# Nonreciprocal Optomechanical Entanglement against Backscattering Losses

Ya-Feng Jiao  Key Laboratory of Low-Dimensional Quantum Structures and Quantum
Control of Ministry of Education, Department of Physics and Synergetic
Innovation Center for Quantum Effects and Applications, Hunan Normal
University, Changsha 410081, China  Sheng-Dian Zhang  Key Laboratory of Low-
Dimensional Quantum Structures and Quantum Control of Ministry of Education,
Department of Physics and Synergetic Innovation Center for Quantum Effects and
Applications, Hunan Normal University, Changsha 410081, China  Yan-Lei Zhang
CAS Key Laboratory of Quantum Information, University of Science and
Technology of China, Hefei 230026, China  CAS Center For Excellence in Quantum
Information and Quantum Physics, University of Science and Technology of
China, Hefei, Anhui 230026, China  Adam Miranowicz  Faculty of Physics, Adam
Mickiewicz University, 61-614 Poznań, Poland  Le-Man Kuang
<EMAIL_ADDRESS>Key Laboratory of Low-Dimensional Quantum Structures and
Quantum Control of Ministry of Education, Department of Physics and Synergetic
Innovation Center for Quantum Effects and Applications, Hunan Normal
University, Changsha 410081, China  Hui Jing<EMAIL_ADDRESS>Key
Laboratory of Low-Dimensional Quantum Structures and Quantum Control of
Ministry of Education, Department of Physics and Synergetic Innovation Center
for Quantum Effects and Applications, Hunan Normal University, Changsha
410081, China

###### Abstract

We propose how to achieve nonreciprocal quantum entanglement of light and
motion and reveal its counterintuitive robustness against random losses. We
find that by splitting the counterpropagating lights of a spinning resonator
via the Sagnac effect, photons and phonons can be entangled strongly in a
chosen direction but fully uncorrelated in the other. This makes it possible
both to realize quantum nonreciprocity even in the absence of any classical
nonreciprocity and also to achieve significant entanglement revival against
backscattering losses in practical devices. Our work provides a way to protect
and engineer quantum resources by utilizing diverse nonreciprocal devices, for
building noise-tolerant quantum processors, realizing chiral networks, and
backaction-immune quantum sensors.

Nonreciprocal physics has witnessed rapid advances in recent years, with
unique applications ranging from backaction-immune signal transfer or
processing, chiral networking, and invisible sensing Shoji and Mizumoto
(2014). By breaking the Lorentz reciprocity, one-way flow of classical
information, i.e., mean photon numbers, has been realized by using atoms
Ramezani  _et al._ (2018); Zhang  _et al._ (2018), solid devices Dong  _et
al._ (2015); Kim  _et al._ (2015); Xu  _et al._ (2020); Manipatruni  _et al._
(2009); Shen  _et al._ (2016); Bernier  _et al._ (2017); Mercier de Lépinay
_et al._ (2019); Fang  _et al._ (2017); Hafezi and Rabl (2012), and synthetic
materials Wang  _et al._ (2013); Yang  _et al._ (2019); Maayani  _et al._
(2018); Sounas and Alù (2017); Peng  _et al._ (2014, 2016); Zhong  _et al._
(2019). Likewise, quantum optical diode or one-way flow of quantum information
can also be achieved. In fact, nonreciprocal control of single photons and
their quantum fluctuations have been demonstrated, such as single-photon
diodes Xia  _et al._ (2018); Dong  _et al._ or circulators Scheucher  _et al._
(2016), and one-way photon blockade Huang  _et al._ (2018); Yang  _et al._ ,
providing key tools for chiral quantum engineering Lodahl  _et al._ (2017);
Gonzalez-Ballestero  _et al._ (2015); Gangaraj  _et al._ (2017); Hu  _et al._
(2019). However, up to now, the possibilities of switching a single
nonreciprocal device between classical and quantum regimes, as well as
protecting quantum entanglement with nonreciprocal devices, have not yet been
revealed.

Figure 1: Nonreciprocal optomechanical entanglement in a spinning resonator.
(a) Frequency spectrum of the spinning COM system. By fixing the CW rotation
of the resonator, we have $\Delta_{F}>0$ ($\Delta_{F}<0$) for the case by
driving the CCW (CW) mode. Besides, the resonator can support a radiation-
pressure-induced mechanical radial breathing mode. The CW and CCW modes are
coupled via backscattering with the strength $J$. (b) The Sagnac-Fizeau shift
$\Delta_{F}$ versus the angular velocity $\Omega$. Increasing the angular
velocity results in a linear opposite frequency shift for the
counterpropagating modes. For the same input light, due to the opposite
frequency shift for the counterpropagating modes, COM entanglement can appear
unidirectionally. See the text for more details.

Here we propose how to achieve nonreciprocal quantum entanglement in cavity
optomechanics (COM), revealing its unique properties which are otherwise
unattainable in conventional devices. COM devices featuring coherent light-
motion coupling Aspelmeyer  _et al._ (2014); Verhagen  _et al._ (2012) have
been widely used for quantum control of massive objects Hong  _et al._ (2017);
Lecocq  _et al._ (2015); Wollman  _et al._ (2015); Pirkkalainen  _et al._
(2015); Stannigel  _et al._ (2010); Mirhosseini  _et al._ , particularly COM
entanglement Vitali  _et al._ (2007); Huang and Agarwal (2009); Genes  _et
al._ (2008); Riedinger  _et al._ (2016); Palomaki  _et al._ (2013); Barzanjeh
_et al._ (2019); Chen  _et al._ (2020); Ockeloen-Korppi  _et al._ (2018);
Riedinger  _et al._ (2018) or COM sensors Massel  _et al._ (2012); McClelland
_et al._ (2011); Qvarfort  _et al._ (2018). Very recently, quantum
correlations at room temperature were observed even between light and
$40\text{\,}\mathrm{kg}$ mirrors Yu  _et al._ (2020). Here we show that COM
entanglement can be manipulated in a highly asymmetric way and the resulting
nonreciprocal entanglement has the counterintuitive ability to preserve its
optimal quality in a chosen direction against losses. This gives a new way to
engineer quantum resources by utilizing diverse nonreciprocal devices, without
the need of any topological or dissipative structure. In a broader view, our
findings shed new light on the marriage of nonreciprocal physics and quantum
technology, which can benefit such a wide range of applications as noise-
tolerant quantum processing Stannigel  _et al._ (2012); Horodecki  _et al._
(2009), chiral quantum networking Lodahl  _et al._ (2017); Gonzalez-Ballestero
_et al._ (2015); Gangaraj  _et al._ (2017); Hu  _et al._ (2019); Kimble
(2008), and backaction-immune quantum sensing Ma  _et al._ (2017); Fleury  _et
al._ (2015); Yang  _et al._ (2015).

As shown in Fig. 1, we consider a spinning COM resonator evanescently coupled
with a tapered fiber. In a recent experiment Maayani  _et al._ (2018),
nonreciprocal propagation of light with 99.6 $\%$ isolation was demonstrated
by using such a spinning device. The optical paths of counterpropagating
lights in the resonator are different due to the rotation, resulting in an
irreversible refractive index for the clockwise (CW) and counterclockwise
(CCW) modes Maayani  _et al._ (2018), i.e.,
$n_{\circlearrowleft,\circlearrowright}\\!=\\!n[1\pm nR\Omega(n^{-2}-1)/c]$,
where $n$ is the refractive index of the material, $\Omega$ is the angular
velocity of the resonator with radius $R$, and $c$ is the speed of light in
the vacuum. Correspondingly, the resonance frequencies of the
counterpropagating modes experience an opposite Sagnac-Fizeau shift; i.e.,
$\omega_{c}\rightarrow\omega_{c}+\Delta_{F}$, with Malykin (2000)

$\small\Delta_{F}=\pm\Omega\dfrac{nR\omega_{c}}{c}\left(1-\dfrac{1}{n^{2}}-\dfrac{\lambda}{n}\dfrac{\mathrm{d}n}{\mathrm{d}\lambda}\right),$
(1)

where $\omega_{c}$ is the resonance frequency for a stationary resonator, and
$\lambda$ is the light wavelength in vacuum. The dispersion term
$dn/d\lambda$, characterizing the relativistic origin of the Sagnac effect, is
small in typical materials (up to $\sim 1\%$) Maayani  _et al._ (2018);
Malykin (2000). By spinning the resonator along the CW direction, we have
$\Delta_{F}>0$ or $\Delta_{F}<0$ for the case with the driving laser on the
left- or right-hand side, and the corresponding effective optical frequencies
are $\omega_{j}\\!\equiv\\!\omega_{c}\pm|\Delta_{F}|$
($j=\circlearrowleft,\circlearrowright$), respectively. In addition, the
resonator can support a mechanical breathing mode with frequency $\omega_{m}$.
In a rotating frame with respect to
$\hat{H}_{0}=\hbar\omega_{l}(\hat{a}_{\circlearrowleft}^{\dagger}\hat{a}_{\circlearrowleft}+\hat{a}_{\circlearrowright}^{\dagger}\hat{a}_{\circlearrowright})$,
the Hamiltonian of this spinning COM system, with the driving laser on the
left-hand side, is this:

