Abstract

We investigate corrections on the cooling limit of high-order Lamb-Dicke (LD) parameters in the double electromagnetically induced transparency (EIT) cooling scheme. Via utilizing quantum interferences, the single-phonon heating mechanism vanishes and the system evolves to a double dark state, from which we will obtain the mechanical occupation on the single-phonon excitation state. In addition, the further correction induced by two-phonon heating transitions is included to achieve a more accurate cooling limit. There exist two pathways of two-phonon heating transitions: direct two-phonon excitation from the dark state and further excitation from the single-phonon excited state. By adding up these two parts of correction, the obtained analytical predictions show a well consistence with numerical results. Moreover, we find that the two pathways can destructively interfere with each other, leading to the elimination of two-phonon heating transitions and achieving a lower cooling limit.

© 2017 Optical Society of America

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  1. K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Bose-Einstein condensation in a gas of sodium atoms,” Phys. Rev. Lett. 75(22), 3969–3973 (1995).
    [Crossref] [PubMed]
  2. F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of Bose-Einstein condensation in trapped gases,” Rev. Mod. Phys. 71(3), 463–512 (1999).
    [Crossref]
  3. P. Barletta, J. Tennyson, and P. F. Barker, “Creating ultracold molecules by collisions with ultracold rare-gas atoms in an optical trap,” Phys. Rev. A 78(5), 052707 (2008).
    [Crossref]
  4. J. Weiner, V. S. Bagnato, S. Zilio, and P. S. Julienne, “Experiments and theory in cold and ultracold collisions,” Rev. Mod. Phys. 71(1), 1–85 (1999).
    [Crossref]
  5. Y. Shin, M. Saba, T. A. Pasquini, W. Ketterle, D. E. Pritchard, and A. E. Leanhardt, “Atom interferometry with Bose-Einstein condensates in a double-well protential,” Phys. Rev. Lett. 92(5), 050405 (2004).
    [Crossref] [PubMed]
  6. R. Stevenson, M. R. Hush, T. Bishop, I. Lesanovsky, and T. Fernholz, “Sagnac interferometry with a single atomic clock,” Phys. Rev. Lett. 115(16), 163001 (2015).
    [Crossref] [PubMed]
  7. F. X. Esnault, D. Holleville, N. Rossetto, S. Guerandel, and N. Dimarcq, “High-stability compact atomic clock based on isotropic laser cooling,” Phys. Rev. A 82(3), 033436 (2010).
    [Crossref]
  8. A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. O. Schmidt, “Optical atomic clocks,” Rev. Mod. Phys. 87(2), 637–701 (2015).
    [Crossref]
  9. V. Zhelyazkova, A. Cournol, T. E. Wall, A. Matsushima, J. J. Hudson, E. A. Hinds, M. R. Tarbutt, and B. E. Sauer, “Laser cooling and slowing of CaF molecules,” Phys. Rev. A 89(5), 053416 (2014).
    [Crossref]
  10. M. Kowalewski, G. Morigi, P. W. H. Pinkse, and R. de Vivie-Riedle, “Cavity sideband cooling of trapped molecules,” Phys. Rev. A 84(3), 033408 (2011).
    [Crossref]
  11. A. H. Safavi-Naeini, J. Chan, J. T. Hill, T. P. M. Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108(3), 033602 (2012).
    [Crossref] [PubMed]
  12. J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
    [Crossref] [PubMed]
  13. F. Diedrich, J. C. Bergquist, W. M. Itano, and D. J. Wineland, “Laser cooling to the zero-point energy of motion,” Phys. Rev. Lett. 62(4), 403–406 (1989).
    [Crossref] [PubMed]
  14. C. E. Wieman, D. E. Pritchard, and D. J. Wineland, “Atom cooling, trapping, and quantum manipulation,” Rev. Mod. Phys. 71(2), s253–s262 (1999).
    [Crossref]
  15. J. Eschner, G. Morigi, F. Schmidt-Kaler, and R. Blatt, “Laser cooling of trapped ions,” J. Opt. Soc. Am. B 20(5), 1003–1015 (2003).
    [Crossref]
  16. S. Stenholm, “The semiclassical theory of laser cooling,” Rev. Mod. Phys. 58(3), 699–739 (1986).
    [Crossref]
  17. S. Zhang, J. Zhang, Q. Duan, C. Guo, C. Wu, W. Wu, and P. Chen, “Ground-state cooling of a trapped ion by quantum interference pathways,” Phys. Rev. A 90(4), 043409 (2014).
    [Crossref]
  18. J. Evers and C. H. Keitel, “Double-EIT ground-state laser cooling without blue-sideband heating,” Europhys. Lett. 68(3), 370–376 (2004).
    [Crossref]
  19. Y. Lu, J. Q. Zhang, J. M. Cui, D. Y. Cao, S. Zhang, Y. F. Huang, C. F. Li, and G. C. Guo, “Dark-state cooling of a trapped ion using microwave coupling,” Phys. Rev. A 92(2), 023420 (2015).
    [Crossref]
  20. J. Cerrillo, A. Retzker, and M. B. Plenio, “Fast and robust laser cooling of trapped systems,” Phys. Rev. Lett. 104(4), 043003 (2010).
    [Crossref] [PubMed]
  21. A. Albrecht, A. Retzker, C. Wunderlich, and M. B. Plenio, “Enhancement of laser cooling by the use of magnetic gradients,” New J. Phys. 13(3), 033009 (2011).
    [Crossref]
  22. S. Zhang, C. W. Wu, and P. X. Chen, “Dark-state laser cooling of a trapped ion using standing waves,” Phys. Rev. A 85(5), 053420 (2012).
    [Crossref]
  23. L. Diósi, “Laser linewidth hazard in optomechanical cooling,” Phys. Rev. A 78(2), 021801(R) (2008).
    [Crossref]
  24. D. Breyer and M. Bienert, “Light scattering in an optomechanical cavity coupled to a single atom,” Phys. Rev. A 86(5), 053819 (2012).
    [Crossref]
  25. M. Bienert and G. Morigi, “Cavity cooling of a trapped atom using electromagnetically induced transparency,” New. J. Phys. 14(2), 023002 (2011).
    [Crossref]
  26. H. C. Nägerl, W. Bechter, J. Eschner, F. Schmidt-Kaler, and R. Blatt, “Ion strings for quantum gates,” Appl. Phys. B 66(5), 603–608 (1998).
    [Crossref]
  27. D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, “Quantum dynamics of single trapped ions,” Rev. Mod. Phys. 75(1), 281–324 (2003).
    [Crossref]
  28. J. I. Cirac, R. Blatt, P. Zoller, and W. D. Phillips, “Laser cooling of trapped ions in a standing wave,” Phys. Rev. A 46(5), 2668–2681 (1992).
    [Crossref] [PubMed]
  29. G. Morigi, J. Eschner, and C. H. Keitel, “Ground state laser cooling using electromagnetically induced transparency,” Phys. Rev. Lett. 85(21), 4458–4461 (2000).
    [Crossref] [PubMed]
  30. S. M. Tan, “A computational toolbox for qantum and atomic optics,” J. Opt. B 1(4), 424–432 (1999).
    [Crossref]
  31. F. Schmidt-Kaler, J. Eschner, G. Morigi, C.F. Roos, D. Leibfried, A. Mundt, and R. Blatt, “Laser cooling with electromagnetically induced transparency: application to trapped samples of ions or neutral atoms,” Appl. Phys. B 73(8), 807–814 (2001).
    [Crossref]

