Information Technology Reference
In-Depth Information
3 Dissipative Coupling to the Heat Bath
A final approximation to be discussed here is the relaxation time approxima-
tion, which is commonly used in simulations of QCA dynamics. In the absence
of energy dissipation and other decohering effects, a QCA array will evolve
coherently according to the Liouville equation shown in Eq. (
14
). Over fairly
short time scales, quantum mechanical systems often fall to a thermal steady
state [
26
]. If the QCA system is weakly-coupled to the environment, and the
energy transfer between the system and environment is well-described by a
Markov process, then at low temperatures, the simplest way to incorporate
energy dissipation into a model of QCA dynamics is via the relaxation time
approximation [
26
,
40
,
42
,
46
,
48
]. This is done by adding a dissipation term to
Eq. (
15
):
H,
ˆ
i
d
dt
i
=
i
1
˄
(
i
−
ss
)
,
−
(20)
where
˄
is a phenomenological time constant and describes the relaxation of the
coherence vector towards the steady-state coherence vector element,
ss
.The
value of
˄
depends on the specific implementation details of the QCA system as
well as the nature of the coupling to the thermal environment, and would need
to be determined experimentally. The
ss
term can be found by evaluating,
=Tr
ʻ
ss
ˆ
i
,
ss
i
(21)
where
ʻ
ss
is the steady-state density matrix and is defined as,
e
− H
(
t
)
/k
B
T
Tr
e
− H
(
t
)
/k
B
T
.
ʻ
ss
↕
(22)
Determining the steady state density matrix exactly is critical to solving the
complete Schrodinger equation for the system, and thus comes up against all the
diculties mentioned above. It is therefore usually calculated using the ICHA
and the two-state approximation.
4 Conclusions
In this chapter, we reviewed the most common methods for modelling the dynam-
ics of large QCA circuits. Due to the inherent di
culties of solving complete
quantum mechanical problems, a number of simplifying assumptions are typi-
cally made when attempting to model the dynamic behaviour of large QCA cir-
cuits. These include a reduction of the Hilbert space to two states per cell (2-state
approximation), treatment of intercell interactions via a mean-field approach
(the ICHA), and an assumption of exponential energy relaxation (relaxation-
time approximation). Collectively, these assumptions form the basis of almost
Search WWH ::
Custom Search