Abstract

We demonstrate the use of the ultrafast spatial coherent-control method to resolve the fine-structure two-photon transitions of atomic rubidium. Counter-propagating ultrafast optical pulses with spectral phase and amplitude programmed with our optimized solutions successfully induced the two-photon transitions through 5S1/2-5P1/2-5D and 5S1/2-5P3/2-5D pathways, both simultaneously and at distinct spatial locations. Three different pulse-shaping solutions are introduced that combine amplitude shaping, which avoids direct intermediate resonances, and phase programming, which enables the remaining spectral components to be coherently interfered through the targeted transition pathways. Experiments were performed with a room-temperature vapor cell, and the results agree well with theoretical analysis.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. K. Bergmann, H. Theuer, and B. W. Shore, “Coherent population transfer among quantum states of atoms and molecules,” Rev. Mod. Phys. 70, 1003–1025 (1998).
    [Crossref]
  2. M. Shapiro and P. Brumer, Principles of the Quantum Control of Molecular Processes (Wiley, New York, 2003).
  3. D.J. Tanner and S. A. Rice, “Control of selectivity of chemical reaction via control of wavepacket evolution,” J. Chem. Phys. 83, 5013–5018 (1985).
    [Crossref]
  4. N. Dudovich, B. Dayan, S. M. Gallagher Faeder, and Y. Silberberg, “Transform-limited pulses are not optimal for resonant multiphoton transitions,” Phys. Rev. Lett. 86, 47–50 (2001).
    [Crossref] [PubMed]
  5. S. Lee, J. Lim, and J. Ahn, “Strong-field two-photon absorption in atomic Cesium: an analytic approach,” Opt. Express 17, 7648 (2009).
    [Crossref] [PubMed]
  6. H. G. Lee, H. Kim, J. Lim, and J. Ahn, “Quantum interference control of four-level diamond-configuration quantum system,” Phys. Rev. A 88, 053427 (2013).
    [Crossref]
  7. M. C. Stowe, A. Pe’er, and J. Ye, “Control of Four-Level Quantum Coherence via Discrete Spectral Shaping of an Optical Frequency Comb,” Phys. Rev. Lett. 100, 203001 (2008).
    [Crossref] [PubMed]
  8. D. Meshulach and Y. Silberberg, “Coherent quantum control of two-photon transitions by a femtosecond laser pulse,” Nature 396, 239 (1998).
    [Crossref]
  9. R. S. Judson and H. Rabitz, “Teaching lasers to control molecules,” Phys. Rev. Lett. 68, 1500–1503 (1992).
    [Crossref] [PubMed]
  10. J. J. García-Ripoll, P. Zoller, and J. I. Cirac, “Speed Optimized Two-Qubit Gates with Laser Coherent Control Techniques for Ion Trap Quantum Computing,” Phys. Rev. Lett. 91, 157901 (2003).
    [Crossref] [PubMed]
  11. X. Wang, C. Jin, and C. D. Lin, “Coherent control of high-harmonic generation using waveform-synthesized chirped laser fields,” Phys. Rev. A 90, 023416 (2014).
    [Crossref]
  12. N. Dudovich, D. Oron, and Y. Silberberg, “Single-pulse coherently controlled nonlinear Raman spectroscopy and microscopy,” Nature 418, 512 (2002).
    [Crossref] [PubMed]
  13. I. Barmes, S. Witte, and K. S. E. Eikema, “Spatial and spectral coherent control with frequency combs,” Nat. Photonics 7, 38 (2013).
    [Crossref]
  14. D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635 (2000).
    [Crossref] [PubMed]
  15. R. Holzwarth, Th. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Optical frequency synthesizer for precision spectroscopy,” Phys. Rev. Lett. 85, 2264 (2000).
    [Crossref] [PubMed]
  16. I. Barmes, S. Witte, and K. S. E. Eikema, “High-precision spectroscopy with counterpropagating femtosecond pulses,” Phys. Rev. Lett. 111, 023007 (2013).
    [Crossref] [PubMed]
  17. C. Wieman and T. W. Hänsch, “Doppler-free laser polarization spectroscopy,” Phys. Rev. Lett. 36, 1170 (1976).
    [Crossref]
  18. W. Lee, H. Kim, K. Kim, and J. Ahn, “Coherent control of resonant two-photon transitions by counter-propagating ultrashort pulse pairs,” Phys. Rev. A 92, 033415 (2015).
    [Crossref]
  19. F. Gao, R. Rey-de-Castro, A.M. Donovan, J. Xu, Y. Wang, H. Rabitz, and F. Shuang, “Pathway dynamics in the optimal quantum control of rubidium: Cooperation and competition,” Phys. Rev. A 89, 023416 (2014).
    [Crossref]
  20. F. Gao, Y. Wang, R. Rey-de-Castro, H. Rabitz, and F. Shuang, “Quantum control and pathway manipulation in rubidium,” Phys. Rev. A 92, 033423 (2015).
    [Crossref]
  21. F. Gao, R. Rey-de-Castro, Y. Wang, H. Rabitz, and F. Shuang, “Identifying a cooperative control mechanism between an applied field and the environment of open quantum systems,” Phys. Rev. A 93, 053407 (2016).
    [Crossref]
  22. A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71, 1929 (2000).
    [Crossref]
  23. D. Cao, L. Yang, Y. Wang, F. Shuang, and F. Gao, “Controlling pathway dynamics of a four-level quantum system with pulse shaping,” J. Phys. A Math. Theor. 49, 285302 (2016).
    [Crossref]
  24. D. Cao, Y. Wang, S. Li, L. Yang, F. Shuang, and F. Gao, “Optimal control of multiple two-photon transitions,” J. Math. Chem. 55, 1053–1066 (2017).
    [Crossref]

