Abstract

In a recent paper [Y. Xiao et al., Phys. Rev. Lett. 96, 043601 (2006)] we characterized diffusion-induced Ramsey narrowing as a general phenomenon, in which diffusion of coherence in-and-out of an interaction region such as a laser beam induces spectral narrowing of the associated resonance lineshape. Here we provide a detailed presentation of the repeated interaction model of diffusion-induced Ramsey narrowing, with particular focus on its application to Electromagnetically Induced Transparency (EIT) of atomic vapor in a buffer gas cell. We compare this model both to experimental data and numerical calculations.

© 2008 Optical Society of America

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References

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  1. W. Happer, “Optical pumping,” Rev. Mod. Phys. 44, 169–249 (1972).
    [Crossref]
  2. E. Arimondo, “Relaxation processes in coherent-population trapping,” Phys. Rev. A  54, 2216–2223 (1996).
    [Crossref] [PubMed]
  3. M. Erhard and H. Helm, “Buffer-gas effects on dark resonances: theory and experiment,” Phys. Rev. A 63, 043813 (2001).
    [Crossref]
  4. Y. Xiao, I. Novikova, D. F. Phillips, and R. L. Walsworth, “Diffusion-induced Ramsey narrowing,” Phys. Rev. Lett. 96, 043601 (2006).
    [Crossref] [PubMed]
  5. N. F. Ramsey, Molecular beams (Clarendon, Oxford, 1956).
  6. A. S. Zibrov, I. Novikova, and A. B. Matsko, “Observation of Ramsey fringes in an atomic cell with buffer gas,” Opt. Lett. 26, 1311–1313 (2001).
    [Crossref]
  7. A. S. Zibrov and A. B. Matsko, “Optical Ramsey fringes induced by Zeeman coherence,” Phys. Rev. A 65, 013814 (2001).
    [Crossref]
  8. E. Alipieva, S. Gateva, E. Taskova, and S. Cartaleva, “Narrow structure in the coherent population trapping resonance in rubidium,” Opt. Lett. 28, 1817–1819 (2003).
    [Crossref] [PubMed]
  9. G. Alzetta, S. Carta1eva, S. Gozzini, T. Karaulanov, A. Lucchesini, C. Marinelli, L. MOi, K. Nasyrov, V. Sarova, and K. Vaseva, “Magnetic Coherence Resonance Profiles in Na and K,” Proc. SPIE,  5830, 181–185 (2005).
    [Crossref]
  10. J. Vanier, “Atomic clocks based on Coherent Population Trapping: a review,” Appl. Phys. B 81(4), 421–442 (2005).
    [Crossref]
  11. D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, “Resonant nonliear magneto-optical effects in atoms,” Rev. Mod. Phys. 74, 1153–1201 (2002).
    [Crossref]
  12. I. Novikova, Y. Xiao, D. F. Phillips, and R. L. Walsworth,“EIT and diffusion of atomic coherence,” J. Mod. Opt.,  52, 2381–2390 (2005).
    [Crossref]
  13. M. D. Lukin, “Colloquium: Trapping and manipulating photon states in atomic ensembles,” Rev. Mod. Phys. 75, 457–472 (2003).
    [Crossref]
  14. T. Zanon, S. Guerandel, E. de Clercq, D. Holleville, N. Dimarcq, and A. Clairon, “High contrast Ramsey fringes with Coherent-Population-Trapping pulses in a double lambda atomic system,” Phys. Rev. Lett. 94, 193002 (2005).
    [Crossref] [PubMed]
  15. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, Cambridge, UK, 1997).
  16. M. S. Shahriar, P. R. Hemmer, D. P. Katz, A. Lee , and M. G. Prentiss, “Dark-state-based three-element vector model for the stimulated Raman interaction,” Phys. Rev. A 55, 2272–2282 (1997).
    [Crossref]
  17. F. Levi, A. Godone, J. Vanier, S. Micalizio, and G. Modugno, “Line-shape of dark line and maser emission profile in CPT,” Eur. Phys. J. D 12, 53(2000).
    [Crossref]
  18. A. V. Taichenachev, A. M. Tumaikin, V. I. Yudin, M. Stahler, R. Wynands, J. Kitching, and L. Hollberg,“Nonliear-resonance line shapes: ependence on the transverse intensity distribution of a light beam,” Phys. Rev. A 69, 024501 (2004).
    [Crossref]
  19. E. Pfleghaar, J. Wurster, S. I. Kanorsky , and A. Weis, “Time of flight effects in nonliear magneto-optical spectroscopy,” Optics Commun. 99, 303–308 (1993).
    [Crossref]
  20. P. R. Hemmer, M. S. Shahriar, V. D. Natoli, and S. Ezekiel,“AC Stark shifts in a two-zone Raman interaction,” J. Opt. Soc. Am. B 6, 1519–1528 (1989).
    [Crossref]
  21. M. D. Lukin, M. Fleischhauer, A. S. Zibrov, H. G. Robinson, V. L. Velichansky, L. Hollberg, and M. O. Scully, “Spectroscopy in dense coherent media: Line narrowing and interference effects,” Phys. Rev. Lett. 79, 2959–2962 (1997).
    [Crossref]
  22. J. Vanier, M. W. Levine, D. Janssen, and M. Delaney, “Contrast and linewidth of the coherent population trapping transmission hyperfine resonance line in 87Rb : Effect of optical pumping,” Phys. Rev. A. 67, 065801 (2003).
    [Crossref]
  23. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Cambridge University Press, New York (1992) Ch. 19.
  24. A typical EIT lineshape, such as that shown in figure 7, takes approximately 12 hours to calculate on a 1.25 GHz PowerPC G4 whereas the repeated interaction model can be evaluated immediately.
  25. W. Zhang and D. G. Cory, “First Direct Measurement of the Spin Diffusion Rate in a Homogenous Solid,” Phys. Rev. Lett. 80, 1324–1327 (1998).
    [Crossref]
  26. J. Taylor.
  27. J. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C. Gossard, “Coherent manipulation of coupled electron spins in semiconductor quantum dots,” Science 309, 2180–2184 (2005).
    [Crossref] [PubMed]

