Abstract

We present a wave-function approach to the study of the evolution of a small system when it is coupled to a large reservoir. Fluctuations and dissipation originate in this approach from quantum jumps that occur randomly during the time evolution of the system. This approach can be applied to a wide class of relaxation operators in the Markovian regime, and it is equivalent to the standard master-equation approach. For systems with a number of states N much larger than unity this Monte Carlo wave-function approach can be less expensive in terms of calculation time than the master-equation treatment. Indeed, a wave function involves only N components, whereas a density matrix is described by N2 terms. We evaluate the gain in computing time that may be expected from such a formalism, and we discuss its applicability to several examples, with particular emphasis on a quantum description of laser cooling.

© 1993 Optical Society of America

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  1. W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973).
  2. F. Haake, Statistical Treatment of Open Systems by Generalized Master Equations, G. Hohler, ed., Vol. 66 of Springer Tracts in Modern Physics (Springer-Verlag, Berlin, 1973).
  3. C. Cohen-Tannoudji, in Les Houches 1975, Frontiers in Laser Spectroscopy, R. Balian, S. Haroche, and S. Liberman, eds. (North-Holland, Amsterdam, 1977).
  4. C. W. Gardiner, Handbook of Stochastic Methods (Springer-Verlag, Berlin, 1983).
    [CrossRef]
  5. M. Lax, Phys. Rev. 172, 350 (1968).
    [CrossRef]
  6. J. Dalibard, Y. Castin, and K. Mølmer, Phys. Rev. Lett. 68, 580 (1992).
    [CrossRef] [PubMed]
  7. H. J. Carmichael, “An open systems approach to quantum optics,” lectures presented at l’Université Libre de Bruxelles, Bruxelles, Belgium, fall 1991.
  8. P. L. Kelley and W. H. Kleiner, Phys. Rev. 136, A316 (1964).
    [CrossRef]
  9. M. D. Srinivas and E. B. Davies, Opt. Acta 28, 981 (1981).
    [CrossRef]
  10. R. Dum, P. Zoller, and H. Ritsch, Phys. Rev. A 45, 4879 (1992).
    [CrossRef] [PubMed]
  11. G. C. Hegerfeldt and T. S. Wilser, in Proceedings of the II International Wigner Symposium, July 1991, Goslar (World Scientific, Singapore, to be published).
  12. N. Gisin, Phys. Rev. Lett. 52, 1657 (1984); Helvetica Phys. Acta 62, 363 (1989).
    [CrossRef]
  13. B. Misra and E. C. G. Sudarshan, J. Math. Phys. 18, 756 (1977).
    [CrossRef]
  14. R. H. Dicke, Am. J. Phys. 49, 925 (1981).
    [CrossRef]
  15. D. T. Pegg and P. L. Knight, Phys. Rev. A 37, 4303 (1988).
    [CrossRef] [PubMed]
  16. R. J. Cook, in Progress in Optics XXVIII, E. Wolf, ed., (Elsevier, New York, 1990), p. 363.
  17. M. Porrati and S. Putterman, Phys. Rev. A 36, 929 (1987).
    [CrossRef] [PubMed]
  18. R. Mollow, Phys. Rev. A 12, 1919 (1975).
    [CrossRef]
  19. C. Cohen-Tannoudji and J. Dalibard, Europhys. Lett. 1, 441 (1986).
    [CrossRef]
  20. P. Zoller, M. Marte, and D. F. Walls, Phys. Rev. A 35, 198 (1987).
    [CrossRef] [PubMed]
  21. H. J. Carmichael, S. Singh, R. Vyas, and P. R. Rice, Phys. Rev. A 39, 1200 (1989).
    [CrossRef] [PubMed]
  22. R. Blatt, W. Ertmer, P. Zoller, and J. L. Hall, Phys. Rev. A 34, 3022 (1986).
    [CrossRef] [PubMed]
  23. C. Cohen-Tannoudji, F. Bardou, and A. Aspect, in Laser Spectroscopy, X. M. Ducloy, E. Giacobino, and G. Camy, eds. (World Scientific, Singapore, to be published).
  24. C. Cohen-Tannoudji, B. Zambon, and E. Arimondo, C. R. Acad. Sci. Paris 314, 1139 (1992); C. R. Acad. Sci. Paris 314, 1293 (1992).
  25. V. P. Belavkin and P. Staszewski, Phys. Rev. A 45, 1347 (1992).
    [CrossRef] [PubMed]
  26. N. Gisin, Group of Applied Physics, University of Geneva, 1211 Geneva 4, Switzerland (personal communication); N. Gisin and I. Percival, Phys. Rev. A 46, 4382 (1992).
    [CrossRef]
  27. A different procedure for calculating spectra has been recently proposed by R. Dum, A. S. Parkins, P. Zoller, and C. W. Gardiner, Phys. Rev. A 46, 4382 (1992). It is based on the probing of the atomic system with either a monochromatic weak laser or a white noise driving field.
    [CrossRef] [PubMed]
  28. R. Loudon, The Quantum Theory of Light (Oxford U. Press, New York, 1983).
  29. J. Javanainen and S. Stenholm, Appl. Phys. 21, 35 (1980).
    [CrossRef]
  30. C. Cohen-Tannoudji, in Les Houches 1990, Fundamental Systems in Quantum Optics, J. Dalibard, J.-M. Raimond, and J. Zinn-Justin, eds. (North-Holland, Amsterdam, 1992).
  31. Y. Castin, H. Wallis, and J. Dalibard, J. Opt. Soc. Am. B 6, 2046 (1989).
    [CrossRef]
  32. T. W. Hänsch and A. Schawlow, Opt. Commun. 13, 68 (1975).
    [CrossRef]
  33. D. Wineland and H. Dehmelt, Bull. Am. Phys. Soc. 20, 637 (1975).
  34. V. G. Minogin, Sov. Phys. JETP 53, 1164 (1981).
  35. K. Berg-Sørensen, E. Bonderup, K. Mølmer, and Y. Castin, J. Phys. B 25, 4195 (1992).
    [CrossRef]
  36. C. Tanguy, S. Reynaud, and C. Cohen-Tannoudji, J. Phys. B 17, 4623 (1984).
    [CrossRef]
  37. M. Wilkens, E. Schumacher, and P. Meystre, Opt. Commun. 86, 34 (1991).
    [CrossRef]
  38. S. Stenholm in Laser Manipulation of Atoms and Ions, E. Arimondo and W. D. Phillips, eds. (North-Holland, Amsterdam, 1992).
  39. Strictly speaking, we should put 1/(n− 1) instead of 1/n as a normalization factor; see S. Brandt, Statistical and Computational Methods in Data Analysis (North-Holland, Amsterdam, 1970). However, we restrict ourselves to the case of large n for which the difference is negligible.
  40. E. Arimondo and G. Orriols, Lett. Nuovo Cimento 17, 333 (1976).
    [CrossRef]
  41. Paul Lett, National Institute of Standards and Technology, Gaithersburg, Md. 20899 (personal communication, 1992).

