Abstract

We investigate the time evolution of Morse coherent states in the potential of the NO molecule. We present animated wave functions and Wigner functions of the system exhibiting spontaneous formation of Schrödinger-cat states at certain stages of the time evolution. These nonclassical states are coherent superpositions of two localized states corresponding to two different positions of the center of mass. We analyze the degree of nonclassicality as the function of the expectation value of the position in the initial state. Our numerical calculations are based on a novel, essentially algebraic treatment of the Morse potential.

© 2002 Optical Society of America

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References

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  1. J. Parker and C. R. Stroud, Jr., �Coherence and decay of Rydberg Wave packets,� Phys. Rev. Lett. 56, 716-719 (1986).
    [CrossRef] [PubMed]
  2. D. L. Aronstein and C. R. Stroud, Jr., �Analytical investigation of revival phenomena in the finite square-well potential,� Phys. Rev. A 62, 022102-1�022102-9 (2000).
    [CrossRef]
  3. S. I. Vetchinkin and V. V. Eryomin, �The structure of wavepacket fractional revivals in a Morselike anharmonic system,� Chem. Phys. Lett. 222, 394-398 (1994).
    [CrossRef]
  4. K. P. Huber and G. Herzberg, Molecular spectra and molecular structure IV. Constants of diatomic molecules, (van Nostrand Reinhold, 1979).
  5. M. G. Benedict and B. Molnar, �Algebraic construction of the coherent states of the Morse potential based on supersymmetric quantum mechanics,� Phys. Rev. A 60 R1737-R1740 (1999).
    [CrossRef]
  6. B. Molnar, P. Foldi, M. G. Benedict and F. Bartha, �Time evolution in the Morse potential using supersymmetry: dissociation of the NO molecule,� quant-ph/0202069.
  7. J. Banerji and G. S. Agarwal, �Non-linear wave packet dynamics of coherent states of various symmetry groups,� Opt. Express 5, 220-229 (1999), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-5-10-220">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-5-10-220</a>.
    [CrossRef] [PubMed]
  8. J. Bertrand and M. Irac-Astaud, �The SU(1,1) coherent states related to the affine group wavelets,� Czech J. Phys. 51 (12), 1272-1278 (2001).
    [CrossRef]
  9. B. Molnar, M. G. Benedict and P. Foldi, �State evolution in the anharmonic Morse potential subjected to an external sinusoidal field,� Fortschr. Phys. 49, 1053-1057 (2001)
    [CrossRef]
  10. E. T. Jaynes and F. W. Cummings, �Comparison of quantum semiclassical radiation theories with application to the beam maser,� Proc. Inst. Elect. Eng. 51, 89-109 (1963).
  11. J. H. Eberly, N. B. Narozhny, J. J. Sanchez-Mondragon �Periodic spontaneous collapse and revival in a simple quantum model,� Phys. Rev. Lett 44, 1323-1327 (1980).
    [CrossRef]
  12. I. Sh. Averbukh and N. F. Perelman, �Fractional revivals: Universality in the long term evolution of quantum wave packets beyond the correspondence principle dynamics,� Phys. Lett. A 139, 449-453 (1989).
  13. C. Leichtle, I. Sh. Averbukh and W. P. Schleich, �Multilevel quantum beats: An analytical approach,� Phys. Rev. A. 54, 5299-5312 (1996).
    [CrossRef] [PubMed]
  14. P. Domokos, T. Kiss, J. Janszky, A. Zucchetti, Z. Kis and W. Vogel, �Collapse and revival in the vibronic dynamics of laser-driven diatomic molecules,� Chem. Phys. Lett. 322 3-4, 255-262 (2000).
    [CrossRef]
  15. Ch. Warmuth, A. Tortschano., F. Milota, M. Shapiro, Y. Prior, I. Sh. Averbukh, W. Schleich, W. Jakubetz and H. F. Kau.mann, �Studying vibrational wavepacket dynamics by measuring fluorescence interference fluctuations,� J. Chem. Phys. 112, 5060-5069 (2000).
    [CrossRef]
  16. Y. S. Kim and M. E. Noz, Phase space picture of quantum mechanics, (World Scientific, 1991).
  17. J. Janszky, An. V. Vinogradov, T. Kobayashi and Z. Kis, �Vibrational Schroedinger-cat states,� Phys. Rev. A 50, 1777-1784 (1994 ), and see also references therein.
    [CrossRef] [PubMed]
  18. J. Eiselt and H. Risken, �Quasiprobability distributions for the Jaynes-Cummings model with cavity damping,� Phys. Rev. A 43, 346-360 (1991).
    [CrossRef] [PubMed]
  19. M. G. Benedict, A. Czirjak, �Wigner functions, squeezing properties, and slow decoherence of a mesoscopic superposition of two-level atoms,� Phys. Rev. A 60, 4034-4044 (1999).
    [CrossRef]