$\displaystyle\hat{H}=\hat{\cal{H}}_{c}+\dfrac{\hbar\omega_{m}}{2}(\hat{p}^{2}+\hat{q}^{2})-\hbar
G_{0}(\hat{a}_{\circlearrowright}^{\dagger}\hat{a}_{\circlearrowright}+\hat{a}_{\circlearrowleft}^{\dagger}\hat{a}_{\circlearrowleft})\hat{q},$
$\displaystyle\hat{\cal{H}}_{c}\\!=\\!\sum_{j=\circlearrowleft,\circlearrowright}\\!\hbar\Delta_{j}\hat{a}_{j}^{\dagger}\hat{a}_{j}\\!+\\!\hbar
J(\hat{a}_{\circlearrowright}^{\dagger}\hat{a}_{\circlearrowleft}+\hat{a}_{\circlearrowleft}^{\dagger}\hat{a}_{\circlearrowright})+i\hbar\varepsilon(\hat{a}_{\circlearrowleft}^{\dagger}\\!-\\!\hat{a}_{\circlearrowleft}),$
(2)

where $\hat{a}_{j}$ ($\hat{a}_{j}^{\dagger}$) is the optical annihilation
(creation) operator, $\Delta_{j}=\omega_{j}-\omega_{l}$, and $\hat{q}$
($\hat{p}$) is the dimensionless mechanical displacement (momentum) operator.
The frame rotating with driving frequency $\omega_{l}$ is obtained by applying
the unitary transformation $\hat{U}=\exp[i\hat{H}_{0}t/\hbar]$ (see, e.g.,
Ref. Aspelmeyer  _et al._ (2014)). The field amplitude of the driving laser is
$|\varepsilon|=\sqrt{2\kappa P/\hbar\omega_{l}}$, where $P$ and $\kappa$ are
the input laser power and the optical decay rate, respectively.
$G_{0}=(\omega_{c}/R)\sqrt{\hbar/m\omega_{m}}$ denotes the single-photon COM
coupling rate Aspelmeyer  _et al._ (2014), with $m$ the mass of the resonator.
Also, imperfections of devices, such as surface roughness or material defect,
can cause optical backscattering, as described by the mode-coupling strength
$J$. In a recent experiment, by breaking the time-reversal symmetry with
Brillouin devices, dynamical suppression of the backscattering was already
observed Kim  _et al._ (2019).

Quantum Langevin equations of this spinning COM system then read:

$\displaystyle\dot{\hat{a}}_{\circlearrowleft}=-(i\Delta_{\circlearrowleft}+\kappa)\hat{a}_{\circlearrowleft}-iJ\hat{a}_{\circlearrowright}+iG_{0}\hat{a}_{\circlearrowleft}\hat{q}+\varepsilon+\sqrt{2\kappa}\hat{a}_{\circlearrowleft}^{\textrm{in}},$
$\displaystyle\dot{\hat{a}}_{\circlearrowright}=-(i\Delta_{\circlearrowright}+\kappa)\hat{a}_{\circlearrowright}-iJ\hat{a}_{\circlearrowleft}+iG_{0}\hat{a}_{\circlearrowright}\hat{q}+\sqrt{2\kappa}\hat{a}_{\circlearrowright}^{\textrm{in}},$
$\displaystyle\dot{\hat{q}}=\omega_{m}\hat{p},$
$\displaystyle\dot{\hat{p}}=-\omega_{m}\hat{q}-\gamma_{m}\hat{p}+G_{0}(\hat{a}_{\circlearrowright}^{\dagger}\hat{a}_{\circlearrowright}+\hat{a}_{\circlearrowleft}^{\dagger}\hat{a}_{\circlearrowleft})+\hat{\xi},$
(3)

where $\gamma_{m}$ is the mechanical damping rate, and
$\hat{a}_{j}^{\textrm{in}}$ ($\hat{\xi}$) is the zero-mean input noise
operator for the optical (mechanical) mode, characterized by the following
correlation functions Gardiner and Zoller (2000):

$\displaystyle\langle\hat{a}_{j}^{\textrm{in}}(t)\hat{a}_{j}^{\textrm{in},\dagger}(t^{\prime})\rangle=\delta(t-t^{\prime}),$
$\displaystyle\langle\hat{\xi}(t)\hat{\xi}(t^{\prime})\rangle\simeq\gamma_{m}(2n_{m}+1)\delta(t-t^{\prime}),~{}\text{for}~{}\omega_{m}/\gamma_{m}\\!\gg\\!1,$
(4)

where $n_{m}\\!=\\![\exp(\hbar\omega_{m}/k_{\textit{B}}\mathrm{T})-1]^{-1}$
denotes the thermal phonon number, $k_{\textit{B}}$ is the Boltzmann constant,
and $\mathrm{T}$ is the bath temperature. Under the condition of strong
optical driving, we can linearize the dynamics by expanding each operator as a
sum of its steady-state value and a small fluctuation around it, i.e.,
$\hat{a}_{j}=\alpha_{j}+\delta\hat{a}_{j},~{}\hat{q}=q_{s}+\delta\hat{q},~{}\hat{p}=p_{s}+\delta\hat{p}$.
By defining the vectors of quadrature fluctuations and input noises as
$u^{\textit{T}}(t)=(\delta\hat{X}_{\circlearrowleft},\delta\hat{Y}_{\circlearrowleft},\delta\hat{X}_{\circlearrowright},\delta\hat{Y}_{\circlearrowright},\delta\hat{q},\delta\hat{p})$,
$v^{\textit{T}}(t)=(\sqrt{2\kappa}\hat{X}_{\circlearrowleft}^{\textrm{in}},\sqrt{2\kappa}\hat{Y}_{\circlearrowleft}^{\textrm{in}},\sqrt{2\kappa}\hat{X}_{\circlearrowright}^{\textrm{in}},\sqrt{2\kappa}\hat{Y}_{\circlearrowright}^{\textrm{in}},0,\hat{\xi})$,
with the components:

$\displaystyle\delta\hat{X}_{j}$
$\displaystyle=\dfrac{1}{\sqrt{2}}(\delta\hat{a}_{j}^{\dagger}+\delta\hat{a}_{j}),$
$\displaystyle\delta\hat{Y}_{j}$
$\displaystyle=\dfrac{i}{\sqrt{2}}(\delta\hat{a}_{j}^{\dagger}-\delta\hat{a}_{j}),$
$\displaystyle\hat{X}_{j}^{\textrm{in}}$
$\displaystyle=\dfrac{1}{\sqrt{2}}(\hat{a}_{j}^{\textrm{in}\dagger}+\hat{a}_{j}^{\textrm{in}}),$
$\displaystyle\hat{Y}_{j}^{\textrm{in}}$
$\displaystyle=\dfrac{i}{\sqrt{2}}(\hat{a}_{j}^{\textrm{in}\dagger}-\hat{a}_{j}^{\textrm{in}}),$
(5)

we obtain a compact form of the linearized equations of quantum fluctuations:
$\dot{u}(t)=Au(t)+v(t),$ where

$A=\begin{pmatrix}-\kappa&\tilde{\Delta}_{\circlearrowleft}&0&J&-G_{\circlearrowleft}^{y}&0\\\
-\tilde{\Delta}_{\circlearrowleft}&-\kappa&-J&0&G_{\circlearrowleft}^{x}&0\\\
0&J&-\kappa&\tilde{\Delta}_{\circlearrowright}&-G_{\circlearrowright}^{y}&0\\\
-J&0&-\tilde{\Delta}_{\circlearrowright}&-\kappa&G_{\circlearrowright}^{x}&0\\\
0&0&0&0&0&\omega_{m}\\\
G_{\circlearrowleft}^{x}&G_{\circlearrowleft}^{y}&G_{\circlearrowright}^{x}&G_{\circlearrowright}^{y}&-\omega_{m}&-\gamma_{m}\end{pmatrix},$
(6)

and $\tilde{\Delta}_{j}=\Delta_{j}-G_{0}q_{s}$ is the effective optical
detuning, $G_{j}^{x}$ ($G_{j}^{y}$) is the real (imaginary) part of the
effective COM coupling rate
($G_{j}\equiv\sqrt{2}G_{0}\alpha_{j}=G_{j}^{x}+iG_{j}^{y}$), and the steady-
state mean values of the dynamical variables are given by:

$\displaystyle\alpha_{\circlearrowleft}=\dfrac{(i\tilde{\Delta}_{\circlearrowright}+\kappa)\varepsilon}{(i\tilde{\Delta}_{\circlearrowright}+\kappa)(i\tilde{\Delta}_{\circlearrowleft}+\kappa)+J^{2}},$
$\displaystyle\alpha_{\circlearrowright}=\dfrac{-iJ}{(i\tilde{\Delta}_{\circlearrowright}+\kappa)}\alpha_{\circlearrowleft},$
$\displaystyle
q_{s}=\dfrac{G_{0}}{\omega_{m}}(|\alpha_{\circlearrowright}|^{2}+|\alpha_{\circlearrowleft}|^{2}),~{}p_{s}=0.$
(7)

The solution of the linearized Langevin equations is given by
$u(t)=\mathcal{M}(t)u(0)+\int_{0}^{t}d\tau\mathcal{M}(\tau)v(t-\tau)$, where
$\mathcal{M}(t)=\exp(At)$. The system is stable and reaches its steady state
when all real parts of the eigenvalues of $A$ are negative, as characterized
by the Routh-Hurwitz criterion DeJesus and Kaufman (1987); SM . When the
stability condition is fulfilled, we have $\mathcal{M}(\infty)=0$ in the
steady state and

$\displaystyle
u_{i}(\infty)=\int_{0}^{\infty}d\tau\sum_{k}\mathcal{M}_{ik}(\tau)v_{k}(t-\tau).$
(8)

Due to the linearized dynamics and the Gaussian nature of the quantum noises,
the steady state of the system, independently of any initial conditions,
finally evolves into a tripartite zero-mean Gaussian state, which is fully
characterized by a $6\times 6$ correlation matrix $V$, with its components

