# Chiral Majorana hinge modes on a curved surface with magnetic impurities

Ilyoun Na<EMAIL_ADDRESS>Department of Physics, University of
California, Berkeley, California 94720, USA  Materials Sciences Division,
Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
Molecular Foundry, Lawrence Berkeley National Laboratory, Berkeley, CA 94720,
USA  James G. McHugh  University of Manchester, School of Physics and
Astronomy, Manchester M13 9PL, United Kingdom  National Graphene Institute,
University of Manchester, Manchester M13 9PL, United Kingdom  Sinéad M.
Griffin  Materials Sciences Division, Lawrence Berkeley National Laboratory,
Berkeley, California 94720, USA  Molecular Foundry, Lawrence Berkeley National
Laboratory, Berkeley, CA 94720, USA  Luca Chirolli  Department of Physics,
University of California, Berkeley, California 94720, USA  NEST, Scuola
Normale Superiore, and Istituto Nanoscienze-CNR, I-56127 Pisa

###### Abstract

Chiral Majorana one-dimensional modes have been proposed as they key component
for topological quantum computing. In this study, we explore their potential
realization as hinge modes in higher-order topological superconductors. To
create such phases, we engineer a sign-changing, time-reversal symmetry-
breaking mass term through an ensemble of magnetic impurities on the surface
of a sphere. The magnetization of this ensemble arises from the competition
between the external magnetic field and the Ruderman-Kittel-Kasuya-Yosida
(RKKY) interaction among the impurities, mediated by the surface Majorana
modes. We determine the magnetic phase diagram and identify the optimal
magnetic field to minimize orbital effects and induce a sign changing mass
term. This term opens a gap in the surface spectrum, resulting in a gapless
one-dimensional chiral Majorana mode along the nodal line of the mass term,
thereby implementing a second-order topological superconductor.

Introduction.—Chiral Majorana one-dimensional (1D) modes are much sought after
for their potential application in topological quantum computing Nayak  _et
al._ (2008); Sarma  _et al._ (2015); Chirolli  _et al._ (2018); Lian  _et al._
(2018); You  _et al._ (2019); Bomantara and Gong (2020); Pahomi  _et al._
(2020) – topological superconductors (TSCs) Kitaev (2001); Qi and Zhang
(2011); Alicea (2012); Flensberg  _et al._ (2021); Das Sarma (2023) offer a
promising platform for isolating these modes. In tandem with the search for
these modes, remarkable progress has been made in higher-order topology Song
_et al._ (2017); Langbehn  _et al._ (2017); Benalcazar  _et al._ (2017);
Schindler  _et al._ (2018); Khalaf (2018); Khalaf  _et al._ (2018); Peng and
Refael (2019); Peng (2020); Yang  _et al._ (2023), which extends the
conventional first-order bulk-boundary correspondence, providing more possible
avenues for realizing Majorana modes.

In this letter, we present a method for creating higher-order TSCs by
introducing symmetry-breaking perturbations to gap out the one-dimensional
lower boundary modes of first-order topological phases. For instsance,
applying a Zeeman field in a two-dimensional (2D) time-reversal invariant
(TRI) TSC results in Majorana corner states Zhu (2018). Extending this
approach to the surface of a three-dimensional (3D) TRI TSC with a magnetic
field generates chiral Majorana hinge modes along lines tangent to the surface
Volovik (2010); Fu and Berg (2010). However, challenges arise in 3D
superconductors due to the presence of Meissner screening, which leads to the
emergence of an orbital supercurrent. This supercurrent couples with surface
Majorana modes, tilting the surface spectrum and becoming dominant over the
Zeeman term as the applied field strength increases Chirolli and Guinea (2018,
2019).

To address this, we propose decorating the surface of a 3D TRI TSC with
magnetic impurities. The net Zeeman field is given by the sum of the external
field and the magnetic moments of the impurities. Aligning the magnetic
impurities with the direction of the field locally enhances the Zeeman term
without affecting the orbital supercurrent, thus increasing the strength of
the nodal time-reversal symmetry-breaking mass term. However, a potential
obstacle to this proposal is the self-ordering of the magnetic impurities
through the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction. Similar to the
case of a topological insulator (TI) Abanin and Pesin (2011), coupling
magnetic impurities with surface Majorana modes in a planar geometry results
in a ferromagnetic RKKY interaction due to the Ising properties of Majorana
surface fermions Chung and Zhang (2009), leading to magnetic order.

We generalize the Ising properties of Majorana surface fermions and the
resulting RKKY interaction, demonstrating its role in promoting a
ferromagnetic radial order that competes with the external magnetic field. In
the absence of an applied field, impurities form a hedgehog configuration
resembling a magnetic monopole. With a sufficiently strong field, the ground
state undergoes a phase transition to a domain wall state. We illustrate that
the critical field for the onset of the domain wall state decreases as the
number of impurities increases. Consequently, in systems with dense enough
magnetic impurity ensembles, the resulting mass term opens a gap in the
surface spectrum, leaving a gapless chiral Majorana mode along the nodal line
at the equator.

Second-order TSC with Zeeman field.—We illustrate the realization of a chiral
Majorana mode along the inversion-symmetric line, which is protected by
intrinsic higher-order bulk topology Khalaf (2018); Trifunovic and Brouwer
(2019). To create a second-order TSC, we start with the massless surface
(flat) states of a first-order 3D TRI odd-parity class DIII TSC Ryu  _et al._
(2010); Fu and Berg (2010); Chiu  _et al._ (2016). By adding a sign-changing
mass term $V(y)=sgn(y)V_{z}$ to the surface Hamiltonian, which is represented
as $H=-iv(\partial_{x}\sigma_{y}-\partial_{y}\sigma_{x})+V(y)\sigma_{z}$ with
Pauli matrices $\sigma_{i}$ denoting the spin degree of freedom, we generate a
chiral mode propagating in the $x$ direction and localized at mass term node.
This mass term can be effectively regarded as a Zeeman term due to its
interaction with the spin degree of freedom. In a spherical geometry, we
introduce the sign changing mass term by applying a uniform Zeeman field along
the polar axis, breaking the time-reversal symmetry while preserving inversion
symmetry. This setup results in the emergence of a single chiral Majorana mode
along the inversion-symmetric equator, where the mass term vanishes, as
illustrated in Fig. 1(a). This realization represents a second-order TSC of
class D with inversion symmetry, topologically classified by $\mathbb{Z}_{2}$
Trifunovic and Brouwer (2019).

Figure 1: (a) Schematic illustration of how second-order TSC can be realized
along the inversion-symmetric curve on the surface of a sphere using a uniform
Zeeman field along the polar axis preserving the inversion symmetry. (b) The
surface energy spectrum of the Dirac Hamiltonian on the surface of sphere
defined in Eq. (1) without, and (c) with a Zeeman field.

