# Gain-loss induced non-Abelian Bloch braids

Bikashkali Midya<EMAIL_ADDRESS>Indian Institute of Science Education
and Research Berhampur, Odisha 760003, India

###### Abstract

Onsite gain-loss induced topological braiding principles of non-Hermitian
energy bands is theoretically formulated in multiband lattice models with
Hermitian hopping amplitudes. Braid phase transition occurs when the gain-loss
parameter is tuned across exceptional point degeneracies. Laboratory
realizable effective-Hamiltonians are proposed to realize braid groups
$\mathbb{B}_{2}$ and $\mathbb{B}_{3}$ of two and three bands respectively.
While $\mathbb{B}_{2}$ is trivially Abelian, the group $\mathbb{B}_{3}$
features non-Abelian braiding and energy permutation. Phase diagrams with
respect to lattice parameters to realize braid group generators and their non-
commutativity are shown. The proposed theory is conducive to synthesize
exceptional materials for applications in topological quantum photonic
computation and information processing.

The idea of braids and knots of elementary excitations in certain interacting
many-body quantum systems has opened extraordinary possibility for building
blocks of a fault-tolerant topological quantum computer [1, 2, 3]. While the
quantum materials with desired real-space braiding dynamics is notoriously
illusive [4], very recently there has been a growing interest on non-trivial
knots and braids of energy bands in reciprocal space of single-particle non-
Hermitian (NH) systems [5, 6, 7, 8, 9, 12, 10, 13, 14, 15, 11]. Periodic open
systems described by effective NH Hamiltonians generally feature complex band
energy manifolds with exceptional topological properties, not accessible in
Hermitian systems, with far reaching practical consequences [17, 19, 27, 25,
28, 21, 16, 18, 22, 20, 30, 29, 24, 23, 26]. Two Bloch bands, for example, of
a one-dimensional (1D) NH lattice can wind around forming nontrivial
topological braid in the 3D space spanned by complex-energy and momentum
$(\mbox{Re}E,\mbox{Im}E,k)$ [7]; this situation is forbidden in a 1D Hermitian
lattice. When lattice parameters are suitably varied, such NH braid structures
undergo phase transitions at exceptional point (EP) band degeneracy at which
two bands become indistinguishable. Topologically distinct braids of $N$ bands
can naturally be classified by the elements of a braid group $\mathbb{B}_{N}$
such that each braid carries a topological invariance [6].

Current progress on 1D reciprocal space braiding, however, is limited mainly
to lattice models based on NH coupling principle [7, 10, 8, 6, 9]. Possibility
of onsite gain-loss induced braid phase transition and generalization to non-
Abelian braiding remain largely unexplored. Here, we address following
question : can onsite gain or loss solely induce braid group and corresponding
topological phase transitions in an otherwise 1D Hermitian lattice? We show by
considering toy examples of two and three bands lattices that it is indeed
possible to create distinct topological braid elements, for the Abelian groups
$\mathbb{B}_{2}$ and non-Abelian group $\mathbb{B}_{3}$ respectively, by
judicious choices of externally controllable gain-loss potentials and other
Hermitian lattice parameters. Non-Abelian braids under $B_{3}$ provides
remarkable inequivalent energy permutations when the order in which two braids
concatenate is altered. While the handedness of a braid is shown to be tunable
by interchanging the roles of gain and loss, the degree of a braid can be
tunable by controlling the order of long-range hopping between lattice cells.
Parameters regimes to realize elementary braids and their non-commutativity
are obtained. Essential topological features of braids are determined by the
spectral winding number.

Before describing gain-loss induced braiding mechanism, we first brief about
general ideas of band spectrum braids. A braid is a disjoint collection of
interlacing trajectories of $N$ non-degenerate complex energy bands
$\\{E_{1},E_{2},\cdots,E_{N}\\}$ as $k$ evolves in the Brillouin zone (BZ)
$k_{0}\leq k\leq k_{0}+2\pi$; bands are ordered here according to their real
parts at an arbitrary $k_{0}$. This can be mathematically defined by the
mapping $E_{n}(k):BZ\rightarrow\mathbb{C}\times BZ$ for
$n\in\\{1,2,\cdots,N\\}$. Every braid can be obtained as a product of $(N-1)$
generators $\tau_{1},\tau_{2},\cdots,\tau_{N-1}$ and their inverses
$\tau_{1}^{-1},\tau_{2}^{-1},\cdots,\tau_{N-1}^{-1}$. These elementary braids
$\tau_{n}$ or $\tau_{n}^{-1}$ consists of only a single twist between
neighboring bands $E_{n}$ and $E_{n+1}$ in the positive or negative direction.
The identity braid with no twist is denoted here by $\tau^{0}$. Collection of
braids forms a braid group $\mathbb{B}_{N}$ under concatenation (the braid
$\tau_{n}\tau_{\ell}$ is obtained by joining end points of $\tau_{n}$ with
initials of $\tau_{\ell}$), such that
$\tau_{n}\tau_{\ell}=\tau_{\ell}\tau_{n}$ for $|n-\ell|>1$ and
$\tau_{n}\tau_{n+1}\neq\tau_{n+1}\tau_{n}$ [31]. Braid group formed by $N>2$
bands is thus non-Abelian. A link or knot is formed when a braid is closed.

Figure 1: (a) A 1D lattice of non-Hermitian dimers for two-band braids. (b)
Phase diagram showing different regimes for topologically distinct braids;
phase boundaries are the lines of EPs. Topological indices for $\tau_{1}^{\pm
m}$ and $\tau^{0}$ are $\nu=\pm m$ and $0$ respectively. (c1,c3,c5) Braids
$\tau_{1}^{-1}$, $\tau^{0}$ and $\tau_{1}$ respectively for two bands ($E_{1}$
in red and $E_{2}$ in blue) as a function of $k$ in $[0,2\pi]$. (c2,c3) Band
evolution in the presence of EP at $E_{1}=E_{2}=0$ for $k=\pi$. The EP is
located by black dot in the $(\mbox{Re}E,\mbox{Im}E)$ plane. Figs (c1-c5) are
obtained by varying $\gamma=-1,-0.5,-0.2,0.5,1$ respectively, and fixed
$\beta=1.5$, $\delta=0.3$, $\alpha=1,m=1$.  Figure 2: Higher-order two band
braids (a) $\tau_{1}^{2}$ (Hopf link) and (b) $\tau_{1}^{3}$ (trefoil knot)
when $m=2$ and $3$ respectively. (c,d) Energy loops on the Riemann sheets of
$H(z)$ are shown corresponding to the braids in (a) and (b) respectively.
Location of EPs (black dots) are shown inside first BZ (black circle) on the
$z$-plane. Here $\beta=1.5,\gamma=1,\delta=0.3,\alpha=1$. Figure 3: (a) A
lattice of non-Hermitian trimers considered for three band braids. (b) Phase
diagram to realize identity braid $\tau^{0}$, $\mathbb{B}_{2}$ generators
$\tau_{1}$, $\tau_{2}$, and braid words $\tau_{1}\tau_{2}$, $\tau_{2}\tau_{1}$
are shown. Different braid regimes are separated by EP lines. White, light
gray and deep gray regions correspond to topological index $\nu=0,1$ and $2$
respectively. (c) Schematic diagrams of few braids. Braiding of $E_{1}$ (red),
$E_{2}$ (blue) and $E_{3}$ (green) bands are shown for (a) $\tau^{0}$, (b)
$\tau_{1}$ and (c) $\tau_{2}$. Figs (c,d,e) are obtained for
$(\beta,\gamma)=(1,0.1),(0.8,0.2),(1.6,0.3)$ respectively, and fixed values of
v$=0.7$, $\delta=0.3,\alpha=1,k_{0}=\pi/4$. Energies at
$(\mbox{EP}_{1},\mbox{EP}_{2})\simeq(-0.7,1.4)$.

