using the linearly polarized light in order to verify the previously reported values and make a comparative study with the values obtained for the circularly polarized light. Most of these wavelengths are found to be in the optical region and could be of immense interest to the experimentalists for carrying out high precision measurements. To obtain these wavelengths, we have calculated dynamic dipole polarizabilities of the ground, $6p ~{^2}P_{1/2}$ and $6p ~{^2}P_{3/2}$ states. We use the available precise values of the electric dipole (E1) matrix elements for the most dominantly contributing transitions from the lifetime measurements of the above excited states and obtain other relevant important E1 matrix elements by employing a relativistic coupled-cluster method approximating at the singles and doubles level. Accuracies in these quantities are substantiated by comparing our evaluated scalar polarizability values with the corresponding experimental results.We demonstrate magic wavelengths for the $6s ~ {^2}S_{1/2}-6p ~ {^2}P_{3/2,1/2}$ transitions in the Cs atom due to the circularly polarized light. In addition, we also obtain magic wavelengths using the linearly polarized light in order to verify the previously reported values and make a comparative study with the values obtained for the circularly polarized light. Most of these wavelengths are found to be in the optical region and could be of immense interest to the experimentalists for carrying out high precision measurements. To obtain these wavelengths, we have calculated dynamic dipole polarizabilities of the ground, $6p ~{^2}P_{1/2}$ and $6p ~{^2}P_{3/2}$ states. We use the available precise values of the electric dipole (E1) matrix elements for the most dominantly contributing transitions from the lifetime measurements of the above excited states and obtain other relevant important E1 matrix elements by employing a relativistic coupled-cluster method approximating at the singles and doubles level. Accuracies in these quantities are substantiated by comparing our evaluated scalar polarizability values with the corresponding experimental results.

Jan 4, 20160 citations

Measurement of the strontium triplet Rydberg series by depletion spectroscopy of ultracold atoms

Luc Couturier, Ingo Nosske, Fachao Hu +5 more

We report on the atom loss spectroscopy of strontium Rydberg atoms in a magnetooptical trap, using a two-photon excitation scheme through the intermediate state $5s5p{\phantom{\rule{0.16em}{0ex}}}^{3}{P}_{1}$ . Energies of the $5sns{\phantom{\rule{0.16em}{0ex}}}^{3}{S}_{1}$ and $5snd{\phantom{\rule{0.16em}{0ex}}}^{3}{D}_{1,2}$ Rydberg series of $^{88}\mathrm{Sr}$ in the range $13\ensuremath{\le}n\ensuremath{\le}50$ are determined with an absolute accuracy of 10 MHz, including the perturbed region where the $5snd{\phantom{\rule{0.16em}{0ex}}}^{3}{D}_{2}$ series couples to the $5snd{\phantom{\rule{0.16em}{0ex}}}^{1}{D}_{2}$ series. This represents an improvement by more than two orders of magnitude compared to previously published data. The quantum defects for each series are determined using the extended Rydberg-Ritz formula in the range where there is no strong perturbation. A value of $1\phantom{\rule{0.16em}{0ex}}377\phantom{\rule{0.16em}{0ex}}012\phantom{\rule{0.16em}{0ex}}721(10)\phantom{\rule{0.28em}{0ex}}\text{MHz}$ for the first ionization limit of $^{88}\mathrm{Sr}$ is extracted.We report on the atom loss spectroscopy of strontium Rydberg atoms in a magnetooptical trap, using a two-photon excitation scheme through the intermediate state $5s5p{\phantom{\rule{0.16em}{0ex}}}^{3}{P}_{1}$. Energies of the $5sns{\phantom{\rule{0.16em}{0ex}}}^{3}{S}_{1}$ and $5snd{\phantom{\rule{0.16em}{0ex}}}^{3}{D}_{1,2}$ Rydberg series of $^{88}\mathrm{Sr}$ in the range $13\ensuremath{\le}n\ensuremath{\le}50$ are determined with an absolute accuracy of 10 MHz, including the perturbed region where the $5snd{\phantom{\rule{0.16em}{0ex}}}^{3}{D}_{2}$ series couples to the $5snd{\phantom{\rule{0.16em}{0ex}}}^{1}{D}_{2}$ series. This represents an improvement by more than two orders of magnitude compared to previously published data. The quantum defects for each series are determined using the extended Rydberg-Ritz formula in the range where there is no strong perturbation. A value of $1\phantom{\rule{0.16em}{0ex}}377\phantom{\rule{0.16em}{0ex}}012\phantom{\rule{0.16em}{0ex}}721(10)\phantom{\rule{0.28em}{0ex}}\text{MHz}$ for the first ionization limit of $^{88}\mathrm{Sr}$ is extracted.

