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Abstract

Strong enough electric field is predicted to spontaneously create electron-positron pairs in vacuum via Schwinger effect. A similar effect is also predicted to occur in sufficiently strong gravitational field. However, due to necessity of very large field strength, these effects were never observed in the experiment. Here we demonstrate that optical analog of a very strong gravitational field (up to ~10^24 g) may be created in electromagnetic metamaterials, leading to an optical analog of Schwinger effect in a metamaterial waveguide. Such waveguide geometries may also potentially be used to search for axion-like particles weakly interacting with electromagnetic field.
1
Optical analog of Schwinger effect
Igor I. Smolyaninov
Department of Electrical and Computer Engineering, University of Maryland, College
Park, MD 20742, USA
e-mail: smoly@umd.edu
Strong enough electric field is predicted to spontaneously create electron-positron
pairs in vacuum via Schwinger effect. A similar effect is also predicted to occur in
sufficiently strong gravitational field. However, due to necessity of very large field
strength, these effects were never observed in the experiment. Here we
demonstrate that optical analog of a very strong gravitational field (up to ~1024 g)
may be created in electromagnetic metamaterials, leading to an optical analog of
Schwinger effect in a metamaterial waveguide. Such waveguide geometries may
also potentially be used to search for axion-like particles weakly interacting with
electromagnetic field.
One of the well-known predictions of quantum electrodynamics (QED) is that electron-
positron pairs may be spontaneously created in vacuum in the presence of a very strong
electric field. This prediction is called the Schwinger effect [1]. Schwinger pair
production in a constant electric field takes place at a constant rate per unit volume

:
 
1223
232
1
4n
eE ncm
e
nc
eE
(1)
2
where m is the electron mass, and E is the electric field strength. Since pair production
takes place exponentially slowly when the electric field strength is much below the
Schwinger limit
18
32 1032.1 ecm
Es
V/m, (2)
Schwinger effect has never been observed in the experiment. For example, various
proposals to observe Schwinger effect in a strong laser field so far fall short of the
required electric field strength [2]. The electric field Es can be understood as the field
which can pull a couple of virtual charged particles of mass m out of quantum vacuum
to a distance of the order of the Compton wavelength of the particle
mc
C
, (3)
so that these particles become real. The corresponding condition for the electric field
strength is
2
mceE Cs
(4)
It was noted by several authors [3-5] that a similar gravitational Schwinger
effect must exist in a strong enough gravitational field (or acceleration). Similar to
Schwinger effect in electric field, a strong enough gravitational field Gs can create a
couple of particles with mass m out of vacuum if
2
mcmG Cs
, (5)
so that the necessary gravitational field can be estimated as
3
3
mc
Gs
(6)
If m is assumed to be equal to the electron mass, Gs~3x1028g is needed to observe the
gravitational Schwinger effect (where g is the free fall acceleration near Earth). If
existence of massive axion-like particles [6] weakly interacting with electromagnetic
field is assumed (axions are considered to be the leading dark matter candidate [7]), the
gravitational field required to observe axion creation due to Schwinger effect becomes
somewhat smaller. According to most estimates, the axion mass ma must fall in between
50 and 1,500 µeV [8], which means that a gravitational field (or acceleration) of the
order of Gs~3x1018 - 1021 g must be necessary for axion creation due to Schwinger
effect. In any case, only in the early universe the gravitational fields were probably
strong enough to lead to observable consequences of the cosmological Schwinger effect
[5].
As noted in [4], the gravitational Schwinger effect is closely related to Unruh
effect [9], which may be derived from Schwinger effect in the massless limit. Unruh
effect predicts that an accelerating object perceives its surroundings as a bath of thermal
radiation, even if it accelerates in vacuum. Similar to Schwinger effect, Unruh effect is
also believed to be very difficult to observe in the experiment, since an observer
accelerating at g=9.8 m/s2 should see vacuum temperature of only 4x10-20 K.
Nevertheless, very recently it was noted that sufficiently strong accelerations for
experimental observation of Unruh effect may be created in specially designed optical
waveguides [10]. Using hyperbolic metamaterials [11] the upper limit on the effective
accelerations may be pushed towards ~1024g [12]. Based on the estimates above, such
4
large magnitudes of effective accelerations may lead to an optical analog of the
gravitational Schwinger effect in electromagnetic metamaterials.
Let us demonstrate how an optical analog of a strong gravitational field may be
created using electromagnetic metamaterials. The equations of electrodynamics in the
presence of static gravitational field are identical to Maxwell equations in an effective
medium in which
2/1
00
g
, where g00 is the tt component of the metric tensor
[13]. It is convenient to describe effects arising in a homogeneous gravitational field (or
in a reference frame moving with constant acceleration) using the Rindler metric [13]:
2222
2
22
2dzdydxdt
cxG
ds
, (7)
where G is the constant proper acceleration measured by a co-moving accelerometer.
The analogy noted above indicates that very large constant gravitational field G may be
emulated using a metamaterial medium exhibiting the following coordinate
dependencies of its dielectric permittivity
and magnetic permeability
:
Gx
c2
(8)
A typical order of magnitude of effective accelerations achievable in such an optical
analog configuration may be estimated as at least
2
c
Geff
, (9)
which reaches ~1022g in the visible frequency range (assuming that optical experiments
are conducted at
~1m). As noted above, even larger magnitudes of effective
accelerations of the order of ~1024g appear to be achievable in some hyperbolic
5
metamaterial geometries [12]. However, in order for this analogy to be precise, one
must assume zero imaginary parts of
and