Gain-loss induced braid group $\mathbb{B}_{2}$. Consider a two band non-
Hermitian lattice with $\alpha$ and $(\beta,\delta)$ are intra-dimer and
$m$-th nearest-neighbor Hermitian hopping amplitudes respectively, $\pm
i\gamma$ represent onsite imaginary potentials (Fig.1a). The effective-
Hamiltonian under periodic boundary condition is given by

$H(k)=\begin{pmatrix}2\delta\sin mk+i\gamma&\alpha+\beta e^{-imk}\\\
\alpha+\beta e^{imk}&-i\gamma\end{pmatrix},\quad$ (1)

with two energy bands

$\begin{array}[]{ll}E_{1,2}(k)=\delta\sin mk\\\
\quad\mp\sqrt{\alpha^{2}+\beta^{2}+2\alpha\beta\cos mk+(i\gamma+2\delta\sin
mk)^{2}}.\end{array}$ (2)

In general $E_{1}$ and $E_{2}$ either braid $m$ times or does not braid at all
such that energy permutes non-trivially
$(E_{1},E_{2})\rightarrow(E_{2},E_{1})$ when $m$ is odd, and trivial
permutation $(E_{1},E_{2})\rightarrow(E_{1},E_{2})$ corresponds to even $m$.
Examples of $\tau^{0}$ and generators $\tau_{1}^{\pm 1}$ of $\mathbb{B}_{2}$,
corresponding to unlink and unknot respectively, are shown in Fig.1c. Higher
order braids $\tau_{1}^{2}=\tau_{1}\tau_{1}$ (Hopf link) and
$\tau_{1}^{3}=\tau_{1}\tau_{1}\tau_{1}$ (trefoil knot) are shown in Fig.2. At
an EP two bands do not braid but undergo a braid phase transition. Here, EPs
are obtained at the vanishing discriminant of the characteristic polynomial
$\mbox{det}(H(k)-E)$, and are given by the lines

$\begin{array}[]{ll}\gamma=\pm(\beta-\alpha)\quad\mbox{at}\quad k=\pi/m,\\\
\gamma=\pm(\beta+\alpha)\quad\mbox{at}\quad k=0,\end{array}$ (3)

for $\delta\neq 0$. At EPs $E_{1}=E_{2}=E^{*}=0$. Different braid diagrams are
distinguished by the topological integer invariant $\nu\in\mathbb{Z}$ defined
by the spectral winding number with respect to the energy at an EP [19]

$\nu=\int\limits_{k\in BZ}\frac{dk}{2\pi
i}\frac{d}{dk}\ln\det\left(H(k)-\mbox{EP}\right).$ (4)

The braid degree $\nu$ describes how many times the two bands wind around an
EP, with the sign indicating the handedness of braids. We obtain $\nu\neq
0~{}(=0)$ for a non-trivial (trivial) braid with $\nu(\tau_{1}^{\pm m})=\pm
m$. Phase diagram in the $(\beta-\gamma)$ plane for braids with distinct $\nu$
are shown in Fig.1b. Different phases are separated by exceptional lines given
in (3). While zero gain-loss always results a trivial braid, a non-zero
critical gain-loss is required to induce non-trivial braids. Nontrivial braids
are also absent whenever the long-range coupling either $\delta$, or $\beta$
is zero. For a fixed non-zero value of $\beta$ the transition from trivial to
non-trivial braid $\tau^{0}\rightarrow\tau_{1}^{m}$ occurs solely by the
tuning of onsite gain/loss in the system (FIg.1b). The braiding direction can
be reversed $(\tau_{1}^{m}\rightarrow\tau_{1}^{-m})$ by switching gain into
loss $(\gamma\rightarrow-\gamma)$ as consequence of
$H^{\dagger}(k;\gamma)=H(k;-\gamma)$. Further, braids with $m$ turns are
obtained by long-range coupling every dimer to its $m$-th neighbor (Fig.1a).

To further understand the role of EPs on the formation of braids and knots we
obtain energy Riemann surfaces of $H(z)$ where $z\rightarrow re^{ik}$, for
$r>0$. The first BZ is now a circle on the complex $z$-plane. In general, for
a $m$-th order braid or knot the two energy bands either encircles in the
vicinity of a single EP $m$-times or $m$ distinct EPs inside the first BZ.
Locations of EPs on the $z$-plane are given by

$z^{*}=\left(\frac{\alpha^{2}+\beta^{2}-\gamma^{2}}{2\alpha\beta}\right)^{1/m}e^{i\frac{(2j-1)\pi}{m}},\quad
j=1\cdots m,$ (5)

with $E_{1,2}(z^{*})=0$. Examples of Hopf link and trefoil knot on energy
Riemann surfaces of $H(z)$ are shown in Fig.2 with two and three EPs
respectively inside the first BZ.

Figure 4: Braids (a) $\tau_{1}\tau_{2}$ and (b) $\tau_{2}\tau_{1}$ showing
non-Abelian energy swapping. (c,d) Corresponding non-Abelian loops are shown
on the energy Riemann surfaces of $H(z)$ in the vicinity of two EPs
$(\mbox{EP}_{1},\mbox{EP}_{2})\simeq(-0.51,1.15)$ lying inside the first BZ.
The $\tau_{1}\tau_{2}$ loop encircles first EP1 then EP2, and the
$\tau_{2}\tau_{1}$ loop encircles first EP2 then EP1 when the BZ is traversed
in counterclockwise. Here $\delta=0.3,\mbox{v}=0.7,\gamma=0.7,\alpha=1$, and
$k_{0}=\pi/4$ but $\beta=-1.2$ in (a,c) and $1.2$ in (b,d) respectively.

