Abstract

A photon blackbody field in Kerr nonlinear crystal is a squeezed thermal radiation state in which there is a new kind of quasi particle, the nonpolariton. A nonpolariton is a condensate of virtual nonpolar phonons, with a bare photon acting as the nucleus of condensation. The propagation velocity of nonpolaritons is a monotonically increasing function of temperature, and the noise of one quadrature phase in the squeezed thermal radiation state can be below the noise level in the vacuum state. The photon system undergoes a second-order phase transition from the normal to the squeezed thermal radiation state.

© 2002 Optical Society of America

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References

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  1. P. D. Drummond, R. M. Shelby, S. R. Friberg, and Y. Yamamoto, “Quantum solitons in optical fibres,” Nature 365, 307–313 (1993), and references therein.
    [CrossRef]
  2. I. H. Deutsch, R. Y. Chiao, and J. C. Garrison, “Two-photon bound states: the diphoton bullet in dispersive self-focusing media,” Phys. Rev. A 47, 3330–3336 (1993).
    [CrossRef] [PubMed]
  3. A. Hasegawa, Optical Solitons in Fibers, 2nd ed. (Springer-Verlag, Berlin, 1990).
  4. Y. Lai and H. A. Haus, “Quantum theory of solitons in optical fibers. I. Time-dependent Hartree approximation,” Phys. Rev. A 40, 844–853 (1989).
    [CrossRef] [PubMed]
  5. Y. Lai and H. A. Haus, “Quantum theory of solitons in optical fibers. II. Exact solution,” Phys. Rev. A 40, 854–866 (1989).
    [CrossRef] [PubMed]
  6. F. X. Kärtner and H. A. Haus, “Quantum-mechanical stability of solitons and the correspondence principle,” Phys. Rev. A 48, 2361–2369 (1993).
    [CrossRef] [PubMed]
  7. Ze Cheng, “Photonic superguiding state in nonlinear polar crystals,” Phys. Rev. A 51, 675–691 (1995).
    [CrossRef] [PubMed]
  8. Z. Cheng and G. Kurizki, “Theory of one-dimensional quantum gap solitons,” Phys. Rev. A 54, 3576–3591 (1996).
    [CrossRef] [PubMed]
  9. W. Hayes and R. Loudon, Scattering of Light by Crystals (Wiley, New York, 1978).
  10. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Chap. 2.
  11. J. Callaway, Quantum Theory of the Solid State, 2nd ed. (Academic, New York, 1991), p. 720.
  12. L. E. Reichl, A Modern Course in Statistical Physics (U. Texas Press, Austin, Tex., 1980).
  13. L. D. Landau and E. M. Lifshitz, Statistical Physics, 3rd ed. (Reed Educational, Oxford, 1980), Part 1, p. 183.
  14. W. Greiner, Quantum Mechanics: An Introduction, 3rd ed. (Springer-Verlag, Berlin, 1994), p. 74.
  15. D. Stoler, “Equivalence classes of minimum uncertainty packets,” Phys. Rev. D 1, 3217–3219 (1970).
    [CrossRef]
  16. D. Stoler, “Equivalence classes of minimum-uncertainty packets. II,” Phys. Rev. D 4, 1925–1926 (1971).
    [CrossRef]
  17. E. Y. C. Lu, “Quantum correlations in two-photon amplification,” Lett. Nuovo Cimento 3, 585–589 (1972).
    [CrossRef]
  18. H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A 13, 2226–2243 (1976).
    [CrossRef]
  19. R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
    [CrossRef] [PubMed]
  20. L.-A. Wu, H. J. Kimble, J. L. Hall, and H.-F. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57, 2520–2523 (1986).
    [CrossRef] [PubMed]
  21. H. J. Kimble and D. F. Walls, eds., feature on squeezed states of the electromagnetic field, J. Opt. Soc. Am. B 4(10), (1987).
  22. R. Loudon and P. L. Knight, eds., special issue on squeezed light, J. Mod. Opt. 34(6/7), (1987).
  23. E. Giacobino and E. C. Fabre, eds., feature on quantum noise reduction in optical systems—experiments, Appl. Phys. B 55(3), (1992).
  24. K. Huang, “On the interaction between the radiation field and ionic crystals,” Proc. R. Soc. London, Ser. A 208, 352–365 (1951).
    [CrossRef]
  25. J. J. Hopfield, “Theory of the contribution of excitons to the complex dielectric constant of crystals,” Phys. Rev. 112, 1555–1567 (1958).
    [CrossRef]
  26. C. H. Henry and J. J. Hopfield, “Raman scattering by polaritons,” Phys. Rev. Lett. 15, 964–966 (1965).
    [CrossRef]
  27. M. Artoni and J. L. Birman, “Quantum-optical properties of polariton waves,” Phys. Rev. B 44, 3736–3756 (1991).
    [CrossRef]
  28. L. N. Cooper, “Bound electron pairs in a degenerate Fermi gas,” Phys. Rev. 104, 1189–1190 (1956).
    [CrossRef]
  29. J. Bardeen, L. N. Cooper, and J. R. Schrieffer, “Theory of superconductivity,” Phys. Rev. 108, 1175–1204 (1957).
    [CrossRef]

