Abstract

We have investigated theoretically the tuning characteristics of a Josephson junction within a microcavity for one-photon spontaneous emission and for one-photon and two-photon stimulated emission. For spontaneous emission, we have established the linear relationship between the magnetic induction and the voltage needed to tune the system to emit at resonant frequencies. For stimulated emission, we have found an oscillatory dependence of the emission rate on the initial Cooper pair phase difference and the phase of the applied field. Under specific conditions, we have also calculated the values of the applied radiation amplitude for the first few emission maxima of the system and for the first five junction–cavity resonances for each process. Because the emission of photons can be controlled, it may be possible to use such a system to produce photons on demand. Such sources will have applications in the fields of quantum cryptography, communications, and computation.

© 2004 Optical Society of America

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References

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  1. H. B. G. Casimir and D. Polder, “The influence of retardation on the London-van der Waals force,” Phys. Rev. 73, 360–372 (1948).
    [CrossRef]
  2. K. H. Drexhage, “Interaction of light with monomolecular dye layers,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1974), Vol. 12, p. 163.
  3. J. R. Ackerhalt and J. H. Eberly, “Quantum electrodynamics and radiation reaction: nonrelativistic atomic frequency shifts and lifetimes,” Phys. Rev. D 10, 3350–3375 (1974).
    [CrossRef]
  4. D. Meschede, H. Walther, and G. Müller, “The one-atom maser,” Phys. Rev. Lett. 54, 551–554 (1985).
    [CrossRef] [PubMed]
  5. R. Arun and G. S. Agarwal, “Dark states and interferences in cascade transitions of ultracold atoms in a cavity,” Phys. Rev. A 66, 043812-(1–9) (2002).
    [CrossRef]
  6. A. Aiello, D. Fargion, and E. Cianci, “Parametric fluorescence and second-harmonic generation in a planar Fabry-Perot microcavity,” Phys. Rev. A 58, 2446–2459 (1998).
    [CrossRef]
  7. Y. B. Xie, “Two-level atom and multichromatic waves in a lossless cavity,” J. Mod. Opt. 44, 359–378 (1997).
    [CrossRef]
  8. A. Carollo, M. F. Santos, and V. Vedral, “Berry’s phase in cavity QED: proposal for observing an effect of field quantisation,” Phys. Rev. A 67, 063804-(1–4) (2003).
    [CrossRef]
  9. D. R. Tilley, “Superradiance in arrays of superconducting weak links,” Phys. Lett. 33A, 205–206 (1970).
    [CrossRef]
  10. D. Rogovin and M. Scully, “Superconductivity and macroscopic quantum phenomena,” Phys. Rep. 25C, 175–291 (1976).
    [CrossRef]
  11. P. Barbara, A. B. Cawthorne, S. V. Shitov, and C. J. Lobb, “Stimulated emission and amplification in Josephson junction arrays,” Phys. Rev. Lett. 82, 1963–1966 (1999).
    [CrossRef]
  12. Y. B. Xie, “Superconducting Josephson junction of finite capacitance in a lossless cavity: a plausible two-level system with a giant coupling constant,” J. Mod. Opt. 45, 2025–2038 (1998).
    [CrossRef]
  13. W. A. Al-Saidi and D. Stroud, “Several small Josephson junctions in a resonant cavity: deviation from the Dicke model,” Phys. Rev. B 65, 224512-(1–10) (2002).
    [CrossRef]
  14. E. Almaas and D. Stroud, “Theory of two-dimensional Josephson arrays in a resonant cavity,” Phys. Rev. B 67, 064511-(1–12) (2003).
    [CrossRef]
  15. D. Rogovin, M. O. Scully, and P. Lee, Quantum Theory of Josephson Radiation (Pergamon, New York, 1973), p. 241.
  16. F. De Martini, M. Marrocco, P. Mataloni, L. Crescentini, and R. Loudon, “Spontaneous emission in the optical microscopic cavity,” Phys. Rev. A 43, 2480–2497 (1991).
    [CrossRef] [PubMed]
  17. D. Rogovin, “The Josephson junction as a macroscopic radiating atom,” Ann. Phys. (Leipzig) 90, 18–47 (1975).
    [CrossRef]
  18. B. D. Josephson, “Possible new effects in superconductive tunnelling,” Phys. Lett. 1, 251–253 (1962).
    [CrossRef]
  19. J. C. Swihart, “Field solution for a thin-film superconducting strip transmission line,” J. Appl. Phys. 32, 461–469 (1961).
    [CrossRef]
  20. R. E. Eck, D. J. Scalapino, and B. N. Taylor, “Self-detection of the ac Josephson current,” Phys. Rev. Lett. 13, 15–18 (1964).
    [CrossRef]
  21. A. Barone and G. Paterno, Physics and Applications of the Josephson Effect (Wiley, New York, 1982), p. 238.
  22. R. J. Glauber, “Optical coherence and photon statistics,” in Quantum Optics and Electronics, C. De Witt, A. Blandin, and C. Cohen-Tannoudji, eds. (Gordon & Breach, New York, 1965), p. 79.
  23. W. A. Al-Saidi and D. Stroud, “Eigenstates of a small Josephson junction coupled to a resonant cavity,” Phys. Rev. B 65, 014512-(1–7) (2001).
    [CrossRef]

