Abstract

We study the force exerted on two-level atoms by short, counterpropagating laser pulses. When the counterpropagating pulses overlap each other partially, multiphoton adiabatic processes are possible in several configurations, which amplify the force exerted on the atoms. We investigate the practical usefulness of such multiphoton adiabatic transitions for the manipulation of the atoms’ mechanical state. In particular, we compare the efficiency of a pair of constant frequency, oppositely detuned laser pulses and that of a pair of frequency-chirped pulses. We also consider the case of prolonged exposure to a sequence of laser pulses for a duration that is comparable to or much larger than the spontaneous lifetime of the atoms. We use numerical methods to calculate the reduction of the force and the heating of the atomic ensemble when spontaneous emission cannot be neglected during the interaction. In addition, we derive simple approximate formulas for the force and the heating, and compare them to the numerical results.

© 2009 Optical Society of America

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  1. H. Metcalf and P. van der Straten, Laser Cooling and Trapping (Springer, 1999).
    [CrossRef]
  2. B. Nölle, H. Nölle, J. Schmand, and H. J. Andrä, “Atomic-beam deflection by double-Pi-pulse laser technique,” Europhys. Lett. 33, 261-266 (1996).
    [CrossRef]
  3. T. G. M. Freegarde, J. Walz, and T. W. Hänsch, “Confinement and manipulation of atoms using short laser pulses,” Opt. Commun. 117, 262-267 (1995).
    [CrossRef]
  4. A. Goepfert, I. Bloch, D. Haubrich, F. Lison, R. Schütze, R. Wynands, and D. Meschede, “Stimulated focusing and deflection of an atomic beam using picosecond laser pulses,” Phys. Rev. A 56, R3354-R3357 (1997).
    [CrossRef]
  5. I. Nebenzahl and A. Szöke, “Deflection of atomic beams by resonance radiation using stimulated emission,” Appl. Phys. Lett. 25, 327-329 (1974).
    [CrossRef]
  6. J. S. Bakos, G. P. Djotyan, G. Demeter, and Zs. Sörlei, “Transient laser cooling of two-level quantum systems with narrow natural linewidths,” Phys. Rev. A 53, 2885-2888 (1996).
    [CrossRef] [PubMed]
  7. G. P. Djotyan, J. S. Bakos, G. Demeter, and Zs. Sörlei, “Manipulation of two-level quantum systems with narrow transition lines by short linearly polarized frequency-chirped laser pulses,” J. Opt. Soc. Am. B 13, 1697-1705 (1996).
    [CrossRef]
  8. T. Freegarde, G. Daniell, and D. Segal, “Coherent amplification in laser cooling and trapping,” Phys. Rev. A 73, 033409 (2006).
    [CrossRef]
  9. M. Cashen, O. Rivoire, L. Yatsenko, and H. Metcalf, “Coherent exchange of momentum between atoms and light,” J. Opt. B: Quantum Semiclassical Opt. 4, 75-79 (2002).
    [CrossRef]
  10. J. S. Bakos, G. P. Djotyan, P. N. Ignacz, M. A. Kedves, M. Serenyi, Z. Sorlei, J. Szigeti, and Z. Toth, “Interaction of frequency modulated light pulses with rubidium atoms in a magneto-optical trap,” Eur. Phys. J. D 39, 59-66 (2006).
    [CrossRef]
  11. X. Miao, E. Wertz, M. G. Cohen, and H. Metcalf, “Strong optical forces from adiabatic rapid passage,” Phys. Rev. A 75, 011402(R) (2007).
    [CrossRef]
  12. J. S. Bakos, G. P. Djotyan, P. Ignacz, M. A. Kedves, M. Serenyi, Z. Sorlei, J. Szigeti, and Z. Toth, “Acceleration of cold Rb atoms by frequency modulated light pulses,” Eur. Phys. J. D 44, 141-149 (2007).
    [CrossRef]
  13. G. Demeter, G. P. Djotyan, and J. S. Bakos, “Deflection and splitting of atomic beams using counter-propagating, short, chirped laser pulses,” J. Opt. Soc. Am. B 15, 16-24 (1998).
    [CrossRef]
  14. V. S. Voitsekhovich, M. V. Danileiko, A. M. Negrijko, V. I. Romanenko, and L. P. Yatsenko, “Observation of a stimulated radiation pressure of amplitude-modulated light on atoms,” JETP Lett. 49, 161-164 (1989).
  15. J. Söding, R. Grimm, Y. B. Ovchinnikov, P. Bouyer, and C. Salomon, “Short-distance atomic beam deceleration with a stimulated light force,” Phys. Rev. Lett. 78, 1420-1423 (1997).
    [CrossRef]
  16. M. R. Williams, F. Chi, M. T. Cashen, and H. Metcalf, “Measurement of the bichromatic optical force on Rb atoms,” Phys. Rev. A 60, R1763-R1766 (1999).
    [CrossRef]
  17. M. R. Williams, F. Chi, M. T. Cashen, and H. Metcalf, “Bichromatic force measurements using atomic beam deflections,” Phys. Rev. A 61, 023408 (2000).
    [CrossRef]
  18. V. I. Romanenko and L. P. Yatsenko, “Scattering of atoms in a bichromatic field of oppositely propagating light pulses,” JETP 90, 407-414 (2000).
    [CrossRef]
  19. G. Demeter, G. P. Djotyan, Zs. Sörlei, and J. S. Bakos, “Mechanical effect of retroreflected frequency-chirped laser pulses on two-level atoms,” Phys. Rev. A 74, 013401 (2006).
    [CrossRef]
  20. J. F. Shirley, “Solution of the Schrödinger equation with a Hamiltonian periodic in time,” Phys. Rev. 138, B979-B987 (1965).
    [CrossRef]
  21. S. Guérin, L. P. Yatsenko, and H. R. Jauslin, “Dynamical resonances and the topology of the multiphoton adiabatic passage,” Phys. Rev. A 63, 031403(R) (2001).
    [CrossRef]
  22. B. W. Shore, The Theory of Coherent Atomic Excitation (Wiley, 1990).
  23. S. Guérin, F. Monti, J.-M. Dupont, and H. R. Jauslin, “On the relation between cavity-dressed states, Floquet states, RWA and semiclassical models,” J. Phys. A 30, 7193-7215 (1997).
    [CrossRef]
  24. C. W. S. Conover, M. C. Doogue, and F. J. Struwe, “Chirped-pulse multiphoton transition between Rydberg states,” Phys. Rev. A 65, 033414 (2002).
    [CrossRef]
  25. H. Maeda, J. H. Gurian, D. V. L. Norum, and T. F. Gallagher, “Coherent population transfer in an atom by multiphoton adiabatic rapid passage,” Phys. Rev. Lett. 96, 073002 (2006).
    [CrossRef] [PubMed]
  26. G. N. Gibson, “Adiabatic passage on high-order multiphoton transitions,” Phys. Rev. A 72, 041404(R) (2005).
    [CrossRef]
  27. G. P. Djotyan, J. S. Bakos, G. Demeter, and Zs. Sörlei, “Population transfer in three-level Λ-atoms with Doppler-broadened transition lines by a single frequency-chirped short laser pulse,” J. Opt. Soc. Am. B 17, 107-113 (2000).
    [CrossRef]
  28. G. P. Djotyan, J. S. Bakos, G. Demeter, Zs. Sörlei, J. Szigeti, and D. Dzsotjan, “Creation of a coherent superposition of quantum states by a single frequency-chirped short laser pulse,” J. Opt. Soc. Am. B 25, 166-174 (2008).
    [CrossRef]