Chiral Majorana hinge modes on sphere.—We introduce gamma matrices
$\gamma^{i}\equiv(\gamma^{0},\gamma^{1},\gamma^{2})=(-i\sigma_{z},\sigma_{x},\sigma_{y})$
in flat $2+1$-dimensional Minkowski space with
$\\{\gamma_{i},\gamma_{j}\\}=2\eta_{ij}$ and $\eta_{ij}={\rm diag}(-1,1,1)$.
In curved space-time, the $\eta_{ij}$ is now replaced the metric tensor
$g_{\mu\nu}$, giving new gamma matrices $\gamma^{\mu}$ satisfying
$\\{\gamma^{\mu},\gamma^{\nu}\\}=2g^{\mu\nu}$. The resulting Dirac equation in
curved space-time takes the form $\hbar
v_{\Delta}\gamma^{0}[-i\gamma^{\mu}(x)D_{\mu}]\psi=0$ Wald (1984); Nakahara
(1990). Here, the covariant derivative $D_{\mu}=\partial_{\mu}+\Gamma_{\mu}$
results, with the spin connection $\Gamma_{\mu}$ originating from local
Lorentz transformations.

On the surface of a sphere with radius $R$, the metric is given by
$g_{\mu\nu}={\rm diag}(-1,R^{2},R^{2}\sin^{2}\theta)$. This results in the
gamma matrices 111The relation between two coordinates
$\left\\{e_{\mu}\right\\}$ and $\left\\{e_{i}\right\\}$ is completely
specified by the tetrad $e^{i}_{~{}\mu}\equiv\partial x^{i}/\partial x^{\mu}$.
Then, $g_{\mu\nu}=\eta_{ij}e^{i}_{~{}\mu}e^{j}_{~{}\nu}$,
$\gamma^{\mu}=\gamma^{i}e_{i}^{~{}\mu}$. The spin connection takes the form as
$\Gamma_{\mu}=\frac{i}{2}\Gamma^{i~{}~{}j}_{~{}\mu}\Sigma_{ij}$ with
$\Sigma_{ij}=\frac{i}{2}[\gamma_{i},\gamma_{j}]$ and
$\Gamma^{i~{}~{}j}_{~{}\mu}=e^{i}_{~{}\nu}\nabla_{\mu}e^{j\nu}$.,
$\gamma^{0}=-i\sigma_{z}$, $\gamma^{1}(x)=\gamma^{1}/R=\sigma_{x}/R$, and
$\gamma^{2}(x)=\gamma^{2}/R\sin\theta=\sigma_{y}/R\sin\theta$. The spin
connection is $\Gamma_{\mu}=-\frac{i}{2}\sigma_{z}\cos\theta~{}\delta_{2\mu}$.
Consequently, the Dirac Hamiltonian on the surface of the sphere is given by
González  _et al._ (1992); Abrikosov (2002); Parente  _et al._ (2011);
Chirolli and Guinea (2018),

$h_{0}^{cur}=\frac{\Delta}{k_{F}R}~{}\gamma^{0}\left[-i\sigma_{x}\left(\partial_{\theta}+\frac{\cot\theta}{2}\right)-i\frac{\sigma_{y}}{\sin\theta}~{}\partial_{\phi}~{}\right].$
(1)

The eigenstates of Eq. (1) can be represented using Jacobi polynomials
$|\Upsilon^{\sigma}_{lm}\rangle$, where $l$ denotes the angular momentum, $m$
is the projection along the polar axis, and $\sigma=\pm 1$ is the particle-
hole index. The corresponding eigenenergies are given by $\epsilon={\rm
sgn}(\sigma)(l+1/2)\frac{\Delta}{k_{F}R}$, with $l$ taking half-integer values
Abrikosov (2002); Zhang and Vishwanath (2010); Imura  _et al._ (2012); Takane
and Imura (2013). For each $l$, there are $2l+1$ degenerate states, as shown
in Fig. 1(b). The presence of the spin connection induces a gap in the surface
energy spectrum of the Majorana Dirac cone, which scales proportionally to
$1/R$.

We now introduce a Zeeman term into the Hamiltonian in Eq. (1). The Ising
properties of surface Majorana modes Chung and Zhang (2009) in a planar
geometry only allow the orthogonal spin component $s_{\bot}$ to the plane to
be a relevant perturbation. $s_{\bot}$ is mapped to $\sigma_{z}$ in the
Majorana representation, and thus, the Zeeman term $\bf{V}\cdot\bf{s}$ can be
written as $h_{Z}=V_{\bot}\sigma_{z}$. The normal to the local tangential
plane at any given point on the surface of a sphere corresponds to the radial
component. Then, the uniform Zeeman field $\bf{V}$ applied in the axial
direction generates the mass term as $V_{\bot}=V\cos\theta$. In accordance
with inversion symmetry requirements, the mass term $V(\bf{r})=V_{\bot}$
satisfies $V(\textbf{r})=-V(-\textbf{r})$ from
$V\cos\theta=-V\cos(\theta+\pi)$. The mass term vanishes along the equator at
$\theta=\pi/2$, where the Zeeman field is tangential to the surface.

The Zeeman term has two consequences. First, it generates finite matrix
elements between positive and negative energy states of the Hamiltonian in Eq.
(1), leading to an additional gap in the spectrum. These matrix elemetns are
given by

$\displaystyle\langle\Upsilon^{+}_{lm}|\sigma_{z}\cos\theta|\Upsilon^{-}_{l^{\prime}m^{\prime}}\rangle=$
$\displaystyle~{}i\delta_{mm^{\prime}}\frac{\sqrt{(l+1)^{2}-m^{2}}}{2(l+1)}\delta_{l^{\prime},l+1}$
(2)
$\displaystyle-i\delta_{mm^{\prime}}\frac{\sqrt{l^{2}-m^{2}}}{2l}\delta_{l^{\prime},l-1}.$

Additionally, the perturbation results in non-trivial diagonal matrix elements
Chirolli  _et al._ (2018),

$\langle\Upsilon^{\pm}_{lm}|\sigma_{z}\cos\theta|\Upsilon^{\pm}_{l^{\prime}m^{\prime}}\rangle=\delta_{m,m^{\prime}}\delta_{l,l^{\prime}}\frac{m}{2l(l+1)}.$
(3)

These matrix elements induce non-trivial results. They cause an upward energy
shift in all states with positive $m$, which is particularly pronounced in
low-energy states with small $l$. Conversely, they induce a downward energy
shift in all states with negative $m$. For states with $l=|m|$, the energy
shift undoes the gapping, resulting in a single chiral mode crossing the gap,
as shown in Fig. 1(c).

Figure 2: The dimensionless static susceptibility $\chi_{1,2}\cdot\Delta$ as a
function of the dimensionless distance $k_{F}\cdot d=k_{F}\cdot R\theta=l_{\rm
max}\theta$, where $l_{\rm max}=k_{F}R$ represents the upper limit for the
summation over $l$ in Eq. (6), and $\theta$ denotes the angle between
neighboring impurities on the sphere with radius $R$. The negative sign of the
RKKY coupling coefficient, $J^{\rm R}=\lambda^{2}\chi_{1,2}$, implies a
ferromagnetic Ising RKKY interaction among the radial components of
impurities. We obtain its asymptotic form in Eq. (7), which serves as a valid
approximation based on the plot.