Gain-loss induced non-Abelian braid group $B_{3}$. We now consider a minimal
model of non-Hermitian lattice of trimers (Fig.3a), with Bloch Hamiltonian
given by

$H(k)=\begin{pmatrix}-2\delta\sin 2k+i\gamma&\alpha&\beta e^{-ik}\\\
\alpha&\mbox{v}&\alpha\\\ \beta e^{ik}&\alpha&-i\gamma\end{pmatrix}.\quad$ (6)

Here $\alpha$ is the intra-cell coupling, $\beta$ is the nearest-neighbor and
$\delta$ is the next-nearest-neighbor Hermitian coupling. $\pm i\gamma$ and v
are imaginary and real onsite potentials respectively. Three energy bands
$E_{1}$, $E_{2}$, and $E_{3}$ as a function of $k$ constitute different braid
elements for the group $\mathbb{B}_{3}$, provided the discriminant of the
characteristic polynomial is non-zero to avoid degeneracy. Phase diagram in
the $(\beta-\gamma)$ plane for few lowest order braids
$\tau^{0},\tau_{1},\tau_{2},\tau_{1}\tau_{2}$ and $\tau_{2}\tau_{1}$ are
computed and shown in Fig.3. Phase boundaries i.e. exceptional lines between
two distinct braids correspond to vanishing discriminant of the cubic
polynomial $\mbox{det}(H-E)$. Numerically computed examples of braid
generators $\tau_{1}$, $\tau_{2}$ and identity braid $\tau^{0}$ are shown in
Fig.3, with energy permutation
$(E_{1},E_{2},E_{3})\rightarrow(E_{2},E_{1},E_{3})$, $(E_{1},E_{3},E_{2})$ and
$(E_{1},E_{2},E_{3})$ respectively. Phase transitions
$\tau^{0}\rightarrow\tau_{1}$ occurs through the coalescence of bands $E_{1}$
and $E_{2}$ at $\mbox{EP}_{1}$, and $\tau^{0}\rightarrow\tau_{2}$ occurs at
the $\mbox{EP}_{2}$ through the coalescence of $E_{2}$ and $E_{3}$.
Determination of topological index for braids (and knots) for more than two
bands is a technically challenging task and usually done by polynomial
invariant [31, 6, 32]. Here we adopt the global winding number approach which
correctly captures the essential braid invariance properties of interests. The
index is given by $\nu=\sum\limits_{j}\nu_{j}(\mbox{EP}_{j}),$ where
$\nu_{j}(\mbox{EP}_{j})$ is the winding number around the energy at
$\mbox{EP}_{j}$ (refer to Eq. (4)). In the absence of exact expressions we
numerically obtain EPs for the examples given in Fig. 3. Both the EPs lie
outside the energy loops for the identity braid $\tau^{0}$, but EP1 lies
inside the $E_{1}-E_{2}$ loop of $\tau_{1}$, and EP2 lies inside the
$E_{2}-E_{3}$ loop of $\tau_{2}$. Correspondingly, we obtain $\nu(\tau^{0})=0$
and $\nu(\tau_{1})=\nu(\tau_{2})=1$. Similar two band cases, the handedness of
a three band braiding can also be reversed by $\gamma\rightarrow-\gamma$, and
higher order braids can be induced by considering long-range coupling.

One of the remarkable features of three bands braiding is that concatenation
of two braids is non-Abelian. For example,
$\tau_{1}\tau_{2}\neq\tau_{2}\tau_{1}$ as the braid diagrams for these two
products are different (particular examples are shown in Fig.4 together with
associated energy loop on three Riemann sheets). The spectral winding number
is given by $\nu(\tau_{1}\tau_{2})=\nu(\tau_{2}\tau_{1})=2$ as a consequence
of two EPs lie inside the energy loops. Inequivalence of $\tau_{1}\tau_{2}$
and $\tau_{2}\tau_{1}$ is, however, stems from the universal non-Abelian
evolution around multiple EP singularities (see ref. [34, 33, 35]). Note the
different orders in which two EPs are encircled for the two braids:
counterclockwise loop $\tau_{1}\tau_{2}$ encircles EP1 first followed by EP2,
the order is reversed for $\tau_{2}\tau_{1}$ (see Fig. 4). Non-Abelian
braiding has important observable effect in terms of energy permutations; one
obtains $(E_{1},E_{2},E_{3})\rightarrow(E_{2},E_{3},E_{1})$ for
$\tau_{1}\tau_{2}$ and $(E_{1},E_{2},E_{3})\rightarrow(E_{3},E_{1},E_{2})$ for
$\tau_{2}\tau_{1}$ (Fig. 4(a,b)). For a fixed gain-loss potential two non-
Abelian braids can be realized by changing sign of the coupling $\beta$ (see
phase diagram in Fig. 3b). Note that engineering of negative coupling was
demonstrated in ref.[36, 37].

Looking ahead, the fundamental interplay of non-Abelian eigenstate braiding,
non-Abelian Wilczek–Zee connection [38], and open-boundary induced non-
Hermitian skin-effect [18], kept outside the scope of current investigation,
merit further exploration in NH multiband lattices. From application point of
view, the proposed braiding and non-Abelian swapping of energy bands opens
pathways to design variety of quantum and classical non-Hermitian fault-
tolerant topological devices where gain and loss are generated by judicious
interaction with the environment [39].

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# Strong coupling of plasmonic bright and dark modes with two eigenmodes of a
photonic crystal cavity

Fanqi Meng<EMAIL_ADDRESS>Physikalisches Institut, Johann
Wolfgang Goethe-Universität, Frankfurt am Main, Germany  Lei Cao
Physikalisches Institut, Johann Wolfgang Goethe-Universität, Frankfurt am
Main, Germany  State Key Laboratory of Advanced Electromagnetic Engineering
and Technology, Huazhong University of Science and Technology, Wuhan 430074,
China  Aristeidis Karalis  Research Laboratory of Electronics, Massachusetts
Institute of Technology, Cambridge,USA  Hantian Gu  Physikalisches Institut,
Johann Wolfgang Goethe-Universität, Frankfurt am Main, Germany  Mark D.
Thomson  Physikalisches Institut, Johann Wolfgang Goethe-Universität,
Frankfurt am Main, Germany  Hartmut G. Roskos<EMAIL_ADDRESS>Physikalisches Institut, Johann Wolfgang Goethe-Universität, Frankfurt am
Main, Germany

###### Abstract

Dark modes represent a class of forbidden transitions or transitions with weak
dipole moments between energy states. Due to their low transition probability,
it is difficult to realize their interaction with light, let alone achieve the
strong interaction of the modes with the photons in a cavity. However, by
mutual coupling with a bright mode, the strong interaction of dark modes with
photons is possible. This type of mediated interaction is widely investigated
in the metamaterials community and is known under the term electromagnetically
induced transparency (EIT). Here, we report strong coupling between a
plasmonic dark mode of an EIT-like metamaterial with the photons of a 1D
photonic crystal cavity in the terahertz frequency range. The coupling between
the dark mode and the cavity photons is mediated by a plasmonic bright mode,
which is proven by the observation of a frequency splitting which depends on
the strength of the inductive interaction between the plasmon bright and dark
modes of the EIT-like metamaterial. In addition, since the plasmonic dark mode
strongly couples with the cavity dark mode, we observes four polariton modes.
The frequency splitting by interaction of the four modes (plasmonic bright and
dark mode and the two eigenmodes of the photonic cavity) can be reproduced in
the framework of a model of four coupled harmonic oscillators.

††preprint: AIP/123-QED

##  I Introduction

Dark states are either excitations of (quasi-)particles coupled with the
ground state by electromagnetic transitions with weak or vanishing dipole
moments or electromagnetic modes of resonators which do not or only weakly
couple to external radiation. As dark states are not easily accessible by
light, it is difficult to achieve strong interaction with the photons of a
cavity Rivera  _et al._ (2016). In recent years, there has been a significant
interest to investigate ways to achieve strong coupling of dark statesWhite
_et al._ (2019); Rousseaux  _et al._ (2020); Cuartero-González and Fernández-
Domínguez (2018). Such investigations are not only of interest for a
fundamental understanding of light-matter and many-particle interactions, but
also have the potential to provide alternative routes for designing new
optoelectronic and memory devicesZhang  _et al._ (2015).

