Chemistry Reference
In-Depth Information
where 1
≤
n
≤
N
and 1
≤
k
≤
N
. The parameter
γ
kn
is the relaxation rate from
state
, which means that
γ
kn
ρ
kk
represents the population loss.
The dynamics of the off-diagonal elements of the density matrix is governed by
the following differential equation:
|
k
to state
|
n
ρ
kn
(
t
)
=−
i
([
H, ρ
])
kn
−
kn
ρ
kn
(
t
)
where
kn
is the dephasing rate between the states
. A part of the
dephasing is due to the relaxation rate, but other sources may exist. The total
dephasing rate can be written as follows:
|
k
and
|
n
N
1
2
˜
kn
+
kn
=
(
γ
mk
+
γ
mn
)
m
=
1
2
m
=
1
(
γ
mk
+
1
where the term
γ
mn
) is the contribution of the relaxation to the
dephasing rate and the term
˜
kn
the contribution of other sources called
pure
dephasing rate
. Note that the Lindblad equation imposes nontrivial constraints
on the different dissipation parameters of the Redfield equation (see [22] for a
derivation for three and four level quantum systems).
B. Construction of the Model
In this section, we discuss the last step of the construction of the model. We consider
the control of a two-level dissipative quantum system whose dynamics is governed
by the Lindblad equation. The evolution equation can be written as:
i
∂ρ
∂t
=
[
H
0
+
H
1
,ρ
]
+
i
L
(
ρ
)
(39)
where
H
0
is the field-free Hamiltonian of the system,
H
1
represents the interaction
with the control field and
L
is the dissipative part of the equation. Here,
H
1
is
assumed to be of the form
H
1
=−
μ
x
E
x
−
μ
y
E
y
where the operators
μ
x
and
μ
y
are proportional to the Pauli matrices
σ
x
and
σ
y
in the
eigenbasis of
H
0
. The electric field is the superposition of two linearly polarized
fields
E
x
and
E
y
. These two fields are in resonance with the Bohr frequency
E
2
−
E
1
. Also, this Hamiltonian describes the dynamics of a spin 1/2 particle in a
magnetic field. In this case, the term
H
0
corresponds to a constant magnetic field
along the
z
- axis and the dynamics is controlled by two magnetic fields polarized,
respectively, along the
x
- and
y
- axes.
