Abstract

We have developed a microwave frequency standard based on the 15.2 GHz ground-stated hyperfine transition of 113Cd+ ions. Using a laser-cooled ion cloud trapped in a linear quadrupole Paul trap, the clock transition frequency is measured to be 15 199 862 855.0125(87) Hz, with an accuracy at the 10−13 level. The main errors are from the microwave frequency reference used in the experiment. The precision is improved by nearly two orders of magnitude than that reported before.

© 2013 OSA

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References

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  1. T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, “Quantum computers,” Nature 464(7285), 45–53 (2010).
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  2. K. Kim, M. S. Chang, S. Korenblit, R. Islam, E. E. Edwards, J. K. Freericks, G. D. Lin, L. M. Duan, and C. Monroe, “Quantum simulation of frustrated Ising spins with trapped ions,” Nature 465(7298), 590–593 (2010).
    [Crossref] [PubMed]
  3. B. B. Blinov, D. L. Moehring, L. M. Duan, and C. Monroe, “Observation of entanglement between a single trapped atom and a single photon,” Nature 428(6979), 153–157 (2004).
    [Crossref] [PubMed]
  4. C. W. Chou, D. B. Hume, J. C. J. Koelemeij, D. J. Wineland, and T. Rosenband, “Frequency comparison of two high-accuracy Al+ optical clocks,” Phys. Rev. Lett. 104(7), 070802 (2010).
    [Crossref] [PubMed]
  5. T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; Metrology at the 17th Decimal Place,” Science 319(5871), 1808–1812 (2008).
    [Crossref] [PubMed]
  6. A. A. Madej, P. Dubé, Z. Zhou, J. E. Bernard, and M. Gertsvolf, “88Sr+ 445-THz single-ion reference at the 10-17 level via control and cancellation of systematic uncertainties and its measurement against the SI second,” Phys. Rev. Lett. 109(20), 203002 (2012).
    [Crossref] [PubMed]
  7. E. Burt, S. Taghavi-Larigani, and R. Tjoelker, “High-resolution spectroscopy of 201Hg+ hyperfine structure: A sensitive probe of nuclear structure and the hyperfine anomaly,” Phys. Rev. A 79(6), 062506 (2009).
    [Crossref]
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    [Crossref] [PubMed]
  10. B. Jelenković, S. Chung, J. Prestage, and L. Maleki, “High-resolution microwave-optical double-resonance spectroscopy of hyperfine splitting of trapped 113Cd+ ions,” Phys. Rev. A 74(2), 022505 (2006).
    [Crossref]
  11. J. W. Zhang, Z. B. Wang, S. G. Wang, K. Miao, B. Wang, and L. J. Wang, “High-resolution laser microwave double-resonance spectroscopy of hyperfine splitting of trapped 113Cd+ and 111Cd+ ions,” Phys. Rev. A 86(2), 022523 (2012).
    [Crossref]
  12. J. Hamel and J. Vienne, “Optical pumping measurement of the hyperfine structure of cadmium ion ground state,” Opt. Commun. 7(1), 83–85 (1973).
    [Crossref]
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  18. R. A. Bernheim and L. M. Kohuth, “Effects of molecular buffer gases on the cesium hyperfine frequency,” J. Chem. Phys. 50(2), 899 (1968).
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    [Crossref]
  20. M. Mizushima, “Theory of resonance frequency shift due to radiation field,” Phys. Rev. 133(2A), A414–A418 (1964).
    [Crossref]
  21. A. Bauch and R. Schröder, “Experimental verification of the shift of the cesium hyperfine transition frequency due to blackbody radiation,” Phys. Rev. Lett. 78(4), 622–625 (1997).
    [Crossref]
  22. W. M. Itano, L. L. Lewis, and D. J. Wineland, “Shift of 2S1/2 hyperfine splittings due to blackbody radiation,” Phys. Rev. A 25(2), 1233–1235 (1982).
  23. R. B. Warrington, P. T. H. Fisk, M. J. Wouters, and M. A. Lawn, “Temperature of laser-cooled 171Yb+ ions and application to a microwave frequency standard,” IEEE Trans. Ultrason., Ferroelec. Freq. Control 49(8), 1166–1174 (2002).
    [Crossref]
  24. G. Dixit, H. S. Nataraj, B. K. Sahoo, R. K. Chaudhuri, and S. Majumder, “Ab initio relativistic many-body calculation of hyperfine splittings of 113Cd+,” Phys. Rev. A 77(1), 012718 (2008).

2013 (2)

S. Wang, J. Zhang, Z. Wang, B. Wang, W. Liu, Y. Zhao, and L. Wang, “Frequency stabilization of 214.5-nm ultraviolet laser,” Chin. Opt. Lett. 11(3), 031401–031403 (2013).
[Crossref]

S.-G. Wang, J.-W. Zhang, K. Miao, Z.-B. Wang, and L.-J. Wang, “Cooling and crystallization of trapped 113Cd+ ions for atomic clock,” Chin. Phys. Lett. 30(1), 013703 (2013).
[Crossref]

2012 (2)

A. A. Madej, P. Dubé, Z. Zhou, J. E. Bernard, and M. Gertsvolf, “88Sr+ 445-THz single-ion reference at the 10-17 level via control and cancellation of systematic uncertainties and its measurement against the SI second,” Phys. Rev. Lett. 109(20), 203002 (2012).
[Crossref] [PubMed]

