# Electrostatic tuning of bilayer graphene edge modes

Hira Ali  Llorenç Serra<EMAIL_ADDRESS>(H.A<EMAIL_ADDRESS>(L.S) Institute for Cross-Disciplinary Physics and Complex Systems IFISC
(CSIC-UIB), E-07122 Palma, Spain
Physics Department, University of the Balearic Islands, E-07122 Palma, Spain

###### Abstract

We study the effect of a local potential shift induced by a side electrode on
the edge modes at the boundary between gapped and ungapped bilayer graphene. A
potential shift close to the gapped-ungapped boundary causes the emergence of
unprotected edge modes, propagating in both directions along the boundary.
These counterpropagating edge modes allow edge backscattering, as opposed to
the case of valley-momentum-locked edge modes. We then calculate the
conductance of a bilayer graphene wire in presence of finger-gate electrodes,
finding strong asymmetries with energy inversion and deviations from
conductance quantization that can be understood with the gate-induced
unprotected edge modes.

††preprint: APS/123-QED

##  I  Introduction

Bilayer graphene (BLG) allows a remarkable mechanism of electronic confinement
by tuning the energy gap with electrostatic gates on the sides of the two
graphene layers.[1, 2, 3, 4, 5, 6, 7, 8, 9, 10] Indeed, an interlayer electric
field opens a gap in the spectrum, thus favouring electronic confinement to
those regions with a vanishing (or small) interlayer field. Electrodes of
carefully chosen shapes, designed with lithographic techniques, allow
different types of BLG nanostructures such as open semi-infinite edges,
quasi-1D wires (electron guides), and fully closed loops, rings and dots. For
instance, the blue/red electrodes in Fig. 1a create an open BLG edge at $y=0$,
separating two half planes, gapped and ungapped, for electronic motion.

Graphene nanostructures can also be made with etching techniques, removing
parts of the graphene system, as opposed to the above mentioned electrostatic
confinement of BLG. With etching, however, the specific atomic arrangement at
the borders as well as the presence of undesired edge roughnesses or
imperfections is usually relevant and methods to reduce or minimize them are
generally desired.[11, 12, 13]

The tuning of the electric gap in BLG using electric fields was demonstrated
in early magnetotransport experiments with bulk BLG [14]. These were followed
by a very intense research activity, as summarized, e.g., in the field reviews
Refs. [1, 3]. More recently, experiments on electrostatic confinement in BLG
nanostructures have been reported for dots [6, 7, 8, 9, 10] and 1D edges [15,
16, 17, 12, 18]. The hallmark of the latter are the observation of conductance
quantization in quantum transport experiments.

Figure 1:  a) Sketch showing the two bilayer graphene planes (gray), the pair
of electrodes for asymmetric potential $V_{a}$ (red and blue) and the
electrode for symmetric potential $V_{s}$ (green). A $y$-displacement of the
symmetric and asymmetric electrodes is indicated by $l_{y}$. b) Model
symmetric $V_{s}(y)$ and asymmetric $V_{a}(y)$ potentials with a displacement
$l_{y}=200$ nm, $V^{0}_{s}=2$ meV, $V^{0}_{a}=20$ meV.

Ungapped BLG hosts bulk propagating electronic modes for any energy, with
characteristic 2D wave numbers, $(k,q)$ in $(x,y)$ directions. States in
translationally invariant edges or wires in only one direction ($x$) have a 1D
wave number ($k$); while closed loops and dots possess a fully discrete
electronic spectrum. In Ref. [19] it was shown that an open electrostatic edge
in BLG is able to bind an edge mode with a characteristic valley-momentum
locking; i.e., with opposite valleys propagating in opposite directions along
the edge. Remarkably, the wave number $k$ of this mode separates from the
continuum band of bulk ungapped modes and yields characteristic transport
signatures in BLG junctions.[19]

In this work we further investigate the properties of electrostatic edge modes
in BLG. In particular, we focus on the effect of a potential shift as induced
by an additional side electrode. We consider an electrostatic edge defined by
two side gates (red and blue in Fig.1a), and an additional gate creating the
potential shift (green in Fig. 1a). We found that a lateral shift of the
electrodes by a small distance $l_{y}$ has a very relevant effect. It causes
additional edge modes in the stripe of width $l_{y}$, running in both
directions along the edge. Therefore, they allow backscattering mediated by
these edge modes alone, without the need to couple with bulk modes. In
presence of disorder, or other inhomogeneities, an additional electrode will
then strongly affect the conductance along the edge. Besides, the electrode
also causes energy-inversion asymmetry, with different conductances for
positive and negative energies (with zero energy being the Dirac-point
reference energy).

Subsequently, we use the results on the open edge to understand the effect of
finger gate electrodes (FGE) across a BLG wire or guide. We consider the cases
of an extended FGE covering both wire edges, or shorter FGE’s affecting one or
the two edges of the wire. We predict conspicuous energy asymmetries and
conductance deviations from quantization that can be explained with the FGE
induced edge modes. Therefore, similarly to the case of semiconductor
wires,[20] FGE’s are a practical way to manipulate electronic transport in BLG
electrostatic wires.

##  II  Theoretical Model

Our analysis is based on a low-energy Hamiltonian for BLG in presence of
electrostatic potentials.[1] We consider two types of potentials: a symmetric
potential $V_{s}$, equal on the two layers, and an asymmetric potential $\pm
V_{a}$, with opposite signs on the two layers. In this work we consider
parameterized model functions for both potentials, as shown in Fig. 1b for the
case of an open edge. These functions read

$V_{s/a}(y)=\frac{V^{0}_{s/a}}{1+e^{(y-y_{s/a})/s}}\;,$ (1)

where parameters $V^{0}_{s/a}$ and $y_{s/a}$ are the asymptotic value and
position of the border for the symmetric/asymmetric potential. Parameter $s$
is a small distance representing the smoothness of the potential steps.
Examples of our model potentials can be seen in Fig.1b.

The low-energy effective Hamiltonian we will use in this work is built on an
underlying tight-binding atomistic description for BLG. The electronic band
struture of unbiased bulk BLG is characterized by gap closings at the six
Dirac points in reciprocal space, three of them corresponding to the valley
$K_{+}$ and the other three to valley $K_{-}$. Near those Dirac points, an
expansion to the leading terms in electronic momenta yields an effective
multiband continuum Hamiltonian. We refer the reader to Ref. [1] for details
on the mathematical derivations and only stress here that we restrict to
graphene layers in AB Bernal stacking. Adding the model potentials of the type
(1) to the resulting effective Hamiltonian describes the specific confinement
mechanisms due to the electrostatic gates of this work. The potential
difference $2V_{a}$ between the two graphene layers opens an energy gap in the
low-energy scale around the Dirac points which is modulated in space by a
position dependent potential.

