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m
v
h
ω
Δ
p
=
*
−
(14)
c
2
2
v
/
c
−
1
where
m
is an absolute value of the mass for the tunneling photon moving at superluminal
speed and
ω
is an angular frequency of the photon, the velocity of the tunneling photon can
be estimated as
⎛
⎞
1
v
≈
c
⎜
⎝
1
+
⎟
⎠
(15)
ω
t
from Eqs.(4) and (10), and
=
E
[Musha, 2006].
If we let the tunneling distance be
d
, the time for a photon tunneling across the barrier
can be roughly estimated as by
ω
Δ
t
=
d
/
v
. Then the velocity of the tunneling photon can be
given by
⎛
⎞
(16)
c
c
c
2
⎜
⎝
⎟
⎠
v
≈
c
1
+
+
+
2
ω
d
ω
d
4
ω
2
d
2
From which, the ratio of the minimum energy required for computation by QTP gates and
the conventional computation can be given as follows by equating Eqs.(12) and (13);
<
E
>
1
(17)
*
R
=
≈
<
E
>
β
(
β
−
1
where
2
c
c
c
(18)
β
≈
1
+
+
+
2
2
2
ω
d
ω
d
4
ω
d
By the Higgs mechanism in quantum field theory, the penetration depth of tunneling
photons can be estimated by [Jibu et al., 1996].
h
(19)
r
=
0
Mc
where
M
is the effective mass which yields
M
10≈
.
From which, the penetration depth of the tunneling photon is estimated as
eV
1 ×
m.
Then the ratio of minimum energy required for the computation by QTP gates to the
conventional computation processes can be estimated as shown in Figure 3, when we let the
tunneling distance of the barrier be
8
.
10
d
≈
10
n
m.