arXiv:1810.07611Physical Review AFeb 4, 201921 citations

Electric dipole polarizabilities of Rydberg states of alkali-metal atoms

Vladimir A. Yerokhin, Stefan Yoshi Buhmann, S. Fritzsche +1 more

Calculations of the static electric-dipole scalar and tensor polarizabilities are presented for two alkali-metal atoms, Rb and Cs, for the $nS, n{P}_{\textonehalf{},\phantom{\rule{0.16em}{0ex}}3/2}$, and $n{D}_{3/2,\phantom{\rule{0.16em}{0ex}}5/2}$ states with large principal quantum numbers up to $n=50$ . The calculations are performed within an effective one-electron approximation, based on the Dirac-Fock Hamiltonian with a semiempirical core-polarization potential. The obtained results are compared with those from a simpler semiempirical approach and with available experimental data.Calculations of the static electric-dipole scalar and tensor polarizabilities are presented for two alkali-metal atoms, Rb and Cs, for the $nS, n{P}_{\textonehalf{},\phantom{\rule{0.16em}{0ex}}3/2}$, and $n{D}_{3/2,\phantom{\rule{0.16em}{0ex}}5/2}$ states with large principal quantum numbers up to $n=50$. The calculations are performed within an effective one-electron approximation, based on the Dirac-Fock Hamiltonian with a semiempirical core-polarization potential. The obtained results are compared with those from a simpler semiempirical approach and with available experimental data.

arXiv:1608.04515Physical Review ASep 6, 201631 citations

Electric dipole matrix elements for the 6p2PJ→7s2S1/2 transition in atomic cesium