2015 (3)

R. Stevenson, M. R. Hush, T. Bishop, I. Lesanovsky, and T. Fernholz, “Sagnac interferometry with a single atomic clock,” Phys. Rev. Lett. 115(16), 163001 (2015).
[Crossref] [PubMed]

A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. O. Schmidt, “Optical atomic clocks,” Rev. Mod. Phys. 87(2), 637–701 (2015).
[Crossref]

Y. Lu, J. Q. Zhang, J. M. Cui, D. Y. Cao, S. Zhang, Y. F. Huang, C. F. Li, and G. C. Guo, “Dark-state cooling of a trapped ion using microwave coupling,” Phys. Rev. A 92(2), 023420 (2015).
[Crossref]

2014 (2)

V. Zhelyazkova, A. Cournol, T. E. Wall, A. Matsushima, J. J. Hudson, E. A. Hinds, M. R. Tarbutt, and B. E. Sauer, “Laser cooling and slowing of CaF molecules,” Phys. Rev. A 89(5), 053416 (2014).
[Crossref]

S. Zhang, J. Zhang, Q. Duan, C. Guo, C. Wu, W. Wu, and P. Chen, “Ground-state cooling of a trapped ion by quantum interference pathways,” Phys. Rev. A 90(4), 043409 (2014).
[Crossref]

2012 (3)

A. H. Safavi-Naeini, J. Chan, J. T. Hill, T. P. M. Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108(3), 033602 (2012).
[Crossref] [PubMed]

S. Zhang, C. W. Wu, and P. X. Chen, “Dark-state laser cooling of a trapped ion using standing waves,” Phys. Rev. A 85(5), 053420 (2012).
[Crossref]

D. Breyer and M. Bienert, “Light scattering in an optomechanical cavity coupled to a single atom,” Phys. Rev. A 86(5), 053819 (2012).
[Crossref]

2011 (4)

M. Bienert and G. Morigi, “Cavity cooling of a trapped atom using electromagnetically induced transparency,” New. J. Phys. 14(2), 023002 (2011).
[Crossref]

A. Albrecht, A. Retzker, C. Wunderlich, and M. B. Plenio, “Enhancement of laser cooling by the use of magnetic gradients,” New J. Phys. 13(3), 033009 (2011).
[Crossref]

J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
[Crossref] [PubMed]

M. Kowalewski, G. Morigi, P. W. H. Pinkse, and R. de Vivie-Riedle, “Cavity sideband cooling of trapped molecules,” Phys. Rev. A 84(3), 033408 (2011).
[Crossref]

2010 (2)

F. X. Esnault, D. Holleville, N. Rossetto, S. Guerandel, and N. Dimarcq, “High-stability compact atomic clock based on isotropic laser cooling,” Phys. Rev. A 82(3), 033436 (2010).
[Crossref]

J. Cerrillo, A. Retzker, and M. B. Plenio, “Fast and robust laser cooling of trapped systems,” Phys. Rev. Lett. 104(4), 043003 (2010).
[Crossref] [PubMed]

2008 (2)

L. Diósi, “Laser linewidth hazard in optomechanical cooling,” Phys. Rev. A 78(2), 021801(R) (2008).
[Crossref]

P. Barletta, J. Tennyson, and P. F. Barker, “Creating ultracold molecules by collisions with ultracold rare-gas atoms in an optical trap,” Phys. Rev. A 78(5), 052707 (2008).
[Crossref]

2004 (2)

Y. Shin, M. Saba, T. A. Pasquini, W. Ketterle, D. E. Pritchard, and A. E. Leanhardt, “Atom interferometry with Bose-Einstein condensates in a double-well protential,” Phys. Rev. Lett. 92(5), 050405 (2004).
[Crossref] [PubMed]

J. Evers and C. H. Keitel, “Double-EIT ground-state laser cooling without blue-sideband heating,” Europhys. Lett. 68(3), 370–376 (2004).
[Crossref]

2003 (2)

J. Eschner, G. Morigi, F. Schmidt-Kaler, and R. Blatt, “Laser cooling of trapped ions,” J. Opt. Soc. Am. B 20(5), 1003–1015 (2003).
[Crossref]

D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, “Quantum dynamics of single trapped ions,” Rev. Mod. Phys. 75(1), 281–324 (2003).
[Crossref]

2001 (1)

F. Schmidt-Kaler, J. Eschner, G. Morigi, C.F. Roos, D. Leibfried, A. Mundt, and R. Blatt, “Laser cooling with electromagnetically induced transparency: application to trapped samples of ions or neutral atoms,” Appl. Phys. B 73(8), 807–814 (2001).
[Crossref]

2000 (1)

G. Morigi, J. Eschner, and C. H. Keitel, “Ground state laser cooling using electromagnetically induced transparency,” Phys. Rev. Lett. 85(21), 4458–4461 (2000).
[Crossref] [PubMed]

1999 (4)

S. M. Tan, “A computational toolbox for qantum and atomic optics,” J. Opt. B 1(4), 424–432 (1999).
[Crossref]

C. E. Wieman, D. E. Pritchard, and D. J. Wineland, “Atom cooling, trapping, and quantum manipulation,” Rev. Mod. Phys. 71(2), s253–s262 (1999).
[Crossref]

J. Weiner, V. S. Bagnato, S. Zilio, and P. S. Julienne, “Experiments and theory in cold and ultracold collisions,” Rev. Mod. Phys. 71(1), 1–85 (1999).
[Crossref]

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of Bose-Einstein condensation in trapped gases,” Rev. Mod. Phys. 71(3), 463–512 (1999).
[Crossref]

1998 (1)

H. C. Nägerl, W. Bechter, J. Eschner, F. Schmidt-Kaler, and R. Blatt, “Ion strings for quantum gates,” Appl. Phys. B 66(5), 603–608 (1998).
[Crossref]

1995 (1)

K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Bose-Einstein condensation in a gas of sodium atoms,” Phys. Rev. Lett. 75(22), 3969–3973 (1995).
[Crossref] [PubMed]

1992 (1)

J. I. Cirac, R. Blatt, P. Zoller, and W. D. Phillips, “Laser cooling of trapped ions in a standing wave,” Phys. Rev. A 46(5), 2668–2681 (1992).
[Crossref] [PubMed]

1989 (1)

F. Diedrich, J. C. Bergquist, W. M. Itano, and D. J. Wineland, “Laser cooling to the zero-point energy of motion,” Phys. Rev. Lett. 62(4), 403–406 (1989).
[Crossref] [PubMed]

1986 (1)

S. Stenholm, “The semiclassical theory of laser cooling,” Rev. Mod. Phys. 58(3), 699–739 (1986).
[Crossref]

Albrecht, A.

A. Albrecht, A. Retzker, C. Wunderlich, and M. B. Plenio, “Enhancement of laser cooling by the use of magnetic gradients,” New J. Phys. 13(3), 033009 (2011).
[Crossref]

Alegre, T. P. M.

A. H. Safavi-Naeini, J. Chan, J. T. Hill, T. P. M. Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108(3), 033602 (2012).
[Crossref] [PubMed]

J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
[Crossref] [PubMed]

Andrews, M. R.