2008 (1)

2007 (2)

X. Miao, E. Wertz, M. G. Cohen, and H. Metcalf, “Strong optical forces from adiabatic rapid passage,” Phys. Rev. A 75, 011402(R) (2007).
[CrossRef]

J. S. Bakos, G. P. Djotyan, P. Ignacz, M. A. Kedves, M. Serenyi, Z. Sorlei, J. Szigeti, and Z. Toth, “Acceleration of cold Rb atoms by frequency modulated light pulses,” Eur. Phys. J. D 44, 141-149 (2007).
[CrossRef]

2006 (4)

J. S. Bakos, G. P. Djotyan, P. N. Ignacz, M. A. Kedves, M. Serenyi, Z. Sorlei, J. Szigeti, and Z. Toth, “Interaction of frequency modulated light pulses with rubidium atoms in a magneto-optical trap,” Eur. Phys. J. D 39, 59-66 (2006).
[CrossRef]

T. Freegarde, G. Daniell, and D. Segal, “Coherent amplification in laser cooling and trapping,” Phys. Rev. A 73, 033409 (2006).
[CrossRef]

G. Demeter, G. P. Djotyan, Zs. Sörlei, and J. S. Bakos, “Mechanical effect of retroreflected frequency-chirped laser pulses on two-level atoms,” Phys. Rev. A 74, 013401 (2006).
[CrossRef]

H. Maeda, J. H. Gurian, D. V. L. Norum, and T. F. Gallagher, “Coherent population transfer in an atom by multiphoton adiabatic rapid passage,” Phys. Rev. Lett. 96, 073002 (2006).
[CrossRef] [PubMed]

2005 (1)

G. N. Gibson, “Adiabatic passage on high-order multiphoton transitions,” Phys. Rev. A 72, 041404(R) (2005).
[CrossRef]

2002 (2)

C. W. S. Conover, M. C. Doogue, and F. J. Struwe, “Chirped-pulse multiphoton transition between Rydberg states,” Phys. Rev. A 65, 033414 (2002).
[CrossRef]

M. Cashen, O. Rivoire, L. Yatsenko, and H. Metcalf, “Coherent exchange of momentum between atoms and light,” J. Opt. B: Quantum Semiclassical Opt. 4, 75-79 (2002).
[CrossRef]

2001 (1)

S. Guérin, L. P. Yatsenko, and H. R. Jauslin, “Dynamical resonances and the topology of the multiphoton adiabatic passage,” Phys. Rev. A 63, 031403(R) (2001).
[CrossRef]

2000 (3)

G. P. Djotyan, J. S. Bakos, G. Demeter, and Zs. Sörlei, “Population transfer in three-level Λ-atoms with Doppler-broadened transition lines by a single frequency-chirped short laser pulse,” J. Opt. Soc. Am. B 17, 107-113 (2000).
[CrossRef]

M. R. Williams, F. Chi, M. T. Cashen, and H. Metcalf, “Bichromatic force measurements using atomic beam deflections,” Phys. Rev. A 61, 023408 (2000).
[CrossRef]

V. I. Romanenko and L. P. Yatsenko, “Scattering of atoms in a bichromatic field of oppositely propagating light pulses,” JETP 90, 407-414 (2000).
[CrossRef]

1999 (2)

H. Metcalf and P. van der Straten, Laser Cooling and Trapping (Springer, 1999).
[CrossRef]

M. R. Williams, F. Chi, M. T. Cashen, and H. Metcalf, “Measurement of the bichromatic optical force on Rb atoms,” Phys. Rev. A 60, R1763-R1766 (1999).
[CrossRef]

1998 (1)

1997 (3)

J. Söding, R. Grimm, Y. B. Ovchinnikov, P. Bouyer, and C. Salomon, “Short-distance atomic beam deceleration with a stimulated light force,” Phys. Rev. Lett. 78, 1420-1423 (1997).
[CrossRef]

A. Goepfert, I. Bloch, D. Haubrich, F. Lison, R. Schütze, R. Wynands, and D. Meschede, “Stimulated focusing and deflection of an atomic beam using picosecond laser pulses,” Phys. Rev. A 56, R3354-R3357 (1997).
[CrossRef]

S. Guérin, F. Monti, J.-M. Dupont, and H. R. Jauslin, “On the relation between cavity-dressed states, Floquet states, RWA and semiclassical models,” J. Phys. A 30, 7193-7215 (1997).
[CrossRef]

1996 (3)

B. Nölle, H. Nölle, J. Schmand, and H. J. Andrä, “Atomic-beam deflection by double-Pi-pulse laser technique,” Europhys. Lett. 33, 261-266 (1996).
[CrossRef]

J. S. Bakos, G. P. Djotyan, G. Demeter, and Zs. Sörlei, “Transient laser cooling of two-level quantum systems with narrow natural linewidths,” Phys. Rev. A 53, 2885-2888 (1996).
[CrossRef] [PubMed]

G. P. Djotyan, J. S. Bakos, G. Demeter, and Zs. Sörlei, “Manipulation of two-level quantum systems with narrow transition lines by short linearly polarized frequency-chirped laser pulses,” J. Opt. Soc. Am. B 13, 1697-1705 (1996).
[CrossRef]

1995 (1)

T. G. M. Freegarde, J. Walz, and T. W. Hänsch, “Confinement and manipulation of atoms using short laser pulses,” Opt. Commun. 117, 262-267 (1995).
[CrossRef]

1990 (1)

B. W. Shore, The Theory of Coherent Atomic Excitation (Wiley, 1990).

1989 (1)

V. S. Voitsekhovich, M. V. Danileiko, A. M. Negrijko, V. I. Romanenko, and L. P. Yatsenko, “Observation of a stimulated radiation pressure of amplitude-modulated light on atoms,” JETP Lett. 49, 161-164 (1989).