EIT-like planar plasmonic metamaterials (MMs) is an intensively studied
research topicZhang  _et al._ (2008, 2016a). It is a classical analogue of EIT
of three-level quantum systems, and manifests as a sharp transparency band in
a broad absorption spectrum, originating from the destructive interference
between a bright and a dark plasmonic mode of the MM. Owing to their highly
dispersive refractive index, EIT-like MMs are ideal candidates for realizing
slow-light devices Gu  _et al._ (2012); Ma  _et al._ (2021) and ultrasensitive
sensorsZhu  _et al._ (2021); Wu  _et al._ (2017). The strong coupling between
plasmonic bright-bright and bright-dark modes was observed to occur in various
MMs structures Singh  _et al._ (2009); Zhang  _et al._ (2016a); Li  _et al._
(2017); Yu  _et al._ (2018); Zhang  _et al._ (2022).

A rotationally symmetric 1D photonic crystal cavity exhibits two degenerate
fundamental eigenmodes with perpendicular polarization directions of the
electric field. Frequency-resonant, linearly polarized incident photons excite
only one of these eigenmodes, which is called the bright photonic eigenmode.
The unexcited mode with perpendicular polarization can be regarded as dark
photonic mode. In previous publications, we reported the strong coupling
between bright plasmonic and bright photonic modesMeng  _et al._ (2019, 2021);
Cao  _et al._ (2022). In this contribution, we investigate the strong coupling
of bright and dark plasmonic modes of an EIT-like MM with bright and dark
cavity modes of a terahertz photonic crystal cavity. The interactions among
plasmonic bright mode, dark mode and the two photonic eigenmodes can be
simulated in a conventional four-particle coupling picture.

##  II Experimental results and electromagnetic simulations

In the experiments, we employed a 1D photonic crystal (PC) cavity made from Si
slabs separated by air gapsMeng  _et al._ (2019); Zhang  _et al._ (2016b). The
slabs, with lateral dimensions of 10$\times$10 mm2, were cut from commercially
available wafers of highly resistive silicon (specific resistance: >20
k$\Omega$cm, two-side polished, thickness as purchased). For their positioning
at a fixed distance, we employed metallic shim rings purchased from MISUMI
Europa GmbH, seeMeng  _et al._ (2021, 2019). The cavity consisted of five Si
slabs. It was composed of a central 100-$\mu$m-thick Si slab as a ‘defect’
layer between two Bragg mirrors. The Bragg mirrors each consisted of two pairs
of thin Si dielectric layers and air layers, with the respective thicknesses
of 50 $\mu$m and 96 $\mu$m, resulting in a fundamental cavity mode at 490 GHz
as confirmed by terahertz transmission measurements. The EIT-like MM consisted
of a 2D periodic array of pairs of square-shaped split-ring resonators (SRRs)
which were fabricated by photo-lithography on the surface of the defect layer,
as shown in Fig. 1a (top). The SRRs was made by evaporating Ti/Au with the
thicknesses of 10/200 nm. The geometrical parameters for the SRRs are given in
the caption of the figure. A schematic view of the cavity, loaded with the
EIT-like MM structure, is shown in Fig. 1a (bottom). A description of the
experimental setup for the terahertz measurements can be found in Refs. Meng
_et al._ (2019, 2021). The electric radiation field was polarized in the
x-direction (for the definition of directions, see the coordinate system in
Fig. 1a (top)).

Figure 1: (a) Top: Unit cell of the EIT-like MM; bottom: Schematic
presentation of the 1D PC cavity loaded with the EIT-like MM. Geometrical
parameters of the EIT-like MM: m = 90 $\mu$m, n = p = 40 $\mu$m, u = 5 $\mu$m,
g = 5 $\mu$m, s = 2 $\mu$m. (b) Power transmittance spectra of the
metamaterial-loaded cavity, both measurements (full lines) and results of CST
simulations (dahed lines). Top panel: MM with a single SRR in the unit cell;
bottom pannel: EIT-like MM.

For comparative measurements, we used a MM with a single square-shaped SRR in
the unit cell. It had the same geometrical period as the SRRs of the EIT-like
MM unit cell, the side with the split was oriented along the x-direction. The
resonance frequency as predicted by simulations using the CST electromagnetic
solver (CST: Computer Simulation Technology from Dassault Systèmes Simulia SE)
is 445 GHz. The measured resonance frequency of the bare cavity is 480 GHz,
which is slightly lower than the simulated frequency due to fabrication
tolerances. With the bright SRR MM in the PC cavity, we measured the
transmission spectrum shown by the blue solid curve in Fig. 1b. One observes
two transmittance resonances which represent the lower polariton (LP) and the
upper polariton (UP) of the coupled system. When the dark SRR MM is positioned
in the PC cavity, the measured transmission spectrum only shows one cavity
mode, with the reduced Q-factor of 23. The measured cavity frequency decreases
to 466 GHz due to additional metal added to the cavity. By taking the
perturbed cavity mode frequency (466 GHz) and simulated resonant frequency of
SRR, we determined the Rabi splitting is of 87 GHz, which corresponds to about
19% of the cavity resonance frequency and hence indicates a mode interaction
in the strong-coupling regime Meng  _et al._ (2019). Measured transmittance
peaks and resonance frequencies (dashed line) obtained by CST simulations are
in good agreement with each other. But the measured spectral linwidths are
broader than that of the simulations.

When instead the EIT-like MM is used in the cavity, one instead observes four
polariton modes. This is shown in the lower panel of Fig. 1b (red solid
curve). The four polariton modes are at 387 GHz (LP1), 423 GHz (LP2), 483 GHz
(UP1) and 520 GHz (UP2), in good agreement with CST simulations (dashed line
in Fig. 1b). The appearance of four polariton modes can be explained as
follows.

Figure 2: Schematic illustration of the four-mode coupling. "E" represents
electric coupling and "M" represents magnetic coupling.

The mutual interaction of the modes of the cavity and the EIT-like MM is
schematically shown in Fig. 2. One can understand the interaction if one
applies a sequential-excitation picture: For an incoming terahertz wave with
an electric field polarized along the x-axis, the photons excite the bright
cavity eigenmode, which couples directly with the plasmonic bright mode of the
left SRR in Fig. 1a. The right SRR (exhibiting the plasmonic dark mode,
frequency-degenerate with the bright mode of the left SRR) is then excited
mainly due to magnetic near-field interaction – albeit also with an electric
contribution Li  _et al._ (2017) – with the left SRR, with a coupling strength
dependent on the lateral distance between them. The right SRR becomes
radiative, with its electric dipole moment oriented along y-direction
contributing emission with an electric field in y-direction which couples to
the dark cavity eigenmode. This interaction chain hence couples all four
quasi-particles (plasmonic bright and dark MM modes, photonic bright and dark
cavity modes) with each other, which produces four new polariton modes as well
as cross-polarization conversion.

