Abstract

We discuss femtosecond Raman-type techniques to control molecular vibrations, which can be implemented for internal-state cooling from Feshbach states with the use of optical frequency combs (OFCs) with and without modulation. The technique makes use of multiple two-photon resonances induced by optical frequencies present in the comb. It provides us with a useful tool to study the details of molecular dynamics at ultracold temperatures. In our theoretical model we take into account decoherence in the form of spontaneous emission and collisional dephasing in order to ascertain an accurate model of the population transfer in the three-level system. We analyze the effects of odd and even chirps of the OFC in the form of sine and cosine functions on the population transfer. We compare the effects of these chirps to the results attained with the standard OFC to see if they increase the population transfer to the final deeply bound state in the presence of decoherence. We also analyze the inherent phase relation that takes place owing to collisional dephasing between molecules in each of the states. This ability to control the rovibrational states of a molecule with an OFC enables us to create deeply bound ultra-cold polar molecules from the Feshbach state.

© 2013 Optical Society of America

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  1. J. Deiglmayr, M. Repp, O. Dulieu, R. Wester, and M. Weidemuller, “Population redistribution in optically trapped polar molecules,” Eur. Phys. J. D 65, 99–104 (2011).
    [CrossRef]
  2. J. Qian, L. Zhou, K. Zhang, and W. Zhang, “Bose–Einstein condensates via an all-optical R-type atom-molecule adiabatic passage,” New J. Phys. 12, 033002 (2010).
    [CrossRef]
  3. I. Manai, R. Horchani, H. Lignier, P. Pillet, D. Comparat, A. Fioretti, and M. Allegrini, “Rovibrational cooling of molecules by optical pumping,” Phys. Rev. Lett. 109, 183001 (2012).
    [CrossRef]
  4. K.-K. Ni, S. Ospelkaus, D. Wang, G. Quemener, B. Neyenhuis, M. H. G. de Miranda, J. L. Bohn, J. Ye, and D. S. Jin, “Dipolar collisions of polar molecules in the quantum regime,” Nature 464, 1324–1328 (2010).
    [CrossRef]
  5. K.-K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Pe’er, B. Neyenhuis, J. J. Zirbel, S. Kotochigova, P. S. Juliene, D. Jin, and J. Ye, “A high phase-space-density gas of polar molecules,” Science 322, 231–235 (2008).
    [CrossRef]
  6. M. C. Stowe, A. Peér, 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]
  7. W. Shi and S. Malinovskaya, “Implementation of a single femtosecond optical frequency comb for molecular cooling,” Phys. Rev. A 82, 013407 (2010).
    [CrossRef]
  8. E. A. Shapiro, A. Pe’er, J. Ye, and M. Shapiro, “Piecewise adiabatic population transfer in a molecule via a wave packet,” Phys. Rev. Lett. 101, 023601 (2008).
    [CrossRef]
  9. J. L. Hall, L. Hollberg, T. Baer, and H. G. Robinson, “Optical heterodine saturation spectroscopy,” Appl. Phys. Lett. 39, 680–682 (1981).
    [CrossRef]
  10. T. Suzuki, M. Hirai, and M. Katsuragawa, “Octave-spanning Raman comb with carrier envelope offset control,” Phys. Rev. Lett. 101, 243602 (2008).
    [CrossRef]
  11. D. Goswami, “Laser pulse modulation approaches towards ensemble quantum computing,” Phys. Rev. Lett. 88, 177901 (2002).
    [CrossRef]
  12. J. Ye and S. T. Cundiff, eds., Femtosecond Optical Frequency Comb: Principle, Operation, and Applications (Springer, 2005).
  13. P. R. Berman and R. C. O’Connell, “Constraints on dephasing widths and shifts in three-level quantum systems,” Phys. Rev. A 71, 022501 (2005).
    [CrossRef]
  14. S. G. Schirmer and A. I. Solomon, “Constraints on relaxation rates for N-level quantum systems,” Phys. Rev. A 70, 022107 (2004).
    [CrossRef]

2012 (1)

I. Manai, R. Horchani, H. Lignier, P. Pillet, D. Comparat, A. Fioretti, and M. Allegrini, “Rovibrational cooling of molecules by optical pumping,” Phys. Rev. Lett. 109, 183001 (2012).
[CrossRef]