RKKY interaction among magnetic impurities on a sphere.—As largely discussed
in Chirolli  _et al._ (2018), applying an external Zeeman field comes at the
price of an orbital tilting effect on Bogoliubov deGennes (BdG) and surface
Majorana spectra that dominates over the Zeeman term upon increasing the
field. To address this issue and enable the realization of Majorana modes with
reduced field strength and enhanced insensitivity to this tilting effect, we
introduce magnetic impurities on the surface. These impurities, denoted as
$\textbf{S}_{i}$ and positioned at locations $\textbf{R}_{i}$, interact with
the surface Majorana modes. For simplicity, we use semiclassical descriptions
of magnetic impurities and focus exclusively on the on-site exchange
interaction between these impurities and the surface Majorana modes. Due to
the locality, the exchange term on a curved surface involves only the radial
component of the magnetic moment $S_{i}^{r}$ at its location, taking the form
as

$h_{\rm
ex}^{cur}=i\gamma_{0}\lambda\sum_{i}S_{i}^{r}\delta(\textbf{x}-\textbf{R}_{i})\frac{1}{\sqrt{\left\lvert
g\right\rvert}},$ (4)

with the interaction strength $\lambda$. Here, $g$ represents the determinant
of the metric $g_{\mu\nu}$, and it can be eliminated through an appropriate
rescaling of the wavefunction, $\Psi\to\chi=\left\lvert
g\right\rvert^{1/4}\Psi$.

The magnetic impurities experience RKKY interactions mediated by propagating
surface Majorana modes. We can evaluate the RKKY interaction on the curved
surface by integrating out the Majorana modes and expanding with their Green’s
function up to second order in $\lambda$. The resulting expression is

$H^{\rm RKKY}=\lambda^{2}\sum_{i,j}\chi_{ij}S^{r}_{i}S^{r}_{j},$ (5)

where $\chi_{ij}\equiv\chi\left(\textbf{R}_{i},\textbf{R}_{j}\right)=-\langle
Ts^{r}_{1}\left(\textbf{R}_{1},\tau\right)s^{r}_{2}\left(\textbf{R}_{2},0\right)\rangle$
represents the out-of-plane spin susceptibility of the surface modes, with
$\tau$ denoting imaginary time. Its one loop approximation is given by
$\chi_{i,j}\left(i\nu_{n}\right)=-1/\beta~{}$
$\sum_{iw_{n}}\operatorname{Tr}[s_{i}^{r}~{}\hat{G}_{0}\left(\textbf{R}_{i},\textbf{R}_{j},i\nu_{n}+iw_{n}\right)s^{r}_{j}~{}\hat{G}_{0}\left(\textbf{R}_{j},\textbf{R}_{i},iw_{n}\right)]$.
Here, $\beta=1/T$ ($k_{B}\equiv 1$) and $w_{n}=2\pi/\beta\cdot(n+1/2)$
represents the Matsubara frequency Kapusta and Landshoff (1989). In this
expression, $\hat{G}_{0}=(iw-\hat{h}^{\rm cur}_{0})^{-1}$, and we take a
partial trace over the particle-hole degree of freedom of eigenstates
$|\Psi^{\pm}_{lm}\rangle$ of $\hat{G}_{0}$. We confine our system to the low-
temperature limit, where $T\ll\Delta/k_{F}R$, simplifying the static
susceptibility in the low-energy regime near the gap as

$\chi_{ij}=-\sum_{lm,l^{\prime}m^{\prime}}\frac{Z_{lm;l^{\prime}m^{\prime}}({\bf
R}_{j})Z^{*}_{l,m;l^{\prime}m^{\prime}}({\bf
R}_{i})}{\mathcal{E}_{l}+\mathcal{E}_{l^{\prime}}},$ (6)

where $Z_{lm;l^{\prime}m^{\prime}}({\bf R})=\Psi^{+}_{lm}({\bf
R})\sigma_{3}\Psi^{-}_{l^{\prime}m^{\prime}}({\bf R})$ involving transitions
between negative and positive energy states with
$\mathcal{E}_{l}=(l+1/2)\Delta/{k_{F}R}$, mediated by the exchange
interaction. The sum over $|m|\leq l<l_{\rm max}\simeq k_{F}R$ is bounded
below the gap of the bulk energy spectrum, that is an order of $\Delta$. We
compute the dimensionless static susceptibility $\chi_{1,2}\cdot\Delta$ as a
function of the dimensionless distance $k_{F}\cdot d=k_{F}\cdot R\theta$,
where $\theta$ is the angle between neighboring impurities. Using the
azimuthal symmetry, we simplify the calculation by fixing one impurity at the
north pole and placing the other at a latitude $\theta$ of radius $R$. As
shown in Fig. 2, a ferromagnetic Ising RKKY interaction emerges among the
radial components of impurities, as indicated by the negative sign of $J^{\rm
R}=\lambda^{2}\chi$.

To characterize the asymptotic behavior of the susceptibility, we calculate it
as a function of azimuthal angle $\phi$ while fixing two impurities along the
equator line, allowing us to derive a valid asymptotic form for large values
of $l_{\rm max}$ and $\phi\ll\pi$

$\chi(\phi)=-\frac{k_{F}}{4\pi^{2}R^{3}\Delta}\sum_{l,l^{\prime}=1/2}^{l_{\rm
max}}\frac{(l+1/2)(l^{\prime}+1/2)}{l+l^{\prime}+1}{\cal
F}_{l,l^{\prime}}(\phi),$ (7)

where ${\cal
F}_{l,l^{\prime}}(\phi)=[J_{0}(l\phi)J_{0}(l^{\prime}\phi)+J_{1}(l\phi)J_{1}(l^{\prime}\phi)]$.
Our comparison with the exact case in Fig. 2 confirms its validity for $l_{\rm
max}>20$. This asymptotic approximation shows a range bounded from above by
$1/R^{3}$, similar to the flat surface case Choy  _et al._ (2013).

Magnetic phase diagram.—With the form and the range of RKKY interaction now
established, our focus turns to the magnetic phase diagram. The magnetic
moment of the $i$-th impurity, denoted as $\mathbf{S}_{i}$, can be
parameterized using polar and azimuthal angles, $\Theta_{i}$ and $\Phi_{i}$.
These angles define the orientation of $\mathbf{S}_{i}$ relative to local
coordinates $z_{i}$ and $x_{i}$ in a tangential plane centered at its position
$\mathbf{R}_{i}=(\theta_{i},\phi_{i})$. The free energy of the system is given
as a sum of the RKKY interaction and the Zeeman term. Its dependence on
$\Phi_{i}$ is solely due to the RKKY interaction, and we can set
$\Phi_{i}=\phi_{i}$ as a solution of the Euler-Lagrange equations with respect
to $\Phi_{i}$, which minimizes the energy functional. Consequently, we are
left with

$H=\sum_{ij}J^{\rm
R}_{ij}\cos(\Theta_{i})\cos(\Theta_{j})-B\sum_{j}\cos(\Theta_{j}+\theta_{j}),$
(8)

where the $J^{\rm R}_{\textit{ij}}=\lambda^{2}\chi_{\textit{ij}}$. In a
simplified representation of the RKKY interaction, illustrated in Fig. 2, when
we confine its spatial influence to nearest neighbor (NN) pairs by setting
$\chi_{ij}=\chi$ for such pairs $(i,j)$. The ferromagnetic Ising RKKY
interaction compels the magnetic impurities to arrange themselves radially,
forming a hedgehog configuration, with $\Theta_{i}$ taking on values of either
$0$ or $\pi$. In contrast, the Zeeman term dictates that impurities align with
the external field, adopting an axial configuration with
$\Theta_{i}=-\theta_{i}$. The ground state is determined by the interplay
between these two competing terms.

