# Performance Analysis of an Interference-Limited RIS-Aided Network

Liang Yang, Yin Yang, Daniel Benevides da Costa, and Imene Trigui L. Yang and
Y. Yang are with the College of Information Science and Engineering, Hunan
University, Changsha 410082, China<EMAIL_ADDRESS>yy19971417@163.com).D. B. da Costa is with the Department of Computer
Engineering, Federal University of Ceará, Sobral, CE, Brazil (email:
danielbcosta@ieee.org).I. Trigui is with the University of Quebec, Montreal,
QC, Canada (e-mail: trigui.imene17@gmail.com).

###### Abstract

In this work, the performance of reconfigurable intelligent surface
(RIS)-aided communication systems corrupted by the co-channel interference
(CCI) at the destination is investigated. Assuming Rayleigh fading and equal-
power CCI, we present the analysis for the outage probability (OP), average
bit error rate (BER), and ergodic capacity. In addition, an asymptotic outage
analysis is carried in order to obtain further insights. Our analysis shows
that the number of reflecting elements as well as the number of interferers
have a great impact on the overall system performance.

###### Index Terms:

Average bit error rate (BER), average channel capacity, co-channel
interference (CCI), outage probability (OP), reconfigurable intelligent
surfaces (RISs).

##  I Introduction

A new disruptive technology, called reconfigurable intelligent surface (RIS),
has recently emerged in the research community. The RIS is an artificial
surface composed of electromagnetic (EM) materials that is electronically
controlled by low-cost electronic devices. Owing to its unique functions, RIS
can efficiently customize the wireless environment, thereby maximizing the
signal quality at the receiver [1]. Compared with other competing
technologies, it is worth noting that RIS does not require encoding and
decoding operations during the signal transmission. In addition, it does not
create new waves and can change the shape of the wireless signal through soft
programming.

Recently, several works have investigated the incorporation of RISs in
wireless systems. Specifically, the authors in [2] studied the joint
optimization of transmitting source and RIS beams in multiple-input single-
output (MISO) communication systems. In [3], the authors studied the
optimization of joint transmission and reflected beamforming in RIS-assisted
multiuser systems. In [4], the application of RISs in downlink multiuser
communications was examined, while [5] provided an accurate analytical results
for the coverage, probability of signal-to-noise ratio (SNR) gain, and delay
outage rate of RIS-assisted wireless communication systems. Moreover, the
analysis in [6] showed that using RISs has a positive effect on improving the
secrecy performance of wireless systems. A performance analysis for the
application of RISs in a dual-hop free-space optical (FSO)/radio-frequency
(RF) system was carried out in [7]. Finally, [8] analyzed the performance of
RIS-aided unmanned aerial vehicle (UAV) communication relaying systems.
However, for all we know, the impact of co-channel interference (CCI) in RIS-
aided communication system remains to be investigated in the literature yet,
which motivates this new work.

In this paper, the performance of RIS-assisted interference-limited
communication systems with CCI at the destination is studied. To this end, we
first derive the statistical distribution of the effective signal-to-
interference ratio (SIR) expression and, based on it, exact analysis for the
outage probability (OP), average bit error rate (BER), and average channel
capacity are presented. To clearly show the effects of the interference and
power levels on the system performance, an asymptotic outage expression is
also derived. Our results reveal that the number of reflecting meta-surfaces
as well as the number of interferers have a significant impact on the overall
system performance.

##  II System and Channel Models

Figure 1: System model.

Our considered system model can be found in Fig. 1, where a single-antenna
source (S) intends to communicate with a single-antenna destination (D) via an
RIS formed by $N$ reflecting elements. The terminal D is affected by $L$
independent equal-power interferers. Such a scenario finds applicability in
cases where the RIS and S are located at the center of a cellular network,
whereas D is located at its border. We assume that direct link transmission
between S and D does not exist due to obstacles. Moreover, it is assumed that
the considered system operates in an interference-limited regime so that the
effect of noise on the overall performance can be negligible. Similar to [1],
we assume that the RIS has the ability to obtain the full channel state
information (CSI). Thus, the resulting signal at D can be written as

$y=\sqrt{P_{s}}\left[\sum_{i=1}^{N}h_{i}\rho_{i}g_{i}\right]x+\sum_{l=1}^{L}\sqrt{P_{I}}h_{l}x_{l},$
(1)

where $P_{s}$ denotes the average transmitted power per symbol at the S, $x$
is the source signal, $P_{I}$ denotes the transmit power of the interferers,
and $h_{l}$ and $x_{l}$ are, respectively, the channel coefficient suffering
from the Rayleigh fading and unit energy signal for the $l\rm th$ interferer.
In (1), $\rho_{i}=\omega_{i}(\phi_{i})e^{j\phi_{i}}$ denotes the reflection
coefficient generated by the $i$th reflecting element of the RIS, with
$\omega_{i}(\phi_{i})=1$ corresponding to ideal phase shifts
($i=1,2,\ldots,N$). In addition, $h_{i}$ and $g_{i}$ refer to the channel
coefficients of the S-RIS and RIS-D links, where
$h_{i}=\alpha_{i}e^{-j\theta_{i}}$ and $g_{i}=\beta_{i}e^{-j\varphi_{i}}$,
with $\theta_{i}$ and $\varphi_{i}$ denoting the respective phases of the
fading channel coefficients, and $\alpha_{i}$ and $\beta_{i}$ denote the
respective channels’ amplitudes, which are independent Rayleigh random
variables (RVs).