which is not typically the case in
electromagnetic metamaterials
On the other hand, this requirement may be met by
incorporating an optically (or electronically) pumped gain medium into the
metamaterial design, as described for example in [14]. External energy pumped into the
metamaterial may enable particle creation due to either Unruh or Schwinger effect,
which would be otherwise prohibited by energy conservation.
Photons in a waveguide behave as massive quasi-particles which may be
characterized by both inertial and gravitational mass, which obey the Einstein
equivalence principle [12,15,16]. Therefore, creation of such massive photons in a
waveguide-based optical analog of a strong gravitational field may be considered as
optical Schwinger effect. The estimates made above strongly indicate that the order of
magnitude of the analog gravity strength looks sufficient for the Schwinger effect
observation.
Let us consider an empty rectangular optical waveguide shown in Fig. 1(a),
which walls are made of an ideal metal, and assume that this waveguide has constant
dimensions (d and b) in the z- and y- directions, respectively. The dispersion law of
photons propagating inside this waveguide coincides with a dispersion law of massive
quasi-particles:
2
22
2
22
2
2
2
bJ
dI
k
cx
, (10)
6
where kx is the photon wave vector in the x-direction,
is the photon frequency, and I
and J are the mode numbers in the z- and y- directions, respectively. The effective
inertial rest mass of the photon in the waveguide is
2/1
2
22
2
22
2
bJ
dI
c
c
mij
eff
(11)
It is equal to its effective gravitational mass [12,16]. Let us assume that this waveguide
is either immersed in a constant gravitational field which is aligned along the x-
direction, or (equivalently) is subjected to accelerated motion. As demonstrated above,
since the gravitational field is static, this geometry may be represented by a static
waveguide filled with a medium, in which
and
gradually change as a function of x-
coordinate (see Fig.1(a)). In the weak gravitational field limit
2
00 2
1c
g
, (12)
where the gravitational potential
Gx
. Therefore, material parameters representing
such a waveguide may be chosen so that ε=μ, and the refractive index n of the
waveguide has a gradient in the x-direction:
 
2
2/1 1c
Gx
n

(13)
This means that the “optical dimensions” of the waveguide in the transverse directions
change as a function of the x-coordinate. Such a waveguide is called a tapered
waveguide. Similar to massive bodies, photons in this waveguide moving against an
external gravitational field eventually stop and turn around near the waveguide cut-off.
The effective waveguide acceleration may be related to the refractive index gradient as
7
dx
dn
cGa 2
(14)
In fact, as demonstrated in [12], the effect of spatial gradients of ε and μ inside the
waveguide and the effect of waveguide tapering (see Fig.1(b)) are similar, so that an
effective waveguide acceleration may be calculated as
dx
dc
ca gr
gr
, (15)
where
xgr dkdc /
is the group velocity of photons in the waveguide.
Optical analog of Schwinger effect in the waveguide geometries is illustrated in
Fig.2. Real massive photons may be created from the virtual (evanescent) ones due to
strong effective gravitational field in both the gradient index metamaterial waveguide
geometry (Fig.2(a)) and the tapered waveguide geometry (Fig.2(b)). In fact, the latter
configuration almost coincides with the well-known photon scanning tunneling
microscopy (PSTM) geometry [17], which means that the optical Schwinger effect
should be relatively straightforward to observe and study, as illustrated in Fig.3.
Fig.3(a) shows a scanning electron microscope image of a PSTM optical fiber
probe which was fabricated by overcoating a tapered optical fiber (its original shape is
indicated by the continuous red lines in Fig.3(a)) with a thick aluminum layer, and
cutting its end using focused ion beam milling [17]. A 100 nm open aperture (visible as
a dark black circle) was left at the tip apex. The approximate positions of the waveguide
cutoff at
0=632 nm and the Rindler horizon, as perceived by the photons passing
through the cutoff region, are shown by the dashed lines. The effective acceleration and
8
the approximate position X of the Rindler horizon for these photons may be estimated
using Eqs. (7) and (15) as
 