2003 (2)

A. Carollo, M. F. Santos, and V. Vedral, “Berry’s phase in cavity QED: proposal for observing an effect of field quantisation,” Phys. Rev. A 67, 063804-(1–4) (2003).
[CrossRef]

E. Almaas and D. Stroud, “Theory of two-dimensional Josephson arrays in a resonant cavity,” Phys. Rev. B 67, 064511-(1–12) (2003).
[CrossRef]

2002 (2)

W. A. Al-Saidi and D. Stroud, “Several small Josephson junctions in a resonant cavity: deviation from the Dicke model,” Phys. Rev. B 65, 224512-(1–10) (2002).
[CrossRef]

R. Arun and G. S. Agarwal, “Dark states and interferences in cascade transitions of ultracold atoms in a cavity,” Phys. Rev. A 66, 043812-(1–9) (2002).
[CrossRef]

2001 (1)

W. A. Al-Saidi and D. Stroud, “Eigenstates of a small Josephson junction coupled to a resonant cavity,” Phys. Rev. B 65, 014512-(1–7) (2001).
[CrossRef]

1999 (1)

P. Barbara, A. B. Cawthorne, S. V. Shitov, and C. J. Lobb, “Stimulated emission and amplification in Josephson junction arrays,” Phys. Rev. Lett. 82, 1963–1966 (1999).
[CrossRef]

1998 (2)

Y. B. Xie, “Superconducting Josephson junction of finite capacitance in a lossless cavity: a plausible two-level system with a giant coupling constant,” J. Mod. Opt. 45, 2025–2038 (1998).
[CrossRef]

A. Aiello, D. Fargion, and E. Cianci, “Parametric fluorescence and second-harmonic generation in a planar Fabry-Perot microcavity,” Phys. Rev. A 58, 2446–2459 (1998).
[CrossRef]

1997 (1)

Y. B. Xie, “Two-level atom and multichromatic waves in a lossless cavity,” J. Mod. Opt. 44, 359–378 (1997).
[CrossRef]

1991 (1)

F. De Martini, M. Marrocco, P. Mataloni, L. Crescentini, and R. Loudon, “Spontaneous emission in the optical microscopic cavity,” Phys. Rev. A 43, 2480–2497 (1991).
[CrossRef] [PubMed]

1985 (1)

D. Meschede, H. Walther, and G. Müller, “The one-atom maser,” Phys. Rev. Lett. 54, 551–554 (1985).
[CrossRef] [PubMed]

1976 (1)

D. Rogovin and M. Scully, “Superconductivity and macroscopic quantum phenomena,” Phys. Rep. 25C, 175–291 (1976).
[CrossRef]

1975 (1)