From [1], in order to achieve the maximum SIR, RIS can fully eliminate the
phase shifts by supposing $\phi_{i}=\theta_{i}+\varphi_{i}$. Therefore, the
maximum SIR is

$\gamma_{\rm
SIR}=\frac{P_{s}\left|\sum_{i=1}^{N}\alpha_{i}\beta_{i}\right|^{2}}{P_{I}\sum_{l=1}^{L}\left|h_{l}\right|^{2}}.$
(2)

Let $Y=P_{s}\left|\sum_{i=1}^{N}\alpha_{i}\beta_{i}\right|^{2}=P_{s}Z^{2}$.
From [9], the probability density function (PDF) of $Y$ can be modeled by a
squared $K_{G}$ distribution, i.e.,

$f_{Y}(y)=\frac{2A^{k+m}}{\Gamma(k)\Gamma(m)P_{s}^{\frac{k+m}{2}}}y^{\left(\frac{k+m}{2}-1\right)}K_{k-m}\left(2A\sqrt{\frac{y}{P_{s}}}\right),$
(3)

where $A=\sqrt{km/\Omega}$, $\Omega\overset{\triangle}{=}E(Z^{2})$ is the mean
power, $k$ and $m$ are the shaping parameters, $K_{\nu}(\cdot)$ is the
modified Bessel function of the second kind with zero order [10], and
$\Gamma(\cdot)$ denotes the gamma function [10]. Now, let
$X=P_{I}\sum_{l=1}^{L}\left|h_{l}\right|^{2}$. From [11], we know that $X$ is
a chi-squared RV with $2L$ degrees of freedom and whose PDF is given by

$f_{X}(x)=\frac{1}{P_{I}^{L}\Gamma(L)}x^{L-1}e^{-\frac{x}{P_{I}}}.$ (4)

Using (3) and (4), the PDF of $\gamma_{\rm SIR}$ can be written as

$\displaystyle f_{\gamma_{\rm SIR}}(\gamma)=$
$\displaystyle\int_{0}^{\infty}xf_{Y}(\gamma x)f_{X}(x)dx$ $\displaystyle=$
$\displaystyle\frac{A^{k+m}}{\Gamma(k)\Gamma(m)P_{s}^{\frac{k+m}{2}}\Gamma(L)}\gamma^{\frac{k+m}{2}-1}P_{I}^{\frac{k+m}{2}}$
$\displaystyle\times$ $\displaystyle
G_{1,2}^{2,1}\left[\left.\frac{A^{2}\gamma
P_{I}}{P_{s}}\right|\begin{matrix}1-L-\frac{k+m}{2}\\\
\frac{k-m}{2},\frac{m-k}{2}\end{matrix}\right],$ (5)

where $G_{c,d}^{a,b}\left[\cdot\right]$ denotes the Meijer G-function defined
in [10]. Furthermore, the cumulative distribution function (CDF) of
$\gamma_{\rm SIR}$ can be shown to be given by

$\displaystyle F_{\gamma_{\rm SIR}}(\gamma)=$
$\displaystyle\frac{A^{k+m}}{\Gamma(k)\Gamma(m)P_{s}^{\frac{k+m}{2}}\Gamma(L)}\gamma^{\frac{k+m}{2}}P_{I}^{\frac{k+m}{2}}$
$\displaystyle\times$ $\displaystyle
G_{2,3}^{2,2}\left[\left.\frac{A^{2}\gamma
P_{I}}{P_{s}}\right|\begin{matrix}1-\frac{k+m}{2},1-L-\frac{k+m}{2}\\\
\frac{k-m}{2},\frac{m-k}{2},-\frac{k+m}{2}\end{matrix}\right].$ (6)

##  III Performance Analysis

In this section, we investigate the OP, average BER, and average channel
capacity of our considered model under the impact of equal-power CCI. In
addition, we carry out an asymptotic analysis by assuming high $P_{s}$ and
$P_{I}$ to get further insights, which may be helpful for the system design.

###  III-A Outage Probability

_1\. Exact Analysis_

From [12], the definition of OP is the probability that the end-to-end SIR
$\gamma_{\rm SIR}$ is lower than a preset threshold $\gamma_{\rm th}$, which
is mathematically written as $P_{out}=\Pr(\gamma_{\rm SIR}<\gamma_{\rm th})$.

By inserting $\gamma=\gamma_{\rm th}$ into (6), the OP can be written as

$P_{out}=F_{\gamma_{\rm SIR}}(\gamma_{\rm th}).$ (7)

_2\. Asymptotic Analysis_

According to [13], note that the PDF of $Y$ can be formulated as
$f_{Y}(y)=\Phi y^{\iota}+o(y)$, where $\Phi$ is a positive constant, $\iota$
represents the quantization of the smoothing order of $f_{Y}(y)$ at the
origin, and $o(y)$ denotes the higher order terms. Therefore, one can obtain
an asymptotic PDF expression for (3) as

$f_{Y}(y)\simeq\frac{2A^{k+m}}{\Gamma(k)\Gamma(m)P_{s}^{\frac{k+m}{2}}}y^{\left(\frac{k+m}{2}-1\right)},$
(8)

and its corresponding asymptotic PDF expression as

$f_{\gamma_{\rm
SIR}}(\gamma)\simeq\frac{2A^{k+m}\Gamma(\frac{k+m}{2}+L)P_{I}^{\frac{k+m}{2}}}{\Gamma(k)\Gamma(m)P_{s}^{\frac{k+m}{2}}\Gamma(L)}\gamma^{\left(\frac{k+m}{2}-1\right)}.$
(9)

Therefore, the asymptotic outage expression can be derived as

$P_{out}\simeq\frac{4A^{k+m}\Gamma(\frac{k+m}{2}+L)P_{I}^{\frac{k+m}{2}}}{\Gamma(k)\Gamma(m)P_{s}^{\frac{k+m}{2}}\Gamma(L)(k+m)}\gamma_{\rm
th}^{\frac{k+m}{2}}.$ (10)

The above expression clearly shows that the diversity order is
$\frac{k+m}{2}$, where $k$ and $m$ denote the pair of conjugate complex
numbers, and are determined by $N$. As expected, by increasing the number of
interferers $L$ or the interference power $P_{I}$ will lead to an increase in
the OP, thereby reducing the system performance. In addition, by increasing
the power of S will make the system performance better.

###  III-B Average BER

The performance metric BER is a main indicator to evaluate the accuracy of the
signal transmission. From [14], the average BER can be expressed as

$P_{e}=\frac{q^{p}}{2\Gamma(p)}\int_{0}^{\infty}e^{-q\gamma}\gamma^{p-1}F_{\gamma_{\rm
SIR}}(\gamma)d\gamma,$ (11)

where different values of $p$ and $q$ correspond to various modulation
schemes. For instance, differential phase shift keying (DPSK) ($q=1$ and
$p=1$), binary phase shift keying (BPSK) ($q=\frac{1}{2}$ and $p=1$), and
binary frequency shift keying (BFSK) ($q=\frac{1}{2}$ and $p=\frac{1}{2}$). In
this work, we consider the DPSK and the BPSK schemes.

