Abstract

We demonstrate a single-photon Rydberg excitation spectroscopy of cesium (Cs) atoms in a room-temperature vapor cell. Cs atoms are excited directly from 6S1/2 ground state to nP3/2 (n = 70 - 100) Rydberg states with a 318.6 nm ultraviolet (UV) laser, and Rydberg excitation spectra are obtained by transmission enhancement of a probe beam resonant to Cs 6S1/2, F = 4 - 6P3/2, F’ = 5 transition as partial population on F = 4 ground state are transferred to Rydberg state. Analysis reveals that the observed spectra are velocity-selective spectroscopy of Rydberg state, from which the amplitude and linewidth influenced by lasers’ Rabi frequency have been investigated. Fitting to energies of Cs nP3/2 (n = 70 −100) states, the determined quantum defect is 3.56671(42). The demodulated spectra can also be employed as frequency references to stabilize the UV laser frequency to specific Cs Rydberg transition.

© 2017 Optical Society of America

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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  30. K. Singer, M. Reetz-Lamour, T. Amthor, L. G. Marcassa, and M. Weidemüller, “Suppression of excitation and spectral broadening induced by interactions in a cold gas of Rydberg atoms,” Phys. Rev. Lett. 93(16), 163001 (2004).
    [Crossref] [PubMed]
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2016 (4)

S. X. Bao, H. Zhang, J. Zhou, L. J. Zhang, J. M. Zhao, L. T. Xiao, and S. T. Jia, “Polarization spectra of Zeeman sublevels in Rydberg electromagnetically induced transparency,” Phys. Rev. A 94(4), 043822 (2016).
[Crossref]

Y. C. Jiao, X. X. Han, Z. W. Yang, J. K. Li, G. Raithel, J. M. Zhao, and S. T. Jia, “Spectroscopy of cesium Rydberg atoms in strong radio-frequency fields,” Phys. Rev. A 94(2), 023832 (2016).
[Crossref]

J. Y. Wang, J. D. Bai, J. He, and J. M. Wang, “Realization and characterization of single-frequency tunable 637.2 nm high-power laser,” Opt. Commun. 370, 150–155 (2016).
[Crossref]

J. Y. Wang, J. D. Bai, J. He, and J. M. Wang, “Development and characterization of a 2.2 W narrow-linewidth 318.6-nm ultraviolet laser,” J. Opt. Soc. Am. B 33(10), 2020–2025 (2016).
[Crossref]

2015 (1)

Y. Y. Jau, A. M. Hankin, T. Keating, I. H. Deutsch, and G. W. Biedermann, “Entangling atomic spins with a Rydberg-dressed spin-flip blockade,” Nat. Phys. 12(1), 71–74 (2015).
[Crossref]

2014 (1)

A. M. Hankin, Y. Y. Jau, L. P. Parazzoli, C. W. Chou, D. J. Armstrong, A. J. Landahl, and G. W. Biedermann, “Two-atom Rydberg blockade using direct 6S to nP excitation,” Phys. Rev. A 89(3), 033416 (2014).
[Crossref]

2013 (2)

T. Keating, K. Goyal, Y. Y. Jau, G. W. Biedermann, A. J. Landahl, and I. H. Deutsch, “Adiabatic quantum computation with Rydberg-dressed atoms,” Phys. Rev. A 87(5), 052314 (2013).
[Crossref]

J. A. Sedlacek, A. Schwettmann, H. Kübler, and J. P. Shaffer, “Atom-based vector microwave electrometry using rubidium Rydberg atoms in a vapor cell,” Phys. Rev. Lett. 111(6), 063001 (2013).
[Crossref] [PubMed]

2012 (3)

J. D. Carter, O. Cherry, and J. D. D. Martin, “Electric-field sensing near the surface microstructure of an atom chip using cold Rydberg atoms,” Phys. Rev. A 86(5), 053401 (2012).
[Crossref]

J. A. Sedlacek, A. Schwettmann, H. Kübler, R. Löw, T. Pfau, and J. P. Shaffer, “Microwave electrometry with Rydberg atoms in a vapor cell using bright atomic resonances,” Nat. Phys. 8(11), 819–824 (2012).
[Crossref]

Y. O. Dudin and A. Kuzmich, “Strongly interacting Rydberg excitations of a cold atomic gas,” Science 336(6083), 887–889 (2012).
[Crossref] [PubMed]

2010 (4)

M. Saffman, T. G. Walker, and K. Mølmer, “Quantum information with Rydberg atoms,” Rev. Mod. Phys. 82(3), 2313–2363 (2010).
[Crossref]

B. Zhao, M. Müller, K. Hammerer, and P. Zoller, “Efficient quantum repeater based on deterministic Rydberg gates,” Phys. Rev. A 81(5), 052329 (2010).
[Crossref]

T. Wilk, A. Gaëtan, C. Evellin, J. Wolters, Y. Miroshnychenko, P. Grangier, and A. Browaeys, “Entanglement of two individual neutral atoms using Rydberg blockade,” Phys. Rev. Lett. 104(1), 010502 (2010).
[Crossref] [PubMed]

L. Isenhower, E. Urban, X. L. Zhang, A. T. Gill, T. Henage, T. A. Johnson, T. G. Walker, and M. Saffman, “Demonstration of a neutral atom controlled-NOT quantum gate,” Phys. Rev. Lett. 104(1), 010503 (2010).
[Crossref] [PubMed]

2009 (3)

E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Observation of Rydberg blockade between two atoms,” Nat. Phys. 5(2), 110–114 (2009).
[Crossref]

A. Gaëtan, Y. Miroshnychenko, T. Wilk, A. Chotia, M. Viteau, D. Comparat, P. Pillet, A. Browaeys, and P. Grangier, “Observation of collective excitation of two individual atoms in the Rydberg blockade regime,” Nat. Phys. 5(2), 115–118 (2009).
[Crossref]

P. Thoumany, T. Hänsch, G. Stania, L. Urbonas, and T. Becker, “Optical spectroscopy of rubidium Rydberg atoms with a 297 nm frequency-doubled dye laser,” Opt. Lett. 34(11), 1621–1623 (2009).
[Crossref] [PubMed]

2007 (1)

A. K. Mohapatra, T. R. Jackson, and C. S. Adams, “Coherent optical detection of highly excited Rydberg states using electromagnetically induced transparency,” Phys. Rev. Lett. 98(11), 113003 (2007).
[Crossref] [PubMed]

