Chemistry Reference
In-Depth Information
and Eq. (104) can be rewritten as:
d
dt
ρ
slow
˜
aa
ρ
slow
=
(107)
a
a
a
where
˜
aa
=
α
(
a
)
,β
(
a
);
α
(
a
)
,β
(
a
)
.
The solution of Eq. (107) is similar to that of Eq. (80). On the other hand, there
are uncoupled DM elemens with
|
α
−
β
| ≥
2
,
as in the secular approximation.
Instead of Eq. (102) one has
˜
αβ
)
t
E
−
1
slow
(0)
A
βα
ρ
αβ
(0)
e
−
(
iω
αβ
+
A
(
t
)
=
A
·
E
·
W
(
t
)
·
·
ρ
+
(108)
|
α
−
β
|
>
1
and instead of Eq. (103) one has
2
A
R
μ
−
μ
L
μ
slow
(0)
+
|
α
−
β
|
>
1
A
βα
ρ
αβ
(0)(
iω
αβ
3
N
−
2
μ
˜
αβ
)
−
1
·
·
ρ
+
=
τ
int
=
=
2
A
R
μ
L
μ
·
ρ
slow
(0)
+
|
α
−
β
|
>
1
A
βα
ρ
αβ
(0)
3
N
−
2
μ
·
(109)
B.
Linear Response
In this section, we consider a small harmonic perturbation
V
(
t
)
V
0
e
−
iωt
V
0
e
iωt
=
+
(110)
acting on the small system. The frequency
ω
of the perturbation can be close or
not to the resonance with any transition
ω
αβ
.
In the former case, the response to
the perturbation is of a resonance character and is effectuated by the presence of
V
(
t
) in the conservative term of the DME. If
ω
ω
αβ
for all
α, β,
the response
of the small system is due to the relaxation and depends on the relation between
ω
and the relaxation rate
.
To take account of this effect, one has to include
V
(
t
) into the relaxation terms of the DME. Temporal change of
V
(
t
) changes the
instantaneous equilibrium to which the system relaxes that gives rise to a dissipative
dynamics. Similar to the case of free evolution, Section III.A, the formalism can
be developed within the secular, semisecular, and nonsecular approximations. The
secular approximation will be used below for simplicity. It will be shown how the
method can be generalized for the nonsecular DME.
It is convenient to write the DME in the diagonal unperturbed basis
χ
(0
α
defined
by Eq. (46) with
H
s
that does not include
V
(
t
)
.
On the other hand, the relaxation
term in the DME naturally describes relaxation toward the quasiequilibrium DME
ρ
e
αβ
(
t
) depending on the instantaneous value of
V
(
t
) and it emerges in the basis
χ
α
(
t
) defined by Eq. (46) with
H
s
that includes
V
(
t
)
.
The general (nonsecular)