J. W. Zhang, Z. B. Wang, S. G. Wang, K. Miao, B. Wang, and L. J. Wang, “High-resolution laser microwave double-resonance spectroscopy of hyperfine splitting of trapped 113Cd+ and 111Cd+ ions,” Phys. Rev. A 86(2), 022523 (2012).
[Crossref]

2010 (3)

T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, “Quantum computers,” Nature 464(7285), 45–53 (2010).
[Crossref] [PubMed]

K. Kim, M. S. Chang, S. Korenblit, R. Islam, E. E. Edwards, J. K. Freericks, G. D. Lin, L. M. Duan, and C. Monroe, “Quantum simulation of frustrated Ising spins with trapped ions,” Nature 465(7298), 590–593 (2010).
[Crossref] [PubMed]

C. W. Chou, D. B. Hume, J. C. J. Koelemeij, D. J. Wineland, and T. Rosenband, “Frequency comparison of two high-accuracy Al+ optical clocks,” Phys. Rev. Lett. 104(7), 070802 (2010).
[Crossref] [PubMed]

2009 (1)

E. Burt, S. Taghavi-Larigani, and R. Tjoelker, “High-resolution spectroscopy of 201Hg+ hyperfine structure: A sensitive probe of nuclear structure and the hyperfine anomaly,” Phys. Rev. A 79(6), 062506 (2009).
[Crossref]

2008 (2)

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; Metrology at the 17th Decimal Place,” Science 319(5871), 1808–1812 (2008).
[Crossref] [PubMed]

G. Dixit, H. S. Nataraj, B. K. Sahoo, R. K. Chaudhuri, and S. Majumder, “Ab initio relativistic many-body calculation of hyperfine splittings of 113Cd+,” Phys. Rev. A 77(1), 012718 (2008).

2006 (1)

B. Jelenković, S. Chung, J. Prestage, and L. Maleki, “High-resolution microwave-optical double-resonance spectroscopy of hyperfine splitting of trapped 113Cd+ ions,” Phys. Rev. A 74(2), 022505 (2006).
[Crossref]

2004 (1)

B. B. Blinov, D. L. Moehring, L. M. Duan, and C. Monroe, “Observation of entanglement between a single trapped atom and a single photon,” Nature 428(6979), 153–157 (2004).
[Crossref] [PubMed]

2002 (1)

R. B. Warrington, P. T. H. Fisk, M. J. Wouters, and M. A. Lawn, “Temperature of laser-cooled 171Yb+ ions and application to a microwave frequency standard,” IEEE Trans. Ultrason., Ferroelec. Freq. Control 49(8), 1166–1174 (2002).
[Crossref]

1997 (1)

A. Bauch and R. Schröder, “Experimental verification of the shift of the cesium hyperfine transition frequency due to blackbody radiation,” Phys. Rev. Lett. 78(4), 622–625 (1997).
[Crossref]

1996 (1)

U. Tanaka, H. Imajo, K. Hayasaka, R. Ohmukai, M. Watanabe, and S. Urabe, “Determination of the ground-state hyperfine splitting of trapped 113Cd+ ions,” Phys. Rev. A 53(6), 3982–3985 (1996).
[Crossref] [PubMed]

1982 (1)

W. M. Itano, L. L. Lewis, and D. J. Wineland, “Shift of 2S1/2 hyperfine splittings due to blackbody radiation,” Phys. Rev. A 25(2), 1233–1235 (1982).

1975 (1)

J. Vetter, M. Stuke, and E. W. Weber, “Hyperfine density shifts of 137Ba+ ions in noble gas buffers,” J. Phys. A 273(2), 129–135 (1975).
[Crossref]

1973 (1)

J. Hamel and J. Vienne, “Optical pumping measurement of the hyperfine structure of cadmium ion ground state,” Opt. Commun. 7(1), 83–85 (1973).
[Crossref]

1971 (1)

D. R. Denison, “Operating parameters of a quadrupole in a grounded cylindrical housing,” J. Vac. Sci. Technol. 8(1), 266–269 (1971).
[Crossref]

1968 (1)

R. A. Bernheim and L. M. Kohuth, “Effects of molecular buffer gases on the cesium hyperfine frequency,” J. Chem. Phys. 50(2), 899 (1968).
[Crossref]

1964 (1)

M. Mizushima, “Theory of resonance frequency shift due to radiation field,” Phys. Rev. 133(2A), A414–A418 (1964).
[Crossref]

Bauch, A.

A. Bauch and R. Schröder, “Experimental verification of the shift of the cesium hyperfine transition frequency due to blackbody radiation,” Phys. Rev. Lett. 78(4), 622–625 (1997).
[Crossref]

Bergquist, J. C.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; Metrology at the 17th Decimal Place,” Science 319(5871), 1808–1812 (2008).
[Crossref] [PubMed]

Bernard, J. E.

A. A. Madej, P. Dubé, Z. Zhou, J. E. Bernard, and M. Gertsvolf, “88Sr+ 445-THz single-ion reference at the 10-17 level via control and cancellation of systematic uncertainties and its measurement against the SI second,” Phys. Rev. Lett. 109(20), 203002 (2012).
[Crossref] [PubMed]

Bernheim, R. A.

R. A. Bernheim and L. M. Kohuth, “Effects of molecular buffer gases on the cesium hyperfine frequency,” J. Chem. Phys. 50(2), 899 (1968).
[Crossref]

Blinov, B. B.