1992 (6)

J. Dalibard, Y. Castin, and K. Mølmer, Phys. Rev. Lett. 68, 580 (1992).
[CrossRef] [PubMed]

R. Dum, P. Zoller, and H. Ritsch, Phys. Rev. A 45, 4879 (1992).
[CrossRef] [PubMed]

C. Cohen-Tannoudji, B. Zambon, and E. Arimondo, C. R. Acad. Sci. Paris 314, 1139 (1992); C. R. Acad. Sci. Paris 314, 1293 (1992).

V. P. Belavkin and P. Staszewski, Phys. Rev. A 45, 1347 (1992).
[CrossRef] [PubMed]

A different procedure for calculating spectra has been recently proposed by R. Dum, A. S. Parkins, P. Zoller, and C. W. Gardiner, Phys. Rev. A 46, 4382 (1992). It is based on the probing of the atomic system with either a monochromatic weak laser or a white noise driving field.
[CrossRef] [PubMed]

K. Berg-Sørensen, E. Bonderup, K. Mølmer, and Y. Castin, J. Phys. B 25, 4195 (1992).
[CrossRef]

1991 (1)

M. Wilkens, E. Schumacher, and P. Meystre, Opt. Commun. 86, 34 (1991).
[CrossRef]

1989 (2)

Y. Castin, H. Wallis, and J. Dalibard, J. Opt. Soc. Am. B 6, 2046 (1989).
[CrossRef]

H. J. Carmichael, S. Singh, R. Vyas, and P. R. Rice, Phys. Rev. A 39, 1200 (1989).
[CrossRef] [PubMed]

1988 (1)

D. T. Pegg and P. L. Knight, Phys. Rev. A 37, 4303 (1988).
[CrossRef] [PubMed]

1987 (2)

M. Porrati and S. Putterman, Phys. Rev. A 36, 929 (1987).
[CrossRef] [PubMed]

P. Zoller, M. Marte, and D. F. Walls, Phys. Rev. A 35, 198 (1987).
[CrossRef] [PubMed]

1986 (2)

R. Blatt, W. Ertmer, P. Zoller, and J. L. Hall, Phys. Rev. A 34, 3022 (1986).
[CrossRef] [PubMed]

C. Cohen-Tannoudji and J. Dalibard, Europhys. Lett. 1, 441 (1986).
[CrossRef]

1984 (2)

N. Gisin, Phys. Rev. Lett. 52, 1657 (1984); Helvetica Phys. Acta 62, 363 (1989).
[CrossRef]

C. Tanguy, S. Reynaud, and C. Cohen-Tannoudji, J. Phys. B 17, 4623 (1984).
[CrossRef]

1981 (3)

V. G. Minogin, Sov. Phys. JETP 53, 1164 (1981).

M. D. Srinivas and E. B. Davies, Opt. Acta 28, 981 (1981).
[CrossRef]

R. H. Dicke, Am. J. Phys. 49, 925 (1981).
[CrossRef]

1980 (1)

J. Javanainen and S. Stenholm, Appl. Phys. 21, 35 (1980).
[CrossRef]

1977 (1)

B. Misra and E. C. G. Sudarshan, J. Math. Phys. 18, 756 (1977).
[CrossRef]

1976 (1)

E. Arimondo and G. Orriols, Lett. Nuovo Cimento 17, 333 (1976).
[CrossRef]

1975 (3)

T. W. Hänsch and A. Schawlow, Opt. Commun. 13, 68 (1975).
[CrossRef]

D. Wineland and H. Dehmelt, Bull. Am. Phys. Soc. 20, 637 (1975).

R. Mollow, Phys. Rev. A 12, 1919 (1975).
[CrossRef]

1968 (1)

M. Lax, Phys. Rev. 172, 350 (1968).
[CrossRef]

1964 (1)

P. L. Kelley and W. H. Kleiner, Phys. Rev. 136, A316 (1964).
[CrossRef]

Arimondo, E.

C. Cohen-Tannoudji, B. Zambon, and E. Arimondo, C. R. Acad. Sci. Paris 314, 1139 (1992); C. R. Acad. Sci. Paris 314, 1293 (1992).

E. Arimondo and G. Orriols, Lett. Nuovo Cimento 17, 333 (1976).
[CrossRef]

Aspect, A.

C. Cohen-Tannoudji, F. Bardou, and A. Aspect, in Laser Spectroscopy, X. M. Ducloy, E. Giacobino, and G. Camy, eds. (World Scientific, Singapore, to be published).

Bardou, F.

C. Cohen-Tannoudji, F. Bardou, and A. Aspect, in Laser Spectroscopy, X. M. Ducloy, E. Giacobino, and G. Camy, eds. (World Scientific, Singapore, to be published).