2006 (1)

Y. H. Wang, H. J. Yang, Z. J. Du, T. C. Zhang, and J. M. Wang, “Autler-Townes doublet in novel sub-Doppler spectra with cesium vapor cell,” Chin. Phys. B 15(1), 138–142 (2006).
[Crossref]

2004 (2)

K. Singer, M. Reetz-Lamour, T. Amthor, L. G. Marcassa, and M. Weidemüller, “Suppression of excitation and spectral broadening induced by interactions in a cold gas of Rydberg atoms,” Phys. Rev. Lett. 93(16), 163001 (2004).
[Crossref] [PubMed]

D. Tong, S. M. Farooqi, J. Stanojevic, S. Krishnan, Y. P. Zhang, R. Côté, E. E. Eyler, and P. L. Gould, “Local blockade of Rydberg excitation in an ultracold gas,” Phys. Rev. Lett. 93(6), 063001 (2004).
[Crossref] [PubMed]

1998 (1)

S. Wielandy and A. L. Gaeta, “Investigation of electromagnetically induced transparency in the strong probe regime,” Phys. Rev. A 58(3), 2500–2505 (1998).
[Crossref]

1997 (1)

G. S. Agarwal, “Nature of the quantum interference in electromagnetic-field-induced control of absorption,” Phys. Rev. A 55(3), 2467–2470 (1997).
[Crossref]

1984 (1)

C. J. Lorenzen and K. Niemax, “Precise quantum defects of nS, nP and nD levels in Cs I,” Z. Phys., A At. Nucl. 315(2), 127–133 (1984).
[Crossref]

1980 (1)

B. P. Stoicheff and E. Weinberger, “Frequency shifts, line broadenings, and phase-interference effects in Rb + Rb collisions, measured by Doppler-free two-photon spectroscopy,” Phys. Rev. Lett. 44(11), 733–736 (1980).
[Crossref]

1979 (1)

M. L. Zimmerman, M. G. Littman, M. M. Kash, and D. Kleppner, “Stark structure of the Rydberg states of alkali-metal atoms,” Phys. Rev. A 20(6), 2251–2275 (1979).
[Crossref]

Adams, C. S.

A. K. Mohapatra, T. R. Jackson, and C. S. Adams, “Coherent optical detection of highly excited Rydberg states using electromagnetically induced transparency,” Phys. Rev. Lett. 98(11), 113003 (2007).
[Crossref] [PubMed]

Agarwal, G. S.

G. S. Agarwal, “Nature of the quantum interference in electromagnetic-field-induced control of absorption,” Phys. Rev. A 55(3), 2467–2470 (1997).
[Crossref]

Amthor, T.

K. Singer, M. Reetz-Lamour, T. Amthor, L. G. Marcassa, and M. Weidemüller, “Suppression of excitation and spectral broadening induced by interactions in a cold gas of Rydberg atoms,” Phys. Rev. Lett. 93(16), 163001 (2004).
[Crossref] [PubMed]

Armstrong, D. J.

A. M. Hankin, Y. Y. Jau, L. P. Parazzoli, C. W. Chou, D. J. Armstrong, A. J. Landahl, and G. W. Biedermann, “Two-atom Rydberg blockade using direct 6S to nP excitation,” Phys. Rev. A 89(3), 033416 (2014).
[Crossref]

Bai, J. D.

J. Y. Wang, J. D. Bai, J. He, and J. M. Wang, “Development and characterization of a 2.2 W narrow-linewidth 318.6-nm ultraviolet laser,” J. Opt. Soc. Am. B 33(10), 2020–2025 (2016).
[Crossref]

J. Y. Wang, J. D. Bai, J. He, and J. M. Wang, “Realization and characterization of single-frequency tunable 637.2 nm high-power laser,” Opt. Commun. 370, 150–155 (2016).
[Crossref]

Bao, S. X.

S. X. Bao, H. Zhang, J. Zhou, L. J. Zhang, J. M. Zhao, L. T. Xiao, and S. T. Jia, “Polarization spectra of Zeeman sublevels in Rydberg electromagnetically induced transparency,” Phys. Rev. A 94(4), 043822 (2016).
[Crossref]

Becker, T.

Biedermann, G. W.

Y. Y. Jau, A. M. Hankin, T. Keating, I. H. Deutsch, and G. W. Biedermann, “Entangling atomic spins with a Rydberg-dressed spin-flip blockade,” Nat. Phys. 12(1), 71–74 (2015).
[Crossref]

A. M. Hankin, Y. Y. Jau, L. P. Parazzoli, C. W. Chou, D. J. Armstrong, A. J. Landahl, and G. W. Biedermann, “Two-atom Rydberg blockade using direct 6S to nP excitation,” Phys. Rev. A 89(3), 033416 (2014).
[Crossref]

T. Keating, K. Goyal, Y. Y. Jau, G. W. Biedermann, A. J. Landahl, and I. H. Deutsch, “Adiabatic quantum computation with Rydberg-dressed atoms,” Phys. Rev. A 87(5), 052314 (2013).
[Crossref]

Browaeys, A.

T. Wilk, A. Gaëtan, C. Evellin, J. Wolters, Y. Miroshnychenko, P. Grangier, and A. Browaeys, “Entanglement of two individual neutral atoms using Rydberg blockade,” Phys. Rev. Lett. 104(1), 010502 (2010).
[Crossref] [PubMed]

A. Gaëtan, Y. Miroshnychenko, T. Wilk, A. Chotia, M. Viteau, D. Comparat, P. Pillet, A. Browaeys, and P. Grangier, “Observation of collective excitation of two individual atoms in the Rydberg blockade regime,” Nat. Phys. 5(2), 115–118 (2009).
[Crossref]

Carter, J. D.

J. D. Carter, O. Cherry, and J. D. D. Martin, “Electric-field sensing near the surface microstructure of an atom chip using cold Rydberg atoms,” Phys. Rev. A 86(5), 053401 (2012).
[Crossref]

Cherry, O.

J. D. Carter, O. Cherry, and J. D. D. Martin, “Electric-field sensing near the surface microstructure of an atom chip using cold Rydberg atoms,” Phys. Rev. A 86(5), 053401 (2012).
[Crossref]

Chotia, A.