2011 (1)

J. Deiglmayr, M. Repp, O. Dulieu, R. Wester, and M. Weidemuller, “Population redistribution in optically trapped polar molecules,” Eur. Phys. J. D 65, 99–104 (2011).
[CrossRef]

2010 (3)

J. Qian, L. Zhou, K. Zhang, and W. Zhang, “Bose–Einstein condensates via an all-optical R-type atom-molecule adiabatic passage,” New J. Phys. 12, 033002 (2010).
[CrossRef]

W. Shi and S. Malinovskaya, “Implementation of a single femtosecond optical frequency comb for molecular cooling,” Phys. Rev. A 82, 013407 (2010).
[CrossRef]

K.-K. Ni, S. Ospelkaus, D. Wang, G. Quemener, B. Neyenhuis, M. H. G. de Miranda, J. L. Bohn, J. Ye, and D. S. Jin, “Dipolar collisions of polar molecules in the quantum regime,” Nature 464, 1324–1328 (2010).
[CrossRef]

2008 (4)

K.-K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Pe’er, B. Neyenhuis, J. J. Zirbel, S. Kotochigova, P. S. Juliene, D. Jin, and J. Ye, “A high phase-space-density gas of polar molecules,” Science 322, 231–235 (2008).
[CrossRef]

M. C. Stowe, A. Peér, 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]

E. A. Shapiro, A. Pe’er, J. Ye, and M. Shapiro, “Piecewise adiabatic population transfer in a molecule via a wave packet,” Phys. Rev. Lett. 101, 023601 (2008).
[CrossRef]

T. Suzuki, M. Hirai, and M. Katsuragawa, “Octave-spanning Raman comb with carrier envelope offset control,” Phys. Rev. Lett. 101, 243602 (2008).
[CrossRef]

2005 (1)

P. R. Berman and R. C. O’Connell, “Constraints on dephasing widths and shifts in three-level quantum systems,” Phys. Rev. A 71, 022501 (2005).
[CrossRef]

2004 (1)

S. G. Schirmer and A. I. Solomon, “Constraints on relaxation rates for N-level quantum systems,” Phys. Rev. A 70, 022107 (2004).
[CrossRef]

2002 (1)

D. Goswami, “Laser pulse modulation approaches towards ensemble quantum computing,” Phys. Rev. Lett. 88, 177901 (2002).
[CrossRef]

1981 (1)

J. L. Hall, L. Hollberg, T. Baer, and H. G. Robinson, “Optical heterodine saturation spectroscopy,” Appl. Phys. Lett. 39, 680–682 (1981).
[CrossRef]

Allegrini, M.

I. Manai, R. Horchani, H. Lignier, P. Pillet, D. Comparat, A. Fioretti, and M. Allegrini, “Rovibrational cooling of molecules by optical pumping,” Phys. Rev. Lett. 109, 183001 (2012).
[CrossRef]

Baer, T.

J. L. Hall, L. Hollberg, T. Baer, and H. G. Robinson, “Optical heterodine saturation spectroscopy,” Appl. Phys. Lett. 39, 680–682 (1981).
[CrossRef]

Berman, P. R.

P. R. Berman and R. C. O’Connell, “Constraints on dephasing widths and shifts in three-level quantum systems,” Phys. Rev. A 71, 022501 (2005).
[CrossRef]

Bohn, J. L.

K.-K. Ni, S. Ospelkaus, D. Wang, G. Quemener, B. Neyenhuis, M. H. G. de Miranda, J. L. Bohn, J. Ye, and D. S. Jin, “Dipolar collisions of polar molecules in the quantum regime,” Nature 464, 1324–1328 (2010).
[CrossRef]

Comparat, D.

I. Manai, R. Horchani, H. Lignier, P. Pillet, D. Comparat, A. Fioretti, and M. Allegrini, “Rovibrational cooling of molecules by optical pumping,” Phys. Rev. Lett. 109, 183001 (2012).
[CrossRef]

de Miranda, M. H. G.