B. B. Blinov, D. L. Moehring, L. M. Duan, and C. Monroe, “Observation of entanglement between a single trapped atom and a single photon,” Nature 428(6979), 153–157 (2004).
[Crossref] [PubMed]

Brusch, A.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; Metrology at the 17th Decimal Place,” Science 319(5871), 1808–1812 (2008).
[Crossref] [PubMed]

Burt, E.

E. Burt, S. Taghavi-Larigani, and R. Tjoelker, “High-resolution spectroscopy of 201Hg+ hyperfine structure: A sensitive probe of nuclear structure and the hyperfine anomaly,” Phys. Rev. A 79(6), 062506 (2009).
[Crossref]

Chang, M. S.

K. Kim, M. S. Chang, S. Korenblit, R. Islam, E. E. Edwards, J. K. Freericks, G. D. Lin, L. M. Duan, and C. Monroe, “Quantum simulation of frustrated Ising spins with trapped ions,” Nature 465(7298), 590–593 (2010).
[Crossref] [PubMed]

Chaudhuri, R. K.

G. Dixit, H. S. Nataraj, B. K. Sahoo, R. K. Chaudhuri, and S. Majumder, “Ab initio relativistic many-body calculation of hyperfine splittings of 113Cd+,” Phys. Rev. A 77(1), 012718 (2008).

Chou, C. W.

C. W. Chou, D. B. Hume, J. C. J. Koelemeij, D. J. Wineland, and T. Rosenband, “Frequency comparison of two high-accuracy Al+ optical clocks,” Phys. Rev. Lett. 104(7), 070802 (2010).
[Crossref] [PubMed]

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; Metrology at the 17th Decimal Place,” Science 319(5871), 1808–1812 (2008).
[Crossref] [PubMed]

Chung, S.

B. Jelenković, S. Chung, J. Prestage, and L. Maleki, “High-resolution microwave-optical double-resonance spectroscopy of hyperfine splitting of trapped 113Cd+ ions,” Phys. Rev. A 74(2), 022505 (2006).
[Crossref]

Denison, D. R.

D. R. Denison, “Operating parameters of a quadrupole in a grounded cylindrical housing,” J. Vac. Sci. Technol. 8(1), 266–269 (1971).
[Crossref]

Diddams, S. A.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; Metrology at the 17th Decimal Place,” Science 319(5871), 1808–1812 (2008).
[Crossref] [PubMed]

Dixit, G.

G. Dixit, H. S. Nataraj, B. K. Sahoo, R. K. Chaudhuri, and S. Majumder, “Ab initio relativistic many-body calculation of hyperfine splittings of 113Cd+,” Phys. Rev. A 77(1), 012718 (2008).

Drullinger, R. E.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; Metrology at the 17th Decimal Place,” Science 319(5871), 1808–1812 (2008).
[Crossref] [PubMed]

Duan, L. M.

K. Kim, M. S. Chang, S. Korenblit, R. Islam, E. E. Edwards, J. K. Freericks, G. D. Lin, L. M. Duan, and C. Monroe, “Quantum simulation of frustrated Ising spins with trapped ions,” Nature 465(7298), 590–593 (2010).
[Crossref] [PubMed]

B. B. Blinov, D. L. Moehring, L. M. Duan, and C. Monroe, “Observation of entanglement between a single trapped atom and a single photon,” Nature 428(6979), 153–157 (2004).
[Crossref] [PubMed]

Dubé, P.

A. A. Madej, P. Dubé, Z. Zhou, J. E. Bernard, and M. Gertsvolf, “88Sr+ 445-THz single-ion reference at the 10-17 level via control and cancellation of systematic uncertainties and its measurement against the SI second,” Phys. Rev. Lett. 109(20), 203002 (2012).
[Crossref] [PubMed]

Edwards, E. E.

K. Kim, M. S. Chang, S. Korenblit, R. Islam, E. E. Edwards, J. K. Freericks, G. D. Lin, L. M. Duan, and C. Monroe, “Quantum simulation of frustrated Ising spins with trapped ions,” Nature 465(7298), 590–593 (2010).
[Crossref] [PubMed]

Fisk, P. T. H.

R. B. Warrington, P. T. H. Fisk, M. J. Wouters, and M. A. Lawn, “Temperature of laser-cooled 171Yb+ ions and application to a microwave frequency standard,” IEEE Trans. Ultrason., Ferroelec. Freq. Control 49(8), 1166–1174 (2002).
[Crossref]

Fortier, T. M.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; Metrology at the 17th Decimal Place,” Science 319(5871), 1808–1812 (2008).
[Crossref] [PubMed]

Freericks, J. K.

K. Kim, M. S. Chang, S. Korenblit, R. Islam, E. E. Edwards, J. K. Freericks, G. D. Lin, L. M. Duan, and C. Monroe, “Quantum simulation of frustrated Ising spins with trapped ions,” Nature 465(7298), 590–593 (2010).
[Crossref] [PubMed]

Gertsvolf, M.

A. A. Madej, P. Dubé, Z. Zhou, J. E. Bernard, and M. Gertsvolf, “88Sr+ 445-THz single-ion reference at the 10-17 level via control and cancellation of systematic uncertainties and its measurement against the SI second,” Phys. Rev. Lett. 109(20), 203002 (2012).
[Crossref] [PubMed]

Hamel, J.