A. Gaëtan, Y. Miroshnychenko, T. Wilk, A. Chotia, M. Viteau, D. Comparat, P. Pillet, A. Browaeys, and P. Grangier, “Observation of collective excitation of two individual atoms in the Rydberg blockade regime,” Nat. Phys. 5(2), 115–118 (2009).
[Crossref]

Chou, C. W.

A. M. Hankin, Y. Y. Jau, L. P. Parazzoli, C. W. Chou, D. J. Armstrong, A. J. Landahl, and G. W. Biedermann, “Two-atom Rydberg blockade using direct 6S to nP excitation,” Phys. Rev. A 89(3), 033416 (2014).
[Crossref]

Comparat, D.

A. Gaëtan, Y. Miroshnychenko, T. Wilk, A. Chotia, M. Viteau, D. Comparat, P. Pillet, A. Browaeys, and P. Grangier, “Observation of collective excitation of two individual atoms in the Rydberg blockade regime,” Nat. Phys. 5(2), 115–118 (2009).
[Crossref]

Côté, R.

D. Tong, S. M. Farooqi, J. Stanojevic, S. Krishnan, Y. P. Zhang, R. Côté, E. E. Eyler, and P. L. Gould, “Local blockade of Rydberg excitation in an ultracold gas,” Phys. Rev. Lett. 93(6), 063001 (2004).
[Crossref] [PubMed]

Deutsch, I. H.

Y. Y. Jau, A. M. Hankin, T. Keating, I. H. Deutsch, and G. W. Biedermann, “Entangling atomic spins with a Rydberg-dressed spin-flip blockade,” Nat. Phys. 12(1), 71–74 (2015).
[Crossref]

T. Keating, K. Goyal, Y. Y. Jau, G. W. Biedermann, A. J. Landahl, and I. H. Deutsch, “Adiabatic quantum computation with Rydberg-dressed atoms,” Phys. Rev. A 87(5), 052314 (2013).
[Crossref]

Du, Z. J.

Y. H. Wang, H. J. Yang, Z. J. Du, T. C. Zhang, and J. M. Wang, “Autler-Townes doublet in novel sub-Doppler spectra with cesium vapor cell,” Chin. Phys. B 15(1), 138–142 (2006).
[Crossref]

Dudin, Y. O.

Y. O. Dudin and A. Kuzmich, “Strongly interacting Rydberg excitations of a cold atomic gas,” Science 336(6083), 887–889 (2012).
[Crossref] [PubMed]

Evellin, C.

T. Wilk, A. Gaëtan, C. Evellin, J. Wolters, Y. Miroshnychenko, P. Grangier, and A. Browaeys, “Entanglement of two individual neutral atoms using Rydberg blockade,” Phys. Rev. Lett. 104(1), 010502 (2010).
[Crossref] [PubMed]

Eyler, E. E.

D. Tong, S. M. Farooqi, J. Stanojevic, S. Krishnan, Y. P. Zhang, R. Côté, E. E. Eyler, and P. L. Gould, “Local blockade of Rydberg excitation in an ultracold gas,” Phys. Rev. Lett. 93(6), 063001 (2004).
[Crossref] [PubMed]

Farooqi, S. M.

D. Tong, S. M. Farooqi, J. Stanojevic, S. Krishnan, Y. P. Zhang, R. Côté, E. E. Eyler, and P. L. Gould, “Local blockade of Rydberg excitation in an ultracold gas,” Phys. Rev. Lett. 93(6), 063001 (2004).
[Crossref] [PubMed]

Gaeta, A. L.

S. Wielandy and A. L. Gaeta, “Investigation of electromagnetically induced transparency in the strong probe regime,” Phys. Rev. A 58(3), 2500–2505 (1998).
[Crossref]

Gaëtan, A.

T. Wilk, A. Gaëtan, C. Evellin, J. Wolters, Y. Miroshnychenko, P. Grangier, and A. Browaeys, “Entanglement of two individual neutral atoms using Rydberg blockade,” Phys. Rev. Lett. 104(1), 010502 (2010).
[Crossref] [PubMed]

A. Gaëtan, Y. Miroshnychenko, T. Wilk, A. Chotia, M. Viteau, D. Comparat, P. Pillet, A. Browaeys, and P. Grangier, “Observation of collective excitation of two individual atoms in the Rydberg blockade regime,” Nat. Phys. 5(2), 115–118 (2009).
[Crossref]

Gill, A. T.

L. Isenhower, E. Urban, X. L. Zhang, A. T. Gill, T. Henage, T. A. Johnson, T. G. Walker, and M. Saffman, “Demonstration of a neutral atom controlled-NOT quantum gate,” Phys. Rev. Lett. 104(1), 010503 (2010).
[Crossref] [PubMed]

Gould, P. L.

D. Tong, S. M. Farooqi, J. Stanojevic, S. Krishnan, Y. P. Zhang, R. Côté, E. E. Eyler, and P. L. Gould, “Local blockade of Rydberg excitation in an ultracold gas,” Phys. Rev. Lett. 93(6), 063001 (2004).
[Crossref] [PubMed]

Goyal, K.

T. Keating, K. Goyal, Y. Y. Jau, G. W. Biedermann, A. J. Landahl, and I. H. Deutsch, “Adiabatic quantum computation with Rydberg-dressed atoms,” Phys. Rev. A 87(5), 052314 (2013).
[Crossref]

Grangier, P.

T. Wilk, A. Gaëtan, C. Evellin, J. Wolters, Y. Miroshnychenko, P. Grangier, and A. Browaeys, “Entanglement of two individual neutral atoms using Rydberg blockade,” Phys. Rev. Lett. 104(1), 010502 (2010).
[Crossref] [PubMed]

A. Gaëtan, Y. Miroshnychenko, T. Wilk, A. Chotia, M. Viteau, D. Comparat, P. Pillet, A. Browaeys, and P. Grangier, “Observation of collective excitation of two individual atoms in the Rydberg blockade regime,” Nat. Phys. 5(2), 115–118 (2009).
[Crossref]

Hammerer, K.

B. Zhao, M. Müller, K. Hammerer, and P. Zoller, “Efficient quantum repeater based on deterministic Rydberg gates,” Phys. Rev. A 81(5), 052329 (2010).
[Crossref]

Han, X. X.