2017 (1)

D. Cao, Y. Wang, S. Li, L. Yang, F. Shuang, and F. Gao, “Optimal control of multiple two-photon transitions,” J. Math. Chem. 55, 1053–1066 (2017).
[Crossref]

2016 (2)

D. Cao, L. Yang, Y. Wang, F. Shuang, and F. Gao, “Controlling pathway dynamics of a four-level quantum system with pulse shaping,” J. Phys. A Math. Theor. 49, 285302 (2016).
[Crossref]

F. Gao, R. Rey-de-Castro, Y. Wang, H. Rabitz, and F. Shuang, “Identifying a cooperative control mechanism between an applied field and the environment of open quantum systems,” Phys. Rev. A 93, 053407 (2016).
[Crossref]

2015 (2)

F. Gao, Y. Wang, R. Rey-de-Castro, H. Rabitz, and F. Shuang, “Quantum control and pathway manipulation in rubidium,” Phys. Rev. A 92, 033423 (2015).
[Crossref]

W. Lee, H. Kim, K. Kim, and J. Ahn, “Coherent control of resonant two-photon transitions by counter-propagating ultrashort pulse pairs,” Phys. Rev. A 92, 033415 (2015).
[Crossref]

2014 (2)

F. Gao, R. Rey-de-Castro, A.M. Donovan, J. Xu, Y. Wang, H. Rabitz, and F. Shuang, “Pathway dynamics in the optimal quantum control of rubidium: Cooperation and competition,” Phys. Rev. A 89, 023416 (2014).
[Crossref]

X. Wang, C. Jin, and C. D. Lin, “Coherent control of high-harmonic generation using waveform-synthesized chirped laser fields,” Phys. Rev. A 90, 023416 (2014).
[Crossref]

2013 (3)

I. Barmes, S. Witte, and K. S. E. Eikema, “Spatial and spectral coherent control with frequency combs,” Nat. Photonics 7, 38 (2013).
[Crossref]

H. G. Lee, H. Kim, J. Lim, and J. Ahn, “Quantum interference control of four-level diamond-configuration quantum system,” Phys. Rev. A 88, 053427 (2013).
[Crossref]

I. Barmes, S. Witte, and K. S. E. Eikema, “High-precision spectroscopy with counterpropagating femtosecond pulses,” Phys. Rev. Lett. 111, 023007 (2013).
[Crossref] [PubMed]

2009 (1)

2008 (1)

M. C. Stowe, A. Pe’er, and J. Ye, “Control of Four-Level Quantum Coherence via Discrete Spectral Shaping of an Optical Frequency Comb,” Phys. Rev. Lett. 100, 203001 (2008).
[Crossref] [PubMed]

2003 (1)

J. J. García-Ripoll, P. Zoller, and J. I. Cirac, “Speed Optimized Two-Qubit Gates with Laser Coherent Control Techniques for Ion Trap Quantum Computing,” Phys. Rev. Lett. 91, 157901 (2003).
[Crossref] [PubMed]

2002 (1)

N. Dudovich, D. Oron, and Y. Silberberg, “Single-pulse coherently controlled nonlinear Raman spectroscopy and microscopy,” Nature 418, 512 (2002).
[Crossref] [PubMed]

2001 (1)

N. Dudovich, B. Dayan, S. M. Gallagher Faeder, and Y. Silberberg, “Transform-limited pulses are not optimal for resonant multiphoton transitions,” Phys. Rev. Lett. 86, 47–50 (2001).
[Crossref] [PubMed]

2000 (3)

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635 (2000).
[Crossref] [PubMed]

R. Holzwarth, Th. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Optical frequency synthesizer for precision spectroscopy,” Phys. Rev. Lett. 85, 2264 (2000).
[Crossref] [PubMed]

A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71, 1929 (2000).
[Crossref]

1998 (2)

D. Meshulach and Y. Silberberg, “Coherent quantum control of two-photon transitions by a femtosecond laser pulse,” Nature 396, 239 (1998).
[Crossref]

K. Bergmann, H. Theuer, and B. W. Shore, “Coherent population transfer among quantum states of atoms and molecules,” Rev. Mod. Phys. 70, 1003–1025 (1998).
[Crossref]

1992 (1)

R. S. Judson and H. Rabitz, “Teaching lasers to control molecules,” Phys. Rev. Lett. 68, 1500–1503 (1992).
[Crossref] [PubMed]

1985 (1)

D.J. Tanner and S. A. Rice, “Control of selectivity of chemical reaction via control of wavepacket evolution,” J. Chem. Phys. 83, 5013–5018 (1985).
[Crossref]

1976 (1)

C. Wieman and T. W. Hänsch, “Doppler-free laser polarization spectroscopy,” Phys. Rev. Lett. 36, 1170 (1976).
[Crossref]

Ahn, J.

