Abstract

Several recent studies have shown that the time evolution of an atom submitted to coherent laser fields and to dissipative processes, such as spontaneous emission of photons or excitation by a broadband incoherent field, can be considered to consist of a sequence of coherent evolution periods separated by quantum jumps occurring at random times. A general statistical analysis of this random sequence is presented for the case in which the number of relevant atomic states is finite and the delay functions giving the distribution of the time intervals between two successive jumps can easily be calculated. These general considerations are then applied to a simple model recently proposed for demonstrating the possibility of amplification without inversion of populations. We show how the quantum-jump approach allows one to calculate the respective contributions of the various physical processes responsible for the amplification or the attenuation of the probe field and to get new insights into the relevant physical mechanisms.

© 1993 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. W. Nagourney, J. Sandberg, H. G. Dehmelt, “Shelved optical electron amplifier: observation of quantum jumps,” Phys. Rev. Lett. 56, 2797–2799 (1986); Th. Sauter, W. Neuhauser, R. Blatt, P. Toschek, “Observation of quantum jumps,” Phys. Rev. Lett. 57, 1696–1698 (1986); J. C. Bergquist, R. G. Hulet, W. M. Itano, D. J. Wineland, “Observation of quantum jumps in a single atom,” Phys. Rev. Lett. 57, 1699–1702 (1986).
    [CrossRef] [PubMed]
  2. F. J. Cook, “Quantum jumps,” in Progress in Optics XXVIII, E. Wolf, ed. (Elsevier, Berlin, 1990), pp. 361–418, and references therein.
    [CrossRef]
  3. A. E. Kaplan, P. Meystre, eds., special issue on the quantum and nonlinear optics of single electrons, atoms and ions, IEEE J. Quantum Electron. 24, 1312–1482 (1988).
    [CrossRef]
  4. C. Cohen-Tannoudji, J. Dalibard, “Single-atom laser spectroscopy. Looking for dark periods in fluorescence light,” Europhys. Lett. 1, 441–448 (1986).
    [CrossRef]
  5. P. Zoller, M. Marte, D. F. Walls, “Quantum jumps in atomic systems,” Phys. Rev. A 35, 198–207 (1987). See also R. Dum, P. Zoller, H. Ritsch, “Monte Carlo simulation of the atomic master equation for spontaneous emission,” Phys. Rev. A 45, 4879–4887 (1992).
    [CrossRef] [PubMed]
  6. C. Cohen-Tannoudji, F. Bardou, A. Aspect, “Review on fundamental processes in laser cooling,” in Laser Spectroscopy X, M. Ducloy, E. Giacobino, G. Camy, eds. (World Scientific, Singapore, 1992), pp. 3–14.
  7. J. Dalibard, Y. Castin, K. Molmer, “Wave-function approach to dissipative processes in quantum optics,” Phys. Rev. Lett. 68, 580–583 (1992).
    [CrossRef] [PubMed]
  8. K. Mölmer, Y. Castin, J. Dalibard, “A Monte-Carlo wave function method in quantum optics,” J. Opt. Soc. Am. B 10, 524–538 (1993).
    [CrossRef]
  9. H. J. Carmichael, “An open systems approach to quantum optics,” lectures presented at the Université Libre de Bruxelles, Bruxelles, Belgium, Fall 1991.
  10. G. C. Hegerfeldt, T. S. Wilser, “Ensemble or individual system, collapse or no collapse: a description of a single radiating atom,” in Proceedings of the Second International Wigner Symposium, H. D. Doebner, W. Schere, F. S. Schroech, eds. (World Scientific, Singapore, to be published).
  11. N. Gisin, “Stochastic quantum dynamics and relativity,” Helv. Phys. Acta 62, 363–371 (1989); “Quantum measurements and stochastic processes,” Phys. Rev. Lett. 52, 1657–1660 (1984).
  12. A. Imamoglũ, J. E. Field, S. E. Harris, “Lasers without inversion: a closed lifetime broadened system,” Phys. Rev. Lett. 66, 1154–1156 (1991).
    [CrossRef]
  13. O. A. Kocharovskaya, Ya. I. Khanin, “Coherent amplification of an ultrashort pulse in a three-level medium without a population inversion,” JEPT Lett. 48, 630–634 (1988); M. O. Scully, S. Y. Zhu, A. Gavrielides, “Degenerate quantum-beat laser: lasing without inversion and inversion without lasing,” Phys. Rev. Lett. 62, 2813–2816 (1989); S. E. Harris, “Lasers without inversion: interference of radiatively broadened resonances,” Phys. Rev. A 40, 2835–2838 (1989); G. S. Agarwal, “Origin of gain in systems without inversions in bare or dressed states,” Phys. Rev. A 44, R28–R30 (1991). For more recent reviews on lasers without inversion, see O. Kocharovskaya, M. O. Scully, in Proceedings of the Twentieth Solvay Conference on Physics on Quantum Optics,” P. Mandel, ed., Phys. Rep.219, 175–212 (1992).
    [CrossRef] [PubMed]
  14. C. Cohen-Tannoudji, B. Zambon, E. Arimondo, “Propriétés statistiques de la suite de sauts quantiques associée des processus dissipatifs,” C. R. Acad. Sci. 314, 1139–1145 (1992); “Modèle simple d’amplification sans inversion de population. Etude par la méthode de sauts quantiques,” C. R. Acad. Sci. 314, 1293–1299 (1992).
  15. See, for example, C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg, Atom-Photon Interaction (Wiley, New York, 1992).
  16. C. Cohen-Tannoudji, “Optical pumping and interaction of atoms with the electromagnetic field,” in Cargèse Lectures in Physics, M. Lévy, ed. (Gordon & Breach, New York, 1968), Vol. 2, pp. 347–393.
  17. A. Imamoglũ, Harvard College Observatory, Institute for Theoretical Atomic and Molecular Physics, Cambridge, Mass. 02138 (personal communication).
  18. E. Arimondo, “Mechanisms in laser without inversion,” in 1992 Shanghai International Symposium on Quantum Optics, D. Wang, Z. Wang, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1726, 484–489 (1992).
    [CrossRef]
  19. J. Dalibard, S. Reynaud, C. Cohen-Tannoudji, “La cascade radiative de l’atom habillé,” in Interaction of Radiation with Matter (Scuola Normale Superiore, Pisa, Italy, 1987), pp. 29–48; A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, “Laser cooling below the one-photon recoil energy by velocity selective coherent population trapping: theoretical analysis,” J. Opt. Soc. Am. B 6, 2112–2124 (1989), and references therein.
    [CrossRef]
  20. B. Lounis, C. Cohen-Tannoudji, “Coherent population trapping and Fano profile,” J. Phys. B 2, 579–592 (1992).
  21. G. Grynberg, C. Cohen-Tannoudji, “Central resonance of the Mollow absorption spectrum. Physical origin of gain without population inversion,” Opt. Commun. 96, 150–163 (1993).
    [CrossRef]
  22. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