Figure 3: (a) The magnetization of a system with $N$ magnetic impurities on
the circular geometry, influenced by the RKKY interaction and the external
Zeeman field along the axial direction, is computed using the transfer matrix
method for the given energy functional $H$ in Eq. (8). We perform this
calculation with $M=100$ and set the temperature to $T/J=0.1$. (b) The
critical field is represented as a function of $N$ consistent with the
analytic prediction of the phase transition scale from the hedgehog to the
domain wall configuration.

It is instructive to begin by examining the one-dimensional (1D) problem on a
circle as a geodesic curve along the sphere. Assuming a uniform distribution
of $N$ impurities with positions set at $\theta_{i}=2\pi i/N$, we can express
the RKKY interaction as $\chi_{N}=\chi[\cos(2\pi/N)]$. The energy for both
hedgehog and axial configurations can be written as follows $E_{\rm
hed}=-N\lambda^{2}\chi_{N}$ and $E_{\rm
ax}=-N\cos(2\pi/N)\lambda^{2}\chi_{N}/2-NB$.

In the hedgehog configuration, the Zeeman term contributions cancel out due to
the exact cancellation of impurities placed at antipodal points. In contrast,
the energy in the axial configuration results from both Zeeman term and RKKY
interactions. The critical field obtained by equating the energies of the
hedgehog and axial configurations, is given by $B_{c}/J=1-\cos(2\pi/N)/2$, and
it saturates at $1/2$ for large $N$. When considering the energy in a domain
wall kink located near the equator, at the intersection of two hemicycles with
opposite signs of the local magnetic moment in the radial component, it can be
expressed as $E_{\rm dw}=-(N-4)\lambda^{2}\chi_{N}-B\cot(\pi/2N)$.

For the domain wall configuration to become the ground state, the Zeeman term
must dominate over the RKKY interaction along the domain wall, that is for
$\cot(\pi/2N)>4\lambda^{2}\chi_{N}/B$. By neglecting the $N$ dependence of
$\chi_{N}$ for large $N$ and defining the RKKY interaction strength as
$J=\lambda^{2}\chi$, the critical field for the phase transition from zero
total magnetization to finite magnetization along the axial direction scales
as $B_{c}/J=4/\cot(\pi/2N)$, revealing an inverse relationship between
increasing $N$ and decreasing critical field.

To validate the results obtained from the approximate analytic estimate, we
calculate the magnetization and study its behavior with an increasing number
of magnetic impurities $N$. The partition function at temperature $T$ is given
by
$Z=\int\prod_{i}^{N}\frac{d\Theta_{i}}{2\pi}\exp\left[-H[\\{\Theta_{i}\\}]/T\right]\equiv\exp\left[-F/T\right]$,
where $F$ represents the free energy. We discretize the magnetic
configurations of $i$-th impurity as $\Theta_{i}=2\pi n_{i}/M$ for a
sufficiently large $M$, and employ the transfer matrix methodto compute the
partition function. The magnetization along the axial direction is determined
as $m_{\rm ax}=-\frac{1}{N}\frac{\partial F}{\partial B}$. In Fig. 3(a), we
present a plot of the magnetization versus the applied field $B$ at $T/J=0.1$
for different numbers of magnetic impurities. Notably, we observe a non-linear
decrease in the critical field as $N$ increases, following a behavior of
approximately $3/\cot(\pi/2N)$ in close agreement with the analytic prediction
for the critical field of the phase transition from the hedgehog to the domain
wall configuration, as shown in Fig. 3(b).

The analysis on the 2D surface can be approached similarly. Assuming a uniform
distribution of impurities on the sphere, we can do a triangulation of the
sphere with $N$ impurities occupying $N_{v}=N$ equally distanced vertices.
Here, our aim is not to provide an exact account of each impurity position,
but rather to offer a qualitative estimate of the phase diagram and an
approximate understanding of how it depends on the number of impurities. For
every triangulation, the number of edges $N_{e}$ and triangles $N_{t}$ are
related through $N_{e}=3(N-2)$, $N_{t}=2(N-2)$. On the surface with a radius
of $R$, each triangle has an area of $4\pi R^{2}/N_{t}=d^{2}\sqrt{3}/4$, where
$d=R\theta_{d}$ represents the average distance between two nearest neighbors,
and the neighboring arch angle $\theta_{d}$ is given by
$\theta_{d}=\sqrt{16\pi/(N_{t}\sqrt{3})}\equiv\theta_{g}/\sqrt{N}$.Neglecting
the dependence on $N$ for $\chi_{N}=\chi[\cos(\theta_{g}/\sqrt{N})]$ when $N$
is large, the energy in a hedgehog configuration is expressed as

$\displaystyle E_{\rm hed}=-3(N-2)J,$ (9)

with $J=\lambda^{2}\chi_{N}$. In the axial configuration, the RKKY
contribution to the energy can be estimated by integrating over the Northern
hemisphere, given as $E_{\rm
ax}^{RKKY}=-(3NJ/2)\int_{0}^{\pi/2}d\theta\sin\theta\cos^{2}\theta$, which
considers the projection along the radial direction. This leads to the energy
of the axial configuration as

$E_{\rm ax}\sim-NJ-NB.$ (10)

In the domain wall configuration, the RKKY contribution is determined by the
energy of hedgehog configuration minus twice the energy of the domain wall
near the equator. The Zeeman term in the continuum limit can be calculated as
$E^{Z}_{\rm dw}=-NB/2\int_{0}^{\pi/2}d\theta\sin\theta\cos\theta-
NB/2\int_{\pi/2}^{\pi}d\theta\sin\theta\cos(\pi-\theta)$ resulting in the
following expression

$\displaystyle E_{\rm dw}\sim-(3N-2\sqrt{10N})J-\frac{NB}{2},$ (11)

where $\sqrt{10N}$ represents the number of edges cutting the equator at the
intersection of two hemispheres with opposite signs of local magnetic moment
in a radial order. As previously discussed in the 1D geodesic case, the ground
state can exhibit a domain wall configuration when the Zeeman term surpasses
the RKKY interaction along the equator, which occurs for $B>4J\sqrt{10/N}$
whereas the critical field scales as $1/N$ for large $N$ in the 1D case.

Figure 4: (a) The magnetic moments of impurities along the geodesic in each
configuration. (b) The energy density of a system of $N=2000$ impurities on
the surface of a sphere subjected to RKKY interaction and an external magnetic
field along the axial direction, as described by Eq. (9), (10), and (11).

Conclusion.—We realize a second-order chiral Majorana hinge mode along the
inversion-symmetric equator by modifying a 3D TRI odd-parity TSC. This
involves introducing a uniform external Zeeman field to break time-reversal
symmetry while preserving inversion symmetry. As the magnetic field increases,
the orbital tilting effect on the surface energy spectrum takes precedence
over the Zeeman term. To counteract this, we introduce surface magnetic
impurities, proposing that the chiral mode emerges in a domain wall
configuration with a reduced magnetic field strength. We analyze the
ferromagnetic Ising RKKY interaction among the radial components of these
impurities, deriving its asymptotic form converging to the interaction form on
a flat surface as the radius of sphere increases. We examine scenarios with 1D
and 2D impurities, determining the critical field for the transition to a
domain wall configuration hosting a chiral Maajorana mode. Detailed
investigations of how these obtained chiral hinge modes can be utilized for
interferometers Li  _et al._ (2021); Chaou  _et al._ (2023) is left for future
work.

