Abstract

The nonlinear transient effects, similar to self-induced transparency and adiabatic following, are studied for a moving two-level atom that is entering into an ideal microwave cavity in a coherent superposition of its states. The atom undergoes a one-photon transition in the cavity, sustaining a spatial field distribution for a single-mode coherent (or thermal or Fock state) field. For some particular choice of parameters of atomic coherence, removal of an appreciable amount of field energy from the cavity could be observed.

© 2004 Optical Society of America

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References

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  1. E. T. Jaynes and F. W. Cummings, “Comparison of quantum and semiclassical radiation theories with a single mode quantized field,” Proc. IEEE 51, 89–109 (1963).
    [CrossRef]
  2. J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon, “Periodic spontaneous collapse and revival in a simple quantum model,” Phys. Rev. Lett. 44, 1323–1326 (1980).
    [CrossRef]
  3. Y. I. Yoo and J. H. Eberly, “Dynamical theory of an atom with two or three levels interacting with quantized cavity field,” Phys. Rep. 118, 239–337 (1985).
    [CrossRef]
  4. A. Joshi and R. R. Puri, “Effect of the binomial field distribution on collapse and revival phenomena in the Jaynes–Cummings model,” J. Mod. Opt. 34, 1421–1431 (1987).
    [CrossRef]
  5. A. Joshi and S. V. Lawande, “The effects of negative binomial field distribution on Rabi oscillations in a two-level atom,” Opt. Commun. 70, 21–24 (1989).
    [CrossRef]
  6. See, for example, a recent topical review by B. W. Shore and P. L. Knight, “The Jaynes–Cummings model,” J. Mod. Opt. 40, 1195–1238 (1993), and references therein.
    [CrossRef]
  7. A. Joshi, “Quantum electrodynamics of two-level atoms incavities: some theoretical studies,” Ph.D. thesis (University of Bombay, India, 1991).
  8. R. R. Schlicher, “Jaynes–Cummings model with atomic motion,” Opt. Commun. 70, 97–102 (1989).
    [CrossRef]
  9. A. Joshi and S. V. Lawande, “Effect of atomic motion on Rydberg atoms undergoing two-photon transitions in a lossless cavity,” Phys. Rev. A 42, 1752–1756 (1990).
    [CrossRef] [PubMed]
  10. A. Joshi, “Two-mode two-photon Jaynes–Cummings model with atomic motion,” Phys. Rev. A 58, 4662–4667 (1998).
    [CrossRef]
  11. L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975).
  12. J. J. Slosser, P. Meystre, and S. L. Braunstein, “Harmonic oscillator driven by a quantum current,” Phys. Rev. Lett. 63, 934–937 (1989).
    [CrossRef] [PubMed]
  13. Another interesting paper where comparison of classical, neoclassical, and quantum theories is presented by G. S. Agarwal and R. R. Puri, “Monitoring quantum effects in cavity electrodynamics by atoms in semiclassical dressed states,” J. Opt. Soc. Am. B 5, 1669–1676 (1988).
    [CrossRef]
  14. J. J. Slosser and P. Meystre, “Tangent and cotangent states of the electromagnetic field,” Phys. Rev. A 41, 3867–3874 (1990).
    [CrossRef] [PubMed]
  15. S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50 (7), 36–42 (1997), and references therein.
    [CrossRef]
  16. M. Xiao, H. Wang, and D. Goorskey, “Light controlling light with enhanced Kerr nonlinearity,” Opt. Photonics News 13 (9), 44–48 (2002), and references therein.
    [CrossRef]
  17. M. Weidinger, B. T. H. Varcoe, R. Heerlein, and H. Walther, “Trapping states in the micromaser,” Phys. Rev. Lett. 82, 3795–3798 (1999).
    [CrossRef]
  18. M. Sargent III, M. O. Scully, and W. E. Lamb, Jr., Laser Physics (Addision-Wesley, Reading, Mass., 1974).
  19. B.-G. Englert, J. Schwinger, A. O. Barut, and M. O. Scully, “Reflection of slow atoms from a micromaser field,” Europhys. Lett. 14, 25–28 (1991).
    [CrossRef]
  20. P. Meystre, Atom Optics, Vol. 33 of Springer Series in Atomic, Optical and Plasma Physics (Springer, Berlin, 2001).
  21. P. R. Berman, ed., Cavity Quantum Electrodynamics, Advances in Atomic and Molecular Physics (Academic, New York, 1994), Suppl. 2.
  22. M. Wilkens and P. Meystre, “Spectrum of spontaneous emission in a Fabry–Perot cavity: the effects of atomic motion,” Opt. Commun. 94, 66–70 (1992).
    [CrossRef]
  23. Y. S. Bai, A. G. Yodh, and T. W. Mossberg, “Selective excitation of dressed atomic states by use of phase-controlled optical fields,” Phys. Rev. Lett. 55, 1277–1280 (1985).
    [CrossRef] [PubMed]