From (6) and (11), and using [15, Eq.(07.34.21.0088.01)], the average BER can
be evaluated as

$\displaystyle P_{e}=$
$\displaystyle\frac{q^{-\frac{k+m}{2}}A^{k+m}P_{I}^{\frac{k+m}{2}}}{2\Gamma(p)\Gamma(k)\Gamma(m)P_{s}^{\frac{k+m}{2}}\Gamma(L)}$
$\displaystyle\times$ $\displaystyle
G_{3,3}^{2,3}\left[\left.\frac{A^{2}P_{I}}{P_{s}q}\right|\begin{matrix}1-p-\frac{k+m}{2},1-\frac{k+m}{2},1-L-\frac{k+m}{2}\\\
\frac{k-m}{2},\frac{m-k}{2},-\frac{k+m}{2}\end{matrix}\right].$ (12)

###  III-C Average Channel Capacity

From [16], we have

$C=\frac{1}{\rm ln(2)}\int_{0}^{\infty}\rm
ln(1+\gamma)\emph{f}_{\gamma_{SIR}}(\gamma)\emph{d}\gamma.$ (13)

For a simple calculation of the capacity, one can apply a useful identity in
[15, Eq.(01.04.26.0003.01)], i.e., $\rm
ln(1+\gamma)=\emph{G}_{2,2}^{1,2}\left[\left.\gamma\right|\begin{matrix}1,1\\\
1,0\end{matrix}\right]$. By combining [15, Eq.(07.34.21.0011.01)] together
with (5) and (13), the average capacity can be evaluated as

$\displaystyle C=$
$\displaystyle\frac{A^{k+m}}{\Gamma(k)\Gamma(m)P_{s}^{\frac{k+m}{2}}\Gamma(L)\rm
ln(2)}\gamma^{\frac{k+m}{2}}P_{I}^{\frac{k+m}{2}}$ $\displaystyle\times$
$\displaystyle
G_{3,4}^{4,2}\left[\left.\frac{A^{2}P_{I}}{P_{s}}\right|\begin{matrix}1-L-\frac{k+m}{2},-\frac{k+m}{2},1-\frac{k+m}{2}\\\
\frac{k-m}{2},\frac{m-k}{2},-\frac{k+m}{2},-\frac{k+m}{2}\end{matrix}\right].$
(14)

##  IV Numerical Results and Discussions

In this section, we select some numerical examples to verify the impact of the
main system parameters on the system performance. Our analysis is confirmed by
Monte Carlo simulations, which generates $10^{5}$ simulation points. In the
next figures, the SIR threshold is set to $\gamma_{\rm th}=20\,\rm dB$.

Figure 2: Outage probability versus $P_{s}$ for different number of $N$.

In Fig. 2, we present the OP performance of our considered system with various
values of $N$, and by setting the number of interferers $L=4$, with all
interferers having the same power, i.e., $P_{I}=1\,\rm dB$. From this figure,
one can find that the analytical results match the simulation results
perfectly. In addition, the simulation results of the ideal and practical
phase shifts are given to observe the performance hit. As mentioned in [3],
one can use $\omega_{i}(\phi_{i})=(1-\varpi_{\rm min})\left(\frac{\rm
sin(\phi_{\emph{i}}-\kappa)+1}{2}\right)^{\varsigma}+\varpi_{\rm min}$ to
represent the performance loss when considering the practical phase shifts,
where $\varpi_{\rm min}$ stands for the minimum amplitude, $\kappa$ represents
the horizontal distance between $\frac{\pi}{2}$ and $\varpi_{\rm min}$, and
$\varsigma$ denotes the steepness of function curve. We consider the setup
$\varpi_{\rm min}=0.8$, $\kappa=0.43\pi$, and $\varsigma=1.6$. One can clearly
see that there is a certain performance difference between the practical and
ideal cases. Finally, it can be observed that applying large values of $N$ can
result in a better system performance.

Figure 3: Outage probability versus $P_{s}$ for different values of $P_{I}$.

In Fig. 3, we plot the OP curves for various values of $P_{I}$ and by setting
$L=4$ and $N=10$. As can be seen, increasing $P_{I}$ leads to a deterioration
of the OP performance. In Fig. 4, the OP is plotted for various values of $L$
and assuming $P_{I}=1\,\rm dB$ and $N=10$. The results show that increasing
the amount of interference reduces the system performance. From both Figs. 3
and 4, one can be clearly seen that all curves have the same slopes, which
means that the existence of interference does not affect the diversity order
of the considered system, corroborating the fact that the diversity order
depends on $N$. Also, at high $P_{s}$, the asymptotic results are close to the
exact values.

In Fig. 5, we plot the BER curves for both DPSK and BPSK schemes when
$P_{I}=1\,\rm dB$, $L=4$ and $N=10$. One can clearly observe that the BPSK
scheme has a better system performance.

Finally, Fig. 6 depicts the average capacity of our considered system for
various values of $N$ and assuming $L=8$ and $P_{I}=1\,\rm dB$. Again, it can
be clearly seen that the analytical results match perfectly the simulation
ones. Moreover, the system performance improves as $N$ increases.

Figure 4: Outage probability versus $P_{s}$ for different numbers of co-
channel interferers. Figure 5: Average BER versus $P_{s}$ for different
modulation schemes. Figure 6: Average capacity versus $P_{s}$ for different
number of $N$.

##  V Conclusion

In this paper, we presented a performance analysis for RIS-assisted
communication systems with equal power CCI. We first performed an accurate
analysis of the OP, average BER, and average capacity. The results showed that
our analysis results match the simulation ones perfectly. In order to clearly
study the effect of parameters on the system performance, we derived an
asymptotically closed-expression for the OP. The results revealed that the
achievable diversity order equals to $\frac{k+m}{2}$, and the number of
interferers has a great influence on the system performance.