J. Hamel and J. Vienne, “Optical pumping measurement of the hyperfine structure of cadmium ion ground state,” Opt. Commun. 7(1), 83–85 (1973).
[Crossref]

Hayasaka, K.

U. Tanaka, H. Imajo, K. Hayasaka, R. Ohmukai, M. Watanabe, and S. Urabe, “Determination of the ground-state hyperfine splitting of trapped 113Cd+ ions,” Phys. Rev. A 53(6), 3982–3985 (1996).
[Crossref] [PubMed]

Hume, D. B.

C. W. Chou, D. B. Hume, J. C. J. Koelemeij, D. J. Wineland, and T. Rosenband, “Frequency comparison of two high-accuracy Al+ optical clocks,” Phys. Rev. Lett. 104(7), 070802 (2010).
[Crossref] [PubMed]

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; Metrology at the 17th Decimal Place,” Science 319(5871), 1808–1812 (2008).
[Crossref] [PubMed]

Imajo, H.

U. Tanaka, H. Imajo, K. Hayasaka, R. Ohmukai, M. Watanabe, and S. Urabe, “Determination of the ground-state hyperfine splitting of trapped 113Cd+ ions,” Phys. Rev. A 53(6), 3982–3985 (1996).
[Crossref] [PubMed]

Islam, R.

K. Kim, M. S. Chang, S. Korenblit, R. Islam, E. E. Edwards, J. K. Freericks, G. D. Lin, L. M. Duan, and C. Monroe, “Quantum simulation of frustrated Ising spins with trapped ions,” Nature 465(7298), 590–593 (2010).
[Crossref] [PubMed]

Itano, W. M.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; Metrology at the 17th Decimal Place,” Science 319(5871), 1808–1812 (2008).
[Crossref] [PubMed]

W. M. Itano, L. L. Lewis, and D. J. Wineland, “Shift of 2S1/2 hyperfine splittings due to blackbody radiation,” Phys. Rev. A 25(2), 1233–1235 (1982).

Jelenkovic, B.

B. Jelenković, S. Chung, J. Prestage, and L. Maleki, “High-resolution microwave-optical double-resonance spectroscopy of hyperfine splitting of trapped 113Cd+ ions,” Phys. Rev. A 74(2), 022505 (2006).
[Crossref]

Jelezko, F.

T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, “Quantum computers,” Nature 464(7285), 45–53 (2010).
[Crossref] [PubMed]

Kim, K.

K. Kim, M. S. Chang, S. Korenblit, R. Islam, E. E. Edwards, J. K. Freericks, G. D. Lin, L. M. Duan, and C. Monroe, “Quantum simulation of frustrated Ising spins with trapped ions,” Nature 465(7298), 590–593 (2010).
[Crossref] [PubMed]

Koelemeij, J. C. J.

C. W. Chou, D. B. Hume, J. C. J. Koelemeij, D. J. Wineland, and T. Rosenband, “Frequency comparison of two high-accuracy Al+ optical clocks,” Phys. Rev. Lett. 104(7), 070802 (2010).
[Crossref] [PubMed]

Kohuth, L. M.

R. A. Bernheim and L. M. Kohuth, “Effects of molecular buffer gases on the cesium hyperfine frequency,” J. Chem. Phys. 50(2), 899 (1968).
[Crossref]

Korenblit, S.

K. Kim, M. S. Chang, S. Korenblit, R. Islam, E. E. Edwards, J. K. Freericks, G. D. Lin, L. M. Duan, and C. Monroe, “Quantum simulation of frustrated Ising spins with trapped ions,” Nature 465(7298), 590–593 (2010).
[Crossref] [PubMed]

Ladd, T. D.

T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, “Quantum computers,” Nature 464(7285), 45–53 (2010).
[Crossref] [PubMed]

Laflamme, R.

T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, “Quantum computers,” Nature 464(7285), 45–53 (2010).
[Crossref] [PubMed]

Lawn, M. A.

R. B. Warrington, P. T. H. Fisk, M. J. Wouters, and M. A. Lawn, “Temperature of laser-cooled 171Yb+ ions and application to a microwave frequency standard,” IEEE Trans. Ultrason., Ferroelec. Freq. Control 49(8), 1166–1174 (2002).
[Crossref]

Lewis, L. L.

W. M. Itano, L. L. Lewis, and D. J. Wineland, “Shift of 2S1/2 hyperfine splittings due to blackbody radiation,” Phys. Rev. A 25(2), 1233–1235 (1982).

Lin, G. D.

K. Kim, M. S. Chang, S. Korenblit, R. Islam, E. E. Edwards, J. K. Freericks, G. D. Lin, L. M. Duan, and C. Monroe, “Quantum simulation of frustrated Ising spins with trapped ions,” Nature 465(7298), 590–593 (2010).
[Crossref] [PubMed]

Liu, W.

Lorini, L.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; Metrology at the 17th Decimal Place,” Science 319(5871), 1808–1812 (2008).
[Crossref] [PubMed]

Madej, A. A.

A. A. Madej, P. Dubé, Z. Zhou, J. E. Bernard, and M. Gertsvolf, “88Sr+ 445-THz single-ion reference at the 10-17 level via control and cancellation of systematic uncertainties and its measurement against the SI second,” Phys. Rev. Lett. 109(20), 203002 (2012).
[Crossref] [PubMed]

Majumder, S.