2002 (1)

M. Xiao, H. Wang, and D. Goorskey, “Light controlling light with enhanced Kerr nonlinearity,” Opt. Photonics News 13 (9), 44–48 (2002), and references therein.
[CrossRef]

1999 (1)

M. Weidinger, B. T. H. Varcoe, R. Heerlein, and H. Walther, “Trapping states in the micromaser,” Phys. Rev. Lett. 82, 3795–3798 (1999).
[CrossRef]

1998 (1)

A. Joshi, “Two-mode two-photon Jaynes–Cummings model with atomic motion,” Phys. Rev. A 58, 4662–4667 (1998).
[CrossRef]

1997 (1)

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50 (7), 36–42 (1997), and references therein.
[CrossRef]

1993 (1)

See, for example, a recent topical review by B. W. Shore and P. L. Knight, “The Jaynes–Cummings model,” J. Mod. Opt. 40, 1195–1238 (1993), and references therein.
[CrossRef]

1992 (1)

M. Wilkens and P. Meystre, “Spectrum of spontaneous emission in a Fabry–Perot cavity: the effects of atomic motion,” Opt. Commun. 94, 66–70 (1992).
[CrossRef]

1991 (1)

B.-G. Englert, J. Schwinger, A. O. Barut, and M. O. Scully, “Reflection of slow atoms from a micromaser field,” Europhys. Lett. 14, 25–28 (1991).
[CrossRef]

1990 (2)

J. J. Slosser and P. Meystre, “Tangent and cotangent states of the electromagnetic field,” Phys. Rev. A 41, 3867–3874 (1990).
[CrossRef] [PubMed]

A. Joshi and S. V. Lawande, “Effect of atomic motion on Rydberg atoms undergoing two-photon transitions in a lossless cavity,” Phys. Rev. A 42, 1752–1756 (1990).
[CrossRef] [PubMed]

1989 (3)

A. Joshi and S. V. Lawande, “The effects of negative binomial field distribution on Rabi oscillations in a two-level atom,” Opt. Commun. 70, 21–24 (1989).
[CrossRef]

R. R. Schlicher, “Jaynes–Cummings model with atomic motion,” Opt. Commun. 70, 97–102 (1989).
[CrossRef]

J. J. Slosser, P. Meystre, and S. L. Braunstein, “Harmonic oscillator driven by a quantum current,” Phys. Rev. Lett. 63, 934–937 (1989).
[CrossRef] [PubMed]

1988 (1)

1987 (1)

A. Joshi and R. R. Puri, “Effect of the binomial field distribution on collapse and revival phenomena in the Jaynes–Cummings model,” J. Mod. Opt. 34, 1421–1431 (1987).
[CrossRef]

1985 (2)

Y. I. Yoo and J. H. Eberly, “Dynamical theory of an atom with two or three levels interacting with quantized cavity field,” Phys. Rep. 118, 239–337 (1985).
[CrossRef]

Y. S. Bai, A. G. Yodh, and T. W. Mossberg, “Selective excitation of dressed atomic states by use of phase-controlled optical fields,” Phys. Rev. Lett. 55, 1277–1280 (1985).
[CrossRef] [PubMed]

1980 (1)

J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon, “Periodic spontaneous collapse and revival in a simple quantum model,” Phys. Rev. Lett. 44, 1323–1326 (1980).
[CrossRef]

1963 (1)

E. T. Jaynes and F. W. Cummings, “Comparison of quantum and semiclassical radiation theories with a single mode quantized field,” Proc. IEEE 51, 89–109 (1963).
[CrossRef]

Agarwal, G. S.