W. Lee, H. Kim, K. Kim, and J. Ahn, “Coherent control of resonant two-photon transitions by counter-propagating ultrashort pulse pairs,” Phys. Rev. A 92, 033415 (2015).
[Crossref]

H. G. Lee, H. Kim, J. Lim, and J. Ahn, “Quantum interference control of four-level diamond-configuration quantum system,” Phys. Rev. A 88, 053427 (2013).
[Crossref]

S. Lee, J. Lim, and J. Ahn, “Strong-field two-photon absorption in atomic Cesium: an analytic approach,” Opt. Express 17, 7648 (2009).
[Crossref] [PubMed]

Barmes, I.

I. Barmes, S. Witte, and K. S. E. Eikema, “Spatial and spectral coherent control with frequency combs,” Nat. Photonics 7, 38 (2013).
[Crossref]

I. Barmes, S. Witte, and K. S. E. Eikema, “High-precision spectroscopy with counterpropagating femtosecond pulses,” Phys. Rev. Lett. 111, 023007 (2013).
[Crossref] [PubMed]

Bergmann, K.

K. Bergmann, H. Theuer, and B. W. Shore, “Coherent population transfer among quantum states of atoms and molecules,” Rev. Mod. Phys. 70, 1003–1025 (1998).
[Crossref]

Brumer, P.

M. Shapiro and P. Brumer, Principles of the Quantum Control of Molecular Processes (Wiley, New York, 2003).

Cao, D.

D. Cao, Y. Wang, S. Li, L. Yang, F. Shuang, and F. Gao, “Optimal control of multiple two-photon transitions,” J. Math. Chem. 55, 1053–1066 (2017).
[Crossref]

D. Cao, L. Yang, Y. Wang, F. Shuang, and F. Gao, “Controlling pathway dynamics of a four-level quantum system with pulse shaping,” J. Phys. A Math. Theor. 49, 285302 (2016).
[Crossref]

Cirac, J. I.

J. J. García-Ripoll, P. Zoller, and J. I. Cirac, “Speed Optimized Two-Qubit Gates with Laser Coherent Control Techniques for Ion Trap Quantum Computing,” Phys. Rev. Lett. 91, 157901 (2003).
[Crossref] [PubMed]

Cundiff, S. T.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635 (2000).
[Crossref] [PubMed]

Dayan, B.

N. Dudovich, B. Dayan, S. M. Gallagher Faeder, and Y. Silberberg, “Transform-limited pulses are not optimal for resonant multiphoton transitions,” Phys. Rev. Lett. 86, 47–50 (2001).
[Crossref] [PubMed]

Diddams, S. A.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635 (2000).
[Crossref] [PubMed]

Donovan, A.M.

F. Gao, R. Rey-de-Castro, A.M. Donovan, J. Xu, Y. Wang, H. Rabitz, and F. Shuang, “Pathway dynamics in the optimal quantum control of rubidium: Cooperation and competition,” Phys. Rev. A 89, 023416 (2014).
[Crossref]

Dudovich, N.

N. Dudovich, D. Oron, and Y. Silberberg, “Single-pulse coherently controlled nonlinear Raman spectroscopy and microscopy,” Nature 418, 512 (2002).
[Crossref] [PubMed]

N. Dudovich, B. Dayan, S. M. Gallagher Faeder, and Y. Silberberg, “Transform-limited pulses are not optimal for resonant multiphoton transitions,” Phys. Rev. Lett. 86, 47–50 (2001).
[Crossref] [PubMed]

Eikema, K. S. E.

I. Barmes, S. Witte, and K. S. E. Eikema, “High-precision spectroscopy with counterpropagating femtosecond pulses,” Phys. Rev. Lett. 111, 023007 (2013).
[Crossref] [PubMed]

I. Barmes, S. Witte, and K. S. E. Eikema, “Spatial and spectral coherent control with frequency combs,” Nat. Photonics 7, 38 (2013).
[Crossref]

Gallagher Faeder, S. M.

N. Dudovich, B. Dayan, S. M. Gallagher Faeder, and Y. Silberberg, “Transform-limited pulses are not optimal for resonant multiphoton transitions,” Phys. Rev. Lett. 86, 47–50 (2001).
[Crossref] [PubMed]

Gao, F.

