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Superluminal propagation of evanescent modes as a quantum
effect
Zhi-Yong Wang1*, Cai-Dong Xiong1, Bing He2
1School of Optoelectronic Information, University of Electronic Science and Technology of China,
Chengdu 610054, CHINA
2Department of Physics and Astronomy, Hunter College of the City University of New York, 695 Park
Avenue, New York, NY 10021, USA
*E-mail: zywang@uestc.edu.cn
Abstract
Contrary to mechanical waves, the two-slit interference experiment of single photons
shows that the behavior of classical electromagnetic waves corresponds to the quantum
mechanical one of single photons, which is also different from the quantum-field-theory
behavior such as the creations and annihilations of photons, the vacuum fluctuations, etc.
Owing to a purely quantum effect, quantum tunneling particles including tunneling photons
(evanescent modes) can propagate over a spacelike interval without destroying causality.
With this picture we conclude that the superluminality of evanescent modes is a quantum
mechanical rather than a classical phenomenon.
Keywords: Evanescent modes, superluminal propagation, quantum effect, virtual photons
PACS: 03.65.Xp, 41.20.Jb, 42.50.Nn, 03.65.Pm
1. Introduction
Nowadays, both theoretical and experimental investigations have presented a
conclusion that the evanescent modes of the electromagnetic field can superluminally
propagate [1-10]. At the level of quantum mechanics, via tunneling analogy the
superluminal propagation of evanescent modes has been described as the quantum tunneling
behavior of photons, which implies that the superluminality of evanescent modes is due to a
quantum effect. In this paper, at the level of quantum field theory, we will further show that
the superluminality of evanescent modes is due to a purely quantum effect, and clarify some
misunderstandings on the physical properties of evanescent modes. As an application, we
conclude that a recent objection [11-13] to the superluminality of evanescent modes is
invalid, because the objection is completely based on classical mechanics by regarding
electromagnetic waves as mechanical waves.
To avoid misunderstanding our argumentation (e.g., in spite of the genuine text of Refs.
[9, 10], in Ref. [13] the author made improper claim that “they mistake a non-zero
1
propagator for a non-zero commutator”), one should not confuse the following two issues:
(1) whether a particle can propagate over a spacelike interval? (2) whether such propagation
destroys causality (i.e., whether it means a measurement performed at one point can affect
another measurement at a point separated from the first with a spacelike interval), if a
particle does propagate over a spacelike interval? According to quantum field theory, a
non-zero propagator or non-zero transition probability amplitude for a spacelike interval
implies that a particle can propagate over the spacelike interval [14], but this spacelike
propagation does not destroy causality provided that the commutator of two observables
with a spacelike interval vanishes, that is, a measurement performed at one point does not
affect another measurement at a point separated from the first with a spacelike interval.
On the other hand, the commutator between two field operators located at spacelike
distance does not always vanish if the field operators are not observables. For example, in
the Coulomb gauge, the commutator between electromagnetic potentials does not vanish for
spacelike distances [15]. Moreover, in the quantization theory of evanescent modes [16], by
assuming the high-frequency behavior of the refractive index, one can find that the
commutator of evanescent field operators between two space-like separated points does not
vanish, whose physical meaning and the related causality problem have been discussed by
Stahlhofen and Nimtz [17].
2. Quantum field theory naturally explains superluminality of particles
A main reason to object all the existing theoretical and experimental investigations on
the superluminality of evanescent modes lies in the fact that such superluminal propagation
is in conflict with special relativity. However, special relativity has been developed on the
basis of classical mechanics without taking into account any quantum-mechanical effect. On
the other hand, because quantum field theory combines quantum mechanics with special
relativity, such that it can give us such a conclusion [14, 18, 19]: owing to
quantum-mechanical effect, a particle can propagate over a spacelike interval (without
destroying Einstein’s causality), which corresponds to the quantum tunneling phenomenon
[18].
For example, just as S. Weinberg discussed [19] (with some different notations and
conventions): “Although the relativity of temporal order raises no problems for classical
physics, it plays a profound role in quantum theories. The uncertainty principle tells us that
when we specify that a particle is at position at time , we cannot also define its
velocity precisely. In consequence there is a certain chance of a particle getting from
1
x1
t
2
11
(, )tx to even if the spacetime interval is spacelike, that is,
22
(, )tx12 12
ct t−> −xx .