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# MROS: A framework for robot self-adaptation

Gustavo Rezende Silva  ,  Darko Bozhinoski  ,  Mario Garzon Oviedo  ,  Mariano
Ramírez Montero  ,  Nadia Hammoudeh Garcia  ,  Harshavardhan Deshpande  ,
Andrzej Wasowski  and  Carlos Hernandez Corbato

###### Abstract.

Self-adaptation can be used in robotics to increase system robustness and
reliability. This work describes the Metacontrol method for self-adaptation in
robotics. Particularly, it details how the MROS (Metacontrol for ROS Systems)
framework implements and packages Metacontrol, and it demonstrate how MROS can
be applied in a navigation scenario where a mobile robot navigates in a
factory floor. Video: https://www.youtube.com/watch?v=ISe9aMskJuE

Self-adaptive systems, Self-adaptation, Metacontrol, MROS, Robotics

##  1\. Introduction

Autonomous robots are usually programmed by integrating and configuring
individual robot capabilities in a governing static control architecture.
However, static control architectures fall short when addressing context
variability in open-ended, dynamic environments, where internal errors also
compromise the quality and autonomy of mission execution. The demand for
higher autonomy in robotic systems requires that robots can decide and switch
between available alternative configurations, acting as a self-adaptive
system. The addition of self-adaptation capabilities to robotic systems could
be performed by including the adaptation logic into the robot’s application
logic. However, this leads to the problems: (1) the adaptation logic being
tangled up with application logic, making it difficult to individually change
any of the separate logic; (2) the lack of reuse of the adaptation reasoning,
requiring it to be reprogrammed for each application. Thus, it is advantageous
to use methods and tools that enable the separation of concerns and promote
reuse. This paper demonstrate how Metacontrol (Corbato, 2013) can be applied
with the MROS (Bozhinoski et al., 2022) tooling to enable self-adaptation in
robotic systems.

Metacontrol (Corbato, 2013) is a framework that incorporates systems with the
capability to self-adapt to maintain their functionalities at an expected
performance. Metacontrol has the design goals of being reusable and
extensible. This is achieved with the design principles: (1) separating the
adaptation and application reasoning; (2) exploiting at runtime the
engineering knowledge of how the system is designed to reason how and when the
system needs to adapt, i.e., by being model-based.

MROS (Bozhinoski et al., 2022) is a tooling that implements the Metacontrol
framework for ROS-based (Quigley et al., 2009) systems. This paper
demonstrates how MROS111This paper address the ROS1 version of MROS, which can
be found at: https://github.com/meta-
control/mc_mros_reasoner/tree/mros1-master lowers the barrier for robot
developers to add self-adaptation to a robot control architecture by
leveraging Metacontrol’s design principles to promote reuse and extensibility.
In this paper, the novel contributions are:

  * •

A methodology for robotic developers to effectively use the MROS tooling to
add self-adaptation capabilities in a robotic system.

  * •

A case study that demonstrates the value of the MROS tooling and the proposed
methodology.

##  2\. Metacontrol

Metacontrol (Corbato, 2013) is a reference architecture that provides systems
with the capability to self-adapt to maintain their functionalities at an
expected performance, despite external disturbances, faults, and unexpected
behavior. For this, it uses engineering knowledge of how the system is
designed, in the form of a runtime model, to reason when and how the system
needs to adapt. Metacontrol is further explored in the works (Hernández et
al., 2015; Hernández et al., 2018; Hernandez Corbato et al., 2019; Aguado et
al., 2021; Bozhinoski et al., 2022; Päßler et al., 2022).

Metacontrol follows the standard model in self-adaptive systems (Figure 2)
that separates the application into a Managed system, which consists of the
robotic domain specific components, and a Managing subsystem, which is a
Metacontroller component that closes a feedback loop with the Robot subsystem
to monitor and adapt it when necessary.

The Metacontroller operates following the MAPE-K loop (Kephart and Chess,
2003), which consists of a Monitor, Analyze, Plan and Execute steps, that are
based on a Knowledge Base (KB). The Monitor step is responsible for measuring
relevant quality attributes for the functionalities of the system. The Analyze
step is responsible for deciding whether the Managed system needs to
reconfigure. When it is necessary to perform reconfiguration, the Plan step is
responsible for selecting a new configuration for the Managed system that
satisfies its requirements. The Execute step is responsible for reconfiguring
the Managed system.

The KB is the main difference of Metacontrol with other self-adaptation
frameworks. It consists of a runtime model based on the TOMASys metamodel.
TOMASys contains concepts to describe the functional and physical architecture
of a system, and its variants, both at design time and at runtime. For the
robotic developer using MROS, only TOMASys design time concepts are needed to
create the Metacontrol KB, while the runtime elements are used by an automatic
reasoner in the Metacontroller. A more detailed description of TOMASys can be
found in (Corbato, 2013; Aguado et al., 2021).

The Function element represents an abstract functionality of the system, such
as navigating from point A to B. A Function Design is an engineering design
solution that solves a specific Function with an expected performance. Quality
Attribute Type and Quality Attribute Value are used to capture the systems
engineering meaning of QAs. A Quality Attribute Type represents a
characteristic of the system that shall be observed, such as energy. And a
Quality Attribute Value represents an amount of a Quality Attribute Type,
e.g., 1 Joule.

Figure 1. Metacontrol

Figure 2. TOMASys

##  3\. MROS framework

MROS is a ROS-based implementation of Metacontrol that enables architectural
self-adaptation in ROS-based robotic systems. MROS monitors the state of the
ROS system and updates its KB according to it, then it uses ontological
reasoning to Analyze and Plan the required adaptations, which are then
executed by reconfiguring the ROS node graph. This section first describes how
the MROS library implements the MAPE-K loop and the Metacontrol KB, and then
it presents the methodology that robot developers can follow to use it to
implement self-adaptation in ROS 1 applications222There is already a beta
version of MROS supporting ROS 2, but it requires a different method.

###  3.1. MROS library

MROS follows the MAPE-K model for runtime adaptation, as shown in in Figure 3.

Figure 3. MROS framework

Monitor: This component is realized with the ROS node _rosgraph_monitor_, it
provides system observers templates. The observer nodes operate at a fixed
frequency and publish the monitored values to the Metacontrol reasoner using
the standard ROS diagnostic mechanism (i.e., a specific channel and data
structure in ROS systems). This means they easily integrate into an existing
ROS system and might use or provide services already needed in the system,
regardless of the presence of a Metacontroller.

Knowledge Base: The Knowledge Base consists of (1) the TOMASys metamodel that
allows modeling the functional architecture of autonomous systems; (2) A
TOMASys model of the Managed subsystem that defines its unique Functions,
Function Designs and its expected Quality Attributes, in other words, all
possible configurations for the Managed subsystem. The KB is implemented as an
ontology using the Ontology Web Language (OWL) in combination with the
Semantic Web Rule Language (SWRL).