D. Cao, Y. Wang, S. Li, L. Yang, F. Shuang, and F. Gao, “Optimal control of multiple two-photon transitions,” J. Math. Chem. 55, 1053–1066 (2017).
[Crossref]

F. Gao, R. Rey-de-Castro, Y. Wang, H. Rabitz, and F. Shuang, “Identifying a cooperative control mechanism between an applied field and the environment of open quantum systems,” Phys. Rev. A 93, 053407 (2016).
[Crossref]

D. Cao, L. Yang, Y. Wang, F. Shuang, and F. Gao, “Controlling pathway dynamics of a four-level quantum system with pulse shaping,” J. Phys. A Math. Theor. 49, 285302 (2016).
[Crossref]

F. Gao, Y. Wang, R. Rey-de-Castro, H. Rabitz, and F. Shuang, “Quantum control and pathway manipulation in rubidium,” Phys. Rev. A 92, 033423 (2015).
[Crossref]

F. Gao, R. Rey-de-Castro, A.M. Donovan, J. Xu, Y. Wang, H. Rabitz, and F. Shuang, “Pathway dynamics in the optimal quantum control of rubidium: Cooperation and competition,” Phys. Rev. A 89, 023416 (2014).
[Crossref]

García-Ripoll, J. J.

J. J. García-Ripoll, P. Zoller, and J. I. Cirac, “Speed Optimized Two-Qubit Gates with Laser Coherent Control Techniques for Ion Trap Quantum Computing,” Phys. Rev. Lett. 91, 157901 (2003).
[Crossref] [PubMed]

Hall, J. L.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635 (2000).
[Crossref] [PubMed]

Hänsch, T. W.

R. Holzwarth, Th. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Optical frequency synthesizer for precision spectroscopy,” Phys. Rev. Lett. 85, 2264 (2000).
[Crossref] [PubMed]

C. Wieman and T. W. Hänsch, “Doppler-free laser polarization spectroscopy,” Phys. Rev. Lett. 36, 1170 (1976).
[Crossref]

Holzwarth, R.

R. Holzwarth, Th. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Optical frequency synthesizer for precision spectroscopy,” Phys. Rev. Lett. 85, 2264 (2000).
[Crossref] [PubMed]

Jin, C.

X. Wang, C. Jin, and C. D. Lin, “Coherent control of high-harmonic generation using waveform-synthesized chirped laser fields,” Phys. Rev. A 90, 023416 (2014).
[Crossref]

Jones, D. J.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635 (2000).
[Crossref] [PubMed]

Judson, R. S.

R. S. Judson and H. Rabitz, “Teaching lasers to control molecules,” Phys. Rev. Lett. 68, 1500–1503 (1992).
[Crossref] [PubMed]

Kim, H.

W. Lee, H. Kim, K. Kim, and J. Ahn, “Coherent control of resonant two-photon transitions by counter-propagating ultrashort pulse pairs,” Phys. Rev. A 92, 033415 (2015).
[Crossref]

H. G. Lee, H. Kim, J. Lim, and J. Ahn, “Quantum interference control of four-level diamond-configuration quantum system,” Phys. Rev. A 88, 053427 (2013).
[Crossref]

Kim, K.

W. Lee, H. Kim, K. Kim, and J. Ahn, “Coherent control of resonant two-photon transitions by counter-propagating ultrashort pulse pairs,” Phys. Rev. A 92, 033415 (2015).
[Crossref]

Knight, J. C.

R. Holzwarth, Th. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Optical frequency synthesizer for precision spectroscopy,” Phys. Rev. Lett. 85, 2264 (2000).
[Crossref] [PubMed]

Lee, H. G.

H. G. Lee, H. Kim, J. Lim, and J. Ahn, “Quantum interference control of four-level diamond-configuration quantum system,” Phys. Rev. A 88, 053427 (2013).
[Crossref]

Lee, S.

Lee, W.

W. Lee, H. Kim, K. Kim, and J. Ahn, “Coherent control of resonant two-photon transitions by counter-propagating ultrashort pulse pairs,” Phys. Rev. A 92, 033415 (2015).
[Crossref]

Li, S.

D. Cao, Y. Wang, S. Li, L. Yang, F. Shuang, and F. Gao, “Optimal control of multiple two-photon transitions,” J. Math. Chem. 55, 1053–1066 (2017).
[Crossref]

Lim, J.

H. G. Lee, H. Kim, J. Lim, and J. Ahn, “Quantum interference control of four-level diamond-configuration quantum system,” Phys. Rev. A 88, 053427 (2013).
[Crossref]

S. Lee, J. Lim, and J. Ahn, “Strong-field two-photon absorption in atomic Cesium: an analytic approach,” Opt. Express 17, 7648 (2009).
[Crossref] [PubMed]

Lin, C. D.

X. Wang, C. Jin, and C. D. Lin, “Coherent control of high-harmonic generation using waveform-synthesized chirped laser fields,” Phys. Rev. A 90, 023416 (2014).
[Crossref]

Meshulach, D.