1993

K. Mölmer, Y. Castin, J. Dalibard, “A Monte-Carlo wave function method in quantum optics,” J. Opt. Soc. Am. B 10, 524–538 (1993).
[CrossRef]

G. Grynberg, C. Cohen-Tannoudji, “Central resonance of the Mollow absorption spectrum. Physical origin of gain without population inversion,” Opt. Commun. 96, 150–163 (1993).
[CrossRef]

1992

J. Dalibard, Y. Castin, K. Molmer, “Wave-function approach to dissipative processes in quantum optics,” Phys. Rev. Lett. 68, 580–583 (1992).
[CrossRef] [PubMed]

C. Cohen-Tannoudji, B. Zambon, E. Arimondo, “Propriétés statistiques de la suite de sauts quantiques associée des processus dissipatifs,” C. R. Acad. Sci. 314, 1139–1145 (1992); “Modèle simple d’amplification sans inversion de population. Etude par la méthode de sauts quantiques,” C. R. Acad. Sci. 314, 1293–1299 (1992).

B. Lounis, C. Cohen-Tannoudji, “Coherent population trapping and Fano profile,” J. Phys. B 2, 579–592 (1992).

1991

A. Imamoglũ, J. E. Field, S. E. Harris, “Lasers without inversion: a closed lifetime broadened system,” Phys. Rev. Lett. 66, 1154–1156 (1991).
[CrossRef]

1989

N. Gisin, “Stochastic quantum dynamics and relativity,” Helv. Phys. Acta 62, 363–371 (1989); “Quantum measurements and stochastic processes,” Phys. Rev. Lett. 52, 1657–1660 (1984).

1988

O. A. Kocharovskaya, Ya. I. Khanin, “Coherent amplification of an ultrashort pulse in a three-level medium without a population inversion,” JEPT Lett. 48, 630–634 (1988); M. O. Scully, S. Y. Zhu, A. Gavrielides, “Degenerate quantum-beat laser: lasing without inversion and inversion without lasing,” Phys. Rev. Lett. 62, 2813–2816 (1989); S. E. Harris, “Lasers without inversion: interference of radiatively broadened resonances,” Phys. Rev. A 40, 2835–2838 (1989); G. S. Agarwal, “Origin of gain in systems without inversions in bare or dressed states,” Phys. Rev. A 44, R28–R30 (1991). For more recent reviews on lasers without inversion, see O. Kocharovskaya, M. O. Scully, in Proceedings of the Twentieth Solvay Conference on Physics on Quantum Optics,” P. Mandel, ed., Phys. Rep.219, 175–212 (1992).
[CrossRef] [PubMed]

A. E. Kaplan, P. Meystre, eds., special issue on the quantum and nonlinear optics of single electrons, atoms and ions, IEEE J. Quantum Electron. 24, 1312–1482 (1988).
[CrossRef]

1987

P. Zoller, M. Marte, D. F. Walls, “Quantum jumps in atomic systems,” Phys. Rev. A 35, 198–207 (1987). See also R. Dum, P. Zoller, H. Ritsch, “Monte Carlo simulation of the atomic master equation for spontaneous emission,” Phys. Rev. A 45, 4879–4887 (1992).
[CrossRef] [PubMed]

1986

W. Nagourney, J. Sandberg, H. G. Dehmelt, “Shelved optical electron amplifier: observation of quantum jumps,” Phys. Rev. Lett. 56, 2797–2799 (1986); Th. Sauter, W. Neuhauser, R. Blatt, P. Toschek, “Observation of quantum jumps,” Phys. Rev. Lett. 57, 1696–1698 (1986); J. C. Bergquist, R. G. Hulet, W. M. Itano, D. J. Wineland, “Observation of quantum jumps in a single atom,” Phys. Rev. Lett. 57, 1699–1702 (1986).
[CrossRef] [PubMed]

C. Cohen-Tannoudji, J. Dalibard, “Single-atom laser spectroscopy. Looking for dark periods in fluorescence light,” Europhys. Lett. 1, 441–448 (1986).
[CrossRef]

Arimondo, E.

C. Cohen-Tannoudji, B. Zambon, E. Arimondo, “Propriétés statistiques de la suite de sauts quantiques associée des processus dissipatifs,” C. R. Acad. Sci. 314, 1139–1145 (1992); “Modèle simple d’amplification sans inversion de population. Etude par la méthode de sauts quantiques,” C. R. Acad. Sci. 314, 1293–1299 (1992).

E. Arimondo, “Mechanisms in laser without inversion,” in 1992 Shanghai International Symposium on Quantum Optics, D. Wang, Z. Wang, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1726, 484–489 (1992).
[CrossRef]

Aspect, A.

C. Cohen-Tannoudji, F. Bardou, A. Aspect, “Review on fundamental processes in laser cooling,” in Laser Spectroscopy X, M. Ducloy, E. Giacobino, G. Camy, eds. (World Scientific, Singapore, 1992), pp. 3–14.

Bardou, F.

C. Cohen-Tannoudji, F. Bardou, A. Aspect, “Review on fundamental processes in laser cooling,” in Laser Spectroscopy X, M. Ducloy, E. Giacobino, G. Camy, eds. (World Scientific, Singapore, 1992), pp. 3–14.

Carmichael, H. J.