1974 (1)

I. Nebenzahl and A. Szöke, “Deflection of atomic beams by resonance radiation using stimulated emission,” Appl. Phys. Lett. 25, 327-329 (1974).
[CrossRef]

1965 (1)

J. F. Shirley, “Solution of the Schrödinger equation with a Hamiltonian periodic in time,” Phys. Rev. 138, B979-B987 (1965).
[CrossRef]

Andrä, H. J.

B. Nölle, H. Nölle, J. Schmand, and H. J. Andrä, “Atomic-beam deflection by double-Pi-pulse laser technique,” Europhys. Lett. 33, 261-266 (1996).
[CrossRef]

Bakos, J. S.

G. P. Djotyan, J. S. Bakos, G. Demeter, Zs. Sörlei, J. Szigeti, and D. Dzsotjan, “Creation of a coherent superposition of quantum states by a single frequency-chirped short laser pulse,” J. Opt. Soc. Am. B 25, 166-174 (2008).
[CrossRef]

J. S. Bakos, G. P. Djotyan, P. Ignacz, M. A. Kedves, M. Serenyi, Z. Sorlei, J. Szigeti, and Z. Toth, “Acceleration of cold Rb atoms by frequency modulated light pulses,” Eur. Phys. J. D 44, 141-149 (2007).
[CrossRef]

J. S. Bakos, G. P. Djotyan, P. N. Ignacz, M. A. Kedves, M. Serenyi, Z. Sorlei, J. Szigeti, and Z. Toth, “Interaction of frequency modulated light pulses with rubidium atoms in a magneto-optical trap,” Eur. Phys. J. D 39, 59-66 (2006).
[CrossRef]

G. Demeter, G. P. Djotyan, Zs. Sörlei, and J. S. Bakos, “Mechanical effect of retroreflected frequency-chirped laser pulses on two-level atoms,” Phys. Rev. A 74, 013401 (2006).
[CrossRef]

G. P. Djotyan, J. S. Bakos, G. Demeter, and Zs. Sörlei, “Population transfer in three-level Λ-atoms with Doppler-broadened transition lines by a single frequency-chirped short laser pulse,” J. Opt. Soc. Am. B 17, 107-113 (2000).
[CrossRef]

G. Demeter, G. P. Djotyan, and J. S. Bakos, “Deflection and splitting of atomic beams using counter-propagating, short, chirped laser pulses,” J. Opt. Soc. Am. B 15, 16-24 (1998).
[CrossRef]

G. P. Djotyan, J. S. Bakos, G. Demeter, and Zs. Sörlei, “Manipulation of two-level quantum systems with narrow transition lines by short linearly polarized frequency-chirped laser pulses,” J. Opt. Soc. Am. B 13, 1697-1705 (1996).
[CrossRef]

J. S. Bakos, G. P. Djotyan, G. Demeter, and Zs. Sörlei, “Transient laser cooling of two-level quantum systems with narrow natural linewidths,” Phys. Rev. A 53, 2885-2888 (1996).
[CrossRef] [PubMed]

Bloch, I.

A. Goepfert, I. Bloch, D. Haubrich, F. Lison, R. Schütze, R. Wynands, and D. Meschede, “Stimulated focusing and deflection of an atomic beam using picosecond laser pulses,” Phys. Rev. A 56, R3354-R3357 (1997).
[CrossRef]

Bouyer, P.

J. Söding, R. Grimm, Y. B. Ovchinnikov, P. Bouyer, and C. Salomon, “Short-distance atomic beam deceleration with a stimulated light force,” Phys. Rev. Lett. 78, 1420-1423 (1997).
[CrossRef]

Cashen, M.

M. Cashen, O. Rivoire, L. Yatsenko, and H. Metcalf, “Coherent exchange of momentum between atoms and light,” J. Opt. B: Quantum Semiclassical Opt. 4, 75-79 (2002).
[CrossRef]

Cashen, M. T.

M. R. Williams, F. Chi, M. T. Cashen, and H. Metcalf, “Bichromatic force measurements using atomic beam deflections,” Phys. Rev. A 61, 023408 (2000).
[CrossRef]

M. R. Williams, F. Chi, M. T. Cashen, and H. Metcalf, “Measurement of the bichromatic optical force on Rb atoms,” Phys. Rev. A 60, R1763-R1766 (1999).
[CrossRef]

Chi, F.

M. R. Williams, F. Chi, M. T. Cashen, and H. Metcalf, “Bichromatic force measurements using atomic beam deflections,” Phys. Rev. A 61, 023408 (2000).
[CrossRef]

M. R. Williams, F. Chi, M. T. Cashen, and H. Metcalf, “Measurement of the bichromatic optical force on Rb atoms,” Phys. Rev. A 60, R1763-R1766 (1999).
[CrossRef]

Cohen, M. G.

X. Miao, E. Wertz, M. G. Cohen, and H. Metcalf, “Strong optical forces from adiabatic rapid passage,” Phys. Rev. A 75, 011402(R) (2007).
[CrossRef]

Conover, C. W. S.

C. W. S. Conover, M. C. Doogue, and F. J. Struwe, “Chirped-pulse multiphoton transition between Rydberg states,” Phys. Rev. A 65, 033414 (2002).
[CrossRef]

Daniell, G.

T. Freegarde, G. Daniell, and D. Segal, “Coherent amplification in laser cooling and trapping,” Phys. Rev. A 73, 033409 (2006).
[CrossRef]

Danileiko, M. V.

V. S. Voitsekhovich, M. V. Danileiko, A. M. Negrijko, V. I. Romanenko, and L. P. Yatsenko, “Observation of a stimulated radiation pressure of amplitude-modulated light on atoms,” JETP Lett. 49, 161-164 (1989).

Demeter, G.

Djotyan, G. P.

