Vol 17 No 11, November 2008
c
° 2008 Chin. Phys. So c.
1674-1056/2008/17(11)/3985-06
Chinese Physics B
and IOP Publishing Ltd
Quantum mechanical description of waveguides
∗
Wang Zhi-Yong(王智勇)
a)†
, Xiong Cai-Dong(熊彩东)
a)
, and He Bing(何 兵)
b)
a)
School of Optoelectronic Information, University of Electronic Science and Technology of China, Chengdu 610054, China
b)
Department of Physics and Astronomy, Hunter College of the City University of New York, 695 Park Avenue,
New York, NY 10021, USA
(Received 15 February 2008; revised manuscript received 12 March 2008)
Applying the spinor representation of the electromagnetic field, this paper present a quantum-mechanical descrip-
tion of waveguides. As an example of application, a potential qubit generated by photon tunnelling is discussed.
Keywords: electromagnetic waves, waveguide, qubit, photon tunnelling
PACC: 0350D, 0365B, 0365C
1. Introduction
Waveguide theory is usually based on the clas-
sical or quantum field theory of the electromagnetic
field, while in the first-quantized sense, the quantum-
mechanical treatment of waveguides is absent. On the
other hand, contrary to mechanical waves, the two-
slit interference experiment of single photons shows
that the behaviour of classical electromagnetic waves
corresponds to the quantum mechanical one of single
photons, which is also different from the quantum-
field-theory behaviour such as the creations and anni-
hilations of photons, the vacuum fluctuations, etc. In
fact, in spite of the traditional conclusion that single
photon cannot be localized,
[1,2]
there have been devel-
oped photon wave mechanics which plays the role of
the first-quantized theory of single photons,
[3−10]
and
some recent studies have shown that photons can be
localized in space.
[11−13]
In this paper, in terms of the spinor represen-
tation of the electromagnetic field, we rewrite the
Maxwell equations of free electromagnetic fields as the
quantum-mechanical equation of single photons, and
basing on which we will present a quantum-mechanical
description of waveguides.
In the following, the natural units of measure-
ment (~ = c = 1) is applied, repeated indices must be
summed according to the Einstein rule, and the four-
dimensional (4D) space–time metric tensor is chosen
as g
µν
= diag(1, −1, −1, −1), µ, ν = 0, 1, 2, 3. For our
convenience, let x
µ
= (t, −x), instead of x
µ
= (t, x),
denote the contravariant position four-vector (and so
on), and then in our case ˆp
µ
= i∂/∂x
µ
≡ i∂
µ
=
i(∂
t
, −∇) denote 4D momentum operators.
2. Dirac-like equation of free pho-
tons
In vacuum the electric field, E = (E
1
, E
2
, E
3
),
and the magnetic field, B = (B
1
, B
2
, B
3
), satisfy the
Maxwell equations (~ = c = 1)
∇ × E = −∂
t
B, ∇ × B = ∂
t
E, (1)
∇ · E = 0, ∇· B = 0. (2)
Let k
µ
= (ω, k) denote the 4D momentum of pho-
tons (~ = c = 1), where ω is also the frequency
and k the wave-number vector. The vectors E and
B can also be expressed as the column-matrix form:
E = (E
1
E
2
E
3
)
T
and B = (B
1
B
2
B
3
)
T
(the super-
script T denotes the matrix transpose), by them one
can define a 6×1 spinor ψ(x) as follows:
ψ =
1
√
2
E
iB
. (3)
Moreover, by means of the 3 ×3 unit matrix I
3×3
and
the matrix vector τ = (τ
1
, τ
2
, τ
3
) with the compo-
nents
τ
1
=
0 0 0
0 0 −i
0 i 0
,
∗
Project supported by the National Natural Science Foundation of China (Grant No 60671030) and by the Scientific Research
Foundation for the Introduced Talents, UESTC (Grant No Y02002010501022).
†
Corresponding author. E-mail: zywang@uestc.edu.cn
http://www.iop.org/journals/cpb
http://cpb.iphy.ac.cn