1996

Z. Cheng and G. Kurizki, “Theory of one-dimensional quantum gap solitons,” Phys. Rev. A 54, 3576–3591 (1996).
[CrossRef] [PubMed]

1995

Ze Cheng, “Photonic superguiding state in nonlinear polar crystals,” Phys. Rev. A 51, 675–691 (1995).
[CrossRef] [PubMed]

1993

F. X. Kärtner and H. A. Haus, “Quantum-mechanical stability of solitons and the correspondence principle,” Phys. Rev. A 48, 2361–2369 (1993).
[CrossRef] [PubMed]

P. D. Drummond, R. M. Shelby, S. R. Friberg, and Y. Yamamoto, “Quantum solitons in optical fibres,” Nature 365, 307–313 (1993), and references therein.
[CrossRef]

I. H. Deutsch, R. Y. Chiao, and J. C. Garrison, “Two-photon bound states: the diphoton bullet in dispersive self-focusing media,” Phys. Rev. A 47, 3330–3336 (1993).
[CrossRef] [PubMed]

1991

M. Artoni and J. L. Birman, “Quantum-optical properties of polariton waves,” Phys. Rev. B 44, 3736–3756 (1991).
[CrossRef]

1989

Y. Lai and H. A. Haus, “Quantum theory of solitons in optical fibers. I. Time-dependent Hartree approximation,” Phys. Rev. A 40, 844–853 (1989).
[CrossRef] [PubMed]

Y. Lai and H. A. Haus, “Quantum theory of solitons in optical fibers. II. Exact solution,” Phys. Rev. A 40, 854–866 (1989).
[CrossRef] [PubMed]

1986

L.-A. Wu, H. J. Kimble, J. L. Hall, and H.-F. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57, 2520–2523 (1986).
[CrossRef] [PubMed]

1985

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
[CrossRef] [PubMed]

1976

H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A 13, 2226–2243 (1976).
[CrossRef]

1972

E. Y. C. Lu, “Quantum correlations in two-photon amplification,” Lett. Nuovo Cimento 3, 585–589 (1972).
[CrossRef]

1971

D. Stoler, “Equivalence classes of minimum-uncertainty packets. II,” Phys. Rev. D 4, 1925–1926 (1971).
[CrossRef]

1970

D. Stoler, “Equivalence classes of minimum uncertainty packets,” Phys. Rev. D 1, 3217–3219 (1970).
[CrossRef]

1965

C. H. Henry and J. J. Hopfield, “Raman scattering by polaritons,” Phys. Rev. Lett. 15, 964–966 (1965).
[CrossRef]

1958

J. J. Hopfield, “Theory of the contribution of excitons to the complex dielectric constant of crystals,” Phys. Rev. 112, 1555–1567 (1958).
[CrossRef]

1957

J. Bardeen, L. N. Cooper, and J. R. Schrieffer, “Theory of superconductivity,” Phys. Rev. 108, 1175–1204 (1957).
[CrossRef]

1956

L. N. Cooper, “Bound electron pairs in a degenerate Fermi gas,” Phys. Rev. 104, 1189–1190 (1956).
[CrossRef]

1951

K. Huang, “On the interaction between the radiation field and ionic crystals,” Proc. R. Soc. London, Ser. A 208, 352–365 (1951).
[CrossRef]

Artoni, M.

M. Artoni and J. L. Birman, “Quantum-optical properties of polariton waves,” Phys. Rev. B 44, 3736–3756 (1991).
[CrossRef]

Bardeen, J.