The BLG low-energy Hamiltonian reads [1]

$\displaystyle H$ $\displaystyle=$ $\displaystyle
v_{F}p_{x}\tau_{z}\sigma_{x}+v_{F}\,p_{y}\sigma_{y}+\frac{t}{2}\,\left(\,\lambda_{x}\sigma_{x}+\lambda_{y}\sigma_{y}\,\right)$
(2) $\displaystyle+$ $\displaystyle V_{s}(x,y)+V_{a}(x,y)\,\lambda_{z}\;,$

with the Fermi velocity $\hbar v_{F}=660\,{\rm meV}\,{\rm nm}$ and the
interlayer coupling $t=380\,{\rm meV}$. In Eq. (2), $\sigma_{x,y,z}$,
$\tau_{x,y,z}$ and $\lambda_{x,y,z}$ are sets of Pauli matrices for
sublattice, valley and layer degrees of freedom, respectively. This
Hamiltonian is valley diagonal and it has been used to study quantum states in
a variety of BLG nanostructures. As mentioned in Sec. I, the use of position-
dependent potentials $V_{s}(x,y)$ and $V_{a}(x,y)$ allow modeling the effect
of potential gates that create electrostatic borders. Notice that Eq. (2) is
for general inhomogenous potentials $V_{s}$ and $V_{a}$ depending on both
coordinates $(x,y)$. See below, however, for the restricted cases considered
in this work of potentials which are uniform along $x$ or piecewise-uniform
along $x$, with each uniform section of the type given in Eq. (1).

We also stress that for the case of sign-changing $V_{a}(x,y)$’s, Hamiltonian
(2) predicts the emergence of topological modes near the sign-change border.
The spectrum becomes gapless in presence of these modes since their energies
$E(k)$ cross from the negative to the positive energy sectors. These modes
also show a characteristic valley-momentum locking and protection from bulk
modes by an energy gap. [21, 22, 23, 24, 25, 26, 27] From a formal Condensed
Matter topology approach, it has been pointed out that specific invariants for
each valley $N_{\tau}=\pm 1$ can be approximately defined in BLG.[28, 2]
However, it has also been stressed that the bulk-boundary correspondence
between those invariants and the edge modes is not general and may depend on
the specific type of interface, such as in BLG-vacuum or BLG-BLG.[28] The
latter type corresponding to the electrostatic boundaries considered in this
work.

Figure 2:  a) Eigenenergies for the (open) single edge with $V_{s}=0$. The
gray region is the bulk continuum while the red line is the discrete branch of
edge states. b) Spatial probability distributions for two selected wave
numbers $k$ indicated in panel a) by the corresponding labels. c,d) Similar
results to a,b) but with $V_{s}=2$ meV and $l_{y}=200$ nm. Other parameters:
$V_{a}=20$ meV, $s=7.5$ nm.

Below, we will discuss a) the eigenstates of the Hamiltonian (2) for fully
translational invariant BLG systems with both potentials $V_{s}(y)$ and
$V_{a}(y)$; and b) the conductance through junctions of different BLG sections
described by $V_{s}(x,y)$ and $V_{a}(x,y)$ having a piecewise-constant
dependence on $x$. They model the effect of a central FGE on a quantum wire
(see device sketches in Figs. 3-5). In all cases our resolution method is
based on a combination of spatial grid discretization and multiple component
wave functions using complex-band-structure theory. More details of the method
can be found in Sec. V.

##  III  Results and Discussion

###  III.0.1  Single Edge

Figure 2 shows the electron eigenenergies for the open BLG edge sketched in
Fig. 1. The gray region in Fig. 2a is the continuum for bulk modes, given by
the condition

$|k|<\frac{1}{\hbar v_{F}}\,\sqrt{|E|\,\left(\,|E|+t\,\right)}\;.$ (3)

See App. A for a derivation of this momentum restriction for bulk propagating
states. The red line in Fig. 2a shows the edge mode in absence of symmetric
potential $V_{s}=0$. This mode spatially decays with the distance to the
boundary (Fig. 2b) and it is characterized by valley-momentum locking;
reversed valleys propagating in reversed directions in a similar way to the
quantum spin Hall effect but replacing spin with valley.

The edge mode becomes damped when it overlaps with the continuum of bulk BLG
modes, indicated in gray colour in Fig. 2a. In this case the localized edge
mode decays into bulk modes with the same $E$ and $k$, thus flying away from
the edge. In Fig. 2a this overlap occurs in the region of vanishing $E$ and
$k$ and, technically, it is not easily resolved by the numerical calculation.

Figure 3: a) Conductance of a quantum wire in presence of a FGE across all the
wire (a), and with a FGE covering only one edge (b). The sketch insets show
the corresponding systems. The number of active modes in the asymptotic leads
${\cal N}_{l}$ and center ${\cal N}_{c}$ are also shown. Parameters:
$L_{y}=600$ nm, $l_{y}=200$ nm, $l_{x}=1$ $\mu$m, $V_{s}^{0}=0.5$ meV.  Figure
4: Similar to Fig. 3 but with two FGE’s with the same $V^{0}_{s}$ (a), and
with opposite $V^{0}_{s}$ (b).

The modification induced by a potential shift of an additional gate with
$l_{y}=200$ nm is shown in Figs. 2cd. The discrete branch of states of Fig. 2a
is now shifted upwards in energy, merging with the continuum for energies
beyond a given maximum value. In addition, new branches of modes emerge at low
and negative energies that are localized to the region of width $l_{y}$ near
the boundary. These modes propagate in both directions, as seen from the
positive and negative slopes of the energy branches. The corresponding
probability densities show substantial overlaps (Fig. 2d), suggesting the
possibility of backscattering mediated by these edge modes in presence of
inhomogeneities along the edge. Most remarkably, the additional side gate (and
potential shift $V_{s}$) yields energy-inversion asymmetry in Fig. 2c, with
the presence of edge-mode branches only in the lower part of the energy
diagram. We stress that the shift $l_{y}$ in Fig. 1a is essential for the
emergence of additional branches of edge states, as well as for the energy
asymmetry of the spectra.

###  III.0.2 Quantum Wire Junctions

Having analyzed the gate-induced modifications in the open edge, we consider
next the role of a FGE on an electrostatic quantum wire of width $L_{y}$. More
specifically, we calculate the total left-to-right transmission $T$ (with
conductance $G=Te^{2}/h$) using the complex-band-structure method for the
double junction system sketched in Figs. 3-5. Firstly, Fig. 3a shows that a
FGE covering all the wire has a negligible effect on the wire conductance:
transmission is perfect and the coductance simply reproduces the staircase
function of the number of active modes. On the contrary, a FGE covering only
one edge of the quantum wire (Fig. 3b) yields relevant modifications. $G$
deviates from the plateau values, with conspicuous minima for energies
$E<V_{s}^{0}$. There is also a clear asymmetry with respect to energy
inversion in Fig. 3b. For $E>V_{s}^{0}$ the conductance is almost perfectly
quantized, while for $E<V_{s}^{0}$ it shows the mentioned deviations.