D. Meshulach and Y. Silberberg, “Coherent quantum control of two-photon transitions by a femtosecond laser pulse,” Nature 396, 239 (1998).
[Crossref]

Oron, D.

N. Dudovich, D. Oron, and Y. Silberberg, “Single-pulse coherently controlled nonlinear Raman spectroscopy and microscopy,” Nature 418, 512 (2002).
[Crossref] [PubMed]

Pe’er, A.

M. C. Stowe, A. Pe’er, and J. Ye, “Control of Four-Level Quantum Coherence via Discrete Spectral Shaping of an Optical Frequency Comb,” Phys. Rev. Lett. 100, 203001 (2008).
[Crossref] [PubMed]

Rabitz, H.

F. Gao, R. Rey-de-Castro, Y. Wang, H. Rabitz, and F. Shuang, “Identifying a cooperative control mechanism between an applied field and the environment of open quantum systems,” Phys. Rev. A 93, 053407 (2016).
[Crossref]

F. Gao, Y. Wang, R. Rey-de-Castro, H. Rabitz, and F. Shuang, “Quantum control and pathway manipulation in rubidium,” Phys. Rev. A 92, 033423 (2015).
[Crossref]

F. Gao, R. Rey-de-Castro, A.M. Donovan, J. Xu, Y. Wang, H. Rabitz, and F. Shuang, “Pathway dynamics in the optimal quantum control of rubidium: Cooperation and competition,” Phys. Rev. A 89, 023416 (2014).
[Crossref]

R. S. Judson and H. Rabitz, “Teaching lasers to control molecules,” Phys. Rev. Lett. 68, 1500–1503 (1992).
[Crossref] [PubMed]

Ranka, J. K.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635 (2000).
[Crossref] [PubMed]

Rey-de-Castro, R.

F. Gao, R. Rey-de-Castro, Y. Wang, H. Rabitz, and F. Shuang, “Identifying a cooperative control mechanism between an applied field and the environment of open quantum systems,” Phys. Rev. A 93, 053407 (2016).
[Crossref]

F. Gao, Y. Wang, R. Rey-de-Castro, H. Rabitz, and F. Shuang, “Quantum control and pathway manipulation in rubidium,” Phys. Rev. A 92, 033423 (2015).
[Crossref]

F. Gao, R. Rey-de-Castro, A.M. Donovan, J. Xu, Y. Wang, H. Rabitz, and F. Shuang, “Pathway dynamics in the optimal quantum control of rubidium: Cooperation and competition,” Phys. Rev. A 89, 023416 (2014).
[Crossref]

Rice, S. A.

D.J. Tanner and S. A. Rice, “Control of selectivity of chemical reaction via control of wavepacket evolution,” J. Chem. Phys. 83, 5013–5018 (1985).
[Crossref]

Russell, P. St. J.

R. Holzwarth, Th. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Optical frequency synthesizer for precision spectroscopy,” Phys. Rev. Lett. 85, 2264 (2000).
[Crossref] [PubMed]

Shapiro, M.

M. Shapiro and P. Brumer, Principles of the Quantum Control of Molecular Processes (Wiley, New York, 2003).

Shore, B. W.

K. Bergmann, H. Theuer, and B. W. Shore, “Coherent population transfer among quantum states of atoms and molecules,” Rev. Mod. Phys. 70, 1003–1025 (1998).
[Crossref]

Shuang, F.

D. Cao, Y. Wang, S. Li, L. Yang, F. Shuang, and F. Gao, “Optimal control of multiple two-photon transitions,” J. Math. Chem. 55, 1053–1066 (2017).
[Crossref]

D. Cao, L. Yang, Y. Wang, F. Shuang, and F. Gao, “Controlling pathway dynamics of a four-level quantum system with pulse shaping,” J. Phys. A Math. Theor. 49, 285302 (2016).
[Crossref]

F. Gao, R. Rey-de-Castro, Y. Wang, H. Rabitz, and F. Shuang, “Identifying a cooperative control mechanism between an applied field and the environment of open quantum systems,” Phys. Rev. A 93, 053407 (2016).
[Crossref]

F. Gao, Y. Wang, R. Rey-de-Castro, H. Rabitz, and F. Shuang, “Quantum control and pathway manipulation in rubidium,” Phys. Rev. A 92, 033423 (2015).
[Crossref]

F. Gao, R. Rey-de-Castro, A.M. Donovan, J. Xu, Y. Wang, H. Rabitz, and F. Shuang, “Pathway dynamics in the optimal quantum control of rubidium: Cooperation and competition,” Phys. Rev. A 89, 023416 (2014).
[Crossref]

Silberberg, Y.

N. Dudovich, D. Oron, and Y. Silberberg, “Single-pulse coherently controlled nonlinear Raman spectroscopy and microscopy,” Nature 418, 512 (2002).
[Crossref] [PubMed]

N. Dudovich, B. Dayan, S. M. Gallagher Faeder, and Y. Silberberg, “Transform-limited pulses are not optimal for resonant multiphoton transitions,” Phys. Rev. Lett. 86, 47–50 (2001).
[Crossref] [PubMed]

D. Meshulach and Y. Silberberg, “Coherent quantum control of two-photon transitions by a femtosecond laser pulse,” Nature 396, 239 (1998).
[Crossref]

Stentz, A.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635 (2000).
[Crossref] [PubMed]

Stowe, M. C.