$\displaystyle V_{kl}=\langle
u_{k}(\infty)u_{l}(\infty)\\!+\\!u_{l}(\infty)u_{k}(\infty)\rangle/2.$ (9)

By substituting Eq. (8) into Eq. (9) and using the fact that the six
components of $v(t)$ are uncorrelated, the steady-state correlation matrix $V$
is obtained as

$\displaystyle
V=\int_{0}^{\infty}d\tau\mathcal{M}(\tau)D\mathcal{M}^{T}(\tau),$ (10)

where
$D\\!=\\!\textrm{Diag}\,[\kappa,\kappa,\kappa,\kappa,0,\gamma_{m}(2n_{m}\\!+\\!1)]$
is the diffusion matrix, defined through $\langle
v_{k}(\tau)v_{l}(\tau^{\prime})\\!+\\!v_{l}(\tau^{\prime})v_{k}(\tau)\rangle/2=D_{kl}\delta(\tau-\tau^{\prime})$.
Under the stability condition, the steady-state correlation matrix $V$
fulfills the Lyapunov equation Vitali  _et al._ (2007):

$\displaystyle AV+VA^{\textit{T}}=-D.$ (11)

This linear Eq. (11) allows us to find $V$ for any values of the relevant
parameters. To quantify the entanglement between the mechanical mode and the
driven optical mode, we adopt the logarithmic negativity, $E_{\mathcal{N}}$,
as a bipartite entanglement measure for continuous variables Adesso  _et al._
(2004)

$\displaystyle E_{\mathcal{N}}=\max\,[0,-\ln(2\nu^{-})],$ (12)

where
$\nu^{-}\\!=\\!2^{-1/2}\\{\Sigma(V^{{}^{\prime}})-[\Sigma(V^{{}^{\prime}})^{2}-4\det\\!V^{{}^{\prime}}]^{1/2}\\}^{1/2}$
, with
$\Sigma(V^{{}^{\prime}})\\!=\\!\det\mathcal{A}+\det\mathcal{B}-2\det\mathcal{C}$,
is the smallest eigenvalue of the partial transpose of the reduced correlation
matrix $V^{{}^{\prime}}$. By tracing out the reflected mode in $V$ and writing
it in a $2\times 2$ block form, we have

$\displaystyle V^{{}^{\prime}}=\left(\begin{matrix}\mathcal{A}&\mathcal{C}\\\
\mathcal{C}^{\textit{T}}&\mathcal{B}\end{matrix}\right).$ (13)

$V^{{}^{\prime}}$ preserves the Gaussian nature, and entanglement emerges in
its corresponding subsystem if and only if $\nu^{-}<1/2$, which is equivalent
to Simon’s necessary and sufficient entanglement criterion (or the related
Peres-Horodecki criterion) for certifying entanglement of two-mode system in
Gaussian states Simon (2000).

Figure 2: Nonreciprocal COM entanglement without backscattering. (a)
$E_{\mathcal{N}}$ versus $\Delta_{c}/\omega_{m}$ for different input
directions. For $\Delta_{F}=0$, COM entanglement exists around the resonance
$\Delta_{c}/\omega_{m}\simeq 1$. The spectral offset is due to the the COM-
induced redshift of the cavity mode. For $\Delta_{F}\neq 0$, the resonance
conditions for the countercirculating modes are modified by the opposite
Sagnac shifts, resulting in the peaks symmetrically shifted for the opposite
input directions. (b)-(c) Tunable quantum nonreciprocity versus classical
nonreciprocity. For $\Delta_{c}/\omega_{m}\sim 0.27$, quantum nonreciprocity
exists even when $N_{\circlearrowleft}=N_{\circlearrowright}$ (without
classical nonreciprocity); in contrast, for $\Delta_{c}/\omega_{m}\sim 1$,
classical nonreciprocity appears for
$E_{\mathcal{N},\circlearrowleft}=E_{\mathcal{N},\circlearrowright}$. The
parameters are chosen as $\Omega=8\,\textrm{kHz}$, $J=0$, and
$P=20\,\textrm{mW}$.  Figure 3: Suppression of COM entanglement due to random-
defect-induced backscattering, and its nonreciprocal revival resulting from
the rotation-induced compensation. (a), (b) The logarithmic negativity
$E_{\mathcal{N}}$ and (c), (d) the effective COM coupling $|G_{j}|/\omega_{m}$
are plotted as a function of the scaled optical detuning
$\Delta_{c}/\omega_{m}$. For $\Omega=23\,\textrm{kHz}$, the value of
$E_{\mathcal{N}}$ is enhanced for $\sim 2.5$ times, reaching almost that as in
an ideal device without backscattering. (e) The effective COM coupling
$|G_{j}|/\omega_{m}$ versus the optical detuning $\Delta_{c}/\omega_{m}$ and
the rotation speed $\Omega$. (f) Density plot of the revival factor $\chi$ as
a function of the optical coupling strength $J$ and the rotation speed
$\Omega$.

In our calculations, for ensuring the stability of the system, we use the
experimentally feasible values Maayani  _et al._ (2018); Righini  _et al._
(2011): $n\\!=\\!1.48$, $m\\!=\\!10\,\textrm{ng}$, $R\\!=\\!1.1\,\textrm{mm}$,
$\lambda\\!=\\!$1.55\text{\,}\mathrm{\SIUnitSymbolMicro m}$$,
$Q\\!=\\!\omega_{c}/\kappa\\!=\\!3.2\times 10^{7}$,
$\omega_{m}\\!=\\!63\,\textrm{MHz}$, $\gamma_{m}\\!=\\!5.2\,\textrm{kHz}$,
$\mathrm{T}\\!=\\!130\,\textrm{mK}$, and $\Omega\\!=\\!8\,\textrm{kHz}$ or
$23\,\textrm{kHz}$. We first consider the case without backscattering, i.e.,
$J=0$. In Fig. 2, we plot the logarithmic negativity $E_{\mathcal{N},j}$ and
the intracavity photon number $N_{j}\equiv|\alpha_{j}|^{2}$ of the driven
mode, as a function of the detuning $\Delta_{c}=\omega_{c}-\omega_{l}$. Here
$j$ denotes the driving direction. For a stationary resonator,
$E_{\mathcal{N},j}$ is independent on the driving direction, while for a
spinning one, it becomes different by reversing the direction. For example,
Figs. 2(a) and 2(b) show that, when the maximal COM entanglement is created by
driving from one side, no entanglement occurs by driving it from the other
side. The underlying physics can be understood as follows. In COM, the driving
laser is scattered by the mechanical mode into the Stokes and anti-Stokes
sidebands. When the cavity mode is resonant with one of the sidebands, COM
correlations are created, as shown in experiments Palomaki  _et al._ (2013);
Riedinger  _et al._ (2016). Now, by spinning the resonator, nonreciprocity
emerges for the created COM entanglement, which is fundamentally different
from classical nonreciprocity of mean photon numbers. In fact, as shown in
Figs. 2(b) and 2(c), nonreciprocal COM entanglement exists even without any
classical nonreciprocity; in contrast, for $\Delta_{c}/\omega_{m}\sim 1$,
significant classical nonreciprocity appears for
$E_{\mathcal{N},\circlearrowleft}\simeq E_{\mathcal{N},\circlearrowright}$.
Hence it is possible to switch a single nonreciprocal device between classical
and quantum regimes. We note that one-way quantum control of photon bunching
and antibunching has been observed very recently Scheucher  _et al._ (2016);
Yang  _et al._ .

More importantly, quantum nonreciprocity provides a feasible way to protect
devices against losses, which is reminiscent of that in topological systems or
chiral environment Gangaraj  _et al._ (2017); Gonzalez-Ballestero  _et al._
(2015). Figure 3(a) shows that, in conventional COM, $E_{\mathcal{N}}$
decreases for $J\neq 0$, while for a spinning device, the maximum value of
$E_{\mathcal{N}}$ is significantly enhanced, approaching to that in an ideal
device [see Fig. 3(b)]. To better understand this counterintuitive effect, we
also plot the effective COM coupling of the driven mode $|G_{j}|$ with respect
to $\Delta_{c}$ in Figs. 3(c)-3(e). We see that the backscattering-induced
reflection can be significantly suppressed in a spinning device. As a result,
one can achieve nearly ideal COM entanglement, which can be clearly shown by
defining a revival factor:

$\displaystyle\chi=\frac{\max\left[E_{\mathcal{N}}(\Omega\neq 0,J\neq
0)\right]}{\max\left[E_{\mathcal{N}}(\Omega=0,J=0)\right]}.$ (14)

Figure 3(f) shows that the maximal factor can reach $99.1\%$; i.e., COM
entanglement in such a nonreciprocal device is immune to random losses. This
provides a new strategy to improve the performance of quantum devices by
harnessing the power of nonreciprocity. Also we have confirmed that
nonreciprocal entanglement can survive even when quantum COM entanglement is
fully destroyed by thermal noises in a conventional reciprocal device (see the
Supplemental Material for more details SM ).

Finally, we remark that in experiments, COM entanglement can be detected by
measuring the correlation matrix $V$ under a proper readout choice via a
filter Genes  _et al._ (2008); Riedinger  _et al._ (2016); Palomaki  _et al._
(2013). The optical quadratures can be measured via a homodyne or heterodyne
detection of the output Palomaki  _et al._ (2013); Barzanjeh  _et al._ (2019);
Chen  _et al._ (2020), and the readout of mechanical ones requires a probe
being resonant with the anti-Stokes sideband, mapping the mechanical motion to
the output field Palomaki  _et al._ (2013); Ockeloen-Korppi  _et al._ (2018).
With the same procedure, nonreciprocal features are observable also in the
output field, including transmission rates Lü  _et al._ (2017) and COM
correlations Palomaki  _et al._ (2013); Barzanjeh  _et al._ (2019); Chen  _et
al._ (2020); Riedinger  _et al._ (2016); Ockeloen-Korppi  _et al._ (2018);
Riedinger  _et al._ (2018), detailed calculations of which will be given
elsewhere.