K.-K. Ni, S. Ospelkaus, D. Wang, G. Quemener, B. Neyenhuis, M. H. G. de Miranda, J. L. Bohn, J. Ye, and D. S. Jin, “Dipolar collisions of polar molecules in the quantum regime,” Nature 464, 1324–1328 (2010).
[CrossRef]

K.-K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Pe’er, B. Neyenhuis, J. J. Zirbel, S. Kotochigova, P. S. Juliene, D. Jin, and J. Ye, “A high phase-space-density gas of polar molecules,” Science 322, 231–235 (2008).
[CrossRef]

Deiglmayr, J.

J. Deiglmayr, M. Repp, O. Dulieu, R. Wester, and M. Weidemuller, “Population redistribution in optically trapped polar molecules,” Eur. Phys. J. D 65, 99–104 (2011).
[CrossRef]

Dulieu, O.

J. Deiglmayr, M. Repp, O. Dulieu, R. Wester, and M. Weidemuller, “Population redistribution in optically trapped polar molecules,” Eur. Phys. J. D 65, 99–104 (2011).
[CrossRef]

Fioretti, A.

I. Manai, R. Horchani, H. Lignier, P. Pillet, D. Comparat, A. Fioretti, and M. Allegrini, “Rovibrational cooling of molecules by optical pumping,” Phys. Rev. Lett. 109, 183001 (2012).
[CrossRef]

Goswami, D.

D. Goswami, “Laser pulse modulation approaches towards ensemble quantum computing,” Phys. Rev. Lett. 88, 177901 (2002).
[CrossRef]

Hall, J. L.

J. L. Hall, L. Hollberg, T. Baer, and H. G. Robinson, “Optical heterodine saturation spectroscopy,” Appl. Phys. Lett. 39, 680–682 (1981).
[CrossRef]

Hirai, M.

T. Suzuki, M. Hirai, and M. Katsuragawa, “Octave-spanning Raman comb with carrier envelope offset control,” Phys. Rev. Lett. 101, 243602 (2008).
[CrossRef]

Hollberg, L.

J. L. Hall, L. Hollberg, T. Baer, and H. G. Robinson, “Optical heterodine saturation spectroscopy,” Appl. Phys. Lett. 39, 680–682 (1981).
[CrossRef]

Horchani, R.

I. Manai, R. Horchani, H. Lignier, P. Pillet, D. Comparat, A. Fioretti, and M. Allegrini, “Rovibrational cooling of molecules by optical pumping,” Phys. Rev. Lett. 109, 183001 (2012).
[CrossRef]

Jin, D.

K.-K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Pe’er, B. Neyenhuis, J. J. Zirbel, S. Kotochigova, P. S. Juliene, D. Jin, and J. Ye, “A high phase-space-density gas of polar molecules,” Science 322, 231–235 (2008).
[CrossRef]

Jin, D. S.

K.-K. Ni, S. Ospelkaus, D. Wang, G. Quemener, B. Neyenhuis, M. H. G. de Miranda, J. L. Bohn, J. Ye, and D. S. Jin, “Dipolar collisions of polar molecules in the quantum regime,” Nature 464, 1324–1328 (2010).
[CrossRef]

Juliene, P. S.

K.-K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Pe’er, B. Neyenhuis, J. J. Zirbel, S. Kotochigova, P. S. Juliene, D. Jin, and J. Ye, “A high phase-space-density gas of polar molecules,” Science 322, 231–235 (2008).
[CrossRef]

Katsuragawa, M.

T. Suzuki, M. Hirai, and M. Katsuragawa, “Octave-spanning Raman comb with carrier envelope offset control,” Phys. Rev. Lett. 101, 243602 (2008).
[CrossRef]

Kotochigova, S.

K.-K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Pe’er, B. Neyenhuis, J. J. Zirbel, S. Kotochigova, P. S. Juliene, D. Jin, and J. Ye, “A high phase-space-density gas of polar molecules,” Science 322, 231–235 (2008).
[CrossRef]

Lignier, H.

I. Manai, R. Horchani, H. Lignier, P. Pillet, D. Comparat, A. Fioretti, and M. Allegrini, “Rovibrational cooling of molecules by optical pumping,” Phys. Rev. Lett. 109, 183001 (2012).
[CrossRef]

Malinovskaya, S.