G. P. Djotyan, J. S. Bakos, G. Demeter, Zs. Sörlei, J. Szigeti, and D. Dzsotjan, “Creation of a coherent superposition of quantum states by a single frequency-chirped short laser pulse,” J. Opt. Soc. Am. B 25, 166-174 (2008).
[CrossRef]

J. S. Bakos, G. P. Djotyan, P. Ignacz, M. A. Kedves, M. Serenyi, Z. Sorlei, J. Szigeti, and Z. Toth, “Acceleration of cold Rb atoms by frequency modulated light pulses,” Eur. Phys. J. D 44, 141-149 (2007).
[CrossRef]

J. S. Bakos, G. P. Djotyan, P. N. Ignacz, M. A. Kedves, M. Serenyi, Z. Sorlei, J. Szigeti, and Z. Toth, “Interaction of frequency modulated light pulses with rubidium atoms in a magneto-optical trap,” Eur. Phys. J. D 39, 59-66 (2006).
[CrossRef]

G. Demeter, G. P. Djotyan, Zs. Sörlei, and J. S. Bakos, “Mechanical effect of retroreflected frequency-chirped laser pulses on two-level atoms,” Phys. Rev. A 74, 013401 (2006).
[CrossRef]

G. P. Djotyan, J. S. Bakos, G. Demeter, and Zs. Sörlei, “Population transfer in three-level Λ-atoms with Doppler-broadened transition lines by a single frequency-chirped short laser pulse,” J. Opt. Soc. Am. B 17, 107-113 (2000).
[CrossRef]

G. Demeter, G. P. Djotyan, and J. S. Bakos, “Deflection and splitting of atomic beams using counter-propagating, short, chirped laser pulses,” J. Opt. Soc. Am. B 15, 16-24 (1998).
[CrossRef]

G. P. Djotyan, J. S. Bakos, G. Demeter, and Zs. Sörlei, “Manipulation of two-level quantum systems with narrow transition lines by short linearly polarized frequency-chirped laser pulses,” J. Opt. Soc. Am. B 13, 1697-1705 (1996).
[CrossRef]

J. S. Bakos, G. P. Djotyan, G. Demeter, and Zs. Sörlei, “Transient laser cooling of two-level quantum systems with narrow natural linewidths,” Phys. Rev. A 53, 2885-2888 (1996).
[CrossRef] [PubMed]

Doogue, M. C.

C. W. S. Conover, M. C. Doogue, and F. J. Struwe, “Chirped-pulse multiphoton transition between Rydberg states,” Phys. Rev. A 65, 033414 (2002).
[CrossRef]

Dupont, J.-M.

S. Guérin, F. Monti, J.-M. Dupont, and H. R. Jauslin, “On the relation between cavity-dressed states, Floquet states, RWA and semiclassical models,” J. Phys. A 30, 7193-7215 (1997).
[CrossRef]

Dzsotjan, D.

Freegarde, T.

T. Freegarde, G. Daniell, and D. Segal, “Coherent amplification in laser cooling and trapping,” Phys. Rev. A 73, 033409 (2006).
[CrossRef]

Freegarde, T. G. M.

T. G. M. Freegarde, J. Walz, and T. W. Hänsch, “Confinement and manipulation of atoms using short laser pulses,” Opt. Commun. 117, 262-267 (1995).
[CrossRef]

Gallagher, T. F.

H. Maeda, J. H. Gurian, D. V. L. Norum, and T. F. Gallagher, “Coherent population transfer in an atom by multiphoton adiabatic rapid passage,” Phys. Rev. Lett. 96, 073002 (2006).
[CrossRef] [PubMed]

Gibson, G. N.

G. N. Gibson, “Adiabatic passage on high-order multiphoton transitions,” Phys. Rev. A 72, 041404(R) (2005).
[CrossRef]

Goepfert, A.

A. Goepfert, I. Bloch, D. Haubrich, F. Lison, R. Schütze, R. Wynands, and D. Meschede, “Stimulated focusing and deflection of an atomic beam using picosecond laser pulses,” Phys. Rev. A 56, R3354-R3357 (1997).
[CrossRef]

Grimm, R.

J. Söding, R. Grimm, Y. B. Ovchinnikov, P. Bouyer, and C. Salomon, “Short-distance atomic beam deceleration with a stimulated light force,” Phys. Rev. Lett. 78, 1420-1423 (1997).
[CrossRef]

Guérin, S.

S. Guérin, L. P. Yatsenko, and H. R. Jauslin, “Dynamical resonances and the topology of the multiphoton adiabatic passage,” Phys. Rev. A 63, 031403(R) (2001).
[CrossRef]

S. Guérin, F. Monti, J.-M. Dupont, and H. R. Jauslin, “On the relation between cavity-dressed states, Floquet states, RWA and semiclassical models,” J. Phys. A 30, 7193-7215 (1997).
[CrossRef]

Gurian, J. H.

H. Maeda, J. H. Gurian, D. V. L. Norum, and T. F. Gallagher, “Coherent population transfer in an atom by multiphoton adiabatic rapid passage,” Phys. Rev. Lett. 96, 073002 (2006).
[CrossRef] [PubMed]

Hänsch, T. W.

T. G. M. Freegarde, J. Walz, and T. W. Hänsch, “Confinement and manipulation of atoms using short laser pulses,” Opt. Commun. 117, 262-267 (1995).
[CrossRef]

Haubrich, D.

A. Goepfert, I. Bloch, D. Haubrich, F. Lison, R. Schütze, R. Wynands, and D. Meschede, “Stimulated focusing and deflection of an atomic beam using picosecond laser pulses,” Phys. Rev. A 56, R3354-R3357 (1997).
[CrossRef]

Ignacz, P.

J. S. Bakos, G. P. Djotyan, P. Ignacz, M. A. Kedves, M. Serenyi, Z. Sorlei, J. Szigeti, and Z. Toth, “Acceleration of cold Rb atoms by frequency modulated light pulses,” Eur. Phys. J. D 44, 141-149 (2007).
[CrossRef]

Ignacz, P. N.

J. S. Bakos, G. P. Djotyan, P. N. Ignacz, M. A. Kedves, M. Serenyi, Z. Sorlei, J. Szigeti, and Z. Toth, “Interaction of frequency modulated light pulses with rubidium atoms in a magneto-optical trap,” Eur. Phys. J. D 39, 59-66 (2006).
[CrossRef]

Jauslin, H. R.