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

Castin, Y.

K. Mölmer, Y. Castin, J. Dalibard, “A Monte-Carlo wave function method in quantum optics,” J. Opt. Soc. Am. B 10, 524–538 (1993).
[CrossRef]

J. Dalibard, Y. Castin, K. Molmer, “Wave-function approach to dissipative processes in quantum optics,” Phys. Rev. Lett. 68, 580–583 (1992).
[CrossRef] [PubMed]

Cohen-Tannoudji, C.

G. Grynberg, C. Cohen-Tannoudji, “Central resonance of the Mollow absorption spectrum. Physical origin of gain without population inversion,” Opt. Commun. 96, 150–163 (1993).
[CrossRef]

B. Lounis, C. Cohen-Tannoudji, “Coherent population trapping and Fano profile,” J. Phys. B 2, 579–592 (1992).

C. Cohen-Tannoudji, B. Zambon, E. Arimondo, “Propriétés statistiques de la suite de sauts quantiques associée des processus dissipatifs,” C. R. Acad. Sci. 314, 1139–1145 (1992); “Modèle simple d’amplification sans inversion de population. Etude par la méthode de sauts quantiques,” C. R. Acad. Sci. 314, 1293–1299 (1992).

C. Cohen-Tannoudji, J. Dalibard, “Single-atom laser spectroscopy. Looking for dark periods in fluorescence light,” Europhys. Lett. 1, 441–448 (1986).
[CrossRef]

C. Cohen-Tannoudji, F. Bardou, A. Aspect, “Review on fundamental processes in laser cooling,” in Laser Spectroscopy X, M. Ducloy, E. Giacobino, G. Camy, eds. (World Scientific, Singapore, 1992), pp. 3–14.

C. Cohen-Tannoudji, “Optical pumping and interaction of atoms with the electromagnetic field,” in Cargèse Lectures in Physics, M. Lévy, ed. (Gordon & Breach, New York, 1968), Vol. 2, pp. 347–393.

See, for example, C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg, Atom-Photon Interaction (Wiley, New York, 1992).

J. Dalibard, S. Reynaud, C. Cohen-Tannoudji, “La cascade radiative de l’atom habillé,” in Interaction of Radiation with Matter (Scuola Normale Superiore, Pisa, Italy, 1987), pp. 29–48; A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, “Laser cooling below the one-photon recoil energy by velocity selective coherent population trapping: theoretical analysis,” J. Opt. Soc. Am. B 6, 2112–2124 (1989), and references therein.
[CrossRef]

Cook, F. J.

F. J. Cook, “Quantum jumps,” in Progress in Optics XXVIII, E. Wolf, ed. (Elsevier, Berlin, 1990), pp. 361–418, and references therein.
[CrossRef]

Dalibard, J.

K. Mölmer, Y. Castin, J. Dalibard, “A Monte-Carlo wave function method in quantum optics,” J. Opt. Soc. Am. B 10, 524–538 (1993).
[CrossRef]

J. Dalibard, Y. Castin, K. Molmer, “Wave-function approach to dissipative processes in quantum optics,” Phys. Rev. Lett. 68, 580–583 (1992).
[CrossRef] [PubMed]

C. Cohen-Tannoudji, J. Dalibard, “Single-atom laser spectroscopy. Looking for dark periods in fluorescence light,” Europhys. Lett. 1, 441–448 (1986).
[CrossRef]

J. Dalibard, S. Reynaud, C. Cohen-Tannoudji, “La cascade radiative de l’atom habillé,” in Interaction of Radiation with Matter (Scuola Normale Superiore, Pisa, Italy, 1987), pp. 29–48; A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, “Laser cooling below the one-photon recoil energy by velocity selective coherent population trapping: theoretical analysis,” J. Opt. Soc. Am. B 6, 2112–2124 (1989), and references therein.
[CrossRef]

Dehmelt, H. G.

W. Nagourney, J. Sandberg, H. G. Dehmelt, “Shelved optical electron amplifier: observation of quantum jumps,” Phys. Rev. Lett. 56, 2797–2799 (1986); Th. Sauter, W. Neuhauser, R. Blatt, P. Toschek, “Observation of quantum jumps,” Phys. Rev. Lett. 57, 1696–1698 (1986); J. C. Bergquist, R. G. Hulet, W. M. Itano, D. J. Wineland, “Observation of quantum jumps in a single atom,” Phys. Rev. Lett. 57, 1699–1702 (1986).
[CrossRef] [PubMed]

Dupont-Roc, J.

See, for example, C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg, Atom-Photon Interaction (Wiley, New York, 1992).

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Field, J. E.

A. Imamoglũ, J. E. Field, S. E. Harris, “Lasers without inversion: a closed lifetime broadened system,” Phys. Rev. Lett. 66, 1154–1156 (1991).
[CrossRef]

Gisin, N.

N. Gisin, “Stochastic quantum dynamics and relativity,” Helv. Phys. Acta 62, 363–371 (1989); “Quantum measurements and stochastic processes,” Phys. Rev. Lett. 52, 1657–1660 (1984).

Grynberg, G.

G. Grynberg, C. Cohen-Tannoudji, “Central resonance of the Mollow absorption spectrum. Physical origin of gain without population inversion,” Opt. Commun. 96, 150–163 (1993).
[CrossRef]

See, for example, C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg, Atom-Photon Interaction (Wiley, New York, 1992).

Harris, S. E.

A. Imamoglũ, J. E. Field, S. E. Harris, “Lasers without inversion: a closed lifetime broadened system,” Phys. Rev. Lett. 66, 1154–1156 (1991).
[CrossRef]

Hegerfeldt, G. C.

G. C. Hegerfeldt, T. S. Wilser, “Ensemble or individual system, collapse or no collapse: a description of a single radiating atom,” in Proceedings of the Second International Wigner Symposium, H. D. Doebner, W. Schere, F. S. Schroech, eds. (World Scientific, Singapore, to be published).

Imamoglu, A.