Figure 3:  (a) Power transmission spectra of radiation polarized in
x-direction. For a cavity loaded with an EIT-like MM, where the SRRs are
physically connected with each other (experiment: red line, CST simulation:
green dashed line), respectively have a distance s of 2 $\mu$m (experiment:
blue line, CST simulation: magenta dashed line). The 2-$\mu$m-data are a
reproduction of those shown already in Fig. 1b. (b) Amplitude transmission
spectra of radiation polarized in x-direction. CST simulations for several
distances between the two SRRs. Each neighboring curve shifts vertically by
0.1 for better comparison. In this set of simulations, m is taken as 110
$\mu$m, and n and p are assumed to be 38 $\mu$m.

As stated above, the Rabi splitting between polariton modes LP1 (UP1) and LP2
(UP2) depends on the distance of the two SRRs. This is confirmed both
experimentally and theoretically with the data shown in Fig. 3. The red solid
curve in Fig. 3a displays the four polariton modes measured when the distance
between the two SRRs is zero and the two SRRs are physically connected with
each other. The LP1 mode lies in the shoulder of the PC’s forbidden band, but
is still recognizable. For comparison, the blue curve in Fig. 3a reproduces
the transmittance curve of Fig. 1a measured with the EIT-like MM in the
cavity, but here with the SRRs separated by 2 $\mu$m. In the case of zero
distance, the mode splitting between LP1 and LP2, respectively UP1 and UP2, is
larger than if the distance of the two SRRs is 2 $\mu$m. This observation is
confirmed by the results of CST simulations shown by the dashed curves in Fig.
3a. When the two SRRs are not physically connected, the coupling between them
is mediated by the light-induced magnetic field, and thus is inductive. When
the two SRRs are physically connected, the coupling between them is both
inductive and conductive, which leads to a stronger interaction and larger
Rabi splitting Liu, Kaiser, and Giessen (2008).

Additional insight is provided by CST simulations. Fig. 3b shows calculated
amplitude transmission spectra for various distances between the two SRRs,
ranging from 0 $\mu$m to 11 $\mu$m. In order to suppress interaction between
them for all separation distances, the period of the MM was increased from 90
$\mu$m in the measurements to 110 $\mu$m in the simulations, and the size of
the SRRs was reduced from 40 $\mu$m to 38 $\mu$m. The polariton splitting is
then always dominated by interactions within each unit cell. The simulation
confirms that the splitting between LP1 and LP2, and UP1 and UP2, increases as
the distance between the SRRs is reduced. Interestingly, the frequency
positions of LP2 and UP1 do not change much with distance. Except for zero
distance, LP2 remains at 420 GHz and UP1 at 500 GHz. As the distance between
the two SRRs increases, LP1 moves upwards in frequency towards LP2, and UP2
downwards towards UP1. One anticipates that ultimately the two LP modes would
merge into one, and the same would occur with the two UP modes, hence the four
polariton modes would converge into the two polariton modes which arise by the
coupling of the bright SRR mode with the bright cavity mode.

Figure 4: Resonance frequencies of the four mixed modes (representing the
polaritons) as obtained by the model of four coupled harmonic oscillators.
Black symbols: Resonance frequencies are extracted from Fig. 3b, and plotted
versus the value of the coupling parameter $V_{2}$ obtained by a fit. The full
lines are the results of a fit with a fixed global coupling parameter $V_{1}$.

##  III Discussions

###  III.1 Model of four coupled oscillators

The frequency dependence of the interaction among the four modes can be
captured with a model of four coupled harmonic oscillators, described by four
differential equations of motion with two coupling constants, $V_{1}$ and
$V_{2}$. $V_{1}$ represents the coupling constant between plasmon and photon,
and $V_{2}$ indicates the coupling constant between bright plasmon and dark
plasmon. The set of differential equations together with the derivation of the
eigenfrequenices of the mixed modes is given in Appendix A, extending the
analysis of Meng  _et al._ (2021).

Each mode is represented by its time-dependent generalized coordinate,
$x_{1}(t)$ to $x_{4}(t)$. $x_{1}$ and $x_{4}$ describe the bright and dark
cavity modes, respectively, assumed to be frequency-degenerate, with the
common resonance frequency $\omega_{c}=2\pi f_{c}$. $x_{2}$ and $x_{3}$ are
the coordinates of the bright and the dark plasmonic mode of the MM – also
assumed to be frequency-degenerate at $\omega_{p}=2\pi f_{p}$. The two bright
modes couple directly with each other, which is expressed by coupling terms in
the equations for $x_{1}$ and $x_{2}$ with the coupling strength $V_{1}$.
Correspondingly, the dark modes couple with each other. The interaction is via
the y-oriented electric dipole moment of the dark plasmonic mode, which has
the same dipole strength as the x-oriented dipole moment of the bright
plasmonic mode. In the harmonic-oscillator model, this interaction finds its
expression in the equations of motion for $x_{3}$ and $x_{4}$ by coupling
terms, which also have the strength $V_{1}$. The third and final interaction
is between the two plasmonic modes. It is represented by coupling terms in the
equations of $x_{2}$ and $x_{3}$, with the strength of this interaction given
by $V_{2}$. A higher value of $V_{2}$ corresponds physically to a smaller
distance between the two SRRs of the unit cell.

The full lines in Fig. 4 show the evolution of the resonance frequencies of
the mixed modes (representing the polaritons), obtained from the set of
differential equations, as a function of the coupling parameter $V_{2}$. The
curves were derived by a fit to the resonance frequencies extracted from the
five CST-determined transmission spectra displayed in Fig. 3b. The coupling
parameter $V_{1}$ was used as a global parameter for all values of the
distance s of the SRRs, with a resultant value of $V_{1}=0.4950\times 10^{12}$
rad/s. The other parameters in the model were obtained as $f_{p}$ = 464.8 GHz
and $f_{c}$ = 474.2 GHz. The mixed-mode branches 1 and 4 are found to exhibit
strong gradients as a function of $V_{2}$, while the branches 2 and 3 are less
sensitive to the variation of $V_{2}$. For $V_{2}\rightarrow 0$, branch 1
merges with branch 2, as does branch 3 with branch 4.

We also performed independent fits for each value of the distance s between
the SRRs, where only the coupling parameter $V_{2}$ was varied. The black
symbols in Fig. 4 represent the resonance frequencies as a function of
$V_{2}$. The fit with a fixed value of $V_{1}$ especially predicts a change of
the resonance frequencies of the mixed modes 2 and 3, when the value s is
changed, while one observes in Fig. 3b (and for the black circles and upwards-
pointing triangles in Fig. 4) that the mixed-mode resonance frequencies remain
nearly constant, except for $s\rightarrow 0~{}\mu$m. One also notes that for
$s\rightarrow 0~{}\mu$m, a fit with a constant $V_{1}$ overestimates the
resonance frequency of the mixed mode 1, and also slightly that of mixed mode
2. This is caused by the conductive coupling which arises in addition to the
inductive coupling of the two SRRs and impacts also $V_{1}$.

Figure 5:  Color plots of the simulated transmission spectra of a EIT-MM-
loaded cavity as a function of the resonance frequency of the dark SRR. The
inset displays the loaded cavity, with the EIT-like MM represented by a two-
SRR unit cell. The white dashed lines represent the results calculated with
the equations of motion of the four-coupled-oscillators model.