G. Dixit, H. S. Nataraj, B. K. Sahoo, R. K. Chaudhuri, and S. Majumder, “Ab initio relativistic many-body calculation of hyperfine splittings of 113Cd+,” Phys. Rev. A 77(1), 012718 (2008).

Maleki, L.

B. Jelenković, S. Chung, J. Prestage, and L. Maleki, “High-resolution microwave-optical double-resonance spectroscopy of hyperfine splitting of trapped 113Cd+ ions,” Phys. Rev. A 74(2), 022505 (2006).
[Crossref]

Miao, K.

S.-G. Wang, J.-W. Zhang, K. Miao, Z.-B. Wang, and L.-J. Wang, “Cooling and crystallization of trapped 113Cd+ ions for atomic clock,” Chin. Phys. Lett. 30(1), 013703 (2013).
[Crossref]

J. W. Zhang, Z. B. Wang, S. G. Wang, K. Miao, B. Wang, and L. J. Wang, “High-resolution laser microwave double-resonance spectroscopy of hyperfine splitting of trapped 113Cd+ and 111Cd+ ions,” Phys. Rev. A 86(2), 022523 (2012).
[Crossref]

Mizushima, M.

M. Mizushima, “Theory of resonance frequency shift due to radiation field,” Phys. Rev. 133(2A), A414–A418 (1964).
[Crossref]

Moehring, D. L.

B. B. Blinov, D. L. Moehring, L. M. Duan, and C. Monroe, “Observation of entanglement between a single trapped atom and a single photon,” Nature 428(6979), 153–157 (2004).
[Crossref] [PubMed]

Monroe, C.

K. Kim, M. S. Chang, S. Korenblit, R. Islam, E. E. Edwards, J. K. Freericks, G. D. Lin, L. M. Duan, and C. Monroe, “Quantum simulation of frustrated Ising spins with trapped ions,” Nature 465(7298), 590–593 (2010).
[Crossref] [PubMed]

T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, “Quantum computers,” Nature 464(7285), 45–53 (2010).
[Crossref] [PubMed]

B. B. Blinov, D. L. Moehring, L. M. Duan, and C. Monroe, “Observation of entanglement between a single trapped atom and a single photon,” Nature 428(6979), 153–157 (2004).
[Crossref] [PubMed]

Nakamura, Y.

T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, “Quantum computers,” Nature 464(7285), 45–53 (2010).
[Crossref] [PubMed]

Nataraj, H. S.

G. Dixit, H. S. Nataraj, B. K. Sahoo, R. K. Chaudhuri, and S. Majumder, “Ab initio relativistic many-body calculation of hyperfine splittings of 113Cd+,” Phys. Rev. A 77(1), 012718 (2008).

Newbury, N. R.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; Metrology at the 17th Decimal Place,” Science 319(5871), 1808–1812 (2008).
[Crossref] [PubMed]

O’Brien, J. L.

T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, “Quantum computers,” Nature 464(7285), 45–53 (2010).
[Crossref] [PubMed]

Ohmukai, R.

U. Tanaka, H. Imajo, K. Hayasaka, R. Ohmukai, M. Watanabe, and S. Urabe, “Determination of the ground-state hyperfine splitting of trapped 113Cd+ ions,” Phys. Rev. A 53(6), 3982–3985 (1996).
[Crossref] [PubMed]

Oskay, W. H.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; Metrology at the 17th Decimal Place,” Science 319(5871), 1808–1812 (2008).
[Crossref] [PubMed]

Prestage, J.

B. Jelenković, S. Chung, J. Prestage, and L. Maleki, “High-resolution microwave-optical double-resonance spectroscopy of hyperfine splitting of trapped 113Cd+ ions,” Phys. Rev. A 74(2), 022505 (2006).
[Crossref]

Rosenband, T.

C. W. Chou, D. B. Hume, J. C. J. Koelemeij, D. J. Wineland, and T. Rosenband, “Frequency comparison of two high-accuracy Al+ optical clocks,” Phys. Rev. Lett. 104(7), 070802 (2010).
[Crossref] [PubMed]

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; Metrology at the 17th Decimal Place,” Science 319(5871), 1808–1812 (2008).
[Crossref] [PubMed]

Sahoo, B. K.

G. Dixit, H. S. Nataraj, B. K. Sahoo, R. K. Chaudhuri, and S. Majumder, “Ab initio relativistic many-body calculation of hyperfine splittings of 113Cd+,” Phys. Rev. A 77(1), 012718 (2008).

Schmidt, P. O.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; Metrology at the 17th Decimal Place,” Science 319(5871), 1808–1812 (2008).
[Crossref] [PubMed]

Schröder, R.

A. Bauch and R. Schröder, “Experimental verification of the shift of the cesium hyperfine transition frequency due to blackbody radiation,” Phys. Rev. Lett. 78(4), 622–625 (1997).
[Crossref]

Stalnaker, J. E.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; Metrology at the 17th Decimal Place,” Science 319(5871), 1808–1812 (2008).
[Crossref] [PubMed]

Stuke, M.

J. Vetter, M. Stuke, and E. W. Weber, “Hyperfine density shifts of 137Ba+ ions in noble gas buffers,” J. Phys. A 273(2), 129–135 (1975).
[Crossref]

Swann, W. C.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; Metrology at the 17th Decimal Place,” Science 319(5871), 1808–1812 (2008).
[Crossref] [PubMed]

Taghavi-Larigani, S.