Analyze and Plan: These components are realized with the ROS node
_mros1_reasoner_, it integrates the OWL ontology, i.e. the KB, with ROS and it
reasons over it to decide when and how to adapt the Managed subsystem. It
consists of a ROS node implemented with Python that makes use of the library
OwlReady2 to bridge the ontology with Python and ROS. And the reasoning is
performed with the off-the-shelf ontological reasoner Pellet.

Execute: This step is realized with the ROS node
_mc_rosgraph_manipulator_. It provides a ROS node that is responsible for
killing and starting new ROS nodes and changing the necessary ROS parameters
to fulfill the desired configuration.

###  3.2. MROS methodology

To use MROS to add self-adaptation to a ROS system, the activities in Figure 4
may be followed.

Figure 4. MROS Design time activities

Step 1: Define the possible adaptations, i.e. set of architectural variants
relevant for the robot to be able to perform a mission. For each architectural
variant, the developer creates a ROS launch file for the automatic deployment
of the corresponding configuration of ROS nodes.

Step 2: Create the KB by modeling the ROS system architectural variants with
TOMASys. MROS provides an implementation of TOMASys in OWL, the developer
needs to create the individuals for the application’s Functions, Function
Designs and its expected Quality Attribute Values and Quality Attribute Types.

Step 3: Create observers using the templates provided in the ROS package
_rosgraph_monitor_ to monitor the status of the active ROS nodes and to
measure the relevant Quality Attributes.

Step 4: Configure the _mros1_reasoner_ and _mc_rosgraph_manipulator_ nodes by
linking the system architectural variants (step 1) with TOMASys Function
Designs (step 2), and connecting them to application-specific reconfiguration
actions (e.g. to store required node states). This is done simply editing a
template Metacontroller configuration file in YAML.

##  4\. Case study

This section describes how MROS can be set up for a robot navigating on a
factory floor following the MROS methodology described in Section 3.2. This
case study comes for the previous work of Bozhinoski et al. (Bozhinoski et
al., 2022). The experiment code can be found at https://github.com/rosin-
project/metacontrol_sim.

#### Case study description

The case study consists of a Clearpath Ridgeback mobile robot that navigates
in a factory floor. The robot is equipped with two laser sensors, one IMU, and
an odometry system. The navigation system is realized with the ROS1 navigation
stack. As the robot navigates, unexpected obstacles may appear in it’s path
and it may get closer or further to objects, which causes its safety quality
attribute level to change. When safety levels are high (lower risk of
collision), the robot can navigate with higher speed and acceleration. When
safety levels are low (higher risk of collision), it needs to use lower
speeds. In addition, throughout the mission, the battery level diminishes and
with this the robot must navigate with lower speed and acceleration to save
energy. MROS is used to adapt at runtime the navigation parameters, such as
maximum speed and acceleration, to satisfy safety and energy constraints.

###  4.1. Application to MROS to the case study

Step 1: Identify the architectural variants corresponding to different
configurations (parameter values) of the main node of the ROS1 navigation
stack, and create its corresponding launch files. In total, 27 different
function designs were defined. A snippet of the ROS launch file of one
Function Design is shown in listing 1. The parameters that have been
specifically defined for this Function Design are: max_vel_x, max_vel_y,
acc_lim_x, acc_lim_y, qa_safety, and qa_energy. The QA values are specified as
ROS parameters to enable, if necessary, to change them at runtime. The QA
values defined in the launchfiles override the ones defined in the ontology.

Listing 1: Function design 1 (out of 27)

⬇

1<launch>

2 <param name="qa_safety" value="0.7"/>

3 <param name="qa_energy" value="0.33"/>

4 <node pkg="move_base" type="move_base" name="move_base"

5 cwd="node" respawn="false" output="screen">

6 <param name="TrajectoryPlannerROS/max_vel_x"

7 value="0.3" />

8 <param name="TrajectoryPlannerROS/max_vel_y"

9 value="0.3" />

10 <param name="TrajectoryPlannerROS/acc_lim_x"

11 value="3.6" />

12 <param name="TrajectoryPlannerROS/acc_lim_y"

13 value="3.6" />

14 </node>

15</launch>

Step 2: Model the architectural variants with the MROS TOMASys ontology. To
create the OWL file for the KB, the graphical tool Protegé can be used to
simplify the process. The TOMASys metamodel is available in the package
mc_mdl_tomasys. It is only necessary to set up the application-specific
ontology by creating individuals of the design time TOMASys classes (Figure
2).

For this case study, the following individuals are created:

  * •

A Function individual for the navigation capability;

  * •

Quality Attribute Type individuals for both safety and energy quality
attributes;

  * •

Function Design individuals for each variant that solves navigation, including
object property individuals of Quality Attribute Value with their Quality
Attribute Type and a data field with the expected QA value.

Step 3: Create observers with the templates provided in the ROS package
rosgraph_monitor. These templates are two Python classes called TopicObserver
and ServiceObserver that implement the general functionalities needed to
monitor ROS topics and services, respectively. For each specific quality
attribute that needs to be monitored, it is necessary to implement a new class
that inherits from one of them.

For the use case, two Observers are implemented: SafetyQualityObserver and
EnergyQualityObserver. A snippet of the implementation of the latter can be
seen in listing 2. The class EnergyQualityObserver inherits from
TopicObserver. In the initialization of the class, the topic to which the
observer needs to subscribe and its message type are defined. In line 4, it is
defined that it needs to subscribe to the topic /power_load to retrieve
information about the battery, and that its message type is a float. For all
observers, the method calculate_attribute must be overloaded, it is
responsible for performing any necessary calculation with the data received
via the topic defined in the initialization, it must return the final data as
a key-value pair structured as a diagnostic_msgs/DiagnosticArray message. In
the EnergyQualityObserver, in line 9, the method calculates the battery level
as a normalized value, and returns it, for example, as $\\{energy,0.9\\}$. The
output of the observers is published in the topic diagnostics.