J. Bardeen, L. N. Cooper, and J. R. Schrieffer, “Theory of superconductivity,” Phys. Rev. 108, 1175–1204 (1957).
[CrossRef]

Birman, J. L.

M. Artoni and J. L. Birman, “Quantum-optical properties of polariton waves,” Phys. Rev. B 44, 3736–3756 (1991).
[CrossRef]

Cheng, Z.

Z. Cheng and G. Kurizki, “Theory of one-dimensional quantum gap solitons,” Phys. Rev. A 54, 3576–3591 (1996).
[CrossRef] [PubMed]

Cheng, Ze

Ze Cheng, “Photonic superguiding state in nonlinear polar crystals,” Phys. Rev. A 51, 675–691 (1995).
[CrossRef] [PubMed]

Chiao, R. Y.

I. H. Deutsch, R. Y. Chiao, and J. C. Garrison, “Two-photon bound states: the diphoton bullet in dispersive self-focusing media,” Phys. Rev. A 47, 3330–3336 (1993).
[CrossRef] [PubMed]

Cooper, L. N.

J. Bardeen, L. N. Cooper, and J. R. Schrieffer, “Theory of superconductivity,” Phys. Rev. 108, 1175–1204 (1957).
[CrossRef]

L. N. Cooper, “Bound electron pairs in a degenerate Fermi gas,” Phys. Rev. 104, 1189–1190 (1956).
[CrossRef]

Deutsch, I. H.

I. H. Deutsch, R. Y. Chiao, and J. C. Garrison, “Two-photon bound states: the diphoton bullet in dispersive self-focusing media,” Phys. Rev. A 47, 3330–3336 (1993).
[CrossRef] [PubMed]

Drummond, P. D.

P. D. Drummond, R. M. Shelby, S. R. Friberg, and Y. Yamamoto, “Quantum solitons in optical fibres,” Nature 365, 307–313 (1993), and references therein.
[CrossRef]

Friberg, S. R.

P. D. Drummond, R. M. Shelby, S. R. Friberg, and Y. Yamamoto, “Quantum solitons in optical fibres,” Nature 365, 307–313 (1993), and references therein.
[CrossRef]

Garrison, J. C.

I. H. Deutsch, R. Y. Chiao, and J. C. Garrison, “Two-photon bound states: the diphoton bullet in dispersive self-focusing media,” Phys. Rev. A 47, 3330–3336 (1993).
[CrossRef] [PubMed]

Hall, J. L.

L.-A. Wu, H. J. Kimble, J. L. Hall, and H.-F. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57, 2520–2523 (1986).
[CrossRef] [PubMed]

Haus, H. A.

F. X. Kärtner and H. A. Haus, “Quantum-mechanical stability of solitons and the correspondence principle,” Phys. Rev. A 48, 2361–2369 (1993).
[CrossRef] [PubMed]

Y. Lai and H. A. Haus, “Quantum theory of solitons in optical fibers. II. Exact solution,” Phys. Rev. A 40, 854–866 (1989).
[CrossRef] [PubMed]

Y. Lai and H. A. Haus, “Quantum theory of solitons in optical fibers. I. Time-dependent Hartree approximation,” Phys. Rev. A 40, 844–853 (1989).
[CrossRef] [PubMed]

Henry, C. H.

C. H. Henry and J. J. Hopfield, “Raman scattering by polaritons,” Phys. Rev. Lett. 15, 964–966 (1965).
[CrossRef]

Hollberg, L. W.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
[CrossRef] [PubMed]

Hopfield, J. J.

C. H. Henry and J. J. Hopfield, “Raman scattering by polaritons,” Phys. Rev. Lett. 15, 964–966 (1965).
[CrossRef]

J. J. Hopfield, “Theory of the contribution of excitons to the complex dielectric constant of crystals,” Phys. Rev. 112, 1555–1567 (1958).
[CrossRef]

Huang, K.

K. Huang, “On the interaction between the radiation field and ionic crystals,” Proc. R. Soc. London, Ser. A 208, 352–365 (1951).
[CrossRef]

Kärtner, F. X.

F. X. Kärtner and H. A. Haus, “Quantum-mechanical stability of solitons and the correspondence principle,” Phys. Rev. A 48, 2361–2369 (1993).
[CrossRef] [PubMed]

Kimble, H. J.