2006 (1)

Y. Xiao, I. Novikova, D. F. Phillips, and R. L. Walsworth, “Diffusion-induced Ramsey narrowing,” Phys. Rev. Lett. 96, 043601 (2006).
[Crossref] [PubMed]

2005 (5)

G. Alzetta, S. Carta1eva, S. Gozzini, T. Karaulanov, A. Lucchesini, C. Marinelli, L. MOi, K. Nasyrov, V. Sarova, and K. Vaseva, “Magnetic Coherence Resonance Profiles in Na and K,” Proc. SPIE,  5830, 181–185 (2005).
[Crossref]

J. Vanier, “Atomic clocks based on Coherent Population Trapping: a review,” Appl. Phys. B 81(4), 421–442 (2005).
[Crossref]

I. Novikova, Y. Xiao, D. F. Phillips, and R. L. Walsworth,“EIT and diffusion of atomic coherence,” J. Mod. Opt.,  52, 2381–2390 (2005).
[Crossref]

T. Zanon, S. Guerandel, E. de Clercq, D. Holleville, N. Dimarcq, and A. Clairon, “High contrast Ramsey fringes with Coherent-Population-Trapping pulses in a double lambda atomic system,” Phys. Rev. Lett. 94, 193002 (2005).
[Crossref] [PubMed]

J. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C. Gossard, “Coherent manipulation of coupled electron spins in semiconductor quantum dots,” Science 309, 2180–2184 (2005).
[Crossref] [PubMed]

2004 (1)

A. V. Taichenachev, A. M. Tumaikin, V. I. Yudin, M. Stahler, R. Wynands, J. Kitching, and L. Hollberg,“Nonliear-resonance line shapes: ependence on the transverse intensity distribution of a light beam,” Phys. Rev. A 69, 024501 (2004).
[Crossref]

2003 (3)

M. D. Lukin, “Colloquium: Trapping and manipulating photon states in atomic ensembles,” Rev. Mod. Phys. 75, 457–472 (2003).
[Crossref]

E. Alipieva, S. Gateva, E. Taskova, and S. Cartaleva, “Narrow structure in the coherent population trapping resonance in rubidium,” Opt. Lett. 28, 1817–1819 (2003).
[Crossref] [PubMed]

J. Vanier, M. W. Levine, D. Janssen, and M. Delaney, “Contrast and linewidth of the coherent population trapping transmission hyperfine resonance line in 87Rb : Effect of optical pumping,” Phys. Rev. A. 67, 065801 (2003).
[Crossref]

2002 (1)

D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, “Resonant nonliear magneto-optical effects in atoms,” Rev. Mod. Phys. 74, 1153–1201 (2002).
[Crossref]

2001 (3)

M. Erhard and H. Helm, “Buffer-gas effects on dark resonances: theory and experiment,” Phys. Rev. A 63, 043813 (2001).
[Crossref]

A. S. Zibrov, I. Novikova, and A. B. Matsko, “Observation of Ramsey fringes in an atomic cell with buffer gas,” Opt. Lett. 26, 1311–1313 (2001).
[Crossref]

A. S. Zibrov and A. B. Matsko, “Optical Ramsey fringes induced by Zeeman coherence,” Phys. Rev. A 65, 013814 (2001).
[Crossref]

2000 (1)

F. Levi, A. Godone, J. Vanier, S. Micalizio, and G. Modugno, “Line-shape of dark line and maser emission profile in CPT,” Eur. Phys. J. D 12, 53(2000).
[Crossref]

1998 (1)

W. Zhang and D. G. Cory, “First Direct Measurement of the Spin Diffusion Rate in a Homogenous Solid,” Phys. Rev. Lett. 80, 1324–1327 (1998).
[Crossref]

1997 (2)

M. D. Lukin, M. Fleischhauer, A. S. Zibrov, H. G. Robinson, V. L. Velichansky, L. Hollberg, and M. O. Scully, “Spectroscopy in dense coherent media: Line narrowing and interference effects,” Phys. Rev. Lett. 79, 2959–2962 (1997).
[Crossref]

M. S. Shahriar, P. R. Hemmer, D. P. Katz, A. Lee , and M. G. Prentiss, “Dark-state-based three-element vector model for the stimulated Raman interaction,” Phys. Rev. A 55, 2272–2282 (1997).
[Crossref]

1996 (1)

E. Arimondo, “Relaxation processes in coherent-population trapping,” Phys. Rev. A  54, 2216–2223 (1996).
[Crossref] [PubMed]

1993 (1)

E. Pfleghaar, J. Wurster, S. I. Kanorsky , and A. Weis, “Time of flight effects in nonliear magneto-optical spectroscopy,” Optics Commun. 99, 303–308 (1993).
[Crossref]

1989 (1)

1972 (1)

W. Happer, “Optical pumping,” Rev. Mod. Phys. 44, 169–249 (1972).
[Crossref]

Alipieva, E.

Alzetta, G.

G. Alzetta, S. Carta1eva, S. Gozzini, T. Karaulanov, A. Lucchesini, C. Marinelli, L. MOi, K. Nasyrov, V. Sarova, and K. Vaseva, “Magnetic Coherence Resonance Profiles in Na and K,” Proc. SPIE,  5830, 181–185 (2005).
[Crossref]

Arimondo, E.

E. Arimondo, “Relaxation processes in coherent-population trapping,” Phys. Rev. A  54, 2216–2223 (1996).
[Crossref] [PubMed]

Budker, D.

D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, “Resonant nonliear magneto-optical effects in atoms,” Rev. Mod. Phys. 74, 1153–1201 (2002).
[Crossref]

Carta1eva, S.

G. Alzetta, S. Carta1eva, S. Gozzini, T. Karaulanov, A. Lucchesini, C. Marinelli, L. MOi, K. Nasyrov, V. Sarova, and K. Vaseva, “Magnetic Coherence Resonance Profiles in Na and K,” Proc. SPIE,  5830, 181–185 (2005).
[Crossref]

Cartaleva, S.

