Optical Bistability pp 367-404 | Cite as

# The Driven Dicke Model and its Macroscopic Extension: Bistability or Bifurcation?

## Abstract

We are concerned with the relation between recent work on the “driven Dicke model” of N two-level atoms, on the same site, driven by a c. w. laser field Ω, and a corresponding theory for the more realistic macroscopically extended system. We review the results on the driven Dicke model: two different decorrelation schemes yield different results; in the steady state at resonance a semi classical approximation without damping best approximates the exact solution of the quantum model also described. The exact solution of the quantum model does not display normal optical bistability (OB):calculation of \({g^{(2)}}(O) = {G^{(2)}}(O)/\left\{ {{G^{(1)}}{{(O)}^2}} \right\}\) (where G^{(n)}(0) = < (S_{+})^{n}(S_{-})^{n} > and S_{±} are collective spin operators) shows g^{(2)}(0)→ 1.2 and there is a simple bifurcation point at θ {∞ limΩN^{−1}, N→∞} = 1. The inversion r_{3} plays the role of the order parameter: r_{3}= ± 1/2 (l− θ^{2}) 1/2, θ < 1; =0, θ > 1. There is a second-order type phase transition, and by moving off-resonance and relating to the decor- related model, we are able to identify one set of equivalent thermodynamic parameters for the model. We find “critical exponents” α = 1/2, β = 1/2, γ = 1.5 and α + 2β + γ > 2 in this manner. Results are compared with the operator theory for the extended system also presented (unlike the Dicke model this model does not have total spin as a constant of the motion). Decorrelation of operator products with self-correlation (radiation damping) leads of course to the c-number theory of cusp catastrophe OB. An operator theory involving a natural power dependent refractive index is sketched and we believe that it is this which should appear as the parameter in the usual treatment of the Fabry-Perot interferometer. But, alternatively, by extracting a single mode theory in the “mean-field” Approximation, we regain both the Bloch equations and the master equation of the driven Dicke model. The spectra and correlation functions shown in Figs. 1–4 are calculated from these in a decorrelation approximation which retains single-particle damping and which differs from the exact solution of the master equation. The hierarchy of different models relates to the realistic extended system model in ways very similar to those of a similar hierarchy in the theory of super fluorescence. It is concluded that mean field theory maltreats the analysis. However, it is expected that the decorrelation scheme adopted for the spectra we have calculated will be adequate to describe their essential features.

## Keywords

Master Equation Extended System Bloch Equation Optical Bistability Exact Quantum## Preview

Unable to display preview. Download preview PDF.