A. Imamoglũ, J. E. Field, S. E. Harris, “Lasers without inversion: a closed lifetime broadened system,” Phys. Rev. Lett. 66, 1154–1156 (1991).
[CrossRef]

A. Imamoglũ, Harvard College Observatory, Institute for Theoretical Atomic and Molecular Physics, Cambridge, Mass. 02138 (personal communication).

Khanin, Ya. I.

O. A. Kocharovskaya, Ya. I. Khanin, “Coherent amplification of an ultrashort pulse in a three-level medium without a population inversion,” JEPT Lett. 48, 630–634 (1988); M. O. Scully, S. Y. Zhu, A. Gavrielides, “Degenerate quantum-beat laser: lasing without inversion and inversion without lasing,” Phys. Rev. Lett. 62, 2813–2816 (1989); S. E. Harris, “Lasers without inversion: interference of radiatively broadened resonances,” Phys. Rev. A 40, 2835–2838 (1989); G. S. Agarwal, “Origin of gain in systems without inversions in bare or dressed states,” Phys. Rev. A 44, R28–R30 (1991). For more recent reviews on lasers without inversion, see O. Kocharovskaya, M. O. Scully, in Proceedings of the Twentieth Solvay Conference on Physics on Quantum Optics,” P. Mandel, ed., Phys. Rep.219, 175–212 (1992).
[CrossRef] [PubMed]

Kocharovskaya, O. A.

O. A. Kocharovskaya, Ya. I. Khanin, “Coherent amplification of an ultrashort pulse in a three-level medium without a population inversion,” JEPT Lett. 48, 630–634 (1988); M. O. Scully, S. Y. Zhu, A. Gavrielides, “Degenerate quantum-beat laser: lasing without inversion and inversion without lasing,” Phys. Rev. Lett. 62, 2813–2816 (1989); S. E. Harris, “Lasers without inversion: interference of radiatively broadened resonances,” Phys. Rev. A 40, 2835–2838 (1989); G. S. Agarwal, “Origin of gain in systems without inversions in bare or dressed states,” Phys. Rev. A 44, R28–R30 (1991). For more recent reviews on lasers without inversion, see O. Kocharovskaya, M. O. Scully, in Proceedings of the Twentieth Solvay Conference on Physics on Quantum Optics,” P. Mandel, ed., Phys. Rep.219, 175–212 (1992).
[CrossRef] [PubMed]

Lounis, B.

B. Lounis, C. Cohen-Tannoudji, “Coherent population trapping and Fano profile,” J. Phys. B 2, 579–592 (1992).

Marte, M.

P. Zoller, M. Marte, D. F. Walls, “Quantum jumps in atomic systems,” Phys. Rev. A 35, 198–207 (1987). See also R. Dum, P. Zoller, H. Ritsch, “Monte Carlo simulation of the atomic master equation for spontaneous emission,” Phys. Rev. A 45, 4879–4887 (1992).
[CrossRef] [PubMed]

Molmer, K.

J. Dalibard, Y. Castin, K. Molmer, “Wave-function approach to dissipative processes in quantum optics,” Phys. Rev. Lett. 68, 580–583 (1992).
[CrossRef] [PubMed]

Mölmer, K.

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Nagourney, W.

W. Nagourney, J. Sandberg, H. G. Dehmelt, “Shelved optical electron amplifier: observation of quantum jumps,” Phys. Rev. Lett. 56, 2797–2799 (1986); Th. Sauter, W. Neuhauser, R. Blatt, P. Toschek, “Observation of quantum jumps,” Phys. Rev. Lett. 57, 1696–1698 (1986); J. C. Bergquist, R. G. Hulet, W. M. Itano, D. J. Wineland, “Observation of quantum jumps in a single atom,” Phys. Rev. Lett. 57, 1699–1702 (1986).
[CrossRef] [PubMed]

Reynaud, S.

J. Dalibard, S. Reynaud, C. Cohen-Tannoudji, “La cascade radiative de l’atom habillé,” in Interaction of Radiation with Matter (Scuola Normale Superiore, Pisa, Italy, 1987), pp. 29–48; A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, “Laser cooling below the one-photon recoil energy by velocity selective coherent population trapping: theoretical analysis,” J. Opt. Soc. Am. B 6, 2112–2124 (1989), and references therein.
[CrossRef]

Sandberg, J.

W. Nagourney, J. Sandberg, H. G. Dehmelt, “Shelved optical electron amplifier: observation of quantum jumps,” Phys. Rev. Lett. 56, 2797–2799 (1986); Th. Sauter, W. Neuhauser, R. Blatt, P. Toschek, “Observation of quantum jumps,” Phys. Rev. Lett. 57, 1696–1698 (1986); J. C. Bergquist, R. G. Hulet, W. M. Itano, D. J. Wineland, “Observation of quantum jumps in a single atom,” Phys. Rev. Lett. 57, 1699–1702 (1986).
[CrossRef] [PubMed]

Walls, D. F.

P. Zoller, M. Marte, D. F. Walls, “Quantum jumps in atomic systems,” Phys. Rev. A 35, 198–207 (1987). See also R. Dum, P. Zoller, H. Ritsch, “Monte Carlo simulation of the atomic master equation for spontaneous emission,” Phys. Rev. A 45, 4879–4887 (1992).
[CrossRef] [PubMed]

Wilser, T. S.

G. C. Hegerfeldt, T. S. Wilser, “Ensemble or individual system, collapse or no collapse: a description of a single radiating atom,” in Proceedings of the Second International Wigner Symposium, H. D. Doebner, W. Schere, F. S. Schroech, eds. (World Scientific, Singapore, to be published).

Zambon, B.

C. Cohen-Tannoudji, B. Zambon, E. Arimondo, “Propriétés statistiques de la suite de sauts quantiques associée des processus dissipatifs,” C. R. Acad. Sci. 314, 1139–1145 (1992); “Modèle simple d’amplification sans inversion de population. Etude par la méthode de sauts quantiques,” C. R. Acad. Sci. 314, 1293–1299 (1992).

Zoller, P.