In summary, we have shown how to achieve quantum nonreciprocal entanglement in
a COM system, how to switch such a single nonreciprocal device between
classical and quantum regimes, and how to keep the optimal entanglement in a
chosen direction against losses. These findings, shedding light on the
marriage of nonreciprocal physics and quantum engineering, open up the way to
control quantum states by utilizing such diverse nonreciprocal devices as in
optics, atomtronics Dong  _et al._ ; Scheucher  _et al._ (2016); Yang  _et
al._ , electronics Kurpiers  _et al._ (2018), and in acoustics Maznev  _et
al._ (2013). In fact, quantum nonreciprocity is achievable in systems well
beyond COM; for instance, one-way control of sub- or super-Poissonian
correlations, as predicted in an optical system Huang  _et al._ (2018), was
demonstrated very recently using cavity atoms Yang  _et al._ . In a broader
view, nonreciprocal entanglement provides an unconventional tool for tasks
that cannot be performed by classical one-way devices, such as building noise-
tolerant quantum processors Stannigel  _et al._ (2012) and achieving
directional quantum sensing Wollman  _et al._ (2015); Pirkkalainen  _et al._
(2015); Stannigel  _et al._ (2010); Mirhosseini  _et al._ or steering
multipartite entanglement Li  _et al._ (2018); DiCarlo  _et al._ (2010); Qin
_et al._ (2018); Karg  _et al._ (2020).

We thank Chang-Ling Zou and Ran Huang for discussions. A. M. is supported by
the Polish National Science Centre under the Maestro Grant No.
DEC-2019/34/A/ST2/00081. L.-M. K. is supported by the National Natural Science
Foundation of China (NSFC, Grant No. 11775075 and No. 11434011), and H. J. is
supported by the NSFC (Grant No. 11774086 and No. 11935006). Y.-L. Z. is
supported by the NSFC (Grants No. 11704370) and the China Postdoctoral Science
Foundation (2019M652181).

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Supplementary Material for “Nonreciprocal Optomechanical Entanglement
Against Backscattering Losses”

Ya-Feng Jiao,1 Sheng-Dian Zhang,1 Yan-Lei Zhang,2, 3 Adam Miranowicz,4 Le-Man
Kuang,${}^{1,\,*}$ and Hui Jing${}^{1,\,\dagger}$

1Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of
Ministry of Education,
Department of Physics and Synergetic Innovation Center for Quantum Effects and
Applications,
Hunan Normal University, Changsha 410081, China
2CAS Key Laboratory of Quantum Information, University of Science and
Technology of China, Hefei 230026, P. R. China
3CAS Center For Excellence in Quantum Information and Quantum Physics,
University of Science and Technology of China, Hefei, Anhui 230026, P. R.
China
4Faculty of Physics, Adam Mickiewicz University, 61-614 Poznań, Poland

Here, we present the technical details on nonreciprocal optomechanical
entanglement, including: (1) a summary of all parameter values used in our
work; (2) experimental feasibility of our system, especially the conditions of
stable coupling between the tapered fiber and the spinning resonator; (3) more
discussions on mechanical or optical stability; (4) the influence of thermal
effects and quality factors on nonreciprocal entanglement.

###### Contents

  1. S1 System parameters
  2. S2 Experimental feasibility
    1. S2.1 Self-adjustment process
    2. S2.2 Intermolecular forces
  3. S3 Stability conditions
    1. S3.1 Mechanical stability
    2. S3.2 Optical stability
  4. S4 The role of thermal effects and quality factors

##  S1 System parameters

Table 1 shows the main symbols and parameters which have been used in this
work.

Symbol | Definition | Name | Value
---|---|---|---
$\omega_{m}$ |  | Mechanical resonance frequency | $63\text{\,}\mathrm{MHz}$
$\gamma_{m}$ |  | Mechanical linewidth | $5.2\text{\,}\mathrm{kHz}$
$Q_{m}$ | $\omega_{m}/\gamma_{m}$ | Mechanical quality factor | $1.21\text{\times}{10}^{4}$
$\mathrm{T}$ |  | Mechanical bath temperature | $130\text{\,}\mathrm{mK}$
$n_{m}$ | $[\exp(\hbar\omega_{m}/k_{B}\mathrm{T})-1]^{-1}$ | Thermal phonon occupation | $269.4$
$m$ |  | Effective mass | $10\text{\,}\mathrm{ng}$
$x_{\mathrm{zp}}$ | $\sqrt{\hbar/(m\omega_{m})}$ | Zero point fluctuations | $0.41\text{\,}\mathrm{fm}$
$\omega_{c}$ |  | Optical resonance frequency | $1.22\text{\,}\mathrm{PHz}$
$\lambda$ |  | Laser wavelength | $1.55\text{\,}\mathrm{\SIUnitSymbolMicro m}$
$\kappa$ |  | Cavity linewidth | $38.0\text{\,}\mathrm{MHz}$
$Q$ | $\omega_{c}/\kappa$ | Optical quality factor | $3.2\text{\times}{10}^{7}$
$P$ |  | Input laser power |  $20\text{\,}\mathrm{mW}$ or $100\text{\,}\mathrm{mW}$
$G_{0}$ | $\omega_{c}x_{\mathrm{zp}}/R$ | Single-photon optomechanical coupling rate | $452.1\text{\,}\mathrm{Hz}$
$R$ |  | Sphere radius | $1.1\text{\,}\mathrm{mm}$
$r$ |  | Fiber radius | $544\text{\,}\mathrm{nm}$
$\Omega$ |  | Spinning frequency |  $8\text{\,}\mathrm{kHz}$ or $23\text{\,}\mathrm{kHz}$
$\mathrm{E}$ |  | Young modulus of silica | $75\text{\,}\mathrm{GPa}$
$\Upsilon$ |  | Elastic limit of silica | $9\text{\,}\mathrm{GPa}$
$\varepsilon_{0}$ |  | Dielectric constant of air | $1$
$\varepsilon_{1}$ ($\varepsilon_{2}$) |  | Dielectric constant of silica | $3.9$
$n_{0}$ |  | Refractive index of air | $1$
$n_{1}$ ($n_{2}$) |  | Refractive index of silica | $1.48$
Table 1: Feasible parameters. Unless specified otherwise, these parameters are
applied to all evaluations in the text.

With these experimentally feasible parameters, we have confirmed that other
quantum effects, such as two-mode squeezing or non-classicality, can also be
manipulated in a highly asymmetric way in this optomechanical system, which
may find applications in chiral quantum optics and technology Lodahl  _et al._
(2017); Kimble (2008); Stannigel  _et al._ (2012).

Figure S1: Analysis of the “self-adjustment” behavior. (a) The strain
$\epsilon$ (red curve) and the displacement $d$ (blue curve) as a function of
the angular velocity $\Omega$ for $h=250\,\textrm{nm}$. (b) The air pressure
$T_{\mathrm{air}}$ and the ratio of intermolecular forces
$|T_{\mathrm{int}}|/T_{\mathrm{air}}$ for varying taper-resonator separation
$h$ at $\Omega=23\,\textrm{kHz}$. (c) Total force between the fiber and the
sphere versus the angular velocity $\Omega$ and the separation $h$. The
resulting force of air pressure and intermolecular forces has a minimal value
of $T_{\mathrm{tot}}=$1.135\text{\,}\mathrm{\SIUnitSymbolMicro N}$$, which
indicates the interactions of the fiber and the spinning sphere are always
repulsive.

##  S2 Experimental feasibility

###  S2.1 Self-adjustment process

As shown in a very recent experiment Maayani  _et al._ (2018), we consider a
whispering-gallery-mode silica sphere, which is mounted on a turbine and spins
along its axis with, e.g., the angular velocity $\Omega=6.6\,\textrm{kHz}$ for
the sphere radius $r=1.1\,\textrm{mm}$. Faster rotations have also been
reported in experiments by using, e.g., levitated nanomechanical rotors
Reimann  _et al._ (2018); Ahn  _et al._ (2018). Then, by positioning the
spinning resonator near a single-mode telecommunication fiber, light can be
coupled into or out of the resonator evanescently.