Acknowledgement.—This work (I.N, S.M.G.) was supported by the US Department of
Energy, Office of Science, National Quantum Information Science Research
Centers, Quantum Systems Accelerator (QSA). Computational resources were
provided by the National Energy Research Scientific Computing Center and the
Molecular Foundry, DOE Office of Science User Facilities supported under
Contract No. DEAC02-05-CH11231. This project (L.C.) has received funding from
the European Union’s Horizon 2020 research and innovation program under the
Marie Skłodowska-Curie Grant Agreement No. 841894 (TOPOCIRCUS). L.C. also
acknowledges EU’s Horizon 2020 Research and Innovation Framework Programme
under Grant No. 964398 (SUPERGATE) and No. 101057977 (SPECTRUM).

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# An electro-optic waveform interconnect based on quantum interference

Li-Guo Qin1111[Email address]<EMAIL_ADDRESS>Zhong-Yang Wang1222[Email
address]<EMAIL_ADDRESS>$\&$ Shang-Qing Gong2 1Shanghai Advanced Research
Institute, Chinese Academy of Sciences, Shanghai, 201210, China
2Department of Physics, East China University of Science and Technology,
Shanghai 200237, China

###### Abstract

The ability to modulate an optical field via an electric field is regarded as
a key function of electro-optic interconnects, which are used in optical
communications and information processing systems. One of the main required
devices for such interconnects is the electro-optic modulator (EOM). Current
EOM based on the electro-optic effect and the electro-absorption effect often
is bulky and power inefficient due to the weak electro-optic properties of its
constituent materials. Here we propose a new mechanism to produce an
arbitrary-waveform EOM based on the quantum interference, in which both the
real and imaginary parts of the susceptibility are engineered coherently with
the superhigh efficiency. Based on this EOM, a waveform interconnect from the
voltage to the modulated optical absorption is realised. We expect that such a
new type of electro-optic interconnect will have a broad range of applications
including the optical communications and network.

###### pacs:

85.85.+j, 42.50.Gy, 84.30.Qi

The trend toward on-chip and off-chip photonics has lasted for decades bce ;
mdar ; sab . Current integrated Silicon photonics technology can support the
design of electronics and photonics on the same chip and enable cost-efficient
optical links that connect racks, modules and chips together sab ; sun ; vya .
This indicates the beginning of an era of chip-scale electronic-photonic
systems sun . However, to realise the electro-optic interconnect, the electro-
optic chip faces the challenges including the electro-optic transformation and
low power consumption bce , etc. Moreover, the synchronous interconnect of the
electro-optic signals meets the difficulties of the match in the propagation
velocity, the high modulation capability and the response speed. To realise
the dynamical transformation from the electrical signals to optical ones, the
electro-optic modulator (EOM) is a potential device as a main required
function of the electro-optic interconnect.

In an EOM, the modulation of the optical field is primarily subject to the
electro-optic properties of the medium chr . For EOMs based on the electro-
optic effect, the refractive index $n$, i.e., the real part of the material
susceptibility, is proportional to the electric field (Pockels effect) or the
square of the electric field (Kerr effect) klc . Although these well-studied
effects have been exploited in numerous EOM devices, they require large
components and/or a high driving voltage owing to the weakness of the electro-
optic effect cpd  (e.g., $\Delta n\sim 3.3\times 10^{-3}$ for the Pockels
effect in LiNbO3 and $\Delta n\sim 10^{-5}$ for the Kerr effect in silicon
with an applied field of $20V/\mu m$ klc ). These requirements lead to higher
energy dissipation than appropriate for off- or on-chip interconnection mda .

Another modulation mechanism is the electro-absorption effects, in which the
absorption spectra are modified by the applied electric field. Electro-
absorption effects include the Franz-Keldysh effect mdab1  in bulk
semiconductor materials and the quantum-confined Stark effect mdab2  in
quantum well structures cpd ; ljb ; these effects originate from the
distortion of the energy bands by an applied electric field, which causes the
modification of the absorption coefficient klc . EOMs based on the electro-
absorption effects, which typically involve III-V direct-band-gap
semiconductors, are compatible with semiconductor technology cpd , which lends
itself to the development of compact devices with low power consumption lae ;
vjs . For on-chip integration and miniaturisation, silicon-based EOMs are more
desirable rgt ; however, their development is hampered by the weak electro-
optic absorption effect of the indirect band gap of silicon cmp . Thus a key
area of EOM-related research is the investigation of novel materials with good
electro-optic properties, such as a graphene-based broadband optical modulator
with the several distinctive advantages including the strong light-graphene
interaction, high-speed operation as well as complementary metal-oxide-
semiconductor compatible capability mlx ; the high performance GeSi electro-
absorption modulator with the ultralow energy consumption offering unique
advantages for use in high-performance electronic-photonic integration with
complementary metal oxide semiconductor circuits cpd ; lje . Here, we report a
new modulation mechanism for EOMs based on quantum interference, in which the
refractive index and the absorption of the material are engineered coherently
with unprecedented efficiency fmi .

Quantum interference between two excitation pathways of the internal quantum
states of a coherent three-level medium has given us the unique capability to
engineer the linear and nonlinear susceptibility of the medium. In the case of
electromagnetically induced transparency (EIT) kjb , when the two metastable
states couple to a common excited state by the control and probe fields, the
absorption of the probe field in the medium is largely modified through the
destructive quantum interference of the amplitudes of the two optical
transitions. The width of the transparency window and the steepness of the
refractive index curves are modulated over a wide range by dynamically
changing the strength of the control field fmi . This ability to modify the
susceptibility has revealed many new phenomena in quantum optics, such as
ultra-slow group velocities lvh  and the storage of light and quantum memory
mfm . Furthermore, this feature has been shown experimentally to be able to
decrease the group velocity of light by many orders of magnitude bdk ; tav .
With the slowing of light, nonlinear optics can extend to the few-photon level
bme ; vvs .

An optical switch based on quantum interference was proposed by Harris and
Yamamoto in a four-level EIT atomic system seh . Subsequently, this optical
switching mechanism was experimentally demonstrated in a four-level
${}^{87}Rb$ atomic system mye . In 2011, researchers have observed that the
transmission of the probe field can be perfectly switched using cavity EIT and
by applying a weak switching field maa . These results demonstrate the ability
of quantum interference to modulate an optical field.

Photons are the fastest and most robust information carriers. As such, the
interconnects used in data communication systems and in long links will
gradually change from electrical-based to light-based. However, photons must
be slowed down to match the speed of the slow electrons in the electro-optic
interconnect. Based on cavity-induced transparency, we previously proposed an
approach to realise the electrical manipulation of the dark-state polariton
and the group velocity of light lgqin . Here, we propose a hybrid EOM system
composed of a high-quality tunable cavity and an electro-mechanical system. By
the interaction between a three-level medium confined inside the cavity and
two optical fields, the group velocity of the modulated optical field in the
medium is electrically slowed to match the slow electric signal. Through the
use of electrically-driven quantum interference, we establish a one-to-one
correspondence between the waveforms of the electrical signal and the optical
field (that is, to create an electro-optic waveform interconnect) and use it
to produce an arbitrary-waveform optical field. To the best of our knowledge,
this work represents the first proposed electro-optic waveform interconnect
based on such an approach.