K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Bose-Einstein condensation in a gas of sodium atoms,” Phys. Rev. Lett. 75(22), 3969–3973 (1995).
[Crossref] [PubMed]

Aspelmeyer, M.

J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
[Crossref] [PubMed]

Bagnato, V. S.

J. Weiner, V. S. Bagnato, S. Zilio, and P. S. Julienne, “Experiments and theory in cold and ultracold collisions,” Rev. Mod. Phys. 71(1), 1–85 (1999).
[Crossref]

Barker, P. F.

P. Barletta, J. Tennyson, and P. F. Barker, “Creating ultracold molecules by collisions with ultracold rare-gas atoms in an optical trap,” Phys. Rev. A 78(5), 052707 (2008).
[Crossref]

Barletta, P.

P. Barletta, J. Tennyson, and P. F. Barker, “Creating ultracold molecules by collisions with ultracold rare-gas atoms in an optical trap,” Phys. Rev. A 78(5), 052707 (2008).
[Crossref]

Bechter, W.

H. C. Nägerl, W. Bechter, J. Eschner, F. Schmidt-Kaler, and R. Blatt, “Ion strings for quantum gates,” Appl. Phys. B 66(5), 603–608 (1998).
[Crossref]

Bergquist, J. C.

F. Diedrich, J. C. Bergquist, W. M. Itano, and D. J. Wineland, “Laser cooling to the zero-point energy of motion,” Phys. Rev. Lett. 62(4), 403–406 (1989).
[Crossref] [PubMed]

Bienert, M.

D. Breyer and M. Bienert, “Light scattering in an optomechanical cavity coupled to a single atom,” Phys. Rev. A 86(5), 053819 (2012).
[Crossref]

M. Bienert and G. Morigi, “Cavity cooling of a trapped atom using electromagnetically induced transparency,” New. J. Phys. 14(2), 023002 (2011).
[Crossref]

Bishop, T.

R. Stevenson, M. R. Hush, T. Bishop, I. Lesanovsky, and T. Fernholz, “Sagnac interferometry with a single atomic clock,” Phys. Rev. Lett. 115(16), 163001 (2015).
[Crossref] [PubMed]

Blatt, R.

J. Eschner, G. Morigi, F. Schmidt-Kaler, and R. Blatt, “Laser cooling of trapped ions,” J. Opt. Soc. Am. B 20(5), 1003–1015 (2003).
[Crossref]

D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, “Quantum dynamics of single trapped ions,” Rev. Mod. Phys. 75(1), 281–324 (2003).
[Crossref]

F. Schmidt-Kaler, J. Eschner, G. Morigi, C.F. Roos, D. Leibfried, A. Mundt, and R. Blatt, “Laser cooling with electromagnetically induced transparency: application to trapped samples of ions or neutral atoms,” Appl. Phys. B 73(8), 807–814 (2001).
[Crossref]

H. C. Nägerl, W. Bechter, J. Eschner, F. Schmidt-Kaler, and R. Blatt, “Ion strings for quantum gates,” Appl. Phys. B 66(5), 603–608 (1998).
[Crossref]

J. I. Cirac, R. Blatt, P. Zoller, and W. D. Phillips, “Laser cooling of trapped ions in a standing wave,” Phys. Rev. A 46(5), 2668–2681 (1992).
[Crossref] [PubMed]

Boyd, M. M.

A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. O. Schmidt, “Optical atomic clocks,” Rev. Mod. Phys. 87(2), 637–701 (2015).
[Crossref]

Breyer, D.

D. Breyer and M. Bienert, “Light scattering in an optomechanical cavity coupled to a single atom,” Phys. Rev. A 86(5), 053819 (2012).
[Crossref]

Cao, D. Y.

Y. Lu, J. Q. Zhang, J. M. Cui, D. Y. Cao, S. Zhang, Y. F. Huang, C. F. Li, and G. C. Guo, “Dark-state cooling of a trapped ion using microwave coupling,” Phys. Rev. A 92(2), 023420 (2015).
[Crossref]

Cerrillo, J.

J. Cerrillo, A. Retzker, and M. B. Plenio, “Fast and robust laser cooling of trapped systems,” Phys. Rev. Lett. 104(4), 043003 (2010).
[Crossref] [PubMed]

Chan, J.

A. H. Safavi-Naeini, J. Chan, J. T. Hill, T. P. M. Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108(3), 033602 (2012).
[Crossref] [PubMed]

J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
[Crossref] [PubMed]

Chen, P.

S. Zhang, J. Zhang, Q. Duan, C. Guo, C. Wu, W. Wu, and P. Chen, “Ground-state cooling of a trapped ion by quantum interference pathways,” Phys. Rev. A 90(4), 043409 (2014).
[Crossref]

Chen, P. X.

S. Zhang, C. W. Wu, and P. X. Chen, “Dark-state laser cooling of a trapped ion using standing waves,” Phys. Rev. A 85(5), 053420 (2012).
[Crossref]

Cirac, J. I.

J. I. Cirac, R. Blatt, P. Zoller, and W. D. Phillips, “Laser cooling of trapped ions in a standing wave,” Phys. Rev. A 46(5), 2668–2681 (1992).
[Crossref] [PubMed]

Cournol, A.

V. Zhelyazkova, A. Cournol, T. E. Wall, A. Matsushima, J. J. Hudson, E. A. Hinds, M. R. Tarbutt, and B. E. Sauer, “Laser cooling and slowing of CaF molecules,” Phys. Rev. A 89(5), 053416 (2014).
[Crossref]

Cui, J. M.

Y. Lu, J. Q. Zhang, J. M. Cui, D. Y. Cao, S. Zhang, Y. F. Huang, C. F. Li, and G. C. Guo, “Dark-state cooling of a trapped ion using microwave coupling,” Phys. Rev. A 92(2), 023420 (2015).
[Crossref]

Dalfovo, F.

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of Bose-Einstein condensation in trapped gases,” Rev. Mod. Phys. 71(3), 463–512 (1999).
[Crossref]

Davis, K. B.

K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Bose-Einstein condensation in a gas of sodium atoms,” Phys. Rev. Lett. 75(22), 3969–3973 (1995).
[Crossref] [PubMed]

de Vivie-Riedle, R.

M. Kowalewski, G. Morigi, P. W. H. Pinkse, and R. de Vivie-Riedle, “Cavity sideband cooling of trapped molecules,” Phys. Rev. A 84(3), 033408 (2011).
[Crossref]

Diedrich, F.

F. Diedrich, J. C. Bergquist, W. M. Itano, and D. J. Wineland, “Laser cooling to the zero-point energy of motion,” Phys. Rev. Lett. 62(4), 403–406 (1989).
[Crossref] [PubMed]

Dimarcq, N.

F. X. Esnault, D. Holleville, N. Rossetto, S. Guerandel, and N. Dimarcq, “High-stability compact atomic clock based on isotropic laser cooling,” Phys. Rev. A 82(3), 033436 (2010).
[Crossref]

Diósi, L.

L. Diósi, “Laser linewidth hazard in optomechanical cooling,” Phys. Rev. A 78(2), 021801(R) (2008).
[Crossref]

Duan, Q.