For such a spinning device, the aerodynamic process plays a key role in the
stable fiber-resonator coupling. Specifically, the fast spinning resonator can
drag air into the region between the tapered fiber and the resonator, thereby
forming a lubrication layer of air in this region. Then the thin film of air,
exerting pressure on the surface of the tapered fiber facing the resonator,
can make the fiber fly above the resonator with the separation of a few
nanometres. Therefore, if any perturbation causes the taper to rise higher
than the stable-equilibrium height, it can float back to its original
position, which is referred to as “self-adjustment”. Hence the fiber will not
touch or stick to the resonator even if it is pushed towards the spinning
sphere Maayani  _et al._ (2018). To see this clearly, we write the air
pressure as $\varDelta
T_{\mathrm{air}}=(\rho\varDelta\theta)T_{\mathrm{air}}/L$, with $\rho$
($\theta$) the radius (angle) of the winding shape of the deformation region.
Then, the total air pressure $T_{\mathrm{air}}$ on the taper can be estimated
analytically as Maayani  _et al._ (2018)

$\displaystyle T_{\mathrm{air}}=6.19\mu
R^{5/2}\Omega\int_{0}^{r}\left(h-\sqrt{r^{2}-x^{2}}+r\right)^{-3/2}\mathrm{d}x,$
(S2.1)

where $\mu$ is the viscosity of air, $R~{}(r)$ is the radius of the sphere
(taper), and $h=h_{0}+d$ represents the taper-resonator separation, with
$h_{0}$ denoting the stationary gap between the fiber and the sphere. The
local deformation can also lead to a tension on the infinitesimal cylinder of
the fiber, which can be calculated by

$\displaystyle\varDelta
T_{\mathrm{ela}}=2\mathcal{F}\sin\left(\varDelta\theta\middle/2\right)\approx\mathcal{F}\varDelta\theta,$
(S2.2)

where $\mathcal{F}$ is the elastic force on the taper, obeying
$\sigma=\mathcal{F}/(\pi r^{2})=\epsilon\mathrm{E}$. Here $\sigma$ is the
uniaxial stress, $\mathrm{E}$ is the Young modulus of silica, and
$\epsilon=\delta_{L}/L$ is the strain, where $\delta_{L}=L^{\prime}-L$ denotes
the variation of the original length $L$ of the deformation region.
Furthermore, $\delta_{L}$ can be straightforwardly derived via the following
relations: $L^{\prime}=\rho\theta$, $(L/2)^{2}+(\rho-d)^{2}=\rho^{2}$, and
$\sin(\theta/2)=L/(2\rho)$. Hence, in the case of equilibrium (i.e.,
$\varDelta T_{\mathrm{air}}=\varDelta T_{\mathrm{ela}}$), $T_{\mathrm{air}}$
can be given in another form:

$\displaystyle T_{\mathrm{air}}=2\pi
r^{2}\mathrm{E}\left[\arcsin\left(\phi\right)-\phi\right]\approx\frac{\pi}{3}r^{2}\phi^{3}\mathrm{E},$
(S2.3)

where $\left.\phi=4Ld\middle/\left.(L^{2}+4d^{2}\right.)\right.$, and the
approximation, $\arcsin(\phi)=\phi+\phi^{3}/6+\cdots$, is made within the
limit of $|\phi|\ll 1$, which physically requires a comparatively small
distortion. As shown in Fig. S1(a), we confirmed that these approximation
conditions can be easily satisfied with experimentally accessible parameters
Maayani  _et al._ (2018); Bellouard  _et al._ (2005):
$\mathrm{E}=$75\text{\,}\mathrm{GPa}$$, $r=$544\text{\,}\mathrm{nm}$$, and
$L=$3\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. In this case, the displacement
$d$ is given by

$\displaystyle\left.d=\frac{L}{2}\left(\beta-\sqrt{\beta^{2}-1}\right),\right.$
(S2.4)

where $\left.\beta=\left[\pi
r^{2}\mathrm{E}\middle/\left(3T_{\mathrm{air}}\right)\right]^{\left.1\middle/3\right.}\right.$.
The strain of the taper thus can be reduced to
$\epsilon=\arcsin\left(\phi\right)/\phi-1\approx\phi^{2}/6$, from which we
find that the strain (i.e., the elastic force) is positively associated with
the taper-resonator separation:

$\displaystyle\frac{\partial\mathcal{F}}{\partial h}=\pi
r^{2}\mathrm{E}\left(\frac{\partial\epsilon}{\partial d}\right)=\frac{16\pi
r^{2}\mathrm{E}L^{2}d\left(L^{2}-4d^{2}\right)}{3\left(L^{2}+4d^{2}\right)^{3}}>0.$
(S2.5)

Equation (S2.5) clearly reveals that the elastic force becomes stronger when
the air gap gets larger than the stable-equilibrium distance. Also, as shown
in Fig. S1(b), the air pressure on the taper is largely suppressed. As a
result, the taper can be dragged back to its original position when any
perturbation causes it away from the spinning resonator, leading to the self-
adjustment behavior. The self-adjustment of the tapered fiber enables the
critical coupling of light into or out of the resonator, by which the counter-
propagating beams can experience an optical drag identical in size, but
opposite in sign.

###  S2.2 Intermolecular forces

The intermolecular forces, such as the Casimir and van der Waals forces, could
also affect the stable fiber-resonator coupling. In our system, such kinds of
intermolecular forces can be described as Maayani  _et al._ (2018):

$T_{\mathrm{int}}=rR\left(-\frac{\mathbb{A}}{6\pi
h^{3}}+\frac{\mathbb{B}}{45\pi h^{9}}-\frac{\pi^{2}c\hbar}{240h^{4}}\right),$
(S2.6)

where the Hamaker constant $\mathbb{A}$ can be calculated by Chen  _et al._
(2009):

$\mathbb{A}=\frac{3\varepsilon_{-}^{\left(1\right)}\varepsilon_{-}^{\left(2\right)}k_{\mathrm{B}}\mathrm{T}}{4\varepsilon_{+}^{\left(1\right)}\varepsilon_{+}^{\left(2\right)}}+\frac{\nu\left[n_{-}^{\left(1\right)}n_{-}^{\left(2\right)}\right]^{2}}{n_{+}^{\left(1\right)}n_{+}^{\left(2\right)}\left[n_{+}^{\left(1\right)}+n_{+}^{\left(2\right)}\right]},$
(S2.7)

with $\left.\nu=3\sqrt{2}\hbar\nu_{e}\middle/16\right.$,
$\varepsilon_{\pm}^{\left(u\right)}=\varepsilon_{u}\pm\varepsilon_{0}$,
$n_{\pm}^{\left(u\right)}=\sqrt{n_{u}^{2}\pm n_{0}^{2}}$, and $u=1$, $2$.
Hereafter, we use $\varepsilon_{0}$ ($n_{0}$), $\varepsilon_{1}$ ($n_{1}$),
and $\varepsilon_{2}$ ($n_{2}$) to represent the dielectric constant (the
refractive index) of the air, the taper and the spinning resonator,
respectively; $k_{\mathrm{B}}$ is the Boltzmann constant, $\mathrm{T}$ is the
mechanical bath temperature, and $\nu_{e}=$3\text{\,}\mathrm{PHz}$$ Chen  _et
al._ (2009). Note that for simplicity, we have replaced $n_{2}$ with $n$ in
the main text. Moreover, the constant $\mathbb{B}$ is typically of the order
of ${10}^{-76}$ $\mathrm{J}\,\mathrm{m}^{6}$ for interactions between
condensed matter phases across the vacuum or air Wu and Bogy (2002). Taking
intermolecular forces into account, the total force between the taper and the
resonator becomes: $T_{\mathrm{tot}}=T_{\mathrm{air}}+T_{\mathrm{int}}$.
Herein, we choose experimentally accessible parameters Joyce  _et al._ (1968):
$\varepsilon_{0}=1$, $\varepsilon_{1}=\varepsilon_{2}=3.9$, $n_{0}=1$,
$n_{1}=n_{2}=1.48$, and $\mathrm{T}=$130\text{\,}\mathrm{mK}$$. As expected,
the intermolecular forces are found to be negligible ($<0.1\%$), and the
taper-resonator interactions remain repulsive [see Figs. S1(b) and S1(c)].
Thus, the effects of the Casimir and van der Waals forces can be safely
omitted on critical coupling. Other factors, such as lubricant
compressibility, tapered-fiber stiffness, and the wrap angle of a fiber, may
also affect critical coupling. However, these factors are confirmed to be
negligible in experiments, which can be also safely ignored in our discussions
Maayani  _et al._ (2018).

##  S3 Stability conditions

###  S3.1 Mechanical stability

The realization of stable fiber-resonator coupling sets a limit to the angular
velocity of spinning devices. Specifically, the requirement $\beta\geq 1$ in
Eq. (S2.4) yields the first limit:

$\Omega_{0}=\frac{\varrho\pi r^{2}\mathrm{E}}{18.57\mu
R^{\left.5\middle/2\right.}},$ (S3.8)

where

$\varrho=\left[\int_{0}^{r}\left(h-\sqrt{r^{2}-x^{2}}+r\right)^{\left.-3\middle/2\right.}\mathrm{d}x\right]^{-1}.$
(S3.9)

Also, the tiny displacement should obey $d=h-h_{0}<h$, which gives the second
limit:

$\Omega_{1}=\frac{\varrho\pi r^{2}\Lambda\mathrm{E}}{18.57\mu
R^{\left.5\middle/2\right.}},$ (S3.10)

where
$\left.\Lambda=\left[4Lh\middle/\left(L^{2}+4h^{2}\right)\right]^{3}\right.$.
Finally, the elastic limit of the tapered fiber provides the third stability
condition ($\sigma=\Upsilon$):

$\Omega_{2}=\frac{\varrho\pi r^{2}\Upsilon}{3.095\mu
R^{\left.5\middle/2\right.}}\sqrt{\frac{6\Upsilon}{\mathrm{E}}},$ (S3.11)

where $\Upsilon$ is typically $9\text{\,}\mathrm{GPa}$ for silica devices
Sugiura  _et al._ (1997). Thus, the mechanical limit to the angular velocity
can be obtained as:

$\Omega_{\max}=\min\left\\{\Omega_{0},\Omega_{1},\Omega_{2}\right\\}.$ (S3.12)

When operating at taper-resonator separations near $250\text{\,}\mathrm{nm}$,
we have $\Omega_{0}=$81.6\text{\,}\mathrm{MHz}$$,
$\Omega_{1}=$2.8\text{\,}\mathrm{MHz}$$,
$\Omega_{2}=$49.9\text{\,}\mathrm{MHz}$$, and thus
$\Omega_{\max}=$2.8\text{\,}\mathrm{MHz}$$. Therefore, it is reasonable to use
$\Omega=$8\text{\,}\mathrm{kHz}$$ or $23\text{\,}\mathrm{kHz}$ in the main
text.