Belavkin, V. P.

V. P. Belavkin and P. Staszewski, Phys. Rev. A 45, 1347 (1992).
[CrossRef] [PubMed]

Berg-Sørensen, K.

K. Berg-Sørensen, E. Bonderup, K. Mølmer, and Y. Castin, J. Phys. B 25, 4195 (1992).
[CrossRef]

Blatt, R.

R. Blatt, W. Ertmer, P. Zoller, and J. L. Hall, Phys. Rev. A 34, 3022 (1986).
[CrossRef] [PubMed]

Bonderup, E.

K. Berg-Sørensen, E. Bonderup, K. Mølmer, and Y. Castin, J. Phys. B 25, 4195 (1992).
[CrossRef]

Brandt, S.

Strictly speaking, we should put 1/(n− 1) instead of 1/n as a normalization factor; see S. Brandt, Statistical and Computational Methods in Data Analysis (North-Holland, Amsterdam, 1970). However, we restrict ourselves to the case of large n for which the difference is negligible.

Carmichael, H. J.

H. J. Carmichael, S. Singh, R. Vyas, and P. R. Rice, Phys. Rev. A 39, 1200 (1989).
[CrossRef] [PubMed]

H. J. Carmichael, “An open systems approach to quantum optics,” lectures presented at l’Université Libre de Bruxelles, Bruxelles, Belgium, fall 1991.

Castin, Y.

J. Dalibard, Y. Castin, and K. Mølmer, Phys. Rev. Lett. 68, 580 (1992).
[CrossRef] [PubMed]

K. Berg-Sørensen, E. Bonderup, K. Mølmer, and Y. Castin, J. Phys. B 25, 4195 (1992).
[CrossRef]

Y. Castin, H. Wallis, and J. Dalibard, J. Opt. Soc. Am. B 6, 2046 (1989).
[CrossRef]

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, B. Zambon, and E. Arimondo, C. R. Acad. Sci. Paris 314, 1139 (1992); C. R. Acad. Sci. Paris 314, 1293 (1992).

C. Cohen-Tannoudji and J. Dalibard, Europhys. Lett. 1, 441 (1986).
[CrossRef]

C. Tanguy, S. Reynaud, and C. Cohen-Tannoudji, J. Phys. B 17, 4623 (1984).
[CrossRef]

C. Cohen-Tannoudji, F. Bardou, and A. Aspect, in Laser Spectroscopy, X. M. Ducloy, E. Giacobino, and G. Camy, eds. (World Scientific, Singapore, to be published).

C. Cohen-Tannoudji, in Les Houches 1990, Fundamental Systems in Quantum Optics, J. Dalibard, J.-M. Raimond, and J. Zinn-Justin, eds. (North-Holland, Amsterdam, 1992).

C. Cohen-Tannoudji, in Les Houches 1975, Frontiers in Laser Spectroscopy, R. Balian, S. Haroche, and S. Liberman, eds. (North-Holland, Amsterdam, 1977).

Cook, R. J.

R. J. Cook, in Progress in Optics XXVIII, E. Wolf, ed., (Elsevier, New York, 1990), p. 363.

Dalibard, J.

J. Dalibard, Y. Castin, and K. Mølmer, Phys. Rev. Lett. 68, 580 (1992).
[CrossRef] [PubMed]

Y. Castin, H. Wallis, and J. Dalibard, J. Opt. Soc. Am. B 6, 2046 (1989).
[CrossRef]

C. Cohen-Tannoudji and J. Dalibard, Europhys. Lett. 1, 441 (1986).
[CrossRef]

Davies, E. B.

M. D. Srinivas and E. B. Davies, Opt. Acta 28, 981 (1981).
[CrossRef]

Dehmelt, H.

D. Wineland and H. Dehmelt, Bull. Am. Phys. Soc. 20, 637 (1975).

Dicke, R. H.

R. H. Dicke, Am. J. Phys. 49, 925 (1981).
[CrossRef]

Dum, R.

R. Dum, P. Zoller, and H. Ritsch, Phys. Rev. A 45, 4879 (1992).
[CrossRef] [PubMed]

A different procedure for calculating spectra has been recently proposed by R. Dum, A. S. Parkins, P. Zoller, and C. W. Gardiner, Phys. Rev. A 46, 4382 (1992). It is based on the probing of the atomic system with either a monochromatic weak laser or a white noise driving field.
[CrossRef] [PubMed]

Ertmer, W.

R. Blatt, W. Ertmer, P. Zoller, and J. L. Hall, Phys. Rev. A 34, 3022 (1986).
[CrossRef] [PubMed]

Gardiner, C. W.

A different procedure for calculating spectra has been recently proposed by R. Dum, A. S. Parkins, P. Zoller, and C. W. Gardiner, Phys. Rev. A 46, 4382 (1992). It is based on the probing of the atomic system with either a monochromatic weak laser or a white noise driving field.
[CrossRef] [PubMed]

C. W. Gardiner, Handbook of Stochastic Methods (Springer-Verlag, Berlin, 1983).
[CrossRef]

Gisin, N.

N. Gisin, Phys. Rev. Lett. 52, 1657 (1984); Helvetica Phys. Acta 62, 363 (1989).
[CrossRef]

N. Gisin, Group of Applied Physics, University of Geneva, 1211 Geneva 4, Switzerland (personal communication); N. Gisin and I. Percival, Phys. Rev. A 46, 4382 (1992).
[CrossRef]

Haake, F.

F. Haake, Statistical Treatment of Open Systems by Generalized Master Equations, G. Hohler, ed., Vol. 66 of Springer Tracts in Modern Physics (Springer-Verlag, Berlin, 1973).

Hall, J. L.