S. Zhang, J. Zhang, Q. Duan, C. Guo, C. Wu, W. Wu, and P. Chen, “Ground-state cooling of a trapped ion by quantum interference pathways,” Phys. Rev. A 90(4), 043409 (2014).
[Crossref]

Durfee, D. S.

K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Bose-Einstein condensation in a gas of sodium atoms,” Phys. Rev. Lett. 75(22), 3969–3973 (1995).
[Crossref] [PubMed]

Eschner, J.

J. Eschner, G. Morigi, F. Schmidt-Kaler, and R. Blatt, “Laser cooling of trapped ions,” J. Opt. Soc. Am. B 20(5), 1003–1015 (2003).
[Crossref]

F. Schmidt-Kaler, J. Eschner, G. Morigi, C.F. Roos, D. Leibfried, A. Mundt, and R. Blatt, “Laser cooling with electromagnetically induced transparency: application to trapped samples of ions or neutral atoms,” Appl. Phys. B 73(8), 807–814 (2001).
[Crossref]

G. Morigi, J. Eschner, and C. H. Keitel, “Ground state laser cooling using electromagnetically induced transparency,” Phys. Rev. Lett. 85(21), 4458–4461 (2000).
[Crossref] [PubMed]

H. C. Nägerl, W. Bechter, J. Eschner, F. Schmidt-Kaler, and R. Blatt, “Ion strings for quantum gates,” Appl. Phys. B 66(5), 603–608 (1998).
[Crossref]

Esnault, F. X.

F. X. Esnault, D. Holleville, N. Rossetto, S. Guerandel, and N. Dimarcq, “High-stability compact atomic clock based on isotropic laser cooling,” Phys. Rev. A 82(3), 033436 (2010).
[Crossref]

Evers, J.

J. Evers and C. H. Keitel, “Double-EIT ground-state laser cooling without blue-sideband heating,” Europhys. Lett. 68(3), 370–376 (2004).
[Crossref]

Fernholz, T.

R. Stevenson, M. R. Hush, T. Bishop, I. Lesanovsky, and T. Fernholz, “Sagnac interferometry with a single atomic clock,” Phys. Rev. Lett. 115(16), 163001 (2015).
[Crossref] [PubMed]

Giorgini, S.

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of Bose-Einstein condensation in trapped gases,” Rev. Mod. Phys. 71(3), 463–512 (1999).
[Crossref]

Gröblacher, S.

J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
[Crossref] [PubMed]

Guerandel, S.

F. X. Esnault, D. Holleville, N. Rossetto, S. Guerandel, and N. Dimarcq, “High-stability compact atomic clock based on isotropic laser cooling,” Phys. Rev. A 82(3), 033436 (2010).
[Crossref]

Guo, C.

S. Zhang, J. Zhang, Q. Duan, C. Guo, C. Wu, W. Wu, and P. Chen, “Ground-state cooling of a trapped ion by quantum interference pathways,” Phys. Rev. A 90(4), 043409 (2014).
[Crossref]

Guo, G. C.

Y. Lu, J. Q. Zhang, J. M. Cui, D. Y. Cao, S. Zhang, Y. F. Huang, C. F. Li, and G. C. Guo, “Dark-state cooling of a trapped ion using microwave coupling,” Phys. Rev. A 92(2), 023420 (2015).
[Crossref]

Hill, J. T.

A. H. Safavi-Naeini, J. Chan, J. T. Hill, T. P. M. Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108(3), 033602 (2012).
[Crossref] [PubMed]

J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
[Crossref] [PubMed]

Hinds, E. A.

V. Zhelyazkova, A. Cournol, T. E. Wall, A. Matsushima, J. J. Hudson, E. A. Hinds, M. R. Tarbutt, and B. E. Sauer, “Laser cooling and slowing of CaF molecules,” Phys. Rev. A 89(5), 053416 (2014).
[Crossref]

Holleville, D.

F. X. Esnault, D. Holleville, N. Rossetto, S. Guerandel, and N. Dimarcq, “High-stability compact atomic clock based on isotropic laser cooling,” Phys. Rev. A 82(3), 033436 (2010).
[Crossref]

Huang, Y. F.

Y. Lu, J. Q. Zhang, J. M. Cui, D. Y. Cao, S. Zhang, Y. F. Huang, C. F. Li, and G. C. Guo, “Dark-state cooling of a trapped ion using microwave coupling,” Phys. Rev. A 92(2), 023420 (2015).
[Crossref]

Hudson, J. J.

V. Zhelyazkova, A. Cournol, T. E. Wall, A. Matsushima, J. J. Hudson, E. A. Hinds, M. R. Tarbutt, and B. E. Sauer, “Laser cooling and slowing of CaF molecules,” Phys. Rev. A 89(5), 053416 (2014).
[Crossref]

Hush, M. R.

R. Stevenson, M. R. Hush, T. Bishop, I. Lesanovsky, and T. Fernholz, “Sagnac interferometry with a single atomic clock,” Phys. Rev. Lett. 115(16), 163001 (2015).
[Crossref] [PubMed]

Itano, W. M.

F. Diedrich, J. C. Bergquist, W. M. Itano, and D. J. Wineland, “Laser cooling to the zero-point energy of motion,” Phys. Rev. Lett. 62(4), 403–406 (1989).
[Crossref] [PubMed]

Julienne, P. S.

J. Weiner, V. S. Bagnato, S. Zilio, and P. S. Julienne, “Experiments and theory in cold and ultracold collisions,” Rev. Mod. Phys. 71(1), 1–85 (1999).
[Crossref]

Keitel, C. H.

J. Evers and C. H. Keitel, “Double-EIT ground-state laser cooling without blue-sideband heating,” Europhys. Lett. 68(3), 370–376 (2004).
[Crossref]

G. Morigi, J. Eschner, and C. H. Keitel, “Ground state laser cooling using electromagnetically induced transparency,” Phys. Rev. Lett. 85(21), 4458–4461 (2000).
[Crossref] [PubMed]

Ketterle, W.

Y. Shin, M. Saba, T. A. Pasquini, W. Ketterle, D. E. Pritchard, and A. E. Leanhardt, “Atom interferometry with Bose-Einstein condensates in a double-well protential,” Phys. Rev. Lett. 92(5), 050405 (2004).
[Crossref] [PubMed]

K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Bose-Einstein condensation in a gas of sodium atoms,” Phys. Rev. Lett. 75(22), 3969–3973 (1995).
[Crossref] [PubMed]

Kowalewski, M.

M. Kowalewski, G. Morigi, P. W. H. Pinkse, and R. de Vivie-Riedle, “Cavity sideband cooling of trapped molecules,” Phys. Rev. A 84(3), 033408 (2011).
[Crossref]

Krause, A.

A. H. Safavi-Naeini, J. Chan, J. T. Hill, T. P. M. Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108(3), 033602 (2012).
[Crossref] [PubMed]

J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
[Crossref] [PubMed]

Kurn, D. M.

K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Bose-Einstein condensation in a gas of sodium atoms,” Phys. Rev. Lett. 75(22), 3969–3973 (1995).
[Crossref] [PubMed]

Leanhardt, A. E.

Y. Shin, M. Saba, T. A. Pasquini, W. Ketterle, D. E. Pritchard, and A. E. Leanhardt, “Atom interferometry with Bose-Einstein condensates in a double-well protential,” Phys. Rev. Lett. 92(5), 050405 (2004).
[Crossref] [PubMed]

Leibfried, D.