Moreover, we also consider the influence of the radial breathing of the
resonator on stable fiber-resonator coupling. In paticular, we compare the
mean mechanical displacement $x_{s}$ with the air-induced displacement $d$,
i.e.,

$\left.\eta=x_{s}\middle/d=q_{s}x_{\mathrm{zp}}\middle/d,\right.$ (S3.13)

where
$x_{\mathrm{zp}}=\sqrt{\left.\hbar\middle/\left(m\omega_{m}\right)\right.}$
denotes the standard deviation of the zero-point motion of the mechanical
mode. As shown in Fig. S2(a), the effect of the mechanical displacement is
negligible indeed ($<1\%$), thereby the radial breathing of the resonator does
not disturb the critical evanescent coupling. Note that, for simplicity, we
have introduced $\Omega_{r}=\pm\Omega$.

Figure S2: The mechanical and optical stability conditions. (a) The ratio of
the mean mechanical displacement $x_{s}$ to the air-induced displacement $d$
as a function of the angular velocity $\Omega_{r}$. In the aerodynamic
process, this ratio is extremely small ($\sim 0.1\%$), meaning that radial
breathing of the resonator is negligible compared to the air-induced
displacement. (b-c) Stability functions $\Theta_{5}$ and $\Theta_{6}$ versus
the angular velocity $\Omega_{r}$ and the scaled optical detuning
$\Delta_{c}/\omega_{m}$ at $J/\kappa=2$ and $P=20\,\textrm{mW}$. The white
contour line in (c) is the boundary between the stability and instability
regions, and a.u. denotes arbitrary units. The parameters are listed in Table
1.

###  S3.2 Optical stability

Figure S3: Thermal effect on nonreciprocal optomechanical entanglement. (a-b)
The logarithmic negativity $E_{\mathcal{N}}$ as a function of the environment
temperature T for different driving directions. (c-d) Density plot of the
logarithmic negativity $E_{\mathcal{N}}$ versus the angular velocity
$\Omega_{r}$ and the environment temperature T. (e-f) The COM entanglement
difference with respect to different driving directions, $\Delta
E_{\mathcal{N}}$, versus the scaled optical detuning $\Delta_{c}/\omega_{m}$
and the environment temperature T. The rotation speed is chosen as
$\Omega=8\,\textrm{kHz}$ in (a) and (e), and $\Omega=23\,\textrm{kHz}$ in (b)
and (f). Other parameters are listed in Table 1.

According to Routh-Hurwitz criterion DeJesus and Kaufman (1987), the system is
stable and reaches its steady state when all eigenvalues of the matrix $A$
have negative real parts. Therefore, we start our analysis by determining the
eigenvalues of the matrix $A$, i.e., $|A-\lambda I|=0$, yielding the following
characteristic equation:

$\displaystyle\lambda^{6}+a_{1}\lambda^{5}+a_{2}\lambda^{4}+a_{3}\lambda^{3}+a_{4}\lambda^{2}+a_{5}\lambda+a_{6}=0,$
(S3.14)

where

$\displaystyle a_{1}$
$\displaystyle=4\kappa+\gamma_{m},~{}~{}~{}a_{2}=\sigma_{0}+\omega_{m}^{2}+4\kappa(\kappa+\gamma_{m}),$
$\displaystyle a_{3}$
$\displaystyle=\sigma_{0}(2\kappa+\gamma_{m})+4\kappa\left(\omega_{m}^{2}+\kappa\gamma_{m}\right),~{}~{}~{}a_{4}=\sigma_{0}\sigma_{1}+\sigma_{2}+4\kappa^{2}\omega_{m}^{2},$
$\displaystyle a_{5}$
$\displaystyle=\gamma_{m}\mu_{2}-2\kappa\mu_{0}\mu_{3}\left(\gamma_{m}\mu_{0}+4\kappa\omega_{m}^{2}\right)+\kappa\mu_{1}\left(\kappa\gamma_{m}+2\omega_{m}^{2}\right),$
$\displaystyle a_{6}$
$\displaystyle=\omega_{m}\left(\sigma_{+}+\sigma_{-}+\sigma_{2}\omega_{m}-\mu_{0}\mu_{4}\right)-\mu_{3}\tilde{\Delta}_{+}\tilde{\Delta}_{-},$
(S3.15)

and

$\displaystyle\sigma_{0}$
$\displaystyle=2\mu_{0}+\mu_{1},~{}~{}~{}\sigma_{1}=\kappa^{2}+2\kappa\gamma_{m}+\omega_{m}^{2},~{}~{}~{}\sigma_{2}=\mu_{0}^{2}+\kappa^{2}\mu_{1}+\mu_{2},$
$\displaystyle\sigma_{2}$
$\displaystyle=\mu_{2}-\mu_{3}+\mu_{0}\left(J^{2}-\kappa^{2}\right),~{}~{}~{}\sigma_{\pm}=(\Delta_{\pm}J^{2}-\Delta_{\mp}\kappa^{2})G_{\mp}^{2},$
$\displaystyle\mu_{0}$
$\displaystyle=J^{2}+\kappa^{2},~{}~{}~{}\mu_{1}=\tilde{\Delta}_{+}^{2}+\tilde{\Delta}_{-}^{2},~{}~{}~{}\mu_{2}=(\tilde{\Delta}_{+}\tilde{\Delta}_{-}-2J^{2})\tilde{\Delta}_{+}\tilde{\Delta}_{-},$
$\displaystyle\mu_{3}$
$\displaystyle=\omega_{m}(\tilde{\Delta}_{+}|G_{\circlearrowleft}|^{2}+\tilde{\Delta}_{-}|G_{\circlearrowright}|^{2}+\mu_{4}),~{}~{}~{}\mu_{4}=2J(G_{\circlearrowright}^{x}G_{\circlearrowleft}^{x}+G_{\circlearrowright}^{y}G_{\circlearrowleft}^{y}).$
(S3.16)

Using the coefficients $a_{k}$, we can form a set of $k\times k$ matrices,
$\theta_{k}$, for $k\leq 6$, with their entries defined as:

$\displaystyle\theta_{ln}=\begin{cases}0,&2l-n<0~{}\text{or}~{}2l-n>k,\\\
a_{2l-n},&\text{otherwise}.\end{cases}$ (S3.17)

The stability conditions can be satisfied when all the determinants of the
matrices $\theta_{k}$ are positive DeJesus and Kaufman (1987). Through careful
analysis, we find only $\theta_{5}$ and $\theta_{6}$ are nontrivial. As shown
in Figs. S2(b) and S2(c), we numerically plot these functions in a logarithmic
form, i.e.,

$\displaystyle\Theta_{k}=\begin{cases}\ln\theta_{k},&\theta_{k}>0,\\\
-\ln\left|\theta_{k}\right|,&\theta_{k}<0.\end{cases}$ (S3.18)

Note that $\theta_{k}$ and $\Theta_{k}$ maintain the same sign within the
parameters used in the main text, thus the contour line in Fig. S2(c) clearly
determines the boundary between the stability and instability regions. In this
case, we can confirm that these experimentally feasible parameters keep this
optomechanical system in a stable zone.

##  S4 The role of thermal effects and quality factors

Thermal noises can destroy fragile quantum correlations in practical devices.
Thus protecting quantum resources from environmental thermal perturbations is
essential for achieving quantum nonreciprocity. To see the influence of the
thermal effect on nonreciprocal optomechanical entanglement, we plot the
logarithmic negativity $E_{\mathcal{N}}$ with respect to the environment
temperature T in Figs. S3(a) - S3(d). We find that nonreciprocal
optomechanical entanglement in a chosen direction can exist at higher
temperature compared to the case of a stationary system. This means that, by
spinning the resonator, optomechanical entanglement can be more robust against
thermal noises. In addition, by defining the difference of COM entanglement
for the opposite driving directions: $\Delta E_{\mathcal{N}}\equiv
E_{\mathcal{N},\circlearrowleft}-E_{\mathcal{N},\circlearrowright}$, we
demonstrate the dependence of quantum nonreciprocity on environmental
temperature in Figs. S3(e) and S3(f). Clearly, the condition $\Delta
E_{\mathcal{N}}\neq 0$ can be satisfied even at $\textrm{T}\sim
600\,\textrm{mK}$. Thereby, quantum nonreciprocity provides a new strategy to
protect quantum entanglement in a noisy environment, i.e., enhancing the
entanglement quality in a chosen (wanted) direction at the price of losing its
quality in the other (unwanted) direction. This new possibility, as far as we
know, has not been revealed in all previous works.

Figure S4: The influence of $Q$-factor on nonreciprocal optomechanical
entanglement. (a) The COM entanglement difference, $\Delta E_{\mathcal{N}}$,
versus the scaled optical detuning $\Delta_{c}/\omega_{m}$ and the quality
factor $Q$ at temperature $\textrm{T}=130\,\textrm{mK}$. (b) Density plot of
the logarithmic negativity $E_{\mathcal{N}}$ versus the environment
temperature T and the quality factor $Q$ at $\Delta_{c}/\omega_{m}=1.2$. (c)
The logarithmic negativity $E_{\mathcal{N}}$ as a function of the quality
factor $Q$ at $\Delta_{c}/\omega_{m}=1.6$. The rotation speed is chosen as
$\Omega=8\,\textrm{kHz}$ in (a)-(c). Other parameters are listed in Table 1.