P. Zoller, M. Marte, D. F. Walls, “Quantum jumps in atomic systems,” Phys. Rev. A 35, 198–207 (1987). See also R. Dum, P. Zoller, H. Ritsch, “Monte Carlo simulation of the atomic master equation for spontaneous emission,” Phys. Rev. A 45, 4879–4887 (1992).
[CrossRef] [PubMed]

C. R. Acad. Sci.

C. Cohen-Tannoudji, B. Zambon, E. Arimondo, “Propriétés statistiques de la suite de sauts quantiques associée des processus dissipatifs,” C. R. Acad. Sci. 314, 1139–1145 (1992); “Modèle simple d’amplification sans inversion de population. Etude par la méthode de sauts quantiques,” C. R. Acad. Sci. 314, 1293–1299 (1992).

Europhys. Lett.

C. Cohen-Tannoudji, J. Dalibard, “Single-atom laser spectroscopy. Looking for dark periods in fluorescence light,” Europhys. Lett. 1, 441–448 (1986).
[CrossRef]

Helv. Phys. Acta

N. Gisin, “Stochastic quantum dynamics and relativity,” Helv. Phys. Acta 62, 363–371 (1989); “Quantum measurements and stochastic processes,” Phys. Rev. Lett. 52, 1657–1660 (1984).

IEEE J. Quantum Electron.

A. E. Kaplan, P. Meystre, eds., special issue on the quantum and nonlinear optics of single electrons, atoms and ions, IEEE J. Quantum Electron. 24, 1312–1482 (1988).
[CrossRef]

J. Opt. Soc. Am. B

J. Phys. B

B. Lounis, C. Cohen-Tannoudji, “Coherent population trapping and Fano profile,” J. Phys. B 2, 579–592 (1992).

JEPT Lett.

O. A. Kocharovskaya, Ya. I. Khanin, “Coherent amplification of an ultrashort pulse in a three-level medium without a population inversion,” JEPT Lett. 48, 630–634 (1988); M. O. Scully, S. Y. Zhu, A. Gavrielides, “Degenerate quantum-beat laser: lasing without inversion and inversion without lasing,” Phys. Rev. Lett. 62, 2813–2816 (1989); S. E. Harris, “Lasers without inversion: interference of radiatively broadened resonances,” Phys. Rev. A 40, 2835–2838 (1989); G. S. Agarwal, “Origin of gain in systems without inversions in bare or dressed states,” Phys. Rev. A 44, R28–R30 (1991). For more recent reviews on lasers without inversion, see O. Kocharovskaya, M. O. Scully, in Proceedings of the Twentieth Solvay Conference on Physics on Quantum Optics,” P. Mandel, ed., Phys. Rep.219, 175–212 (1992).
[CrossRef] [PubMed]

Opt. Commun.

G. Grynberg, C. Cohen-Tannoudji, “Central resonance of the Mollow absorption spectrum. Physical origin of gain without population inversion,” Opt. Commun. 96, 150–163 (1993).
[CrossRef]

Phys. Rev. A

P. Zoller, M. Marte, D. F. Walls, “Quantum jumps in atomic systems,” Phys. Rev. A 35, 198–207 (1987). See also R. Dum, P. Zoller, H. Ritsch, “Monte Carlo simulation of the atomic master equation for spontaneous emission,” Phys. Rev. A 45, 4879–4887 (1992).
[CrossRef] [PubMed]

Phys. Rev. Lett.

W. Nagourney, J. Sandberg, H. G. Dehmelt, “Shelved optical electron amplifier: observation of quantum jumps,” Phys. Rev. Lett. 56, 2797–2799 (1986); Th. Sauter, W. Neuhauser, R. Blatt, P. Toschek, “Observation of quantum jumps,” Phys. Rev. Lett. 57, 1696–1698 (1986); J. C. Bergquist, R. G. Hulet, W. M. Itano, D. J. Wineland, “Observation of quantum jumps in a single atom,” Phys. Rev. Lett. 57, 1699–1702 (1986).
[CrossRef] [PubMed]

J. Dalibard, Y. Castin, K. Molmer, “Wave-function approach to dissipative processes in quantum optics,” Phys. Rev. Lett. 68, 580–583 (1992).
[CrossRef] [PubMed]

A. Imamoglũ, J. E. Field, S. E. Harris, “Lasers without inversion: a closed lifetime broadened system,” Phys. Rev. Lett. 66, 1154–1156 (1991).
[CrossRef]

Other

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

See, for example, C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg, Atom-Photon Interaction (Wiley, New York, 1992).

C. Cohen-Tannoudji, “Optical pumping and interaction of atoms with the electromagnetic field,” in Cargèse Lectures in Physics, M. Lévy, ed. (Gordon & Breach, New York, 1968), Vol. 2, pp. 347–393.

A. Imamoglũ, Harvard College Observatory, Institute for Theoretical Atomic and Molecular Physics, Cambridge, Mass. 02138 (personal communication).

E. Arimondo, “Mechanisms in laser without inversion,” in 1992 Shanghai International Symposium on Quantum Optics, D. Wang, Z. Wang, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1726, 484–489 (1992).
[CrossRef]

J. Dalibard, S. Reynaud, C. Cohen-Tannoudji, “La cascade radiative de l’atom habillé,” in Interaction of Radiation with Matter (Scuola Normale Superiore, Pisa, Italy, 1987), pp. 29–48; A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, “Laser cooling below the one-photon recoil energy by velocity selective coherent population trapping: theoretical analysis,” J. Opt. Soc. Am. B 6, 2112–2124 (1989), and references therein.
[CrossRef]

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

G. C. Hegerfeldt, T. S. Wilser, “Ensemble or individual system, collapse or no collapse: a description of a single radiating atom,” in Proceedings of the Second International Wigner Symposium, H. D. Doebner, W. Schere, F. S. Schroech, eds. (World Scientific, Singapore, to be published).

F. J. Cook, “Quantum jumps,” in Progress in Optics XXVIII, E. Wolf, ed. (Elsevier, Berlin, 1990), pp. 361–418, and references therein.
[CrossRef]

C. Cohen-Tannoudji, F. Bardou, A. Aspect, “Review on fundamental processes in laser cooling,” in Laser Spectroscopy X, M. Ducloy, E. Giacobino, G. Camy, eds. (World Scientific, Singapore, 1992), pp. 3–14.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

Three-level atom forming a Λ configuration and subjected to dissipative processes, inducing transitions between the three levels e, g1, g2 with rates Γ1, Γ2, R1, R2 (arrows). The atom is also driven by two laser fields with frequencies ωL1 and ωL2 that are close, respectively, to the frequencies ωe1 and ωe2 of the two transitions g1e and g2e.