S. Guérin, L. P. Yatsenko, and H. R. Jauslin, “Dynamical resonances and the topology of the multiphoton adiabatic passage,” Phys. Rev. A 63, 031403(R) (2001).
[CrossRef]

S. Guérin, F. Monti, J.-M. Dupont, and H. R. Jauslin, “On the relation between cavity-dressed states, Floquet states, RWA and semiclassical models,” J. Phys. A 30, 7193-7215 (1997).
[CrossRef]

Kedves, M. A.

J. S. Bakos, G. P. Djotyan, P. Ignacz, M. A. Kedves, M. Serenyi, Z. Sorlei, J. Szigeti, and Z. Toth, “Acceleration of cold Rb atoms by frequency modulated light pulses,” Eur. Phys. J. D 44, 141-149 (2007).
[CrossRef]

J. S. Bakos, G. P. Djotyan, P. N. Ignacz, M. A. Kedves, M. Serenyi, Z. Sorlei, J. Szigeti, and Z. Toth, “Interaction of frequency modulated light pulses with rubidium atoms in a magneto-optical trap,” Eur. Phys. J. D 39, 59-66 (2006).
[CrossRef]

Lison, F.

A. Goepfert, I. Bloch, D. Haubrich, F. Lison, R. Schütze, R. Wynands, and D. Meschede, “Stimulated focusing and deflection of an atomic beam using picosecond laser pulses,” Phys. Rev. A 56, R3354-R3357 (1997).
[CrossRef]

Maeda, H.

H. Maeda, J. H. Gurian, D. V. L. Norum, and T. F. Gallagher, “Coherent population transfer in an atom by multiphoton adiabatic rapid passage,” Phys. Rev. Lett. 96, 073002 (2006).
[CrossRef] [PubMed]

Meschede, D.

A. Goepfert, I. Bloch, D. Haubrich, F. Lison, R. Schütze, R. Wynands, and D. Meschede, “Stimulated focusing and deflection of an atomic beam using picosecond laser pulses,” Phys. Rev. A 56, R3354-R3357 (1997).
[CrossRef]

Metcalf, H.

X. Miao, E. Wertz, M. G. Cohen, and H. Metcalf, “Strong optical forces from adiabatic rapid passage,” Phys. Rev. A 75, 011402(R) (2007).
[CrossRef]

M. Cashen, O. Rivoire, L. Yatsenko, and H. Metcalf, “Coherent exchange of momentum between atoms and light,” J. Opt. B: Quantum Semiclassical Opt. 4, 75-79 (2002).
[CrossRef]

M. R. Williams, F. Chi, M. T. Cashen, and H. Metcalf, “Bichromatic force measurements using atomic beam deflections,” Phys. Rev. A 61, 023408 (2000).
[CrossRef]

M. R. Williams, F. Chi, M. T. Cashen, and H. Metcalf, “Measurement of the bichromatic optical force on Rb atoms,” Phys. Rev. A 60, R1763-R1766 (1999).
[CrossRef]

H. Metcalf and P. van der Straten, Laser Cooling and Trapping (Springer, 1999).
[CrossRef]

Miao, X.

X. Miao, E. Wertz, M. G. Cohen, and H. Metcalf, “Strong optical forces from adiabatic rapid passage,” Phys. Rev. A 75, 011402(R) (2007).
[CrossRef]

Monti, F.

S. Guérin, F. Monti, J.-M. Dupont, and H. R. Jauslin, “On the relation between cavity-dressed states, Floquet states, RWA and semiclassical models,” J. Phys. A 30, 7193-7215 (1997).
[CrossRef]

Nebenzahl, I.

I. Nebenzahl and A. Szöke, “Deflection of atomic beams by resonance radiation using stimulated emission,” Appl. Phys. Lett. 25, 327-329 (1974).
[CrossRef]

Negrijko, A. M.

V. S. Voitsekhovich, M. V. Danileiko, A. M. Negrijko, V. I. Romanenko, and L. P. Yatsenko, “Observation of a stimulated radiation pressure of amplitude-modulated light on atoms,” JETP Lett. 49, 161-164 (1989).

Nölle, B.

B. Nölle, H. Nölle, J. Schmand, and H. J. Andrä, “Atomic-beam deflection by double-Pi-pulse laser technique,” Europhys. Lett. 33, 261-266 (1996).
[CrossRef]

Nölle, H.

B. Nölle, H. Nölle, J. Schmand, and H. J. Andrä, “Atomic-beam deflection by double-Pi-pulse laser technique,” Europhys. Lett. 33, 261-266 (1996).
[CrossRef]

Norum, D. V. L.

H. Maeda, J. H. Gurian, D. V. L. Norum, and T. F. Gallagher, “Coherent population transfer in an atom by multiphoton adiabatic rapid passage,” Phys. Rev. Lett. 96, 073002 (2006).
[CrossRef] [PubMed]

Ovchinnikov, Y. B.

J. Söding, R. Grimm, Y. B. Ovchinnikov, P. Bouyer, and C. Salomon, “Short-distance atomic beam deceleration with a stimulated light force,” Phys. Rev. Lett. 78, 1420-1423 (1997).
[CrossRef]

Rivoire, O.

M. Cashen, O. Rivoire, L. Yatsenko, and H. Metcalf, “Coherent exchange of momentum between atoms and light,” J. Opt. B: Quantum Semiclassical Opt. 4, 75-79 (2002).
[CrossRef]

Romanenko, V. I.

V. I. Romanenko and L. P. Yatsenko, “Scattering of atoms in a bichromatic field of oppositely propagating light pulses,” JETP 90, 407-414 (2000).
[CrossRef]

V. S. Voitsekhovich, M. V. Danileiko, A. M. Negrijko, V. I. Romanenko, and L. P. Yatsenko, “Observation of a stimulated radiation pressure of amplitude-modulated light on atoms,” JETP Lett. 49, 161-164 (1989).

Salomon, C.

J. Söding, R. Grimm, Y. B. Ovchinnikov, P. Bouyer, and C. Salomon, “Short-distance atomic beam deceleration with a stimulated light force,” Phys. Rev. Lett. 78, 1420-1423 (1997).
[CrossRef]

Schmand, J.

B. Nölle, H. Nölle, J. Schmand, and H. J. Andrä, “Atomic-beam deflection by double-Pi-pulse laser technique,” Europhys. Lett. 33, 261-266 (1996).
[CrossRef]

Schütze, R.