W. Shi and S. Malinovskaya, “Implementation of a single femtosecond optical frequency comb for molecular cooling,” Phys. Rev. A 82, 013407 (2010).
[CrossRef]

Manai, I.

I. Manai, R. Horchani, H. Lignier, P. Pillet, D. Comparat, A. Fioretti, and M. Allegrini, “Rovibrational cooling of molecules by optical pumping,” Phys. Rev. Lett. 109, 183001 (2012).
[CrossRef]

Neyenhuis, B.

K.-K. Ni, S. Ospelkaus, D. Wang, G. Quemener, B. Neyenhuis, M. H. G. de Miranda, J. L. Bohn, J. Ye, and D. S. Jin, “Dipolar collisions of polar molecules in the quantum regime,” Nature 464, 1324–1328 (2010).
[CrossRef]

K.-K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Pe’er, B. Neyenhuis, J. J. Zirbel, S. Kotochigova, P. S. Juliene, D. Jin, and J. Ye, “A high phase-space-density gas of polar molecules,” Science 322, 231–235 (2008).
[CrossRef]

Ni, K.-K.

K.-K. Ni, S. Ospelkaus, D. Wang, G. Quemener, B. Neyenhuis, M. H. G. de Miranda, J. L. Bohn, J. Ye, and D. S. Jin, “Dipolar collisions of polar molecules in the quantum regime,” Nature 464, 1324–1328 (2010).
[CrossRef]

K.-K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Pe’er, B. Neyenhuis, J. J. Zirbel, S. Kotochigova, P. S. Juliene, D. Jin, and J. Ye, “A high phase-space-density gas of polar molecules,” Science 322, 231–235 (2008).
[CrossRef]

O’Connell, R. C.

P. R. Berman and R. C. O’Connell, “Constraints on dephasing widths and shifts in three-level quantum systems,” Phys. Rev. A 71, 022501 (2005).
[CrossRef]

Ospelkaus, S.

K.-K. Ni, S. Ospelkaus, D. Wang, G. Quemener, B. Neyenhuis, M. H. G. de Miranda, J. L. Bohn, J. Ye, and D. S. Jin, “Dipolar collisions of polar molecules in the quantum regime,” Nature 464, 1324–1328 (2010).
[CrossRef]

K.-K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Pe’er, B. Neyenhuis, J. J. Zirbel, S. Kotochigova, P. S. Juliene, D. Jin, and J. Ye, “A high phase-space-density gas of polar molecules,” Science 322, 231–235 (2008).
[CrossRef]

Pe’er, A.

K.-K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Pe’er, B. Neyenhuis, J. J. Zirbel, S. Kotochigova, P. S. Juliene, D. Jin, and J. Ye, “A high phase-space-density gas of polar molecules,” Science 322, 231–235 (2008).
[CrossRef]

E. A. Shapiro, A. Pe’er, J. Ye, and M. Shapiro, “Piecewise adiabatic population transfer in a molecule via a wave packet,” Phys. Rev. Lett. 101, 023601 (2008).
[CrossRef]

Peér, A.

M. C. Stowe, A. Peér, 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]

Pillet, P.

I. Manai, R. Horchani, H. Lignier, P. Pillet, D. Comparat, A. Fioretti, and M. Allegrini, “Rovibrational cooling of molecules by optical pumping,” Phys. Rev. Lett. 109, 183001 (2012).
[CrossRef]

Qian, J.

J. Qian, L. Zhou, K. Zhang, and W. Zhang, “Bose–Einstein condensates via an all-optical R-type atom-molecule adiabatic passage,” New J. Phys. 12, 033002 (2010).
[CrossRef]

Quemener, G.

K.-K. Ni, S. Ospelkaus, D. Wang, G. Quemener, B. Neyenhuis, M. H. G. de Miranda, J. L. Bohn, J. Ye, and D. S. Jin, “Dipolar collisions of polar molecules in the quantum regime,” Nature 464, 1324–1328 (2010).
[CrossRef]

Repp, M.

J. Deiglmayr, M. Repp, O. Dulieu, R. Wester, and M. Weidemuller, “Population redistribution in optically trapped polar molecules,” Eur. Phys. J. D 65, 99–104 (2011).
[CrossRef]

Robinson, H. G.

