Chemistry Reference
In-Depth Information
the rate
S
−
k
+
1
,S
−
k
.
Since the exact eigenstates at the tunneling resonance are
|±
(see Section IV.F), both of
these eigenstates are damped with the rate of order
S
−
k
+
1
,S
−
k
(in fact, half of
it). The secular DME uses rate equations for
ρ
that are linear combinations of
|−
S
and
|
S
−
k
,ρ
,
and so on, and the ini-
++
−−
|
−
tial condition spin in the state
S
gives rise to the initial conditions
ρ
(0)
=
++
ρ
relax with a rate of order
S
−
k
+
1
,S
−
k
,
the spin quickly leaves the metastable state, even in the case of a vanishing tunnel
splitting,
(0)
=
1
/
2
.
Since both
ρ
and
ρ
−−
++
−−
0
.
Indeed, such an unphysical behavior follows from the analyti-
cal and numerical solution of the secular DME at weak tunneling resonances. In
contrast, coupling of
ρ
→
and
ρ
to the slow nondiagonal DM elements
ρ
++
−−
+−
and
ρ
in the nonsecular DME leads to the physically expected vanishing of the
escape rate in the limit
−+
0
.
Below we will use the nonsecular DME, Eq. (52), in the development of the
formalism. The secular and semisecular reductions of it can be obtained later. The
relaxation tensor
R
αβ,α
β
→
0 and
T
→
is a sum of two contributions,
R
(1)
R
(2)
αβ,α
β
R
αβ,α
β
=
αβ,α
β
+
(159)
that are due to the first- and second-order phonon processes. These contributions
will be calculated separately below.
2.
Initial Condition for Free Relaxation
Let us consider the question of the initial state of the spin in the case of free
evolution. Typically, in resonance experiments it is the first excited state. Although
in these experiments only a small portion of the population is being transferred
from the ground to the excited state, one can consider the system prepared fully
in the excited state because of the linearity of the DME. Preparing the spin in the
metastable energy minimum, one can study its thermal activation over the barrier
and tunneling under the barrier. In general, it is not easy to find the quantum
mechanical state realizing or approximating this classical state, and in the case
of a tunneling resonance, such a state does not exist. A good practical way to
create such an initial condition is to prepare the spin in the coherent state
pointing in the direction of the metastable minimum found classically. The spin
coherent state is given by
|
n
(
θ, ϕ
)
S
|
n
(
θ, ϕ
)
=
C
m
|
m
(160)
m
=−
S
where
2
S
S
1
/
2
cos
θ
2
S
+
m
sin
θ
2
S
−
m
e
−
imϕ
C
m
=
(161)
+
m