Fig. 2
Fig. 2

Manifolds E ( N 1 , N 2 ) of states of the atom + laser photon system. The coherent couplings within a given manifold (horizontal arrows) are characterized by the Rabi frequencies Ω1 and Ω2. The oblique arrows represent quantum jumps bringing the system from one manifold to another.

Fig. 3
Fig. 3

Sequence of coherent evolution periods …(i, j), (k, l), (m, n)… separated by quantum jumps from j to k, from l to m, and so on. Each coherent evolution period (i, j) is characterized by the state of entry i and the state of exit j. The duration of each period is a random variable whose distribution is given by the delay functions introduced in the text.

Fig. 4
Fig. 4

Stochastic evolution of the number of probe photons obtained by the Monte Carlo, simulation explained in the text. The parameters are Γ ˜ 1 = 0.25 Γ ˜ , Γ ˜ 2 = 0.75 Γ ˜ , R 1 = R 2 = ( 2.5 × 10 3 ) Γ ˜ , Ω 1 = ( 2.5 × 10 2 ) Γ ˜ , Ω 2 = 0.15 Γ ˜, δ1 = δ2 = 0. Changes in probe photon number are tagged by the processes that produce them; ↑ indicates stimulated Raman gain; ↓ stimulated Raman absorption; ⊤ one-photon gain; ⊥ one-photon loss. (a) Probe photon number versus normalized time Γ ˜ t. The dashed line represents the mean rate of variation of N1 computed from Eq. (4.16) below. (b) Enlarged part of the time evolution, with the number of probe photons being represented only at the time of the quantum jumps, where it has a well-defined value. One sees that between two successive changes of N1 there are several quantum jumps during which N1 does not change. (c) Further time enlargement allows one to distinguish the coherent evolution periods, which are represented by oblique lines joining their state in and their state out |g1〉 |g2〉, or |e〉.

Fig. 5
Fig. 5

Probabilities P ( 2 , 1 ) and P ( 1 , e ) for the coherent evolution periods (2, 1) and (1, e) contributing to the gain or loss of the probe photon number versus the detuning δ1 of the probe laser at frequency ωL1. The parameters are the same as in Fig. 4, with (a) δ1 = 0 and (b) δ 1 = 0.5 Γ ˜.

Equations (123)

