Abstract

At low temperatures steady-state macroscopic superpositions can be generated in the field of a micromaser pumped by a stream of two-level atoms injected in a coherent superposition of their upper and lower states. The onset of these superpositions resembles a second-order phase transition, with the control parameter being the ratio of the atomic injection rate to the cavity damping rate.

© 1990 Optical Society of America

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References

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  1. See, e.g., A. J. Leggett, Prog. Theor. Phys. Suppl. 69, 80 (1980).
    [CrossRef]
  2. J. M. Martinis, M. H. Devoret, J. Clarke, Phys. Rev. B 35, 4682 (1988).
    [CrossRef]
  3. A. O. Caldeira, A. J. Leggett, Phys. Rev. A 31, 1059 (1985);D. F. Walls, G. J. Milburn, Phys. Rev. A 31, 2403 (1985);C. M. Savage, D. F. Walls, Phys. Rev. A 32, 2316 (1985).
    [CrossRef] [PubMed]
  4. B. Yurke, D. Stoler, Phys. Rev. Lett. 57, 13 (1986).
    [CrossRef] [PubMed]
  5. D. Meschede, H. Walther, N. Klein, Phys. Rev. Lett. 54, 551 (1985).
    [CrossRef] [PubMed]
  6. J. J. Slosser, P. Meystre, S. L. Braunstein, Phys. Rev. Lett. 63, 934 (1989).
    [CrossRef] [PubMed]
  7. P. Filipowicz, J. Javanainen, P. Meystre, J. Opt. Soc. Am. B 3, 906 (1986).
    [CrossRef]
  8. P. Meystre, G. Rempe, H. Walther, Opt. Lett. 13, 1078 (1988).
    [CrossRef] [PubMed]
  9. See, e.g., W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973).
  10. P. Filipowicz, J. Javanainen, P. Meystre, Phys. Rev. A 34, 3077 (1986).
    [CrossRef] [PubMed]
  11. For the two-photon maser case, see L. Davidovich, J. M. Raimond, M. Brune, S. Haroche, Phys. Rev. A 36, 3771 (1987).
    [CrossRef] [PubMed]
  12. E. T. Jaynes, F. W. Cummings, Proc. IEEE 51, 89 (1963).
    [CrossRef]
  13. See also J. Bergou, L. Davidovich, M. Orszag, C. Ben-kert, M. Hillery, M. O. Scully, Opt. Commun. 72, 82 (1989).
    [CrossRef]
  14. J. J. Slosser, P. Meystre, “Tangent and cotangent states of the electromagnetic field,” submitted to Phys. Rev. A.

1989

J. J. Slosser, P. Meystre, S. L. Braunstein, Phys. Rev. Lett. 63, 934 (1989).
[CrossRef] [PubMed]

See also J. Bergou, L. Davidovich, M. Orszag, C. Ben-kert, M. Hillery, M. O. Scully, Opt. Commun. 72, 82 (1989).
[CrossRef]

1988

P. Meystre, G. Rempe, H. Walther, Opt. Lett. 13, 1078 (1988).
[CrossRef] [PubMed]

J. M. Martinis, M. H. Devoret, J. Clarke, Phys. Rev. B 35, 4682 (1988).
[CrossRef]

1987

For the two-photon maser case, see L. Davidovich, J. M. Raimond, M. Brune, S. Haroche, Phys. Rev. A 36, 3771 (1987).
[CrossRef] [PubMed]

1986

B. Yurke, D. Stoler, Phys. Rev. Lett. 57, 13 (1986).
[CrossRef] [PubMed]

P. Filipowicz, J. Javanainen, P. Meystre, Phys. Rev. A 34, 3077 (1986).
[CrossRef] [PubMed]

P. Filipowicz, J. Javanainen, P. Meystre, J. Opt. Soc. Am. B 3, 906 (1986).
[CrossRef]

1985

D. Meschede, H. Walther, N. Klein, Phys. Rev. Lett. 54, 551 (1985).
[CrossRef] [PubMed]

A. O. Caldeira, A. J. Leggett, Phys. Rev. A 31, 1059 (1985);D. F. Walls, G. J. Milburn, Phys. Rev. A 31, 2403 (1985);C. M. Savage, D. F. Walls, Phys. Rev. A 32, 2316 (1985).
[CrossRef] [PubMed]

1980

See, e.g., A. J. Leggett, Prog. Theor. Phys. Suppl. 69, 80 (1980).
[CrossRef]

1963

E. T. Jaynes, F. W. Cummings, Proc. IEEE 51, 89 (1963).
[CrossRef]

Ben-kert, C.

See also J. Bergou, L. Davidovich, M. Orszag, C. Ben-kert, M. Hillery, M. O. Scully, Opt. Commun. 72, 82 (1989).
[CrossRef]

Bergou, J.

See also J. Bergou, L. Davidovich, M. Orszag, C. Ben-kert, M. Hillery, M. O. Scully, Opt. Commun. 72, 82 (1989).
[CrossRef]

Braunstein, S. L.