Specifically, our scheme proposes implementing a coherent three-level medium
that is confined in a compact device composed of a tunable cavity with a high
quality factor and an electro-mechanical system, as shown in Fig. 1. The
cavity consists of a moveable mirror, which is charged to operate as a charged
mechanical oscillator (CMO), and a fixed mirror. The CMO has positive charge Q
and capacitively couples to a fixed conductive plate that is supplied with
negative charge -Q by a voltage waveform with time regime $T$ from a waveform
generator. Thus the CMO and the fixed conductive plate form a mechanically
variable capacitor.

In this model, the control field injected into the cavity is a constant
optical field with an amplitude
$\varepsilon_{c}=\sqrt{2P_{c}\kappa/\hbar\omega_{0}}$, where the parameters
represent respectively the frequency $\omega_{0}$, the power $P_{c}$, the
Planck constant $\hbar$ and the cavity decay rate $\kappa$. In this structure,
the displacement $q$ of the CMO, with a mass $m$ and frequency $\omega_{m}$,
is driven by the radiation pressure force from the optical field and the
Coulomb force from the capacitor. The Coulomb force and the radiation pressure
forces on the CMO have the same direction and the Coulomb interactions can be
approximated as $V_{c}\approx-U^{2}\eta(q+r)$ for $q\ll r$, where
$\eta=\frac{\varepsilon_{0}S}{2r^{2}}$, $\varepsilon_{0}$ is the vacuum
permittivity, $U$ is the voltage, $S$ is the area and $r$ is the distance
between two capacitive plates lgqin . Because of the motion of the CMO, the
resonance frequency in the cavity is modified as
$\omega_{c}(q)=\omega_{c}(0)+q\partial\omega_{c}(q)/\partial q+O(q)$, where
the high order terms $O(q)$ are neglected amt . This process can lead to a
detuning between the cavity field ($a$ ($a^{{\dagger}}$) is the the
annihilation (creation) operator of the photons in the cavity) and the control
field $\Delta_{cmo}=\omega_{c}(q)-\omega_{0}$, which induces the leakage of
photons in the cavity. The dynamical equations of the CMO and the cavity field
are given by

Figure 1: (Color online)The proposed opto- and electro-mechanical hybrid
system is composed of a tuneable cavity with a charged mirror operating as a
CMO and a mechanically variable capacitor. A $\Lambda$-type three-level medium
confined inside the cavity interacts with two optical fields: a constant
optical field $\varepsilon_{c}$, which is resonantly injected into the cavity
along the $x$ axis to form the cavity field, and the probe field $E_{p}$,
which is externally injects into the cavity along the $z$ axis at frequency
$\omega_{p}$.

$\displaystyle\ddot{q}+\gamma_{m}\dot{q}+\omega^{2}_{m}q=(\hbar
G_{0}n(t)+U^{2}\eta)/m,$
$\displaystyle\dot{a}=(-\kappa-i\Delta_{cmo})a+\varepsilon_{c},$ (1)

where $\gamma_{m}$ is the damping rate of the CMO, $n_{c}=a^{{\dagger}}a$ is
the photon number in the cavity and $G_{0}=-\partial\omega_{c}(q)/\partial
q\approx\omega_{c}/L$ is the opto- and electro-mechanical coupling strength
for a cavity of length $L$ amt . Compared with the Coulomb force, the
radiation pressure force can be neglected in our model and the motion of the
CMO only depends on the Coulomb force. When the charges are slowly injected
into the capacitor with a duration that satisfies $T\gg 1/\gamma_{m}$ and
$1/\kappa_{eff}$, the evolution of the displacement of the CMO and the cavity
field can be considered adiabatically and the solutions of equation (1) can be
derived as $q=\frac{U^{2}\eta}{m\omega_{m}^{2}}$ and
$a=\frac{\varepsilon_{c}}{\kappa+i\Delta_{cmo}}$. When
$\kappa\ll\Delta_{cmo}$, the photon number $n_{c}$ in the cavity is only
controlled by the voltage, hence, the cavity field can be electrically and
adiabatically switched. Therefore, we can define the effective cavity decay
rate as $\kappa_{eff}=\sqrt{\kappa^{2}+\Delta_{cmo}^{2}}$ and quality factor
of the cavity as $Q_{eff}=\omega_{c}/\kappa_{eff}$, which is dynamically
modulated by changing the voltage on the capacitor.

To realise the modulation of a probe field by the voltage, i.e., EOM, an
interactive medium with a three-level configuration must be confined at an
antinode of the standing wave inside the cavity, in which the probe field
couples with the atomic dipole transition between levels $|b\rangle$ and
$|a\rangle$ with a coupling strength $g_{p}$ and detuning $\Delta_{p}$; and
the quantised cavity field couples the atomic dipole transition between levels
$|c\rangle$ and $|a\rangle$ with a coupling strength $g$ and detuning
$\delta$. We assume that the probe field propagates in a direction
perpendicular to that of the cavity field. The amplitude of the cavity field
at this antinode depends only on time. Here we choose an ensemble of $N$
identical $\Lambda$-type cold atoms with a volume $V$ that interacts with the
probe field and cavity field, as shown in Fig. 1, however, any types EIT-media
can be used, such as $V$-type and the ladder type. The interaction Hamiltonian
of the system is given by $H=-\hbar
G_{0}a^{\dagger}aq+i\hbar\varepsilon_{c}(a^{\dagger}e^{-i\omega_{0}t}-ae^{i\omega_{0}t})-\hbar
N(g_{p}\varepsilon_{p}\sigma_{ab}+ga\sigma_{ac}+H.c.)-U^{2}\eta(q+r)$, where
$\sigma_{\alpha\beta}=\frac{1}{N}\sum_{i=1}^{N}|\alpha_{i}\rangle\langle\beta_{i}|$
are the slowly varying collective operators of the atomic ensemble.