In addition, the $Q$-factor of the resonator also affects the realization of
quantum nonreciprocity. In case of $J=0$, as shown in Fig. S4(a), it is found
that quantum nonreciprocity can persist for $Q\geq 10^{7}$ at
$\textrm{T}=130\,\textrm{mK}$ but its degree is lowered by decreasing the
values of $Q$. Fig. S4(b) shows that both lower $Q$-factor and higher
temperature are harmful for the robustness of COM entanglement. In addition,
it is seen that COM entanglement even completely vanishes at high temperatures
regardless of $Q$-factors. Moreover, in the presence of backscattering, e.g.,
$J=50\,\textrm{MHz}$, it is found that for the same values of $Q$ and T, by
spinning the resonator, nonreciprocal entanglement can always be better than
that in a nonspinning device [see Fig. S4(c)]. These results indicate that
considerable entanglement revival can be achieved with high $Q$-factor
spinning resonators.

## References

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11institutetext:  Sapienza University of Rome
11email<EMAIL_ADDRESS>22institutetext: Università degli Studi di Udine
22email<EMAIL_ADDRESS>
# Analyzing EEG Data with Machine and Deep Learning: A Benchmark

Danilo Avola 11 Marco Cascio 11 Luigi Cinque 11 Alessio Fagioli 11
Gian Luca Foresti 22 Marco Raoul Marini 11 Daniele Pannone 11

###### Abstract

Nowadays, machine and deep learning techniques are widely used in different
areas, ranging from economics to biology. In general, these techniques can be
used in two ways: trying to adapt well-known models and architectures to the
available data, or designing custom architectures. In both cases, to speed up
the research process, it is useful to know which type of models work best for
a specific problem and/or data type. By focusing on EEG signal analysis, and
for the first time in literature, in this paper a benchmark of machine and
deep learning for EEG signal classification is proposed. For our experiments
we used the four most widespread models, i.e., multilayer perceptron,
convolutional neural network, long short-term memory, and gated recurrent
unit, highlighting which one can be a good starting point for developing EEG
classification models.

###### Keywords:

Brain computer interfaces (BCI) Electroencephalography (EEG) Deep learning
Benchmark Classification

##  1 Introduction

In the last years, machine learning (ML) and deep learning (DL) techniques
have been used to overcome the limitation of classical image processing
approaches in several fields [1, 9]. The prominent results allowed researchers
and industries to get outstanding outcomes in different tasks, such as
environmental monitoring [7, 17, 5], medical and rehabilitation [18, 4, 3],
deception detection [6], and more [2]. In addition, ML and DL have also found
application for cross-domain problems, such as trading, biology, and
neuroscience. Despite these very different contexts, an ML/DL practitioner
commonly follows a well-defined pipeline. The latter usually consists in: 1)
choosing a model (or designing one from scratch), 2) setting its
hyperparameters, 3) training the chosen model, and finally 4) evaluating it.
Then, if the observed results are not convincing, the process is re-iterated
from 2) and, in the worst case, even from 1). Hence, to avoid changing or
redesigning the model, in order to speed up the process, it is useful to know
a priori the class of models that works the best for a certain problem or
typology of data.

For this reason, in this paper a benchmark on ML and DL models for EEG
classification is proposed. The motivation of providing a benchmark for such a
task is twofold. Firstly, EEG signals classification is increasingly used for
different applications, such as security and smart prosthesis [19, 21].
Secondly, to the best of our knowledge, there are no benchmarks available that
highlight which class of models works best with EEG data. The comparisons have
been performed by using four well-known ML and DL techniques, namely, the
multilayer perceptron (MLP) [26], the convolutional neural network (CNN) [30],
the long short-term memory (LSTM) [28], and the gated recurrent unit (GRU)
[8]. These approaches have been tested on the EEG motor movement/imagery
dataset [23], which contains EEG data of 109 volunteers performing different
tasks.

The remainder of this paper is organized as follows. Section 2 reports the
current state-of-the-art in EEG signal classification techniques. Section 3
provides a background on EEG signals and on the methods used for this
benchmark. Section 4 presents the performed experiments on the chosen dataset,
highlighting the best typology of models to perform EEG signals
classification. Finally, Section 5 concludes the paper.

##  2 Related Work

In the last decades, several methods have been developed to analyze EEG
signals and try to address different problems. Effective pipelines generally
perform data pre-processing procedures, to reduce noise and to clean signals,
a feature extraction step, to obtain a meaningful input representation, and an
inference phase to correctly associate signals to the addressed task.

Concerning data pre-processing procedures, several techniques can be employed
to remove artifacts from signals, such as filtering methods or wavelet
transforms [12]. In [28], for instance, MI-BCI signals are improved via the
fast Fourier transform (FFT) and stacked restricted Boltzmann machines (RBM).
The refined signals can then be fed to a CNN to extract meaningful features,
which are in turn analyzed and classified via an LSTM network. Differently, in
[20], the authors explore various procedures, e.g., discrete wavelet or
Fourier transforms, to polish the input signals. Subsequently, other
techniques, i.e., PCA and fuzzification, are also applied as pre-processing
to, respectively, reduce the input dimensionality and prepare the data for a
fuzzy decision tree (FDT) classifier to detect epileptic seizures. While these
approaches can be beneficial when analyzing EEG signals, other works
concentrate on raw data by defining specific architectures. In [31], for
example, an autoencoder is designed to reduce signal noise by exploiting brain
activity of multiple persons. The XGBoost algorithm is then applied to these
improved signals for MI classification.

Regarding the feature extraction step, it enables to capture diverse aspects
from BCI signals and can be performed in many different ways. In [27], for
example, an autoregressive discrete variational autoencoder (dVAE) is
implemented to retain dynamic signal information. A Gaussian mixture hidden
Markov model (GM-HMM) is then used to check for open or closed eyes. The
authors of [14], instead, detect epileptic seizures using an SVM by means of
multiscale radial basis functions (MRBF), and a modified particle swarm
optimization (MPSO), that can simultaneously improve the EEG signals
time–frequency extracted features associated to seizures. Differently, the
authors of [15] employ a filter bank common spatial pattern (FB-CSP) in
conjunction with a sliding window to extract spatial-frequency-sequential
features that are used to classify MI by a gated recurrent unit (GRU) network.

In relation to the inference phase, it can be performed using various
approaches such as machine or deep learning algorithms. In [16], for instance,
a relevance vector machine (RVM) is used to predict speech from BCI signals
represented through a Riemannian manifold covariance matrix. In [32], instead,
a support vector machine (SVM) is employed to classify gender discrimination,
alcholism and epilepsy from k-means clustered inter/intra BCI channels
dependencies, in the latter. As for DL methods, the authors of [29] address
the MI task exploiting the transfer learning paradigm, and fine-tune a pre-
trained architecture using short time Fourier transformed BCI signals. The
latter transform is also used in [25] to tackle the same MI task, although the
authors introduce a custom model called CNN-SAE to automatically extract and
combine time, frequency and location information from the input data. In [13],
finally, a compact depth-wise and separable CNN, which is one of the chosen
models used in this work benchmark, is designed to be effective on separate
tasks such as event related potential, feedback error-related negativity,
movement related cortical potential, as well as sensory motor rhythm
classifications.

To conclude this overview, although many works tend to develop a specific
methodology, others try to present a common ground by comparing existing
approaches on complex BCI EEG signals analysis tasks. The authors of [11], for
example, examine linear discriminant analysis (LDA), a MLP with a single
hidden layer, and SVM performances on $5$ different mental tasks, i.e., rest,
math, letter, rotate, and count. In [10], instead, the best combination to
detect epileptic seizures is searched among several wavelet transforms, i.e.,
Haar, Daubechies, Coiflets, and Biorthogonal, and either a SVM or
probabilistic neural network classifier (PNN). Therefore, inspired by these
works, and following their rationale, we present a benchmark with more
advanced deep learning algorithms to address another task, i.e., the MI-BCI
classification.

##  3 Background and Method

To provide a clearer overview concerning the context of the benchmark, this
section provides the necessary background needed to understand EEG signals. In
addition, a summary of ML and DL approaches used in the benchmark is provided.

###  3.1 Background on EEG Signals

The first step regarding the analysis of EEG data is acquiring the latter
during the execution of a specific task, i.e. a trial, from the subject
involved in the experiment. This acquisition is usually performed by placing
electrodes, which capture electrical signals emitted by human neurons, on
specific positions of the subject scalp, as shown in Fig. 1.

(a)

(b)

Figure 1: The a) different areas of the human brain, and b) the corresponding
10-20 map system.

These signals have an extremely low voltage (between $-70mV$ and $+40mV$),
where some peaks could determine if an action occurred. Indeed, the brain
activity is determined by sequences of such peaks. The entire EEG spectrum is
comprised in the range $[1Hz,50Hz]$ and can be divided in several frequency
bandwidths, i.e., the rhythms. Such rhythm, indicating different brain
activities, are defined as:

  * •

$\delta$ waves: representing deep sleep and pathological states, in the range
of $[0.5Hz,3Hz]$;

  * •

$\theta$ waves: indicating sleepiness and childhood memories, in the range of
$[3Hz,7Hz]$;

  * •

$\alpha$ waves: representing adult relaxing state (rest status), in the range
of $[8Hz,13Hz]$;

  * •

$\beta$ waves: which identify the active state, involving calculus, focus, and
muscular contraction, in the range of $[14Hz,30Hz]$;

  * •

$\gamma$ waves: indicating tension or strong emotional states, over $30Hz$.