A. Goepfert, I. Bloch, D. Haubrich, F. Lison, R. Schütze, R. Wynands, and D. Meschede, “Stimulated focusing and deflection of an atomic beam using picosecond laser pulses,” Phys. Rev. A 56, R3354-R3357 (1997).
[CrossRef]

Segal, D.

T. Freegarde, G. Daniell, and D. Segal, “Coherent amplification in laser cooling and trapping,” Phys. Rev. A 73, 033409 (2006).
[CrossRef]

Serenyi, M.

J. S. Bakos, G. P. Djotyan, P. Ignacz, M. A. Kedves, M. Serenyi, Z. Sorlei, J. Szigeti, and Z. Toth, “Acceleration of cold Rb atoms by frequency modulated light pulses,” Eur. Phys. J. D 44, 141-149 (2007).
[CrossRef]

J. S. Bakos, G. P. Djotyan, P. N. Ignacz, M. A. Kedves, M. Serenyi, Z. Sorlei, J. Szigeti, and Z. Toth, “Interaction of frequency modulated light pulses with rubidium atoms in a magneto-optical trap,” Eur. Phys. J. D 39, 59-66 (2006).
[CrossRef]

Shirley, J. F.

J. F. Shirley, “Solution of the Schrödinger equation with a Hamiltonian periodic in time,” Phys. Rev. 138, B979-B987 (1965).
[CrossRef]

Shore, B. W.

B. W. Shore, The Theory of Coherent Atomic Excitation (Wiley, 1990).

Söding, J.

J. Söding, R. Grimm, Y. B. Ovchinnikov, P. Bouyer, and C. Salomon, “Short-distance atomic beam deceleration with a stimulated light force,” Phys. Rev. Lett. 78, 1420-1423 (1997).
[CrossRef]

Sorlei, Z.

J. S. Bakos, G. P. Djotyan, P. Ignacz, M. A. Kedves, M. Serenyi, Z. Sorlei, J. Szigeti, and Z. Toth, “Acceleration of cold Rb atoms by frequency modulated light pulses,” Eur. Phys. J. D 44, 141-149 (2007).
[CrossRef]

J. S. Bakos, G. P. Djotyan, P. N. Ignacz, M. A. Kedves, M. Serenyi, Z. Sorlei, J. Szigeti, and Z. Toth, “Interaction of frequency modulated light pulses with rubidium atoms in a magneto-optical trap,” Eur. Phys. J. D 39, 59-66 (2006).
[CrossRef]

Sörlei, Zs.

Struwe, F. J.

C. W. S. Conover, M. C. Doogue, and F. J. Struwe, “Chirped-pulse multiphoton transition between Rydberg states,” Phys. Rev. A 65, 033414 (2002).
[CrossRef]

Szigeti, J.

G. P. Djotyan, J. S. Bakos, G. Demeter, Zs. Sörlei, J. Szigeti, and D. Dzsotjan, “Creation of a coherent superposition of quantum states by a single frequency-chirped short laser pulse,” J. Opt. Soc. Am. B 25, 166-174 (2008).
[CrossRef]

J. S. Bakos, G. P. Djotyan, P. Ignacz, M. A. Kedves, M. Serenyi, Z. Sorlei, J. Szigeti, and Z. Toth, “Acceleration of cold Rb atoms by frequency modulated light pulses,” Eur. Phys. J. D 44, 141-149 (2007).
[CrossRef]

J. S. Bakos, G. P. Djotyan, P. N. Ignacz, M. A. Kedves, M. Serenyi, Z. Sorlei, J. Szigeti, and Z. Toth, “Interaction of frequency modulated light pulses with rubidium atoms in a magneto-optical trap,” Eur. Phys. J. D 39, 59-66 (2006).
[CrossRef]

Szöke, A.

I. Nebenzahl and A. Szöke, “Deflection of atomic beams by resonance radiation using stimulated emission,” Appl. Phys. Lett. 25, 327-329 (1974).
[CrossRef]

Toth, Z.

J. S. Bakos, G. P. Djotyan, P. Ignacz, M. A. Kedves, M. Serenyi, Z. Sorlei, J. Szigeti, and Z. Toth, “Acceleration of cold Rb atoms by frequency modulated light pulses,” Eur. Phys. J. D 44, 141-149 (2007).
[CrossRef]

J. S. Bakos, G. P. Djotyan, P. N. Ignacz, M. A. Kedves, M. Serenyi, Z. Sorlei, J. Szigeti, and Z. Toth, “Interaction of frequency modulated light pulses with rubidium atoms in a magneto-optical trap,” Eur. Phys. J. D 39, 59-66 (2006).
[CrossRef]

van der Straten, P.

H. Metcalf and P. van der Straten, Laser Cooling and Trapping (Springer, 1999).
[CrossRef]

Voitsekhovich, V. S.

V. S. Voitsekhovich, M. V. Danileiko, A. M. Negrijko, V. I. Romanenko, and L. P. Yatsenko, “Observation of a stimulated radiation pressure of amplitude-modulated light on atoms,” JETP Lett. 49, 161-164 (1989).

Walz, J.

T. G. M. Freegarde, J. Walz, and T. W. Hänsch, “Confinement and manipulation of atoms using short laser pulses,” Opt. Commun. 117, 262-267 (1995).
[CrossRef]

Wertz, E.

X. Miao, E. Wertz, M. G. Cohen, and H. Metcalf, “Strong optical forces from adiabatic rapid passage,” Phys. Rev. A 75, 011402(R) (2007).
[CrossRef]

Williams, M. R.

M. R. Williams, F. Chi, M. T. Cashen, and H. Metcalf, “Bichromatic force measurements using atomic beam deflections,” Phys. Rev. A 61, 023408 (2000).
[CrossRef]

M. R. Williams, F. Chi, M. T. Cashen, and H. Metcalf, “Measurement of the bichromatic optical force on Rb atoms,” Phys. Rev. A 60, R1763-R1766 (1999).
[CrossRef]

Wynands, R.

A. Goepfert, I. Bloch, D. Haubrich, F. Lison, R. Schütze, R. Wynands, and D. Meschede, “Stimulated focusing and deflection of an atomic beam using picosecond laser pulses,” Phys. Rev. A 56, R3354-R3357 (1997).
[CrossRef]

Yatsenko, L.