The conductance non-quantization and asymmetry of Fig. 3b can be understood as
effects of the edge modes induced by the FGE, as discussed above. Quasi bound
states, allowed by edge mode backscatterings at the interfaces, lead to
conductance dips for specific (resonant) energies. This mechanism is only
present for $E<V_{s}^{0}$, thereby explaining the asymmetry in conductance.

The case of two FGE’s, one on each edge of the wire, is presented in Fig. 4.
We studied this configuration using the same shift $V_{s}^{0}$ on the two
FGE’s (Fig. 4a), and with opposite signs of the shift $\pm V_{s}^{0}$ on the
two FGE’s (Fig. 4b). The case of identical shifts is very similar to the
preceding case with just a single FGE (Fig. 3b). However, with opposite signs
the results change markedly; the conductance becoming again symmetric with
energy inversion and the deviations from quantization are enhanced.

Figure 5:  Similar to Figs. 3 and 4 but for a quantum wire with topological confinement; i.e., with reversed blue and red electrodes on the two wire edges, as shown in the inset sketches. Confinement | FGE1 x FGE2 | conductance
---|---|---
Trivial | + | A
| - | S
Topological | + | A
| - | S
Table 1:  Symmetric/antisymmetric (S/A) character of the conductances in Figs.
4 and 5. The column FGE1xFGE2 indicates the sign product of the FGE potentials
covering the two wire edges.

As a final case, we consider a topological inversion in the asymmetric
potential $V_{a}$,[29, 30, 31, 32, 33] with the red/blue electrodes being
reverted on the two edges of the quantum wire (Fig. 5) and with two FGE’s. The
results in Fig. 5ab are very similar to those in Fig. 4ab but with a notable
difference near zero energy. Namely, in the topological cases the conductance
is perfectly quantized to $4e^{2}/h$ in a small energy plateau around zero,
while it vanishes in Figs. 3 and 4. This is explained by the gapless character
of the topological wire, which hosts two valley-momentum-locked branches
crossing zero energy. On the contrary, the nontopological (trivial)
confinement in a finite $L_{y}$ wire is always characterized by a zero energy
gap due to the finite size.

The energy inversion symmetries of the different configurations of FGE
electrodes considered in Figs. 4 and 5 is summarized in Table 1. Notice that
this symmetry is fixed by the product of the signs of FGE potentials on
opposite edges of the wire, irrespectively of the trivial or topological
character of the wire confinement. This result illustrates how conductance
measurements could be used to observe tuning of the edge modes using FGE’s.

###  III.0.3  Further discussion

All results presented above are for a single valley, $K_{+}$. The
corresponding results for the reversed valley $K_{-}$ can be inferred simply
reverting $k\to-k$ in Fig. 2ab for a single edge, while the results remain
invariant in the cases of wires with FGE’s of Figs. 3-5. Therefore, we do not
find any valley polarization induced by a FGE in the quantum wire.

An important underlying aspect, however beyond our present analysis, is the
role of random imperfections and disorder in the device. While it can be
reasonable to assume that BLG is relatively free of such disorder effects, the
additional processing required for the electrostatic electrodes could
introduce such random disorder. Therefore, this is a relevant aspect to
consider in the future. We may expect, however, that the conductance asymmetry
and the nonquantization induced by FGE’s in a quantum wire would be enhanced
in presence of random disorder.

##  IV Conclusions

We have studied the role of an electrode creating a potential shift near an
electrostatic BLG edge. We found that in presence of a displacement $l_{y}$
between the BLG edge and the additional electrode, new modes emerge near the
edge, in the region of width $l_{y}$, that propagate in both directions.
Furthermore, the valley-momentum-locked branch of the single edge is shifted
in energy by the electrode and the spectrum becomes asymmetric with energy
inversion.

We also investigated the more practical case of a BLG quantum wire in presence
of transverse FGE’s. If a FGE is covering the two edges of a quantum wire, the
system’s conductance is almost unchanged and remains nearly perfectly
quantized. However, if the FGE covers only one edge, or there are two
different FGE’s covering the two edges, then the conductance displays strong
non quantizations and asymmetries with respect to energy inversion. These
changes are in good agreement with the modifications expected from the single-
edge spectrum in presence and a displaced electrode. The energy inversion
symmetry in the quantum wire is restored with two FGE’s having opposite
potential signs on the two edges. In summary, our work shows that FGE’s can be
a practical way to manipulate transport properties of BLG quantum wires by the
electrostatic tuning the electronic modes at the wire edges.

##  V Methods

We solved the eigen-problem with Hamiltonian (2) using finite-difference
discretization and matrix diagonalization routines. With translational
invariance, in the cases of a single edge and quantum wire, a matrix
diagonalization for each (real) wave number $k$ yields the band structure
$E_{n}(k)$ as well as it corresponding eigenstates. An important aspect is the
filtering of spurious modes emerging due to an artificial Fermion doubling of
the physical eigenstates. In practice the filtering is done by eliminating
those states with large oscillations in neigbouring grid points, such that
spatially averaging on a small neigbourhood strongly modifies the wave
function. We found this simple technique to be quite effective and robust.[32]

The transport problem for the junctions of piece wise homogenous sections in
the transport direction ($x$) was solved using the complex-band-structure
approach discussed in Ref. [34]. Here, it is important to include complex wave
numbers $k$ in order to describe evanescent-state behavior in the proximity of
the junction interfaces. The wave-function matching at the junction interfaces
is transformed into a large set of linear equations whose solution determines
the quantum transmissions $T_{kk^{\prime}}$ and its corresponding Landauer
conductance $G=\frac{e^{2}}{h}\sum_{kk^{\prime}}{T_{kk^{\prime}}}$.

We refer to Ref. [34] for more details of the complex band structure approach
and to Refs. [32, 19] for its specific application to BLG structures.