R. Blatt, W. Ertmer, P. Zoller, and J. L. Hall, Phys. Rev. A 34, 3022 (1986).
[CrossRef] [PubMed]

Hänsch, T. W.

T. W. Hänsch and A. Schawlow, Opt. Commun. 13, 68 (1975).
[CrossRef]

Hegerfeldt, G. C.

G. C. Hegerfeldt and T. S. Wilser, in Proceedings of the II International Wigner Symposium, July 1991, Goslar (World Scientific, Singapore, to be published).

Javanainen, J.

J. Javanainen and S. Stenholm, Appl. Phys. 21, 35 (1980).
[CrossRef]

Kelley, P. L.

P. L. Kelley and W. H. Kleiner, Phys. Rev. 136, A316 (1964).
[CrossRef]

Kleiner, W. H.

P. L. Kelley and W. H. Kleiner, Phys. Rev. 136, A316 (1964).
[CrossRef]

Knight, P. L.

D. T. Pegg and P. L. Knight, Phys. Rev. A 37, 4303 (1988).
[CrossRef] [PubMed]

Lax, M.

M. Lax, Phys. Rev. 172, 350 (1968).
[CrossRef]

Lett, Paul

Paul Lett, National Institute of Standards and Technology, Gaithersburg, Md. 20899 (personal communication, 1992).

Loudon, R.

R. Loudon, The Quantum Theory of Light (Oxford U. Press, New York, 1983).

Louisell, W. H.

W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973).

Marte, M.

P. Zoller, M. Marte, and D. F. Walls, Phys. Rev. A 35, 198 (1987).
[CrossRef] [PubMed]

Meystre, P.

M. Wilkens, E. Schumacher, and P. Meystre, Opt. Commun. 86, 34 (1991).
[CrossRef]

Minogin, V. G.

V. G. Minogin, Sov. Phys. JETP 53, 1164 (1981).

Misra, B.

B. Misra and E. C. G. Sudarshan, J. Math. Phys. 18, 756 (1977).
[CrossRef]

Mollow, R.

R. Mollow, Phys. Rev. A 12, 1919 (1975).
[CrossRef]

Mølmer, K.

J. Dalibard, Y. Castin, and K. Mølmer, Phys. Rev. Lett. 68, 580 (1992).
[CrossRef] [PubMed]

K. Berg-Sørensen, E. Bonderup, K. Mølmer, and Y. Castin, J. Phys. B 25, 4195 (1992).
[CrossRef]

Orriols, G.

E. Arimondo and G. Orriols, Lett. Nuovo Cimento 17, 333 (1976).
[CrossRef]

Parkins, A. S.

A different procedure for calculating spectra has been recently proposed by R. Dum, A. S. Parkins, P. Zoller, and C. W. Gardiner, Phys. Rev. A 46, 4382 (1992). It is based on the probing of the atomic system with either a monochromatic weak laser or a white noise driving field.
[CrossRef] [PubMed]

Pegg, D. T.

D. T. Pegg and P. L. Knight, Phys. Rev. A 37, 4303 (1988).
[CrossRef] [PubMed]

Porrati, M.

M. Porrati and S. Putterman, Phys. Rev. A 36, 929 (1987).
[CrossRef] [PubMed]

Putterman, S.

M. Porrati and S. Putterman, Phys. Rev. A 36, 929 (1987).
[CrossRef] [PubMed]

Reynaud, S.

C. Tanguy, S. Reynaud, and C. Cohen-Tannoudji, J. Phys. B 17, 4623 (1984).
[CrossRef]

Rice, P. R.

H. J. Carmichael, S. Singh, R. Vyas, and P. R. Rice, Phys. Rev. A 39, 1200 (1989).
[CrossRef] [PubMed]

Ritsch, H.

R. Dum, P. Zoller, and H. Ritsch, Phys. Rev. A 45, 4879 (1992).
[CrossRef] [PubMed]

Schawlow, A.

T. W. Hänsch and A. Schawlow, Opt. Commun. 13, 68 (1975).
[CrossRef]

Schumacher, E.

M. Wilkens, E. Schumacher, and P. Meystre, Opt. Commun. 86, 34 (1991).
[CrossRef]

Singh, S.

H. J. Carmichael, S. Singh, R. Vyas, and P. R. Rice, Phys. Rev. A 39, 1200 (1989).
[CrossRef] [PubMed]

Srinivas, M. D.

M. D. Srinivas and E. B. Davies, Opt. Acta 28, 981 (1981).
[CrossRef]

Staszewski, P.

V. P. Belavkin and P. Staszewski, Phys. Rev. A 45, 1347 (1992).
[CrossRef] [PubMed]

Stenholm, S.

J. Javanainen and S. Stenholm, Appl. Phys. 21, 35 (1980).
[CrossRef]

S. Stenholm in Laser Manipulation of Atoms and Ions, E. Arimondo and W. D. Phillips, eds. (North-Holland, Amsterdam, 1992).

Sudarshan, E. C. G.

B. Misra and E. C. G. Sudarshan, J. Math. Phys. 18, 756 (1977).
[CrossRef]

Tanguy, C.

C. Tanguy, S. Reynaud, and C. Cohen-Tannoudji, J. Phys. B 17, 4623 (1984).
[CrossRef]

Vyas, R.

H. J. Carmichael, S. Singh, R. Vyas, and P. R. Rice, Phys. Rev. A 39, 1200 (1989).
[CrossRef] [PubMed]

Wallis, H.

Walls, D. F.

P. Zoller, M. Marte, and D. F. Walls, Phys. Rev. A 35, 198 (1987).
[CrossRef] [PubMed]

Wilkens, M.

M. Wilkens, E. Schumacher, and P. Meystre, Opt. Commun. 86, 34 (1991).
[CrossRef]

Wilser, T. S.

G. C. Hegerfeldt and T. S. Wilser, in Proceedings of the II International Wigner Symposium, July 1991, Goslar (World Scientific, Singapore, to be published).