The plasmonic dark mode plays a key role in bringing the cavity dark mode into
the coupling scenario. We study now how this mediating effect is influenced
when the cavity dark mode is detuned from the resonance frequencies of the
cavity modes and the plasmonic bright mode. We simulated the transmission of
the EIT-SRR-loaded cavity with the CST Maxwell solver, changing the
geometrical dimensions of the SRR located at the right side in the sketch of
Fig. 1a (top). For the MM, we otherwise used the same geometrical parameter
values as for the plots in Fig. 3b, namely m = 110 $\mu$m, n = 38 $\mu$m, and
s = 2 $\mu$m. Only the value of p was changed from 32 $\mu$m to 42 $\mu$m. The
color graph in Fig. 5 presents the transmission spectra for x-polarized
radiation as a function of the changing resonance frequency of that SRR. When
the frequency is strongly detuned (both at the low-frequency limit and the
high-frequency one), both the plasmonic dark mode and the cavity dark mode are
no longer involved in the strong coupling, and only the bright modes of the
cavity and the MM participate in the strong interaction. One thus expects only
two polariton modes to remain, which lie around the frequencies marked as “LP”
and “UP” (written in white lettering) in Fig. 5. In fact, for detuning to
large frequencies, the UP2 and the LP2 mixed-mode branches die out, leaving
oscillator strength only in UP1 and LP1. For detuning to low frequencies, the
opposite occurs: Oscillator strength survives in UP2 and LP2, while UP1 and
LP1 die out. Inbetween these extremes, one observes anti-crossing behavior of
the UP1 and UP2 branches, and the LP1 and LP2 branches, the respective mode
splittings amount to about 8 GHz, are larger than the spectral linewidths of
the polariton modes, and correspond to 6% and 5% of the respective LP and UP
mode frequencies. This indicates that the interactions are in the strong-
coupling regime.

We also calculated the mixed-mode frequencies with the equations of motion of
the four coupled harmonic oscillators, using the values $f_{c}=462$ GHz,
$f_{p}=436$ GHz for the resonance of the unmodified SRR, $V_{1}=0.7024\times
10^{12}$ rad/s and $V_{2}=0.2903\times 10^{12}$ rad/s. The results are plotted
as white dashed curves in Fig. 5. They reproduce the resonances obtained with
the full-wave simulations reasonably well. The deviations are probably due to
the fact that we used fixed values for the coupling constants $V_{1}$ and
$V_{2}$.

We end this chapter with two remarks and refer the reader for further
information to the Appendix. First, we point out the equivalent-circuit model
addressed in Appendix B. The model is based on lumped elements (capacitors,
inductors and resistors) from which the various coupled resonators are built.
The model can provide a qualitative and even quantitative analysis of the
system of coupled resonators if the coupling mechanisms (capacitive, inductive
or both) are clearPoo  _et al._ (2014). In this regard, it is of a more
microscopic nature than the phenomenological model of four coupled equations
of motion.

Second, we point out that if both SRRs are bright (i.e., have parallel
electric dipole moments), there will be three polariton modes instead of four,
when coupled to the cavity (see Appendix C with Fig. 10). In this case, the MM
modes only interact with the bright cavity mode and not with the dark one. The
mixed-mode frequencies are well reproduced by the equations-of-motion model of
three coupled oscillatorsMeng  _et al._ (2021).

Figure 6:  Amplitude transmission spectrum of a metallic-grating-type Fabry-
Perot cavity loaded with an EIT-like MM (dark-grey line). Amplitude
transmission spectrum of empty Fabry-Perot cavity (red line), of bare MM with
single bright SRR in each unit cell (blue line), of bare MM with EIT SRRs in
each units cell (green line). The sketches indicate the four structures of
which the transmission spectra are displayed, which were calculated with CST
for radiation polarized perpendicular to the metal wires of the grating.

###  III.2 Coupling in a metallic grating-type Fabry-Perot cavity

A way to suppress the coupling to the cavity dark mode is to employ a cavity
that does not support modes with a polarization orthogonal to that of the
bright mode. Such a type of cavity is a Fabry-Perot resonator made with
metallic gratings as reflectors. Ohmic losses strongly suppress standing waves
with a polarization parallel to the metal stripes of the grating, but the
losses are much lower for radiation polarized perpendicular to the stripes.
(We note in passing that a fundamental-mode cavity made with reflectors
consisting of unpatterned sheets of gold, or silver or any other good metallic
conductor would have a poor Q-factor for all polarization directions. In a
fundamental mode cavity, the radiation field penetrates into the metal where
it experiences substantial losses even in the case of good conductors. For
that reason, we employ cavities made from dielectric materials for studies
like those of this publication.)

We investigated mode coupling of SRR MMs in a grating-type cavity by CST
simulations. The gratings were assumed to be made from rectangular gold wires
(conductivity of $4.56\times 10^{7}$ S/m) with a width of 1.4 $\mu$m and a
thickness of 0.2 $\mu$m, and separated from each other by 0.6 $\mu$m. The
wires were assumed to be free-standing in air. The cavity consisted simply of
two grating reflectors, with a distance of 180 $\mu$m between them. The red
line in Fig. 6 displays the transmission spectrum of the empty cavity for the
amplitude of radiation polarized perpendicular to the grating wires. The
transmittance exhibits a peak at 784 GHz, the resonance frequency of the
fundamental cavity mode.

Into the cavity, centered between the reflectors, we placed an EIT-like MM
structure (see schematic drawings in Fig. 6). It consisted of two substrate-
free metallic SRRs, each having a fundamental resonance at 754 GHz (see blue
line in Fig. 6), split into two polariton modes at 701 GHz and 794 GHz (see
green line in Fig. 6). The parameters of the MM in the simulations (using the
nomenclature of Fig. 1a (bottom)) were as follows. Material: gold stripes with
a thickness of 0.2 $\mu$m, u = 5 $\mu$m, g = 5 $\mu$m, s = 2 $\mu$m, n = p =
40 $\mu$m, m = 90 $\mu$m. With the EIT-like MM structure in the cavity, one
observes three transmission peaks at 618 GHz, 751 GHz and 862 GHz, the middle
one lying close to the resonance frequency of the individual SRRs and being
considerably weaker than the transmission peaks above and below. The
appearance of three resonances is consistent with the strong coupling among
three modes (cavity bright mode, plasmonic bright mode and plasmonic dark
mode). The suppressed cavity dark mode does not contribute noticeably in the
interaction. This result supports the notion that it is indeed the cavity dark
mode which is responsible for the appearance of the fourth resonance in the
experiments and simulations with the EIT-like MM in the dielectric cavity
discussed in the previous sections of this publication.