Y. C. Jiao, X. X. Han, Z. W. Yang, J. K. Li, G. Raithel, J. M. Zhao, and S. T. Jia, “Spectroscopy of cesium Rydberg atoms in strong radio-frequency fields,” Phys. Rev. A 94(2), 023832 (2016).
[Crossref]

Hankin, A. M.

Y. Y. Jau, A. M. Hankin, T. Keating, I. H. Deutsch, and G. W. Biedermann, “Entangling atomic spins with a Rydberg-dressed spin-flip blockade,” Nat. Phys. 12(1), 71–74 (2015).
[Crossref]

A. M. Hankin, Y. Y. Jau, L. P. Parazzoli, C. W. Chou, D. J. Armstrong, A. J. Landahl, and G. W. Biedermann, “Two-atom Rydberg blockade using direct 6S to nP excitation,” Phys. Rev. A 89(3), 033416 (2014).
[Crossref]

Hänsch, T.

He, J.

J. Y. Wang, J. D. Bai, J. He, and J. M. Wang, “Development and characterization of a 2.2 W narrow-linewidth 318.6-nm ultraviolet laser,” J. Opt. Soc. Am. B 33(10), 2020–2025 (2016).
[Crossref]

J. Y. Wang, J. D. Bai, J. He, and J. M. Wang, “Realization and characterization of single-frequency tunable 637.2 nm high-power laser,” Opt. Commun. 370, 150–155 (2016).
[Crossref]

Henage, T.

L. Isenhower, E. Urban, X. L. Zhang, A. T. Gill, T. Henage, T. A. Johnson, T. G. Walker, and M. Saffman, “Demonstration of a neutral atom controlled-NOT quantum gate,” Phys. Rev. Lett. 104(1), 010503 (2010).
[Crossref] [PubMed]

E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Observation of Rydberg blockade between two atoms,” Nat. Phys. 5(2), 110–114 (2009).
[Crossref]

Isenhower, L.

L. Isenhower, E. Urban, X. L. Zhang, A. T. Gill, T. Henage, T. A. Johnson, T. G. Walker, and M. Saffman, “Demonstration of a neutral atom controlled-NOT quantum gate,” Phys. Rev. Lett. 104(1), 010503 (2010).
[Crossref] [PubMed]

E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Observation of Rydberg blockade between two atoms,” Nat. Phys. 5(2), 110–114 (2009).
[Crossref]

Jackson, T. R.

A. K. Mohapatra, T. R. Jackson, and C. S. Adams, “Coherent optical detection of highly excited Rydberg states using electromagnetically induced transparency,” Phys. Rev. Lett. 98(11), 113003 (2007).
[Crossref] [PubMed]

Jau, Y. Y.

Y. Y. Jau, A. M. Hankin, T. Keating, I. H. Deutsch, and G. W. Biedermann, “Entangling atomic spins with a Rydberg-dressed spin-flip blockade,” Nat. Phys. 12(1), 71–74 (2015).
[Crossref]

A. M. Hankin, Y. Y. Jau, L. P. Parazzoli, C. W. Chou, D. J. Armstrong, A. J. Landahl, and G. W. Biedermann, “Two-atom Rydberg blockade using direct 6S to nP excitation,” Phys. Rev. A 89(3), 033416 (2014).
[Crossref]

T. Keating, K. Goyal, Y. Y. Jau, G. W. Biedermann, A. J. Landahl, and I. H. Deutsch, “Adiabatic quantum computation with Rydberg-dressed atoms,” Phys. Rev. A 87(5), 052314 (2013).
[Crossref]

Jia, S. T.

Y. C. Jiao, X. X. Han, Z. W. Yang, J. K. Li, G. Raithel, J. M. Zhao, and S. T. Jia, “Spectroscopy of cesium Rydberg atoms in strong radio-frequency fields,” Phys. Rev. A 94(2), 023832 (2016).
[Crossref]

S. X. Bao, H. Zhang, J. Zhou, L. J. Zhang, J. M. Zhao, L. T. Xiao, and S. T. Jia, “Polarization spectra of Zeeman sublevels in Rydberg electromagnetically induced transparency,” Phys. Rev. A 94(4), 043822 (2016).
[Crossref]

Jiao, Y. C.

Y. C. Jiao, X. X. Han, Z. W. Yang, J. K. Li, G. Raithel, J. M. Zhao, and S. T. Jia, “Spectroscopy of cesium Rydberg atoms in strong radio-frequency fields,” Phys. Rev. A 94(2), 023832 (2016).
[Crossref]

Johnson, T. A.

L. Isenhower, E. Urban, X. L. Zhang, A. T. Gill, T. Henage, T. A. Johnson, T. G. Walker, and M. Saffman, “Demonstration of a neutral atom controlled-NOT quantum gate,” Phys. Rev. Lett. 104(1), 010503 (2010).
[Crossref] [PubMed]

E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Observation of Rydberg blockade between two atoms,” Nat. Phys. 5(2), 110–114 (2009).
[Crossref]

Kash, M. M.

M. L. Zimmerman, M. G. Littman, M. M. Kash, and D. Kleppner, “Stark structure of the Rydberg states of alkali-metal atoms,” Phys. Rev. A 20(6), 2251–2275 (1979).
[Crossref]

Keating, T.

Y. Y. Jau, A. M. Hankin, T. Keating, I. H. Deutsch, and G. W. Biedermann, “Entangling atomic spins with a Rydberg-dressed spin-flip blockade,” Nat. Phys. 12(1), 71–74 (2015).
[Crossref]

T. Keating, K. Goyal, Y. Y. Jau, G. W. Biedermann, A. J. Landahl, and I. H. Deutsch, “Adiabatic quantum computation with Rydberg-dressed atoms,” Phys. Rev. A 87(5), 052314 (2013).
[Crossref]

Kleppner, D.

M. L. Zimmerman, M. G. Littman, M. M. Kash, and D. Kleppner, “Stark structure of the Rydberg states of alkali-metal atoms,” Phys. Rev. A 20(6), 2251–2275 (1979).
[Crossref]

Krishnan, S.

D. Tong, S. M. Farooqi, J. Stanojevic, S. Krishnan, Y. P. Zhang, R. Côté, E. E. Eyler, and P. L. Gould, “Local blockade of Rydberg excitation in an ultracold gas,” Phys. Rev. Lett. 93(6), 063001 (2004).
[Crossref] [PubMed]

Kübler, H.