E. Burt, S. Taghavi-Larigani, and R. Tjoelker, “High-resolution spectroscopy of 201Hg+ hyperfine structure: A sensitive probe of nuclear structure and the hyperfine anomaly,” Phys. Rev. A 79(6), 062506 (2009).
[Crossref]

Tanaka, U.

U. Tanaka, H. Imajo, K. Hayasaka, R. Ohmukai, M. Watanabe, and S. Urabe, “Determination of the ground-state hyperfine splitting of trapped 113Cd+ ions,” Phys. Rev. A 53(6), 3982–3985 (1996).
[Crossref] [PubMed]

Tjoelker, R.

E. Burt, S. Taghavi-Larigani, and R. Tjoelker, “High-resolution spectroscopy of 201Hg+ hyperfine structure: A sensitive probe of nuclear structure and the hyperfine anomaly,” Phys. Rev. A 79(6), 062506 (2009).
[Crossref]

Urabe, S.

U. Tanaka, H. Imajo, K. Hayasaka, R. Ohmukai, M. Watanabe, and S. Urabe, “Determination of the ground-state hyperfine splitting of trapped 113Cd+ ions,” Phys. Rev. A 53(6), 3982–3985 (1996).
[Crossref] [PubMed]

Vetter, J.

J. Vetter, M. Stuke, and E. W. Weber, “Hyperfine density shifts of 137Ba+ ions in noble gas buffers,” J. Phys. A 273(2), 129–135 (1975).
[Crossref]

Vienne, J.

J. Hamel and J. Vienne, “Optical pumping measurement of the hyperfine structure of cadmium ion ground state,” Opt. Commun. 7(1), 83–85 (1973).
[Crossref]

Wang, B.

S. Wang, J. Zhang, Z. Wang, B. Wang, W. Liu, Y. Zhao, and L. Wang, “Frequency stabilization of 214.5-nm ultraviolet laser,” Chin. Opt. Lett. 11(3), 031401–031403 (2013).
[Crossref]

J. W. Zhang, Z. B. Wang, S. G. Wang, K. Miao, B. Wang, and L. J. Wang, “High-resolution laser microwave double-resonance spectroscopy of hyperfine splitting of trapped 113Cd+ and 111Cd+ ions,” Phys. Rev. A 86(2), 022523 (2012).
[Crossref]

Wang, L.

Wang, L. J.

J. W. Zhang, Z. B. Wang, S. G. Wang, K. Miao, B. Wang, and L. J. Wang, “High-resolution laser microwave double-resonance spectroscopy of hyperfine splitting of trapped 113Cd+ and 111Cd+ ions,” Phys. Rev. A 86(2), 022523 (2012).
[Crossref]

Wang, L.-J.

S.-G. Wang, J.-W. Zhang, K. Miao, Z.-B. Wang, and L.-J. Wang, “Cooling and crystallization of trapped 113Cd+ ions for atomic clock,” Chin. Phys. Lett. 30(1), 013703 (2013).
[Crossref]

Wang, S.

Wang, S. G.

J. W. Zhang, Z. B. Wang, S. G. Wang, K. Miao, B. Wang, and L. J. Wang, “High-resolution laser microwave double-resonance spectroscopy of hyperfine splitting of trapped 113Cd+ and 111Cd+ ions,” Phys. Rev. A 86(2), 022523 (2012).
[Crossref]

Wang, S.-G.

S.-G. Wang, J.-W. Zhang, K. Miao, Z.-B. Wang, and L.-J. Wang, “Cooling and crystallization of trapped 113Cd+ ions for atomic clock,” Chin. Phys. Lett. 30(1), 013703 (2013).
[Crossref]

Wang, Z.

Wang, Z. B.

J. W. Zhang, Z. B. Wang, S. G. Wang, K. Miao, B. Wang, and L. J. Wang, “High-resolution laser microwave double-resonance spectroscopy of hyperfine splitting of trapped 113Cd+ and 111Cd+ ions,” Phys. Rev. A 86(2), 022523 (2012).
[Crossref]

Wang, Z.-B.

S.-G. Wang, J.-W. Zhang, K. Miao, Z.-B. Wang, and L.-J. Wang, “Cooling and crystallization of trapped 113Cd+ ions for atomic clock,” Chin. Phys. Lett. 30(1), 013703 (2013).
[Crossref]

Warrington, R. B.

R. B. Warrington, P. T. H. Fisk, M. J. Wouters, and M. A. Lawn, “Temperature of laser-cooled 171Yb+ ions and application to a microwave frequency standard,” IEEE Trans. Ultrason., Ferroelec. Freq. Control 49(8), 1166–1174 (2002).
[Crossref]

Watanabe, M.

U. Tanaka, H. Imajo, K. Hayasaka, R. Ohmukai, M. Watanabe, and S. Urabe, “Determination of the ground-state hyperfine splitting of trapped 113Cd+ ions,” Phys. Rev. A 53(6), 3982–3985 (1996).
[Crossref] [PubMed]

Weber, E. W.

J. Vetter, M. Stuke, and E. W. Weber, “Hyperfine density shifts of 137Ba+ ions in noble gas buffers,” J. Phys. A 273(2), 129–135 (1975).
[Crossref]

Wineland, D. J.