Clairon, A.

T. Zanon, S. Guerandel, E. de Clercq, D. Holleville, N. Dimarcq, and A. Clairon, “High contrast Ramsey fringes with Coherent-Population-Trapping pulses in a double lambda atomic system,” Phys. Rev. Lett. 94, 193002 (2005).
[Crossref] [PubMed]

Cory, D. G.

W. Zhang and D. G. Cory, “First Direct Measurement of the Spin Diffusion Rate in a Homogenous Solid,” Phys. Rev. Lett. 80, 1324–1327 (1998).
[Crossref]

de Clercq, E.

T. Zanon, S. Guerandel, E. de Clercq, D. Holleville, N. Dimarcq, and A. Clairon, “High contrast Ramsey fringes with Coherent-Population-Trapping pulses in a double lambda atomic system,” Phys. Rev. Lett. 94, 193002 (2005).
[Crossref] [PubMed]

Delaney, M.

J. Vanier, M. W. Levine, D. Janssen, and M. Delaney, “Contrast and linewidth of the coherent population trapping transmission hyperfine resonance line in 87Rb : Effect of optical pumping,” Phys. Rev. A. 67, 065801 (2003).
[Crossref]

Dimarcq, N.

T. Zanon, S. Guerandel, E. de Clercq, D. Holleville, N. Dimarcq, and A. Clairon, “High contrast Ramsey fringes with Coherent-Population-Trapping pulses in a double lambda atomic system,” Phys. Rev. Lett. 94, 193002 (2005).
[Crossref] [PubMed]

Erhard, M.

M. Erhard and H. Helm, “Buffer-gas effects on dark resonances: theory and experiment,” Phys. Rev. A 63, 043813 (2001).
[Crossref]

Ezekiel, S.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Cambridge University Press, New York (1992) Ch. 19.

Fleischhauer, M.

M. D. Lukin, M. Fleischhauer, A. S. Zibrov, H. G. Robinson, V. L. Velichansky, L. Hollberg, and M. O. Scully, “Spectroscopy in dense coherent media: Line narrowing and interference effects,” Phys. Rev. Lett. 79, 2959–2962 (1997).
[Crossref]

Gateva, S.

Gawlik, W.

D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, “Resonant nonliear magneto-optical effects in atoms,” Rev. Mod. Phys. 74, 1153–1201 (2002).
[Crossref]

Godone, A.

F. Levi, A. Godone, J. Vanier, S. Micalizio, and G. Modugno, “Line-shape of dark line and maser emission profile in CPT,” Eur. Phys. J. D 12, 53(2000).
[Crossref]

Gossard, A. C.

J. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C. Gossard, “Coherent manipulation of coupled electron spins in semiconductor quantum dots,” Science 309, 2180–2184 (2005).
[Crossref] [PubMed]

Gozzini, S.

G. Alzetta, S. Carta1eva, S. Gozzini, T. Karaulanov, A. Lucchesini, C. Marinelli, L. MOi, K. Nasyrov, V. Sarova, and K. Vaseva, “Magnetic Coherence Resonance Profiles in Na and K,” Proc. SPIE,  5830, 181–185 (2005).
[Crossref]

Guerandel, S.

T. Zanon, S. Guerandel, E. de Clercq, D. Holleville, N. Dimarcq, and A. Clairon, “High contrast Ramsey fringes with Coherent-Population-Trapping pulses in a double lambda atomic system,” Phys. Rev. Lett. 94, 193002 (2005).
[Crossref] [PubMed]

Hanson, M. P.

J. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C. Gossard, “Coherent manipulation of coupled electron spins in semiconductor quantum dots,” Science 309, 2180–2184 (2005).
[Crossref] [PubMed]

Happer, W.

W. Happer, “Optical pumping,” Rev. Mod. Phys. 44, 169–249 (1972).
[Crossref]

Helm, H.

M. Erhard and H. Helm, “Buffer-gas effects on dark resonances: theory and experiment,” Phys. Rev. A 63, 043813 (2001).
[Crossref]

Hemmer, P. R.

M. S. Shahriar, P. R. Hemmer, D. P. Katz, A. Lee , and M. G. Prentiss, “Dark-state-based three-element vector model for the stimulated Raman interaction,” Phys. Rev. A 55, 2272–2282 (1997).
[Crossref]

P. R. Hemmer, M. S. Shahriar, V. D. Natoli, and S. Ezekiel,“AC Stark shifts in a two-zone Raman interaction,” J. Opt. Soc. Am. B 6, 1519–1528 (1989).
[Crossref]

Hollberg, L.

A. V. Taichenachev, A. M. Tumaikin, V. I. Yudin, M. Stahler, R. Wynands, J. Kitching, and L. Hollberg,“Nonliear-resonance line shapes: ependence on the transverse intensity distribution of a light beam,” Phys. Rev. A 69, 024501 (2004).
[Crossref]

M. D. Lukin, M. Fleischhauer, A. S. Zibrov, H. G. Robinson, V. L. Velichansky, L. Hollberg, and M. O. Scully, “Spectroscopy in dense coherent media: Line narrowing and interference effects,” Phys. Rev. Lett. 79, 2959–2962 (1997).
[Crossref]

Holleville, D.

T. Zanon, S. Guerandel, E. de Clercq, D. Holleville, N. Dimarcq, and A. Clairon, “High contrast Ramsey fringes with Coherent-Population-Trapping pulses in a double lambda atomic system,” Phys. Rev. Lett. 94, 193002 (2005).
[Crossref] [PubMed]

Janssen, D.

J. Vanier, M. W. Levine, D. Janssen, and M. Delaney, “Contrast and linewidth of the coherent population trapping transmission hyperfine resonance line in 87Rb : Effect of optical pumping,” Phys. Rev. A. 67, 065801 (2003).
[Crossref]

Johnson, A. C.

J. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C. Gossard, “Coherent manipulation of coupled electron spins in semiconductor quantum dots,” Science 309, 2180–2184 (2005).
[Crossref] [PubMed]

Kanorsky, S. I.