Listing 2: EnergyQualityObserver Observer snippet

⬇

1class EnergyQualityObserver(TopicObserver):

2 def __init__(self, name):

3 #topic to observe and msg type

4 topics = [("/power_load", Float32)]

5 super(EnergyQualityObserver,

6 self).__init__(name, 10, topics)

7

8 def calculate_attr(self, msgs):

9 status_msg = DiagnosticStatus()

10

11 #normalized calculus for energy

12 attr =(msgs[0].data - 0.2)/(5.0-0.2)

13 print("normalized energy: {0}".format(str(attr)))

14

15 status_msg = DiagnosticStatus()

16 status_msg.level = DiagnosticStatus.OK

17 status_msg.name = self._id

18 status_msg.values.append(KeyValue("energy", str(attr)))

19 status_msg.message = "QA status"

20 return status_msg

Step 4: Configure Metacontrol through a yaml file. This is used to map each
Function Design defined in the ontology (step 2) to its respective launch file
(step 1). Additionally, to indicate which ROS nodes are killed and spawned
during the reconfiguration process, as well, what actions and goals must be
saved to be reset when the reconfiguration is performed. A snippet of the
configuration file of this use case can be seen in listing 3.

Listing 3: Metacontrol configuration file

⬇

1reconfiguration_action_name: 'rosgraph_manipulator_action_server'

2configurations:

3 f1_v1_r1:

4 command: roslaunch f1_v1_r1 f1_v1_r1.launch

5 f1_v1_r2:

6 command: roslaunch f1_v1_r2 f1_v1_r2.launch

7 f1_v1_r3:

8 command: roslaunch f1_v1_r3 f1_v1_r3.launch

9

10kill_nodes: ['/move_base']

11save_action: 'move_base'

12goal_msg_type: move_base_msgs.msg.MoveBaseAction

###  4.2. Results

Bozhinoski et al. (Bozhinoski et al., 2022) show that by adding self-
adaptation with MROS to this use case, the robot performance increases
regarding the amount of time it violates its required safety and energy
quality attributes, and the overall mission success. In average, safety
violations decreases from 2.5% to 0.96%, energy violations from 2.98% to
1.86%, and mission success increases from 65.20% to 78.50%.

##  5\. Related Work

Aldrich et al. leverages predictive data models to enable automated robot
adaptation to changes in the environment at run-time (Aldrich et al., 2019).
While the approach depicts the benefits of using models by capturing high-
level artifacts, it makes it extremely challenging for a ROS developer to make
use of them in robotic scenarios because: (1) it does not introduce models
that can be reused for a different application; (2) it does not give insights
on how to build similar models; (3) it does not provide infrastructure to
leverage those models.

Cheng et al. propose a framework that uses GSN assurance case models to manage
run-time adaptations for ROS systems (Cheng et al., 2020). The framework
integrates assurance information from GSN models to ROS specific information
to guide runtime monitoring and adaptation. It uses custom-developed libraries
specific to the approach, rather than standard libraries in ROS (such as ROS
Diagnostics) raising the entry barrier for ROS developers to effectively use
it.

##  6\. Discussion and future works

This paper describes MROS - a tool that enables robots to perform self-
adaptation at runtime based on ontological reasoning. MROS establishes generic
self-adaptation mechanisms that drive self-adaptation through the MAPE-K
reference feedback loop. This eases the process of designing self-adaptation
for robots since it only requires users to define the proper observers, the
Managed system ontological model conforming to TOMASys, a few configuration
files, and the launch files for each architectural variant. Due to its
reusability and extensibility, MROS has been used to handle different
adaptation concerns in different robotic applications, such as reliable
propulsion and motion control in underwater robots (Aguado et al., 2021),
contingency handling in mobile manipulators (Bozhinoski et al., 2022), and
enhanced safety and energy saving in the navigation of a mobile robot
(Bozhinoski and Wijkhuizen, 2021).

#### Acknowledgments

This work was supported by the European Union’s Horizon 2020 Framework
Programme through the MSCA network REMARO (Grant Agreement No 956200), the
grant RobMoSys-ITP-MROS (Grant Agreement No. 732410) and the ROSIN project
(Grant Agreement No. 732287).

## References

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  * Aguado et al. (2021) Esther Aguado, Zorana Milosevic, Carlos Hernández, Ricardo Sanz, Mario Garzon, Darko Bozhinoski, and Claudio Rossi. 2021.  Functional self-awareness and metacontrol for underwater robot autonomy.  _Sensors_ 21, 4 (2021), 1210.
  * Aldrich et al. (2019) Jonathan Aldrich, David Garlan, Christian Kästner, Claire Le Goues, Anahita Mohseni-Kabir, Ivan Ruchkin, Selva Samuel, Bradley Schmerl, Christopher Steven Timperley, Manuela Veloso, et al. 2019\.  Model-based adaptation for robotics software.  _IEEE Software_ 36, 2 (2019), 83–90.
  * Bozhinoski et al. (2022) Darko Bozhinoski, Mario Garzon Oviedo, Nadia Hammoudeh Garcia, Harshavardhan Deshpande, Gijs van der Hoorn, Jon Tjerngren, Andrzej Wasowski, and Carlos Hernandez Corbato. 2022\.  MROS: runtime adaptation for robot control architectures.  _Advanced Robotics_ 36, 11 (2022), 502–518.
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# Quantized Nonlinear Transport with Ultracold Atoms

Fan Yang  Institute for Advanced Study, Tsinghua University, Beijing 100084,
China  Hui Zhai<EMAIL_ADDRESS>Institute for Advanced Study, Tsinghua
University, Beijing 100084, China

###### Abstract

In this work we propose a protocol to measure the quantized nonlinear
transport using two-dimensional ultracold atomic Fermi gases in a harmonic
trap. This scheme requires successively applying two optical pulses in the
left and lower half-planes and then measuring the number of extra atoms in the
first quadrant. In ideal situations, this nonlinear density response to two
successive pulses is quantized, and the quantization value probes the Euler
characteristic of the local Fermi sea at the trap center. We investigate the
practical effects in experiments, including finite pulse duration, finite edge
width of pulses, and finite temperature, which can lead to deviation from
quantization. We propose a method to reduce the deviation by averaging
measurements performed at the first and third quadrants, inspired by symmetry
considerations. With this method, the quantized nonlinear response can be
observed reasonably well with experimental conditions readily achieved with
ultracold atoms.