C. W. Chou, D. B. Hume, J. C. J. Koelemeij, D. J. Wineland, and T. Rosenband, “Frequency comparison of two high-accuracy Al+ optical clocks,” Phys. Rev. Lett. 104(7), 070802 (2010).
[Crossref] [PubMed]

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; Metrology at the 17th Decimal Place,” Science 319(5871), 1808–1812 (2008).
[Crossref] [PubMed]

W. M. Itano, L. L. Lewis, and D. J. Wineland, “Shift of 2S1/2 hyperfine splittings due to blackbody radiation,” Phys. Rev. A 25(2), 1233–1235 (1982).

Wouters, M. J.

R. B. Warrington, P. T. H. Fisk, M. J. Wouters, and M. A. Lawn, “Temperature of laser-cooled 171Yb+ ions and application to a microwave frequency standard,” IEEE Trans. Ultrason., Ferroelec. Freq. Control 49(8), 1166–1174 (2002).
[Crossref]

Zhang, J.

Zhang, J. W.

J. W. Zhang, Z. B. Wang, S. G. Wang, K. Miao, B. Wang, and L. J. Wang, “High-resolution laser microwave double-resonance spectroscopy of hyperfine splitting of trapped 113Cd+ and 111Cd+ ions,” Phys. Rev. A 86(2), 022523 (2012).
[Crossref]

Zhang, J.-W.

S.-G. Wang, J.-W. Zhang, K. Miao, Z.-B. Wang, and L.-J. Wang, “Cooling and crystallization of trapped 113Cd+ ions for atomic clock,” Chin. Phys. Lett. 30(1), 013703 (2013).
[Crossref]

Zhao, Y.

Zhou, Z.

A. A. Madej, P. Dubé, Z. Zhou, J. E. Bernard, and M. Gertsvolf, “88Sr+ 445-THz single-ion reference at the 10-17 level via control and cancellation of systematic uncertainties and its measurement against the SI second,” Phys. Rev. Lett. 109(20), 203002 (2012).
[Crossref] [PubMed]

Chin. Opt. Lett. (1)

Chin. Phys. Lett. (1)

S.-G. Wang, J.-W. Zhang, K. Miao, Z.-B. Wang, and L.-J. Wang, “Cooling and crystallization of trapped 113Cd+ ions for atomic clock,” Chin. Phys. Lett. 30(1), 013703 (2013).
[Crossref]

IEEE Trans. Ultrason., Ferroelec. Freq. Control (1)

R. B. Warrington, P. T. H. Fisk, M. J. Wouters, and M. A. Lawn, “Temperature of laser-cooled 171Yb+ ions and application to a microwave frequency standard,” IEEE Trans. Ultrason., Ferroelec. Freq. Control 49(8), 1166–1174 (2002).
[Crossref]

J. Chem. Phys. (1)

R. A. Bernheim and L. M. Kohuth, “Effects of molecular buffer gases on the cesium hyperfine frequency,” J. Chem. Phys. 50(2), 899 (1968).
[Crossref]

J. Phys. A (1)

J. Vetter, M. Stuke, and E. W. Weber, “Hyperfine density shifts of 137Ba+ ions in noble gas buffers,” J. Phys. A 273(2), 129–135 (1975).
[Crossref]

J. Vac. Sci. Technol. (1)

D. R. Denison, “Operating parameters of a quadrupole in a grounded cylindrical housing,” J. Vac. Sci. Technol. 8(1), 266–269 (1971).
[Crossref]

Nature (3)

T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, “Quantum computers,” Nature 464(7285), 45–53 (2010).
[Crossref] [PubMed]

K. Kim, M. S. Chang, S. Korenblit, R. Islam, E. E. Edwards, J. K. Freericks, G. D. Lin, L. M. Duan, and C. Monroe, “Quantum simulation of frustrated Ising spins with trapped ions,” Nature 465(7298), 590–593 (2010).
[Crossref] [PubMed]

B. B. Blinov, D. L. Moehring, L. M. Duan, and C. Monroe, “Observation of entanglement between a single trapped atom and a single photon,” Nature 428(6979), 153–157 (2004).
[Crossref] [PubMed]

Opt. Commun. (1)

J. Hamel and J. Vienne, “Optical pumping measurement of the hyperfine structure of cadmium ion ground state,” Opt. Commun. 7(1), 83–85 (1973).
[Crossref]

Phys. Rev. (1)

M. Mizushima, “Theory of resonance frequency shift due to radiation field,” Phys. Rev. 133(2A), A414–A418 (1964).
[Crossref]

Phys. Rev. A (6)

G. Dixit, H. S. Nataraj, B. K. Sahoo, R. K. Chaudhuri, and S. Majumder, “Ab initio relativistic many-body calculation of hyperfine splittings of 113Cd+,” Phys. Rev. A 77(1), 012718 (2008).

W. M. Itano, L. L. Lewis, and D. J. Wineland, “Shift of 2S1/2 hyperfine splittings due to blackbody radiation,” Phys. Rev. A 25(2), 1233–1235 (1982).