##  Appendix A Bulk continuum

For constant $V_{s/a}$ potentials the eigenstates of the Hamiltonian (2) are
plane waves, with momenta $(k,q)$ along $(x,y)$,

$\Psi\equiv\Phi_{\eta_{\sigma}\eta_{\tau}\eta_{\lambda}}\,e^{i(kx+qy)}\;,$ (4)

with $\eta_{\sigma},\eta_{\tau},\eta_{\lambda}=1,2$ indicating the different
spinorial components of the wave function. Assuming a given real $k$, we can
determine the corresponding $q$’s by transforming the eigenvalue equation as
follows

$H\Psi=E\Psi\quad\Rightarrow\quad\sigma_{y}H\Psi=E\sigma_{y}\Psi\;.$ (5)

The purpose of the above transformation is that Eq. (5) can be easily
rewritten as an eigenvalue equation for $q$,

$\displaystyle\frac{1}{\hbar
v_{F}}\left[\rule{0.0pt}{14.22636pt}E\,\sigma_{y}\right.$ $\displaystyle+$
$\displaystyle i\,\hbar
v_{F}\,k\,\sigma_{z}\tau_{z}+\frac{t}{2}\left(i\,\lambda_{x}\sigma_{z}-\lambda_{y}\right)$
(6) $\displaystyle-$ $\displaystyle
V_{s}\,\sigma_{y}-\left.V_{a}\,\sigma_{y}\lambda_{z}\rule{0.0pt}{14.22636pt}\right]\Phi=q\,\Phi\;.$

After some algebra the eigenvalues of Eq. (6), assuming free BLG for which
$V_{s/a}=0$, can be analytically determined with the diagonalization of an
algebraic matrix. The $q$ eigenvalues read

$q=\pm\frac{1}{\hbar v_{F}}\sqrt{-\hbar^{2}v_{F}^{2}\,k^{2}+|E|\left(\,|E|\pm
t\,\right)}\;.$ (7)

Notice that Eq. (7) already proves the existence of a critical value
$k_{c}=\sqrt{|E|\left(\,|E|+t\,\right)}/\hbar v_{F}$, as given in Eq. (3).
Indeed, the propagating-mode condition requires $q$ to be real which, from the
square root in Eq. (7), requires $|k|<k_{c}$.

###### Acknowledgements.

We acknowledge support from Grant No. PDR2020-12 funded by GOIB; and from
Grant No. MDM2017-0711 and Grant No. PID2020-117347GB-I00 funded by
MCIN/AEI/10.13039/501100011033. H.A. was supported by GOIB program “SOIB
Recerca i Innovació”

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# Stationary Two-State System in Optics using Layered Materials

Ken-ichi Sasaki<EMAIL_ADDRESS>NTT Research Center for Theoretical Quantum
Physics and NTT Basic Research Laboratories, NTT Corporation, 3-1 Morinosato
Wakamiya, Atsugi, Kanagawa 243-0198, Japan

###### Abstract

When electrodynamics is quantized in a situation where electrons are confined
to a flat surface, as in the case of graphene, one of the Maxwell’s equations
emerges as a local component of the Hamiltonian. We demonstrate that, owing to
the residual gauge invariance for the local Hamiltonian, nontrivial
constraints on physical states arise. We construct two stationary quantum
states with a zero energy expectation value: one replicates the scattering and
absorption of light, a phenomenon familiar in classical optics, while the
other is more fundamentally associated with photon creation. According to the
Hamiltonian, these two states are inseparable, forming a two-state system.
However, there exists a specific number of surfaces for which the two states
become decoupled. This number is $2/\pi\alpha$, where $\pi\alpha$ is the
absorption probability of a single surface. To explore the physics that can
emerge in such decoupled cases, we investigate certain perturbations that can
influence a two-state system based on symmetry considerations, utilizing
concepts such as parity, axial (pseudo) gauge fields, and surface deformation.

Two-state systems represent the simplest cases that exhibit the essential
features of quantum physics. Examples include spin-$1/2$, the two base states
for the ammonia molecule, (clockwise and counterclockwise) persistent currents
of superconducting qubit, and the hydrogen molecular ion. [1, 2] In optical
science, the two-state polarization of the radiation field plays a fundamental
role in uncovering the laws of physics, such as the non-cloning theorem, [3]
serving as a source of quantum entanglement, [4, 5, 6, 7] and contributing to
modern optical transmission technology, allowing for the simultaneous
transmission of double the amount of information. Recently, the two-state
coherent phase of a pulsed field has been related to an artificial spin. [8,
9] There may be more two-state systems in optics beyond these examples. Even
when a system appears to have many degrees of freedom, it is sometimes
apparent that, from a certain perspective, only two crucial states play an
essential role in the mechanism of the phenomenon under investigation. [10,
11, 12, 13] In this paper, we employ quantum electrodynamics [14] to
illustrate the existence of two stationary quantum states in optics involving
layered materials. Our formulation reveals that the appearance of these two
states is intricately connected to gauge invariance, and the origin of the
two-state degree of freedom is inherently linked to the chiral nature of
photons, specifically the right- and left-going components. We show that the
antisymmetric combinations of counter-propagating photons are connected to the
axial gauge field, suggesting a potential link to the geometrical deformation
of layered materials and the Raman effect.

The space is divided into two regions, $x<0$ (left) and $x>0$ (right), by an
absorbing surface located at $x=0$. Some fraction of a continuum incident
light coming from $x=-\infty$ is absorbed only by the surface where the
electronic current ${\bf j}$ exists, and the remaining is either reflected
back to $x=-\infty$ or transmitted forward to $x=+\infty$. In the classical
theory of optics, the reflection and transmission coefficients are derived
from Ampére’s circuital law, given by $c^{2}\nabla\times{\bf B}=\frac{\bf
j}{\epsilon_{0}}+\dot{\bf E}$. Integrating this equation with respect to an
infinitesimal element around the surface, the magnetic field experiences a
discontinuity at the surface, expressed as
$c^{2}\\{B_{y}(0_{+})-B_{y}(0_{-})\\}=\int_{0_{-}}^{0_{+}}\frac{j_{z}}{\epsilon_{0}}dx$,
due to the localization of the current $j_{z}=J_{z}\delta(x)$ about the
surface. [15, 16] Meanwhile, the electric field $E_{z}$ remains continuous at
the surface, and the last term of Ampére’s circuital law vanishes upon
integration. Since the magnetic field is expressed in terms of the radiation
gauge field by $B_{y}=-\partial_{x}A_{z}$, we obtain the boundary condition

$\displaystyle\partial_{x}A_{z}|_{x=0_{+}}-\partial_{x}A_{z}|_{x=0_{-}}+\frac{J_{z}}{\epsilon_{0}c^{2}}=0,$
(1)

showing that the first derivative of $A_{z}(x)$ is discontinuous at the
surface, meanwhile, $A_{z}(x)$ must be continuous. Equation (1) is also
derived as a characteristic of stationary states within the Poynting theory.
[1] The continuity equation is expressed as
$E_{z}j_{z}+\partial_{x}S_{x}+\dot{U}_{em}=0$, where
$S_{x}=-\epsilon_{0}c^{2}E_{z}B_{y}$ represents the Poynting vector, and
$U_{em}=\frac{\epsilon_{0}}{2}(E_{z}^{2}+c^{2}B_{y}^{2})$ is the energy
density of the electromagnetic field. In the context of stationary states,
$\dot{U}_{em}=0$, allowing us to reproduce Eq. (1) by integrating
$E_{z}j_{z}+\partial_{x}S_{x}=0$ with respect to an infinitesimal element
around $x=0$. Now, in quantum field theory, $A_{z}(x)$ as well as $J_{z}$ is a
field operator consisting of particle creation and annihilation operators.

