Abstract

We investigate the analogy between exponential decay of a quantum system into a continuum, and laser-induced excitation of a molecular wave packet. We find that the analogy exists, but it is not as clear-cut for the excited vibrational states of the electronic molecular ground state, as it is for the corresponding vibrational ground state.

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References

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  1. U. Fano, "Effects of conguration interaction on intensities and phase shifts", Phys. Rev. 124, 1866-1878 (1961).
    [CrossRef]
  2. V. Weisskopf and E. Wigner, "Berechnung der nat urlichen Linienbreite auf Grund der Diracschen Lichttheorie", Z. Phys. 63, 54 (1930).
    [CrossRef]
  3. R. E. Peierls, in Proceedings of the 1954 Glasgow Conference (Pergamon Press, New York, 1955), 296.
  4. B. Zumino, in Lectures on Field Theory and the Many-Body Problem, ed. E. R. Caianello (Academic Press, New York, 1961), 27.
  5. M. Levy, in Lectures on Field Theory and the Many-Body Problem, ed. E. R. Caianello (Academic Press, New York, 1961), 47.
  6. M. L. Goldberger and K. M. Watson, Collision Theory, (J. Wiley & Sons, New York, 1964), Chap. 8.
  7. J. Jortner, S. A. Rice and R. M. Hochstrasser, Adv. Photochem., 7, 149 (1969).
  8. M. Gruebele and A. H. Zewail, "Ultrafast reaction dynamics", Physics Today 43(5), 24-33, 1990.
    [CrossRef]
  9. B. M. Garraway and K.-A. Suominen, "Wave-packet dynamics: new physics and chemistry in femto-time", Rep. Prog. Phys. 58, 365-419 (1995).
    [CrossRef]
  10. A. Paloviita, K.-A. Suominen and S. Stenholm, "Weisskopf-Wigner model for wavepacket excitation", J. Phys. B 30, 2623-2633 (1997), http://xxx.lanl.gov/abs/quant-ph/9703011
    [CrossRef]
  11. K.-A. Suominen, B. M. Garraway and S. Stenholm, "Wave-packet model for excitation by ultra-short pulses", Phys. Rev. A 45, 3060-3070 (1992).
    [CrossRef] [PubMed]
  12. E. U. Condon, "Nuclear motions associated with electron transitions in diatomic molecules", Phys. Rev. 32, 858-872 (1928).
    [CrossRef]
  13. M. S. Child, Semiclassical Mechanics with Molecular Applications, (Clarendon Press, Oxford, 1991), 122.
  14. K. Burnett, P. S. Julienne and K.-A. Suominen, "Laser-driven collisions between atoms in Bose-Einstein condensed gas", Phys. Rev. Lett. 77, 1416-1419 (1996).
    [CrossRef] [PubMed]

Other

U. Fano, "Effects of conguration interaction on intensities and phase shifts", Phys. Rev. 124, 1866-1878 (1961).
[CrossRef]

V. Weisskopf and E. Wigner, "Berechnung der nat urlichen Linienbreite auf Grund der Diracschen Lichttheorie", Z. Phys. 63, 54 (1930).
[CrossRef]

R. E. Peierls, in Proceedings of the 1954 Glasgow Conference (Pergamon Press, New York, 1955), 296.

B. Zumino, in Lectures on Field Theory and the Many-Body Problem, ed. E. R. Caianello (Academic Press, New York, 1961), 27.

M. Levy, in Lectures on Field Theory and the Many-Body Problem, ed. E. R. Caianello (Academic Press, New York, 1961), 47.

M. L. Goldberger and K. M. Watson, Collision Theory, (J. Wiley & Sons, New York, 1964), Chap. 8.

J. Jortner, S. A. Rice and R. M. Hochstrasser, Adv. Photochem., 7, 149 (1969).

M. Gruebele and A. H. Zewail, "Ultrafast reaction dynamics", Physics Today 43(5), 24-33, 1990.
[CrossRef]

B. M. Garraway and K.-A. Suominen, "Wave-packet dynamics: new physics and chemistry in femto-time", Rep. Prog. Phys. 58, 365-419 (1995).
[CrossRef]

A. Paloviita, K.-A. Suominen and S. Stenholm, "Weisskopf-Wigner model for wavepacket excitation", J. Phys. B 30, 2623-2633 (1997), http://xxx.lanl.gov/abs/quant-ph/9703011
[CrossRef]

K.-A. Suominen, B. M. Garraway and S. Stenholm, "Wave-packet model for excitation by ultra-short pulses", Phys. Rev. A 45, 3060-3070 (1992).
[CrossRef] [PubMed]

E. U. Condon, "Nuclear motions associated with electron transitions in diatomic molecules", Phys. Rev. 32, 858-872 (1928).
[CrossRef]

M. S. Child, Semiclassical Mechanics with Molecular Applications, (Clarendon Press, Oxford, 1991), 122.

K. Burnett, P. S. Julienne and K.-A. Suominen, "Laser-driven collisions between atoms in Bose-Einstein condensed gas", Phys. Rev. Lett. 77, 1416-1419 (1996).
[CrossRef] [PubMed]

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Figures (12)

Fig. 1.
Fig. 1.