Wineland, D.

D. Wineland and H. Dehmelt, Bull. Am. Phys. Soc. 20, 637 (1975).

Zambon, B.

C. Cohen-Tannoudji, B. Zambon, and E. Arimondo, C. R. Acad. Sci. Paris 314, 1139 (1992); C. R. Acad. Sci. Paris 314, 1293 (1992).

Zoller, P.

A different procedure for calculating spectra has been recently proposed by R. Dum, A. S. Parkins, P. Zoller, and C. W. Gardiner, Phys. Rev. A 46, 4382 (1992). It is based on the probing of the atomic system with either a monochromatic weak laser or a white noise driving field.
[CrossRef] [PubMed]

R. Dum, P. Zoller, and H. Ritsch, Phys. Rev. A 45, 4879 (1992).
[CrossRef] [PubMed]

P. Zoller, M. Marte, and D. F. Walls, Phys. Rev. A 35, 198 (1987).
[CrossRef] [PubMed]

R. Blatt, W. Ertmer, P. Zoller, and J. L. Hall, Phys. Rev. A 34, 3022 (1986).
[CrossRef] [PubMed]

Am. J. Phys. (1)

R. H. Dicke, Am. J. Phys. 49, 925 (1981).
[CrossRef]

Appl. Phys. (1)

J. Javanainen and S. Stenholm, Appl. Phys. 21, 35 (1980).
[CrossRef]

Bull. Am. Phys. Soc. (1)

D. Wineland and H. Dehmelt, Bull. Am. Phys. Soc. 20, 637 (1975).

C. R. Acad. Sci. Paris (1)

C. Cohen-Tannoudji, B. Zambon, and E. Arimondo, C. R. Acad. Sci. Paris 314, 1139 (1992); C. R. Acad. Sci. Paris 314, 1293 (1992).

Europhys. Lett. (1)

C. Cohen-Tannoudji and J. Dalibard, Europhys. Lett. 1, 441 (1986).
[CrossRef]

J. Math. Phys. (1)

B. Misra and E. C. G. Sudarshan, J. Math. Phys. 18, 756 (1977).
[CrossRef]

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

J. Phys. B (2)

K. Berg-Sørensen, E. Bonderup, K. Mølmer, and Y. Castin, J. Phys. B 25, 4195 (1992).
[CrossRef]

C. Tanguy, S. Reynaud, and C. Cohen-Tannoudji, J. Phys. B 17, 4623 (1984).
[CrossRef]

Lett. Nuovo Cimento (1)

E. Arimondo and G. Orriols, Lett. Nuovo Cimento 17, 333 (1976).
[CrossRef]

Opt. Acta (1)

M. D. Srinivas and E. B. Davies, Opt. Acta 28, 981 (1981).
[CrossRef]

Opt. Commun. (2)

M. Wilkens, E. Schumacher, and P. Meystre, Opt. Commun. 86, 34 (1991).
[CrossRef]

T. W. Hänsch and A. Schawlow, Opt. Commun. 13, 68 (1975).
[CrossRef]

Phys. Rev. (2)

M. Lax, Phys. Rev. 172, 350 (1968).
[CrossRef]

P. L. Kelley and W. H. Kleiner, Phys. Rev. 136, A316 (1964).
[CrossRef]

Phys. Rev. A (9)

D. T. Pegg and P. L. Knight, Phys. Rev. A 37, 4303 (1988).
[CrossRef] [PubMed]

P. Zoller, M. Marte, and D. F. Walls, Phys. Rev. A 35, 198 (1987).
[CrossRef] [PubMed]

H. J. Carmichael, S. Singh, R. Vyas, and P. R. Rice, Phys. Rev. A 39, 1200 (1989).
[CrossRef] [PubMed]

R. Blatt, W. Ertmer, P. Zoller, and J. L. Hall, Phys. Rev. A 34, 3022 (1986).
[CrossRef] [PubMed]

R. Dum, P. Zoller, and H. Ritsch, Phys. Rev. A 45, 4879 (1992).
[CrossRef] [PubMed]

V. P. Belavkin and P. Staszewski, Phys. Rev. A 45, 1347 (1992).
[CrossRef] [PubMed]

M. Porrati and S. Putterman, Phys. Rev. A 36, 929 (1987).
[CrossRef] [PubMed]

R. Mollow, Phys. Rev. A 12, 1919 (1975).
[CrossRef]

A different procedure for calculating spectra has been recently proposed by R. Dum, A. S. Parkins, P. Zoller, and C. W. Gardiner, Phys. Rev. A 46, 4382 (1992). It is based on the probing of the atomic system with either a monochromatic weak laser or a white noise driving field.
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[CrossRef] [PubMed]

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Other (14)

S. Stenholm in Laser Manipulation of Atoms and Ions, E. Arimondo and W. D. Phillips, eds. (North-Holland, Amsterdam, 1992).

Strictly speaking, we should put 1/(n− 1) instead of 1/n as a normalization factor; see S. Brandt, Statistical and Computational Methods in Data Analysis (North-Holland, Amsterdam, 1970). However, we restrict ourselves to the case of large n for which the difference is negligible.

R. Loudon, The Quantum Theory of Light (Oxford U. Press, New York, 1983).

C. Cohen-Tannoudji, in Les Houches 1990, Fundamental Systems in Quantum Optics, J. Dalibard, J.-M. Raimond, and J. Zinn-Justin, eds. (North-Holland, Amsterdam, 1992).

N. Gisin, Group of Applied Physics, University of Geneva, 1211 Geneva 4, Switzerland (personal communication); N. Gisin and I. Percival, Phys. Rev. A 46, 4382 (1992).
[CrossRef]

R. J. Cook, in Progress in Optics XXVIII, E. Wolf, ed., (Elsevier, New York, 1990), p. 363.

C. Cohen-Tannoudji, F. Bardou, and A. Aspect, in Laser Spectroscopy, X. M. Ducloy, E. Giacobino, and G. Camy, eds. (World Scientific, Singapore, to be published).