E. Pfleghaar, J. Wurster, S. I. Kanorsky , and A. Weis, “Time of flight effects in nonliear magneto-optical spectroscopy,” Optics Commun. 99, 303–308 (1993).
[Crossref]

Karaulanov, T.

G. Alzetta, S. Carta1eva, S. Gozzini, T. Karaulanov, A. Lucchesini, C. Marinelli, L. MOi, K. Nasyrov, V. Sarova, and K. Vaseva, “Magnetic Coherence Resonance Profiles in Na and K,” Proc. SPIE,  5830, 181–185 (2005).
[Crossref]

Katz, D. P.

M. S. Shahriar, P. R. Hemmer, D. P. Katz, A. Lee , and M. G. Prentiss, “Dark-state-based three-element vector model for the stimulated Raman interaction,” Phys. Rev. A 55, 2272–2282 (1997).
[Crossref]

Kimball, D. F.

D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, “Resonant nonliear magneto-optical effects in atoms,” Rev. Mod. Phys. 74, 1153–1201 (2002).
[Crossref]

Kitching, J.

A. V. Taichenachev, A. M. Tumaikin, V. I. Yudin, M. Stahler, R. Wynands, J. Kitching, and L. Hollberg,“Nonliear-resonance line shapes: ependence on the transverse intensity distribution of a light beam,” Phys. Rev. A 69, 024501 (2004).
[Crossref]

Laird, E. A.

J. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C. Gossard, “Coherent manipulation of coupled electron spins in semiconductor quantum dots,” Science 309, 2180–2184 (2005).
[Crossref] [PubMed]

Lee, A.

M. S. Shahriar, P. R. Hemmer, D. P. Katz, A. Lee , and M. G. Prentiss, “Dark-state-based three-element vector model for the stimulated Raman interaction,” Phys. Rev. A 55, 2272–2282 (1997).
[Crossref]

Levi, F.

F. Levi, A. Godone, J. Vanier, S. Micalizio, and G. Modugno, “Line-shape of dark line and maser emission profile in CPT,” Eur. Phys. J. D 12, 53(2000).
[Crossref]

Levine, M. W.

J. Vanier, M. W. Levine, D. Janssen, and M. Delaney, “Contrast and linewidth of the coherent population trapping transmission hyperfine resonance line in 87Rb : Effect of optical pumping,” Phys. Rev. A. 67, 065801 (2003).
[Crossref]

Lucchesini, A.

G. Alzetta, S. Carta1eva, S. Gozzini, T. Karaulanov, A. Lucchesini, C. Marinelli, L. MOi, K. Nasyrov, V. Sarova, and K. Vaseva, “Magnetic Coherence Resonance Profiles in Na and K,” Proc. SPIE,  5830, 181–185 (2005).
[Crossref]

Lukin, M. D.

J. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C. Gossard, “Coherent manipulation of coupled electron spins in semiconductor quantum dots,” Science 309, 2180–2184 (2005).
[Crossref] [PubMed]

M. D. Lukin, “Colloquium: Trapping and manipulating photon states in atomic ensembles,” Rev. Mod. Phys. 75, 457–472 (2003).
[Crossref]

M. D. Lukin, M. Fleischhauer, A. S. Zibrov, H. G. Robinson, V. L. Velichansky, L. Hollberg, and M. O. Scully, “Spectroscopy in dense coherent media: Line narrowing and interference effects,” Phys. Rev. Lett. 79, 2959–2962 (1997).
[Crossref]

Marcus, C. M.

J. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C. Gossard, “Coherent manipulation of coupled electron spins in semiconductor quantum dots,” Science 309, 2180–2184 (2005).
[Crossref] [PubMed]

Marinelli, C.

G. Alzetta, S. Carta1eva, S. Gozzini, T. Karaulanov, A. Lucchesini, C. Marinelli, L. MOi, K. Nasyrov, V. Sarova, and K. Vaseva, “Magnetic Coherence Resonance Profiles in Na and K,” Proc. SPIE,  5830, 181–185 (2005).
[Crossref]

Matsko, A. B.

A. S. Zibrov and A. B. Matsko, “Optical Ramsey fringes induced by Zeeman coherence,” Phys. Rev. A 65, 013814 (2001).
[Crossref]

A. S. Zibrov, I. Novikova, and A. B. Matsko, “Observation of Ramsey fringes in an atomic cell with buffer gas,” Opt. Lett. 26, 1311–1313 (2001).
[Crossref]

Micalizio, S.

F. Levi, A. Godone, J. Vanier, S. Micalizio, and G. Modugno, “Line-shape of dark line and maser emission profile in CPT,” Eur. Phys. J. D 12, 53(2000).
[Crossref]

Modugno, G.

F. Levi, A. Godone, J. Vanier, S. Micalizio, and G. Modugno, “Line-shape of dark line and maser emission profile in CPT,” Eur. Phys. J. D 12, 53(2000).
[Crossref]

MOi, L.

G. Alzetta, S. Carta1eva, S. Gozzini, T. Karaulanov, A. Lucchesini, C. Marinelli, L. MOi, K. Nasyrov, V. Sarova, and K. Vaseva, “Magnetic Coherence Resonance Profiles in Na and K,” Proc. SPIE,  5830, 181–185 (2005).
[Crossref]

Nasyrov, K.

G. Alzetta, S. Carta1eva, S. Gozzini, T. Karaulanov, A. Lucchesini, C. Marinelli, L. MOi, K. Nasyrov, V. Sarova, and K. Vaseva, “Magnetic Coherence Resonance Profiles in Na and K,” Proc. SPIE,  5830, 181–185 (2005).
[Crossref]

Natoli, V. D.

Novikova, I.