Bai, Y. S.

Y. S. Bai, A. G. Yodh, and T. W. Mossberg, “Selective excitation of dressed atomic states by use of phase-controlled optical fields,” Phys. Rev. Lett. 55, 1277–1280 (1985).
[CrossRef] [PubMed]

Barut, A. O.

B.-G. Englert, J. Schwinger, A. O. Barut, and M. O. Scully, “Reflection of slow atoms from a micromaser field,” Europhys. Lett. 14, 25–28 (1991).
[CrossRef]

Braunstein, S. L.

J. J. Slosser, P. Meystre, and S. L. Braunstein, “Harmonic oscillator driven by a quantum current,” Phys. Rev. Lett. 63, 934–937 (1989).
[CrossRef] [PubMed]

Cummings, F. W.

E. T. Jaynes and F. W. Cummings, “Comparison of quantum and semiclassical radiation theories with a single mode quantized field,” Proc. IEEE 51, 89–109 (1963).
[CrossRef]

Eberly, J. H.

Y. I. Yoo and J. H. Eberly, “Dynamical theory of an atom with two or three levels interacting with quantized cavity field,” Phys. Rep. 118, 239–337 (1985).
[CrossRef]

J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon, “Periodic spontaneous collapse and revival in a simple quantum model,” Phys. Rev. Lett. 44, 1323–1326 (1980).
[CrossRef]

Englert, B.-G.

B.-G. Englert, J. Schwinger, A. O. Barut, and M. O. Scully, “Reflection of slow atoms from a micromaser field,” Europhys. Lett. 14, 25–28 (1991).
[CrossRef]

Goorskey, D.

M. Xiao, H. Wang, and D. Goorskey, “Light controlling light with enhanced Kerr nonlinearity,” Opt. Photonics News 13 (9), 44–48 (2002), and references therein.
[CrossRef]

Harris, S. E.

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50 (7), 36–42 (1997), and references therein.
[CrossRef]

Heerlein, R.

M. Weidinger, B. T. H. Varcoe, R. Heerlein, and H. Walther, “Trapping states in the micromaser,” Phys. Rev. Lett. 82, 3795–3798 (1999).
[CrossRef]

Jaynes, E. T.

E. T. Jaynes and F. W. Cummings, “Comparison of quantum and semiclassical radiation theories with a single mode quantized field,” Proc. IEEE 51, 89–109 (1963).
[CrossRef]

Joshi, A.

A. Joshi, “Two-mode two-photon Jaynes–Cummings model with atomic motion,” Phys. Rev. A 58, 4662–4667 (1998).
[CrossRef]

A. Joshi and S. V. Lawande, “Effect of atomic motion on Rydberg atoms undergoing two-photon transitions in a lossless cavity,” Phys. Rev. A 42, 1752–1756 (1990).
[CrossRef] [PubMed]

A. Joshi and S. V. Lawande, “The effects of negative binomial field distribution on Rabi oscillations in a two-level atom,” Opt. Commun. 70, 21–24 (1989).
[CrossRef]

A. Joshi and R. R. Puri, “Effect of the binomial field distribution on collapse and revival phenomena in the Jaynes–Cummings model,” J. Mod. Opt. 34, 1421–1431 (1987).
[CrossRef]

Knight, P. L.

See, for example, a recent topical review by B. W. Shore and P. L. Knight, “The Jaynes–Cummings model,” J. Mod. Opt. 40, 1195–1238 (1993), and references therein.
[CrossRef]

Lawande, S. V.

A. Joshi and S. V. Lawande, “Effect of atomic motion on Rydberg atoms undergoing two-photon transitions in a lossless cavity,” Phys. Rev. A 42, 1752–1756 (1990).
[CrossRef] [PubMed]

A. Joshi and S. V. Lawande, “The effects of negative binomial field distribution on Rabi oscillations in a two-level atom,” Opt. Commun. 70, 21–24 (1989).
[CrossRef]

Meystre, P.