To be more precise, the probability of a particle reaching if it starts at is
nonnegligible as long as (we call Eq. (1) Weinberg’s formula)
22
(, )tx11
(, )tx
22 2
12 12
0( ) ( ) ( )ct t mc<− − − ≤xx =2
, (1)
where is Planck’s constant (divided by ), c is the velocity of light in vacuum, and
is the particle’s mass, then
=2π
mmc= is the particle’s Compton wavelength. We are thus
faced with our paradox: if one observer sees a particle emitted at , and absorbed at
, and if is positive (but less than or equal to
11
(, )tx
22
(, )tx22
12 12
()(ct t−−−xx 2
)2
(mc=)), then
a second observer may see the particle absorbed at at a time before the time it
is emitted at . There is only one known way out of this paradox. The second observer
must see a particle emitted at and absorbed at . But in general the particle seen by
the second observer will then necessarily be different from that seen by the first observer (it
is the antiparticle of the particle seen by the first observer)”. In other words, to avoid a
possible causality paradox, one can resort to the particle-antiparticle symmetry. The process
of a particle created at and annihilated at as observed in a frame of
reference, is identical with that of an antiparticle created at and annihilated at
as observed in another frame of reference.
2
x2
t1
t
1
x
2
x1
x
11
(, )tx22
(, )tx
22
(, )tx
11
(, )tx
In fact, Weinberg’s argument is equivalent to the usual argument in
quantum-field-theory textbooks: let ()
x
φ
stand for a scalar field operator, 0 denote the
vacuum state, then 0()()0xy
φφ
represents the transition probability amplitude from the
state ()0y
φ
to the state ()0x
φ
[14], such that 2
0()()0xy
φφ
corresponds to the
probability for a scalar particle to propagate over the spacetime interval 2
()
x
y−. In
particular, if the probability amplitude for a scalar particle propagating over a spacelike
interval is denoted as
2
()xy−<0()0()()Dx y x y
φφ
−= 0
, then according to quantum
3
field theory, ()0()()0Dy x y x
φφ
−= represents the probability amplitude for the
corresponding antiparticle propagating backwards over the spacelike interval. Because
for , the two spacelike processes are undistinguishable
and the commutator
()(Dx y Dy x−= −)0
2
()xy−<
[(), ()] 0[(),()]0 0xy xy
φφ φφ
=
=, such that the causality is
maintained. Therefore, Weinberg has provided another way of looking at the statement that
“a measurement performed at one point does not affect another measurement at a point
separated from the first with a spacelike interval”. Studying a quantum Lorentz
transformation one can also obtain Eq. (1) [20].
In fact, Eq. (1) is just an approximation of a more rigorous result. For our purpose, let
us derive the rigorous result within the framework of quantum field theory. The transition
probability amplitude for any particle to propagate over the spacetime interval 2
()
x
y− can
be expressed in terms of ()0()()0Dx y x y
φφ
−= , and then we can ignore the spin
degree of freedom and take the scalar field ()
x
φ
for example. For convenience let
, , and denote
(0,0,0,0)y=( , ,0,0)xtr=(, ) ( )Dtr Dx y
=
−, according to quantum field
theory, one has (up to a constant factor)
d
( , ) exp[ i ( ) ]
2π2p
p
pc
Dtr Et pr
E
+∞
−∞
=−−
∫=, (2)
where 22 24
p
Epcm=+c
. Let (2)
0()
H
z denote the zero-order Hankel function of the
second kind, as the spacetime interval is spacelike (i.e., 22 2 0ct r
−
<), one can prove that,
(2) 2 2 2
0
(, ) ( i4) ( i )Dtr H r ct=− − − , (3)
where mc== is the Compton wavelength. Therefore, the asymptotic behaviors of
are governed by the Hankel function of imaginary argument: falls off like
(, )Dtr (, )Dtr
1 exp( )
z
z− for 222
zrct=− →+∞, while falls off faster than 1 exp( )
z
z
−
for
the other 222
zrct=− . In the observable sense, is always ignored for
(, )Dtr
222 1zrct=− >, that is, one always takes the approximate as follows:
4
22 2 2 2
22 2 2 2
0, for ( )
(, ) 0, for 0 ( )
ct r mc
Dtr ct r mc
⎧=−<−=−
⎪
⎨
≠
>−≥− =−
⎪
⎩
=
=
. (4)
According to the approximate given by Eq. (4), for the spacelike interval , the
probability amplitude for the particle to propagate from to
is nonnegligible as long as the Weinberg’s formula given by Eq. (1) is
satisfied. In other words, the Weinberg’s formula given by Eq. (1) is just an approximate of
the rigorous result given by Eq. (3). The rigorous result (3) implies that there are in
principle no limitations to the spacelike interval of
22 2 0ct r−<
(, )Dtr (0,0,0,0)y=
( , ,0,0)xtr=
222
rct−, but the probability 2
(, )Dtr
falls off rapidly for large 222
rct−, and the spacelike process cannot be observed
provided that the probability is too small.