D. Rogovin, “The Josephson junction as a macroscopic radiating atom,” Ann. Phys. (Leipzig) 90, 18–47 (1975).
[CrossRef]

1974 (1)

J. R. Ackerhalt and J. H. Eberly, “Quantum electrodynamics and radiation reaction: nonrelativistic atomic frequency shifts and lifetimes,” Phys. Rev. D 10, 3350–3375 (1974).
[CrossRef]

1970 (1)

D. R. Tilley, “Superradiance in arrays of superconducting weak links,” Phys. Lett. 33A, 205–206 (1970).
[CrossRef]

1964 (1)

R. E. Eck, D. J. Scalapino, and B. N. Taylor, “Self-detection of the ac Josephson current,” Phys. Rev. Lett. 13, 15–18 (1964).
[CrossRef]

1962 (1)

B. D. Josephson, “Possible new effects in superconductive tunnelling,” Phys. Lett. 1, 251–253 (1962).
[CrossRef]

1961 (1)

J. C. Swihart, “Field solution for a thin-film superconducting strip transmission line,” J. Appl. Phys. 32, 461–469 (1961).
[CrossRef]

1948 (1)

H. B. G. Casimir and D. Polder, “The influence of retardation on the London-van der Waals force,” Phys. Rev. 73, 360–372 (1948).
[CrossRef]

Ackerhalt, J. R.

J. R. Ackerhalt and J. H. Eberly, “Quantum electrodynamics and radiation reaction: nonrelativistic atomic frequency shifts and lifetimes,” Phys. Rev. D 10, 3350–3375 (1974).
[CrossRef]

Agarwal, G. S.

R. Arun and G. S. Agarwal, “Dark states and interferences in cascade transitions of ultracold atoms in a cavity,” Phys. Rev. A 66, 043812-(1–9) (2002).
[CrossRef]

Aiello, A.

A. Aiello, D. Fargion, and E. Cianci, “Parametric fluorescence and second-harmonic generation in a planar Fabry-Perot microcavity,” Phys. Rev. A 58, 2446–2459 (1998).
[CrossRef]

Almaas, E.

E. Almaas and D. Stroud, “Theory of two-dimensional Josephson arrays in a resonant cavity,” Phys. Rev. B 67, 064511-(1–12) (2003).
[CrossRef]

Al-Saidi, W. A.

W. A. Al-Saidi and D. Stroud, “Several small Josephson junctions in a resonant cavity: deviation from the Dicke model,” Phys. Rev. B 65, 224512-(1–10) (2002).
[CrossRef]

W. A. Al-Saidi and D. Stroud, “Eigenstates of a small Josephson junction coupled to a resonant cavity,” Phys. Rev. B 65, 014512-(1–7) (2001).
[CrossRef]

Arun, R.

R. Arun and G. S. Agarwal, “Dark states and interferences in cascade transitions of ultracold atoms in a cavity,” Phys. Rev. A 66, 043812-(1–9) (2002).
[CrossRef]

Barbara, P.

P. Barbara, A. B. Cawthorne, S. V. Shitov, and C. J. Lobb, “Stimulated emission and amplification in Josephson junction arrays,” Phys. Rev. Lett. 82, 1963–1966 (1999).
[CrossRef]

Carollo, A.

A. Carollo, M. F. Santos, and V. Vedral, “Berry’s phase in cavity QED: proposal for observing an effect of field quantisation,” Phys. Rev. A 67, 063804-(1–4) (2003).
[CrossRef]

Casimir, H. B. G.

H. B. G. Casimir and D. Polder, “The influence of retardation on the London-van der Waals force,” Phys. Rev. 73, 360–372 (1948).
[CrossRef]

Cawthorne, A. B.

P. Barbara, A. B. Cawthorne, S. V. Shitov, and C. J. Lobb, “Stimulated emission and amplification in Josephson junction arrays,” Phys. Rev. Lett. 82, 1963–1966 (1999).
[CrossRef]

Cianci, E.