M. Wilkens and P. Meystre, “Spectrum of spontaneous emission in a Fabry–Perot cavity: the effects of atomic motion,” Opt. Commun. 94, 66–70 (1992).
[CrossRef]

J. J. Slosser and P. Meystre, “Tangent and cotangent states of the electromagnetic field,” Phys. Rev. A 41, 3867–3874 (1990).
[CrossRef] [PubMed]

J. J. Slosser, P. Meystre, and S. L. Braunstein, “Harmonic oscillator driven by a quantum current,” Phys. Rev. Lett. 63, 934–937 (1989).
[CrossRef] [PubMed]

Mossberg, T. W.

Y. S. Bai, A. G. Yodh, and T. W. Mossberg, “Selective excitation of dressed atomic states by use of phase-controlled optical fields,” Phys. Rev. Lett. 55, 1277–1280 (1985).
[CrossRef] [PubMed]

Narozhny, N. B.

J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon, “Periodic spontaneous collapse and revival in a simple quantum model,” Phys. Rev. Lett. 44, 1323–1326 (1980).
[CrossRef]

Puri, R. R.

Sanchez-Mondragon, J. J.

J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon, “Periodic spontaneous collapse and revival in a simple quantum model,” Phys. Rev. Lett. 44, 1323–1326 (1980).
[CrossRef]

Schlicher, R. R.

R. R. Schlicher, “Jaynes–Cummings model with atomic motion,” Opt. Commun. 70, 97–102 (1989).
[CrossRef]

Schwinger, J.

B.-G. Englert, J. Schwinger, A. O. Barut, and M. O. Scully, “Reflection of slow atoms from a micromaser field,” Europhys. Lett. 14, 25–28 (1991).
[CrossRef]

Scully, M. O.

B.-G. Englert, J. Schwinger, A. O. Barut, and M. O. Scully, “Reflection of slow atoms from a micromaser field,” Europhys. Lett. 14, 25–28 (1991).
[CrossRef]

Shore, B. W.

See, for example, a recent topical review by B. W. Shore and P. L. Knight, “The Jaynes–Cummings model,” J. Mod. Opt. 40, 1195–1238 (1993), and references therein.
[CrossRef]

Slosser, J. J.

J. J. Slosser and P. Meystre, “Tangent and cotangent states of the electromagnetic field,” Phys. Rev. A 41, 3867–3874 (1990).
[CrossRef] [PubMed]

J. J. Slosser, P. Meystre, and S. L. Braunstein, “Harmonic oscillator driven by a quantum current,” Phys. Rev. Lett. 63, 934–937 (1989).
[CrossRef] [PubMed]

Varcoe, B. T. H.

M. Weidinger, B. T. H. Varcoe, R. Heerlein, and H. Walther, “Trapping states in the micromaser,” Phys. Rev. Lett. 82, 3795–3798 (1999).
[CrossRef]

Walther, H.

M. Weidinger, B. T. H. Varcoe, R. Heerlein, and H. Walther, “Trapping states in the micromaser,” Phys. Rev. Lett. 82, 3795–3798 (1999).
[CrossRef]

Wang, H.

M. Xiao, H. Wang, and D. Goorskey, “Light controlling light with enhanced Kerr nonlinearity,” Opt. Photonics News 13 (9), 44–48 (2002), and references therein.
[CrossRef]

Weidinger, M.

M. Weidinger, B. T. H. Varcoe, R. Heerlein, and H. Walther, “Trapping states in the micromaser,” Phys. Rev. Lett. 82, 3795–3798 (1999).
[CrossRef]

Wilkens, M.

M. Wilkens and P. Meystre, “Spectrum of spontaneous emission in a Fabry–Perot cavity: the effects of atomic motion,” Opt. Commun. 94, 66–70 (1992).
[CrossRef]

Xiao, M.

M. Xiao, H. Wang, and D. Goorskey, “Light controlling light with enhanced Kerr nonlinearity,” Opt. Photonics News 13 (9), 44–48 (2002), and references therein.
[CrossRef]

Yodh, A. G.

Y. S. Bai, A. G. Yodh, and T. W. Mossberg, “Selective excitation of dressed atomic states by use of phase-controlled optical fields,” Phys. Rev. Lett. 55, 1277–1280 (1985).
[CrossRef] [PubMed]

Yoo, Y. I.