Y. Xiao, I. Novikova, D. F. Phillips, and R. L. Walsworth, “Diffusion-induced Ramsey narrowing,” Phys. Rev. Lett. 96, 043601 (2006).
[Crossref] [PubMed]

I. Novikova, Y. Xiao, D. F. Phillips, and R. L. Walsworth,“EIT and diffusion of atomic coherence,” J. Mod. Opt.,  52, 2381–2390 (2005).
[Crossref]

A. S. Zibrov, I. Novikova, and A. B. Matsko, “Observation of Ramsey fringes in an atomic cell with buffer gas,” Opt. Lett. 26, 1311–1313 (2001).
[Crossref]

Petta, J.

J. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C. Gossard, “Coherent manipulation of coupled electron spins in semiconductor quantum dots,” Science 309, 2180–2184 (2005).
[Crossref] [PubMed]

Pfleghaar, E.

E. Pfleghaar, J. Wurster, S. I. Kanorsky , and A. Weis, “Time of flight effects in nonliear magneto-optical spectroscopy,” Optics Commun. 99, 303–308 (1993).
[Crossref]

Phillips, D. F.

Y. Xiao, I. Novikova, D. F. Phillips, and R. L. Walsworth, “Diffusion-induced Ramsey narrowing,” Phys. Rev. Lett. 96, 043601 (2006).
[Crossref] [PubMed]

I. Novikova, Y. Xiao, D. F. Phillips, and R. L. Walsworth,“EIT and diffusion of atomic coherence,” J. Mod. Opt.,  52, 2381–2390 (2005).
[Crossref]

Prentiss, M. G.

M. S. Shahriar, P. R. Hemmer, D. P. Katz, A. Lee , and M. G. Prentiss, “Dark-state-based three-element vector model for the stimulated Raman interaction,” Phys. Rev. A 55, 2272–2282 (1997).
[Crossref]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Cambridge University Press, New York (1992) Ch. 19.

Ramsey, N. F.

N. F. Ramsey, Molecular beams (Clarendon, Oxford, 1956).

Robinson, H. G.

M. D. Lukin, M. Fleischhauer, A. S. Zibrov, H. G. Robinson, V. L. Velichansky, L. Hollberg, and M. O. Scully, “Spectroscopy in dense coherent media: Line narrowing and interference effects,” Phys. Rev. Lett. 79, 2959–2962 (1997).
[Crossref]

Rochester, S. M.

D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, “Resonant nonliear magneto-optical effects in atoms,” Rev. Mod. Phys. 74, 1153–1201 (2002).
[Crossref]

Sarova, V.

G. Alzetta, S. Carta1eva, S. Gozzini, T. Karaulanov, A. Lucchesini, C. Marinelli, L. MOi, K. Nasyrov, V. Sarova, and K. Vaseva, “Magnetic Coherence Resonance Profiles in Na and K,” Proc. SPIE,  5830, 181–185 (2005).
[Crossref]

Scully, M. O.

M. D. Lukin, M. Fleischhauer, A. S. Zibrov, H. G. Robinson, V. L. Velichansky, L. Hollberg, and M. O. Scully, “Spectroscopy in dense coherent media: Line narrowing and interference effects,” Phys. Rev. Lett. 79, 2959–2962 (1997).
[Crossref]

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, Cambridge, UK, 1997).

Shahriar, M. S.

M. S. Shahriar, P. R. Hemmer, D. P. Katz, A. Lee , and M. G. Prentiss, “Dark-state-based three-element vector model for the stimulated Raman interaction,” Phys. Rev. A 55, 2272–2282 (1997).
[Crossref]

P. R. Hemmer, M. S. Shahriar, V. D. Natoli, and S. Ezekiel,“AC Stark shifts in a two-zone Raman interaction,” J. Opt. Soc. Am. B 6, 1519–1528 (1989).
[Crossref]

Stahler, M.

A. V. Taichenachev, A. M. Tumaikin, V. I. Yudin, M. Stahler, R. Wynands, J. Kitching, and L. Hollberg,“Nonliear-resonance line shapes: ependence on the transverse intensity distribution of a light beam,” Phys. Rev. A 69, 024501 (2004).
[Crossref]

Taichenachev, A. V.

A. V. Taichenachev, A. M. Tumaikin, V. I. Yudin, M. Stahler, R. Wynands, J. Kitching, and L. Hollberg,“Nonliear-resonance line shapes: ependence on the transverse intensity distribution of a light beam,” Phys. Rev. A 69, 024501 (2004).
[Crossref]

Taskova, E.

Taylor, J.

J. Taylor.

Taylor, J. M.

J. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C. Gossard, “Coherent manipulation of coupled electron spins in semiconductor quantum dots,” Science 309, 2180–2184 (2005).
[Crossref] [PubMed]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Cambridge University Press, New York (1992) Ch. 19.

Tumaikin, A. M.

A. V. Taichenachev, A. M. Tumaikin, V. I. Yudin, M. Stahler, R. Wynands, J. Kitching, and L. Hollberg,“Nonliear-resonance line shapes: ependence on the transverse intensity distribution of a light beam,” Phys. Rev. A 69, 024501 (2004).
[Crossref]

Vanier, J.

J. Vanier, “Atomic clocks based on Coherent Population Trapping: a review,” Appl. Phys. B 81(4), 421–442 (2005).
[Crossref]

J. Vanier, M. W. Levine, D. Janssen, and M. Delaney, “Contrast and linewidth of the coherent population trapping transmission hyperfine resonance line in 87Rb : Effect of optical pumping,” Phys. Rev. A. 67, 065801 (2003).
[Crossref]

F. Levi, A. Godone, J. Vanier, S. Micalizio, and G. Modugno, “Line-shape of dark line and maser emission profile in CPT,” Eur. Phys. J. D 12, 53(2000).
[Crossref]

Vaseva, K.

G. Alzetta, S. Carta1eva, S. Gozzini, T. Karaulanov, A. Lucchesini, C. Marinelli, L. MOi, K. Nasyrov, V. Sarova, and K. Vaseva, “Magnetic Coherence Resonance Profiles in Na and K,” Proc. SPIE,  5830, 181–185 (2005).
[Crossref]

Velichansky, V. L.