D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, “Quantum dynamics of single trapped ions,” Rev. Mod. Phys. 75(1), 281–324 (2003).
[Crossref]

F. Schmidt-Kaler, J. Eschner, G. Morigi, C.F. Roos, D. Leibfried, A. Mundt, and R. Blatt, “Laser cooling with electromagnetically induced transparency: application to trapped samples of ions or neutral atoms,” Appl. Phys. B 73(8), 807–814 (2001).
[Crossref]

Lesanovsky, I.

R. Stevenson, M. R. Hush, T. Bishop, I. Lesanovsky, and T. Fernholz, “Sagnac interferometry with a single atomic clock,” Phys. Rev. Lett. 115(16), 163001 (2015).
[Crossref] [PubMed]

Li, C. F.

Y. Lu, J. Q. Zhang, J. M. Cui, D. Y. Cao, S. Zhang, Y. F. Huang, C. F. Li, and G. C. Guo, “Dark-state cooling of a trapped ion using microwave coupling,” Phys. Rev. A 92(2), 023420 (2015).
[Crossref]

Lu, Y.

Y. Lu, J. Q. Zhang, J. M. Cui, D. Y. Cao, S. Zhang, Y. F. Huang, C. F. Li, and G. C. Guo, “Dark-state cooling of a trapped ion using microwave coupling,” Phys. Rev. A 92(2), 023420 (2015).
[Crossref]

Ludlow, A. D.

A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. O. Schmidt, “Optical atomic clocks,” Rev. Mod. Phys. 87(2), 637–701 (2015).
[Crossref]

Matsushima, A.

V. Zhelyazkova, A. Cournol, T. E. Wall, A. Matsushima, J. J. Hudson, E. A. Hinds, M. R. Tarbutt, and B. E. Sauer, “Laser cooling and slowing of CaF molecules,” Phys. Rev. A 89(5), 053416 (2014).
[Crossref]

Mewes, M. O.

K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Bose-Einstein condensation in a gas of sodium atoms,” Phys. Rev. Lett. 75(22), 3969–3973 (1995).
[Crossref] [PubMed]

Monroe, C.

D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, “Quantum dynamics of single trapped ions,” Rev. Mod. Phys. 75(1), 281–324 (2003).
[Crossref]

Morigi, G.

M. Bienert and G. Morigi, “Cavity cooling of a trapped atom using electromagnetically induced transparency,” New. J. Phys. 14(2), 023002 (2011).
[Crossref]

M. Kowalewski, G. Morigi, P. W. H. Pinkse, and R. de Vivie-Riedle, “Cavity sideband cooling of trapped molecules,” Phys. Rev. A 84(3), 033408 (2011).
[Crossref]

J. Eschner, G. Morigi, F. Schmidt-Kaler, and R. Blatt, “Laser cooling of trapped ions,” J. Opt. Soc. Am. B 20(5), 1003–1015 (2003).
[Crossref]

F. Schmidt-Kaler, J. Eschner, G. Morigi, C.F. Roos, D. Leibfried, A. Mundt, and R. Blatt, “Laser cooling with electromagnetically induced transparency: application to trapped samples of ions or neutral atoms,” Appl. Phys. B 73(8), 807–814 (2001).
[Crossref]

G. Morigi, J. Eschner, and C. H. Keitel, “Ground state laser cooling using electromagnetically induced transparency,” Phys. Rev. Lett. 85(21), 4458–4461 (2000).
[Crossref] [PubMed]

Mundt, A.

F. Schmidt-Kaler, J. Eschner, G. Morigi, C.F. Roos, D. Leibfried, A. Mundt, and R. Blatt, “Laser cooling with electromagnetically induced transparency: application to trapped samples of ions or neutral atoms,” Appl. Phys. B 73(8), 807–814 (2001).
[Crossref]

Nägerl, H. C.

H. C. Nägerl, W. Bechter, J. Eschner, F. Schmidt-Kaler, and R. Blatt, “Ion strings for quantum gates,” Appl. Phys. B 66(5), 603–608 (1998).
[Crossref]

Painter, O.

A. H. Safavi-Naeini, J. Chan, J. T. Hill, T. P. M. Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108(3), 033602 (2012).
[Crossref] [PubMed]

J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
[Crossref] [PubMed]

Pasquini, T. A.

Y. Shin, M. Saba, T. A. Pasquini, W. Ketterle, D. E. Pritchard, and A. E. Leanhardt, “Atom interferometry with Bose-Einstein condensates in a double-well protential,” Phys. Rev. Lett. 92(5), 050405 (2004).
[Crossref] [PubMed]

Peik, E.

A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. O. Schmidt, “Optical atomic clocks,” Rev. Mod. Phys. 87(2), 637–701 (2015).
[Crossref]

Phillips, W. D.

J. I. Cirac, R. Blatt, P. Zoller, and W. D. Phillips, “Laser cooling of trapped ions in a standing wave,” Phys. Rev. A 46(5), 2668–2681 (1992).
[Crossref] [PubMed]

Pinkse, P. W. H.

M. Kowalewski, G. Morigi, P. W. H. Pinkse, and R. de Vivie-Riedle, “Cavity sideband cooling of trapped molecules,” Phys. Rev. A 84(3), 033408 (2011).
[Crossref]

Pitaevskii, L. P.

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of Bose-Einstein condensation in trapped gases,” Rev. Mod. Phys. 71(3), 463–512 (1999).
[Crossref]

Plenio, M. B.

A. Albrecht, A. Retzker, C. Wunderlich, and M. B. Plenio, “Enhancement of laser cooling by the use of magnetic gradients,” New J. Phys. 13(3), 033009 (2011).
[Crossref]

J. Cerrillo, A. Retzker, and M. B. Plenio, “Fast and robust laser cooling of trapped systems,” Phys. Rev. Lett. 104(4), 043003 (2010).
[Crossref] [PubMed]

Pritchard, D. E.

Y. Shin, M. Saba, T. A. Pasquini, W. Ketterle, D. E. Pritchard, and A. E. Leanhardt, “Atom interferometry with Bose-Einstein condensates in a double-well protential,” Phys. Rev. Lett. 92(5), 050405 (2004).
[Crossref] [PubMed]

C. E. Wieman, D. E. Pritchard, and D. J. Wineland, “Atom cooling, trapping, and quantum manipulation,” Rev. Mod. Phys. 71(2), s253–s262 (1999).
[Crossref]

Retzker, A.

A. Albrecht, A. Retzker, C. Wunderlich, and M. B. Plenio, “Enhancement of laser cooling by the use of magnetic gradients,” New J. Phys. 13(3), 033009 (2011).
[Crossref]

J. Cerrillo, A. Retzker, and M. B. Plenio, “Fast and robust laser cooling of trapped systems,” Phys. Rev. Lett. 104(4), 043003 (2010).
[Crossref] [PubMed]

Roos, C.F.

F. Schmidt-Kaler, J. Eschner, G. Morigi, C.F. Roos, D. Leibfried, A. Mundt, and R. Blatt, “Laser cooling with electromagnetically induced transparency: application to trapped samples of ions or neutral atoms,” Appl. Phys. B 73(8), 807–814 (2001).
[Crossref]

Rossetto, N.