J. J. Slosser, P. Meystre, S. L. Braunstein, Phys. Rev. Lett. 63, 934 (1989).
[CrossRef] [PubMed]

Brune, M.

For the two-photon maser case, see L. Davidovich, J. M. Raimond, M. Brune, S. Haroche, Phys. Rev. A 36, 3771 (1987).
[CrossRef] [PubMed]

Caldeira, A. O.

A. O. Caldeira, A. J. Leggett, Phys. Rev. A 31, 1059 (1985);D. F. Walls, G. J. Milburn, Phys. Rev. A 31, 2403 (1985);C. M. Savage, D. F. Walls, Phys. Rev. A 32, 2316 (1985).
[CrossRef] [PubMed]

Clarke, J.

J. M. Martinis, M. H. Devoret, J. Clarke, Phys. Rev. B 35, 4682 (1988).
[CrossRef]

Cummings, F. W.

E. T. Jaynes, F. W. Cummings, Proc. IEEE 51, 89 (1963).
[CrossRef]

Davidovich, L.

See also J. Bergou, L. Davidovich, M. Orszag, C. Ben-kert, M. Hillery, M. O. Scully, Opt. Commun. 72, 82 (1989).
[CrossRef]

For the two-photon maser case, see L. Davidovich, J. M. Raimond, M. Brune, S. Haroche, Phys. Rev. A 36, 3771 (1987).
[CrossRef] [PubMed]

Devoret, M. H.

J. M. Martinis, M. H. Devoret, J. Clarke, Phys. Rev. B 35, 4682 (1988).
[CrossRef]

Filipowicz, P.

P. Filipowicz, J. Javanainen, P. Meystre, Phys. Rev. A 34, 3077 (1986).
[CrossRef] [PubMed]

P. Filipowicz, J. Javanainen, P. Meystre, J. Opt. Soc. Am. B 3, 906 (1986).
[CrossRef]

Haroche, S.

For the two-photon maser case, see L. Davidovich, J. M. Raimond, M. Brune, S. Haroche, Phys. Rev. A 36, 3771 (1987).
[CrossRef] [PubMed]

Hillery, M.

See also J. Bergou, L. Davidovich, M. Orszag, C. Ben-kert, M. Hillery, M. O. Scully, Opt. Commun. 72, 82 (1989).
[CrossRef]

Javanainen, J.

P. Filipowicz, J. Javanainen, P. Meystre, Phys. Rev. A 34, 3077 (1986).
[CrossRef] [PubMed]

P. Filipowicz, J. Javanainen, P. Meystre, J. Opt. Soc. Am. B 3, 906 (1986).
[CrossRef]

Jaynes, E. T.

E. T. Jaynes, F. W. Cummings, Proc. IEEE 51, 89 (1963).
[CrossRef]

Klein, N.

D. Meschede, H. Walther, N. Klein, Phys. Rev. Lett. 54, 551 (1985).
[CrossRef] [PubMed]

Leggett, A. J.

A. O. Caldeira, A. J. Leggett, Phys. Rev. A 31, 1059 (1985);D. F. Walls, G. J. Milburn, Phys. Rev. A 31, 2403 (1985);C. M. Savage, D. F. Walls, Phys. Rev. A 32, 2316 (1985).
[CrossRef] [PubMed]

See, e.g., A. J. Leggett, Prog. Theor. Phys. Suppl. 69, 80 (1980).
[CrossRef]

Louisell, W. H.

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

Martinis, J. M.

J. M. Martinis, M. H. Devoret, J. Clarke, Phys. Rev. B 35, 4682 (1988).
[CrossRef]

Meschede, D.

D. Meschede, H. Walther, N. Klein, Phys. Rev. Lett. 54, 551 (1985).
[CrossRef] [PubMed]

Meystre, P.

J. J. Slosser, P. Meystre, S. L. Braunstein, Phys. Rev. Lett. 63, 934 (1989).
[CrossRef] [PubMed]

P. Meystre, G. Rempe, H. Walther, Opt. Lett. 13, 1078 (1988).
[CrossRef] [PubMed]

P. Filipowicz, J. Javanainen, P. Meystre, J. Opt. Soc. Am. B 3, 906 (1986).
[CrossRef]

P. Filipowicz, J. Javanainen, P. Meystre, Phys. Rev. A 34, 3077 (1986).
[CrossRef] [PubMed]

J. J. Slosser, P. Meystre, “Tangent and cotangent states of the electromagnetic field,” submitted to Phys. Rev. A.

Orszag, M.

See also J. Bergou, L. Davidovich, M. Orszag, C. Ben-kert, M. Hillery, M. O. Scully, Opt. Commun. 72, 82 (1989).
[CrossRef]

Raimond, J. M.

For the two-photon maser case, see L. Davidovich, J. M. Raimond, M. Brune, S. Haroche, Phys. Rev. A 36, 3771 (1987).
[CrossRef] [PubMed]

Rempe, G.

Scully, M. O.

See also J. Bergou, L. Davidovich, M. Orszag, C. Ben-kert, M. Hillery, M. O. Scully, Opt. Commun. 72, 82 (1989).
[CrossRef]

Slosser, J. J.