M. D. Lukin, M. Fleischhauer, A. S. Zibrov, H. G. Robinson, V. L. Velichansky, L. Hollberg, and M. O. Scully, “Spectroscopy in dense coherent media: Line narrowing and interference effects,” Phys. Rev. Lett. 79, 2959–2962 (1997).
[Crossref]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Cambridge University Press, New York (1992) Ch. 19.

Walsworth, R. L.

Y. Xiao, I. Novikova, D. F. Phillips, and R. L. Walsworth, “Diffusion-induced Ramsey narrowing,” Phys. Rev. Lett. 96, 043601 (2006).
[Crossref] [PubMed]

I. Novikova, Y. Xiao, D. F. Phillips, and R. L. Walsworth,“EIT and diffusion of atomic coherence,” J. Mod. Opt.,  52, 2381–2390 (2005).
[Crossref]

Weis, A.

D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, “Resonant nonliear magneto-optical effects in atoms,” Rev. Mod. Phys. 74, 1153–1201 (2002).
[Crossref]

E. Pfleghaar, J. Wurster, S. I. Kanorsky , and A. Weis, “Time of flight effects in nonliear magneto-optical spectroscopy,” Optics Commun. 99, 303–308 (1993).
[Crossref]

Wurster, J.

E. Pfleghaar, J. Wurster, S. I. Kanorsky , and A. Weis, “Time of flight effects in nonliear magneto-optical spectroscopy,” Optics Commun. 99, 303–308 (1993).
[Crossref]

Wynands, R.

A. V. Taichenachev, A. M. Tumaikin, V. I. Yudin, M. Stahler, R. Wynands, J. Kitching, and L. Hollberg,“Nonliear-resonance line shapes: ependence on the transverse intensity distribution of a light beam,” Phys. Rev. A 69, 024501 (2004).
[Crossref]

Xiao, Y.

Y. Xiao, I. Novikova, D. F. Phillips, and R. L. Walsworth, “Diffusion-induced Ramsey narrowing,” Phys. Rev. Lett. 96, 043601 (2006).
[Crossref] [PubMed]

I. Novikova, Y. Xiao, D. F. Phillips, and R. L. Walsworth,“EIT and diffusion of atomic coherence,” J. Mod. Opt.,  52, 2381–2390 (2005).
[Crossref]

Yacoby, A.

J. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C. Gossard, “Coherent manipulation of coupled electron spins in semiconductor quantum dots,” Science 309, 2180–2184 (2005).
[Crossref] [PubMed]

Yashchuk, V. V.

D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, “Resonant nonliear magneto-optical effects in atoms,” Rev. Mod. Phys. 74, 1153–1201 (2002).
[Crossref]

Yudin, V. I.

A. V. Taichenachev, A. M. Tumaikin, V. I. Yudin, M. Stahler, R. Wynands, J. Kitching, and L. Hollberg,“Nonliear-resonance line shapes: ependence on the transverse intensity distribution of a light beam,” Phys. Rev. A 69, 024501 (2004).
[Crossref]

Zanon, T.

T. Zanon, S. Guerandel, E. de Clercq, D. Holleville, N. Dimarcq, and A. Clairon, “High contrast Ramsey fringes with Coherent-Population-Trapping pulses in a double lambda atomic system,” Phys. Rev. Lett. 94, 193002 (2005).
[Crossref] [PubMed]

Zhang, W.

W. Zhang and D. G. Cory, “First Direct Measurement of the Spin Diffusion Rate in a Homogenous Solid,” Phys. Rev. Lett. 80, 1324–1327 (1998).
[Crossref]

Zibrov, A. S.

A. S. Zibrov and A. B. Matsko, “Optical Ramsey fringes induced by Zeeman coherence,” Phys. Rev. A 65, 013814 (2001).
[Crossref]

A. S. Zibrov, I. Novikova, and A. B. Matsko, “Observation of Ramsey fringes in an atomic cell with buffer gas,” Opt. Lett. 26, 1311–1313 (2001).
[Crossref]

M. D. Lukin, M. Fleischhauer, A. S. Zibrov, H. G. Robinson, V. L. Velichansky, L. Hollberg, and M. O. Scully, “Spectroscopy in dense coherent media: Line narrowing and interference effects,” Phys. Rev. Lett. 79, 2959–2962 (1997).
[Crossref]

Zubairy, M. S.

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, Cambridge, UK, 1997).

Appl. Phys. B (1)

J. Vanier, “Atomic clocks based on Coherent Population Trapping: a review,” Appl. Phys. B 81(4), 421–442 (2005).
[Crossref]

Eur. Phys. J. D (1)

F. Levi, A. Godone, J. Vanier, S. Micalizio, and G. Modugno, “Line-shape of dark line and maser emission profile in CPT,” Eur. Phys. J. D 12, 53(2000).
[Crossref]

J. Mod. Opt. (1)

I. Novikova, Y. Xiao, D. F. Phillips, and R. L. Walsworth,“EIT and diffusion of atomic coherence,” J. Mod. Opt.,  52, 2381–2390 (2005).
[Crossref]

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

Opt. Lett. (2)

Optics Commun. (1)

E. Pfleghaar, J. Wurster, S. I. Kanorsky , and A. Weis, “Time of flight effects in nonliear magneto-optical spectroscopy,” Optics Commun. 99, 303–308 (1993).
[Crossref]

Phys. Rev. A (5)

A. V. Taichenachev, A. M. Tumaikin, V. I. Yudin, M. Stahler, R. Wynands, J. Kitching, and L. Hollberg,“Nonliear-resonance line shapes: ependence on the transverse intensity distribution of a light beam,” Phys. Rev. A 69, 024501 (2004).
[Crossref]

A. S. Zibrov and A. B. Matsko, “Optical Ramsey fringes induced by Zeeman coherence,” Phys. Rev. A 65, 013814 (2001).
[Crossref]

E. Arimondo, “Relaxation processes in coherent-population trapping,” Phys. Rev. A  54, 2216–2223 (1996).
[Crossref] [PubMed]

M. Erhard and H. Helm, “Buffer-gas effects on dark resonances: theory and experiment,” Phys. Rev. A 63, 043813 (2001).
[Crossref]