Y. I. Yoo and J. H. Eberly, “Dynamical theory of an atom with two or three levels interacting with quantized cavity field,” Phys. Rep. 118, 239–337 (1985).
[CrossRef]

Europhys. Lett. (1)

B.-G. Englert, J. Schwinger, A. O. Barut, and M. O. Scully, “Reflection of slow atoms from a micromaser field,” Europhys. Lett. 14, 25–28 (1991).
[CrossRef]

J. Mod. Opt. (2)

A. Joshi and R. R. Puri, “Effect of the binomial field distribution on collapse and revival phenomena in the Jaynes–Cummings model,” J. Mod. Opt. 34, 1421–1431 (1987).
[CrossRef]

See, for example, a recent topical review by B. W. Shore and P. L. Knight, “The Jaynes–Cummings model,” J. Mod. Opt. 40, 1195–1238 (1993), and references therein.
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Commun. (3)

M. Wilkens and P. Meystre, “Spectrum of spontaneous emission in a Fabry–Perot cavity: the effects of atomic motion,” Opt. Commun. 94, 66–70 (1992).
[CrossRef]

A. Joshi and S. V. Lawande, “The effects of negative binomial field distribution on Rabi oscillations in a two-level atom,” Opt. Commun. 70, 21–24 (1989).
[CrossRef]

R. R. Schlicher, “Jaynes–Cummings model with atomic motion,” Opt. Commun. 70, 97–102 (1989).
[CrossRef]

Opt. Photonics News (1)

M. Xiao, H. Wang, and D. Goorskey, “Light controlling light with enhanced Kerr nonlinearity,” Opt. Photonics News 13 (9), 44–48 (2002), and references therein.
[CrossRef]

Phys. Rep. (1)

Y. I. Yoo and J. H. Eberly, “Dynamical theory of an atom with two or three levels interacting with quantized cavity field,” Phys. Rep. 118, 239–337 (1985).
[CrossRef]

Phys. Rev. A (3)

J. J. Slosser and P. Meystre, “Tangent and cotangent states of the electromagnetic field,” Phys. Rev. A 41, 3867–3874 (1990).
[CrossRef] [PubMed]

A. Joshi and S. V. Lawande, “Effect of atomic motion on Rydberg atoms undergoing two-photon transitions in a lossless cavity,” Phys. Rev. A 42, 1752–1756 (1990).
[CrossRef] [PubMed]

A. Joshi, “Two-mode two-photon Jaynes–Cummings model with atomic motion,” Phys. Rev. A 58, 4662–4667 (1998).
[CrossRef]

Phys. Rev. Lett. (4)

J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon, “Periodic spontaneous collapse and revival in a simple quantum model,” Phys. Rev. Lett. 44, 1323–1326 (1980).
[CrossRef]

M. Weidinger, B. T. H. Varcoe, R. Heerlein, and H. Walther, “Trapping states in the micromaser,” Phys. Rev. Lett. 82, 3795–3798 (1999).
[CrossRef]

J. J. Slosser, P. Meystre, and S. L. Braunstein, “Harmonic oscillator driven by a quantum current,” Phys. Rev. Lett. 63, 934–937 (1989).
[CrossRef] [PubMed]

Y. S. Bai, A. G. Yodh, and T. W. Mossberg, “Selective excitation of dressed atomic states by use of phase-controlled optical fields,” Phys. Rev. Lett. 55, 1277–1280 (1985).
[CrossRef] [PubMed]

Phys. Today (1)

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50 (7), 36–42 (1997), and references therein.
[CrossRef]

Proc. IEEE (1)

E. T. Jaynes and F. W. Cummings, “Comparison of quantum and semiclassical radiation theories with a single mode quantized field,” Proc. IEEE 51, 89–109 (1963).
[CrossRef]

Other (5)

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975).

M. Sargent III, M. O. Scully, and W. E. Lamb, Jr., Laser Physics (Addision-Wesley, Reading, Mass., 1974).

A. Joshi, “Quantum electrodynamics of two-level atoms incavities: some theoretical studies,” Ph.D. thesis (University of Bombay, India, 1991).