Taking an electron for example, the Compton wavelength of the electron is
10
3.87 10mc −
≈×= (millimeter, mm). For any spacelike interval there is always an inertial
reference in which one has , using Eq. (1) one has
1
tt=2
22 2 10
12 12 12
( ) ( ) 3.87 10 mmct t mc −
−−−=−≤ ≈×xx xx =. (5)
The spacelike process occurring within such a spatial region is difficult to be observed. In
fact, within the Compton wavelength of the electron, the many-particle effects arising from
the creations and annihilations of virtual electron-positron pairs cannot be ignored. On the
other hand, for large spacelike interval the corresponding probability becomes so small that
the spacelike process cannot be observed. In a word, Eq. (1) as the approximate of the
rigorous result (3), describes a spacelike process with a sufficiently large probability (in the
observable sense).
3. Superluminal propagation of evanescent modes as a quantum effect
Photons inside a hollow waveguide can be treated as free massive particles with an
effective mass 2
eff c
m
ω
==c
[9, 21], where c
ω
is the cut-off frequency of the waveguide,
then the argument in Section 2 is also valid for the guided photons. In fact, quantum field
theory tells us that [9, 10]: owing to a quantum effect, photons inside a waveguide can
5
propagate over a spacelike interval, which corresponds to the fact that the evanescent modes
can propagate superluminally through an undersized waveguide (i.e., the photonic tunneling
phenomenon). Likewise, here the Einstein causality is preserved via the particle-antiparticle
symmetry, but for the moment the antiparticle of a photon is the photon itself, such that the
process that a photon propagates superluminally from A to B as observed in an inertial
frame of reference, is equivalent to that the photon propagates superluminally from B to A
as observed in another inertial frame of reference. Moreover, via quantum Lorentz
transformation [20] or by developing special relativity on the basis of quantum mechanics
[21], another theoretical evidence for the superluminality of evanescent modes can be
obtained, which also shows that the superluminal behavior of evanescent modes arises from
a quantum effect, i.e., the Heisenberg's uncertainty. However, here the theoretical evidence
is obtained at the level of quantum mechanics, it is just an approximate of the more rigorous
result given by Eq. (3), i.e., the related spacelike interval in a spacelike process is just the
one with a sufficiently large propagation probability (i.e., with
2(2) 2
0
( , ) (1 16)[ ( i)]Dtr H≥−
).
As an example, let the cut-off frequency of a waveguide be c9.49GHz
ω
=, the
effective Compton wavelength of photons inside the waveguide is
eff c 31.6mmmc c
ω
=≈=, which is far too larger than (i.e., the Compton
wavelength of the electron). That is, for tunneling photons, the spacelike interval with a
sufficiently large propagation probability is
10
3.87 10 mm
−
×
22 2
12 12 eff c
( ) ( ) 31.6mmct t mcc
ω
−−−≤ =≈xx =. (6)
Therefore, contrary to the superluminal behavior of tunneling electrons, the superluminal
behavior of evanescent modes can be easily observed experimentally. It is important to
mention that, Eq. (6) does not conflict with those experimental results with the largest
tunneling distance larger than 31.6 mm, because: 1) for a given spacelike interval
22
rct−2
, the propagation distance 12
r=−xx
is related to the propagation time
; 2) Eq. (6) as an approximate of the more rigorous result given by Eq. (3), just
corresponds to the result with a sufficiently large propagation probability (i.e., satisfying
12
(ttt=−)
6
2(2) 2
0
( , ) (1 16)[ ( i)]Dtr H≥−
), while Eq. (3) tells us that there are in principle no limitations
to the spacelike interval, though the probability 2
(, )Dtr falls off rapidly for large
spacelike interval.
Moreover, contrary to electrons, photons are bosons and do not carry any charge, by
increasing the number of tunneling photons the observable spacelike interval can be
augmented ad lib. Eq. (3) shows that the probability amplitude falls off
exponentially (but does not vanish) as the spacelike interval
(, )Dtr
222
rct
−
→+∞; on the other
hand, for the spacelike interval there is always an inertial reference in which one has 0t
=
.