H. J. Carmichael, “An open systems approach to quantum optics,” lectures presented at l’Université Libre de Bruxelles, Bruxelles, Belgium, fall 1991.

G. C. Hegerfeldt and T. S. Wilser, in Proceedings of the II International Wigner Symposium, July 1991, Goslar (World Scientific, Singapore, to be published).

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[CrossRef]

Paul Lett, National Institute of Standards and Technology, Gaithersburg, Md. 20899 (personal communication, 1992).

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Figures (7)

Fig. 1
Fig. 1

Solid curves: real part (upper curve) and imaginary part of the dipole correlation function for a two-level atom 〈S+(t + τ)S(t)〉/〈S+(t)S(t)〉, for various choices of the numbers n1 and n2 (see text). Dotted curves: we have indicated the exact result obtained by using optical Bloch equations and the quantum regression theorem. The field parameters of the calculations are Ω = 10Γ, δ = 0.

Fig. 2
Fig. 2

Time evolution of 〈P2〉 in Doppler cooling. Time is measured in units of the excited-state lifetime Γ−1, and momentum in units of ħk. The detuning δ and the Rabi frequency Ω are given by Ω = −δ = Γ/2. The atomic mass is such that Γ = 200ħk2/M. The points represent the Monte Carlo results, obtained by averaging n = 500 MCWF evolutions. The error bars correspond to the statistical error δ P ( n ) 2. We have also indicated by a curve the results of the density-matrix approach. Both calculations involve 200 quantum levels and take approximately the same computing time on a scalar machine. In the inset we have detailed the short-time regime corresponding to the diffraction of the atomic de Broglie wave by the laser standing wave.

Fig. 3
Fig. 3

Time evolution of the momentum distribution for a single Monte Carlo wave function for the Doppler cooling situation described in Fig. 2. The MCWF extends over approximately 5ħk and explores as time goes on all significant parts of the momentum space.

Fig. 4
Fig. 4

Time evolution of the atomic momentum distributions, obtained for the Doppler cooling situation described in Fig. 2. a, Result of the master-equation approach. b, Result of the average of 500 MCWF’s.

Fig. 5
Fig. 5

Configuration schemes leading to a dark resonance: a g, Jg = 1 ↔ e, Je = 1 transition is irradiated by two waves respectively σ+ and σ polarized along the z axis. a, If angular momentum is quantized along the z axis, the dark resonance appears as the formation of a nonabsorbing state, which is a linear combination of |g, mz = −1〉 and |g, mz = 1〉. b, If angular momentum is quantized along the axis parallel with the resulting linear polarization of the light (y axis), the dark resonance corresponds to an optical pumping to the |g, my = 0〉 state, which is not coupled to the light because the Clebsch–Gordan coefficient connecting |g, Jg = 1, my = 0〉 and |e, Je = 1, my = 0〉 is zero.

Fig. 6
Fig. 6

MCWF simulation of a dark resonance. The population of the uncoupled state in a single MCWF evolution, corresponding to a measurement of the photon angular momentum along the z axis, a, or along the y axis, b. The two types of evolution are clearly different. Average of 100 MCWF evolutions, with a measurement of the photon angular momentum along the z axis, c, or along the y axis, d. Apart from fluctuations, the two simulations lead to the same result, as expected.

Fig. 7
Fig. 7

Time evolution of the variance ΔA(ρS)(t) and of the sample variances ΔA(n)(t) obtained for the two simulations with the y and z quantization axis for the dark resonance problem (n = 1000). The sample variance for the y axis choice is close to its upper bound ΔA(ρS)(t), whereas the simulation with the choice of the z axis leads to a noise that is more than two times smaller (also compare Figs. 6c and 6d).

Equations (107)