The dynamic equations of the system can be simplified with the slowly varying
amplitude and weak-field approximations mfm . We then introduce the
dimensionless slowly-varying field amplitude
$E_{p}=\varepsilon_{p}\sqrt{\hbar\omega_{p}/2\epsilon_{0}V}$ and assume that
almost all atoms are in the ground state $|b\rangle$ and that the number of
probe photons $n_{p}$ is considerably smaller than the number of atoms, i.e.,
$\epsilon=\sqrt{n_{p}/N}\ll 1$ (the weak-field approximation). Considering
these approximations and keeping the zeroth-order terms of $\epsilon$, we
determine that $\sigma_{bb}\approx 1$,
$\sigma_{aa}\approx\sigma_{cc}\approx\sigma_{ac}\approx 0$ gnm  and that the
remaining atomic evolutions $\sigma_{ba}$ and $\sigma_{bc}$ can be derived
from

$\displaystyle\dot{R}=-MR+A,$ (2)

where

$\displaystyle R=\left(\begin{array}[]{c}\sigma_{ba}\\\ \sigma_{bc}\\\
\end{array}\right),M=\left(\begin{array}[]{cc}\gamma+i\Delta_{p}&-iga\\\
-iga^{{\dagger}}&\gamma_{s}+i(\Delta_{p}-\delta)\\\
\end{array}\right),A=\left(\begin{array}[]{c}ig_{p}\varepsilon_{p}\\\ 0\\\
\end{array}\right).$ (9)

Here, $\gamma$ ($\gamma_{s}$) is the decay rate of the higher energy level
$|a\rangle$ (the metastable state $|c\rangle$) to the ground state $|b\rangle$
with $\gamma_{s}\ll\gamma$. For a sufficiently long time, the solution of
equation (2) can be generally provided by

$\displaystyle R=e^{-Mt}R_{0}+(1-e^{-Mt})M^{-1}A\approx M^{-1}A.$ (10)

This yields

$\displaystyle\sigma^{s}_{ba}=\frac{ig_{p}\varepsilon_{p}[\gamma_{s}+i(\Delta_{p}-\delta)]}{(\gamma+i\Delta_{p})[\gamma_{s}+i(\Delta_{p}-\delta)]+g^{2}(n_{c}+1)}.$
(11)

Thus the first-order susceptibility of the medium can be derived as
$\chi=\frac{\mu_{ba}N}{\epsilon_{0}E_{p}V}\sigma_{ba}^{s}$, where $\mu_{ba}$
is the dipole moment of the probe transition. The imaginary part of this
susceptibility, $Im(\chi)$, determines the dissipation of the probe field,
i.e., absorption, whereas the real part, $Re(\chi)$, determines the refractive
index fmi . Equation (11) indicates that the width of the transparency window
($\Delta\omega_{EIT}$) and the dispersion of the probe field depend on the
intensity of the control field ($g^{2}(n_{c}+1)$) jao , which is regulated by
the voltage. In Fig. 2, we observe that the width of the transparency window
decreases and even disappears when the voltage $U^{2}$ increases. In this
process, the probe field is gradually absorbed by the medium and can even be
completely absorbed with a large voltage. Meanwhile, the slope of the steep
dispersion is also significantly modified by the voltage (Fig. 2 inset).
Because we use a high-Q cavity and cavity induced transparency, the strong
control field required in conventional EIT fmi  has been reduced to an
assistant optical field with a low and constant intensity, that is, for the
observable features of EIT, provided that the cavity field satisfies
$g^{2}(n_{c}+1)\gg\gamma\gamma_{s}$. Based on the mechanism of cavity-induced
transparency gnm , quantum interference can be then manipulated by the
electric field in this integrated opto- and electro-mechanical system. In the
following we will show that the absorptive waveform of the modulated probe
field has a one-to-one correspondence with the waveform of the voltage.

For the non-resonant case $\Delta_{p}\neq 0$, the width of the transparency
window determines the bandwidth of the modulation, which will be discussed
later. Now to simplify the expression of $\chi$, we assume that the system
operates are in the resonant regime, i.e., $\Delta_{p}=\delta=0$. The
imaginary part of the susceptibility is then given by

$\displaystyle
Im(\chi)^{s}=\chi_{0}\frac{\gamma_{s}}{\gamma\gamma_{s}+g^{2}[\frac{\varepsilon^{2}_{c}}{\kappa^{2}+(\frac{2G_{0}\eta}{mw_{m}^{2}}U^{2})^{2}}+1]},$
(12)

where $\chi_{0}=\frac{\mu^{2}_{ba}N}{\hbar\epsilon_{0}V}$. Thus we obtain an
analytic expression of the relationship between the absorption of the medium
$Im(\chi)$ and the voltage $U^{2}$. When $U^{2}=0$, the photon number in the
cavity field is maximum, described by
$n_{cmax}=\frac{|\varepsilon_{c}|^{2}}{\kappa^{2}}$. The strong cavity field
induces a probe field transparent in the coherent medium. From equation (12),
the minimum absorption in the transparency window is given by
$Im(\chi)_{min}=\frac{\chi_{0}\gamma_{s}}{g^{2}(n_{cmax}+1)+\gamma\gamma_{s}}$.
In real atomic systems, because the coherence decay rate $\gamma_{s}$ with the
forbidden $|b\rangle$ to $|c\rangle$ transition is nonzero, the transparency
is never perfect. When $\gamma\gamma_{s}\ll g^{2}(n_{cmax}+1)$, we can
consider that the medium is transparent. When $n_{c}$ approaches zero for a
large voltage, the maximum absorption is
$Im(\chi)_{max}=\frac{\chi_{0}\gamma_{s}}{g^{2}+\gamma\gamma_{s}}$. The
inverse transformation of the corresponding square of the voltage as a
function of $Im(\chi)$ can also be easily derived from equation (12) as

$\displaystyle
U^{2}=\frac{m\omega^{2}_{m}}{2G_{0}\eta}\sqrt{\frac{g^{2}\varepsilon^{2}_{c}}{\frac{\chi_{0}\gamma_{s}}{Im(\chi)^{s}}-\gamma\gamma_{s}-g^{2}}-\kappa^{2}}.$
(13)

where $\frac{\chi_{0}\gamma_{s}}{g^{2}(n_{cmax}+1)+\gamma\gamma_{s}}\leq
Im(\chi)^{s}<\frac{\chi_{0}\gamma_{s}}{\gamma\gamma_{s}+g^{2}}$. From
equations (12) and (13), we can find a one-to-one correspondence between the
voltage and the absorption of the probe field, that is, the change of the
voltage can lead to the modulation of the absorption of the probe field.
Therefore the EOM based on the electro-optic absorption is achieved. Using
this interconnect, we can generate an arbitrary waveform of the optical field
by adjusting the corresponding voltage waveform.

Figure 2: Imaginary part of the susceptibility of the probe field in the
medium as a function of the square of the voltage $U^{2}$ and the detuning
$\Delta_{p}$. The inset shows the real part of the susceptibility. Here the
units of the voltage and the detuning axis are the square of volt ($V^{2}$)
and the hertz ($Hz$), respectively. The parameters are used from the
experiments in Ref.sgk  as $\varepsilon_{c}=4\times 10^{10}Hz;$
$\gamma=2\pi\times 5.75$MHz;
$\gamma_{s}=0.0001\gamma;g=0.001\gamma;\kappa=0.2\gamma;\omega_{m}=\gamma;m=145$ng;
$G_{0}=2\pi\times 1.5\times 10^{16}Hz/m;S=0.6mm^{2};r=0.21\mu m$; $\delta=0$
and atomic density $\sim 10^{19}m^{-3}$. The modulative material is the 87Rb
with the $\Lambda$-type three levels configuration.

If the change of the cavity field caused by the voltage satisfies the
adiabatic conditions described in references mfm ; gnm ; fml , the
electrically controlled dark-state polariton and group velocity can be
obtained as lgqin

$\displaystyle\Psi=\cos\theta\varepsilon_{p}-\sin\theta\sqrt{N}\sigma_{bc},$
$\displaystyle\nu_{g}=c\;\cos^{2}\theta,$ (14)

where the mixed angle $\theta$ is defined as
$\tan^{2}\theta=\frac{g_{p}^{2}N}{g^{2}(\hat{n}+1)}$ and depends on the
voltage on the capacitor. Equation (An electro-optic waveform interconnect
based on quantum interference) shows that the properties of the dark-state
polariton and the group velocity are determined by the mixing angle $\theta$
between the probe field and the atomic spin states. By modulating the voltage,
the mixing angle can be changed in the range $0-\pi/2$, which can lead to
store and retrieve the probe field in the medium, and the slowing of the group
velocity. Therefore the absorption rate and the group velocity of the probe
field are coherently and simultaneously modulated.