M. S. Shahriar, P. R. Hemmer, D. P. Katz, A. Lee , and M. G. Prentiss, “Dark-state-based three-element vector model for the stimulated Raman interaction,” Phys. Rev. A 55, 2272–2282 (1997).
[Crossref]

Phys. Rev. A. (1)

J. Vanier, M. W. Levine, D. Janssen, and M. Delaney, “Contrast and linewidth of the coherent population trapping transmission hyperfine resonance line in 87Rb : Effect of optical pumping,” Phys. Rev. A. 67, 065801 (2003).
[Crossref]

Phys. Rev. Lett. (4)

W. Zhang and D. G. Cory, “First Direct Measurement of the Spin Diffusion Rate in a Homogenous Solid,” Phys. Rev. Lett. 80, 1324–1327 (1998).
[Crossref]

M. D. Lukin, M. Fleischhauer, A. S. Zibrov, H. G. Robinson, V. L. Velichansky, L. Hollberg, and M. O. Scully, “Spectroscopy in dense coherent media: Line narrowing and interference effects,” Phys. Rev. Lett. 79, 2959–2962 (1997).
[Crossref]

T. Zanon, S. Guerandel, E. de Clercq, D. Holleville, N. Dimarcq, and A. Clairon, “High contrast Ramsey fringes with Coherent-Population-Trapping pulses in a double lambda atomic system,” Phys. Rev. Lett. 94, 193002 (2005).
[Crossref] [PubMed]

Y. Xiao, I. Novikova, D. F. Phillips, and R. L. Walsworth, “Diffusion-induced Ramsey narrowing,” Phys. Rev. Lett. 96, 043601 (2006).
[Crossref] [PubMed]

Proc. SPIE (1)

G. Alzetta, S. Carta1eva, S. Gozzini, T. Karaulanov, A. Lucchesini, C. Marinelli, L. MOi, K. Nasyrov, V. Sarova, and K. Vaseva, “Magnetic Coherence Resonance Profiles in Na and K,” Proc. SPIE,  5830, 181–185 (2005).
[Crossref]

Rev. Mod. Phys. (3)

W. Happer, “Optical pumping,” Rev. Mod. Phys. 44, 169–249 (1972).
[Crossref]

M. D. Lukin, “Colloquium: Trapping and manipulating photon states in atomic ensembles,” Rev. Mod. Phys. 75, 457–472 (2003).
[Crossref]

D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, “Resonant nonliear magneto-optical effects in atoms,” Rev. Mod. Phys. 74, 1153–1201 (2002).
[Crossref]

Science (1)

J. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C. Gossard, “Coherent manipulation of coupled electron spins in semiconductor quantum dots,” Science 309, 2180–2184 (2005).
[Crossref] [PubMed]

Other (5)

J. Taylor.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Cambridge University Press, New York (1992) Ch. 19.

A typical EIT lineshape, such as that shown in figure 7, takes approximately 12 hours to calculate on a 1.25 GHz PowerPC G4 whereas the repeated interaction model can be evaluated immediately.

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, Cambridge, UK, 1997).

N. F. Ramsey, Molecular beams (Clarendon, Oxford, 1956).

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Figures (8)

Fig. 1.
Fig. 1.

(a) Cross-section schematic of a vapor cell showing the vapor cell wall at r=d, a step-like laser beam of radius a, and a cartoon of an atom’s random walk in, out, and back into the laser beam. (b) Atoms diffusing in and out of the laser beam experience a Ramsey-like sequence of alternating periods of interaction with the laser fields and evolution in the dark. The simplest in-out-in Ramsey sequence is indicated with the corresponding times t 1, t 2, and t 3.

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Fig. 2.
Fig. 2.

Comparison of experimental data for the EIT lineshape in Rb vapor with predictions from the repeated interaction model and a simple Lorentzian lineshape with width given by the lowest order mode for atomic diffusion out of the laser beam. Red solid line: experimental data for a Rb vapor cell with 5 torr Ne buffer gas and a laser beam with 22 µW power and a Gaussian transverse profile with 0.8 mm separation between half-intensity points. Blue dotted line: prediction from the repeated interaction model for a step-like laser beam with 0.96 mm diameter, 30 cm2/s Rb diffusion coefficient [12], 2π×95 kHz Rabi frequency of the drive field, 100 MHz FWHM excited state linewidth, and 100 Hz FWHM groundstate linewidth). Black dotted-dashed line: Lorentzian lineshape with width given by the lowest order diffusion mode for a laser beam with 0.8 mm diameter and 30 cm2/s Rb diffusion coefficient. In all cases, EIT peak amplitude and baseline offset have been scaled to match data.

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Fig. 3.
Fig. 3.

Three-level L system coupled by probe and drive fields with one-photon detunings Δ1=ωp -ωea and Δ2=ωd -ωeb , respectively. Two-photon detuning: Δ=Δ1-Δ2. Average one-photon detuning: δ=(Δ1+Δ2)/2.

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Fig. 4.
Fig. 4.

Ramsey-EIT (solid line) calculated for a particular history (Ramsey sequence) with t 1=t 3=τD and t 2=5τD , where τD =19 µs, 1/τD =52 kHz, αD =2π×6.5 kHz, Γ0=2π×50 Hz, γ=2π×50 MHz. The transit-time-limited envelope (dashed line) is calculated with t 1=t 2=0, t 3=τD .

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Fig. 5.
Fig. 5.

Probability distributions of times (a) in the beam and (b) out of the beam in units of τD =a 2/4D with a ratio of cell diameter to step-like beam diameter, d/a=30. The dashed lines are calculated from Eqs. (3) and (4) while the solid lines are numerical evaluations of two-dimensional random walks. At long times these calculations agree. However, at short times, the numerical evaluations show much larger (but finite) probability than predicted by the analytical model which underestimates the number of atoms leaving the beam at short times.

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Fig. 6.
Fig. 6.