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

δ i = ω L i ω e i , i = 1 , 2.
E ( N 1 , N 2 ) = { | e , N 1 , N 2 , | g 1 , N 1 + 1 , N 2 , | g 2 , N 1 , N 2 + 1 } .
e , N 1 , N 2 | V AL | g 1 , N 1 + 1 , N 2 = ћ Ω 1 / 2 ,
e , N 1 , N 2 | V AL | g 2 , N 1 , N 2 + 1 = ћ Ω 2 / 2 ,
Γ ˜ 1 = Γ 1 + R 1 .
Γ ˜ 2 = Γ 2 + R 2 .
period ( 1 , 2 ) Δ N 1 = + 1 , Δ N 2 = 1.
period ( 1 , 2 ) Δ N 1 = 1 , Δ N 2 = + 1.
period ( e , 1 ) Δ N 1 = + 1 , Δ N 2 = 0 ,
period ( 1 , e ) Δ N 1 = 1 , Δ N 2 = 0.
G 1 = R 1 , G 2 = R 2 , G e = Γ ˜ 1 + Γ ˜ 2 = Γ ˜ ,
H eff = ћ [ i G 1 / 2 + δ 1 0 Ω 1 / 2 0 i G 2 / 2 + δ 2 Ω 2 / 2 Ω 1 / 2 Ω 2 / 2 i G e / 2 ] .
c i j ( τ ) = j | exp ( i H eff τ / ћ ) | i
W i j ( τ ) d τ = G j | c i j ( τ ) | 2 d τ ,
j W i j ( τ ) d τ = 1 for all i ,
π 1 j = δ e j , π 2 j = δ e j , π e j = δ 1 j Γ ˜ 1 Γ ˜ + δ 2 j Γ ˜ 2 Γ ˜ .
j π k j = 1 for all k .
| ψ 1 ( t + τ ) = k c i k ( τ ) | k ( k | c i k ( τ ) | 2 ) 1 / 2 .
Δ N 1 = P ( 2 , 1 ) + P ( e , 1 ) P ( 1 , 2 ) P ( 1 , e ) ,
P ( i , j ) = P ( i ) P ( j / i ) ,
P ( j / i ) = 0 W i j ( τ ) d τ = G j 0 d τ | c i j ( τ ) | 2 ,
j P ( j / i ) = 1 for all i ,
P ( j / i ) G j = P ( i / j ) G i .
P ( i , j ) P ( j , i ) = P ( i ) P ( j ) G j G i .
P ( j ) = i P ( i ) Q ( in : j / in i ) .
Q ( in : j / in : i ) = k P ( k / i ) π k j ,
j Q ( in : j / in : i ) = 1 for all i .
i P ( i ) = 1.
T ( i , j ) = 0 τ W i j ( τ ) d τ 0 W i j ( τ ) d τ = G j 0 τ | c i j ( τ ) | 2 d τ P ( j / i ) ,
T = i , j T ( i , j ) P ( i , j ) ,
Π ( i , j ) = P ( i , j ) T ( i , j ) T .
| ψ i ( τ ) = k ( c i k ( τ ) | k [ N i ( τ ) ] 1 / 2 ,
N i ( τ ) = k | c i k ( τ ) | 2 = 1 0 τ k W i k ( τ ) d τ .
Π k = i , j P ( i ) T 0 W i j ( τ ) [ 0 τ | k | ψ i ( τ ) | 2 d τ ] d τ = i P ( i ) T 0 | c i , k ( τ ) | 2 d τ
d N 1 d t = n Δ N 1 T = Δ N 1 T = P ( 2 , 1 ) + P ( e , 1 ) P ( 1 , 2 ) P ( 1 , e ) i , j T ( i , j ) P ( i , j ) .
Ω 1 , Ω 2 Γ ˜ ,
R 1 , R 2 Γ 2 = Ω 2 2 / Γ ,
Ω 1 Ω 2 , R 1 , R 2 , Γ 2 .
R 1 , R 2 Γ 2 Γ ˜ .
δ 2 = 0 ,
0 | δ 1 | Γ ,
| ψ i ( τ ) = c i e ( τ ) | e , N 1 , N 2 + c i 1 ( τ ) | g 1 , N 1 + 1 , N 2 + c i 2 ( τ ) | g 2 , N 1 , N 2 + 1 ,
c ˙ i e ( τ ) = Γ ˜ 2 c i e ( τ ) i Ω 1 2 c i 1 ( τ ) i Ω 2 2 c i 2 ( τ ) ,
c ˙ i 1 ( τ ) = i Ω 1 2 c i e ( τ ) ( R 1 2 + i δ 1 ) c i 1 ( τ ) ,
c ˙ i 2 ( τ ) = i Ω 2 2 c i e ( τ ) R 2 2 c i 2 ( τ ) ,
c i k ( τ = 0 ) = δ i k .
c ˙ i e ( τ ) = i Ω 1 Γ ˜ c i 1 ( τ ) i Ω 2 Γ ˜ c i 2 ( τ ) ,
c ˙ i 1 ( τ ) = ( R 1 2 + i δ 1 ) c i 1 ( τ ) Ω 1 Ω 2 2 Γ ˜ c i 2 ( τ ) ,
c ˙ i 2 ( τ ) = Ω 1 Ω 2 2 Γ ˜ c i 1 ( τ ) Γ ˜ 2 2 c i 2 ( τ ) .
P ( 1 / 2 ) = R 1 0 d τ | c 21 ( τ ) | 2 ,
P ( 1 / 2 ) = Ω 1 2 Ω 2 2 1 1 + ( 2 δ 1 / Γ 2 ) 2 .
P ( 2 / 1 ) = R 2 0 d τ | c 12 ( τ ) | 2 .
P ( 1 / 2 ) = R 2 R 1 Ω 1 2 Ω 2 2 1 1 + ( 2 δ 1 / Γ 2 ) 2 .
P ( 1 / e ) = R 1 0 d τ | c e 1 ( τ ) | 2 .
P ( 1 / e ) = ( R 1 / Γ ˜ ) P ( e / 1 ) ,
P ( 1 / e ) P ( e / 1 ) .
P ( e / 1 ) = Γ ˜ 0 d τ | c 1 e ( τ ) | 2 .
c 1 e ( τ ) = Ω 1 Γ ˜ 1 δ 1 + i ( Γ 2 / 2 ) { Γ 2 2 exp ( Γ 2 2 τ ) i δ 1 exp [ ( R 1 2 + i δ 1 ) τ ] } .
P ( e / 1 ) = Ω 1 2 Ω 2 2 ( Γ 2 / 2 ) 2 ( Γ 2 / 2 ) 2 + δ 1 2 [ 1 + 4 δ 1 2 R 1 Γ 2 4 δ 1 2 ( Γ 2 / 2 ) 2 + δ 1 2 ] ,
P ( e / 1 ) = Ω 1 2 Ω 2 2 ,
P ( e / 1 ) = Ω 1 2 Γ ˜ R 1 Ω 1 2 Ω 2 2 Γ 2 R 1 .
P ( 1 ) P ( 2 ) = Γ ˜ 1 Γ ˜ 2 .
Q ( in : j / in : 1 ) = δ e j .
Q ( in : j / in : e ) Γ ˜ 1 Γ ˜ δ 1 j + Γ ˜ 2 Γ ˜ δ 2 j .
Q ( in : j / in : 2 ) Γ ˜ 1 Γ ˜ δ 1 j + Γ ˜ 2 Γ ˜ δ 2 j .