E. Burt, S. Taghavi-Larigani, and R. Tjoelker, “High-resolution spectroscopy of 201Hg+ hyperfine structure: A sensitive probe of nuclear structure and the hyperfine anomaly,” Phys. Rev. A 79(6), 062506 (2009).
[Crossref]

U. Tanaka, H. Imajo, K. Hayasaka, R. Ohmukai, M. Watanabe, and S. Urabe, “Determination of the ground-state hyperfine splitting of trapped 113Cd+ ions,” Phys. Rev. A 53(6), 3982–3985 (1996).
[Crossref] [PubMed]

B. Jelenković, S. Chung, J. Prestage, and L. Maleki, “High-resolution microwave-optical double-resonance spectroscopy of hyperfine splitting of trapped 113Cd+ ions,” Phys. Rev. A 74(2), 022505 (2006).
[Crossref]

J. W. Zhang, Z. B. Wang, S. G. Wang, K. Miao, B. Wang, and L. J. Wang, “High-resolution laser microwave double-resonance spectroscopy of hyperfine splitting of trapped 113Cd+ and 111Cd+ ions,” Phys. Rev. A 86(2), 022523 (2012).
[Crossref]

Phys. Rev. Lett. (3)

C. W. Chou, D. B. Hume, J. C. J. Koelemeij, D. J. Wineland, and T. Rosenband, “Frequency comparison of two high-accuracy Al+ optical clocks,” Phys. Rev. Lett. 104(7), 070802 (2010).
[Crossref] [PubMed]

A. A. Madej, P. Dubé, Z. Zhou, J. E. Bernard, and M. Gertsvolf, “88Sr+ 445-THz single-ion reference at the 10-17 level via control and cancellation of systematic uncertainties and its measurement against the SI second,” Phys. Rev. Lett. 109(20), 203002 (2012).
[Crossref] [PubMed]

A. Bauch and R. Schröder, “Experimental verification of the shift of the cesium hyperfine transition frequency due to blackbody radiation,” Phys. Rev. Lett. 78(4), 622–625 (1997).
[Crossref]

Science (1)

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; Metrology at the 17th Decimal Place,” Science 319(5871), 1808–1812 (2008).
[Crossref] [PubMed]

Other (3)

S. J. Park, P. J. Manson, M. J. Wouters, R. B. Warrington, M. A. Lawn, and P. T. H. Fisk, “171Yb+ microwave frequency standard,” in Frequency Control Symposium, 2007 Joint with the 21st European Frequency and Time Forum. IEEE International(2007), pp. 613–616.
[Crossref]

J. D. Prestage, R. L. Tjoelker, and L. Maleki, “Higher pole linear traps for atomic clock applications,” in Frequency and Time Forum, 1999 and the IEEE International Frequency Control Symposium, 1999., Proceedings of the 1999 Joint Meeting of the European(1999), pp. 121–124.
[Crossref]

S. K. Chung, J. D. Prestage, R. L. Tjoelker, and L. Maleki, “Buffer gas experiments in mecury (Hg+) ion clock,” in Frequency Control Symposium and Exposition, 2004. Proceedings of the 2004 IEEE International (2004), pp. 130–133.
[Crossref]

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Figures (5)

Fig. 1
Fig. 1

The schematic diagram of the JMI-2 experimental setup. HWP: half-wave plate; QWP: quarter-wave plate; L: lens; AOM: acousto-optic modulator; M: mirror; OS: optical shutter; PBS: polarizing beamsplitter; EMCCD: electron-multiplying CCD; PMT: photo-multiplier tube detector.

Fig. 2
Fig. 2

113Cd+ relevant energy levels and transitions (not to scale).

Fig. 3
Fig. 3

(a) Spectroscopy of ΔF = 1, ΔmF = 1 Zeeman line of the 113Cd+ ions. The rf pulse width is 0.1 ms. (b) Ramsey fringe of the clock transiton of the 113Cd+ ions with a free precession period of 2 s. The rf pulses’ widths are 400 ms. The gate time of photon counting for both measurements are set at 15 ms.

Fig. 4
Fig. 4

Clock transition ν0,0 versus the difference of the frequency of ΔmF = ± 1 Zeeman transitions, ν0,1-ν0,-1. The measured data is fitted to Eq. (3), with residuals also shown.

Fig. 5
Fig. 5

Measurements of the ground-state hyperfine splitting of 113Cd+ ions. 1, 2, 3, and 4 represent the results from Tanaka et al., Jelenkovic et al., the JMI-1 setup and the JMI-2 trap, respectively. The inset shows the comparison of the last three measurements.

Tables (1)

Tables Icon

Table 1 Estimated Fractional Systematic Frequency Shifts and Uncertainties.

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

E(F, m F )= E HFS 2(2I+1) g I μ B B 0 m F ± 1 2 E HFS (1+ 4 m F 2I+1 x+ x 2 ) 1/2 ,
x= ( g J + g I ) μ B B 0 / E HFS .
v 0,0 = v HFS + 1 2 ( g J g I g J + g I ) 2 ( v 0,1 v 0,1 ) 2 v HFS ,
P Rabi ( 2πν )= b 2 Ω 2 sin 2 Ω 2 τ
Ω= ( 2πν2π ν 0 ) 2 + b 2 ,
P Ramsey (2πν)= 4 b 2 Ω 2 sin 2 Ω 2 τ (cos Ω 2 τcos Ω 0 2 T Ω 0 Ω sin Ω 2 τsin Ω 0 2 T) 2 ,
Δν ν | total SODFS =- 3 2 kT m c 2 [ 1+ 2 3 ( N d K ) ],
ν HFS =15199862855.0125 (87) Hz,

Metrics