J. A. Sedlacek, A. Schwettmann, H. Kübler, and J. P. Shaffer, “Atom-based vector microwave electrometry using rubidium Rydberg atoms in a vapor cell,” Phys. Rev. Lett. 111(6), 063001 (2013).
[Crossref] [PubMed]

J. A. Sedlacek, A. Schwettmann, H. Kübler, R. Löw, T. Pfau, and J. P. Shaffer, “Microwave electrometry with Rydberg atoms in a vapor cell using bright atomic resonances,” Nat. Phys. 8(11), 819–824 (2012).
[Crossref]

Kuzmich, A.

Y. O. Dudin and A. Kuzmich, “Strongly interacting Rydberg excitations of a cold atomic gas,” Science 336(6083), 887–889 (2012).
[Crossref] [PubMed]

Landahl, A. J.

A. M. Hankin, Y. Y. Jau, L. P. Parazzoli, C. W. Chou, D. J. Armstrong, A. J. Landahl, and G. W. Biedermann, “Two-atom Rydberg blockade using direct 6S to nP excitation,” Phys. Rev. A 89(3), 033416 (2014).
[Crossref]

T. Keating, K. Goyal, Y. Y. Jau, G. W. Biedermann, A. J. Landahl, and I. H. Deutsch, “Adiabatic quantum computation with Rydberg-dressed atoms,” Phys. Rev. A 87(5), 052314 (2013).
[Crossref]

Li, J. K.

Y. C. Jiao, X. X. Han, Z. W. Yang, J. K. Li, G. Raithel, J. M. Zhao, and S. T. Jia, “Spectroscopy of cesium Rydberg atoms in strong radio-frequency fields,” Phys. Rev. A 94(2), 023832 (2016).
[Crossref]

Littman, M. G.

M. L. Zimmerman, M. G. Littman, M. M. Kash, and D. Kleppner, “Stark structure of the Rydberg states of alkali-metal atoms,” Phys. Rev. A 20(6), 2251–2275 (1979).
[Crossref]

Lorenzen, C. J.

C. J. Lorenzen and K. Niemax, “Precise quantum defects of nS, nP and nD levels in Cs I,” Z. Phys., A At. Nucl. 315(2), 127–133 (1984).
[Crossref]

Löw, R.

J. A. Sedlacek, A. Schwettmann, H. Kübler, R. Löw, T. Pfau, and J. P. Shaffer, “Microwave electrometry with Rydberg atoms in a vapor cell using bright atomic resonances,” Nat. Phys. 8(11), 819–824 (2012).
[Crossref]

Marcassa, L. G.

K. Singer, M. Reetz-Lamour, T. Amthor, L. G. Marcassa, and M. Weidemüller, “Suppression of excitation and spectral broadening induced by interactions in a cold gas of Rydberg atoms,” Phys. Rev. Lett. 93(16), 163001 (2004).
[Crossref] [PubMed]

Martin, J. D. D.

J. D. Carter, O. Cherry, and J. D. D. Martin, “Electric-field sensing near the surface microstructure of an atom chip using cold Rydberg atoms,” Phys. Rev. A 86(5), 053401 (2012).
[Crossref]

Miroshnychenko, Y.

T. Wilk, A. Gaëtan, C. Evellin, J. Wolters, Y. Miroshnychenko, P. Grangier, and A. Browaeys, “Entanglement of two individual neutral atoms using Rydberg blockade,” Phys. Rev. Lett. 104(1), 010502 (2010).
[Crossref] [PubMed]

A. Gaëtan, Y. Miroshnychenko, T. Wilk, A. Chotia, M. Viteau, D. Comparat, P. Pillet, A. Browaeys, and P. Grangier, “Observation of collective excitation of two individual atoms in the Rydberg blockade regime,” Nat. Phys. 5(2), 115–118 (2009).
[Crossref]

Mohapatra, A. K.

A. K. Mohapatra, T. R. Jackson, and C. S. Adams, “Coherent optical detection of highly excited Rydberg states using electromagnetically induced transparency,” Phys. Rev. Lett. 98(11), 113003 (2007).
[Crossref] [PubMed]

Mølmer, K.

M. Saffman, T. G. Walker, and K. Mølmer, “Quantum information with Rydberg atoms,” Rev. Mod. Phys. 82(3), 2313–2363 (2010).
[Crossref]

Müller, M.

B. Zhao, M. Müller, K. Hammerer, and P. Zoller, “Efficient quantum repeater based on deterministic Rydberg gates,” Phys. Rev. A 81(5), 052329 (2010).
[Crossref]

Niemax, K.

C. J. Lorenzen and K. Niemax, “Precise quantum defects of nS, nP and nD levels in Cs I,” Z. Phys., A At. Nucl. 315(2), 127–133 (1984).
[Crossref]

Parazzoli, L. P.

A. M. Hankin, Y. Y. Jau, L. P. Parazzoli, C. W. Chou, D. J. Armstrong, A. J. Landahl, and G. W. Biedermann, “Two-atom Rydberg blockade using direct 6S to nP excitation,” Phys. Rev. A 89(3), 033416 (2014).
[Crossref]

Pfau, T.

J. A. Sedlacek, A. Schwettmann, H. Kübler, R. Löw, T. Pfau, and J. P. Shaffer, “Microwave electrometry with Rydberg atoms in a vapor cell using bright atomic resonances,” Nat. Phys. 8(11), 819–824 (2012).
[Crossref]

Pillet, P.

A. Gaëtan, Y. Miroshnychenko, T. Wilk, A. Chotia, M. Viteau, D. Comparat, P. Pillet, A. Browaeys, and P. Grangier, “Observation of collective excitation of two individual atoms in the Rydberg blockade regime,” Nat. Phys. 5(2), 115–118 (2009).
[Crossref]

Raithel, G.

Y. C. Jiao, X. X. Han, Z. W. Yang, J. K. Li, G. Raithel, J. M. Zhao, and S. T. Jia, “Spectroscopy of cesium Rydberg atoms in strong radio-frequency fields,” Phys. Rev. A 94(2), 023832 (2016).
[Crossref]

Reetz-Lamour, M.

K. Singer, M. Reetz-Lamour, T. Amthor, L. G. Marcassa, and M. Weidemüller, “Suppression of excitation and spectral broadening induced by interactions in a cold gas of Rydberg atoms,” Phys. Rev. Lett. 93(16), 163001 (2004).
[Crossref] [PubMed]

Saffman, M.