Topological physics lies at the center of modern condensed matter physics [1,
2, 3, 4, 5, 6, 7]. Transport is the most direct tool to detect topological
numbers. So far, quantized conductance is mainly limited to one-dimensional
metallic systems, and the conductance is quantized to the number of tunneling
channels [8, 9]. This quantized conductance has been observed in both
condensed matter systems [10, 11, 12, 14, 13] and ultracold atomic gases [15,
16, 17]. A two-dimensional band insulator with nontrivial topology hosts
stable metallic edge states, whose quantized conductance reveals the
topological invariant of the insulating bulk [1, 2, 3]. These include the
quantum Hall effect [18, 19], the quantum anomalous Hall effect [20, 21], and
the quantum spin Hall effect [22, 23, 24].

In a recent paper, Kane proposed a novel effect, which measures a topological
invariant of the Fermi sea through transport [25, 26]. The novelty of this
effect comes from the following aspects. First, this effect concerns a two-
dimensional metal instead of a one-dimensional one. Secondly, this effect
considers the nonlinear conductance instead of the linear conductance.
Thirdly, this effect probes the Euler characteristic $\chi$ of the Fermi sea,
a topological invariant that has not been extensively discussed in dimension
higher than one before.

Formally, the Euler characteristic $\chi$ is defined as the alternating sum of
the Betti numbers [27]

$\chi=\sum_{n=0}^{d}(-1)^{n}b_{n},$ (1)

where the $n$th Betti number $b_{n}$ is the rank of the $n$th homology group,
and $d$ is the dimension of the manifold. In Morse theory, the Euler
characteristic can be computed by evaluating the Hessian at each critical
point. In the case of the Fermi sea, we consider a dispersion $E({\bf k})$
filled up to the Fermi energy, $\chi$ can be expressed more concretely using
Morse theory as [25, 27] (This expression will be used in the Appendix.)

$\chi=\sum\text{sgn}\det\Bigg{[}\frac{\partial^{2}E}{\partial k_{i}\partial
k_{j}}\Bigg{]}\Bigg{\lvert}_{{\bf v}=0}.$ (2)

Here sgn is the sign function, $\text{det}[\cdot]$ denotes the determinant,
and ${\bf v}=\nabla_{\bf k}E/\hbar$. ${\bf v}=0$ denotes critical points (i.e.
local minima, maxima, or saddle points of the dispersion). The summation runs
over all these critical points below the Fermi surface. In one dimension,
$\chi$ counts the number of disconnected components of the Fermi sea. In two
dimensions, Each electron-like, hole-like, and open Fermi surface contributes
$+1$, $-1$, and 0 to $\chi$, respectively. For instance, as shown in Fig.
1(b), $\chi$ always equals one for a simple quadratic dispersion. Adding spin-
orbit coupling along one spatial direction changes the dispersion, resulting
in $\chi=2$ at low density. As density increases, the system undergoes a
Lifshitz transition to $\chi=1$ [28, 29]. Another nontrivial $\chi$ realized
in ultracold atom system is the Dirac dispersion in the honeycomb lattice
[30]. When the chemical potential is above the Dirac point, there are two
electron-like Fermi pockets, giving rise to $\chi=2$, and when the chemical
potential is below the Dirac point, there are two hole-like Fermi pockets,
giving rise to $\chi=-2$. In one-dimension, the famous Landauer formula states
that the linear conductance of the ballistic transport is quantized to the
Euler characteristic. However, for dimensions higher than one, only a recent
work [25] discovered that the nonlinear conductance of the ballistic transport
is quantized to the Euler characteristic.

Figure 1: (a) Schematic of the experimental scheme. The green area (left half-
plane) and the blue area (lower half-plane) are the areas where two optical
pulses are applied. The yellow area (the first quadrant) is the regime where
the number of excess atoms is measured. We set $t_{1}<t_{2}<t_{3}$. (b)
Schematic of various dispersions and Fermi seas with different Euler
characteristics. The circles/ellipses illustrate the topology of the Fermi
sea.

Two methods have been proposed to detect the quantized nonlinear conductance
in Ref. [25]. However, neither has been implemented so far. The first method
is based on measuring frequency-dependent ac transport, which is, in
principle, doable in condensed matter systems. However, it is still an open
question whether this effect is robust enough against local disorders and
electron interactions that are inevitably present in solids. The second method
is a thought experiment, meaning it is nearly impossible to implement in
solid-state materials. Here we point out that the ultracold Fermi gases offer
an ideal platform to realize this thought experiment. Moreover, since
ultracold atom systems can be disorder-free, and the $s$-wave scattering
length between atoms can also be tuned to zero to minimize the interaction
effect, this fulfills the requirement of ballistic transport. However, the
critical issue is whether this effect is robust enough against practical
effects in ultracold atom systems, such as the external confinement potential,
the temporal and spatial profiles of the optical pulses, and the thermal
effect. Addressing these effects becomes the key for ultracold atom
observation of this quantization. After considering all these practical
effects, we conclude that this quantization can be readily observed within the
current capability of ultracold atom experiments.

Experimental Scheme. Before entering the detailed derivations, we first
describe our measurement scheme as follows. We start with a non-interacting
spinless degenerate Fermi gas in a two-dimensional harmonic trap

$V({\bf r})=\frac{1}{2}m(\omega^{2}_{x}x^{2}+\omega^{2}_{y}y^{2}),$ (3)

where $m$ is the mass of atoms, and $\omega_{x}$ and $\omega_{y}$ are harmonic
trap frequencies along $\hat{x}$\- and $\hat{y}$-directions, respectively.

This proposal contains applying two pulses, one applied on the left-half plane
with $x<0$ at time $t_{1}$ and the other applied on the lower-half plane with
$y<0$ at time $t_{2}$, as shown in Fig. 1. Both pulses apply a uniform
potential, which read

$\displaystyle V_{1}$ $\displaystyle=\xi h\delta(t-t_{1})\Theta(-x),$ (4)
$\displaystyle V_{2}$ $\displaystyle=\xi h\delta(t-t_{2})\Theta(-y).$ (5)

Here $\Theta(\cdot)$ is the Heaviside step function, and $h$ is the Planck
constant. $\xi$ is an integer. For typical ultracold atom experiments, one can
choose $\xi$ from several to tens. Ideally, these pulses are $\delta$
functions in the temporal domain and step functions across $x=0$ or $y=0$. In
practice, the pulse should have a finite duration time $\sigma_{t}$, and we
require that the temporal integration of the potential strength equals $\xi
h$. The step function is also smoothed into a domain with edge width
$\sigma_{r}$. Here we first present the general scheme and we will investigate
the effect of finite $\sigma_{t}$ and $\sigma_{r}$ later.