L.-A. Wu, H. J. Kimble, J. L. Hall, and H.-F. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57, 2520–2523 (1986).
[CrossRef] [PubMed]

Kurizki, G.

Z. Cheng and G. Kurizki, “Theory of one-dimensional quantum gap solitons,” Phys. Rev. A 54, 3576–3591 (1996).
[CrossRef] [PubMed]

Lai, Y.

Y. Lai and H. A. Haus, “Quantum theory of solitons in optical fibers. II. Exact solution,” Phys. Rev. A 40, 854–866 (1989).
[CrossRef] [PubMed]

Y. Lai and H. A. Haus, “Quantum theory of solitons in optical fibers. I. Time-dependent Hartree approximation,” Phys. Rev. A 40, 844–853 (1989).
[CrossRef] [PubMed]

Lu, E. Y. C.

E. Y. C. Lu, “Quantum correlations in two-photon amplification,” Lett. Nuovo Cimento 3, 585–589 (1972).
[CrossRef]

Mertz, J. C.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
[CrossRef] [PubMed]

Schrieffer, J. R.

J. Bardeen, L. N. Cooper, and J. R. Schrieffer, “Theory of superconductivity,” Phys. Rev. 108, 1175–1204 (1957).
[CrossRef]

Shelby, R. M.

P. D. Drummond, R. M. Shelby, S. R. Friberg, and Y. Yamamoto, “Quantum solitons in optical fibres,” Nature 365, 307–313 (1993), and references therein.
[CrossRef]

Slusher, R. E.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
[CrossRef] [PubMed]

Stoler, D.

D. Stoler, “Equivalence classes of minimum-uncertainty packets. II,” Phys. Rev. D 4, 1925–1926 (1971).
[CrossRef]

D. Stoler, “Equivalence classes of minimum uncertainty packets,” Phys. Rev. D 1, 3217–3219 (1970).
[CrossRef]

Valley, J. F.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
[CrossRef] [PubMed]

Wu, H.-F.

L.-A. Wu, H. J. Kimble, J. L. Hall, and H.-F. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57, 2520–2523 (1986).
[CrossRef] [PubMed]

Wu, L.-A.

L.-A. Wu, H. J. Kimble, J. L. Hall, and H.-F. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57, 2520–2523 (1986).
[CrossRef] [PubMed]

Yamamoto, Y.

P. D. Drummond, R. M. Shelby, S. R. Friberg, and Y. Yamamoto, “Quantum solitons in optical fibres,” Nature 365, 307–313 (1993), and references therein.
[CrossRef]

Yuen, H. P.

H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A 13, 2226–2243 (1976).
[CrossRef]

Yurke, B.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
[CrossRef] [PubMed]

Lett. Nuovo Cimento

E. Y. C. Lu, “Quantum correlations in two-photon amplification,” Lett. Nuovo Cimento 3, 585–589 (1972).
[CrossRef]

Nature

P. D. Drummond, R. M. Shelby, S. R. Friberg, and Y. Yamamoto, “Quantum solitons in optical fibres,” Nature 365, 307–313 (1993), and references therein.
[CrossRef]

Phys. Rev.

J. J. Hopfield, “Theory of the contribution of excitons to the complex dielectric constant of crystals,” Phys. Rev. 112, 1555–1567 (1958).
[CrossRef]

L. N. Cooper, “Bound electron pairs in a degenerate Fermi gas,” Phys. Rev. 104, 1189–1190 (1956).
[CrossRef]

J. Bardeen, L. N. Cooper, and J. R. Schrieffer, “Theory of superconductivity,” Phys. Rev. 108, 1175–1204 (1957).
[CrossRef]

Phys. Rev. A

I. H. Deutsch, R. Y. Chiao, and J. C. Garrison, “Two-photon bound states: the diphoton bullet in dispersive self-focusing media,” Phys. Rev. A 47, 3330–3336 (1993).
[CrossRef] [PubMed]

H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A 13, 2226–2243 (1976).
[CrossRef]

Y. Lai and H. A. Haus, “Quantum theory of solitons in optical fibers. I. Time-dependent Hartree approximation,” Phys. Rev. A 40, 844–853 (1989).
[CrossRef] [PubMed]

Y. Lai and H. A. Haus, “Quantum theory of solitons in optical fibers. II. Exact solution,” Phys. Rev. A 40, 854–866 (1989).
[CrossRef] [PubMed]