It is possible to show that Eq. (1) appears as a part of Hamiltonian and is
related to the gauge invariance in the fundamental level. For the photon’s
Hamiltonian of

$\displaystyle
H_{ph}=\frac{\epsilon_{0}}{2}\int_{-\infty}^{+\infty}dx\left\\{\dot{A}_{z}(x)^{2}+c^{2}(\partial_{x}A_{z}(x))^{2}\right\\},$
(2)

the integral may be divided into a left region, right region, and the surface
$[0_{-},0_{+}]$. Then, the Hamiltonian of the radiation field at the surface
is given by $\epsilon_{0}c^{2}A_{z}(x)\partial_{x}A_{z}(x)|_{0_{-}}^{0_{+}}$.
By combining it with the gauge coupling of the interaction Hamiltonian,
$H_{int}=A_{z}(0)J_{z}$, we obtain the Hamiltonian at the surface
$H_{ph}|_{x=0}+H_{int}(\equiv H_{local})$ as

$\displaystyle\epsilon_{0}c^{2}A_{z}(0)\left\\{\partial_{x}A_{z}(x)|_{x=0_{+}}-\partial_{x}A_{z}(x)|_{x=0_{-}}+\frac{J_{z}}{\epsilon_{0}c^{2}}\right\\}.$
(3)

To simplify the notation, we use ${\cal B}$ to denote the operator of the
boundary condition, ${\cal
B}\equiv\partial_{x}A_{z}(x)|_{x=0_{+}}-\partial_{x}A_{z}(x)|_{x=0_{-}}+\frac{J_{z}}{\epsilon_{0}c^{2}}$,
and the local Hamiltonian is rewritten as

$\displaystyle H_{local}=\epsilon_{0}c^{2}A_{z}(0){\cal B}.$ (4)

The Hamiltonian must be invariant with respect to a residual gauge
transformation, given by $A_{z}(0)\to
A_{z}(0)+\partial_{z}\lambda(x,z)|_{x=0}$, where $\lambda(x,z)=Cz$ satisfies
$\partial^{2}_{z}\lambda(x,z)=0$. However, the local Hamiltonian changes under
such a residual gauge transformation as $H_{local}\to
H_{local}+\epsilon_{0}c^{2}C{\cal B}$. Thus, to ensure that a residual gauge
transformation does not alter the energy of physical states, any physical
state $|\Phi\rangle$ must satisfy the condition $\langle\Phi|{\cal
B}|\Phi\rangle=0$. Instead of stating that Eq. (1) is satisfied as the
expectation value (Ehrenfest’s theorem), Eq. (1) is satisfied as a consequence
of the residual gauge symmetry. [17] Additionally, any physical state should
not overlap with a state generated by acting the vacuum with ${\cal B}$, i.e.,
$\langle\Phi|{\cal B}\left(|0\rangle\otimes|vac\rangle\right)=0$, where the
ground state of the photon (matter) is expressed by $|0\rangle$
($|vac\rangle$). Failure to meet this condition could allow a residual gauge
transformation to introduce an arbitrary amount of energy, which is a scenario
not observed in nature. We note that the result of the Poynting theory,
$E_{z}{\cal B}=0$, can be derived from $\dot{H}_{local}=0$ and $\dot{\cal
B}=0$, because $E_{z}=-\dot{A}_{z}$.

We will show that the Hilbert space holds two quantum states denoted by
$|\Phi_{a}\rangle$ and $|\Phi_{b}\rangle$. Each state is not single quantum
but a specific combination of light and matter quanta. The former reproduces
scattering and absorption of light by matter which is consistent with the
classical description of optics, and the latter is more fundamentally related
to the photon creation or light emission. We use the Schrödinger picture
(representation), [2, 14] and the operators are time-independent. Moreover,
the state vectors we construct are stationary states with zero-energy of the
local Hamiltonian, the variable of time is concealed. The gauge field operator
is a sum of right- and left-moving components as $A_{z}(x)=A_{R}(x)+A_{L}(x)$.
The electric field operator is denoted by $E_{z}(x)$ which forms canonical
equal-time commutation relations with $A_{z}(x)$, such as
$\left[A_{z}(x),E_{z}(x^{\prime})\right]=-i\frac{\hbar
c}{\epsilon_{0}}\delta(x-x^{\prime})$,
$\left[A_{R}(x),E_{R}(x^{\prime})\right]=-i\frac{\hbar
c}{2\epsilon_{0}}\delta(x-x^{\prime})$, and
$\left[A^{+}(x),E^{-}(x^{\prime})\right]=-i\frac{\hbar
c}{2\epsilon_{0}}\delta(x-x^{\prime})$, where $-$ and $+$ of the superscript
stands for the annihilation and creation part of the field operators,
respectively. Note that $A_{z}(x)$ commutes with $B_{y}(x^{\prime})$ and so
with ${\cal B}$. However, each component satisfies
$[\partial_{x}A_{R}(x),A_{R}(x^{\prime})]=+i\frac{\hbar}{2\epsilon_{0}}\delta(x-x^{\prime})$
and
$[\partial_{x}A_{L}(x),A_{L}(x^{\prime})]=-i\frac{\hbar}{2\epsilon_{0}}\delta(x-x^{\prime})$,
which are canceled in $[B_{y}(x),A_{z}(x^{\prime})]=0$.

First, we define an entangled state that is a superposition of right- and
left-going photons counter-propagating along the $x$-axis. The spatially
uniformity of the light state is perturbed by the absorbing surface as
$|a\rangle=N_{a}\int_{-\infty}^{+\infty}a(x)dx|0\rangle$ and

$\displaystyle
a(x)=\left[\theta(-x)+t\theta(x)\right]e^{i\frac{\omega}{c}x}A_{R}(x)+r\theta(-x)e^{-i\frac{\omega}{c}x}A_{L}(x).$
(5)

Here, $t$ ($r$) is the transmission (reflection) coefficient, $N_{a}$ is a
normalization constant of the state $|a\rangle$ (space is taken to be
infinity), and $\theta(x)$ is the step function; $\theta(x)=1$ for $x>0$ and
vanishes otherwise. We show below that the following quantum state reproduces
the result obtained in classical optics, [18]

$\displaystyle|\Phi_{a}\rangle\equiv|a\rangle\otimes|vac\rangle+\frac{\langle
0|E_{z}(0)|a\rangle}{g}|0\rangle\otimes J_{z}|vac\rangle.$ (6)

Since the electric field is continuous at the surface (as we see below), we
can also take either $x=0_{+}$ or $0_{-}$ for the argument of $E_{z}$. A
constant $g$ is introduced for dimensional consistency whose value is not of
particular importance here. The classical physics is reproduced by replacing
$|a\rangle$ with a coherent state
$|A\rangle=N_{A}e^{i\int_{-\infty}^{+\infty}a(x)dx}|0\rangle$.