J. L. Hall, L. Hollberg, T. Baer, and H. G. Robinson, “Optical heterodine saturation spectroscopy,” Appl. Phys. Lett. 39, 680–682 (1981).
[CrossRef]

Schirmer, S. G.

S. G. Schirmer and A. I. Solomon, “Constraints on relaxation rates for N-level quantum systems,” Phys. Rev. A 70, 022107 (2004).
[CrossRef]

Shapiro, E. A.

E. A. Shapiro, A. Pe’er, J. Ye, and M. Shapiro, “Piecewise adiabatic population transfer in a molecule via a wave packet,” Phys. Rev. Lett. 101, 023601 (2008).
[CrossRef]

Shapiro, M.

E. A. Shapiro, A. Pe’er, J. Ye, and M. Shapiro, “Piecewise adiabatic population transfer in a molecule via a wave packet,” Phys. Rev. Lett. 101, 023601 (2008).
[CrossRef]

Shi, W.

W. Shi and S. Malinovskaya, “Implementation of a single femtosecond optical frequency comb for molecular cooling,” Phys. Rev. A 82, 013407 (2010).
[CrossRef]

Solomon, A. I.

S. G. Schirmer and A. I. Solomon, “Constraints on relaxation rates for N-level quantum systems,” Phys. Rev. A 70, 022107 (2004).
[CrossRef]

Stowe, M. C.

M. C. Stowe, A. Peér, 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]

Suzuki, T.

T. Suzuki, M. Hirai, and M. Katsuragawa, “Octave-spanning Raman comb with carrier envelope offset control,” Phys. Rev. Lett. 101, 243602 (2008).
[CrossRef]

Wang, D.

K.-K. Ni, S. Ospelkaus, D. Wang, G. Quemener, B. Neyenhuis, M. H. G. de Miranda, J. L. Bohn, J. Ye, and D. S. Jin, “Dipolar collisions of polar molecules in the quantum regime,” Nature 464, 1324–1328 (2010).
[CrossRef]

Weidemuller, M.

J. Deiglmayr, M. Repp, O. Dulieu, R. Wester, and M. Weidemuller, “Population redistribution in optically trapped polar molecules,” Eur. Phys. J. D 65, 99–104 (2011).
[CrossRef]

Wester, R.

J. Deiglmayr, M. Repp, O. Dulieu, R. Wester, and M. Weidemuller, “Population redistribution in optically trapped polar molecules,” Eur. Phys. J. D 65, 99–104 (2011).
[CrossRef]

Ye, J.

K.-K. Ni, S. Ospelkaus, D. Wang, G. Quemener, B. Neyenhuis, M. H. G. de Miranda, J. L. Bohn, J. Ye, and D. S. Jin, “Dipolar collisions of polar molecules in the quantum regime,” Nature 464, 1324–1328 (2010).
[CrossRef]

E. A. Shapiro, A. Pe’er, J. Ye, and M. Shapiro, “Piecewise adiabatic population transfer in a molecule via a wave packet,” Phys. Rev. Lett. 101, 023601 (2008).
[CrossRef]

M. C. Stowe, A. Peér, 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]

K.-K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Pe’er, B. Neyenhuis, J. J. Zirbel, S. Kotochigova, P. S. Juliene, D. Jin, and J. Ye, “A high phase-space-density gas of polar molecules,” Science 322, 231–235 (2008).
[CrossRef]

Zhang, K.

J. Qian, L. Zhou, K. Zhang, and W. Zhang, “Bose–Einstein condensates via an all-optical R-type atom-molecule adiabatic passage,” New J. Phys. 12, 033002 (2010).
[CrossRef]

Zhang, W.

J. Qian, L. Zhou, K. Zhang, and W. Zhang, “Bose–Einstein condensates via an all-optical R-type atom-molecule adiabatic passage,” New J. Phys. 12, 033002 (2010).
[CrossRef]

Zhou, L.

J. Qian, L. Zhou, K. Zhang, and W. Zhang, “Bose–Einstein condensates via an all-optical R-type atom-molecule adiabatic passage,” New J. Phys. 12, 033002 (2010).
[CrossRef]

Zirbel, J. J.