To demonstrate the ability of the proposed scheme to modulate the optical
field, we give the simulation results. For the modulative waveforms, we choose
that the standard and general sine, sawtooth and square waveforms are the
target waveforms, shown by the solid lines in Figs. 3(a1), (b1) and (c1). In
order to obtain the regular target waveforms, the corresponding voltage
waveforms of Figs. 3(a2), (b2) and (c2) with the amplitude $U_{m}$ were
applied to the capacitor by the waveform generator. Then, using numerical
simulations, we obtained the corresponding modulated waveforms, indicated by
the dashed lines in Figs. 3(a1), (b1) and (c1). The differences between the
target and the numerical results mainly originate from the transient
properties of the EIT, which can lead to negative absorption, that is, the
transient gain lyq ; srd . The gains appear in the transparency window of the
probe, therefore the modulation amplitude can be enhanced. The numerical
results show that the modulated absorptive sine, sawtooth and square waveforms
all can follow the voltage waveform, and that an arbitrary-waveform modulator
can be produced based on our approach.

Figure 3: Numerical results of the EOM. (a) The modulation of the sine wave:
(a1) the target absorptive waveform and numerical results, (a2) the square of
voltage waveform $U^{2}$ applied to the capacitor. (b) the modulation of the
sawtooth wave: (b1) The target absorptive waveform and numerical results, (b2)
The square of voltage waveform $U^{2}$ applied to the capacitor. (c) the
modulation of the square wave: (c1) The target absorptive waveform and
numerical results, (b2) The square of voltage waveform $U^{2}$ applied to the
capacitor. The simulation parameters are $\varepsilon_{c}=0.5\times
10^{10}Hz$, $\gamma_{m}=3\gamma$ and $\kappa=0.4\gamma$, The other parameters
are same in Fig. 2.

The extinction ratio ( also known as the modulation depth) is used to describe
the modulation efficiency of the transmitted light and is defined as the ratio
of the maximum transmission $I_{max}$ to the minimum transmission $I_{min}$,
in other words $10log(I_{max}/I_{min})$ rgt . To measure the efficiency of the
electrically controlled absorption, we can define the extinction ratio as

$\displaystyle R_{dB}=10log\frac{Im(\chi)_{max}}{Im(\chi)_{min}},$ (15)

where the maximum absorption $Im(\chi)_{max}$ is obtained without the cavity
field when the voltage is maximum; the minimum absorption $Im(\chi)_{min}$
corresponds to the strongest cavity field, which is obtained when the voltage
is zero. From the analytical equation (12), the extinction ratio is calculated
as
$R_{dB}=10log\frac{\gamma\gamma_{s}+g^{2}(n_{cmax}+1)}{\gamma\gamma_{s}+g^{2}(n_{cmin}+1)}$,
where
$n_{cmin}=\frac{\varepsilon^{2}_{c}}{\kappa^{2}+(\frac{2G_{0}\eta}{mw_{m}^{2}}U_{m}^{2})^{2}}$.
Therefore we can enhance the extinction ratio by increasing the difference
between $n_{cmax}$ and $n_{cmin}$.

The modulation speed of the EOM depends on the following factors. First, the
motion of the CMO must follow the change of the voltage, that is, it is
operates as a driven damped mechanical oscillator. The frequency of the
oscillation follows the frequency of the strong driving field and its
amplitude and phase are determined by the mass, frequency and damping of CMO
and by the frequency and the amplitude of the strong driving field wwn .
Second, the cavity field is switched dynamically by the CMO based on the
cavity detuning. Therefore, the quality factor of the cavity must match the
cavity detuning, whose magnitude can be modulated by the area and distance of
the two capacitive plates. For high frequency modulation, the nonadiabatic
changes of the cavity field are caused by the difference between the quick
mechanical detuning and the leakage of the cavity itself. Third, for a high-
frequency driving voltage, the interaction between the optical fields and the
medium can occur under nonadiabatic conditions and the nonadiabatic transfer
of the dark-state polariton between the probe field and the atomic spin state
can be obtained, as discussions in mab . Therefore, a high frequency of
modulation is possible when the above conditions are met. In addition, the
modulation bandwidth, the range of the probe frequency in the transparency
window ($\Delta_{p}$), depends on the strength of the cavity field, which can
be increased by improving the quality factor and strengthening the optical
field $\varepsilon_{c}$.

In the proposed scheme, it is a key that the absorption rate in the EIT can be
modulated by controlling electrically the control field based on quantum
interference, therefore any coherent medium with an energy level configuration
that enables EIT can serve as the modulation medium. The energy level
configuration can be extended from $\Lambda$-type to $V$ \- and cascade-type
three-level systems, even to general $N$\- ykn  or $M$\- type dcl
configurations. Moverover, the medium could include a wide range of systems,
for example, a semiconductor quantum dot bdj  or quantum well wzy , etc.

Compared to conventional EOM technology, which is based on the electro-optic
properties of the constituent materials, our proposed quantum-interference-
based EOM to form electric-optic interconnect offers a number of advantages.
First, the steep EIT-induced dispersion can lead to a modification of the
refractive index over a wide range and even change the refractive index from
positive to negative omo ; tqm . Second, the absorption rate can be greatly
modulated by adjusting the transparency window, in which the complete
absorption and transparency can be switched. Third, in the process of the
coherent interaction, the refractive index and absorption rate are modulated
simultaneously. Fourth, in principle, the proposed scheme can overcome the
dependence of conventional EOMs on the electro-optic properties of materials.

In conclusion, we propose a new mechanism to realise an electro-optic waveform
interconnect by EOM based on quantum interference. The proposed new mechanism
maybe open a new field to study EOM and the waveform transfer from the
electrical signal to the optical one. Owing to quantum interference mechanism
as one of the core in quantum information, the electro-optic modulation,
especially, the electro-optic waveform transfer, may be used in the photonic
modulation at the quantum level. This new mechanism exhibits unprecedented
capacity for EOM in the waveform electro-optic interconnects and maybe have
potential applications in the future communication and signal processing
systems.

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Acknowledgements
We acknowledge Dr. H. Shen, J. Lu and Simon-Pierre Gorza for language editing
and programming supper. This work is supported by the Strategic Priority
Research Program (No. XDB01010200), the Hundred Talents Program of the Chinese
Academy of Sciences (No. Y321311401), National Natural Sciences Foundation of
China (Nos. 11347147, 11274112, 11547035 and 91321101) and Shanghai Advanced
Research Institute (Grant 141004).

Author contributions
L. G. Qin and Z. Y. Wang contributed to the conception and design of this
work. S. Q. Gong participated in the discussion of the results. All authors
contributed in the article preparation. Z. Y. Wang supervised this project.

Additional Information
Competing financial interests: The authors declare no competing financial
interests.