3.10 Predissociation and Fermi’s Golden Rule
109
˙
b(t) −
1
2 |V B (ε B )|
2
t
0
b(t
)
∞
−∞
e
−i(ε−ε B )(t−t
)/ dε
dt
=
=
|2π V B (ε B )|
2
t
0
b(t
) δ(t − t
) dt
=
π |V B (ε B )|
2
b(t)
(3.120)
(see Appendix C for the properties of the δ function). The solution of this differential
equation is any combination of the two exponentials exp(±t/2τ ). Actually, we must
choose exp(−t/2τ ) for t > 0 and exp(t/2τ ) for t < 0 to avoid a divergence (we
shall see later why we are also interested in negative times). So, we can write
b(t) = e
−|t|/2τ
(3.121)
and
|b(t)|
2
= e
−|t|/τ
(3.122)
where
τ =
2π |V B (ε B )|
2
.
(3.123)
We see that the population of the bound state decays exponentially, with a rate constant τ
−1 that is proportional to |V B (ε B )|
2 , i.e., the squared module of the interaction
between |B and the dissociative state degenerate with it. This result is the celebrated
Fermi Golden Rule.
The autocorrelation function of |Ψ (t) is
A(t) = e
−|t|/2τ +iε B t/
(3.124)
and its Fourier transform yields the spectrum
S(ω) = ||B |E |
2
= (2π)
−1
+∞
−∞
e
−|t|/2τ e
−i(ω−ε B /)t dt =
=
(2πτ )
−1
(ω − ε B /) 2 + (2τ ) −2 =
|V B (ε B )|
2
(E − ε B ) 2 + π 2 |V B (ε B )| 4 .
(3.125)
This is a Lorentzian function, centered at E = ω = ε B , with the linewidth FWHM ω =
τ
−1
= 2π |V B (ε B )|
2
/. We see that the linewidth is inversely proportional to the
lifetime and both are determined by |V B (ε B )|
2 . For instance, a linewidth of 1 cm
−1
corresponds to τ = 5.3 ps. A light pulse much shorter than τ produces an excited
state without appreciable interference with the decay process. On the other hand, its
frequency width is large enough as to guarantee the approximation of constant ˜
E 0
within a range of several linewidths.
In Fig. 3.7 we see the results of two simulations for a system with V B (ε)
2
=
0.5 cm
−1 , which corresponds to a linewidth FWHM ω = 3.14 cm
−1 . According to
Fermi’s rule the lifetime is τ = 1890 fs. The excitation from the ground state is
done with a constant amplitude pulse as in the Rabi model, tuned to the G → B
109
˙
b(t) −
1
2 |V B (ε B )|
2
t
0
b(t
)
∞
−∞
e
−i(ε−ε B )(t−t
)/ dε
dt
=
=
|2π V B (ε B )|
2
t
0
b(t
) δ(t − t
) dt
=
π |V B (ε B )|
2
b(t)
(3.120)
(see Appendix C for the properties of the δ function). The solution of this differential
equation is any combination of the two exponentials exp(±t/2τ ). Actually, we must
choose exp(−t/2τ ) for t > 0 and exp(t/2τ ) for t < 0 to avoid a divergence (we
shall see later why we are also interested in negative times). So, we can write
b(t) = e
−|t|/2τ
(3.121)
and
|b(t)|
2
= e
−|t|/τ
(3.122)
where
τ =
2π |V B (ε B )|
2
.
(3.123)
We see that the population of the bound state decays exponentially, with a rate constant τ
−1 that is proportional to |V B (ε B )|
2 , i.e., the squared module of the interaction
between |B and the dissociative state degenerate with it. This result is the celebrated
Fermi Golden Rule.
The autocorrelation function of |Ψ (t) is
A(t) = e
−|t|/2τ +iε B t/
(3.124)
and its Fourier transform yields the spectrum
S(ω) = ||B |E |
2
= (2π)
−1
+∞
−∞
e
−|t|/2τ e
−i(ω−ε B /)t dt =
=
(2πτ )
−1
(ω − ε B /) 2 + (2τ ) −2 =
|V B (ε B )|
2
(E − ε B ) 2 + π 2 |V B (ε B )| 4 .
(3.125)
This is a Lorentzian function, centered at E = ω = ε B , with the linewidth FWHM ω =
τ
−1
= 2π |V B (ε B )|
2
/. We see that the linewidth is inversely proportional to the
lifetime and both are determined by |V B (ε B )|
2 . For instance, a linewidth of 1 cm
−1
corresponds to τ = 5.3 ps. A light pulse much shorter than τ produces an excited
state without appreciable interference with the decay process. On the other hand, its
frequency width is large enough as to guarantee the approximation of constant ˜
E 0
within a range of several linewidths.
In Fig. 3.7 we see the results of two simulations for a system with V B (ε)
2
=
0.5 cm
−1 , which corresponds to a linewidth FWHM ω = 3.14 cm
−1 . According to
Fermi’s rule the lifetime is τ = 1890 fs. The excitation from the ground state is
done with a constant amplitude pulse as in the Rabi model, tuned to the G → B