86
T. Szidarovszky et al.
where |α kn and |β k(n−1) represent the two manifolds of rovibronic states with Fourier
indices n and n − 1, respectively. If the nonresonant coupling terms are neglected in
the Floquet approach, the Floquet Hamiltonian becomes block diagonal [see (4.19)],
|α kn and |β k(n−1) become the same for all n, and (4.36) is simplified to
| k (t) = (|α k + |β k e
−iω 1 t
)e
inω 1 t
.
(4.37)
In (4.37), | k (t) is a Floquet state obtained from the nth two-by-two block of the
Floquet Hamiltonian, and the quasienergy associated with | k (t) may be written
as ε k + nω 1 . As explained under (4.6), if the quasienergy is shifted by −nω 1 and
the Floquet state is multiplied by e
−inω 1 t , one obtains an equivalent physical state.
Therefore, (4.37) can be rewritten as
| k (t) = |α k + |β k e
−iω 1 t
,
(4.38)
with the corresponding quasienergy ε k . Using (4.38) instead of (4.7) leads to a simplified version of (4.35),
(F)
I | I (t) =
l,k
a ∗
l b k e −
i
(ε k −ε l )t 0 l (t 0 )| k (t 0 )
−
1
2i
l,k
a ∗
l b k
t
t 0
β l |E 2 ˆ
μ|α k e −
i
(ε k −ε l −ω 1 ±ω 2 )t
dt
−
1
2i
l,k
a ∗
l b k
t
t 0
α l |E 2 ˆ
μ|β k e −
i
(ε k −ε l +ω 1 ±ω 2 )t
dt , (4.39)
where we use the fact that α l |E 2 ˆ
μ|α k = =β l |E 2 ˆ
μ|β k = 0 for a molecule with no
permanent dipole. By following the standard TDPT procedure, and assuming that
the initial and final states of the transition in (4.39) are the kth and lth light-dressed
states, the T l←k transition amplitude becomes
T l←k ∝
v,J
v ,J
C
(l)∗
Av J C
(k)
Xv J Av
J
|E 2 ˆ
μ|Xv J δ(ε k − ε l − ω 1 ± ω 2 )
+
v,J
v ,J
C
(l)∗
Xv J C
(k)
Av J Xv
J
|E 2 ˆ
μ|Av J δ(ε k − ε l + ω 1 ± ω 2 ). (4.40)
Based on the arguments of the delta functions in (4.40), the first term can be interpreted as a transition between the light-dressed states having quasienergies ε k and
ε l + ω 1 , that is, in the notation of (4.20), a transition between | k (n) and | l (n
)
with n = n
− 1. Similarly, the second term in (4.40) can be interpreted as a transition
between | k (n) and | l (n
) with n = n
+ 1.
T. Szidarovszky et al.
where |α kn and |β k(n−1) represent the two manifolds of rovibronic states with Fourier
indices n and n − 1, respectively. If the nonresonant coupling terms are neglected in
the Floquet approach, the Floquet Hamiltonian becomes block diagonal [see (4.19)],
|α kn and |β k(n−1) become the same for all n, and (4.36) is simplified to
| k (t) = (|α k + |β k e
−iω 1 t
)e
inω 1 t
.
(4.37)
In (4.37), | k (t) is a Floquet state obtained from the nth two-by-two block of the
Floquet Hamiltonian, and the quasienergy associated with | k (t) may be written
as ε k + nω 1 . As explained under (4.6), if the quasienergy is shifted by −nω 1 and
the Floquet state is multiplied by e
−inω 1 t , one obtains an equivalent physical state.
Therefore, (4.37) can be rewritten as
| k (t) = |α k + |β k e
−iω 1 t
,
(4.38)
with the corresponding quasienergy ε k . Using (4.38) instead of (4.7) leads to a simplified version of (4.35),
(F)
I | I (t) =
l,k
a ∗
l b k e −
i
(ε k −ε l )t 0 l (t 0 )| k (t 0 )
−
1
2i
l,k
a ∗
l b k
t
t 0
β l |E 2 ˆ
μ|α k e −
i
(ε k −ε l −ω 1 ±ω 2 )t
dt
−
1
2i
l,k
a ∗
l b k
t
t 0
α l |E 2 ˆ
μ|β k e −
i
(ε k −ε l +ω 1 ±ω 2 )t
dt , (4.39)
where we use the fact that α l |E 2 ˆ
μ|α k = =β l |E 2 ˆ
μ|β k = 0 for a molecule with no
permanent dipole. By following the standard TDPT procedure, and assuming that
the initial and final states of the transition in (4.39) are the kth and lth light-dressed
states, the T l←k transition amplitude becomes
T l←k ∝
v,J
v ,J
C
(l)∗
Av J C
(k)
Xv J Av
J
|E 2 ˆ
μ|Xv J δ(ε k − ε l − ω 1 ± ω 2 )
+
v,J
v ,J
C
(l)∗
Xv J C
(k)
Av J Xv
J
|E 2 ˆ
μ|Av J δ(ε k − ε l + ω 1 ± ω 2 ). (4.40)
Based on the arguments of the delta functions in (4.40), the first term can be interpreted as a transition between the light-dressed states having quasienergies ε k and
ε l + ω 1 , that is, in the notation of (4.20), a transition between | k (n) and | l (n
)
with n = n
− 1. Similarly, the second term in (4.40) can be interpreted as a transition
between | k (n) and | l (n
) with n = n
+ 1.