To specify $|\Phi_{a}\rangle$ uniquely, we use two physical conditions. First
condition is the continuity of the electric field $\langle
0|E_{z}(0_{-})|a\rangle=\langle 0|E_{z}(0_{+})|a\rangle$ giving $1+r=t$.
Second is the necessary condition with respect to a physical state,

$\displaystyle\langle 0|\otimes\langle vac|{\cal B}|\Phi_{a}\rangle=0.$ (7)

Using commutation relations, we obtain from Eq. (7)

$\displaystyle
i\frac{\hbar}{2\epsilon_{0}}t-i\frac{\hbar}{2\epsilon_{0}}(1-r)+\frac{\sigma
g}{\epsilon_{0}c^{2}}i\frac{1}{g}\frac{\hbar c}{2\epsilon_{0}}t=0.$ (8)

We note that $\langle vac|J_{z}J_{z}|vac\rangle=g\sigma$, where $\sigma$ is
dynamical conductivity (or vacuum polarization). In the case of graphene, the
dynamical conductivity is given by $\sigma=\pi\alpha\epsilon_{0}c$, where
$\alpha$ is the fine-structure constant (the absorption probability is
$\pi\alpha\sim$ 2.3 percent). [19, 20, 21] Thus, these two conditions lead to
$t=\frac{2}{2+\pi\alpha}$ and $r=\frac{-\pi\alpha}{2+\pi\alpha}$. Moreover, we
obtain $\langle\Phi_{a}|H_{local}|\Phi_{a}\rangle=0$ which also reproduces the
energy (or probability) conservation given by Poynting vector,
$1-r^{2}-t^{2}=\pi\alpha t^{2}(\equiv A_{abs})$. [15, 16] To calculate
$\langle a|A_{z}(0_{+})$, we take care that two-point correlation function
such as $\langle 0|[A_{z}(x+\varepsilon),A_{z}(x)]|0\rangle$ has an anomalous
commutator part when $\varepsilon$ is tiny (the length scale of wave function
spreading). It is actually defined by $\lim_{\varepsilon\to 0}\langle
0|[(A_{R}(x+\varepsilon)+A_{L}(x-\varepsilon)),A_{z}(x)]|0\rangle$ which gives
$\lim_{\varepsilon\to 0}\varepsilon\langle
0|[\partial_{x}A_{R}(x)-\partial_{x}A_{L}(x),A_{z}(x)]|0\rangle=\lim_{\varepsilon\to
0}\varepsilon\frac{i\hbar}{\epsilon_{0}}\delta(0)$.

Figure 1: Schematic of the two-state system: It consists of $|\Psi_{a}\rangle$
and $|\Psi_{b}\rangle$, having a zero energy expectation value of $H_{local}$
due to the residual gauge symmtry. (Above) The arrow with “1” from left to
right expresses the incident light ray, which is reflected or transmitted by a
thin film (graphene). The dashed line represents the zero photon state.
(Below) The state of light emission from the excited state is symmetrical in
left and right directions. Only the off-diagonal matrix element of
$H_{local}$, $\langle\Phi_{b}|H_{local}|\Phi_{a}\rangle$, is non-zero and
proportional to $b_{R}^{*}t+b_{L}^{*}r$. The states of graphene are
represented by the electron-hole pairs ($\bullet$ and $\circ$) in the Dirac
cone (zigzag line denotes light). The electron-hole pairs in
$|\Psi_{a}\rangle$ are virtual states of photons, while those in
$|\Psi_{b}\rangle$ are real states that are constantly supplied by
$|\Psi_{a}\rangle$.

Next we construct the other state vector,

$\displaystyle|\Phi_{b}\rangle\equiv|b\rangle\otimes|vac\rangle+\frac{\langle
0|E_{z}(0)|a\rangle}{g}|0\rangle\otimes|exc\rangle,$ (9)

where $|b\rangle=N_{a}\int_{-\infty}^{+\infty}b(x)dx|0\rangle$ and
$b(x)=b_{R}\theta(x)e^{i\frac{\omega}{c}x}A_{R}(x)+b_{L}\theta(-x)e^{-i\frac{\omega}{c}x}A_{L}(x)$.
It represents light emission from the surface. The presence of $\langle
0|E_{z}(0)|a\rangle$ on the last term shows that the emission is the
spontaneous one (stimulated one when $|a\rangle\to|A\rangle$). The state
vector of the matter $|exc\rangle$ denotes the excited state. This state is
specified primarily by $J_{z}^{-}|exc\rangle=gF|vac\rangle$. In the same way
as $|\Phi_{a}\rangle$, the continuity of the electric field with respect to
$|b\rangle$, $\langle 0|E_{z}(0_{+})|b\rangle=\langle
0|E_{z}(0_{-})|b\rangle$, and the physical state condition $\langle
0|\otimes\langle vac|{\cal B}|\Phi_{b}\rangle=0$ are sufficient to obtain that
$b_{R}=b_{L}(\equiv b)$ and

$\displaystyle b=-\frac{F}{2\epsilon_{0}c}t.$ (10)

This state automatically satisfies
$\langle\Phi_{b}|H_{local}|\Phi_{b}\rangle=0$. Thus, we found that absorption
means the two state-vectors and that the strength of emitted light depends on
the electric field at the graphene layer ($t$).

Since $J_{z}^{+}J_{z}^{-}$ contains the number operator of electron-hole pairs
$N_{eh}$, the expression $g^{2}|F|^{2}=g\sigma\langle exc|N_{eh}|exc\rangle$
is derived, representing the proportionality to the absorbed photon energy.
The magnitude of $F$ is expressed as $|F|\propto\sqrt{A_{abs}}$, with the sign
of $F$ depending on the specific characteristics of the matter system. [22,
23, 24] It’s worth noting that $|F|$ is influenced by the surrounding
environment. For example, substrates supporting graphene can change the Fermi
energy due to (unintentional) charge doping, affecting the nature of
relaxations through Coulomb interactions, 111The Coulomb interactions are
expressed as  $\displaystyle H_{Coulomb}=\frac{1}{2}\int d{\bf r}^{2}\int
d{\bf r^{\prime}}^{2}\frac{J_{0}({\bf r})J_{0}({\bf
r^{\prime}})}{4\pi\epsilon_{0}|{\bf r}-{\bf r^{\prime}}|},$ (11) where
$J_{0}({\bf r})$ represents the charge density operator. This interaction
aligns with our choice of the radiation gauge in the canonical formulation of
quantum electrodynamics. The consequences of the impact excitations are
elucidated by considering the screening effect of Coulomb interactions, which
are influenced by the position of the Fermi energy. known as impact
excitations. [26] On the other hand, when graphene is suspended in the air,
lattice vibrations do not decay into substrates, potentially reducing the
relative significance of the paths from the excited states to light emission.
To address these complexities, it is advisable to treat the effective
magnitude of relaxations as a branching ratio ${\cal B}$, leading to the
expression $|F|=\epsilon_{0}c\sqrt{2{\cal B}A_{abs}}$ or $|b|=\sqrt{({\cal
B}/2)A_{abs}}|t|$. Physically, a large value of ${\cal B}$ implies that the
excited states exhibit prolonged lifetimes, maintaining pathways for
relaxation toward light emission.