K.-K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Pe’er, B. Neyenhuis, J. J. Zirbel, S. Kotochigova, P. S. Juliene, D. Jin, and J. Ye, “A high phase-space-density gas of polar molecules,” Science 322, 231–235 (2008).
[CrossRef]

Appl. Phys. Lett. (1)

J. L. Hall, L. Hollberg, T. Baer, and H. G. Robinson, “Optical heterodine saturation spectroscopy,” Appl. Phys. Lett. 39, 680–682 (1981).
[CrossRef]

Eur. Phys. J. D (1)

J. Deiglmayr, M. Repp, O. Dulieu, R. Wester, and M. Weidemuller, “Population redistribution in optically trapped polar molecules,” Eur. Phys. J. D 65, 99–104 (2011).
[CrossRef]

Nature (1)

K.-K. Ni, S. Ospelkaus, D. Wang, G. Quemener, B. Neyenhuis, M. H. G. de Miranda, J. L. Bohn, J. Ye, and D. S. Jin, “Dipolar collisions of polar molecules in the quantum regime,” Nature 464, 1324–1328 (2010).
[CrossRef]

New J. Phys. (1)

J. Qian, L. Zhou, K. Zhang, and W. Zhang, “Bose–Einstein condensates via an all-optical R-type atom-molecule adiabatic passage,” New J. Phys. 12, 033002 (2010).
[CrossRef]

Phys. Rev. A (3)

W. Shi and S. Malinovskaya, “Implementation of a single femtosecond optical frequency comb for molecular cooling,” Phys. Rev. A 82, 013407 (2010).
[CrossRef]

P. R. Berman and R. C. O’Connell, “Constraints on dephasing widths and shifts in three-level quantum systems,” Phys. Rev. A 71, 022501 (2005).
[CrossRef]

S. G. Schirmer and A. I. Solomon, “Constraints on relaxation rates for N-level quantum systems,” Phys. Rev. A 70, 022107 (2004).
[CrossRef]

Phys. Rev. Lett. (5)

E. A. Shapiro, A. Pe’er, J. Ye, and M. Shapiro, “Piecewise adiabatic population transfer in a molecule via a wave packet,” Phys. Rev. Lett. 101, 023601 (2008).
[CrossRef]

M. C. Stowe, A. Peér, 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]

I. Manai, R. Horchani, H. Lignier, P. Pillet, D. Comparat, A. Fioretti, and M. Allegrini, “Rovibrational cooling of molecules by optical pumping,” Phys. Rev. Lett. 109, 183001 (2012).
[CrossRef]

T. Suzuki, M. Hirai, and M. Katsuragawa, “Octave-spanning Raman comb with carrier envelope offset control,” Phys. Rev. Lett. 101, 243602 (2008).
[CrossRef]

D. Goswami, “Laser pulse modulation approaches towards ensemble quantum computing,” Phys. Rev. Lett. 88, 177901 (2002).
[CrossRef]

Science (1)

K.-K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Pe’er, B. Neyenhuis, J. J. Zirbel, S. Kotochigova, P. S. Juliene, D. Jin, and J. Ye, “A high phase-space-density gas of polar molecules,” Science 322, 231–235 (2008).
[CrossRef]

Other (1)

J. Ye and S. T. Cundiff, eds., Femtosecond Optical Frequency Comb: Principle, Operation, and Applications (Springer, 2005).

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Figures (10)

Fig. 1.
Fig. 1.

(a) Pulse train in Eqs. (1) and (2); black, sine modulated; red dashed, cosine modulated; green, not modulated. (b), (c), and (d) are the power spectra of the OFCs obtained by taking the fast Fourier transform of the pulse train with sine modulation, cosine modulation, or no modulation across an individual pulse, respectively. The pulse duration is τ = 0.25 [ ω 31 1 ] , the modulation frequency is Ω = 4.9 [ ω 31 ] , the modulation amplitude is 4, the peak Rabi frequency is 1 [ ω 31 ] , and the pulse-train period is taken out of scale for graphical purpose. The value of ω 31 is chosen to be 70 THz as in [8]. The power spectra presented in (b), (c), and (d) have visually unresolved fine-mode structure manifesting itself as a solid color under the envelope.

Fig. 2.
Fig. 2.