Therefore, even if the propagation distance , the propagation time can be
arbitrarily small, which is in agreement with the Hartman effect [22].
r→+∞
In frustrated total internal reflection, evanescent modes as near field consist of virtual
photons [23-24], these virtual photons correspond to the elementary excitations of
electromagnetic interactions. Now we show that evanescent modes inside an undersized
waveguide are also identical with virtual photons. As we know, the near fields of a dipole
antenna fall off with the distance from the antenna like r1n
r ( ). However, if we
assume that an aerial array formed by an infinite set of infinite-length line sources arranging
in a periodic manner (with the period ), then the near fields of the aerial array falls off
like
2n≥
0
r
0
exp( )rr−. With respect to the TE10 mode, an undersized waveguide is equivalent to
such an aerial array [25] and evanescent modes inside the undersized waveguide are
equivalent to the near fields of the aerial array, which implies that evanescent fields inside
the waveguide can also be described by virtual photons. As we know, the propagation of
virtual photons is due to a purely quantum-mechanical effect, which also implies that one
cannot understand the propagation of evanescent waves via classical mechanics. To show
the evanescent TE10 mode (with the frequency c
ω
ω
<
, where c
ω
is the cut-off frequency)
is equivalent to the near field of the aerial array, basing on Ref. [25], one ought to make the
following replacements:
22
cc
0
ωω
=−→22
c
ω
ω
−, exp(i 0 ) 1t
⋅
⋅=→exp(i )t
ω
, (7)
22
22
()(,)0
→
y
Exz
xz
∂∂
+=
∂∂
22 2
2222
()(,
y
Exzt
xzct
∂∂ ∂ ,)0
+
−=
∂∂ ∂ , (8)
7
c
0
π
(,) sin( )exp( )
yz
x
Exz E ac
ω
=−
→0
π
( , , ) sin( )exp(i )
yx
E
xzt E t z
a
ω
κ
=−
, (9)
where 0c
πza c
ω
== , and 22
cc
κωω
=− . The presence of the decay factor exp( )z
κ
−
implies that the field (,,)
y
E
xzt mainly exists in the neighborhood of the aerial array, and
then is the near field of the aerial array.
4. Some misunderstandings about the physical properties of evanescent modes
To show that the superluminal propagation of evanescent modes is due to a purely
quantum effect, in addition to the argument presented above, some misunderstandings on
the physics properties of evanescent modes should be clarified. In particular, these
misunderstandings appear in the objection [11-13] to the superluminality of evanescent
modes.
Firstly, classical electromagnetic waves are conceptually different from mechanical
waves such as water waves, acoustic waves, waves on a string, etc. (e.g., only via media can
mechanical waves propagate, while electromagnetic waves can propagate in vacuum). For
example, the wave nature of mechanical waves is usually described by classical mechanics,
while the wave-particle dualism of light tells us that the wave nature of classical
electromagnetic waves is essentially a quantum mechanical issue, and historically quantum
mechanics arises from extending the wave-particle dualism of light to that of massive
particles. In terms of the spinor representation of electromagnetic field [26, 27], one can
obtain the quantum-mechanical theory of single photons (note that the usual quantum theory
of electromagnetic field is referred to the field-quantized theory rather than quantum
mechanics). In fact, to interpret the two-slit interference experiment of single photons
one-by-one emitted from a light source, one has to regard the behaviors of classical
electromagnetic waves as the quantum-mechanical ones of single photons, and here have
nothing to do with the quantum-field-theory effects such as the creations and annihilations
of photons, or the zero-point fluctuations of quantum electromagnetic field.
From the point of view of classical mechanics, inside an undersized waveguide
evanescent waves have support everywhere (through exponential damping) along the
undersized waveguide, and "the propagation of evanescent modes" is not a well-defined
concept. However, this classical picture just provides us with a phenomenological
description. From the point of view of quantum mechanics, now that some fraction of an
electromagnetic wave beam entering in the input side of an undersized waveguide with
8
finite length will come out of the exit of the waveguide, it indicates that there must have a
physical process that some photons in the evanescent wave beam propagate through the
undersized waveguide. As a quantum tunneling phenomenon, this physical process is due to
a purely quantum-mechanical effect without any classical correspondence, such that only
via quantum theory can one explain the propagation of evanescent modes. In other words, if
evanescent waves could not propagate, likewise any other quantum tunneling phenomenon
could not occur, because the wavefunctions of tunneling particles within a potential barrier
are similar to evanescent electromagnetic waves: they possess imaginary wave-numbers;
they do not describe propagating waves but evanescent waves.