Though the zero-energy conditions
$\langle\Phi_{a}|H_{local}|\Phi_{a}\rangle=0$ and
$\langle\Phi_{b}|H_{local}|\Phi_{b}\rangle=0$ are both satisfied, the off-
diagonal element $\langle\Phi_{b}|H_{local}|\Phi_{a}\rangle$ is generally non-
zero as

$\displaystyle\langle\Phi_{b}|H_{local}|\Phi_{a}\rangle=\epsilon_{0}|E_{in}|^{2}\varepsilon
b^{*}(t+r),$ (12)

and therefore the two states evolve inseparably by the Hamiltonian. We shall
introduce two states written as
$|\Phi_{\theta}^{\pm}\rangle=|\Phi_{a}\rangle\pm e^{i\theta}|\Phi_{b}\rangle$,
where the relative phase $\theta$ is determined so that these states are
decoupled; $\langle\Phi^{+}_{\theta}|H_{local}|\Phi^{-}_{\theta}\rangle=0$ or
${\rm Im}[e^{i\theta}(t+r)^{*}b]=0$. Thus,
$e^{i\theta}=\pm\frac{t+r}{|t+r|}\frac{b^{*}}{|b|}$. Then, their energy
expectation values become non-zero as
$\langle\Phi_{\theta}^{\pm}|H_{local}|\Phi_{\theta}^{\pm}\rangle=\pm
2(\epsilon_{0}|E_{in}|^{2}\varepsilon){\rm Re}[b^{*}(t+r)]$. An analogy to the
electron spin is useful in understanding the physics. Suppose we have up and
down spin states, $|\uparrow\rangle$ and $|\downarrow\rangle$, which are the
eigenstates of $\sigma_{z}$. Let us assume that each spin state is in a
magnetic field along the $y$-axis. The energy expectation value vanishes. Now,
$|\Phi_{\theta}^{\pm}\rangle$ corresponds to
$|\pm\rangle\equiv(|\uparrow\rangle\pm
e^{i\theta}|\downarrow\rangle)/\sqrt{2}$. When $\theta$ is determined so that
$\langle+|\sigma_{y}|-\rangle=0$, we have
$\langle\pm|\sigma_{y}|\pm\rangle=\pm 1$. Note that the general expression of
$H_{local}$ in terms of the spin is
$H_{spin}=a_{+}\sigma_{+}+a_{-}\sigma_{-}$, where $\sigma_{\pm}=\sigma_{x}\pm
i\sigma_{y}$ is the ladder operators and $a_{\pm}$ is the amplitudes. It
satisfies $\langle\uparrow|H_{spin}|\uparrow\rangle=0$ and
$\langle\downarrow|H_{spin}|\downarrow\rangle=0$.

Because the reflection coefficient is calculated by $\langle
0|E_{L}(-\infty)|\Phi^{\pm}_{\theta}\rangle$, $|\Phi_{b}\rangle$ appears as a
correction to the reflection coefficient, and there are two possibilities of
the correct reflection coefficient as

$\displaystyle r\pm e^{i\theta}b=r\pm\left(\frac{t+r}{|t+r|}\right)|b|.$ (13)

When the lower energy state $|\Phi^{-}_{\theta}\rangle$ is realized, the
actual reflection is given by minus sign. Recently, we have applied a slight
extension of this reflection formula to elucidate experiments conducted on
multilayer graphene placed on substrates. [24] Our findings indicate that for
visible range, substrates play a significant role in stabilizing
$|\Phi^{-}_{\theta}\rangle$.

Due to absorption, the two states slightly overlap, i.e.,
$\langle\Phi_{b}|\Phi_{a}\rangle\neq 0$, and a simple interpretation of a two-
state system is difficult to be applied. Apparently, from the reflected (or
transmitted) light we will observe, the signal is inseparable between
$|\Phi_{a}\rangle$ and $|\Phi_{b}\rangle$. If the two states were truly
indistinguishable from each other, the situation is something like an electron
spin in the absence of a magnetic field, and we need to face the issue of
control. However, at least when $t+r=0$ is satisfied,
$\langle\Phi_{b}|H_{local}|\Phi_{a}\rangle$ vanishes. Then, we define new base
states, $|\Phi_{A}\rangle=|\Phi_{a}\rangle$ and
$|\Phi_{B}\rangle=|\Phi_{b}\rangle-|\Phi_{a}\rangle\langle\Phi_{a}|\Phi_{b}\rangle$,
which become orthogonal and are not mixed by the local Hamiltonian. Such an
interesting situation is indeed possible to realize for multilayer graphene.
[15] Let the layer number is $N$. The corresponding transmission and
reflection coefficients are given by the replacement $\pi\alpha\to N\pi\alpha$
in $t$ and $r$. Thus, $t+r=0$ when $N=2/\pi\alpha$. Quite recently, we found
also that one can access $|\Phi_{b}\rangle$ by suppressing the contribution of
$|\Phi_{a}\rangle$ to the reflection ($r\sim 0$) using multilayer graphene
(about 20 layers) and destructive interface effect of SiO2/Si substrates. [24]

Perturbations should be incorporated into $H_{local}$, as they have the
potential to disrupt stabilization and counteract corrections through
interference. These perturbations arise from lattice vibrations and electron-
phonon interactions, particularly in relation to the Raman effect. In this
context, we examine some of these perturbations on symmetry considerations.
Firstly, we observe that $H_{local}$ remains invariant under parity, as the
fields transform according to $A_{z}(x)\to-A_{z}(-x),B_{y}(x)\to B_{y}(-x)$,
and $J_{z}\to-J_{z}$. Let us introduce the axial (pseudo) gauge field
$A^{ax}_{z}\equiv A_{R}-A_{L}$, which is even under parity as
$A^{ax}_{z}(x)\to A^{ax}_{z}(-x)$. [27] This is in contrast to the polar one
$A_{z}=A_{R}+A_{L}$. Since $A_{z}(x)$ and $A_{z}^{ax}(x)$ are regarded as the
bonding and anti-bonding orbitals of counter-propagating photons,
$A_{z}^{ax}(x)$ is useful in discussing energy stabilization governed by
graphene. Secondary, we point out that the current $J_{z}$ in graphene
comprises two currents of different valleys $K$ and $K^{\prime}$, as
$J_{z}=J^{K}_{z}+J^{K^{\prime}}_{z}$, which is even under the interchange of
$K$ and $K^{\prime}$. It is known that the axial current $J^{ax}_{z}$, which
is odd under the valley degrees of freedom as
$J^{ax}_{z}=J^{K}_{z}-J^{K^{\prime}}_{z}$, couples to a field $D_{z}^{ax}(0)$
expressing distortions in graphene’s hexagonal lattice. [12, 13] For example,
the primary Raman band in graphene, the $G$ band, results from their (phonon-
electron) coupling: $D_{z}^{ax}(0)J^{ax}_{z}$. When the system is invariant
under parity transformation, the perturbation in the form of
$A_{z}^{ax}(0)J^{ax}_{z}$ is allowed, but $A_{z}(0)J^{ax}_{z}$ and
$A^{ax}_{z}(0)J_{z}$ are not. These are possible perturbations that make the
system not invariant under parity transformation. Thus, there is a potential
connection between the Raman effect and the axial gauge field, as $J_{z}^{ax}$
can act as the source of $A_{z}^{ax}$ and $A_{z}$. To further elucidate this
point, we rewrite the photon wavefunctions in terms of $A_{z}$ and
$A_{z}^{ax}$ as