L. Isenhower, E. Urban, X. L. Zhang, A. T. Gill, T. Henage, T. A. Johnson, T. G. Walker, and M. Saffman, “Demonstration of a neutral atom controlled-NOT quantum gate,” Phys. Rev. Lett. 104(1), 010503 (2010).
[Crossref] [PubMed]

M. Saffman, T. G. Walker, and K. Mølmer, “Quantum information with Rydberg atoms,” Rev. Mod. Phys. 82(3), 2313–2363 (2010).
[Crossref]

E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Observation of Rydberg blockade between two atoms,” Nat. Phys. 5(2), 110–114 (2009).
[Crossref]

Schwettmann, A.

J. A. Sedlacek, A. Schwettmann, H. Kübler, and J. P. Shaffer, “Atom-based vector microwave electrometry using rubidium Rydberg atoms in a vapor cell,” Phys. Rev. Lett. 111(6), 063001 (2013).
[Crossref] [PubMed]

J. A. Sedlacek, A. Schwettmann, H. Kübler, R. Löw, T. Pfau, and J. P. Shaffer, “Microwave electrometry with Rydberg atoms in a vapor cell using bright atomic resonances,” Nat. Phys. 8(11), 819–824 (2012).
[Crossref]

Sedlacek, J. A.

J. A. Sedlacek, A. Schwettmann, H. Kübler, and J. P. Shaffer, “Atom-based vector microwave electrometry using rubidium Rydberg atoms in a vapor cell,” Phys. Rev. Lett. 111(6), 063001 (2013).
[Crossref] [PubMed]

J. A. Sedlacek, A. Schwettmann, H. Kübler, R. Löw, T. Pfau, and J. P. Shaffer, “Microwave electrometry with Rydberg atoms in a vapor cell using bright atomic resonances,” Nat. Phys. 8(11), 819–824 (2012).
[Crossref]

Shaffer, J. P.

J. A. Sedlacek, A. Schwettmann, H. Kübler, and J. P. Shaffer, “Atom-based vector microwave electrometry using rubidium Rydberg atoms in a vapor cell,” Phys. Rev. Lett. 111(6), 063001 (2013).
[Crossref] [PubMed]

J. A. Sedlacek, A. Schwettmann, H. Kübler, R. Löw, T. Pfau, and J. P. Shaffer, “Microwave electrometry with Rydberg atoms in a vapor cell using bright atomic resonances,” Nat. Phys. 8(11), 819–824 (2012).
[Crossref]

Singer, K.

K. Singer, M. Reetz-Lamour, T. Amthor, L. G. Marcassa, and M. Weidemüller, “Suppression of excitation and spectral broadening induced by interactions in a cold gas of Rydberg atoms,” Phys. Rev. Lett. 93(16), 163001 (2004).
[Crossref] [PubMed]

Stania, G.

Stanojevic, J.

D. Tong, S. M. Farooqi, J. Stanojevic, S. Krishnan, Y. P. Zhang, R. Côté, E. E. Eyler, and P. L. Gould, “Local blockade of Rydberg excitation in an ultracold gas,” Phys. Rev. Lett. 93(6), 063001 (2004).
[Crossref] [PubMed]

Stoicheff, B. P.

B. P. Stoicheff and E. Weinberger, “Frequency shifts, line broadenings, and phase-interference effects in Rb + Rb collisions, measured by Doppler-free two-photon spectroscopy,” Phys. Rev. Lett. 44(11), 733–736 (1980).
[Crossref]

Thoumany, P.

Tong, D.

D. Tong, S. M. Farooqi, J. Stanojevic, S. Krishnan, Y. P. Zhang, R. Côté, E. E. Eyler, and P. L. Gould, “Local blockade of Rydberg excitation in an ultracold gas,” Phys. Rev. Lett. 93(6), 063001 (2004).
[Crossref] [PubMed]

Urban, E.

L. Isenhower, E. Urban, X. L. Zhang, A. T. Gill, T. Henage, T. A. Johnson, T. G. Walker, and M. Saffman, “Demonstration of a neutral atom controlled-NOT quantum gate,” Phys. Rev. Lett. 104(1), 010503 (2010).
[Crossref] [PubMed]

E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Observation of Rydberg blockade between two atoms,” Nat. Phys. 5(2), 110–114 (2009).
[Crossref]

Urbonas, L.

Viteau, M.

A. Gaëtan, Y. Miroshnychenko, T. Wilk, A. Chotia, M. Viteau, D. Comparat, P. Pillet, A. Browaeys, and P. Grangier, “Observation of collective excitation of two individual atoms in the Rydberg blockade regime,” Nat. Phys. 5(2), 115–118 (2009).
[Crossref]

Walker, T. G.

L. Isenhower, E. Urban, X. L. Zhang, A. T. Gill, T. Henage, T. A. Johnson, T. G. Walker, and M. Saffman, “Demonstration of a neutral atom controlled-NOT quantum gate,” Phys. Rev. Lett. 104(1), 010503 (2010).
[Crossref] [PubMed]

M. Saffman, T. G. Walker, and K. Mølmer, “Quantum information with Rydberg atoms,” Rev. Mod. Phys. 82(3), 2313–2363 (2010).
[Crossref]

E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Observation of Rydberg blockade between two atoms,” Nat. Phys. 5(2), 110–114 (2009).
[Crossref]

Wang, J. M.

J. Y. Wang, J. D. Bai, J. He, and J. M. Wang, “Realization and characterization of single-frequency tunable 637.2 nm high-power laser,” Opt. Commun. 370, 150–155 (2016).
[Crossref]

J. Y. Wang, J. D. Bai, J. He, and J. M. Wang, “Development and characterization of a 2.2 W narrow-linewidth 318.6-nm ultraviolet laser,” J. Opt. Soc. Am. B 33(10), 2020–2025 (2016).
[Crossref]

Y. H. Wang, H. J. Yang, Z. J. Du, T. C. Zhang, and J. M. Wang, “Autler-Townes doublet in novel sub-Doppler spectra with cesium vapor cell,” Chin. Phys. B 15(1), 138–142 (2006).
[Crossref]

Wang, J. Y.