J. J. Slosser, P. Meystre, S. L. Braunstein, Phys. Rev. Lett. 63, 934 (1989).
[CrossRef] [PubMed]

J. J. Slosser, P. Meystre, “Tangent and cotangent states of the electromagnetic field,” submitted to Phys. Rev. A.

Stoler, D.

B. Yurke, D. Stoler, Phys. Rev. Lett. 57, 13 (1986).
[CrossRef] [PubMed]

Walther, H.

P. Meystre, G. Rempe, H. Walther, Opt. Lett. 13, 1078 (1988).
[CrossRef] [PubMed]

D. Meschede, H. Walther, N. Klein, Phys. Rev. Lett. 54, 551 (1985).
[CrossRef] [PubMed]

Yurke, B.

B. Yurke, D. Stoler, Phys. Rev. Lett. 57, 13 (1986).
[CrossRef] [PubMed]

J. Opt. Soc. Am. B

Opt. Commun.

See also J. Bergou, L. Davidovich, M. Orszag, C. Ben-kert, M. Hillery, M. O. Scully, Opt. Commun. 72, 82 (1989).
[CrossRef]

Opt. Lett.

Phys. Rev. A

A. O. Caldeira, A. J. Leggett, Phys. Rev. A 31, 1059 (1985);D. F. Walls, G. J. Milburn, Phys. Rev. A 31, 2403 (1985);C. M. Savage, D. F. Walls, Phys. Rev. A 32, 2316 (1985).
[CrossRef] [PubMed]

P. Filipowicz, J. Javanainen, P. Meystre, Phys. Rev. A 34, 3077 (1986).
[CrossRef] [PubMed]

For the two-photon maser case, see L. Davidovich, J. M. Raimond, M. Brune, S. Haroche, Phys. Rev. A 36, 3771 (1987).
[CrossRef] [PubMed]

Phys. Rev. B

J. M. Martinis, M. H. Devoret, J. Clarke, Phys. Rev. B 35, 4682 (1988).
[CrossRef]

Phys. Rev. Lett.

B. Yurke, D. Stoler, Phys. Rev. Lett. 57, 13 (1986).
[CrossRef] [PubMed]

D. Meschede, H. Walther, N. Klein, Phys. Rev. Lett. 54, 551 (1985).
[CrossRef] [PubMed]

J. J. Slosser, P. Meystre, S. L. Braunstein, Phys. Rev. Lett. 63, 934 (1989).
[CrossRef] [PubMed]

Proc. IEEE

E. T. Jaynes, F. W. Cummings, Proc. IEEE 51, 89 (1963).
[CrossRef]

Prog. Theor. Phys. Suppl.

See, e.g., A. J. Leggett, Prog. Theor. Phys. Suppl. 69, 80 (1980).
[CrossRef]

Other

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

J. J. Slosser, P. Meystre, “Tangent and cotangent states of the electromagnetic field,” submitted to Phys. Rev. A.

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Figures (3)

Fig. 1
Fig. 1

Log–log plot of the entropy S of the final micromaser field state as a function of Nex. The parameters are N3 = 10, α = 0.51 (the solid curve) and N3 = 15, α = 0.53 (the dashed curve).

Fig. 2
Fig. 2

Mean photon number 〈n〉 (the solid curve) and the normalized variance σ2 (the dashed curve) as a function of log10 Nex for Nex = 15, α = 0.53.

Fig. 3
Fig. 3

Modulii of the field density matrix elements 〈n|ρ|m〉 for (a) Nex = 15, (b) Nex = 106, and (c) Nex = 109. Here N3 = 15, α = 0.53.

Equations (9)

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

| ψ at = α | a + β | b .
( N q + 1 ) 1 / 2 κ τ = q π , q an integer ,
n ϕ n | n ( α | a + β | b ) n ϕ n { α cos [ ( n + 1 ) 1 / 2 κ τ ] | n + i β sin [ ( n ) 1 / 2 κ τ ] | n 1 } | a + n ϕ n { β cos [ ( n ) 1 / 2 κ τ ] | n + i α sin [ ( n + 1 ) 1 / 2 κ τ ] | n + 1 } | b .
ρ ˙ ( t ) = γ 2 ( n ̅ + 1 ) [ a a ρ ( t ) a ρ ( t ) a ] γ 2 n ̅ [ ρ ( t ) a a a ρ ( t ) a ] + adj L ρ ,
ρ ˙ n ( t ) = γ ( n ̅ + 1 ) [ ( n + 1 ) p n + 1 n p n ] + γ n ̅ [ n p n 1 ( n + 1 ) p n ] .
p ˙ n ( t ) = γ [ ( n + 1 ) p n + 1 n p n ] ,
ρ t = L ρ + R [ F ( τ ) I ] ρ ,
F ( τ ) ρ ( t ) T r at [ U ( τ ) ρ at field U ( τ ) ] ,
ρ ss = ( R L ) 1 R F ( τ ) ρ ss .

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