F. X. Kärtner and H. A. Haus, “Quantum-mechanical stability of solitons and the correspondence principle,” Phys. Rev. A 48, 2361–2369 (1993).
[CrossRef] [PubMed]

Ze Cheng, “Photonic superguiding state in nonlinear polar crystals,” Phys. Rev. A 51, 675–691 (1995).
[CrossRef] [PubMed]

Z. Cheng and G. Kurizki, “Theory of one-dimensional quantum gap solitons,” Phys. Rev. A 54, 3576–3591 (1996).
[CrossRef] [PubMed]

Phys. Rev. B

M. Artoni and J. L. Birman, “Quantum-optical properties of polariton waves,” Phys. Rev. B 44, 3736–3756 (1991).
[CrossRef]

Phys. Rev. D

D. Stoler, “Equivalence classes of minimum uncertainty packets,” Phys. Rev. D 1, 3217–3219 (1970).
[CrossRef]

D. Stoler, “Equivalence classes of minimum-uncertainty packets. II,” Phys. Rev. D 4, 1925–1926 (1971).
[CrossRef]

Phys. Rev. Lett.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
[CrossRef] [PubMed]

L.-A. Wu, H. J. Kimble, J. L. Hall, and H.-F. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57, 2520–2523 (1986).
[CrossRef] [PubMed]

C. H. Henry and J. J. Hopfield, “Raman scattering by polaritons,” Phys. Rev. Lett. 15, 964–966 (1965).
[CrossRef]

Proc. R. Soc. London, Ser. A

K. Huang, “On the interaction between the radiation field and ionic crystals,” Proc. R. Soc. London, Ser. A 208, 352–365 (1951).
[CrossRef]

Other

H. J. Kimble and D. F. Walls, eds., feature on squeezed states of the electromagnetic field, J. Opt. Soc. Am. B 4(10), (1987).

R. Loudon and P. L. Knight, eds., special issue on squeezed light, J. Mod. Opt. 34(6/7), (1987).

E. Giacobino and E. C. Fabre, eds., feature on quantum noise reduction in optical systems—experiments, Appl. Phys. B 55(3), (1992).

A. Hasegawa, Optical Solitons in Fibers, 2nd ed. (Springer-Verlag, Berlin, 1990).

W. Hayes and R. Loudon, Scattering of Light by Crystals (Wiley, New York, 1978).

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Chap. 2.

J. Callaway, Quantum Theory of the Solid State, 2nd ed. (Academic, New York, 1991), p. 720.

L. E. Reichl, A Modern Course in Statistical Physics (U. Texas Press, Austin, Tex., 1980).

L. D. Landau and E. M. Lifshitz, Statistical Physics, 3rd ed. (Reed Educational, Oxford, 1980), Part 1, p. 183.

W. Greiner, Quantum Mechanics: An Introduction, 3rd ed. (Springer-Verlag, Berlin, 1994), p. 74.

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Figures (6)

Fig. 1
Fig. 1

A Kerr nonlinear blackbody: a rectangular Kerr nonlinear crystal enclosed by perfectly conducting walls and kept at a constant temperature; there is a very small hole in a wall.

Fig. 2
Fig. 2

According to Eq. (58), variation of relative transition temperature Tc*=kBTc/ωR with parameter γ.

Fig. 3
Fig. 3

For three values of γ, variation of relative velocity y=v(T)/v(Tc) with relative temperature x=kBT/ωR, where temperature T varies from zero to transition temperature Tc.

Fig. 4
Fig. 4

For three values of γ, variation of order parameter Δ(T) with relative temperature x=kBT/ωR, where temperature T varies from zero to transition temperature Tc.

Fig. 5
Fig. 5

For three values of γ, variation of squeeze factor r(T) with relative temperature x=kBT/ωR, where temperature T varies from zero to transition temperature Tc.

Fig. 6
Fig. 6

Variation of quadrature variances V1 and V2 in the squeezed thermal radiation state with relative temperature x=kBT/ωR. V0 denotes the quadrature variances in the vacuum state, and V3 designates the quadrature variances in the normal thermal radiation state.