Chem. Phys. Lett. (2)

S. I. Vetchinkin and V. V. Eryomin, �The structure of wavepacket fractional revivals in a Morselike anharmonic system,� Chem. Phys. Lett. 222, 394-398 (1994).
[CrossRef]

P. Domokos, T. Kiss, J. Janszky, A. Zucchetti, Z. Kis and W. Vogel, �Collapse and revival in the vibronic dynamics of laser-driven diatomic molecules,� Chem. Phys. Lett. 322 3-4, 255-262 (2000).
[CrossRef]

Czech J. Phys. (1)

J. Bertrand and M. Irac-Astaud, �The SU(1,1) coherent states related to the affine group wavelets,� Czech J. Phys. 51 (12), 1272-1278 (2001).
[CrossRef]

Fortschr. Phys. (1)

B. Molnar, M. G. Benedict and P. Foldi, �State evolution in the anharmonic Morse potential subjected to an external sinusoidal field,� Fortschr. Phys. 49, 1053-1057 (2001)
[CrossRef]

J. Chem. Phys. (1)

Ch. Warmuth, A. Tortschano., F. Milota, M. Shapiro, Y. Prior, I. Sh. Averbukh, W. Schleich, W. Jakubetz and H. F. Kau.mann, �Studying vibrational wavepacket dynamics by measuring fluorescence interference fluctuations,� J. Chem. Phys. 112, 5060-5069 (2000).
[CrossRef]

Opt. Express (1)

Phys. Lett. A (1)

I. Sh. Averbukh and N. F. Perelman, �Fractional revivals: Universality in the long term evolution of quantum wave packets beyond the correspondence principle dynamics,� Phys. Lett. A 139, 449-453 (1989).

Phys. Rev. A (6)

C. Leichtle, I. Sh. Averbukh and W. P. Schleich, �Multilevel quantum beats: An analytical approach,� Phys. Rev. A. 54, 5299-5312 (1996).
[CrossRef] [PubMed]

J. Janszky, An. V. Vinogradov, T. Kobayashi and Z. Kis, �Vibrational Schroedinger-cat states,� Phys. Rev. A 50, 1777-1784 (1994 ), and see also references therein.
[CrossRef] [PubMed]

J. Eiselt and H. Risken, �Quasiprobability distributions for the Jaynes-Cummings model with cavity damping,� Phys. Rev. A 43, 346-360 (1991).
[CrossRef] [PubMed]

M. G. Benedict, A. Czirjak, �Wigner functions, squeezing properties, and slow decoherence of a mesoscopic superposition of two-level atoms,� Phys. Rev. A 60, 4034-4044 (1999).
[CrossRef]

M. G. Benedict and B. Molnar, �Algebraic construction of the coherent states of the Morse potential based on supersymmetric quantum mechanics,� Phys. Rev. A 60 R1737-R1740 (1999).
[CrossRef]

D. L. Aronstein and C. R. Stroud, Jr., �Analytical investigation of revival phenomena in the finite square-well potential,� Phys. Rev. A 62, 022102-1�022102-9 (2000).
[CrossRef]

Phys. Rev. Lett. (2)