F. X. Esnault, D. Holleville, N. Rossetto, S. Guerandel, and N. Dimarcq, “High-stability compact atomic clock based on isotropic laser cooling,” Phys. Rev. A 82(3), 033436 (2010).
[Crossref]

Saba, M.

Y. Shin, M. Saba, T. A. Pasquini, W. Ketterle, D. E. Pritchard, and A. E. Leanhardt, “Atom interferometry with Bose-Einstein condensates in a double-well protential,” Phys. Rev. Lett. 92(5), 050405 (2004).
[Crossref] [PubMed]

Safavi-Naeini, A. H.

A. H. Safavi-Naeini, J. Chan, J. T. Hill, T. P. M. Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108(3), 033602 (2012).
[Crossref] [PubMed]

J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
[Crossref] [PubMed]

Sauer, B. E.

V. Zhelyazkova, A. Cournol, T. E. Wall, A. Matsushima, J. J. Hudson, E. A. Hinds, M. R. Tarbutt, and B. E. Sauer, “Laser cooling and slowing of CaF molecules,” Phys. Rev. A 89(5), 053416 (2014).
[Crossref]

Schmidt, P. O.

A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. O. Schmidt, “Optical atomic clocks,” Rev. Mod. Phys. 87(2), 637–701 (2015).
[Crossref]

Schmidt-Kaler, F.

J. Eschner, G. Morigi, F. Schmidt-Kaler, and R. Blatt, “Laser cooling of trapped ions,” J. Opt. Soc. Am. B 20(5), 1003–1015 (2003).
[Crossref]

F. Schmidt-Kaler, J. Eschner, G. Morigi, C.F. Roos, D. Leibfried, A. Mundt, and R. Blatt, “Laser cooling with electromagnetically induced transparency: application to trapped samples of ions or neutral atoms,” Appl. Phys. B 73(8), 807–814 (2001).
[Crossref]

H. C. Nägerl, W. Bechter, J. Eschner, F. Schmidt-Kaler, and R. Blatt, “Ion strings for quantum gates,” Appl. Phys. B 66(5), 603–608 (1998).
[Crossref]

Shin, Y.

Y. Shin, M. Saba, T. A. Pasquini, W. Ketterle, D. E. Pritchard, and A. E. Leanhardt, “Atom interferometry with Bose-Einstein condensates in a double-well protential,” Phys. Rev. Lett. 92(5), 050405 (2004).
[Crossref] [PubMed]

Stenholm, S.

S. Stenholm, “The semiclassical theory of laser cooling,” Rev. Mod. Phys. 58(3), 699–739 (1986).
[Crossref]

Stevenson, R.

R. Stevenson, M. R. Hush, T. Bishop, I. Lesanovsky, and T. Fernholz, “Sagnac interferometry with a single atomic clock,” Phys. Rev. Lett. 115(16), 163001 (2015).
[Crossref] [PubMed]

Stringari, S.

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of Bose-Einstein condensation in trapped gases,” Rev. Mod. Phys. 71(3), 463–512 (1999).
[Crossref]

Tan, S. M.

S. M. Tan, “A computational toolbox for qantum and atomic optics,” J. Opt. B 1(4), 424–432 (1999).
[Crossref]

Tarbutt, M. R.

V. Zhelyazkova, A. Cournol, T. E. Wall, A. Matsushima, J. J. Hudson, E. A. Hinds, M. R. Tarbutt, and B. E. Sauer, “Laser cooling and slowing of CaF molecules,” Phys. Rev. A 89(5), 053416 (2014).
[Crossref]

Tennyson, J.

P. Barletta, J. Tennyson, and P. F. Barker, “Creating ultracold molecules by collisions with ultracold rare-gas atoms in an optical trap,” Phys. Rev. A 78(5), 052707 (2008).
[Crossref]

van Druten, N. J.

K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Bose-Einstein condensation in a gas of sodium atoms,” Phys. Rev. Lett. 75(22), 3969–3973 (1995).
[Crossref] [PubMed]

Wall, T. E.

V. Zhelyazkova, A. Cournol, T. E. Wall, A. Matsushima, J. J. Hudson, E. A. Hinds, M. R. Tarbutt, and B. E. Sauer, “Laser cooling and slowing of CaF molecules,” Phys. Rev. A 89(5), 053416 (2014).
[Crossref]

Weiner, J.

J. Weiner, V. S. Bagnato, S. Zilio, and P. S. Julienne, “Experiments and theory in cold and ultracold collisions,” Rev. Mod. Phys. 71(1), 1–85 (1999).
[Crossref]

Wieman, C. E.

C. E. Wieman, D. E. Pritchard, and D. J. Wineland, “Atom cooling, trapping, and quantum manipulation,” Rev. Mod. Phys. 71(2), s253–s262 (1999).
[Crossref]

Wineland, D.

D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, “Quantum dynamics of single trapped ions,” Rev. Mod. Phys. 75(1), 281–324 (2003).
[Crossref]

Wineland, D. J.

C. E. Wieman, D. E. Pritchard, and D. J. Wineland, “Atom cooling, trapping, and quantum manipulation,” Rev. Mod. Phys. 71(2), s253–s262 (1999).
[Crossref]

F. Diedrich, J. C. Bergquist, W. M. Itano, and D. J. Wineland, “Laser cooling to the zero-point energy of motion,” Phys. Rev. Lett. 62(4), 403–406 (1989).
[Crossref] [PubMed]

Wu, C.

S. Zhang, J. Zhang, Q. Duan, C. Guo, C. Wu, W. Wu, and P. Chen, “Ground-state cooling of a trapped ion by quantum interference pathways,” Phys. Rev. A 90(4), 043409 (2014).
[Crossref]

Wu, C. W.

S. Zhang, C. W. Wu, and P. X. Chen, “Dark-state laser cooling of a trapped ion using standing waves,” Phys. Rev. A 85(5), 053420 (2012).
[Crossref]

Wu, W.

S. Zhang, J. Zhang, Q. Duan, C. Guo, C. Wu, W. Wu, and P. Chen, “Ground-state cooling of a trapped ion by quantum interference pathways,” Phys. Rev. A 90(4), 043409 (2014).
[Crossref]

Wunderlich, C.

A. Albrecht, A. Retzker, C. Wunderlich, and M. B. Plenio, “Enhancement of laser cooling by the use of magnetic gradients,” New J. Phys. 13(3), 033009 (2011).
[Crossref]

Ye, J.

A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. O. Schmidt, “Optical atomic clocks,” Rev. Mod. Phys. 87(2), 637–701 (2015).
[Crossref]

Zhang, J.

S. Zhang, J. Zhang, Q. Duan, C. Guo, C. Wu, W. Wu, and P. Chen, “Ground-state cooling of a trapped ion by quantum interference pathways,” Phys. Rev. A 90(4), 043409 (2014).
[Crossref]

Zhang, J. Q.

Y. Lu, J. Q. Zhang, J. M. Cui, D. Y. Cao, S. Zhang, Y. F. Huang, C. F. Li, and G. C. Guo, “Dark-state cooling of a trapped ion using microwave coupling,” Phys. Rev. A 92(2), 023420 (2015).
[Crossref]

Zhang, S.

