Chemistry Reference
In-Depth Information
H
1
and
H
2
read in the eigenbasis of
H
0
as:
⎛
⎝
⎞
⎠
⎛
⎝
⎞
⎠
010
101
010
0
i
0
H
1
=
d
H
2
=
d
−
i
0
i
(11)
−
i
0
0
where
d
is a real constant. Equation (10) is written in units such that
=
1. A
basis of
of
H
0
. Here,
E
1
(
t
) and
E
2
(
t
) are two real components of the electric field along two orthogonal directions
of polarization. Both
E
1
(
t
) and
E
2
(
t
) are assumed to be in resonance with the
frequency
E
0
. In the rotating wave approximation [33], the equation for the time
evolution of
H
is given by the eigenvectors
|
1
,
|
2
, and
|
3
|
ψ
(
t
)
can be written as:
⎛
⎝
⎞
⎠
|
ue
iE
0
t
0
0
i
∂
u
∗
e
−
iE
0
t
ue
iE
0
t
∂t
|
ψ
(
t
)
=
E
0
ψ
(
t
)
(12)
u
∗
e
−
iE
0
t
0
2
E
0
where
u
is the complex Rabi frequency. In the interaction representation, Eq. (12)
becomes
⎛
⎝
⎞
⎠|
0
u
1
+
iu
2
0
i
∂
u
1
−
iu
2
u
1
+
iu
2
∂t
|
ψ
(
t
)
=
0
ψ
(
t
)
(13)
0
u
1
−
iu
2
0
where
u
1
and
u
2
are, respectively, the real and imaginary parts of the complex Rabi
frequency. We keep the same notation for the state
after this transformation.
The interaction representation here means that we have performed the unitary
transformation
U
to the state
|
ψ
(
t
)
|
ψ
(
t
)
:
⎛
⎝
⎞
⎠
10 0
0
e
−
iE
0
t
0
00
e
−
2
iE
0
t
U
=
Note that this transformation allows us to eliminate the drift term due to the field-
free Hamiltonian
H
0
[14].
We denote by
c
1
,
c
2
, and
c
3
the complex coefficients of the state
|
ψ
(
t
)
in
the basis
{|
1
,
|
2
,
|
3
}
. We introduce the real coefficients
x
i
(
i
∈{
1
,
2
,...
6
}
) de-
fined by
⎧
⎨
c
1
=
x
1
+
ix
2
c
2
=
x
3
+
ix
4
(14)
⎩
c
3
=
x
5
+
ix
6