J. Y. Wang, J. D. Bai, J. He, and J. M. Wang, “Development and characterization of a 2.2 W narrow-linewidth 318.6-nm ultraviolet laser,” J. Opt. Soc. Am. B 33(10), 2020–2025 (2016).
[Crossref]

J. Y. Wang, J. D. Bai, J. He, and J. M. Wang, “Realization and characterization of single-frequency tunable 637.2 nm high-power laser,” Opt. Commun. 370, 150–155 (2016).
[Crossref]

Wang, Y. H.

Y. H. Wang, H. J. Yang, Z. J. Du, T. C. Zhang, and J. M. Wang, “Autler-Townes doublet in novel sub-Doppler spectra with cesium vapor cell,” Chin. Phys. B 15(1), 138–142 (2006).
[Crossref]

Weidemüller, M.

K. Singer, M. Reetz-Lamour, T. Amthor, L. G. Marcassa, and M. Weidemüller, “Suppression of excitation and spectral broadening induced by interactions in a cold gas of Rydberg atoms,” Phys. Rev. Lett. 93(16), 163001 (2004).
[Crossref] [PubMed]

Weinberger, E.

B. P. Stoicheff and E. Weinberger, “Frequency shifts, line broadenings, and phase-interference effects in Rb + Rb collisions, measured by Doppler-free two-photon spectroscopy,” Phys. Rev. Lett. 44(11), 733–736 (1980).
[Crossref]

Wielandy, S.

S. Wielandy and A. L. Gaeta, “Investigation of electromagnetically induced transparency in the strong probe regime,” Phys. Rev. A 58(3), 2500–2505 (1998).
[Crossref]

Wilk, T.

T. Wilk, A. Gaëtan, C. Evellin, J. Wolters, Y. Miroshnychenko, P. Grangier, and A. Browaeys, “Entanglement of two individual neutral atoms using Rydberg blockade,” Phys. Rev. Lett. 104(1), 010502 (2010).
[Crossref] [PubMed]

A. Gaëtan, Y. Miroshnychenko, T. Wilk, A. Chotia, M. Viteau, D. Comparat, P. Pillet, A. Browaeys, and P. Grangier, “Observation of collective excitation of two individual atoms in the Rydberg blockade regime,” Nat. Phys. 5(2), 115–118 (2009).
[Crossref]

Wolters, J.

T. Wilk, A. Gaëtan, C. Evellin, J. Wolters, Y. Miroshnychenko, P. Grangier, and A. Browaeys, “Entanglement of two individual neutral atoms using Rydberg blockade,” Phys. Rev. Lett. 104(1), 010502 (2010).
[Crossref] [PubMed]

Xiao, L. T.

S. X. Bao, H. Zhang, J. Zhou, L. J. Zhang, J. M. Zhao, L. T. Xiao, and S. T. Jia, “Polarization spectra of Zeeman sublevels in Rydberg electromagnetically induced transparency,” Phys. Rev. A 94(4), 043822 (2016).
[Crossref]

Yang, H. J.

Y. H. Wang, H. J. Yang, Z. J. Du, T. C. Zhang, and J. M. Wang, “Autler-Townes doublet in novel sub-Doppler spectra with cesium vapor cell,” Chin. Phys. B 15(1), 138–142 (2006).
[Crossref]

Yang, Z. W.

Y. C. Jiao, X. X. Han, Z. W. Yang, J. K. Li, G. Raithel, J. M. Zhao, and S. T. Jia, “Spectroscopy of cesium Rydberg atoms in strong radio-frequency fields,” Phys. Rev. A 94(2), 023832 (2016).
[Crossref]

Yavuz, D. D.

E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Observation of Rydberg blockade between two atoms,” Nat. Phys. 5(2), 110–114 (2009).
[Crossref]

Zhang, H.

S. X. Bao, H. Zhang, J. Zhou, L. J. Zhang, J. M. Zhao, L. T. Xiao, and S. T. Jia, “Polarization spectra of Zeeman sublevels in Rydberg electromagnetically induced transparency,” Phys. Rev. A 94(4), 043822 (2016).
[Crossref]

Zhang, L. J.

S. X. Bao, H. Zhang, J. Zhou, L. J. Zhang, J. M. Zhao, L. T. Xiao, and S. T. Jia, “Polarization spectra of Zeeman sublevels in Rydberg electromagnetically induced transparency,” Phys. Rev. A 94(4), 043822 (2016).
[Crossref]

Zhang, T. C.

Y. H. Wang, H. J. Yang, Z. J. Du, T. C. Zhang, and J. M. Wang, “Autler-Townes doublet in novel sub-Doppler spectra with cesium vapor cell,” Chin. Phys. B 15(1), 138–142 (2006).
[Crossref]

Zhang, X. L.

L. Isenhower, E. Urban, X. L. Zhang, A. T. Gill, T. Henage, T. A. Johnson, T. G. Walker, and M. Saffman, “Demonstration of a neutral atom controlled-NOT quantum gate,” Phys. Rev. Lett. 104(1), 010503 (2010).
[Crossref] [PubMed]

Zhang, Y. P.

D. Tong, S. M. Farooqi, J. Stanojevic, S. Krishnan, Y. P. Zhang, R. Côté, E. E. Eyler, and P. L. Gould, “Local blockade of Rydberg excitation in an ultracold gas,” Phys. Rev. Lett. 93(6), 063001 (2004).
[Crossref] [PubMed]

Zhao, B.

B. Zhao, M. Müller, K. Hammerer, and P. Zoller, “Efficient quantum repeater based on deterministic Rydberg gates,” Phys. Rev. A 81(5), 052329 (2010).
[Crossref]

Zhao, J. M.

S. X. Bao, H. Zhang, J. Zhou, L. J. Zhang, J. M. Zhao, L. T. Xiao, and S. T. Jia, “Polarization spectra of Zeeman sublevels in Rydberg electromagnetically induced transparency,” Phys. Rev. A 94(4), 043822 (2016).
[Crossref]

Y. C. Jiao, X. X. Han, Z. W. Yang, J. K. Li, G. Raithel, J. M. Zhao, and S. T. Jia, “Spectroscopy of cesium Rydberg atoms in strong radio-frequency fields,” Phys. Rev. A 94(2), 023832 (2016).
[Crossref]

Zhou, J.