Left column: decrease of Ramsey fringe contrast under high buffer gas pressure or laser intensity leads to a reduction in the sharp, central EIT feature. Solid curves are Ramsey-EIT resonances calculated from a single Ramsey sequence. Dashed curves are weighted averages over all histories. Right column: results of repeated interaction model compared to experimental data for various buffer gas pressures and laser intensities. Dashed curves are model fits and are the same for both the left and right column.

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Fig. 7.
Fig. 7.

Comparison between repeated interaction model (red solid line) and pure numerical calculation (green dashed line). Red solid line: prediction from the repeated interaction model for a step-like laser beam with 0.96 mm diameter and 100 Hz FWHM groundstate linewidth. This is the same curve as the “model evaluation” in Fig. 2. Green dashed line: prediction from the numerical calculation for a 0.8 mm diameter step-like beam and 200 Hz FWHM groundstate linewidth. Common parameters: Rabi frequency of Ω D =2π×95kHz, cell diameter 2d=2.54 cm; excited state full width γ/π=100 MHz; 30 cm2/s Rb diffusion coefficient. A smaller ground state coherence decay rate is used in the repeated interaction model to match the pure numerical calculation because the former underestimates short interaction times (see Fig. 5) hence overestimating the linewidth of the Ramsey-EIT envelope (see Fig. 4).

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Fig. 8.
Fig. 8.

Numerical integration of the steady-state solution to the Bloch equations in the presence of diffusion shown for a transverse laser profile that is step-like (filled squares) and Gaussian (open circles) with mean Rabi frequencies of (a) Ω D =2π×95 kHz and (b) 2π×1.5 MHz. Common parameters: beam diameter 2a=0.8 mm; cell diameter 2d=2.54 cm; 300×300 grid; excited state full width γ/π=100 MHz; ground state full width Γ0/π=100 Hz.

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Equations (24)

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

Im ( ρ ea ) Ω P =
α D ( Δ 2 + Γ 2 ) γ 2 · Re { [ γ Γ − Δ 2 + i Δ ( γ + Γ ) ]
( − 1 + e − t 3 ( Γ − i Δ ) − e − t 3 ( Γ − i Δ ) − t 2 ( Γ 0 − i Δ ) + e − ( t 1 + t 3 ) ( Γ − i Δ ) − t 2 ( Γ 0 − i Δ ) ) }
− α D 2 Γ γ · ( − 1 + e − t 3 Γ − e − t 3 Γ − t 2 Γ 0 + e − ( t 1 + t 3 ) Γ − t 2 Γ 0 ) + 1 2 γ
P ( t 1 , t 2 , t 3 ) = P in ( t 1 ) P out ( t 2 ) P in ( t 3 )
P in ( t ) = 4 Σ m = 1 ∞ 1 χ m 2 · e − ( χ m 2 ) 2 · t τ D
P out ( t ) = 4 Σ m = 1 ∞ e − D χ m 2 t d 2 J 1 ( a χ m d ) 2 χ m 2 J 1 ( χ m ) 2 .
− D ∇ 2 R → ( r ) = [ − ( α ( r ) + Γ 0 ) − β ′ ( r ) 0 β ′ ( r ) − ( α ( r ) + Γ 0 ) Δ sin ⁡ ( 2 θ ) 0 − Δ sin ⁡ ( 2 θ ) − ( α ( r ) + Γ 0 ) ] R → ( r ) + [ 0 0 α ( r ) ] ,
| − 〉 = cos θ | a ∼ 〉 − sin θ | b ∼ 〉 , and | + 〉 = sin θ | a ∼ 〉 + cos θ | b ∼ 〉
ρ ˙ = i ħ ( ρ H † − H ρ ) + ( ρ ee γ 2 + Γ 0 2 ) [ 1 0 0 0 1 0 0 0 0 ] ,
H = ħ 2 [ C 2 Δ − i Γ 0 S 2 Δ 0 S 2 Δ − C 2 Δ − i Γ 0 − Ω 0 − Ω − 2 δ − i γ ]
R → ˙ = [ − ( α + Γ 0 ) − β ′ 0 β ′ − ( α + Γ 0 ) S 2 Δ 0 − S 2 Δ − ( α + Γ 0 ) ] R → + [ 0 0 α ] ,
β ′ = Ω 2 δ γ 2 + 4 δ 2 − C 2 Δ
α = Ω 2 γ 2 ( γ 2 + 4 δ 2 ) .
R → ( t ) = R → s + e − ( α + Γ 0 ) t P − 1 e Λ t P ( R → ( 0 ) − R → s )
Λ = [ 0 0 0 0 − i Ω eff 0 0 0 i Ω eff ] and P − 1 = [ − S 2 Δ β ′ β ′ 0 i Ω eff − i Ω eff β ′ S 2 Δ S 2 Δ ] .
R → ( t 1 , t 2 , t 3 ) = { 1 − e − ( α + Γ 0 ) t 3 P − 1 e Λ t 3 P
+ e − ( α + Γ 0 ) t 3 − Γ 0 t 2 P − 1 e Λ ( t 3 + t 2 ) P ( 1 − e − ( α + Γ 0 ) t 1 P − 1 e Λ t 1 P ) } R → S
ρ e ˜ a ˜ = i 2 ( γ − i 2 δ ) { Ω D ( R 1 − i R 2 ) + Ω P ( 1 − R 3 ) } ,
Im ( ρ e ˜ a ˜ ) Ω P =
α D ( Δ 2 + Γ 2 ) ( γ 2 + Δ 2 ) · Re { [ γ Γ − Δ 2 + i Δ ( γ + Γ ) ] ·
( − 1 + e − t 3 ( Γ − i Δ ) − e − t 3 ( Γ − i Δ ) − t 2 ( Γ 0 − i Δ ) + e − ( t 1 + t 3 ) ( Γ − i Δ ) − t 2 ( Γ 0 − i Δ ) + … ) }
− α D γ 2 Γ ( γ 2 + Δ 2 ) · ( − 1 + e − t 3 Γ − e − t 3 Γ − t 2 Γ 0 + e − ( t 1 + t 3 ) Γ − t 2 Γ 0 + … ) +
γ 2 ( γ 2 + Δ 2 )

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