## References

- 1.
- 2.C. R. Stroud, Jr., and E. T. Jaynes, Phys. Rev. A1, 106 (1970).ADSGoogle Scholar
- 3.C. R. Stroud, Jr., Phys. Rev. A3, 1044 (1971).ADSGoogle Scholar
- 4.S. S. Hassan and R. K. Bullough, J. Phys. B8, L147 (1975).ADSGoogle Scholar
- 5.G. S. Agarwal, Springer Tracts in Modern Physics, 70, (Springer-Verlag: Heidelberg, N.Y., 1974); M. E. Smithers and H. S. Freedhoff, J. Phys. B7, L432 (1974); S. Swain, J. Phys. B8, L437 (1975); H. J. Carmichael and D. F. Walls, J. Phys. B9, 1199 (1976); H. J. Kimble and L. Mandel, Phys. Rev. A13, 2123 (1976); C. Cohen-Tannoudji and S. Reynaud, J. Phys. B10, 345 (1977).Google Scholar
- 6.Proc. 3rd Conf. on Coherence and Quantum Optics, Rochester, (Plenum, N.Y., 1972), Eds: L. Mandel and E, Wolf — see articles by: R. K. Bullough, pp. 121–56; J. R. Ackerhalt, J. H. Eberly and P. L. Knight, pp. 635–44.Google Scholar
- 7.R. Saunders, R. K. Bullough and F. Ahmad, J. Phys. A8, 759 (1975).ADSGoogle Scholar
- 8.J. R. Ackerhalt and J. H. Eberly, Phys. Rev. D10, 3350 (1974).ADSGoogle Scholar
- 9.“IV Rochester Conf. on Coherence and Quantum Optics,” (Plenum, N. Y., 1977 ), Eds: L. Mandel and E. Wolf.Google Scholar
- 10.F. Schuda, C. R. Stroud, Jr. and M. Hercher, J. Phys. B7, L198 (1974); also, W. Hartig, W. Rasmurren, R. Schieder and H. Walther, Z. Physik A278, 205 (1976); F. Y. Wu, R. E. Grove and S. Ezekiel, Phys. Rev. Lett. 35, 1426 (1975).ADSGoogle Scholar
- 11.R. H. Dicke, Phys. Rev. 93, 99 (1954).ADSMATHCrossRefGoogle Scholar
- 12.a) R. Bonifacio, P. Schwendimann and F. Haake, Phys. Rev. A4, 302, 854 (1971); (b) F. Haake and R. Glauber, Phys. Rev. A5, 1457 (1972); 13, 357 (1976); 20, 2047 (1979); “Cooperative Phenomena” (North-Holland, Amsterdam, 1974), Ed. H. Haken, p. 71; M. F. Schuurmans, D. Polder and Q. Vrehen, Phys. Rev. A19, 1192 (1979); (c) G. Banfi and R. Bonifacio, Phys. Rev. Lett. 33, 1259 (1974); Phys. Rev. A12, 2068 (1975); (d) R. Bonifacio and L. Lugiato, Phys. Rev. A11, 1507 (1975); 12, 587 (1975); (e) R. Saunders, S. S. Hassan and R. K. Bullough, J. Phys. A9, 1725 (1976); R. K. Bullough Et al., in Ref. 9, p. 263; J. MacGillivray and M. S. Feld, Phys. Rev. A14, 1169 (1976); (f) “Cooperative effects in matter and radiation,” (Plenum, N. Y., 1977 ), Eds. C. M. Bowden, D. W. Howgate and H. R. Robl — see especially articles by: J. MacGillivray and M. S. Feld, pp. 1–14; R. Bonifacio Et al., pp. 193–208; R. Saunders and R. K. Bullough, pp. 209–256.ADSGoogle Scholar
- 13.a) N. Skribanowitz, I. P. Herman, J. C. MacGillivray and M.S. Feld, Phys. Rev. Lett. 30, 309 (1973); (b) M. Gross, C. Fabre, P. Pillet and S. Haroche, Phys. Rev. Lett. 36, 1035 (1976); H. M. Gibbs, in Ref. 12f, pp. 61-78; Q.H.F. Vrehen, in Ref. 12f, pp. 79–100.ADSCrossRefGoogle Scholar
- 14.G. S. Agarwal, A. C. Brown, L. M. Narducci and G. Vetri, Phys. Rev. A15, 1613 (1977); G. S. Agarwal, D. H. Feng, L. M. Narducci, R. Gilmore and R. A. Tuft, Phys. Rev. A20, 2040 (1979); G. S. Agarwal, R. Saxena, L. M. Narducci, D. H. Feng and R. Gilmore, Phys. Rev. A21, 257 (1980).ADSGoogle Scholar
- 15.A. S. Amin and J. G. Cordes, Phys. Rev. A18, 1298 (1978); C. Mavroyannis, Phys. Rev. A18, 185 (1978); Opt. Comm. 33, 42 (1980); H. J. Carmichael, Phys. Rev. Lett. 43, 1106 (1979).ADSGoogle Scholar
- 16.H. J. Carmichael and D. F. Walls, J. Phys. B10, L685 (1977).ADSGoogle Scholar
- 17.S. S. Hassan and D. F. Walls, J. Phys. A11, L87 (1978).ADSGoogle Scholar
- 18.C. M. Bowden and C. C. Sung, Phys. Rev. A19, 2392 (1979); also C. M. Bowden in this volume.ADSGoogle Scholar
- 19.I. R. Senitzky, Phys. Rev. Lett. 40, 1334 (1978); Phys. Rev. A6, 1171, 1175 (1972).ADSCrossRefGoogle Scholar
- 20.P. D. Drummond and S. S. Hassan, Phys. Rev. A22, 662, (1980).ADSGoogle Scholar
- 21.L. M. Narducci, D. H. Feng, R. Gilmore and G. S. Agarwal, Phys. Rev. A18, 1571 (1978).ADSGoogle Scholar
- 22.P. D. Drummond and H. J. Carmichael, Opt. Comm. 27, 160 (1978); D. F. Walls, P. D. Drummond, S. S. Hassan and H. J. Carmichael, Prog. Theoret. Phys. Suppl. 64, 307 (1978).ADSCrossRefGoogle Scholar