Y. Lu, J. Q. Zhang, J. M. Cui, D. Y. Cao, S. Zhang, Y. F. Huang, C. F. Li, and G. C. Guo, “Dark-state cooling of a trapped ion using microwave coupling,” Phys. Rev. A 92(2), 023420 (2015).
[Crossref]

S. Zhang, J. Zhang, Q. Duan, C. Guo, C. Wu, W. Wu, and P. Chen, “Ground-state cooling of a trapped ion by quantum interference pathways,” Phys. Rev. A 90(4), 043409 (2014).
[Crossref]

S. Zhang, C. W. Wu, and P. X. Chen, “Dark-state laser cooling of a trapped ion using standing waves,” Phys. Rev. A 85(5), 053420 (2012).
[Crossref]

Zhelyazkova, V.

V. Zhelyazkova, A. Cournol, T. E. Wall, A. Matsushima, J. J. Hudson, E. A. Hinds, M. R. Tarbutt, and B. E. Sauer, “Laser cooling and slowing of CaF molecules,” Phys. Rev. A 89(5), 053416 (2014).
[Crossref]

Zilio, S.

J. Weiner, V. S. Bagnato, S. Zilio, and P. S. Julienne, “Experiments and theory in cold and ultracold collisions,” Rev. Mod. Phys. 71(1), 1–85 (1999).
[Crossref]

Zoller, P.

J. I. Cirac, R. Blatt, P. Zoller, and W. D. Phillips, “Laser cooling of trapped ions in a standing wave,” Phys. Rev. A 46(5), 2668–2681 (1992).
[Crossref] [PubMed]

Appl. Phys. B (2)

H. C. Nägerl, W. Bechter, J. Eschner, F. Schmidt-Kaler, and R. Blatt, “Ion strings for quantum gates,” Appl. Phys. B 66(5), 603–608 (1998).
[Crossref]

F. Schmidt-Kaler, J. Eschner, G. Morigi, C.F. Roos, D. Leibfried, A. Mundt, and R. Blatt, “Laser cooling with electromagnetically induced transparency: application to trapped samples of ions or neutral atoms,” Appl. Phys. B 73(8), 807–814 (2001).
[Crossref]

Europhys. Lett. (1)

J. Evers and C. H. Keitel, “Double-EIT ground-state laser cooling without blue-sideband heating,” Europhys. Lett. 68(3), 370–376 (2004).
[Crossref]

J. Opt. B (1)

S. M. Tan, “A computational toolbox for qantum and atomic optics,” J. Opt. B 1(4), 424–432 (1999).
[Crossref]

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

Nature (1)

J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
[Crossref] [PubMed]

New J. Phys. (1)

A. Albrecht, A. Retzker, C. Wunderlich, and M. B. Plenio, “Enhancement of laser cooling by the use of magnetic gradients,” New J. Phys. 13(3), 033009 (2011).
[Crossref]

New. J. Phys. (1)

M. Bienert and G. Morigi, “Cavity cooling of a trapped atom using electromagnetically induced transparency,” New. J. Phys. 14(2), 023002 (2011).
[Crossref]

Phys. Rev. A (10)

S. Zhang, J. Zhang, Q. Duan, C. Guo, C. Wu, W. Wu, and P. Chen, “Ground-state cooling of a trapped ion by quantum interference pathways,” Phys. Rev. A 90(4), 043409 (2014).
[Crossref]

S. Zhang, C. W. Wu, and P. X. Chen, “Dark-state laser cooling of a trapped ion using standing waves,” Phys. Rev. A 85(5), 053420 (2012).
[Crossref]

L. Diósi, “Laser linewidth hazard in optomechanical cooling,” Phys. Rev. A 78(2), 021801(R) (2008).
[Crossref]

D. Breyer and M. Bienert, “Light scattering in an optomechanical cavity coupled to a single atom,” Phys. Rev. A 86(5), 053819 (2012).
[Crossref]

J. I. Cirac, R. Blatt, P. Zoller, and W. D. Phillips, “Laser cooling of trapped ions in a standing wave,” Phys. Rev. A 46(5), 2668–2681 (1992).
[Crossref] [PubMed]

Y. Lu, J. Q. Zhang, J. M. Cui, D. Y. Cao, S. Zhang, Y. F. Huang, C. F. Li, and G. C. Guo, “Dark-state cooling of a trapped ion using microwave coupling,” Phys. Rev. A 92(2), 023420 (2015).
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V. Zhelyazkova, A. Cournol, T. E. Wall, A. Matsushima, J. J. Hudson, E. A. Hinds, M. R. Tarbutt, and B. E. Sauer, “Laser cooling and slowing of CaF molecules,” Phys. Rev. A 89(5), 053416 (2014).
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Figures (4)

Fig. 1
Fig. 1 The level configuration of double EIT system. The excited state |e〉 couples to the ground states |g〉 (j = 1, 2, 3) by three lasers with strengths Ωj and detunings Δj, and γj are the spontaneous decay rates.
Fig. 2
Fig. 2 Single-phonon heating and cooling transitions governed by the interaction V1 in Eq. (14). Starting from the atomic dark state with nth-vibration |d, n〉, heating and cooling processes follow by |d, n〉 → |e, n ± 1〉 and subsequent laser-mediated transitions.
Fig. 3
Fig. 3 Two pathways of two-phonon heating transitions: (a) two-phonon excitation that starts from the initial dark states |d, n〉 governed by the interaction V2 in Eq. (14); (b) the further one-phonon transitions from the first-order perturbation state |g2, n + 1〉.
Fig. 4
Fig. 4 Numerical simulations of the cooling dynamics with different LD parameters of the second laser η̃2 = 0 (red-dotted line) and η̃2 = 0.13 (blue-solide line). The other parameters are γ = 69MHz, γ1 = γ2 = γ3 = γ/3, ν = 1.5MHz, Δ1 = Δ3 = Δ = 80MHz, η̃1 = 0.13, η̃3 = −0.13, Ω1 = 21MHz, Ω2 = 8MHz, Ω3 = 4MHz, and Δ2 = Δ − ν.

Tables (2)

Tables Icon

Table 1 Numerical simulations for the occupation of zero-, one-, and two-phonon excitation states with different values of LD parameter η̃2. The other parameters are γ = 69MHz, γ1 = γ2 = γ3 = γ/3, ν = 1.5MHz, Δ1 = Δ3 = Δ = 80MHz, η̃1 = 0.13, η̃3 = −0.13, Ω1 = 21MHz, Ω2 = 8MHz, Ω3 = 4MHz, and Δ2 = Δ − ν.

Tables Icon

Table 2 Numerical simulations of phonon occupations versus analytical results with different values of η̃2. The other parameters are γ = 69MHz, γ1 = γ2 = γ3 = γ/3, ν = 1.5MHz, Δ1 = Δ3 = Δ = 80MHz, η̃1 = 0.2, η̃3 = −0.1, Ω1 = 21MHz, Ω2 = 8MHz, Ω3 = 4MHz, and Δ2 = Δ − ν. Footnotes (a) and (b) denote numerical and analytical results.