A. Aiello, D. Fargion, and E. Cianci, “Parametric fluorescence and second-harmonic generation in a planar Fabry-Perot microcavity,” Phys. Rev. A 58, 2446–2459 (1998).
[CrossRef]

Crescentini, L.

F. De Martini, M. Marrocco, P. Mataloni, L. Crescentini, and R. Loudon, “Spontaneous emission in the optical microscopic cavity,” Phys. Rev. A 43, 2480–2497 (1991).
[CrossRef] [PubMed]

De Martini, F.

F. De Martini, M. Marrocco, P. Mataloni, L. Crescentini, and R. Loudon, “Spontaneous emission in the optical microscopic cavity,” Phys. Rev. A 43, 2480–2497 (1991).
[CrossRef] [PubMed]

Eberly, J. H.

J. R. Ackerhalt and J. H. Eberly, “Quantum electrodynamics and radiation reaction: nonrelativistic atomic frequency shifts and lifetimes,” Phys. Rev. D 10, 3350–3375 (1974).
[CrossRef]

Eck, R. E.

R. E. Eck, D. J. Scalapino, and B. N. Taylor, “Self-detection of the ac Josephson current,” Phys. Rev. Lett. 13, 15–18 (1964).
[CrossRef]

Fargion, D.

A. Aiello, D. Fargion, and E. Cianci, “Parametric fluorescence and second-harmonic generation in a planar Fabry-Perot microcavity,” Phys. Rev. A 58, 2446–2459 (1998).
[CrossRef]

Josephson, B. D.

B. D. Josephson, “Possible new effects in superconductive tunnelling,” Phys. Lett. 1, 251–253 (1962).
[CrossRef]

Lobb, C. J.

P. Barbara, A. B. Cawthorne, S. V. Shitov, and C. J. Lobb, “Stimulated emission and amplification in Josephson junction arrays,” Phys. Rev. Lett. 82, 1963–1966 (1999).
[CrossRef]

Loudon, R.

F. De Martini, M. Marrocco, P. Mataloni, L. Crescentini, and R. Loudon, “Spontaneous emission in the optical microscopic cavity,” Phys. Rev. A 43, 2480–2497 (1991).
[CrossRef] [PubMed]

Marrocco, M.

F. De Martini, M. Marrocco, P. Mataloni, L. Crescentini, and R. Loudon, “Spontaneous emission in the optical microscopic cavity,” Phys. Rev. A 43, 2480–2497 (1991).
[CrossRef] [PubMed]

Mataloni, P.

F. De Martini, M. Marrocco, P. Mataloni, L. Crescentini, and R. Loudon, “Spontaneous emission in the optical microscopic cavity,” Phys. Rev. A 43, 2480–2497 (1991).
[CrossRef] [PubMed]

Meschede, D.

D. Meschede, H. Walther, and G. Müller, “The one-atom maser,” Phys. Rev. Lett. 54, 551–554 (1985).
[CrossRef] [PubMed]

Müller, G.

D. Meschede, H. Walther, and G. Müller, “The one-atom maser,” Phys. Rev. Lett. 54, 551–554 (1985).
[CrossRef] [PubMed]

Polder, D.

H. B. G. Casimir and D. Polder, “The influence of retardation on the London-van der Waals force,” Phys. Rev. 73, 360–372 (1948).
[CrossRef]

Rogovin, D.

D. Rogovin and M. Scully, “Superconductivity and macroscopic quantum phenomena,” Phys. Rep. 25C, 175–291 (1976).
[CrossRef]

D. Rogovin, “The Josephson junction as a macroscopic radiating atom,” Ann. Phys. (Leipzig) 90, 18–47 (1975).
[CrossRef]

Santos, M. F.

A. Carollo, M. F. Santos, and V. Vedral, “Berry’s phase in cavity QED: proposal for observing an effect of field quantisation,” Phys. Rev. A 67, 063804-(1–4) (2003).
[CrossRef]

Scalapino, D. J.