###  III.3 Cross-polarization conversion and polarization tuning

The coupling of the bright and dark modes of the EIT-like MM can be employed
for cross-polarization conversion. Linearly polarized incident radiation is
converted to elliptically polarized oneLi  _et al._ (2017). We briefly
consider here, how the orientation of the incoming electric field influences
the polarization of the outgoing field. Fig. 7 displays the co-polarized and
cross-polarized transmission spectra of the radiation passing the EIT-like-MM-
loaded cavity, as the polarization angle $\phi$ of the incident electric field
is varied from 0∘ to 150∘. Co-polarization (cross-polarization) means that the
field orientation of the outgoing wave parallel (perpendicular) to the linear
field orientation of the incoming wave is displayed. We do not analyze the
phases of the two outgoing waves relative to the incoming one. The structural
parameters of the EIT-like MM and the cavity are the same as those used for
Fig. 1b, and an orientation of $\phi=0^{\circ}$ corresponds to the situation
shown there.

In Fig. 7, for most values of $\phi$ both the co- and cross-polarized
transmission spectra exhibit four resonance peaks, and their frequencies do
not change with the variation of $\phi$, only the amplitudes. The co-polarized
transmission exhibits a $180^{\circ}$ rotational symmetry, the cross-polarized
transmission, however, a $90^{\circ}$ symmetry. In co-polarized transmission,
one observes at specific angles only three polariton modes. If we number the
modes with rising frequency as $\\#1$ to $\\#4$, then within the range
$0^{\circ}\leq\phi<180^{\circ}$, mode $\\#2$ disappears at about $30^{\circ}$,
mode $\\#4$ at about $60^{\circ}$, mode $\\#1$ at about $100^{\circ}$, and
mode $\\#3$ at about $150^{\circ}$. A similar reduction of the number of
observed modes occurs at specific angles in cross-polarized transmission. The
details of these phenomenological findings are intriguing and invite future
studies.

Figure 7:  Amplitude transmission spectra of linearly polarized radiation
impinging onto the EIT-like-MM-loaded cavity as a function of the polarization
direction of the incident electric field (CST simulations). (a) Co-polarized
transmitted field component, the spectra for different angles are displaced
vertically by 0.2. (b) cross-polarized transmitted field. For clarity, the
spectra for different angles are displaced vertically by 0.1.

##  IV Conclusion

In conclusion, we have studied experimentally and theoretically the
transmission properties of a metamaterial, consisting of coupled split-ring
resonators and exhibiting electromagnetically-induced-transparency-like
behavior, in a rotationally symmetric photonic crystal cavity at sub-terahertz
frequencies. We have observed the appearance of four transmission peaks. They
have been identified as the signatures of polariton modes arising from strong
mutual coupling of four modes, the bright and dark fundamental modes of the
metamaterial with the two orthogonally polarized fundamental cavity modes. In
a quasi-particle picture, the mutual coupling occurs between bright and dark
plasmons of the metamaterial and the orthogonally polarized photons of the
cavity. Incoming linearly polarized radiation is converted to elliptically
polarized radiation. The coupling to both cavity modes can be reduced to
coupling to only one (and the appearance of only three instead of four
polariton modes), if either the rotational symmetry of the cavity is destroyed
(e.g. by using a cavity made from metallic-grating reflectors) or if the
metamaterial exhibits a total dipole moment which is parallel to the
polarization of the incoming wave.

## Acknowledgements

This research work was funded by DFG projects RO 770/46-1 and RO 770/50-1 (the
latter being part of the DFG-Schwerpunkt "Integrierte Terahertz-Systeme mit
neuartiger Funktionalität (INTEREST)"). Lei Cao acknowledges support from the
HUST Overseas Training Program for Outstanding Young Teachers.

## AUTHOR DECLARATIONS

### Conflict of interest

The authors have no conflicts to disclose.

### Author contributions

Fanqi Meng:Conceptualization (lead); Investigation (lead); Writing-original
draft (equal); Formal analysis (equal). Lei Cao: Writing-original draft
(equal); Formal analysis (equal); Validation (equal). Aristeidis Karalis:
Validation (equal). Hantian Gu: Investigation (equal). Mark D. Thomson:
Conceptualization (equal). Hartmut G. Roskos: Conceptualization (equal);
Writing-review & editing(equal); Funding acquisition (lead); Supervision
(lead).

## Data Availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

##  Appendix A Coupled differential equations of the four-oscillators model

We assume four harmonic oscillators, two having a resonance frequency
$\omega_{1}$ = $\omega_{4}$ = $\omega_{c}$ (cavity mode frequency) and the
other two being resonant at $\omega_{2}$ = $\omega_{3}$ = $\omega_{p}$ (MM
plasmonic mode frequency). Oscillators 1 and 2, respectively 3 and 4, are
coupled, with the coupling strength being determined by the parameter $V_{1}$.
Also, oscillators 2 and 3 are coupled, with a strength given by $V_{2}$. For
an explanation of this choice of coupling, see Sec. III.1. The differential
equations of motion for the four generalized coordinates $x_{1}(t)$ to
$x_{4}(t)$ of the oscillators read as follows:

$\displaystyle\ddot{x}_{1}+\omega_{c}^{2}x_{1}+V_{1}\dot{x}_{2}=0\,,$ (1)
$\displaystyle\ddot{x}_{2}+\omega_{p}^{2}x_{2}-V_{1}\dot{x}_{1}-V_{2}\dot{x}_{3}=0\,,$
$\displaystyle\ddot{x}_{3}+\omega_{p}^{2}x_{3}+V_{2}\dot{x}_{2}+V_{1}\dot{x}_{4}=0\,,$
$\displaystyle\ddot{x}_{4}+\omega_{c}^{2}x_{4}-V_{1}\dot{x}_{3}=0\,.$

If the time-dependent factor is taken as $x_{i}(t)$ =
$x_{i}^{0}e^{-j{\omega}t}$, with $\,i=1,2,3,4$, the frequency-domain
expressions of the differential equations can be written in matrix form as

$\begin{bmatrix}{\omega}^{2}-{\omega}_{c}^{2}&j{\omega}V_{1}&0&0\\\
-j{\omega}V_{1}&{\omega}^{2}-{\omega}_{p}^{2}&-j{\omega}V_{2}&0\\\
0&j{\omega}V_{2}&{\omega}^{2}-{\omega}_{p}^{2}&j{\omega}V_{3}\\\
0&0&-j{\omega}V_{3}&{\omega}^{2}-{\omega}_{c}^{2}\end{bmatrix}\begin{bmatrix}\;x_{1}^{0}\;\\\
x_{2}^{0}\\\ x_{3}^{0}\\\ x_{4}^{0}\end{bmatrix}=0\,.$ (2)

Letting the determinant be zero, we obtain the function for the
eigenfrequencies in general polynomial form:

$ax^{4}+bx^{3}+cx^{2}+dx+e=0\,,$ (3)

with $x=\omega^{2}$. The coefficients are given explicitly by

$\left\\{\begin{aligned} &a=1\,,\\\
&b=-(2{\omega}_{c}^{2}+2{\omega}_{p}^{2}+2V_{1}^{2}+V_{2}^{2})\,,\\\
&c=2{\omega}_{c}^{2}{\omega}_{p}^{2}+({\omega}_{c}^{2}+{\omega}_{p}^{2})({\omega}_{c}^{2}+{\omega}_{p}^{2}+2V_{1}^{2})+2{\omega}_{c}^{2}V_{2}^{2}+V_{1}^{4}\,,\\\
&d=-[2{\omega}_{p}^{2}{\omega}_{c}^{2}({\omega}_{p}^{2}+{\omega}_{c}^{2})+2{\omega}_{p}^{2}{\omega}_{c}^{2}V_{1}^{2}+{\omega}_{c}^{4}V_{2}^{2}]\,,\\\
&e={\omega}_{c}^{4}{\omega}_{p}^{4}\,.\end{aligned}\right.$ (4)

Numerical calculation is used to solve the equation and obtain the four
eigenfrequencies of the coupled harmonic oscillators.