Three-level Λ system modeling the Feshbach state, the excited-state manifold, and the final state of molecules interacting with a phase-locked pulse train. The transitional frequencies used in our calculations are ω 21 = 340.7 THz , ω 32 = 410.7 THz , and ω 31 = 70 THz [8], or ω 21 = 309.3 THz , ω 32 = 434.8 THz , and ω 31 = 125.5 THz [5].

Fig. 3.
Fig. 3.

Population dynamics in the three-level Λ system using an OFC having f r = 5 GHz and zero offset frequency. Parameters of the pulse train are the carrier frequency ω L = 434.8 THz , the pulse duration τ 0 = 3 fs , and the peak Rabi frequency Ω R = 1.26 THz ; the system parameters are ω 21 = 309.3 THz , ω 32 = 434.8 THz , and ω 31 = 125.5 THz [5]. The black curve shows the population of the ground state, the red curve that of the transitional state, and the green curve that of the final state. Time is given in the units of [ ω 1 ] , where ω = ω 31 = 125.5 THz .

Fig. 4.
Fig. 4.

Population transfer in the three-level Λ system achieved via the resonant Raman transitions using a sinusoidally modulated OFC described by Eq. (2) ( ϕ = 0 ). The values of the system parameters are ω 32 = 410.7 THz and ω 21 = 340.7 THz [8]. The carrier frequency is ω L = ω 32 , the modulation frequency is Ω = ω 21 , the modulation amplitude is Φ 0 = 4 , and the peak Rabi frequency is Ω R = 70 THz . The pulse duration is τ = 3 fs and the pulse-train period is (a)  6400 τ (20 ps), giving ω r = 50 GHz , and (b)  640 τ (2 ps), giving ω r = 500 GHz . Stepwise, adiabatic accumulation of the population is observed in state | 3 (green), which is the ultracold KRb state. The population of the Feshbach state | 1 is in black, and the excited-state manifold | 2 is in red. Time is given in the units of [ ω 1 ] , where ω = ω 31 = 70 THz .

Fig. 5.
Fig. 5.

Population transfer in the three-level Λ system achieved via the resonant Raman transitions using a phase-modulated OFC described by Eq. (2) with cosine modulation; blue: state | 1 , orange: state | 2 , indigo: state | 3 . For comparison, the results from the previous figure for the sinusoidally modulated comb are presented. The values of the system parameters are ω 32 = 410.7 THz and ω 21 = 340.7 THz [8]. The carrier frequency is ω L = ω 32 , the modulation frequency is Ω = ω 21 , the modulation amplitude is Φ 0 = 4 , and the peak Rabi frequency is Ω R = 70 THz . The pulse duration is τ = 3 fs , and the pulse-train period is 2 ps. The cosine modulation induces much faster Rabi oscillations and populates the excited state up to 50%.

Fig. 6.
Fig. 6.

Population transfer in the three-level Λ system achieved using an OFC as in Eq. (2) with the field parameters detuned off one-photon resonance with the frequencies of the Λ system. The detuning δ is equal to ω 31 / 2 . The carrier frequency is ω L = 5.4 , the modulation frequency is Ω = 4.4 , the Rabi frequency is Ω R = 1 , and the modulation amplitude is Φ 0 = 4 . The pulse duration is 3 fs, and the pulse-train period is 20 ps. The Λ -system transition frequencies are ω 21 = 4.9 , ω 32 = 5.9 (in units of frequency ω 31 = 70 THz ). Full population transfer to the final, cold state (green) occurs within 42 pulses. During this time, population of the initial Feshbach state (black) reduces to zero and the excited state (red) gets substantially populated during the transitional time. The population is reversed by the next 42 pulses. Time is given in the units of [ ω 1 ] , where ω = ω 31 = 70 THz .

Fig. 7.
Fig. 7.