dxdbna
c
X/2 0
2
(16)
leading to a~1022 g (note however that the photon acceleration is not constant). A
3.25x3.25 m2 PSTM image of a test sample prepared by milling 100 nm wide linear
apertures through a 50 nm thick aluminum film deposited onto a glass slide surface is
shown in Fig.3(b). This image was collected using the tapered waveguide shown in
Fig.3(a), which was raster scanned over the sample surface. The sample was illuminated
from the bottom with 632 nm light. This simple experiment demonstrates that real
massive photons (contributing to the image in Fig.3(b)) may be created from the virtual
(evanescent) ones. The only difference between this demonstration and the optical
Schwinger effect is that the photons contributing to the PSTM image are not created
spontaneously. Indeed, the photon acceleration estimated above using Eq.(16) is of the
same order of magnitude as the Schwinger gravitational field Gs determined by Eqs. (6)
(using the effective photon mass defined by Eq.(11)).
Thus, the proper experimental geometries for the optical Schwinger effect
observation may be summarized in Fig.4. In the gradient index metamaterial waveguide
configuration depicted in Fig.4(a), optical pumping of the gain medium component of
the metamaterial-filled waveguide (which is necessary to achieve Im(
)=Im(
)=0
conditions) leads to directional flux of the generated Schwinger photons along the
direction of the effective gravitational field Gs. These photons are created spontaneously
through the volume of the metamaterial medium. The effective gravitational field Gs
must pull these virtual massive photons to a distance of the order of their Compton
9
wavelength, at which point they become real. A similar tapered waveguide
configuration is shown in Fig.4(b).
The optical Schwinger effect may be observed directly via measurements of the
directional flux of the Schwinger photons. Alternatively, it may be observed via
measurements of the reactive “push” exerted by the directional flux of the massive
Schwinger photons onto the tapered waveguide. In the G>>Gs limit this push may be
estimated based on Eq.(1) as
 
2
23
2
2
2424
Gm
V
c
mG
VmcF
, (17)
where V is the waveguide volume. Let us evaluate the order of magnitude of this effect,
since in principle it may be used to detect hypothesized axion-like particles which may
be also created in addition to photons due to the optical Schwinger effect. As we have
discussed above, Gs~3x1018 - 1021 g must be necessary for axion creation due to
Schwinger effect, while the range of effective accelerations, which may be created using
the proposed electromagnetic waveguide geometries is considerably larger. Moreover,
while in typical axion electrodynamics models the axion field is relatively weakly
coupled to the other electromagnetic degrees of freedom [6,7], this coupling is still
believed to be strong enough, so that several microwave cavity-based “axion
haloscopes” have been suggested to look for the hypothesized dark matter axions
[18,19].
The equations of macroscopic axion electrodynamics [20] are typically
introduced in such a way that the hypothetic axion contributions are incorporated into
the macroscopic D and H fields:
10
 
BgDD aa

, (18)
 