R. E. Eck, D. J. Scalapino, and B. N. Taylor, “Self-detection of the ac Josephson current,” Phys. Rev. Lett. 13, 15–18 (1964).
[CrossRef]

Scully, M.

D. Rogovin and M. Scully, “Superconductivity and macroscopic quantum phenomena,” Phys. Rep. 25C, 175–291 (1976).
[CrossRef]

Shitov, S. V.

P. Barbara, A. B. Cawthorne, S. V. Shitov, and C. J. Lobb, “Stimulated emission and amplification in Josephson junction arrays,” Phys. Rev. Lett. 82, 1963–1966 (1999).
[CrossRef]

Stroud, D.

E. Almaas and D. Stroud, “Theory of two-dimensional Josephson arrays in a resonant cavity,” Phys. Rev. B 67, 064511-(1–12) (2003).
[CrossRef]

W. A. Al-Saidi and D. Stroud, “Several small Josephson junctions in a resonant cavity: deviation from the Dicke model,” Phys. Rev. B 65, 224512-(1–10) (2002).
[CrossRef]

W. A. Al-Saidi and D. Stroud, “Eigenstates of a small Josephson junction coupled to a resonant cavity,” Phys. Rev. B 65, 014512-(1–7) (2001).
[CrossRef]

Swihart, J. C.

J. C. Swihart, “Field solution for a thin-film superconducting strip transmission line,” J. Appl. Phys. 32, 461–469 (1961).
[CrossRef]

Taylor, B. N.

R. E. Eck, D. J. Scalapino, and B. N. Taylor, “Self-detection of the ac Josephson current,” Phys. Rev. Lett. 13, 15–18 (1964).
[CrossRef]

Tilley, D. R.

D. R. Tilley, “Superradiance in arrays of superconducting weak links,” Phys. Lett. 33A, 205–206 (1970).
[CrossRef]

Vedral, V.

A. Carollo, M. F. Santos, and V. Vedral, “Berry’s phase in cavity QED: proposal for observing an effect of field quantisation,” Phys. Rev. A 67, 063804-(1–4) (2003).
[CrossRef]

Walther, H.

D. Meschede, H. Walther, and G. Müller, “The one-atom maser,” Phys. Rev. Lett. 54, 551–554 (1985).
[CrossRef] [PubMed]

Xie, Y. B.

Y. B. Xie, “Superconducting Josephson junction of finite capacitance in a lossless cavity: a plausible two-level system with a giant coupling constant,” J. Mod. Opt. 45, 2025–2038 (1998).
[CrossRef]

Y. B. Xie, “Two-level atom and multichromatic waves in a lossless cavity,” J. Mod. Opt. 44, 359–378 (1997).
[CrossRef]

Ann. Phys. (Leipzig) (1)

D. Rogovin, “The Josephson junction as a macroscopic radiating atom,” Ann. Phys. (Leipzig) 90, 18–47 (1975).
[CrossRef]

J. Appl. Phys. (1)

J. C. Swihart, “Field solution for a thin-film superconducting strip transmission line,” J. Appl. Phys. 32, 461–469 (1961).
[CrossRef]

J. Mod. Opt. (2)

Y. B. Xie, “Superconducting Josephson junction of finite capacitance in a lossless cavity: a plausible two-level system with a giant coupling constant,” J. Mod. Opt. 45, 2025–2038 (1998).
[CrossRef]

Y. B. Xie, “Two-level atom and multichromatic waves in a lossless cavity,” J. Mod. Opt. 44, 359–378 (1997).
[CrossRef]

Phys. Lett. (2)

B. D. Josephson, “Possible new effects in superconductive tunnelling,” Phys. Lett. 1, 251–253 (1962).
[CrossRef]

D. R. Tilley, “Superradiance in arrays of superconducting weak links,” Phys. Lett. 33A, 205–206 (1970).
[CrossRef]

Phys. Rep. (1)

D. Rogovin and M. Scully, “Superconductivity and macroscopic quantum phenomena,” Phys. Rep. 25C, 175–291 (1976).
[CrossRef]