Equations (98)

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

E=-At,B=×A.
Hem= dr02 E2+12μ0 B2,
A(r, t)=kσ2V0ωk1/2{akσ(t)ekσ exp[i(kr)]+akσ(t)ekσ* exp[-i(kr)]},
[akσ(t), akσ(t)]-=δk,kδσσ,
[akσ(t), akσ(t)]-=0.
Hem=kσ ωkakσakσ,
|{nkσ}=kσ1nkσ!(akσ)nkσ|0,
Hion=12 nli mls˙nli2+12 nlinli Φnlinlisnlisnli,
snli=qI 1Nmleli(qI)QI(q)exp[i(qRn)],
Dlili(q)=n 1mlmlΦlili(n)exp[i(qRn)],
ωI2(q)eli(qI)=li Dlili(q)eli(qI).
QI(q)=2ωI(q)1/2(bqI+b-q,I),
PI(q)=i-1ωI(q)21/2(b-q,I-bqI),
Hion=qI ωI(q)bqIbqI,
snl(t)=qI2NmlωI(q)1/2el(qI)×(bqI+b-q,I)exp[i(qRn)].
HI=-½ j,j αjj(s)[E(Rj)E(Rj)].
αjj(S)=αll(0)+j1(Bjjj1sj1).
0χ=-ll Lllαll(0)/Ω,
HI=-½ j,jj1(Bjjj1sj1)[E(Rj)E(Rj)].
E(Rn)=-kσiωk2V01/2{-akσekσ exp[i(kRn)]+akσekσ* exp[-i(kRn)]},
HI=kσ,qσ Mkσ(qσ)ak+q,σakσ(bq+b-q).
Mkσ(qσ)=-Nωk+qωk2ωR1/22V0P(q)(ek+q,σ*ekσ),
P(q)=lln2l1 ml1-1/2[Blll1(n2)el1(q)]exp[i(qRn2)],
[akσ(t),bq(t)]-=[akσ(t), bq(t)]-=0.
S=kσ,qσ iMkσ(qσ)×ak+q,σakσbq(ωk+q-ωk-ωR)+ak+q,σakσb-q(ωk+q-ωk+ωR).
HT=Hem+Hion+½i[HI, S],
ρ=exp(-Hion/kBT)Tr exp(-Hion/kBT),
HI=½i[HI, S]
=k,k,qσσ1σσ2 ωRMkσ(-q, σ)Mkσ1(qσ2)[(ωk-ωk-q)2-ωR2]×ak-q,σak+q,σ2akσ1akσ.
Hem=½ kσ ωk(akσakσ+a-k,-σa-k,-σ)+kσ,kσ Vkσ,kσakσa-k,-σa-k,-σakσ,
Vkσ,kσ=ωR|Mkσ(k-k, σ)|2[(ωk-ωk)2-ωR2].
ckσ=UakσU=akσ cosh φkσ-a-k,-σ sinh φkσ,
ckσ=UakσU=akσ cosh φkσ-a-k,-σ sinh φkσ,
U=exp½ kσ φkσ(akσa-k,-σ-a-k,-σakσ).
|{nkσ}=kσ1nkσ!(ckσ)nkσ|G.
akσ=ckσ cosh φkσ+c-k,-σ sinh φkσ,
akσ=ckσ cosh φkσ+c-k,-σ sinh φkσ.
Hem=Ep+½ kσωk cosh 2φkσ+sinh 2φkσ kσ Vkσ,kσ sinh 2φkσ×(1+½Nkσ+½N-k,-σ)(Nkσ+N-k,-σ)+½ kσωk sinh 2φkσ+cosh 2φkσ kσ Vkσ,kσ sinh 2φkσ×(1+Nkσ+N-k,-σ)×(ckσc-k,-σ+c-k,-σckσ),
Ep=kσωk sinh2 φkσ+¼ sinh 2φkσ kσ Vkσ,kσ sinh 2φkσ.
(NkσNkσ)NkσNkσ.
ω˜k(T)=ωk cosh 2φkσ+sinh 2φkσ kσ Vkσ,kσ sinh 2φkσ×(1+Nkσ+N-k,-σ).
Nkσ=1exp[(ω˜k)(T)/kBT]-1,
ωk sinh 2φkσ+cosh 2φkσ kσ Vkσ,kσ sinh 2φkσ(1