Equations (38)

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ρ = | dark dark | | 0 0 | + O ( η 2 ) ,
H = ν b b + j [ Δ j σ g j g j + Ω j 2 ( e i k j cos ϕ j x σ e g j + H . c . ) ] ,
d d t ρ = i [ H , ρ ] + j j ρ ,
j ρ = γ j 2 ( 2 σ g j e ρ ˜ σ e g j σ e e ρ ρ σ e e ) ,
ρ ˜ = 1 2 1 1 𝒩 j ( cos θ ) e i k j x cos θ ρ e i k j x cos θ d cos θ ,
H = H 0 + V 1 + V 2 ,
H 0 = ν b b + j [ Δ j σ g j g j + Ω j 2 ( σ e g j + H . c . ) ] ,
V 1 = i j Ω j 2 η ˜ j ( σ e g j σ g j e ) ( b + b ) ,
V 2 = 1 2 j Ω j 2 η ˜ j 2 ( σ e g j + σ g j e ) ( b + b ) 2 .
j ( 0 ) ρ = γ j 2 ( 2 σ g j e ρ σ e g j σ e e ρ ρ σ e e ) ,
j ( 2 ) ρ = γ j 2 α j η j 2 [ 2 ( b + b ) ϱ j ( b + b ) ( b + b ) 2 ϱ j ϱ j ( b + b ) 2 ]
| c = 1 Ω ( Ω 1 | g 1 + Ω 3 | g 3 ) , | d = 1 Ω ( Ω 3 | g 1 Ω 1 | g 3 ) ,
H 0 = v b b + Δ ( σ c c + σ d d ) + Δ 2 σ g 2 g 2 + Ω 2 ( σ e c + σ c e ) + Ω 2 2 ( σ e g 2 + σ g 2 e ) ,
V 1 = i 2 [ Ω 2 η ˜ 2 ( σ e g 2 σ g 2 e ) + Ω 1 2 η ˜ 1 + Ω 3 2 η ˜ 3 Ω ( σ e c σ c e ) + Ω 1 Ω 3 Ω ( η ˜ 1 η ˜ 3 ) ( σ e d σ d e ) ] ( b + b ) , V 2 = [ Ω 2 η ˜ 2 2 4 ( σ e g 2 + σ g 2 e ) + Ω 1 2 η ˜ 1 2 + Ω 3 2 η ˜ 3 2 4 Ω ( σ e c + σ c e ) + Ω 1 Ω 3 ( η ˜ 1 2 η ˜ 3 2 ) 4 Ω ( σ e d + σ d e ) ] ( b + b ) 2 .
| ψ 1 = | ψ 1 n + 1 + | ψ 1 n 1 ,
| ψ 1 n ± 1 = C e n ± 1 | e , n ± 1 + C c n ± 1 | c , n ± 1 + C g 2 n ± 1 | g 2 , n ± 1 .
i d d t | ψ 1 = H eff | ψ 1 + V 1 | ψ 0 ,
i C ˙ e n ± 1 = [ ( n ± 1 ) ν i γ 2 ] C e n ± 1 + Ω 2 C c n ± 1 + Ω 2 2 C g 2 n ± 1 i Ω 1 Ω 3 2 Ω ( η ˜ 1 η ˜ 3 ) e i ( Δ + n ν ) t n + δ ± , i C ˙ g 2 n ± 1 = [ Δ 2 + ( n ± 1 ) ν ] C g 2 n ± 1 + Ω 2 2 C e n ± 1 , i C ˙ c n ± 1 = [ Δ + ( n ± 1 ) ν ] C c n ± 1 + Ω 2 C e n ± 1 ,
i C ˜ e n ± 1 = [ Δ + ν i γ 2 ] C ˜ e n ± 1 + Ω 2 C ˜ c n ± 1 + Ω 2 2 C ˜ g 2 n ± 1 i Ω 1 Ω 3 2 Ω ( η ˜ 1 η ˜ 3 ) n + δ ± , i C ˜ ˙ g 2 n ± 1 = [ Δ 2 Δ ± ν ] C ˜ g 2 n ± 1 + Ω 2 2 C ˜ e n ± 1 , i C ˜ ˙ c n ± 1 = ± ν C ˜ c n ± 1 + Ω 2 C ˜ e n ± 1 .
Γ n n ± 1 = γ | C ˜ e n ± 1 | 2 .
Δ 2 Δ + ν = 0 ,
C ˜ c n + 1 = 0 , C ˜ g 2 n + 1 = i Ω 1 Ω 3 Ω 2 Ω ( η ˜ 1 η ˜ 3 ) n + 1 .
Γ n n 1 = n A = n γ ( η ˜ 1 η ˜ 3 ) 2 ν 2 Ω 1 2 Ω 3 2 / Ω 2 γ 2 ν 2 + 4 [ ν ( Δ + ν ) + Ω 1 2 4 + Ω 2 2 8 + Ω 3 2 4 ] 2 ,
ν = 1 2 ( Δ 2 + Ω 1 2 + Ω 2 2 / 2 + Ω 3 2 Δ ) ,
A = ( η ˜ 1 η ˜ 3 ) 2 Ω 1 2 Ω 2 Ω 3 2 γ ,
| Ψ f = | d , 0 + i Ω 1 Ω 3 Ω 2 Ω ( η ˜ 1 η ˜ 3 ) | g 2 , 1 + O ( η 2 ) .
P 1 = Ω 1 2 Ω 3 2 Ω 2 2 Ω 2 ( η ˜ 1 η ˜ 3 ) 2 ,
| ψ 2 n + 2 = C e n + 2 | e , n + 2 + C g 2 n + 2 | g 2 , n + 2 + C c n + 2 | c , n + 2 ,
i d d t | ψ 2 n + 2 = H eff | ψ 2 n + 2 + V 1 | ψ 1 n + 1 + V 2 | ψ 0 .
i C ˜ ˙ e n + 2 = [ Δ + 2 ν i γ 2 ] C ˜ e n + 2 + Ω 2 C ˜ c n + 2 + Ω 2 2 C ˜ g 2 n + 2 Ω 1 Ω 3 4 Ω ( η ˜ 1 η ˜ 3 ) η ˜ ( n + 1 ) ( n + 2 ) , i C ˜ ˙ g 2 n + 2 = ν C ˜ g 2 n + 2 + Ω 2 2 C ˜ e n + 2 , i C ˜ ˙ c n + 2 = 2 ν C ˜ c n + 2 + Ω 2 C ˜ e n + 2 ,
C ˜ e n + 2 = ν Ω 1 Ω 3 ( η ˜ 1 η ˜ 3 ) η ˜ ( n + 1 ) ( n + 2 ) 4 Ω [ i γ 2 ν + ν ( Δ 2 ν ) + Ω 2 8 + Ω 2 2 4 ] ,
Γ n n + 2 = ( n + 1 ) ( n + 2 ) A + ( 2 ) = γ | C ˜ e n + 2 | 2 ,
A + ( 2 ) = γ ν 2 Ω 1 2 Ω 3 2 ( η ˜ 1 η ˜ 3 ) 2 η ˜ 2 16 Ω 2 { γ 2 4 ν 2 + [ ν ( Δ 2 ν ) + Ω 2 8 + Ω 2 2 4 ] 2 }
d d t P n = Γ n 2 n P n 2 Γ n n + 2 P n + Γ n + 1 n P n + 1 Γ n n 1 P n ,
d d t n = A n + 2 A + ( 2 ) ( n 2 + 3 n + 2 ) .
n c 4 A + ( 2 ) / A ,
n s s = P 1 + n c .
η ˜ = η ˜ 1 + η ˜ 3 2 η ˜ 2 = 0 ,

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