M. Cashen, O. Rivoire, L. Yatsenko, and H. Metcalf, “Coherent exchange of momentum between atoms and light,” J. Opt. B: Quantum Semiclassical Opt. 4, 75-79 (2002).
[CrossRef]

Yatsenko, L. P.

S. Guérin, L. P. Yatsenko, and H. R. Jauslin, “Dynamical resonances and the topology of the multiphoton adiabatic passage,” Phys. Rev. A 63, 031403(R) (2001).
[CrossRef]

V. I. Romanenko and L. P. Yatsenko, “Scattering of atoms in a bichromatic field of oppositely propagating light pulses,” JETP 90, 407-414 (2000).
[CrossRef]

V. S. Voitsekhovich, M. V. Danileiko, A. M. Negrijko, V. I. Romanenko, and L. P. Yatsenko, “Observation of a stimulated radiation pressure of amplitude-modulated light on atoms,” JETP Lett. 49, 161-164 (1989).

Appl. Phys. Lett. (1)

I. Nebenzahl and A. Szöke, “Deflection of atomic beams by resonance radiation using stimulated emission,” Appl. Phys. Lett. 25, 327-329 (1974).
[CrossRef]

Eur. Phys. J. D (2)

J. S. Bakos, G. P. Djotyan, P. Ignacz, M. A. Kedves, M. Serenyi, Z. Sorlei, J. Szigeti, and Z. Toth, “Acceleration of cold Rb atoms by frequency modulated light pulses,” Eur. Phys. J. D 44, 141-149 (2007).
[CrossRef]

J. S. Bakos, G. P. Djotyan, P. N. Ignacz, M. A. Kedves, M. Serenyi, Z. Sorlei, J. Szigeti, and Z. Toth, “Interaction of frequency modulated light pulses with rubidium atoms in a magneto-optical trap,” Eur. Phys. J. D 39, 59-66 (2006).
[CrossRef]

Europhys. Lett. (1)

B. Nölle, H. Nölle, J. Schmand, and H. J. Andrä, “Atomic-beam deflection by double-Pi-pulse laser technique,” Europhys. Lett. 33, 261-266 (1996).
[CrossRef]

J. Opt. B: Quantum Semiclassical Opt. (1)

M. Cashen, O. Rivoire, L. Yatsenko, and H. Metcalf, “Coherent exchange of momentum between atoms and light,” J. Opt. B: Quantum Semiclassical Opt. 4, 75-79 (2002).
[CrossRef]

J. Opt. Soc. Am. B (4)

J. Phys. A (1)

S. Guérin, F. Monti, J.-M. Dupont, and H. R. Jauslin, “On the relation between cavity-dressed states, Floquet states, RWA and semiclassical models,” J. Phys. A 30, 7193-7215 (1997).
[CrossRef]

JETP (1)

V. I. Romanenko and L. P. Yatsenko, “Scattering of atoms in a bichromatic field of oppositely propagating light pulses,” JETP 90, 407-414 (2000).
[CrossRef]

JETP Lett. (1)

V. S. Voitsekhovich, M. V. Danileiko, A. M. Negrijko, V. I. Romanenko, and L. P. Yatsenko, “Observation of a stimulated radiation pressure of amplitude-modulated light on atoms,” JETP Lett. 49, 161-164 (1989).

Opt. Commun. (1)

T. G. M. Freegarde, J. Walz, and T. W. Hänsch, “Confinement and manipulation of atoms using short laser pulses,” Opt. Commun. 117, 262-267 (1995).
[CrossRef]

Phys. Rev. (1)

J. F. Shirley, “Solution of the Schrödinger equation with a Hamiltonian periodic in time,” Phys. Rev. 138, B979-B987 (1965).
[CrossRef]

Phys. Rev. A (10)

S. Guérin, L. P. Yatsenko, and H. R. Jauslin, “Dynamical resonances and the topology of the multiphoton adiabatic passage,” Phys. Rev. A 63, 031403(R) (2001).
[CrossRef]

G. Demeter, G. P. Djotyan, Zs. Sörlei, and J. S. Bakos, “Mechanical effect of retroreflected frequency-chirped laser pulses on two-level atoms,” Phys. Rev. A 74, 013401 (2006).
[CrossRef]

C. W. S. Conover, M. C. Doogue, and F. J. Struwe, “Chirped-pulse multiphoton transition between Rydberg states,” Phys. Rev. A 65, 033414 (2002).
[CrossRef]

G. N. Gibson, “Adiabatic passage on high-order multiphoton transitions,” Phys. Rev. A 72, 041404(R) (2005).
[CrossRef]

A. Goepfert, I. Bloch, D. Haubrich, F. Lison, R. Schütze, R. Wynands, and D. Meschede, “Stimulated focusing and deflection of an atomic beam using picosecond laser pulses,” Phys. Rev. A 56, R3354-R3357 (1997).
[CrossRef]

J. S. Bakos, G. P. Djotyan, G. Demeter, and Zs. Sörlei, “Transient laser cooling of two-level quantum systems with narrow natural linewidths,” Phys. Rev. A 53, 2885-2888 (1996).
[CrossRef] [PubMed]

T. Freegarde, G. Daniell, and D. Segal, “Coherent amplification in laser cooling and trapping,” Phys. Rev. A 73, 033409 (2006).
[CrossRef]

X. Miao, E. Wertz, M. G. Cohen, and H. Metcalf, “Strong optical forces from adiabatic rapid passage,” Phys. Rev. A 75, 011402(R) (2007).
[CrossRef]

M. R. Williams, F. Chi, M. T. Cashen, and H. Metcalf, “Measurement of the bichromatic optical force on Rb atoms,” Phys. Rev. A 60, R1763-R1766 (1999).
[CrossRef]

M. R. Williams, F. Chi, M. T. Cashen, and H. Metcalf, “Bichromatic force measurements using atomic beam deflections,” Phys. Rev. A 61, 023408 (2000).
[CrossRef]

Phys. Rev. Lett. (2)

J. Söding, R. Grimm, Y. B. Ovchinnikov, P. Bouyer, and C. Salomon, “Short-distance atomic beam deceleration with a stimulated light force,” Phys. Rev. Lett. 78, 1420-1423 (1997).
[CrossRef]

H. Maeda, J. H. Gurian, D. V. L. Norum, and T. F. Gallagher, “Coherent population transfer in an atom by multiphoton adiabatic rapid passage,” Phys. Rev. Lett. 96, 073002 (2006).
[CrossRef] [PubMed]

Other (2)

B. W. Shore, The Theory of Coherent Atomic Excitation (Wiley, 1990).

H. Metcalf and P. van der Straten, Laser Cooling and Trapping (Springer, 1999).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Timing of the laser pulse pairs. The second pulse of the pair, which counterpropagates the first one arrives with a delay T 1 , which is the same order of magnitude as the pulse length τ. The cycle then repeats itself after T 2 = T c T 1 T 1 . (Arrows above the pulses symbolize the propagation direction.)