Q ( in : e / in : i ) δ i 1 .
P ( e ) P ( 1 ) .
P ( 1 ) Γ ˜ 1 = P ( 2 ) Γ ˜ 2 = P ( e ) Γ ˜ 1 = 1 2 Γ ˜ 1 + Γ ˜ 2 ,
P ( 1 ) = Γ ˜ 1 2 Γ ˜ 1 + Γ ˜ 2 , P ( 2 ) = Γ ˜ 2 2 Γ ˜ 1 + Γ ˜ 2 , P ( e ) = Γ ˜ 1 2 Γ ˜ 1 + Γ ˜ 2 .
P ( 2 , 1 ) = P ( 2 ) P ( 1 / 2 ) = Ω 1 2 Ω 2 2 1 1 + ( 2 δ 1 / Γ 2 ) 2 Γ ˜ 2 2 Γ ˜ 1 + Γ ˜ 2 ,
P ( 1 , 2 ) = P ( 1 ) P ( 2 / 1 ) = R 2 R 1 Ω 1 2 Ω 2 2 1 1 + ( 2 δ 1 / Γ 2 ) 2 Γ ˜ 1 2 Γ ˜ 1 + Γ ˜ 2 ,
P ( e , 1 ) = P ( e ) P ( 1 / e ) = R 1 Γ ˜ Ω 1 2 Ω 2 2 Γ ˜ 1 2 Γ ˜ 1 + Γ ˜ 2 ,
P ( 1 , e ) = P ( 1 ) P ( e / 1 ) = Ω 1 2 Ω 2 2 Γ ˜ 1 2 Γ ˜ 1 + Γ ˜ 2 ,
P ( e , 1 ) = P ( e ) P ( 1 / e ) = Ω 1 2 Γ ˜ 2 Γ ˜ 1 2 Γ ˜ 1 + Γ ˜ 2 ,
P ( 1 , e ) = P ( 1 ) P ( e / 1 ) = Ω 1 2 Γ ˜ R 1 Γ ˜ 1 2 Γ ˜ 1 + Γ ˜ 2 .
P ( 1 , 2 ) Γ ˜ 1 R 2 = P ( 2 , 1 ) Γ ˜ 2 R 1 .
Δ N 1 = 1 2 Γ ˜ 1 + Γ ˜ 2 Ω 1 2 Ω 2 2 1 R 1 × ( Γ ˜ 2 R 1 + Γ ˜ 1 R 1 R 1 Γ ˜ Γ ˜ 1 R 2 Γ ˜ 1 R 1 ) .
Γ ˜ 2 Γ ˜ 1 > 1 + R 2 R 1 .
Γ ˜ 2 Γ ˜ 1 > R 2 R 1 .
Π 1 = P ( 1 ) T R 1 ; Π 2 = P ( 2 ) T R 2 ; Π e = P ( e ) T Γ ˜ .
k Π k = 1 ,
Π 1 ( Γ ˜ 1 / R 1 ) = Π 2 ( Γ ˜ 2 / R 2 ) = Π e ( Γ ˜ 1 / Γ ˜ ) = 1 ( Γ ˜ 1 / R 1 ) + ( Γ ˜ 2 / R 2 ) + ( Γ ˜ 1 / Γ ˜ ) .
Π 1 = P ( 1 ) T R 1 , Π 2 = P ( 2 ) T Γ 2 , Π e = P ( e ) T Γ ˜ .
Π 1 ( Γ ˜ 1 / R 1 ) = Π 2 ( Γ ˜ 2 / Γ 2 ) = Π e ( Γ ˜ 1 / Γ ˜ ) = 1 ( Γ ˜ 1 / R 1 ) + ( Γ ˜ 2 / Γ 2 ) + ( Γ ˜ 1 / Γ ˜ ) .
T = P ( 1 ) Π 1 R 1 = P ( 2 ) Π 1 Γ 2 = P ( e ) Π e Γ ˜ = Γ ˜ 1 2 Γ ˜ 1 + Γ ˜ 2 ( 1 R 1 + 1 Γ ˜ ) + Γ ˜ 2 2 Γ ˜ 1 + Γ ˜ 2 1 Γ 2 .
Γ 2 R 1 > Γ 2 Γ 1 .
R 2 R 1 > Γ 2 Γ 1 .
Γ 2 R 1 > Γ 2 Γ 1 > 1 + R 2 R 1 .
P ( 2 , 1 ) P ( 1 , 2 ) = R 2 R 1 Γ ˜ 1 Γ ˜ 2 = ( rate out of 2 ) × ( rate in 1 ) ( rate out of 1 ) × ( rate in 2 ) .
R i = Γ i exp ( ћ ω e i / k B Θ i ) 1 ,
[ P ( e / 1 ) ] na ï ve = Ω 1 2 Γ ˜ 1 R 1 = Ω 1 2 Ω 2 2 Γ 2 R 1 ,
| ψ NC = ( Ω 2 / Ω ) | g 1 , N 1 + 1 , N 2 ( Ω 1 / Ω ) | g 2 , N 1 , N 2 + 1 ,
| ψ C = ( Ω 1 / Ω ) | g 1 , N 1 + 1 , N 2 + ( Ω 2 / Ω ) | g 2 , N 1 , N 2 + 1 ,
Ω = ( Ω 1 2 + Ω 2 2 ) 1 / 2 Ω 2 ,
e , N 1 , N 2 | V AL | ψ NC = 0.
e , N 1 , N 2 | V AL | ψ C = ћ Ω / 2 ћ Ω 2 / 2 .
| g 1 , N 1 + 1 , N 2 = ( Ω 2 / Ω ) | ψ NC + ( Ω 1 / Ω ) | ψ C .
| g 2 , N 1 , N 2 + 1 = ( Ω 2 / Ω ) | ψ NC ,
j 0 W i j ( τ ) d τ = j G j 0 | c i j ( τ ) | 2 d τ = 1 for all i .
j | H eff ћ | k = Ω j k 2 = k | H eff ћ | j * = Ω k j * 2 .
c ˙ i j ( τ ) = ( G j 2 + i δ j ) c i j i k Ω j k 2 c i k with j = 1 , 2 , e ,
j | c i j ( 0 ) | 2 = 1.
d d τ j | c i j ( τ ) | 2 = j G j | c i j ( τ ) | 2 i j , k [ c i j * ( τ ) Ω j k 2 c i k ( τ ) c i k * ( τ ) Ω k j * 2 c i j ( τ ) ] = j G j | c i j ( τ ) | 2 .
j | c i j ( τ ) | 2 = j | c i j ( 0 ) | 2 j G j 0 τ | c i j ( τ ) | 2 d τ = 1 j G j 0 τ | c i j ( τ ) | 2 d τ ,
| ψ ( t ) = j c i j ( t ) | j ,
c i j ( t = 0 ) = δ i j
( H ) * = ( H ) ,
c i j ( t ) = c j i ( t ) .
j | exp ( i H t / ћ ) | i = i | exp ( i H t / ћ ) | j .
i | H eff ћ | i = δ i i ћ G i 2
H | ϕ α = E α | ϕ α .
| ϕ α = i c α i | i .
| ϕ ¯ α = i ( c α i ) * | i .
H * | ϕ ¯ α = E α * | ϕ ¯ α .
ϕ ¯ α | H = E α ϕ ¯ α | .
ϕ ¯ α | ϕ α = 1.
ϕ ¯ β | ϕ α = δ α β .
ϕ ¯ β | H | ϕ α = E α ϕ ¯ β | ϕ α = E β ϕ ¯ β | ϕ α .
( β | ϕ β ϕ ¯ β | ) | ϕ α = β | ϕ β δ α β = | ϕ α .
β | ϕ β ϕ ¯ β | = I .
H = α E α | ϕ α ϕ ¯ α | ,
exp ( i H t / ћ ) = α exp ( i E α t / ћ ) | ϕ α ϕ ¯ α | .
j | exp ( i H t / ћ ) | i = α exp ( i E α t / ћ ) j | ϕ α ϕ ¯ α | i = α exp ( i E α t / ћ ) c α j c α i = i | exp ( i H t / ћ ) | j ,

Metrics