Phys. Rev. (1)

H. B. G. Casimir and D. Polder, “The influence of retardation on the London-van der Waals force,” Phys. Rev. 73, 360–372 (1948).
[CrossRef]

Phys. Rev. A (4)

A. Carollo, M. F. Santos, and V. Vedral, “Berry’s phase in cavity QED: proposal for observing an effect of field quantisation,” Phys. Rev. A 67, 063804-(1–4) (2003).
[CrossRef]

R. Arun and G. S. Agarwal, “Dark states and interferences in cascade transitions of ultracold atoms in a cavity,” Phys. Rev. A 66, 043812-(1–9) (2002).
[CrossRef]

A. Aiello, D. Fargion, and E. Cianci, “Parametric fluorescence and second-harmonic generation in a planar Fabry-Perot microcavity,” Phys. Rev. A 58, 2446–2459 (1998).
[CrossRef]

F. De Martini, M. Marrocco, P. Mataloni, L. Crescentini, and R. Loudon, “Spontaneous emission in the optical microscopic cavity,” Phys. Rev. A 43, 2480–2497 (1991).
[CrossRef] [PubMed]

Phys. Rev. B (3)

W. A. Al-Saidi and D. Stroud, “Eigenstates of a small Josephson junction coupled to a resonant cavity,” Phys. Rev. B 65, 014512-(1–7) (2001).
[CrossRef]

W. A. Al-Saidi and D. Stroud, “Several small Josephson junctions in a resonant cavity: deviation from the Dicke model,” Phys. Rev. B 65, 224512-(1–10) (2002).
[CrossRef]

E. Almaas and D. Stroud, “Theory of two-dimensional Josephson arrays in a resonant cavity,” Phys. Rev. B 67, 064511-(1–12) (2003).
[CrossRef]

Phys. Rev. D (1)

J. R. Ackerhalt and J. H. Eberly, “Quantum electrodynamics and radiation reaction: nonrelativistic atomic frequency shifts and lifetimes,” Phys. Rev. D 10, 3350–3375 (1974).
[CrossRef]

Phys. Rev. Lett. (3)

D. Meschede, H. Walther, and G. Müller, “The one-atom maser,” Phys. Rev. Lett. 54, 551–554 (1985).
[CrossRef] [PubMed]

P. Barbara, A. B. Cawthorne, S. V. Shitov, and C. J. Lobb, “Stimulated emission and amplification in Josephson junction arrays,” Phys. Rev. Lett. 82, 1963–1966 (1999).
[CrossRef]

R. E. Eck, D. J. Scalapino, and B. N. Taylor, “Self-detection of the ac Josephson current,” Phys. Rev. Lett. 13, 15–18 (1964).
[CrossRef]

Other (4)

A. Barone and G. Paterno, Physics and Applications of the Josephson Effect (Wiley, New York, 1982), p. 238.

R. J. Glauber, “Optical coherence and photon statistics,” in Quantum Optics and Electronics, C. De Witt, A. Blandin, and C. Cohen-Tannoudji, eds. (Gordon & Breach, New York, 1965), p. 79.

D. Rogovin, M. O. Scully, and P. Lee, Quantum Theory of Josephson Radiation (Pergamon, New York, 1973), p. 241.

K. H. Drexhage, “Interaction of light with monomolecular dye layers,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1974), Vol. 12, p. 163.

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Figures (5)

Fig. 1
Fig. 1

Schematic of a tunnel or a superconductor–insulator–superconductor Josephson junction in a microcavity.

Fig. 2
Fig. 2

Geometry of the Fabry–Perot microcavity. The numbers 1 and 2 represent the left and right mirrors, respectively, of the microcavity.

Fig. 3
Fig. 3

Graphs illustrating the numerical values for the one-photon process of (a) X for the first three positions of maximum count rate for Δφ=0, and (b) V1 for the first five cavity resonances of the system for X1=3.518, X2=6.866, and X3=10.073 and Δφ=0.