George Toh, Amy Damitz, Nathan Glotzbach +5 more

We report a measurement of the ratio of electric dipole transition matrix elements of cesium for the $6p{\phantom{\rule{0.16em}{0ex}}}^{2}{P}_{1/2}\ensuremath{\rightarrow}7s{\phantom{\rule{0.16em}{0ex}}}^{2}{S}_{1/2}$ and $6p{\phantom{\rule{0.16em}{0ex}}}^{2}{P}_{3/2}\ensuremath{\rightarrow}7s{\phantom{\rule{0.16em}{0ex}}}^{2}{S}_{1/2}$ transitions. We determine this ratio of matrix elements through comparisons of two-color, two-photon excitation rates of the $7s{\phantom{\rule{0.16em}{0ex}}}^{2}{S}_{1/2}$ state using laser beams with polarizations parallel to one another vs perpendicular to one another. Our result of $R\ensuremath{\equiv}\ensuremath{\langle}7s{\phantom{\rule{4pt}{0ex}}}^{2}{S}_{1/2}||r||6p{\phantom{\rule{4pt}{0ex}}}^{2}{P}_{3/2}\ensuremath{\rangle}/\ensuremath{\langle}7s{\phantom{\rule{4pt}{0ex}}}^{2}{S}_{1/2}||r||6p{\phantom{\rule{4pt}{0ex}}}^{2}{P}_{1/2}\ensuremath{\rangle}=1.5272(17)$ is in excellent agreement with a theoretical prediction of $R=1.5270(27)$ . Moreover, the accuracy of the experimental ratio is sufficiently high to differentiate between various theoretical approaches. To our knowledge, there are no prior experimental measurements of $R$ . Combined with our recent determination of the lifetime of the $7s{\phantom{\rule{0.16em}{0ex}}}^{2}{S}_{1/2}$ state, we determine reduced matrix elements for these two transitions, $\ensuremath{\langle}7s{\phantom{\rule{4pt}{0ex}}}^{2}{S}_{1/2}||r||6p{\phantom{\rule{4pt}{0ex}}}^{2}{P}_{3/2}\ensuremath{\rangle}=\ensuremath{-}6.489(5)\phantom{\rule{4pt}{0ex}}{a}_{0}$ and $\ensuremath{\langle}7s{\phantom{\rule{4pt}{0ex}}}^{2}{S}_{1/2}||r||6p{\phantom{\rule{4pt}{0ex}}}^{2}{P}_{1/2}\ensuremath{\rangle}=\ensuremath{-}4.249(4)\phantom{\rule{4pt}{0ex}}{a}_{0}$. These matrix elements are also in excellent agreement with theoretical calculations. These measurements improve knowledge of Cs properties needed for parity violation studies and provide benchmarks for tests of high-precision theory.We report a measurement of the ratio of electric dipole transition matrix elements of cesium for the $6p{\phantom{\rule{0.16em}{0ex}}}^{2}{P}_{1/2}\ensuremath{\rightarrow}7s{\phantom{\rule{0.16em}{0ex}}}^{2}{S}_{1/2}$ and $6p{\phantom{\rule{0.16em}{0ex}}}^{2}{P}_{3/2}\ensuremath{\rightarrow}7s{\phantom{\rule{0.16em}{0ex}}}^{2}{S}_{1/2}$ transitions. We determine this ratio of matrix elements through comparisons of two-color, two-photon excitation rates of the $7s{\phantom{\rule{0.16em}{0ex}}}^{2}{S}_{1/2}$ state using laser beams with polarizations parallel to one another vs perpendicular to one another. Our result of $R\ensuremath{\equiv}\ensuremath{\langle}7s{\phantom{\rule{4pt}{0ex}}}^{2}{S}_{1/2}||r||6p{\phantom{\rule{4pt}{0ex}}}^{2}{P}_{3/2}\ensuremath{\rangle}/\ensuremath{\langle}7s{\phantom{\rule{4pt}{0ex}}}^{2}{S}_{1/2}||r||6p{\phantom{\rule{4pt}{0ex}}}^{2}{P}_{1/2}\ensuremath{\rangle}=1.5272(17)$ is in excellent agreement with a theoretical prediction of $R=1.5270(27)$. Moreover, the accuracy of the experimental ratio is sufficiently high to differentiate between various theoretical approaches. To our knowledge, there are no prior experimental measurements of $R$. Combined with our recent determination of the lifetime of the $7s{\phantom{\rule{0.16em}{0ex}}}^{2}{S}_{1/2}$ state, we determine reduced matrix elements for these two transitions, $\ensuremath{\langle}7s{\phantom{\rule{4pt}{0ex}}}^{2}{S}_{1/2}||r||6p{\phantom{\rule{4pt}{0ex}}}^{2}{P}_{3/2}\ensuremath{\rangle}=\ensuremath{-}6.489(5)\phantom{\rule{4pt}{0ex}}{a}_{0}$ and $\ensuremath{\langle}7s{\phantom{\rule{4pt}{0ex}}}^{2}{S}_{1/2}||r||6p{\phantom{\rule{4pt}{0ex}}}^{2}{P}_{1/2}\ensuremath{\rangle}=\ensuremath{-}4.249(4)\phantom{\rule{4pt}{0ex}}{a}_{0}$. These matrix elements are also in excellent agreement with theoretical calculations. These measurements improve knowledge of Cs properties needed for parity violation studies and provide benchmarks for tests of high-precision theory.

arXiv:1901.00480Physical Review AMar 5, 201922 citations

In this paper, we extend our recent work on cesium

S

and

D

states [P. Shen et al., Phys. Rev. Lett. 133, 233005 (2024)] to the

F

states. We present absolute frequency measurements of the

|6{S}_{1/2},F=3\ensuremath{\rangle}\ensuremath{\rightarrow}n{F}_{5/2,7/2}(n=28--68)

Rydberg series to measure the spectrum of

^{133}\mathrm{Cs}

. Atomic spectra are obtained using a three-photon excitation scheme referenced to an optical frequency comb in a sample of ultracold

^{133}\mathrm{Cs}

. By globally fitting the absolute-frequency measurements to the modified Ritz formula, we determine the quantum defects of the

n{F}_{5/2}

and

n{F}_{7/2}

series. The ionization potential extracted for both series from the modified Ritz formula agrees with our measurements based on the

S

and

D

n{S}_{1/2}

n{P}_{J}

n{D}_{J}

, and

n{F}_{J}

^{133}\mathrm{Cs}

such as collision and decay rates, polarizabilities, and magic wavelengths.