M. C. Stowe, A. Pe’er, and J. Ye, “Control of Four-Level Quantum Coherence via Discrete Spectral Shaping of an Optical Frequency Comb,” Phys. Rev. Lett. 100, 203001 (2008).
[Crossref] [PubMed]

Tanner, D.J.

D.J. Tanner and S. A. Rice, “Control of selectivity of chemical reaction via control of wavepacket evolution,” J. Chem. Phys. 83, 5013–5018 (1985).
[Crossref]

Theuer, H.

K. Bergmann, H. Theuer, and B. W. Shore, “Coherent population transfer among quantum states of atoms and molecules,” Rev. Mod. Phys. 70, 1003–1025 (1998).
[Crossref]

Udem, Th.

R. Holzwarth, Th. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Optical frequency synthesizer for precision spectroscopy,” Phys. Rev. Lett. 85, 2264 (2000).
[Crossref] [PubMed]

Wadsworth, W. J.

R. Holzwarth, Th. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Optical frequency synthesizer for precision spectroscopy,” Phys. Rev. Lett. 85, 2264 (2000).
[Crossref] [PubMed]

Wang, X.

X. Wang, C. Jin, and C. D. Lin, “Coherent control of high-harmonic generation using waveform-synthesized chirped laser fields,” Phys. Rev. A 90, 023416 (2014).
[Crossref]

Wang, Y.

D. Cao, Y. Wang, S. Li, L. Yang, F. Shuang, and F. Gao, “Optimal control of multiple two-photon transitions,” J. Math. Chem. 55, 1053–1066 (2017).
[Crossref]

D. Cao, L. Yang, Y. Wang, F. Shuang, and F. Gao, “Controlling pathway dynamics of a four-level quantum system with pulse shaping,” J. Phys. A Math. Theor. 49, 285302 (2016).
[Crossref]

F. Gao, R. Rey-de-Castro, Y. Wang, H. Rabitz, and F. Shuang, “Identifying a cooperative control mechanism between an applied field and the environment of open quantum systems,” Phys. Rev. A 93, 053407 (2016).
[Crossref]

F. Gao, Y. Wang, R. Rey-de-Castro, H. Rabitz, and F. Shuang, “Quantum control and pathway manipulation in rubidium,” Phys. Rev. A 92, 033423 (2015).
[Crossref]

F. Gao, R. Rey-de-Castro, A.M. Donovan, J. Xu, Y. Wang, H. Rabitz, and F. Shuang, “Pathway dynamics in the optimal quantum control of rubidium: Cooperation and competition,” Phys. Rev. A 89, 023416 (2014).
[Crossref]

Weiner, A. M.

A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71, 1929 (2000).
[Crossref]

Wieman, C.

C. Wieman and T. W. Hänsch, “Doppler-free laser polarization spectroscopy,” Phys. Rev. Lett. 36, 1170 (1976).
[Crossref]

Windeler, R. S.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635 (2000).
[Crossref] [PubMed]

Witte, S.

I. Barmes, S. Witte, and K. S. E. Eikema, “Spatial and spectral coherent control with frequency combs,” Nat. Photonics 7, 38 (2013).
[Crossref]

I. Barmes, S. Witte, and K. S. E. Eikema, “High-precision spectroscopy with counterpropagating femtosecond pulses,” Phys. Rev. Lett. 111, 023007 (2013).
[Crossref] [PubMed]

Xu, J.

F. Gao, R. Rey-de-Castro, A.M. Donovan, J. Xu, Y. Wang, H. Rabitz, and F. Shuang, “Pathway dynamics in the optimal quantum control of rubidium: Cooperation and competition,” Phys. Rev. A 89, 023416 (2014).
[Crossref]

Yang, L.

D. Cao, Y. Wang, S. Li, L. Yang, F. Shuang, and F. Gao, “Optimal control of multiple two-photon transitions,” J. Math. Chem. 55, 1053–1066 (2017).
[Crossref]

D. Cao, L. Yang, Y. Wang, F. Shuang, and F. Gao, “Controlling pathway dynamics of a four-level quantum system with pulse shaping,” J. Phys. A Math. Theor. 49, 285302 (2016).
[Crossref]

Ye, J.

M. C. Stowe, A. Pe’er, and J. Ye, “Control of Four-Level Quantum Coherence via Discrete Spectral Shaping of an Optical Frequency Comb,” Phys. Rev. Lett. 100, 203001 (2008).
[Crossref] [PubMed]

Zoller, P.