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ρ ˙ S = ( i / ) [ ρ S , H S ] + L relax ( ρ S ) .
L relax ( ρ S ) = - 1 2 m ( C m C m ρ S + ρ S C m C m ) + m C m ρ S C m .
L relax ( ρ S ) = - ( Γ / 2 ) ( σ + σ - ρ S + ρ S σ + σ - ) + Γ σ - ρ S σ + .
σ + = S + = e g ,             σ - = S - = g e .
σ - = b ,             σ + = b .
H = H S - i 2 m C m C m .
ϕ ( 1 ) ( t + δ t ) = ( 1 - i H δ t ) ϕ ( t ) .
ϕ ( 1 ) ( t + δ t ) ϕ ( 1 ) ( t + δ t ) = ϕ ( t ) ( 1 + i H δ t ) ( 1 - i H δ t ) ϕ ( t ) = 1 - δ p ,
δ p = δ t i ϕ ( t ) H - H ϕ ( t ) = m δ p m ,
δ p m = δ t ϕ ( t ) C m C m ϕ ( t ) 0.
ϕ ( t + δ t ) = ϕ ( 1 ) ( t + δ t ) / ( 1 - δ p ) 1 / 2 ,             δ p < .
ϕ ( t + δ t ) = C m ϕ ( t ) / ( δ p m / δ t ) 1 / 2 with a probability Π m = δ p m / δ p ,             δ p > .
σ ( t + δ t ) ¯ = ( 1 - δ p ) ϕ ( 1 ) ( t + δ t ) ( 1 - δ p ) 1 / 2 ϕ ( 1 ) ( t + δ t ) ( 1 - δ p ) 1 / 2 + δ p m Π m C m ϕ ( t ) ( δ p m / δ t ) 1 / 2 ϕ ( t ) C m ( δ p m / δ t ) 1 / 2 ,
σ ( t + δ t ) ¯ = σ ( t ) + ( i δ t / ) [ σ ( t ) , H S ] + δ t L relax [ σ ( t ) ] .
d σ ¯ d t = i h [ σ ¯ , H S ] + L relax ( σ ¯ ) .
A ( n ) ( t ) = 1 n i = 1 n ϕ ( i ) ( t ) A ϕ ( i ) ( t ) .
ϕ ( 0 ) = α 0 0 + β 0 1 .
ϕ ( 1 ) ( δ t ) = α 0 0 + β 0 exp ( - i ω 0 δ t ) exp ( - Γ δ t / 2 ) 1 .
δ p = Γ β 0 2 δ t ,
ϕ ( δ t ) = α 0 ( 1 + Γ δ t 2 β 0 2 ) 0 + β 0 ( 1 - Γ δ t 2 α 0 2 ) exp ( - i ω 0 δ t ) 1 ,             δ p < .
ϕ ( δ t ) = α 0 0 + β 0 exp ( - i ω 0 δ t ) 1 ,             δ p < ,
ϕ ( t ) = α ( t ) 0 + β ( t ) exp ( - i ω 0 t ) 1 ,
α ˙ = Γ α β 2 / 2 , β ˙ = - Γ β α 2 / 2.
α ( t ) = α 0 [ α 0 2 + β 0 2 exp ( - Γ t ) ] - 1 / 2 , β ( t ) = β 0 exp ( - Γ t / 2 ) [ α 0 2 + β 0 2 exp ( - Γ t ) ] - 1 / 2 .
P ( t ) = α 0 2 + β 0 2 exp ( - Γ t ) .
ϕ ( t ) = α ( t ) 0 + β ( t ) exp ( - ω 0 t ) 1 ;
ϕ ( t ) = 0 ,
i d ϕ d t = ( H + ϕ H - H 2 ϕ ) ϕ ,
ϕ ( t ) = ϕ ( 0 ) exp ( i H t / ) exp ( - i H t / ) ϕ ( 0 ) - 1 / 2 × exp ( - i H t / ) ϕ ( 0 ) ,
i d ϕ d t = H ϕ
d σ ( n ) d t = i [ σ ( n ) , H ] + feeding term ( σ ( n - 1 ) ) ,
C ( t , τ ) = A ( t + τ ) B ( t ) .
C i j ( t , τ ) = X i j ( t + τ ) B ( t ) .
C i j τ ( t , τ ) = k l L i j k l C k l ( t , τ ) ,
d X i j ( t ) d t = k l L i j k l X k l ( t ) .
χ ± ( 0 ) = 1 μ ± ( 1 ± B ) ϕ ( t ) ,
χ ± ( 0 ) = 1 μ ± ( 1 ± i B ) ϕ ( t ) ,
c ± ( τ ) = χ ± ( τ ) A χ ± ( τ ) ,
c ± ( τ ) = χ ± ( τ ) A χ ± ( τ ) ,
C ( t , τ ) = ¼ [ μ + c + ( τ ) ¯ - μ - c - ( τ ) ¯ - i μ + c + ( τ ) ¯ + i μ - c - ( τ ) ¯ ] .
κ i j ( τ ) = ¼ [ μ + χ + ( τ ) X i j χ + ( τ ) - μ - χ - ( τ ) X i j χ - ( τ ) - i μ + χ + ( τ ) X i j χ + ( τ ) + i μ - χ - ( τ ) X i j χ - ( τ ) ] ,
κ i j ( 0 ) = ¼ [ ϕ ( t ) ( 1 + B ) X i j ( 1 + B ) ϕ ( t ) - ϕ ( t ) ( 1 - B ) X i j ( 1 - B ) ϕ ( t ) - i ϕ ( t ) ( 1 - i B ) X i j ( 1 + i B ) ϕ ( t ) + i ϕ ( t ) ( 1 + B ) X i j ( 1 - i B ) ϕ ( t ) ] = ϕ ( t ) X i j B ϕ ( t ) ,
κ i j ¯ ( 0 ) = C i j ( t , 0 ) .
C s ( t , τ ) = X ( t + τ ) X ( t ) + X ( t ) X ( t + τ ) X = ( b + b ) / 2 .
χ ± ( 0 ) = 2 / 3 0 ± ( 1 / 3 ) 1 .
c + ( τ ) = - c - ( τ ) = P ( τ ) ( 1 / 2 ) [ α * ( τ ) β ( τ ) exp ( - i ω 0 τ ) + α ( τ ) + α ( τ ) β * ( τ ) exp ( i ω 0 τ ) ] .
C s ( t , τ ) = cos ( ω 0 τ ) exp ( - Γ τ / 2 ) ,
H 0 = ( Ω / 2 ) ( S + + S - ) ,
C ( t , τ ) = S + ( t + τ ) S - ( t ) .
L relax ( ρ S ) = - ( Γ / 2 ) [ 1 + n ( ω 0 ) ] × ( σ + σ - ρ S + ρ S σ + σ - - 2 σ - ρ S σ + ) - ( Γ / 2 ) n ( ω 0 ) ( σ - σ + ρ S + ρ S σ - σ + - 2 σ + ρ S σ - ) ,