We present absolute-frequency measurements in ultracold

^{39}\mathrm{K}

samples of the transitions from the

4{s}_{1/2}

ground state to

n{p}_{1/2}

and

n{p}_{3/2}

Rydberg states. A global nonlinear regression of the

n{p}_{1/2}

and

n{p}_{3/2}

term values yields an improved wave number of 35009.8139710(22)${}_{\mathrm{sys}}{(3)}_{\mathrm{stat}}\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1}$ for the first ionization threshold of

^{39}\mathrm{K}

and the quantum defects of the

n{p}_{1/2}

and

n{p}_{3/2}

series. In addition, we report the frequencies of selected one-photon transitions ${n}^{\ensuremath{'}}{s}_{1/2}\ensuremath{\leftarrow}n{p}_{3/2}, {n}^{\ensuremath{'}}{d}_{j}\ensuremath{\leftarrow}n{p}_{3/2}, {n}^{\ensuremath{'}}{f}_{{j}^{\ensuremath{'}}}\ensuremath{\leftarrow}n{d}_{j}$, and ${n}^{\ensuremath{'}}{g}_{{j}^{\ensuremath{'}}}\ensuremath{\leftarrow}n{f}_{j}$ and two-photon transitions $n{f}_{{j}^{\ensuremath{'}}}\ensuremath{\leftarrow}n{p}_{j}$ determined by millimeter-wave spectroscopy, where

j

is the total angular-momentum quantum number. By combining the results from the laser and millimeter-wave spectroscopic experiments, we obtain improved values for the quantum defects of the

{s}_{1/2}, {d}_{3/2}, {d}_{5/2}, {f}_{j}

, and

{g}_{j}

states. For the

{d}_{j}

series, the inverted fine structure was confirmed for $n\ensuremath{\ge}32$. The fine-structure splitting of the

f

series is less than 100 kHz at

n=31

, significantly smaller than the hydrogenic splitting, and the fine structure of the

g

series is regular for $n\ensuremath{\ge}30$, with a fine-structure splitting compatible with the hydrogenic prediction. From the measured quantum defects of the

f

and

g

series we derive an estimate for the static dipole ${\ensuremath{\alpha}}_{\mathrm{d}}$ and quadrupole ${\ensuremath{\alpha}}_{\mathrm{q}}$ polarizabilities of the

{\mathrm{K}}^{+}

ion core. Additionally, the hyperfine splitting of the

4{s}_{1/2}

ground state of

^{39}\mathrm{K}

was determined to be 461.719700(5) MHz using radio-frequency spectroscopy and Ramsey-type interferometry.

We present precise measurements of the quantum defects of cesium

n{F}_{J}

Rydberg levels. We employ high-precision microwave spectroscopy of $(n+2){D}_{5/2}\ensuremath{\rightarrow}n{F}_{5/2,7/2}$ transitions for

n=45

to 50 in a cold-atom setup. Cold cesium

(n+2){D}_{5/2}

atoms, prepared via two-photon laser excitation, are probed by scanning weak microwave fields interacting with the atoms across the

n{F}_{5/2,7/2}

resonances. Transition spectra are acquired using state-selective electric-field ionization and time-gated ion detection. Transition-frequency intervals are obtained by Lorentzian fits to the measured spectral lines, which have linewidths ranging between 70 and 190 kHz, corresponding to about one to three times the Fourier limit. A comprehensive analysis of relevant line-shift uncertainties and line-broadening effects is conducted. We find quantum defect parameters ${\ensuremath{\delta}}_{0}({F}_{5/2})=0.033\phantom{\rule{0.16em}{0ex}}415\phantom{\rule{0.16em}{0ex}}37(70)$ and ${\ensuremath{\delta}}_{2}({F}_{5/2})=\ensuremath{-}0.2014(16)$, as well as ${\ensuremath{\delta}}_{0}({F}_{7/2})=0.033\phantom{\rule{0.16em}{0ex}}564\phantom{\rule{0.16em}{0ex}}6(13)$ and ${\ensuremath{\delta}}_{2}({F}_{7/2})=\ensuremath{-}0.2052(29)$, for

J=5/2

and

7/2

, respectively. Fine-structure parameters

{A}_{\mathrm{FS}}

and

{B}_{\mathrm{FS}}

for Cs

n{F}_{J}

are also obtained. Results are discussed in context with previous works, and the significance of the results is discussed.