J. J. García-Ripoll, P. Zoller, and J. I. Cirac, “Speed Optimized Two-Qubit Gates with Laser Coherent Control Techniques for Ion Trap Quantum Computing,” Phys. Rev. Lett. 91, 157901 (2003).
[Crossref] [PubMed]

J. Chem. Phys. (1)

D.J. Tanner and S. A. Rice, “Control of selectivity of chemical reaction via control of wavepacket evolution,” J. Chem. Phys. 83, 5013–5018 (1985).
[Crossref]

J. Math. Chem. (1)

D. Cao, Y. Wang, S. Li, L. Yang, F. Shuang, and F. Gao, “Optimal control of multiple two-photon transitions,” J. Math. Chem. 55, 1053–1066 (2017).
[Crossref]

J. Phys. A Math. Theor. (1)

D. Cao, L. Yang, Y. Wang, F. Shuang, and F. Gao, “Controlling pathway dynamics of a four-level quantum system with pulse shaping,” J. Phys. A Math. Theor. 49, 285302 (2016).
[Crossref]

Nat. Photonics (1)

I. Barmes, S. Witte, and K. S. E. Eikema, “Spatial and spectral coherent control with frequency combs,” Nat. Photonics 7, 38 (2013).
[Crossref]

Nature (2)

N. Dudovich, D. Oron, and Y. Silberberg, “Single-pulse coherently controlled nonlinear Raman spectroscopy and microscopy,” Nature 418, 512 (2002).
[Crossref] [PubMed]

D. Meshulach and Y. Silberberg, “Coherent quantum control of two-photon transitions by a femtosecond laser pulse,” Nature 396, 239 (1998).
[Crossref]

Opt. Express (1)

Phys. Rev. A (6)

H. G. Lee, H. Kim, J. Lim, and J. Ahn, “Quantum interference control of four-level diamond-configuration quantum system,” Phys. Rev. A 88, 053427 (2013).
[Crossref]

X. Wang, C. Jin, and C. D. Lin, “Coherent control of high-harmonic generation using waveform-synthesized chirped laser fields,” Phys. Rev. A 90, 023416 (2014).
[Crossref]

W. Lee, H. Kim, K. Kim, and J. Ahn, “Coherent control of resonant two-photon transitions by counter-propagating ultrashort pulse pairs,” Phys. Rev. A 92, 033415 (2015).
[Crossref]

F. Gao, R. Rey-de-Castro, A.M. Donovan, J. Xu, Y. Wang, H. Rabitz, and F. Shuang, “Pathway dynamics in the optimal quantum control of rubidium: Cooperation and competition,” Phys. Rev. A 89, 023416 (2014).
[Crossref]

F. Gao, Y. Wang, R. Rey-de-Castro, H. Rabitz, and F. Shuang, “Quantum control and pathway manipulation in rubidium,” Phys. Rev. A 92, 033423 (2015).
[Crossref]

F. Gao, R. Rey-de-Castro, Y. Wang, H. Rabitz, and F. Shuang, “Identifying a cooperative control mechanism between an applied field and the environment of open quantum systems,” Phys. Rev. A 93, 053407 (2016).
[Crossref]

Phys. Rev. Lett. (7)

M. C. Stowe, A. Pe’er, and J. Ye, “Control of Four-Level Quantum Coherence via Discrete Spectral Shaping of an Optical Frequency Comb,” Phys. Rev. Lett. 100, 203001 (2008).
[Crossref] [PubMed]

R. S. Judson and H. Rabitz, “Teaching lasers to control molecules,” Phys. Rev. Lett. 68, 1500–1503 (1992).
[Crossref] [PubMed]

J. J. García-Ripoll, P. Zoller, and J. I. Cirac, “Speed Optimized Two-Qubit Gates with Laser Coherent Control Techniques for Ion Trap Quantum Computing,” Phys. Rev. Lett. 91, 157901 (2003).
[Crossref] [PubMed]

N. Dudovich, B. Dayan, S. M. Gallagher Faeder, and Y. Silberberg, “Transform-limited pulses are not optimal for resonant multiphoton transitions,” Phys. Rev. Lett. 86, 47–50 (2001).
[Crossref] [PubMed]

R. Holzwarth, Th. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Optical frequency synthesizer for precision spectroscopy,” Phys. Rev. Lett. 85, 2264 (2000).
[Crossref] [PubMed]

I. Barmes, S. Witte, and K. S. E. Eikema, “High-precision spectroscopy with counterpropagating femtosecond pulses,” Phys. Rev. Lett. 111, 023007 (2013).
[Crossref] [PubMed]

C. Wieman and T. W. Hänsch, “Doppler-free laser polarization spectroscopy,” Phys. Rev. Lett. 36, 1170 (1976).
[Crossref]

Rev. Mod. Phys. (1)

K. Bergmann, H. Theuer, and B. W. Shore, “Coherent population transfer among quantum states of atoms and molecules,” Rev. Mod. Phys. 70, 1003–1025 (1998).
[Crossref]

Rev. Sci. Instrum. (1)

A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71, 1929 (2000).
[Crossref]

Science (1)