+Nkσ+N-k,-σ)=0.
Δk(T)=-kσ Vkσ,kσ sinh 2φkσ coth[ω˜k(T)/2kBT].
tanh 2φkσ=Δk(T)ωk.
cosh 2φkσ=ωk[2ωk2-Δk2(T)]1/2
sinh 2φkσ=Δk(T)[2ωk2-Δk2(T)]1/2.
ω˜k(T)=[2ωk2-Δk2(T)]1/2,
Δk(T)=-kσ Vkσ,kσ Δk(T)[2ωk2-Δk2(T)]1/2×coth [2ωk2-Δk2(T)]1/22kBT,
Hem=Ep+kσ ω˜k(T)ckσckσ.
v(T)=(c/n)[1-Δ2(T)]1/2.
Vkσ,kσ=-|Mkσ(k-k, σ)|2ωR
Mkσ(k-k, σ)=-Nωkωk2ωR1/22V0P(0).
Vkσ,kσ=-V0ωkωkωk, ωk<ωR,0otherwise,
v(T)=2(c/n)V0 k ωk coth v(T)|k|2kBT,
Es(T)=Hem=Ep+kσ ω˜k(T)Nkσ,
En(T)=Hem=kσ ωk sinh2 φkσ+kσ ωk cosh 2φkσNkσ,
δE(T)=En(T)-Es(T)=-kσ,kσ Vkσ,kσ sinh 2φkσ sinh 2φkσ×[1/4+Nkσ(1+2Nkσ)],
δE(0)=Δ2(0)2[1-Δ2(0)]1/2 k ωk,
v(0)=2(c/n)V0 k ωk.
v(0)=2(n/c)2V0 V(2π)3 0ωR 4πω3dω=γ cn,
γ=2c3Ωn P(0)ωR4π02.
1=4γωR4 0ωR ω3 coth ω2kBTc dω.
1=4γxc4 0xc x3 coth xdx,
v(T)=2(c/n)V0 k ωk+4(c/n)V0 k ωkexp v(T)|k|kBT-1-1.
2(c/n)V0 k ωk=γ(c/n).
v(T)v(Tc)-γ=8γωR4 0ωR ω3dωexp{[v(T)/v(Tc)](ω/kBT)}-1,
y-γ=8γ 01 t3dtexp(yt/x)-1.
v(T)v(Tc)-γ=8γωR4 0 ω3dωexp(γω/kBT)-1.
0 x3dxex-1=(2π)4240.
v(T)v(Tc)-γ=γ30 2πkBTγωR4.
v(T)v(Tc)γ1-Δ(0)γ2[Δ(T)-Δ(0)].
Δ(T)Δ(0)1-130γ2(1-γ2) 2πkBTωR4,
TTc.
v(T)v(Tc)=4γxt4 0xt x3 cothx v(T)v(Tc)dx,
v(T)v(Tc)=TTc 4γxc4 0xc x3 coth xdx.
v(T)v(Tc)=TTc.
v(T)v(Tc)1-½Δ2(T).
TTc=1-1-TTc.
Δ(T)21-TTc1/2,Tc-TTc.
ρ=exp(-Hem/kBT)Tr exp(-Hem/kBT).
Nkσ=1exp(ωk/kBT-1).
Ekσ(r, t)=-½iEkσ(r)[akσ exp(iωkt)-akσ exp(-iωkt)],
Ekσ(r)=2ωkV01/2 exp[i(kr)]ekσ.
[X1,kσ, X2,kσ]=-i(1/2).
Ekσ(r, t)=Ekσ(r)(X1,kσ sin ωkt+X2,kσ cos ωkt).
ΔX1,kσΔX2,kσ1/4.
X1,kσ=½(akσ+akσ),X2,kσ=½i(akσ-akσ).
V(Xi,kσ)=Tr(ρXi,kσ2)=¼(2Nkσ+1),
X1,kσ=½ exp(φkσ)(ckσ+ckσ),
X2,kσ=½i exp(-φkσ)(ckσ-ckσ).
ρ=exp(-Hem/kBT)Tr exp(-Hem/kBT).
V(X1,kσ)=Tr(ρX1,kσ2)=¼ exp(2φkσ)(2Nkσ+1),
V(X2,kσ)=Tr(ρX2,kσ2)=¼ exp(-2φkσ)(2Nkσ+1),
G|akσakσ|G=sinh2 φkσ(0),
r(T)=12 ln 1+Δ(T)1-Δ(T),
ω2=c2|k|2() ωT2-ω2ωL2-ω2,

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