Population dynamics induced by the standard OFC in the presence of spontaneous decay and collisions. Parameters of the pulse train are the carrier frequency ω L = 434.8 THz , the pulse duration τ 0 = 3 fs , and the peak Rabi frequency Ω R = 12.6 THz ; the system parameters are ω 21 = 309.3 THz , ω 32 = 434.8 THz , and ω 31 = 125.5 THz . Solid curves represent the case of pure collisional dephasing, Γ 31 = 0 , Γ 21 = Γ 32 = 0.001 ; dashed curves correspond to the same values of Γ , and γ 21 = γ 32 = 0.001 ; and Γ i j and γ i j are given in units of ω 31 . Note that the dashed red curve is visible in the vicinity of the coordinate origin, showing that the population of the excited state decays almost instantaneously within the presented time scale.

Fig. 8.
Fig. 8.

Population dynamics induced by the sinusoidally modulated OFC (solid curves) versus the standard one (dashed curves) in the presence of decoherence. The system parameters are (a) as in [5], ω 21 = 309.3 THz , ω 32 = 434.8 THz , and Ω R = 12.5 THz ; (b) as in [8], ω 21 = 340.7 THz , ω 32 = 410.7 THz , and Ω R = 70 THz . Decoherence parameters are γ 21 = γ 32 = 0.001 and Γ 21 = Γ 32 = 0.001 , in ω 31 units. Sinusoidal modulation provides almost full population transfer to the ultracold state, while the cosine modulation leads to a stationary solution with equal population distribution between the Feshbach and ultracold state, thus creating the maximum coherence.

Fig. 9.
Fig. 9.

Population dynamics induced by (a) the sine-modulated OFC, (b) the cosine-modulated OFC, and (c) the standard OFC, in the presence of decoherence. The system parameters are ω 21 = 309.3 THz , ω 32 = 434.8 THz , and Ω R = 12.5 THz [5]. Decoherence parameters are γ 21 = γ 32 = 10 6 and Γ 21 = Γ 32 = 10 10 in ω 31 units. The carrier frequency of the field is ω L = ω 32 , the modulation frequency is Ω = ω 21 , the modulation amplitude is Φ 0 = 4 for the sine- /cosine-modulated combs, and the peak Rabi frequency is Ω R = .1 [ ω 31 ] , a weak field regime. The sine-modulated OFC provides almost full population transfer to the ultracold state; the cosine-modulated OFC and unmodulated OFC lead to stationary solutions with the population distribution between the Feshbach and ultracold state close to 50%.

Fig. 10.
Fig. 10.

Phasor diagram analyzing collisional dephasing of the system.

Equations (5)

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E ( t ) = k = 0 N 1 E 0 exp [ ( t k T ) 2 / ( 2 τ 2 ) ] cos { ω L t k ω L T + ϕ } .
E ( t , z ) = ( 1 / 2 ) k = 0 N 1 E 0 exp [ ( t k T ) 2 / ( 2 τ 2 ) ] cos { ω L ( t k T ) + Φ 0 sin ( Ω ( t k T ) + ϕ ) } .
E ( ω ) = ( ( E 0 τ ) / 2 ) n J n ( Φ 0 ) exp [ ( 1 / 2 ) ( ω L + n Ω ω ) 2 τ 2 ] · k exp ( i ω k T ) .
ρ ˙ 11 = 2 Im [ H 12 ρ 21 + H 13 ρ 31 ] , ρ ˙ 22 = 2 Im [ H 21 ρ 12 + H 23 ρ 32 ] , ρ ˙ 33 = 2 Im [ H 31 ρ 13 + H 32 ρ 23 ] , ρ ˙ 12 = i H 12 ( ρ 22 ρ 11 ) i H 13 ρ 32 + i H 32 ρ 13 , ρ ˙ 13 = i H 13 ( ρ 33 ρ 11 ) i H 12 ρ 23 + i H 23 ρ 12 , ρ ˙ 23 = i H 23 ( ρ 33 ρ 22 ) i H 21 ρ 13 + i H 13 ρ 21 .
ρ ˙ 11 ) sp = γ 21 ρ 22 ρ ˙ 12 ) sp , col = ( γ 21 2 + γ 23 2 + Γ 21 ) ρ 12 ρ ˙ 22 ) sp = γ 21 ρ 22 γ 23 ρ 22 ρ ˙ 13 ) sp , col = Γ 31 ρ 13 ρ ˙ 33 ) sp = γ 23 ρ 22 ρ ˙ 23 ) sp , col = ( γ 21 2 + γ 23 2 + Γ 23 ) ρ 23 ,

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