n ( ω 0 ) = [ exp ( ω 0 k T ) - 1 ] - 1 .
C 1 = { Γ [ 1 + n ( ω 0 ) ] } 1 / 2 σ - ,
C 2 = [ Γ n ( ω 0 ) ] 1 / 2 σ + .
L relax ( ρ S ) = - ( 1 / T 2 ) ( P e ρ S P g + P g ρ S P e )
L relax ( ρ S ) = - ( 1 / 4 T 2 ) [ ( P e - P g ) 2 ρ S + ρ S ( P e - P g ) 2 ] + ( 1 / 2 T 2 ) ( P e - P g ) ρ S ( P e - P g ) .
C 1 = 1 / 2 T 2 ( P e - P g ) .
L relax ( ρ S ) = - ( Γ / 2 ) ( P e ρ S + ρ S P e ) + Γ q ( q * · S - ) ρ S ( q · S + ) ,
± = 2 - 1 / 2 ( u x ± i u y ) ,
0 = u z ,
q · S + J g , m g z = ( 1 , J g , q , m g ; J e , m e = m g + q ) × J e , m e = m g + q z , q · S + J e , m e z = 0 , q * · S - = ( q · S + ) .
C q = ( Γ ) 1 / 2 ( q * · S - ) ,             q = 0 , ± 1.
q = - 1 1 C q C q = Γ S + · S - = Γ P e
H S = P 2 / 2 M .
L relax ( ρ S ) = - γ ρ S + γ d 3 q N ( q ) exp ( i q · R / ) ρ S exp ( - i q · R / ) .
Π ˙ ( p ) = - γ Π ( p ) + γ d 3 q N ( q ) Π ( p - q ) .
C q = [ γ N ( q ) ] 1 / 2 exp ( i q · R / ) .
ϕ ( 0 ) = i α i ( 0 ) p i 0 .
L relax ( ρ S ) = - ( Γ / 2 ) ( P e ρ S + ρ S P e ) + ( 3 Γ / 8 π ) d 2 Ω k exp ( - i k · R ) ( * · S - ) × ρ S ( · S + ) exp ( i k · R ) ,
C Ω , = ( 3 Γ / 8 π ) 1 / 2 exp ( - i k · R ) ( * · S - ) ,
d 2 Ω k C Ω , C Ω , = ( 3 Γ / 8 π ) d 2 Ω k ( · S + ) ( * · S - ) = ( 3 Γ / 8 π ) d 2 Ω ( S + · S - - [ ( S + · k ) × ( S - · k ) ] / k 2 ) = Γ S + · S - = Γ P e .
P ( Ω ) = P ( Ω , 1 ) + P ( Ω , 1 ) + P ( Ω , 2 ) ,
P ( Ω , i ) = ( 3 / 8 π ) ( 1 / Π e ) ϕ ( t ) ( i · S + ) ( i * · S - ) ϕ ( t ) ,             i = 1 , 2.
( Γ / 2 ) = u x , u y , u z k exp ( - i k · R ) × ( · S - ) ρ S ( · S + ) exp ( i k · R ) .
L relax ( ρ S ) = - ( Γ / 2 ) ( P e ρ S + ρ S P e ) + Γ q = - 1 1 - k k d k N q ( k ) exp ( - i k Z ) ( q * · S - ) × ρ S ( q · S + ) exp ( i k Z ) .
N ± 1 ( k ) = ( 3 / 8 k ) [ 1 + ( k k ) 2 ] ,
N 0 ( k ) = ( 3 / 4 k ) [ 1 - ( k k ) 2 ] .
C k , q = [ Γ N q ( k ) ] 1 / 2 exp ( - i k Z ) ( * · S - ) .
H S = ( P 2 / 2 M ) + Ω cos ( k Z ) ( S + + S - ) - δ P e ,
ϕ ( t ) = n α n ( t ) g , p = p 0 + 2 n k + β n ( t ) e , p = p 0 + ( 2 n + 1 ) k ,
i α ˙ n = ( p 0 + 2 n k ) 2 2 M α n + Ω 2 ( β n + β n - 1 ) ,
i β ˙ n = { [ p 0 + ( 2 n + 1 ) k ] 2 2 M - δ - i Γ 2 } β n + Ω 2 ( α n + α n + 1 ) .
δ p = Γ n β n 2 δ t .
C k = [ Γ N + ( k ) ] 1 / 2 exp ( - i k Z ) S -
α n ( t + δ t ) = μ β n ( t ) ,
β n ( t + δ t ) = 0 ,
p 0 p 0 + k - k ,
P 2 ( n ) ( t ) = 1 n i = 1 n ϕ ( i ) ( t ) P 2 ϕ ( i ) ( t ) ,
T [ L relax ( ρ S ) ] T = L relax ( T ρ S T ) .
L relax ( ρ S ) = T L relax ( T ρ S T ) T ,
L relax ( ρ S ) = - 1 2 m ( D m D m ρ S + ρ S D m D m ) + m D m ρ S D m ,
D m = T C m T .
ϕ N C = ( g , m z = - 1 + g , m z = 1 ) / 2 .
ϕ N C = g , m y = 0 ,
A ( n ) ( t ) = 1 n i = 1 n ϕ ( i ) ( t ) A ϕ ( i ) ( t ) ,
δ A ( n ) = Δ A ( n ) / n ,
( Δ A ) ( n ) 2 ( t ) = 1 n ( i = 1 n ϕ ( i ) ( t ) A ϕ ( i ) ( t ) 2 ) - A ( n ) 2 ( t ) .
n Δ A ( ) ( t ) / [ A ( t ) ] .
ϕ A ϕ 2 ϕ A 2 ϕ .
( Δ A ) ( n ) 2 ( t ) 1 n ( i = 1 n ϕ ( i ) ( t ) A 2 ϕ ( i ) ( t ) ) - A ( n ) 2 ( t ) .
( Δ A ) ( ρ S ) 2 ( t ) = Tr [ ρ S ( t ) A 2 ] - [ Tr ( ρ S A ) ] 2 .
( Δ A ) ( ) 2 ( t ) ( Δ A ) ( ρ S ) 2 ( t ) .
n Δ A ( ρ S ) ( t ) / [ A ( t ) ] .
A ( t ) ~ 1 N ,             ( Δ A ) ( ρ S ) 2 ( t ) ~ 1 N ,
local operators             n N .
Δ A ( ρ S ) ( t ) ~ A ( t ) .
A = ( 3 / 2 ) k B T ,             Δ A ( ρ S ) = 3 / 2 k B T .
global operators             n 1.

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