S. X. Bao, H. Zhang, J. Zhou, L. J. Zhang, J. M. Zhao, L. T. Xiao, and S. T. Jia, “Polarization spectra of Zeeman sublevels in Rydberg electromagnetically induced transparency,” Phys. Rev. A 94(4), 043822 (2016).
[Crossref]

Zimmerman, M. L.

M. L. Zimmerman, M. G. Littman, M. M. Kash, and D. Kleppner, “Stark structure of the Rydberg states of alkali-metal atoms,” Phys. Rev. A 20(6), 2251–2275 (1979).
[Crossref]

Zoller, P.

B. Zhao, M. Müller, K. Hammerer, and P. Zoller, “Efficient quantum repeater based on deterministic Rydberg gates,” Phys. Rev. A 81(5), 052329 (2010).
[Crossref]

Chin. Phys. B (1)

Y. H. Wang, H. J. Yang, Z. J. Du, T. C. Zhang, and J. M. Wang, “Autler-Townes doublet in novel sub-Doppler spectra with cesium vapor cell,” Chin. Phys. B 15(1), 138–142 (2006).
[Crossref]

J. Opt. Soc. Am. B (1)

Nat. Phys. (4)

J. A. Sedlacek, A. Schwettmann, H. Kübler, R. Löw, T. Pfau, and J. P. Shaffer, “Microwave electrometry with Rydberg atoms in a vapor cell using bright atomic resonances,” Nat. Phys. 8(11), 819–824 (2012).
[Crossref]

E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Observation of Rydberg blockade between two atoms,” Nat. Phys. 5(2), 110–114 (2009).
[Crossref]

A. Gaëtan, Y. Miroshnychenko, T. Wilk, A. Chotia, M. Viteau, D. Comparat, P. Pillet, A. Browaeys, and P. Grangier, “Observation of collective excitation of two individual atoms in the Rydberg blockade regime,” Nat. Phys. 5(2), 115–118 (2009).
[Crossref]

Y. Y. Jau, A. M. Hankin, T. Keating, I. H. Deutsch, and G. W. Biedermann, “Entangling atomic spins with a Rydberg-dressed spin-flip blockade,” Nat. Phys. 12(1), 71–74 (2015).
[Crossref]

Opt. Commun. (1)

J. Y. Wang, J. D. Bai, J. He, and J. M. Wang, “Realization and characterization of single-frequency tunable 637.2 nm high-power laser,” Opt. Commun. 370, 150–155 (2016).
[Crossref]

Opt. Lett. (1)

Phys. Rev. A (9)

G. S. Agarwal, “Nature of the quantum interference in electromagnetic-field-induced control of absorption,” Phys. Rev. A 55(3), 2467–2470 (1997).
[Crossref]

S. Wielandy and A. L. Gaeta, “Investigation of electromagnetically induced transparency in the strong probe regime,” Phys. Rev. A 58(3), 2500–2505 (1998).
[Crossref]

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Figures (5)

Fig. 1
Fig. 1

(a) Relevant hyperfine levels for Cs atomic single-photon Rydberg excitation. The 852.3 nm probe laser is resonant on the transition 6S1/2, F = 4 - 6P3/2, F’ = 5, and the 318.6 nm coupling laser is scanned over the Rydberg transition 6S1/2, F = 4 - nP3/2. (b) Schematic of the experimental setup. The 318.6 nm coupling laser co-propagating with the 852.3 nm probe laser in a 10-cm-long Cs vapor cell. DBR: distributed-Bragg-reflector diode laser; OI: optical isolator; λ/2: half-wave plate; PBS: polarization beam splitter cube; DM: dichroic mirror; BS: 50:50 beam splitter; AOM: acousto-optic modulator; PI: proportion and integration amplifier; EOM: electro-optic modulator; DPD: differential photodiode; PS: polarization spectroscopy.

Fig. 2
Fig. 2

(a) The excitation spectra of 6S1/2, F = 4 - 71P3/2 Rydberg transition in a Cs vapor cell when the 852.3 nm probe laser is locked to Cs 6S1/2, F = 4 - 6P3/2, F’ = 5 cycling transition. The Rabi frequencies of coupling and probe beams are ~0.30 and ~8.53 MHz, respectively. (b) Sideband calibration result with a frequency modulation of 70 MHz for 852.3 nm probe laser, considering the Doppler factor of λp / λc ≈2.675, the observed hyperfine interval becomes ~671 MHz. Red curve is a multi-peak Lorentzian fitting.

Fig. 3
Fig. 3

Amplitudes (a) and linewidths (b) of Cs 6S1/2, F = 4 - 71P3/2 single-photon Rydberg transmission spectra as a function of probe Rabi frequency for indicated coupling Rabi frequencies. Scatters represent the experimental data, while the solid lines are the fitting results. The measured narrowest linewidth is 24 MHz for Ω319 ~0.11 MHz and Ω852 ~6.77 MHz.

Fig. 4
Fig. 4

Energies of Cs nP3/2 Rydberg states (n = 70 - 100). The dots are experimental data, and the solid line is the fitting curve. The quantum defect value of Cs nP3/2 states is 3.56671(42) is obtained, where the error is statistical error.

Fig. 5
Fig. 5

(a) The frequency-modulated Rydberg spectra with Ω319 ~0.11 MHz and Ω852 ~9.50 MHz, and the corresponding error signal from the lock-in amplifier with a modulation frequency and an integration time of 17 kHz and 300 μs, respectively. The red curve in the upper part is a multi-peak Lorentzian fitting. (b) Relative Allan standard deviation plots show the relative frequency instability of the 318.6 nm UV laser (dots).

Equations (6)

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ν= ν 0 ( 1+ υ z c )= ν 0 + Δ p
ρ bc =i Ω p ( γ ac Γ ac +2 Ω c 2 )(i Δ p + γ ac ) [(i Δ p + Γ bc )(i Δ p + Γ ab )+ Ω c 2 ]( γ ac Γ ac +4 Ω c 2 )
A min = Ω p γ ac ( γ ac Γ ac +2 Ω c 2 ) ( Γ bc Γ ab + Ω c 2 )( γ ac Γ ac +4 Ω c 2 )
w w 0 + Ω p 2 γ ac + γ bc
E( n,l )= E R Cs ( n δ n,l ) 2 E f
δ n,l = δ 0 + δ 1 ( n δ 0 ) 2 + δ 2 ( n δ 0 ) 4 +

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