On the other hand, plug in a signal into a tunnel and as long as it can be read out at the
other end, Einstein causality is violated [17].
In the objection to the superluminality of evanescent modes, evanescent waves are by
mistake regarded as “exponentially attenuated standing waves”. To clarify such a
misunderstanding, let us assume that a hollow waveguide is placed along the direction of
z-axes, and the waveguide is a straight rectangular pipe with the transversal dimensions
and (a, the cross-section of the waveguide lies in
a
bb>0
x
a
≤
≤ and ). Inside
the waveguide, for electromagnetic waves with the frequency
0yb≤≤
ω
and wave-number vector
(, , )
x
yz
kkk=k, take the electric field component
x
E for example, it can be written as
( , , , ) cos( )sin( )exp(i i )
xxyz
E
xyzt A kx ky t kz
ω
=
−, (10)
where A is a constant factor, π
x
kna
=
and π
y
klb
=
( ) are the
wavenumbers along the x-axis and y-axis directions, respectively. In Eq. (10),
, 0,1, 2,...nl=
(, ) cos( )sin( )
xy
f
xy A kx ky≡ represents a standing-wave factor (in which the wavenumbers
x
k and
y
k are real), which implies that the electromagnetic waves inside the waveguide
form a standing-wave structure along the
x
y plane (i.e., along the cross-section of the
waveguide) for both propagation and evanescent modes. On the other hand, in Eq. (10), for
a real ,
z
kexp(i i )
z
tkz
ω
− is a propagation factor and then represents
propagation modes; while for an imaginary
( , , , )
x
Exyzt
i
z
k
κ
=
− (
κ
is a real number),
exp(i i )
z
tkz
ω
− becomes the attenuation factor of exp(i )tz
ω
κ
−
and then
represents evanescent modes. Therefore, along the z-axis direction (i.e., along direction of
the waveguide), the electromagnetic waves form propagating- and evanescent-wave
structures for the propagation and evanescent modes, respectively. In other words, along the
( , , , )
x
Exyzt
9
direction of the waveguide, there is no standing-wave structure, such that the evanescent
modes cannot be regarded as “exponentially attenuated standing waves”. If one insists on
regarding the evanescent modes as “exponentially attenuated standing waves” for the reason
that they contain the standing-wave factor (, ) cos( )sin( )
xy
f
xy A kx ky
≡
, he would have to
call the propagation modes “propagating standing waves”, which is a self-contradictory
appellation.
A reason for denying that evanescent modes have quantum mechanical behaviors is
that “evanescent modes can completely be described by Maxwell’s equations”. However, it
is well known that Maxwell’s equations not only describe the classical electromagnetic field,
but also the quantum one (at the level of quantum field theory). On the other hand, just as
mentioned above, the two-slit interference experiment of single photons shows that the
behavior of classical electromagnetic waves corresponds to the quantum mechanical one of
single photons (in the first-quantized sense).
5. Conclusions
The two-slit interference experiment of single photons tells us that, contrary to
mechanical waves described by classical mechanics, the wave nature of electromagnetic
waves is essentially a quantum mechanical issue. Just as all other quantum tunneling
phenomena, the propagation of evanescent modes attributes to the quantum-mechanical
behavior of photons and cannot be understood via classical mechanics. The superluminal
propagation of evanescent modes can be interpreted by the quantum-mechanical behavior of
single photons (in terms of photonic quantum tunneling), or by the quantum-field-theory
behavior of the electromagnetic field (in terms of a non-zero transition probability
amplitude for a spacelike interval, or in terms of spacelike virtual photons). In a word, the
superluminal propagation of evanescent modes is a purely quantum mechanical
phenomenon without any classical correspondence. As a result, any objection to the
superluminality of evanescent modes is invalid provided that the objection is completely
based on classical mechanics.
Acknowledgments
The first author (Z. Y. Wang) would like to thank Professor G. Nimtz for his many
helpful discussions. This work was supported by the National Natural Science Foundation
of China (Grant No. 60671030) and Project supported by the Scientific Research Starting
Foundation for Outstanding Graduate, UESTC , China (Grant No. Y02002010501022).
10
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