EgHH aa

, (19)
where
is the pseudo-scalar axion field and ga

is the small axion-photon coupling
constant. The resulting set of macroscopic Maxwell equations appears to be unchanged
[20]:
fa
D
, (20a)
0B
, (20b)
t
B
E
, (20c)
t
D
JH a
fa
, (20d)
where
f and Jf are the densities of free charges and currents. In fact, several solid state
systems, such as magnetoelectric antiferromagnet chromia [21], exhibit axion-like
quasiparticles which indeed follow this description. If the metamaterial waveguide
shown in Fig.4(a) is made using the magnetoelectric antiferromagnet chromia as one of
its component, these axion-like quasiparticles will be also generated in addition to
photons inside the waveguide due to the optical Schwinger effect. Since the equations of
macroscopic electrodynamics are known to be identical to the equations of
electrodynamics in vacuum in the presence of gravitational field (see [13] and a very
detailed discussion in [22]), emission of real vacuum axions due to the optical
Schwinger effect is also to be expected if strong enough optical analog of the
gravitational field is created.
11
The photon contribution to the Schwinger “push” defined by Eq.(17) may be
eliminated by closing the waveguide on both sides with sufficiently opaque metal
layers, thus making it into a tapered waveguide cavity. However, weakly interacting
real vacuum axions (if any) created due to Schwinger effect may still be able to leave
the cavity, thus pushing it in the opposite direction.
Since the resulting experimental geometry would look somewhat similar to the
controversial EM Drive configuration, and there were already several attempts to
measure its “thrust” (see [23] and the references therein), we may use the reported
sensitivity of these experiments to evaluate experimental feasibility of axion searches
using the optical Schwinger effect. No thrust has been reliably detected in these
experiments, while the measured noise floor in these experiments appears to fall into the
1N range. Given the reported cavity dimensions and its eigenfrequency [23], it appears
based on Eq.(17) that these kinds of measurements would be sensitive to the axion mass
range of the order of ma~10-2 eV. While higher than the estimated ma values between 50
and 1,500 µeV [8], it is not too far from the reported upper boundary, which means that
more refined thrust measurement techniques may become useful in experimental
searches for the axion-like particles. In particular, application of strong DC magnetic
field is predicted to considerably enhance axion-photon coupling in the microwave
cavity-based “axion haloscopes” [18,19]. Similar approach may be implemented in the
thrust-based experimental geometries.
In conclusion, the record high accelerations, which may be created using
metamaterial waveguides in terrestrial laboratories, appear to enable experimental
studies of Schwinger effect. Such experiments may also potentially be used to search for
axion-like particles weakly interacting with electromagnetic field.
12
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(2008).
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Physics, London, 2018).
13
[12] I. I. Smolyaninov, Giant Unruh effect in hyperbolic metamaterial waveguides,
arXiv:1811.08555 [physics.optics] (Optics Letters, in press)
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14
[23] M. Tajmar, G. Fiedler, Direct thrust measurements of an EM Drive and evaluation
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15
Figure Captions
Figure 1. Photon in a waveguide behaves as a massive quasi-particle: (a) Similar to
massive bodies, photons in a waveguide moving against an external gravitational field
eventually stop and turn around near the waveguide cut-off. Gradient of the effective
refractive index n in the waveguide is illustrated by shading. (b) The effect of external
gravitational field on a photon in a waveguide may be emulated by waveguide tapering,
which also leads to accelerated motion of massive photons.
Figure 2. Creation of real massive photons from the virtual (evanescent) ones due to
strong effective gravitational field (a) in a gradient index metamaterial waveguide, and
(b) in a tapered waveguide. The latter configuration almost coincides with the photon
scanning tunneling microscopy (PSTM) geometry.
Figure 3. (a) SEM photograph of a PSTM optical fiber probe which was fabricated by
overcoating a tapered optical fiber (indicated by the continuous red lines) with a thick
aluminum layer, and cutting its end using focused ion beam milling [17]. A 100 nm
open aperture (visible as a dark black circle) was left at the tip apex. The approximate
positions of the waveguide cutoff at
0=632 nm and the Rindler horizon, as perceived
by the photons passing through the cutoff region, are shown by the dashed lines. Note
however that the photon acceleration is not constant. (b) 3.25x3.25 m2 PSTM image of
a test sample prepared by milling 100 nm wide linear apertures through a 50 nm thick
aluminum film deposited onto a glass slide surface. The image was collected using a
probe shown in (a), which was raster scanned over the sample surface. The sample was
illuminated from the bottom with 632 nm light.
Figure 4. (a) Schematic geometry of the optical Schwinger effect in the gradient index
metamaterial waveguide configuration. Optical pumping of the gain medium component
16
of the metamaterial-filled waveguide, which is necessary to achieve Im(
)=Im(
)=0
conditions, leads to directional flux of the generated Schwinger photons. (b) Schematic
geometry of a similar tapered waveguide configuration.
17
Fig. 1
18
Fig. 2
19
Fig. 3
20
Fig. 4
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We pass in review the mechanisms of pair production in a static electric field and in black hole evaporation. New methods are developed in the electric case to show up the similarities with the black hole problem. The latter is then handled both through mode analysis and functional integration techniques wherein the situation of collapse is couched in terms of Eddington-Finkelstein coordinates. As in the electric case, the initial vacuum, identified with vacuum seen by a Schwarzschild observer before collapse, is unstable. We display in detail how it passes over to the Unruh vacuum, the quasi-stationary state which the Schwarzschild observer characterizes by the emitted thermal flux of Hawking.