$\displaystyle
a(x)=\frac{e^{i\frac{\omega}{c}x}}{2}(A_{z}(x)+A^{ax}_{z}(x))+\frac{r}{2}\left[e^{i\frac{\omega}{c}x}\theta(x)+e^{-i\frac{\omega}{c}x}\theta(-x)\right]A_{z}(x)+\frac{r}{2}\left[e^{i\frac{\omega}{c}x}\theta(x)-e^{-i\frac{\omega}{c}x}\theta(-x)\right]A^{ax}_{z}(x),$
(14) $\displaystyle
b(x)=\frac{1}{2}\left[b_{R}e^{i\frac{\omega}{c}x}\theta(x)+b_{L}e^{-i\frac{\omega}{c}x}\theta(-x)\right]A_{z}(x)+\frac{1}{2}\left[b_{R}e^{i\frac{\omega}{c}x}\theta(x)-b_{L}e^{-i\frac{\omega}{c}x}\theta(-x)\right]A^{ax}_{z}(x).$
(15)

Since wavefunctions are continuous, $A^{ax}_{z}(x)$ must vanish at the surface
($x=0$) for both $a(x)$ and $b(x)$. However, $b(x)$ allows a nonzero
$A^{ax}_{z}(x)$ if $b_{R}=-b_{L}$. Thus, if graphene can excite
$A^{ax}_{z}(x)$, the effect appears as a special light emission, whose
wavefunction is orthogonal to $|\Psi_{b}\rangle$. The light emission is a
quantum effect because $A^{ax}_{z}(x)$ satisfies
$[B_{y}(x),A^{ax}_{z}(x^{\prime})]=-i\frac{\hbar}{\epsilon_{0}}\delta(x-x^{\prime})$
and
$[B^{ax}_{y}(x),A_{z}(x^{\prime})]=+i\frac{\hbar}{\epsilon_{0}}\delta(x-x^{\prime})$,
where $B_{y}^{ax}(x)\equiv\partial_{x}A_{z}^{ax}(x)$ (the latter is what we
have used for the anomalous commutator). In the spin analogy, these
perturbations are akin to those proportional to $\sigma_{x}$ (that cause a
change in $\theta$) and $\sigma_{z}$ (that mixes $|+\rangle$ and $|-\rangle$).

The application of the theoretical framework developed for a surface to
$N$-layer is straightforward, when the electronic current flowing between
layers is negligible. The total Hamiltonian is a sum of the local Hamiltonian
as $H_{total}=\sum_{j=1}^{N}\epsilon_{0}c^{2}A_{z}(x_{j}){\cal B}(x_{j})$. The
solution is built as a superposition of $|\Phi_{a_{N}}\rangle$ and
$|\Phi_{b_{j}}\rangle$, which can be known using a transfer matrix method.
[24] The reflectance is given by
$R_{N}=|r_{N}+\sum_{j=1}^{N}e^{i\theta_{j}}b_{j}|^{2}$, where $\theta_{j}$
includes the phase of $\pm$ for the two energy levels. In terms of the spin
analogy, the problem is approached as a spin-chain. This viewpoint may be
useful in analysis of real systems since unexpected perturbations can exist.
For example, when perturbations flip the spin of a layer, the reflection and
transmission coefficients (near the layer) undergo changes, and
$|\Psi_{a_{N}}\rangle$ is consequently modified. This modification results in
correlations among spins at different layers. This aspect appears to be
another important issue connected to super-radiance [28, 29] and the
synchronization of weakly coupled oscillators. [30]

It is also straightforward to include the degrees of freedom of photon
polarization. Namely, we will have additional two states for $A_{y}$ as well
as $A_{z}$. Thus, the system holds four states in total. Because the
characteristics of the matter system enters through the current operator only,
our formulation is applicable to various layered systems besides graphene by
taking the corresponding current operator $J$. A fundamental issue is that
$A_{y}$ is just a copy of $A_{z}$, or there is some sort of correlation
between them. In the case of pristine graphene, the current operators $J_{y}$
and $J_{z}$ excite different electron-hole pairs and therefore such a
correlation is suppressed by optical selection rule. [31]

In conclusion, we have formulated a two-state system that arises from the
distinct states of matter in layered materials: $J_{z}|vac\rangle$ and
$|exc\rangle$. Considering that photon states directly emerge from constraints
imposed by residual gauge invariance, we propose that these two states of
matter can be effectively characterized by distinct configurations of the
light field. The state $|\Phi_{a}\rangle$ appears asymmetrical with respect to
$A_{R}$ and $A_{L}$, yet it is solely described by the polar gauge field
$A_{z}$. On the other hand, $|\Phi_{b}\rangle$ while symmetrical with
$b_{R}=b_{L}$, may incorporate the axial gauge field $A_{z}^{ax}$, which holds
significance in phenomena such as the Raman effect. In the absence of
perturbations, the two-state system is degenerate, particularly for multilayer
graphene with special layer numbers $N\simeq 2/\pi\alpha$, where the matrix
element of $\langle\Phi_{b}|H_{local}|\Phi_{a}\rangle$ vanishes. Then,
$|\Phi_{a}\rangle$ exhibits a perfect asymmetrical combination of $t=1/2$ and
$r=-1/2$, and $|\Phi_{b}\rangle$ exhibits a perfect symmetrical combination of
$b_{R}$ and $b_{L}$. Therefore, it is reasonable to understood that detecting
a left-going or right-going mode does not provide information about which
state, $|\Phi_{a}\rangle$ or $|\Phi_{b}\rangle$, is realized.

## Acknowledgments

The author thank T. Matsui (Anritsu Corporation), M. Kamada (Anritsu
Corporation), and K. Hitachi for helpful discussions.

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