We demonstrate magic wavelengths for the

6s ~ {^2}S_{1/2}-6p ~ {^2}P_{3/2,1/2}

transitions in the Cs atom due to the circularly polarized light. In addition, we also obtain magic wavelengths using the linearly polarized light in order to verify the previously reported values and make a comparative study with the values obtained for the circularly polarized light. Most of these wavelengths are found to be in the optical region and could be of immense interest to the experimentalists for carrying out high precision measurements. To obtain these wavelengths, we have calculated dynamic dipole polarizabilities of the ground,

6p ~{^2}P_{1/2}

and

6p ~{^2}P_{3/2}

states. We use the available precise values of the electric dipole (E1) matrix elements for the most dominantly contributing transitions from the lifetime measurements of the above excited states and obtain other relevant important E1 matrix elements by employing a relativistic coupled-cluster method approximating at the singles and doubles level. Accuracies in these quantities are substantiated by comparing our evaluated scalar polarizability values with the corresponding experimental results.

We report on the atom loss spectroscopy of strontium Rydberg atoms in a magnetooptical trap, using a two-photon excitation scheme through the intermediate state

5s5p{\phantom{\rule{0.16em}{0ex}}}^{3}{P}_{1}

. Energies of the

5sns{\phantom{\rule{0.16em}{0ex}}}^{3}{S}_{1}

and

5snd{\phantom{\rule{0.16em}{0ex}}}^{3}{D}_{1,2}

Rydberg series of

^{88}\mathrm{Sr}

in the range $13\ensuremath{\le}n\ensuremath{\le}50$ are determined with an absolute accuracy of 10 MHz, including the perturbed region where the

5snd{\phantom{\rule{0.16em}{0ex}}}^{3}{D}_{2}

series couples to the

5snd{\phantom{\rule{0.16em}{0ex}}}^{1}{D}_{2}

series. This represents an improvement by more than two orders of magnitude compared to previously published data. The quantum defects for each series are determined using the extended Rydberg-Ritz formula in the range where there is no strong perturbation. A value of

1\phantom{\rule{0.16em}{0ex}}377\phantom{\rule{0.16em}{0ex}}012\phantom{\rule{0.16em}{0ex}}721(10)\phantom{\rule{0.28em}{0ex}}\text{MHz}

for the first ionization limit of

^{88}\mathrm{Sr}

is extracted.

n{D}_{3/2,\phantom{\rule{0.16em}{0ex}}5/2}

states with large principal quantum numbers up to

n=50

. The calculations are performed within an effective one-electron approximation, based on the Dirac-Fock Hamiltonian with a semiempirical core-polarization potential. The obtained results are compared with those from a simpler semiempirical approach and with available experimental data.

7s{\phantom{\rule{0.16em}{0ex}}}^{2}{S}_{1/2}

state using laser beams with polarizations parallel to one another vs perpendicular to one another. Our result of $R\ensuremath{\equiv}\ensuremath{\langle}7s{\phantom{\rule{4pt}{0ex}}}^{2}{S}_{1/2}||r||6p{\phantom{\rule{4pt}{0ex}}}^{2}{P}_{3/2}\ensuremath{\rangle}/\ensuremath{\langle}7s{\phantom{\rule{4pt}{0ex}}}^{2}{S}_{1/2}||r||6p{\phantom{\rule{4pt}{0ex}}}^{2}{P}_{1/2}\ensuremath{\rangle}=1.5272(17)$ is in excellent agreement with a theoretical prediction of

R=1.5270(27)

. Moreover, the accuracy of the experimental ratio is sufficiently high to differentiate between various theoretical approaches. To our knowledge, there are no prior experimental measurements of

R

. Combined with our recent determination of the lifetime of the

7s{\phantom{\rule{0.16em}{0ex}}}^{2}{S}_{1/2}

state, we determine reduced matrix elements for these two transitions, $\ensuremath{\langle}7s{\phantom{\rule{4pt}{0ex}}}^{2}{S}_{1/2}||r||6p{\phantom{\rule{4pt}{0ex}}}^{2}{P}_{3/2}\ensuremath{\rangle}=\ensuremath{-}6.489(5)\phantom{\rule{4pt}{0ex}}{a}_{0}$ and $\ensuremath{\langle}7s{\phantom{\rule{4pt}{0ex}}}^{2}{S}_{1/2}||r||6p{\phantom{\rule{4pt}{0ex}}}^{2}{P}_{1/2}\ensuremath{\rangle}=\ensuremath{-}4.249(4)\phantom{\rule{4pt}{0ex}}{a}_{0}$. These matrix elements are also in excellent agreement with theoretical calculations. These measurements improve knowledge of Cs properties needed for parity violation studies and provide benchmarks for tests of high-precision theory.