Fig. 2
Fig. 2

Solid black curves: average momentum obtained by atoms in the ground state Δ p ¯ during the interaction with a pair of counterpropagating pulses as a function of the normalized pulse amplitude Ω 0 τ . (a) δ, δ configuration, with δ = 100 τ and T 1 = 2 τ . (b) β, β configuration with β = 20 τ 2 and T 1 = 1.5 τ . Dashed curves: population of the excited state after the interaction multiplied by 5 to be visible. (Note that Δ p ¯ has been plotted for the δ, δ case; see the text for an explanation.)

Fig. 3
Fig. 3

Floquet eigenvalues as a function of time during interaction with overlapping pulses. (a) δ, δ configuration for Ω ± τ = 420 , δ ± τ = ± 70 and T 1 = 2 τ and (b) β, β configuration for Ω ± τ = 60 , β ± τ 2 = 20 and T 1 = 1.5 τ . Some levels are labeled with the index of the corresponding Floquet state at the beginning and at the end of the interaction. In particular, the evolution of the Floquet state that is connected with the 0 , g state before the interaction is marked with a thick black curve.

Fig. 4
Fig. 4

(a) Momentum-space distribution functions before (dashed curve) and after (solid curve) the interaction with 30 pairs of overlapping frequency-chirped laser pulses. The parameters used for the calculation are Γ = 1 97 ns , τ = 2 ns , Ω 0 = 100 τ , β = 20 τ 2 , T 1 = 1.714 τ (which corresponds to the N = 4 plateau), T c = 80 ns . (b) The value of Δ p ¯ (crosses) and Δ σ p 2 (diamonds) in units of k after each cycle.

Fig. 5
Fig. 5

(a) Comparison of the simulation results (squares) and the simple model (stars): (a) and (c) depict the steady state momentum transfer per cycle Δ p ¯ s t for T c = 80 and 40 ns , respectively. (b) and (d) depict the heating per cycle Δ σ p 2 for T c = 80 and 40 ns , respectively. All quantities are plotted as a function of N, so the delay T 1 is different in each case. Other parameters are the same as for Fig. 4: Γ = 1 97 ns , τ = 2 ns , Ω 0 = 100 τ , β = 20 τ 2 .

Equations (27)

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

H ̂ = p ̂ 2 2 m + H ̂ a d ̂ E ,
E ( x , t ) = E + ( t ) cos [ k x ω ge t ϕ + ( t ) ] + E ( t ) cos [ k x ω ge t ϕ ( t ) ] ,
Ψ = [ α ( p , t ) e i ω g t g , p + β ( p , t ) e i ω e t e , p ] d p ,
i t α ( p , t ) = p 2 2 m α ( p , t ) Ω + ( t ) 2 e i ϕ + ( t ) β ( p + k , t ) Ω ( t ) 2 e i ϕ ( t ) β ( p k , t ) ,
i t β ( p , t ) = p 2 2 m β ( p , t ) Ω + ( t ) 2 e i ϕ + ( t ) α ( p k , t ) Ω ( t ) 2 e i ϕ ( t ) α ( p + k , t ) .
i t ρ ̂ = [ H ̂ , ρ ̂ ] i Γ ̂ .
Ω + ( t ) = Ω 0 π 2 exp [ t 2 2 τ 2 ] , Ω ( t ) = Ω + ( t T 1 ) ,
ϕ + ( t ) = δ t + ϕ 0 + , ϕ ( t ) = δ ( t T 1 ) + ϕ 0 ,
ϕ + ( t ) = β 2 t 2 + ϕ 0 + , ϕ ( t ) = β 2 ( t T 1 ) 2 + ϕ 0 ,
a ( p , t ) = a ( p 2 k , t ) ,
b ( p , t ) = b ( p + 2 k , t ) .
a ( p , t ) = a ( p 2 N k , t ) ,
b ( p , t ) = b ( p + 2 N k , t ) ,
i t α ( t ) = β ( t ) 2 [ Ω + ( t ) e i ϕ + ( t ) + Ω ( t ) e i ϕ ( t ) ] ,
i t β ( t ) = α ( t ) 2 [ Ω + ( t ) e i ϕ + ( t ) + Ω ( t ) e i ϕ ( t ) ] .
α ( t ) = m A m ( t ) exp ( i m Δ t ) ,
β ( t ) = m B m ( t ) exp ( i m Δ t ) .
i t Ψ = H Floq Ψ ,
H Floq = ( ( m 1 ) Δ + δ + Ω + 2 Ω + 2 ( m 1 ) Δ Ω 2 Ω 2 m Δ + δ + Ω + 2 Ω + 2 m Δ Ω 2 Ω 2 ( m + 1 ) Δ + δ + Ω + 2 Ω + 2 ( m + 1 ) Δ ) .
a n + 1 ( p ) = ( 1 q 1 ) a n ( p 2 N k ) + q 1 q 2 a n ( p ) + q 2 b n ( p + 2 N k ) ,
b n + 1 ( p ) = ( 1 q 2 ) b n ( p + 2 N k ) + q 1 ( 1 q 2 ) a n ( p ) ,
P i , n + 1 = P i st + ( P i , n P i st ) ( 1 q c ) i { a , b } ,
P a st = q 2 q c , P b st = q 1 ( 1 q 2 ) q c .
Δ p ¯ = p ¯ n + 1 p ¯ n = 2 N k [ ( 1 q 1 ) P a , n P b , n ] .
Δ p ¯ st = 2 N k q 2 q 1 q c .
Δ σ p 2 = σ p , n + 1 2 σ p , n 2 = K st ,
K st = 8 N 2 2 k 2 q 1 q 2 ( 1 q 1 ) ( 1 q 2 ) ( 2 q 1 ) ( 2 q 2 ) q c 3 + 4 N 2 2 k 2 q 1 q 2 ( 4 q 1 q 2 2 q c ) q c 2 .

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