J. Parker and C. R. Stroud, Jr., �Coherence and decay of Rydberg Wave packets,� Phys. Rev. Lett. 56, 716-719 (1986).
[CrossRef] [PubMed]

J. H. Eberly, N. B. Narozhny, J. J. Sanchez-Mondragon �Periodic spontaneous collapse and revival in a simple quantum model,� Phys. Rev. Lett 44, 1323-1327 (1980).
[CrossRef]

Proc. Inst. Elect. Eng. (1)

E. T. Jaynes and F. W. Cummings, �Comparison of quantum semiclassical radiation theories with application to the beam maser,� Proc. Inst. Elect. Eng. 51, 89-109 (1963).

Other (3)

B. Molnar, P. Foldi, M. G. Benedict and F. Bartha, �Time evolution in the Morse potential using supersymmetry: dissociation of the NO molecule,� quant-ph/0202069.

K. P. Huber and G. Herzberg, Molecular spectra and molecular structure IV. Constants of diatomic molecules, (van Nostrand Reinhold, 1979).

Y. S. Kim and M. E. Noz, Phase space picture of quantum mechanics, (World Scientific, 1991).

Supplementary Material (2)

» Media 1: MOV (2071 KB)     
» Media 2: MOV (2181 KB)     

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Figures (4)

Fig. 1.
Fig. 1.

[2.1 MB] The absolute square of the wave functions corresponding to the Morse coherent states |x 0,0〉, with x 0 = 0.0 (ground state) and x 0 = 0.5. The time evolution of the latter initial wave function is shown in the attached movie file. These plots correspond to the case of the NO molecule, where s = 54.54.

Fig. 2.
Fig. 2.

The expectation value of the dimensionless position operator as a function of time. The initial states were |ϕ(t = 0)〉 = |x 0,0〉, with x 0 = 1.0, x 0 = 0.5 and x 0 = 0.06.

Fig. 3.
Fig. 3.

A) [1.3 MB] and B) [2.2 MB]. Frames of two movie files, showing Wigner functions of the Morse system at the initial stage of the time evolution and the formation of a Schrödinger-cat state. The plots correspond to t/T = 0 and t/T = 30, respectively. The initial state was |ϕ(t = 0)〉 = |x 0,0), with x 0 = 0.5.

Fig. 4.
Fig. 4.

Nonclassicality as a function of time. The initial state was |ϕ(t = 0)〉 = |x 0, 0〉, with x 0 = 1.0, x 0 = 0.5 and x 0 = 0.06.

Equations (9)

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H = P 2 + ( s + 1 2 ) 2 [ exp ( 2 X ) 2 exp ( X ) ] ,
d dt ϕ = i 2 π 2 s + 1 H ϕ ,
y β = ( 1 β 2 ) s Γ ( 2 s ) ( 1 β ) 2 s y s exp ( y 2 1 + β 1 β ) .
β = n = 0 N c n ψ n = n = 0 [ s ] [ ( 2 s 2 n ) Γ ( 2 s n + 1 ) n ! Γ ( 2 s ) Γ ( 2 s n ) Γ ( 2 s 2 n + 1 ) ( 1 β 2 ) s ( 1 β ) n
× 2 F 1 ( n , 2 s n ; 2 s 2 n + 1 ; 1 β ) ψ n ] + n = [ s ] + 1 N c n ψ n ,
X β = ln ( Re 1 + β 1 β ) , P β = s Re [ ( 1 + β ) ( 1 β ) ] Im [ ( 1 + β ) ( 1 β ) ] ,
X ( t ) = n , k = 0 N c n ( x ) c k * ( x ) ψ k X ψ n exp [ it 2 π 2 s + 1 ( E k ( s ) E n ( s ) ) ]
W x p t = 1 2 π ϕ * ( x + u 2 , t ) ϕ ( x u 2 , t ) e iup du .
M nc ( ϕ ) = 1 I + ( ϕ ) I ( ϕ ) I + ( ϕ ) + I ( ϕ ) ,

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