Usually, the acquired signal presents different types of noise. Examples of
noise recurrent in EEG data is the muscular activity (such as eye blinks and
movements), external sources of electricity, electrodes movements or even
sounds. To sanitize the data, some well-known techniques, such as Individual
Component Analysis (ICA) [24], can be used. ICA decomposes the signals
obtained from multiple electrodes during time in a sum of temporally
independent (and fixed in terms of space) components. In particular, given an
input matrix $X$ of EEG signals, where the rows represent the electrodes and
the columns the time, the algorithm finds a matrix $W$ that decomposes and
linearly smooths the data. The rows $U=WX$ represent the activation times of
the ICA components, and have the form $components\times time$. The columns of
the inverse matrix $W^{-1}$ provide the projection forces related to their
respective components on each of the electrodes. In other words, these weights
provide a topographical image of the scalp for each electrode. In this way, it
is possible to locate brain areas involved in specific actions or feelings.

After data cleaning, it is easier to find the so-called event related
potential (ERP), which is the response to an internal or external event,
namely, a sensory, a cognitive, or a motor event. In addition, these events
can induce temporary modulations in certain frequency bandwidths, e.g. the
increment or decrement of the bandwidth oscillation power. These modulations
are called event related synchronization (ERS) and event related
desynchronization (ERD). Synchronization and desynchronization terms refer to
the fact that the temporary modulations are due to an increment or a decrement
of the synchronization of a neuron population. An example of ERD and ERS is
shown in Fig. 2.

Figure 2: Example of ERS and ERD. With an ERS, we have an increment of the
bandwidth power (filled in blue), while with an ERD we have a decrement of the
bandwidth power (filled in orange).

Since the acquired EEG signal is a summary of the several brain activities, to
get the signal regarding a specific stimulus, it is common practice to average
the EEG signals obtained from the application of the same stimulus several
times (e.g., 100 or more), allowing to remove disturbances not related to the
stimulus itself.

Remember that EEG signals are continuously recorded during the experiments.
Hence, the resulting data is organized as a 2D matrix having the form
$electrodes\times time$. To ease the analysis of EEG data in task-related
experiments, usually the process that goes under the name of epoching is
performed. The latter consists in re-arranging the 2D data in a 3D data
matrix, having the form $(electrodes,time,events)$.

Figure 3: Simplified examples of the used architectures. From top left to
bottom right: multilayer perceptron (MLP), convolutional neural network (CNN),
long short-term memory (LSTM) and the gate recurrent units (GRU). The depicted
architectures represent the base structure of each model, meaning that only
the main elements are present.

###  3.2 Summary on used Machine and Deep Learning Approaches

For the proposed benchmark, four well-known approaches were tested, namely,
MLP, CNN, LSTM, and GRU. These approaches were chosen since they are the most
representative of their category: MLP for classical ML, a standard CNN for DL
approaches that are not designed to work with temporal data, and LSTM and GRU
for DL approaches designed to work with temporal data. As it is possible to
see, two models that can correctly handle temporal data were chosen. This is
due to the fact that the final accuracy of those models strongly depends on
the input data, therefore there is no better model between them. In Fig. 3,
examples of the chosen approaches are shown.

The first approach we have chosen is MLP, which belongs to the class of
feedforward artificial neural networks, and is the most basic neural network
approach that can be used in classification task. The simplest MLP consists of
three layers, namely, an input layer, a hidden layer, and the output layer. In
the proposed benchmark, the used MLP has 1 input layer, 3 hidden layers, and 1
output layer.

CNN, originally designed for image analysis, can be used for analyzing EEG
without efforts. As the name suggests, CNN uses convolutions in order to
extract features from the input data. Depending on the data type, CNN may use
1D, 2D, or even 3D convolutions. In our benchmark, we used 1D convolutions
with kernels of size $(channels,1)$. This kernel size allowed analyzing
contemporary all the channels at each time instant. The used CNN consists of 4
convolutional layers and 1 dense layer for classification.

LSTM, differently from MLP and CNN, has the purpose of handling temporal data.
Belonging to the family of recurrent neural networks (RNN) [22], a single LSTM
unit is composed of a cell, that contains values over time intervals, and
several gates, namely, an input gate, an output gate, and a forget gate. The
aim of these gates is to control the flow of the information into and out of
the cell. This structure allows to overcome the vanishing gradient problem
that affects the training of standard RNNs. The LSTM used in the proposed
benchmark simply consists of 2 stacked LSTM units.

The last approach we are going to summarize is the GRU. As for LSTM, GRU
belongs to RNNs family. In fact, GRU can be seen as an optimized version of
the LSTM due to its different internal structure. In detail, GRU has gating
units allowing to control the information flow, but it lacks the memory cell.
The less complex structure makes GRU computationally more efficient with
respect to LSTM. As for the latter, in our benchmark we used 2 stacked GRU
units.

##  4 Experiments

In this section, the results of the benchmark, together with the used dataset,
are presented. Concerning the data, the EEG signals were handled with MNE-
Python, while all the tested methods were implemented in Pytorch. The machine
used for training the model consisted in an AMD Ryzen 7 CPU, 16GB DDR4 of RAM,
and an NVidia RTX 2080 GPU.

(a)

(b)

Figure 4: Examples of signals corresponding to an event. In Figure a), the
signal corresponding to the left hand is shown, while in b) the signal
corresponding to the right hand is depicted.

###  4.1 Dataset

For the experiments, the EEG-BCI Motor Imagery Dataset [23] has been used. It
consists of more than $1500$ records in European Data Format (EDF) files from
$109$ participants. The latter performed $14$ runs while wearing a BCI2000
device. At each run, the subject could perform one of the following actions:
open or close right or left hand, imagine opening or closing right or left
hand, open or close both hands or feet, imagine opening or closing both hands
or feet. The EDF files contain $64$ EEG signals, with a frequency of 160
samples per second. In the experiments, we have chosen arbitrarily to classify
signals related to right or left hand rather than distinguish signals of hands
from feet. Before training the several models, the data has been cleaned from
noise, and the data related to subjects having id $(43,88,89,92,100,104)$ was
removed. This is due to the fact that in some EEG signals the sampling
frequency was different, or there was too much noise in the data. The final
dimension of the dataset consisted in 3136 events to be classified. These
events are then splitted as follows: 1980 events for training, 216 events for
validation, and 940 for testing. Each event is fed as input in the form of a
2D matrix having the form $[channels,samples]$.

###  4.2 Results

In Fig. 4, it is possible to see an example of signals related to the left
(Fig. 4a) and right (Fig. 4b) hands. As it is possible to see, there are
interesting patterns within the acquired signal. In detail, it is possible to
observe that we have ERS in the intervals $[0,1]$ and $[4.5,5.4]$. Since these
patterns are present in all the considered data, we choose to perform the
experiments on different time intervals, which are the entire event, having
length of $[0,6]$, and the two ERS specified above. In Table 1, the results
obtained with the different models are shown. As it is possible to see, the
model that perform the best is the CNN, obtaining an accuracy of $90.4\%$ on
the interval $[0,6]$, followed by MLP, with an accuracy of $85.2\%$. The worst
results are obtained with the RNN models.

Figure 5: Activations of the electrodes during the runs. As it is possible to see, not every electrode provide relevant data. In this example, the electrodes on the top, i.e. the ones placed on the forehead, can be discarded since they acquire noisy data such as eye blinks. Table 1: Accuracy values obtained with the different models on the different intervals. In detail, we have that Interval 1 = $[0,1]$, Interval 2 = $[4.5,5.4]$, and Interval 3 = $[0,6]$. | Interval 1 | Interval 2 | Interval 3
---|---|---|---
MLP | 74.8% | 84.6% | 85.2%
CNN | 78.8% | 84.6% | 90.4%
LSTM | 65.4% | 78.6% | 57.9%
GRU | 65.7% | 85% | 76.9%

In detail, we have that GRU has obtained an accuracy of $76.9\%$, while the
LSTM has obtained an accuracy of $57.9\%$, resulting as the worst model in our
benchmark. Usually, RNNs performs better with respect to other methods when
dealing with temporal or sequential data. This is amenable to the fact that
the latter usually present some correlation among the several time instants.
It is clear from the reported results that this aspect cannot be assumed in
the case of brain signals. Instead, the several filters applied by the CNN
seem to perform a better abstraction of the data, allowing to obtain the
highest accuracy. This could be related to the fact that convolutions allow to
better identify specific patterns, e.g. peaks, within the data.

During the experiments, we noticed that some channels provided more reliable
data with respect to others. These channels are F7,F8, FT7, FT8, T9, TP7, TP8,
and FC, as shown in the example in Fig. 5. Hence, we re-trained the chosen
models by using only the above-mentioned channels, obtaining the results shown
in Table 2.

Table 2: Accuracy values obtained by using only a subset of the channels. Also in this case, we have that Interval 1 = $[0,1]$, Interval 2 = $[4.5,5.4]$, and Interval 3 = $[0,6]$. | Interval 1 | Interval 2 | Interval 3
---|---|---|---
MLP | 77.3% | 85.5% | 85.7%
CNN | 76.3% | 85.2% | 88.8%
LSTM | 60.3% | 82.2% | 65.2%
GRU | 67.1% | 84% | 78.7%

In this case, there is a general accuracy improvement among the several models
and intervals. Despite this, we have that the CNN still results the best
performing model with an accuracy score of $88.8\%$. Again, MLP follows with
an accuracy of $85.7\%$. Finally, we have GRU and LSTM with an accuracy of
$78.7\%$ and $65.2\%$, respectively.

##  5 Conclusions

In this paper, a benchmark of ML and DL models for EEG classification is
presented. Four well-known models in ML and DL are compared, namely MLP, CNN,
LSTM, and GRU. Extensive experiments have been performed on a dataset
containing EEG data of 109 volunteers performing different tasks, highlighting
that CNN is the best category of models for dealing with EEG data.

## Acknowledgments

This work was supported by the MIUR under grant “Departments of Excellence
2018–2022” of the Sapienza University Computer Science Department and the ERC
Starting Grant no. 802554 (SPECGEO).

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