The potentials (U 1 and U 2) and initial wave packets |φn (x)|2 for our model. We show the harmonic ground state potential U 1(x) (blue solid line), the corresponding energies of the eigenstates εn (dashed lines) and the squared eigenfunctions (solid lines) for n = 1 to n = 4. We show also the corresponding excited state potentials U 2(x) (dotted lines) for each n when α = 1.

Fig. 2.
Fig. 2.

The perturbation theory result for the Franck-Condon factors, as defined in Eq. (23), and calculated numerically for the 6 lowest harmonic oscillator states as a function of the slope parameter α.

Fig. 3.
Fig. 3.

The decay of the vibrational ground state (n = 0) population P 1 for various values of α. For small times exp[-Γ(t - t 0)] ≃ 1 - Γ(t - t 0) when tt 0, and the exponential decay appears as a linear change in P 1(t). The results are obtained using the wave packet approach. Here we have set V 0 = 0.1.

Fig. 4.
Fig. 4.

As Fig. 3, but now the decaying state is the excited vibrational state n = 3.

Fig. 5.
Fig. 5.

The excited electronic state population P 2 as a function of time t and coupling strength V 0. Here n = 0 and α = 2. In the decay model for wave packet excitation this describes the occupation of the continuum modes.

Fig. 6.
Fig. 6.

The excited electronic state population P 2 as a function of time t and coupling strength V 0. Here n = 4 and α = 2.

Fig. 7.
Fig. 7.

The excited electronic state population P 2 as a function of time t and coupling strength V 0. Here n = 4 and α = 3.

Fig. 8.
Fig. 8.

The excited state decay rate Γ(α) for case n = 0, obtained a) using wave packet calculations, and the perturbation theory result with b) numerically evaluated overlap integral, c) modified Condon reflection principle and d) simple Condon reflection principle.

Fig. 9.
Fig. 9.

The excited state decay rate Γ(α) for case n = 1, obtained a) using wave packet calculations, and the perturbation theory result with b) numerically evaluated overlap integral and c) modified Condon reflection principle.

Fig. 10.
Fig. 10.

The excited state decay rate Γ(α) for case n = 2, obtained a) using wave packet calculations, and the perturbation theory result with b) numerically evaluated overlap integral, c) modified Condon reflection principle and d) simple Condon reflection principle.

Fig. 11.
Fig. 11.

The excited state decay rate Γ(α) for case n = 3, obtained a) using wave packet calculations, and the perturbation theory result with b) numerically evaluated overlap integral and c) modified Condon reflection principle.

Fig. 12.
Fig. 12.

The excited state decay rate Γ(α) for case n = 4, obtained a) using wave packet calculations, and the perturbation theory result with b) numerically evaluated overlap integral, c) modified Condon reflection principle and d) simple Condon reflection principle.

Equations (24)

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i t [ ψ 1 ψ 2 ] = [ ( 2 x 2 + U 1 ( x ) ) V V ( 2 x 2 + U 2 ( x ) ) ] [ ψ 1 ψ 2 ] ,
U 2 ( x ) = β αx ,
U 1 ( x ) = 1 2 x 2 .
m = 1 2 ; ω = 2 .
ε n = ω ( 1 2 + n ) = 1 2 + 2 n .
ψ 1 ( x , t = 0 ) = φ n ( x )
ψ ˜ E ( k ) dx 2 π e ikx ψ E ( x ) .
ψ E ( x ) = dk 2 π α exp [ i ( x + E β α ) k i k 3 3 α ] .
ψ E * ( x ) ψ E ( x ) dx = δ ( E E ) .
ψ E * ( x ) ψ E ( x ) dE = δ ( x x ) .
V 2 α ω
P 1 ( t ) = ψ 1 ( x , t ) 2 dx
i = φ n ( x ) 1 ,
Γ = 2 π i H f 2 = 2 π V 2 φ n | ψ E 0 2 ,
E 0 = ε n = 1 2 + 2 n
S n = φ n | ψ E 0 ,
Γ 2 π V 2 α Γ 0 ,
φ n ( k ) = N n H n ( 2 1 4 k ) e k 2 2 .
N n = ( 2 1 4 π 1 2 2 n n ! ) 1 2 .
S n = N n dk 2 πα H n ( 2 1 4 k ) exp ( i k 3 3 α k 2 2 ) .
lim α S n 2 = d ( U 2 U 1 ) dx x = x 0 1 φ n ( x = x 0 ) 2 ,
V ( t ) = { V 0 sech [ ( t t 0 ) T ) ] , t t 0 V 0 t > t 0 ,
lim α Γ V 0 2 = 2 π S n 2 .
P 2 ( t ) = ψ 2 ( x , t ) 2 dx

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