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635 (2000).
[Crossref] [PubMed]

Other (1)

M. Shapiro and P. Brumer, Principles of the Quantum Control of Molecular Processes (Wiley, New York, 2003).

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Figures (4)

Fig. 1
Fig. 1 Ultrafast spatial coherent-control scheme: Laser pulses are phase-modulated to spatially resolve the Doppler-free excitation of 5S1/2-5P1/2-5D and 5S1/2-5P3/2-5D of atomic rubidium.
Fig. 2
Fig. 2 (a) The plot of double-V shape spectral phase modulation (solid red line), modulated spectral amplitude (dotted blue line), and fi(ω) from Eq. (9) (dotted green line). (b) Spectrogram of a pulse having the double-V shape spectral phase from Eq. (10). (c) Composite map of numerical calculation results of Eq. (9) with the spectral phase modulation Eq. (10), where α1 = 0.4 ps is fixed and α2 increases from 0.4 ps to 1.6 ps. (d) Composite map from experiment. (e) Experimental result for α1 = α2 = 0.4 ps (upper) and for α1 = 0.4 ps, α2 = 1.5 ps (lower). The positions of z1 (solid black lines) and z2 (dashed black lines) are illustrated.
Fig. 3
Fig. 3 (a) The plot of three phase slopes phase modulation (solid red line). (b) The spectrogram of a pulse having three phase slopes spectral phase from Eq. (12). (c) Composite map of numerical calculation results of Eq. (9) with the spectral phase modulation Eq. (12), where α1 = 0.1 ps and α2 = 1.5 ps were fixed and α3 increased from 0.1 ps to 1.6 ps. (d) Composite map from experiment. (e) Experimental result for α3 = 0.1 ps and α3 = 1.5 ps, respectively. The positions of z1(solid black lines) and z2(dashed black lines) are illustrated.
Fig. 4
Fig. 4 (a) Plot of periodic square spectral phase modulation. (b) Numerical calculation result of Eq. (9) with the spectral phase modulation Eq. (13), changing the modulation depth from A1 = A2 = 0 to π. (c) Composite map from experiment. (d) Numerical calculation including finite SLM pixel size and intensity distribution by beam focusing. (e) Reconstructed result by eliminating the atomic motion effect.

Equations (13)

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c fg = i = a , b ( c fg , i r + c fg , i nr ) .
c fg , i r = π μ fi μ ig 2 E ( ω ig ) E ( ω fi ) ,
c fg , i nr = i μ fi μ ig 2 E ( ω ) E ( ω fg ω ) ω ig ω d ω ,
E s ( t ) = E s ( ω ) e i ω t d ω = A ( ω ) e i Φ ( ω ) e i ω t d ω ,
A ( ω ) = A 0 ( ω ) ( 1 e ( ω ω a g ) 2 / δ a 2 e ( ω ω b g ) 2 / δ b 2 ) .
c fg = c fg , a nr + c fg , b nr .
E ( ω ) = A ( ω ) e i Φ ( ω ) ( e i ω z / c + e i ω z / c ) ,
c fg ( z ) = i = a , b i f i ( ω ) A ( ω ) A ( ω ^ ) e i [ Φ ( ω ) + Φ ( ω ^ ) ] × [ 1 + e 2 i ω fg z / c + e 2 i ω ^ z / c + e 2 i ω z / c ] d ω ,
| c fg ( z ) | 2 = | i = a , b f i ( ω ) A ( ω ) A ( ω ^ ) e i [ Φ ( ω ) + Φ ( ω ^ ) ] [ cos ( ( ω ^ ω ) z / c ) + 1 + e 2 i ω fg z / c ] d ω | 2 ,
Φ ( ω ) R 1 = α 1 ( ω ω 0 ) + π Θ ( ω ω a g ) Φ ( ω ) R 2 = α 2 ( ω ω 0 ) + π Θ ( ω ω b g ) Φ ( ω ) B 2 = α 2 ( ω ω 0 ) + π Φ ( ω ) B 1 = α 1 ( ω ω 0 ) + π
ω cg = k ω ag + ω bg k + 1 , k = μ fb μ bg μ fa μ ag ,
Φ ( ω ) R 1 = α 1 ( ω ω 0 ) + π Θ ( ω ω a g ) Φ ( ω ) R 2 = α 2 ( ω ω 0 ) + π Θ ( ω ω b g ) Φ ( ω ) B 12 = α 3 ( ω ω 0 ) + π
Φ R 1 ( ω ) = A 1 sgn [ cos ( β 1 ( ω ω 0 ) ) ] + π Φ ( ω ω a g ) Φ R 2 ( ω ) = A 2 sgn [ cos ( β 2 ( ω ω 0 ) ) ] + π Φ ( ω ω b g ) Φ B 2 ( ω ) = A 2 sgn [ cos ( β 2 ( ω ω 0 ) ) ] + π Φ B 1 ( ω ) = A 1 sgn [ cos ( β 1 ( ω ω 0 ) ) ] + π

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