##  Appendix B Equivalent-circuit model

An alternative way to physically simulate the interaction of the photonic and
plasmonic resonators is to represent them by LC oscillators of an electronic
circuit whose wiring expresses the interaction between the oscillators Poo
_et al._ (2014). If damping is to be included, one could employ LCR
oscillators, but we will only considered the undamped limit here. Fig. 8 shows
an equivalent circuit which implements the model of Fig. 2 for the description
of the four-mode coupling phenomenon in the PC cavity. Four LC sub-circuits
are used to model the four resonators. The coupling between the cavity
photonic modes and the plasmonic MM modes is assumed to be capacitive
(mediated by the electric radiation field), while the coupling between the
plasmonic bright and dark modes is considered as inductive (mediated by the
magnetic field of the MM).

Figure 8: Equivalent-circuit model for the four-mode coupling phenomenon in
the photonic cavity.

With the formula for approximate inductance calculations of Ref. Sydoruk  _et
al._ , 2009; Zhu, Bosman, and Yang, 2014, we obtain for each SRR, with the
dimensions as given in Fig. 1, an equivalent inductance of Lpb = Lpd = 0.14
nH. The equivalent inductance of the photonic cavity is chosen arbitrarily as
Lcb = Lcd = 1 nH. The equivalent capacitances of the SRRs are determined as
Cpb = Cpd = 0.91 fF to yield a resonance frequency of the SRRs of $f_{p}=445$
GHz, similarly the equivalent capacitances of the cavity are chosen to be Ccb
= Ccd = 0.11 fF to yield a cavity resonance at $f_{c}=480$ GHz. For the
coupling capacitance, we obtain Cbb1 = Cbb2 = Cdd1 = Cdd2 = 0.09 fF so that
the resonance frequencies obtained from the circuit model match closely with
those obtained from CST simulation, while the dimensionless inductive coupling
coefficient k between Lcb and Lcd can be varied from 0 to 1, depending on the
lateral distance of the two SRRs. The losses in the system are neglected.

Fig. 9a shows the calculated scattering parameter S21 for the value k = 0.3 of
the inductive coupling coefficient. One observes four distinct resonances
corresponding to the four eigenmodes. If the coupling coefficient is varied
between 0.25 and 0.35, one obtains the evolution of the four resonance
frequencies as shown in Fig. 9b. With the increase of k, corresponding to a
decrease of the lateral distance between the two SRRs, the highest branch
(mixed mode 4) goes up in frequency, while the lowest branch (mixed mode 1)
moves down. The middle two branches remain almost unchanged. These results
reproduce the general trends of Figs. 3b and 4, which suggests that the simple
equivalent circuit can serve as a tool to model the four-mode coupling
process. The predictive power of the model is, however limited to semi-
quantitative statements, because the exact values of the eigenfrequencies
cannot be predicted due to the uncertainty of the values of many circuit
parameters and the omission of additional capacitive coupling between the two
SRRsPoo  _et al._ (2014).The precise determination of element parameters in
the circuit model is beyond the scope of this publication.

Figure 9: (a) Calculated transmission coefficient of the circuit shown in the
preceding Fig. 8, for an inductive coupling coefficient k = 0.3. (b)
Frequencies of the four eigenmodes as a function of the inductive coupling
coefficient k, with $0.25\leq\mathrm{k}\leq 0.35$. The port impedance was
chosen to be 100 $\Omega$.

##  Appendix C The case of two bright SRRs in the 1D PC cavity

If a MM with two SRRs, which have the same orientation in the unit cell, is
placed into the PC cavity, and the MM is excited with a polarization along the
gapped sides of the SRRs, then the four-mode coupling phenomenon disappears.
This is shown in Fig. 10, which displays simulated transmittance spectra for
the case of two bright SRRs per unit cell, as indicated in the inset of Fig.
10a. Both SRRs have their sides with the gap oriented in x-direction, which is
also the direction of polarization of the incoming terahertz wave and the
direction of the detected electric field. The red and blue curves in Fig. 10a
display spectra for SRRs of different size (red curve), respectively the same
size (blue curve). The separation distance of the SRRS is in both cases the
same, with a value of s = 2 $\mu$m. The red curve was obtained for resonance
frequencies of the two bright SRRs of 468 and 537 GHz. One observes only three
polariton modes. If the resonance frequencies of the SRRs are degenerate (both
at 468 GHz), then one even observes only two polariton modes, as shown by the
blue curve.

For more insight into the coupling between the MM with two bright SRRs in the
unit cell and the cavity, we show in Fig. 10b a color plot of transmittance
spectra as a function of the resonance frequency of one of the SRRs. The
resonance frequencies of the other one and of the cavity were kept fixed at
468 and 480 GHz, respectively. Also the distance between the two bright SRRs
was kept fixed at 2 $\mu$m. The graph shows an anti-crossing behavior of two
modes, an upper (UP) and a lower (LP) polariton mode, with an additional third
polariton mode (MP) between them. Approaching the waist region of the anti-
crossing, where the resonances of the two SRRs become degenerate, from either
above or below the waist, the third mode looses oscillator strength and
vanishes at the degeneracy point. A fourth polariton mode does not appear,
which is evidence that the second cavity mode is not excited and hence does
not take part in the coupling process. The interaction is limited to the two
MM eigenmodes and a single cavity mode. We can fit the UP, middle polariton
(MP) and LP with the coupled three harmonic oscillators modelMeng  _et al._
(2021). The dash lines derived from the coupled three harmonic oscillators
model nicely fit the three polariton branches of the simulation. The open
circles represent the varying resonance frequency of the bare SRR, derived
from CST simulation. It is interesting to note that the MP diminishes as the
resonances of SRRs superimpose with each other, which also represents a
polariton dark modeZhang  _et al._ (2015).

Figure 10: (a) the simulated co-polarized transmittance spectra of two bright
SRRs loaded 1D PC cavity. The resonance frequency of the cavity lies at 480
GHz, and the resonance frequency of one SRR is fixed at 468 GHz. The resonance
frequencies of another SRR lie at 468 GHz (blue curve) and 537 GHz (red
curve). The inset displays the SRRs-loaded cavity, with the SRRs represented
by a two-SRR unit structure. (b) Color plots of the simulated co-polarized
transmission spectra of two bright SRRs loaded 1D PC cavity, as a function of
the resonance frequency of one SRR. In these simulations, m is taken as 110
$\mu$m, n is assumed as 38 $\mu$m. The resonance frequencies of the cavity and
one SRR are fixed at 0.468 and 480 GHz, respectively. The resonance
frequencies of another SRR change form 370 GHz to 600 GHz, which are presented
by the open circles. The dashed lines are the results derived by using three
coupled harmonic oscillators model. The coupling strength between each bright
mode and the cavity photon is assumed as 84 GHz.

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