- 23.R. R. Puri and S. V. Lawande, Phys. Lett. 72A, 200 (1979); Physica A 101, 599 (1980).MathSciNetADSGoogle Scholar
- 24.S. S. Hassan, R. K. Bullough, R. R. Puri and S. V. Lawande, Physica A (in press); P. D. Drummond also obtained the value g
^{(2)}(0) = 1.2 using the solution in Ref. 23 (preprint).Google Scholar - 25.R. R. Puri, S. V. Lawande and S. S. Hassan, Opt. Comm. (to appear).Google Scholar
- 26.S. Ja. Kilin, J. Applied Spect. (USSR) 28, 255 (1978).Google Scholar
- 27.S. S. Hassan Et al., (in preparation).Google Scholar
- 28.R. Bonifacio and L. Lugiato, Opt. Comm. 19, 172 (1976); Phys. Rev. A18, 1129 (1978).ADSCrossRefGoogle Scholar
- 29.G. S. Agarwal, L. M. Narducci, D. H. Feng and R. Gilmore in Ref. 9, pp. 281–292; Phys. Rev. A18, 620 (1978). Notice that “mean field” arises in two distinct contexts in this paper: one is the “mean field” theory of Ref. 28 in which all atomic operator densities are averaged on the space variable (in the context of §III operate with V−1 ∫ dx) with the prescription that an operator or c-number product is replaced by the product of averages (on A(x) B(x) form V −2 ∫ dx dx’ A(x) B(x’)). The other usage arises in the theory of phase transitions and is essentially a decorrelation approximation of Hartree type. It is the first usage which is referred to in the discussion of the work of Ref, 18 (§11.C); but it is the second one which is referred to at the beginning of §II.E. We shall use double quotes to indicate the first usage throughout the paper.Google Scholar
- 30.S. S. Hassan, P. D. Drummond and D. F, Walls, Opt. Comm. 27, 480 (1978); R. Bonifacio and L. Lugiato, Lett. Nuovo. Cim. 21, 517 (1978).ADSCrossRefGoogle Scholar
- 31.H. M. Gibbs, S. L. McCall and T. N. Venkatesan, Phys. Rev. Lett. 36, 1135 (1976); also see S. L. McCall, Phys. Rev. A9, 1515 (1974).ADSCrossRefGoogle Scholar
- 32.B. R. Mollow, Phys. Rev. A5, 2217 (1972); F. Y. Wu, S. Ezekiel, M. Ducloy and B. R. Mollow, Phys. Rev. Lett. 38, 1077 (1977).ADSGoogle Scholar
- 33.D. A. Miller and S. D. Smith, Opt, Comm. 31, 101 (1979).ADSCrossRefGoogle Scholar
- 34.G. S. Agarwal, L. M.. Narducci, D. H. Feng and R. Gilmore, Phys. Rev. Lett. 42, 1260 (1979); 43, 238 (1979).ADSCrossRefGoogle Scholar
- 35.V. Degiorgio, Physics Today, 29, (October, 1976); N. Corti and V. Degiorgio, Phys. Rev. Lett. 36, 1173 (1976).Google Scholar
- 36.S. S. Hassan, Ph.D. Thesis (U. of Manchester, 1976 ); R. Saunders, Ph.D. Thesis (U. of Manchester, 1973 ).Google Scholar
- 37.R. K. Bullough, J. Phys. A1, 409 (1968); 2, 477 (1969); 3, 708, 726, 751 (1970); R. K. Bullough, A. S. F. Obada, B. V. Thompson and F. Hynne, Chem. Phys. Lett. 2, 293 (1968); R. K. Bullough and F. Hynne, Chem. Phys. Lett. 2, 307 (1968); F. Hynne and R. K. Bullough, J. Phys. A5, 1272 (1972); and “The Scattering of Light ” (to be published, 1980 ).Google Scholar
- 38.S. S. Hassan and R. K. Bullough (to be published); also see Phil. Trans. Roy. Soc. London A293, 232 (1979).Google Scholar
- 39.E. Abraham, R. K. Bullough and S. S. Hassan, Opt. Comm. 28, 109 (1979); 33, 93 (1980); E. Abraham and S. S. Hassan, Opt. Comm. (to appear); P. Meystre, Opt. Comm. 26, 277 (1978); R. Bonifacio and L. Lugiato, Lett. Nuovo Cim. 21, 505 (1978); J. A. Herman, Opt. Acta 27, 159 (1980).ADSCrossRefGoogle Scholar
- 40.R. Saunders and R. K. Bullough, J. Phys. A6, 1348 (1973).ADSGoogle Scholar
- 41.M. Lax, Phys. Rev. 157, 213 (1967).MathSciNetADSCrossRefGoogle Scholar
- 42.G. S. Agarwal, L. M. Narducci, R. Gilmore and D. H. Feng, Phys. Rev. A18, 620 (1978); 20, 545 (1979); L. A. Lugiato, Nuovo Cim. B50, 89 (1978), and references therein; G. S. Agarwal and S. P. Tewari, Phys. Rev. A21, 1638 (1980); S. P. Tewari, Opt. Comm. 34, 273 (1980).ADSGoogle Scholar
- 43.H. M. Gibbs, Q. H. F. Vrehen and H. M. Hikspoors, Phys. Rev. Lett. 39, 547 (1977); and in Ref. 12f.ADSCrossRefGoogle Scholar