112
3 Electronic Excitation and Decay
The treatment of this problem is quite analogous to that of a true continuum, except
that instead of integrals for the continuum states |D ε we have here summations for
the discrete states |D K . In particular, Eq. (3.119) is replaced by
˙
b(t) = −
1
2
K
t
0
b(t
) e
−i(ε K −ε B )(t−t
)/ dt
V B,K
2
(3.126)
where V B,K =
B
ˆ
V
D K
. If the density of states is very high, one can define an
average
V B,K
2 for the |D K states having ε K ∈ [ε, ε + δε]. The (large) number of
states in this (small) interval is ρ(ε)δε. We shall indicate the average as |V B (ε)|
2
and we can go back to the integral formulation (3.119) by replacing |V B (ε)|
2 with
|V B (ε)|
2
ρ(ε):
˙
b(t) = −
1
2
∞
ε min
t
0
b(t
) e
−i(ε−ε B )(t−t
)/ dt
|V B (ε)|
2
ρ(ε) dε .
(3.127)
We now apply the same approximations as in the previous section and in particular
we assume the product of squared coupling times state density to be constant. In
this way we get Fermi’s Golden Rule for a quasi-continuum of states, where the
exponential lifetime is
τ =
2π |V B (ε B )|
2
ρ(ε B )
(3.128)
and the lineshape is
S(ω) =
(2πτ )
−1
(ω − ε B /) 2 + (2τ ) −2 =
|V B (ε B )|
2
ρ(ε B )
(E − ε B ) 2 + π 2
|V B (ε B )|
2
ρ(ε B )
2 . (3.129)
In one of the simplest models to which these equations can be applied, proposed
by Bixon and Jortner [9], the dark levels are equispaced (ε K +1 = ε K + Δε) and
V B,K = V is independent on K . Actually the numerical simulations of Fig. 3.7 were
performed by using this model, with a sufficiently high density of states ρ = 1/Δε.
It is interesting to see what happens when the density ρ is lower, i.e., the separation between consecutive levels is comparable with the linewidth FWHM ω =
2π |V B (ε B )|
2
ρ(ε B ). Within the Bixon-Jortner model this means Δε ≈ V . Figure 3.9
shows the results obtained with Δε = 2 cm
−1 and V = 1 cm
−1 . With such a low
density of states, another parameter of the Bixon-Jortner model becomes important, namely the difference ε C − ε B , where ε C is the dark level closest to ε B . In our
example, the two closest levels differ from ε B by ±Δε.
The upper panel of Fig. 3.9 shows the time dependence of the bright-state probability with such parameters. After an exponential decay in agreement with Fermi’s
rule down to very low values of P B (t), we see a sudden revival of the population.
3 Electronic Excitation and Decay
The treatment of this problem is quite analogous to that of a true continuum, except
that instead of integrals for the continuum states |D ε we have here summations for
the discrete states |D K . In particular, Eq. (3.119) is replaced by
˙
b(t) = −
1
2
K
t
0
b(t
) e
−i(ε K −ε B )(t−t
)/ dt
V B,K
2
(3.126)
where V B,K =
B
ˆ
V
D K
. If the density of states is very high, one can define an
average
V B,K
2 for the |D K states having ε K ∈ [ε, ε + δε]. The (large) number of
states in this (small) interval is ρ(ε)δε. We shall indicate the average as |V B (ε)|
2
and we can go back to the integral formulation (3.119) by replacing |V B (ε)|
2 with
|V B (ε)|
2
ρ(ε):
˙
b(t) = −
1
2
∞
ε min
t
0
b(t
) e
−i(ε−ε B )(t−t
)/ dt
|V B (ε)|
2
ρ(ε) dε .
(3.127)
We now apply the same approximations as in the previous section and in particular
we assume the product of squared coupling times state density to be constant. In
this way we get Fermi’s Golden Rule for a quasi-continuum of states, where the
exponential lifetime is
τ =
2π |V B (ε B )|
2
ρ(ε B )
(3.128)
and the lineshape is
S(ω) =
(2πτ )
−1
(ω − ε B /) 2 + (2τ ) −2 =
|V B (ε B )|
2
ρ(ε B )
(E − ε B ) 2 + π 2
|V B (ε B )|
2
ρ(ε B )
2 . (3.129)
In one of the simplest models to which these equations can be applied, proposed
by Bixon and Jortner [9], the dark levels are equispaced (ε K +1 = ε K + Δε) and
V B,K = V is independent on K . Actually the numerical simulations of Fig. 3.7 were
performed by using this model, with a sufficiently high density of states ρ = 1/Δε.
It is interesting to see what happens when the density ρ is lower, i.e., the separation between consecutive levels is comparable with the linewidth FWHM ω =
2π |V B (ε B )|
2
ρ(ε B ). Within the Bixon-Jortner model this means Δε ≈ V . Figure 3.9
shows the results obtained with Δε = 2 cm
−1 and V = 1 cm
−1 . With such a low
density of states, another parameter of the Bixon-Jortner model becomes important, namely the difference ε C − ε B , where ε C is the dark level closest to ε B . In our
example, the two closest levels differ from ε B by ±Δε.
The upper panel of Fig. 3.9 shows the time dependence of the bright-state probability with such parameters. After an exponential decay in agreement with Fermi’s
rule down to very low values of P B (t), we see a sudden revival of the population.