Fig. 4
Fig. 4

Graphs illustrating the numerical values for the two-photon process of (a) X for the first four positions of maximum count rate for Δφ=0, and (b) V1 for the first five cavity resonances of the system for X1=1.558, X2=4.888, X3=8.289, and X4=11.530 and Δφ=0.

Fig. 5
Fig. 5

Graph illustrating the variation of the count rate with (a) Δφ where Δφ=ϕ0-θ for the one-photon process and (b) Δφ where Δφ=ϕ0-2θ for the two-photon process.

Equations (44)

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

Hˆi=1c0d3rj(r)·Aˆ(r, t),
Aˆ+(r, t)=d3kj16π3c0k1/2(k, j)[Ukj(r)aˆkj+Ukj(r)aˆkj]exp(-ic0kt),
k±=k(sin θ cos ϕ, sin θ sin ϕ, ±cos θ),
|ψn=|q, -q=q(uq+νqCq+C-q+)|0,
|ψm=|s, -s=s(us+νsCs+C-s+)|0,
|ψJ=s12(Cs,,m+C-s,,m++Cs,,n+C-s,,n+)|0.
|Isp=|ψJI, 0,
|Fsp=|ψJF, 1,
-i0tF|Hˆi|Idt,
-ic00td3r[ψJF|j(r)|ψJI×1|Aˆ(r, t)|0]dt.
j1 sin(ω0t-k0z),
ω0=2eV0/,
k0=2ec0(2λL+l)H0
c0=l(2λL+l)1/2c.
16π3c01/21|d3kjk-1/2(k, j)[Ukj*(z)aˆkj++Ukj*(z)aˆkj+]exp(ic0kt)|0,
Uk+j(z)=t1j exp(ikz cos θ)Dj,
Uk-j(z)=t1jr2j exp[ik(d-z)cos θ]Dj,
Uk+j(z)=t2jr1j exp[ik(d+z)cos θ]Dj,
Uk-j(z)=t2j exp(-ikz cos θ)Dj,
Dj1-r1jr2j exp (2ikd cos θ).
r1j=-1,r2j-1,t1j=0,t2j0.
d3kk-1/2exp(ikz)-exp[-ik(z+d)]1+r21*exp(-2ikd)exp(ic0kt).
ωn3/2 sink0-ωnc0L21+r21*exp-2iωndc0k0-ωnc0L2exp-iωndc0.
ωn=nπc0L
k0ωnc0.
Bn=ωnμ02e(2λL+l),
Vn=ωn2e.
j=J1 sinω0t+2eωV1 sin(ωt+θ)+ϕ0,
j=J1s=-Js2eV1ωsin[(ω0+sω)t+ϕ0+sθ],
|Ist=|ψJI, 1,
|Fst=|ψJF, 2,
-ic00td3r[ψJF|j|ψJI2|Aˆx(z, t)|1]dt.
ωn1/2 sinωnL2c01-exp-iωndc0exp(-iθ)1+r21*exp-i2ωndc0×J-22eV1ωnexp(iΔφ)-J02eV1ωnexp(-iΔφ),
Δφ=ϕ0-θ.
J-22eV1ωnexp(iΔφ)-J02eV1ωnexp(-iΔφ),
|Ist=|ψJI, 1,
|Fst=|ψJF, 3.
Fst|12-i0tHˆi(t)dt2|Ist.
0tdt0tdtd3rd3r[ψJF|j(t)j(t)|ψJI×3|Aˆx-(z, t)Aˆx-(z, t)|1].
ω0=2ω=2ω,
ωn1/2 sinωnL4c01-exp-iωnd2c01+r21*exp-iωndc02
×exp(-2iθ)×J-34eV1ωn2×exp(i2Δφ)+J-14eV1ωn2×exp(-i2Δφ)-2J-34eV1ωnJ-14eV1ωn.
Δφ=ϕ0-2θ.
J-34eV1ωn2exp(i2Δφ)+J-14eV1ωn2×exp(-i2Δφ)-2J-34eV1ωnJ-14eV1ωn,

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