P. Meystre, Atom Optics, Vol. 33 of Springer Series in Atomic, Optical and Plasma Physics (Springer, Berlin, 2001).

P. R. Berman, ed., Cavity Quantum Electrodynamics, Advances in Atomic and Molecular Physics (Academic, New York, 1994), Suppl. 2.

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Figures (7)

Fig. 1
Fig. 1

Atomic inversion at the cavity exit versus detuning Δ/g for a fixed transit time tT=π/g. Curve A is for a TEmnp mode with p=1, curve B is for p=2, and curve C is for the standard JCM. In all the cases, the field is initially in a coherent state with n¯=10 (all Cn’s are real) and (a) wn(0)=1/2(Cn2-Cn+12), vn(0)=CnCn+1, un(0)=0; (b) wn(0)=1, vn(0)=0, and un(0)=0.

Fig. 2
Fig. 2

Same as Fig. 1 but for (a) vn(0)=-CnCn+1 and (b) wn(0)=-1.

Fig. 3
Fig. 3

Atomic inversion at the cavity exit versus detuning Δ/g for a fixed transit time tT=π/g: (a) with a TEmnp, p=1 mode, and (b) with a TEmnp, p=2 mode. The field is initially in a coherent state (all Cn’s are real) with vn(0)=-CnCn+1, un(0)=0, wn(0)=1/2(Cn2-Cn+12). Curves A, B, and C are for n¯=20, 25, and 30, respectively.

Fig. 4
Fig. 4

Atomic inversion at the cavity exit versus detuning Δ/g for a fixed transit time tT=π/g: (a) with a TEmnp, p=1 mode and (b) with a TEmnp, p=2 mode. The field is initially in a thermal state with vn(0)=0, un(0)=0, wn(0)=1/2(Cn2-Cn+12). Curves A, B, and C are for n¯=10, 20, and 30, respectively.

Fig. 5
Fig. 5

Effect of transit-time fluctuations on the atomic inversion plotted as a function of Δ/g. The cavity field is in the thermal state with n¯=20, and the initial conditions are vn(0)=0, un(0)=0, etc., and p=2. Curve A is for the transit time tT=π/g. Curves B and C represent ±5% variation over tT while curves D and E represent ±10% variation over tT.

Fig. 6
Fig. 6

Atomic inversion at the cavity exit versus detuning Δ/g for a fixed transit time tT=π/g: (a) with a TEmnp, p=1 mode, and (b) with a TEmnp, p=2 mode. The initial atom–field state is 1/2(|a, n+i|b, n+1). Curves A, B, and C are for n=20, 25, and 30, respectively.

Fig. 7
Fig. 7

Atomic inversion at the cavity exit versus detuning Δ/g for a fixed transit time tT=π/g: (a) with a TEmnp, p=1 mode, and (b) with a TEmnp, p=2 mode. The field is initially in a Fock state |n with wn(0)=0.5, wn-1(0)=-0.5, vn(0)=0, and un(0)=0. Curves A, B, and C are for n=20, 25, and 30, respectively.

Equations (22)

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

H=ωg|gg|+ωe|ee|+ωaa+g(S-a+S+a),
f(z)f(vt),
f(vt)=sin(pπvt/L),
H=ωg|gg|+ωe|ee|+ωaa+gf(z)(S-a+S+a).
ψ(t)=n[an(t)|n, e+bn(t)|n, g].
un=2 Re(anbn+1*),
vn=-2 Im(anbn+1*),
wn=|an|2-|bn+1|2,
ddtwn=2g(n+1) sin(pπvt/L)vn,
ddtvn=Δun-2g(n+1) sin(pπvt/L)wn,
ddtun=-Δvn.
ψ(0)=nCn 11+|μ|2(|e+μ|g)|n,
wn=vn(0)sin[θn(t)]+wn(0)cos[θn(t)],
vn=vn(0)cos[θn(t)]-wn(0)sin[θn(t)],
un=0.
wn=wn(0)cos[θn(t)],
vn=-wn(0)sin[θn(t)],
un=0.
θn(t)=2g(n+1)0tdτf(vτ).
θn(t)=2g(n+1) L2πv[1-cos(πvpt/L)],
θn(tT)=2πq,tT=L/v.
(Δ/g)(gL/pπv)1.

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