Step One. Starting from the equilibrium state, we apply a pulse on the left-
half plane with $x<0$ at time $t_{1}$, and then measure the atom number in the
first quadrant at time $t_{3}$, subtracted by $\overline{N}=N/4$ with $N$
being the total number of atoms,

$\int_{0}^{\infty}dx\int_{0}^{\infty}dyn(x,y,t_{3})-\overline{N}.$ (6)

The result denoted by $N_{x}$.

Step Two. Starting again from the equilibrium state, we apply a pulse on the
lower-half plane with $y<0$ at time $t_{2}$, and then measure the atom number
in the first quadrant at time $t_{3}$, subtracted by $\overline{N}$ using the
same expression as Eq. 6. We denote this result as $N_{y}$.

Step Three. Starting again from the equilibrium state, we first apply a pulse
on the left-half plane with $x<0$ at time $t_{1}$, and then apply another
pulse on the lower-half plane with $y<0$ at time $t_{2}$. We measure the atom
number in the first quadrant at time $t_{3}$, subtract by $\overline{N}$ to
obtain $N_{xy}$ using Eq. 6.

Then, we can obtain a value

$\chi=\frac{N_{xy}-N_{x}-N_{y}}{\xi^{2}}.$ (7)

Ideally, $\chi$ should be quantized and the quantization value equals the
Euler characteristic of the Fermi sea. The practical effects, such as finite
temperature, finite $\sigma_{r}$, and finite $\sigma_{t}$, lead to derivation
from the quantized value. To reduce this derivation, we can proceed to Step
Four.

Step Four. Repeating Step One to Step Three with the same pulses, the
difference is to measure $N_{x}$, $N_{y}$, and $N_{xy}$ in the third quadrant
instead of the first quadrant. This yields another $\chi^{\prime}$ by using
Eq. 7. Then, we average $\chi$ and $\chi^{\prime}$, and this leads to a value
closer to quantized Euler characteristic.

We note that, according to Eq. 7, although $\chi$ should be quantized to a
value of the order of unity, the error bar of measured atom numbers $N_{x}$,
$N_{y}$ and $N_{xy}$ do not need to be of the order of unity. Instead, the
error bar only needs to be smaller than $\xi^{2}$. At the end of this work, we
will show that $\xi\sim 5$ is good enough to reach good quantization. Such a
high accuracy of measuring atom number is challenging, but it is already
within the capability of current ultracold atomic experiments [31].

Proof of the Scheme. We first prove the scheme for ideal situations. We
consider the collisionless Boltzmann equation of noninteracting fermions in a
harmonic trap,

$\partial_{t}f+{\bf v}_{\bf k}\cdot\nabla_{\bf r}f+\frac{1}{\hbar}({\bf
F}-\nabla_{\bf r}V)\cdot\nabla_{\bf k}f=0,$ (8)

where $f({\bf r},{\bf k})$ is the distribution function. ${\bf v}_{\bf k}$ is
the velocity given by $(1/\hbar)\nabla_{\bf k}\epsilon_{\bf k}$. For the
purpose of illustration, we first consider the simple situation of the
quadratic dispersion $\epsilon_{\bf k}=\hbar^{2}{\bf k}^{2}/(2m)$. ${\bf F}$
is the force given by these two pulses, and ${\bf F}={\bf F}_{1}$ or ${\bf
F}_{2}$ for the first two steps, and ${\bf F}={\bf F}_{1}+{\bf F}_{2}$ for the
third step. ${\bf F}_{1}=\xi h\delta(t-t_{1})\delta(x)\hat{{\bf x}}$, and
${\bf F}_{2}=\xi h\delta(t-t_{2})\delta(y)\hat{{\bf y}}$. At $t=0$, $f({\bf
r},{\bf k})$ is the semi-classical Fermi distribution

$f_{0}({\bf r},{\bf k})=n_{\text{F}}(\epsilon_{{\bf k}}+V({\bf r})-\mu),$ (9)

where $n_{\text{F}}(E)=1/(e^{\beta E}+1)$ is the Fermi distribution function
with $\beta=1/(k_{\text{B}}T)$ being the inverse temperature. It is easy to
show that Eq. 9 satisfies Eq. 8 when ${\bf F}=0$.

To solve Eq. 8, we first introduce a mapping between $({\bf r}^{\prime},{\bf
k}^{\prime})$ and $({\bf r},{\bf k})$ as

$\displaystyle{\bf r}^{\prime}_{i}={\bf r}_{i}\cos(\omega_{i}\delta
t)-\frac{\hbar{\bf k}_{i}}{m\omega_{i}}\sin(\omega_{i}\delta t);$ (10)
$\displaystyle{\bf k}^{\prime}_{i}={\bf k}_{i}\cos(\omega_{i}\delta
t)+\frac{m\omega_{i}{\bf r}_{i}}{\hbar}\sin(\omega_{i}\delta t),$ (11)

where $i=x,y$ and $\delta t=t-t^{\prime}$. Below we will repeatedly use this
mapping with different choices of $\delta t$. We will utilize the fact that
when ${\bf F}=0$, $f({\bf r},{\bf k},t)=f({\bf r}^{\prime},{\bf
k}^{\prime},t^{\prime})$. This is ensured by the Liouville’s theorem, and the
energy conservation $\epsilon_{{\bf k}}+V({\bf r})=\epsilon_{{\bf
k}^{\prime}}+V({\bf r}^{\prime})$. Hence, when $t<t_{1}$, $f({\bf r},{\bf
k})=f_{0}({\bf r}^{\prime},{\bf k}^{\prime})=f_{0}({\bf r},{\bf k})$, and the
distribution function does not evolve.

Figure 2: $\chi$ computed from Eq. 7 following the first three steps for the
ideal situation, i.e. pulses with temporal profile of $\delta$-functions and
spatial profile of $\Theta$-functions applied to Fermi gas in a harmonic trap
at zero temperature. We choose $\omega_{x}=\omega_{y}$ for (a) and
$\omega_{x}=0.8\omega_{y}$ for (b). $t_{1}$, $t_{2}$ and $t_{3}$ are labelled
in Fig. 1.

For Step One, note that the force ${\bf F}_{1}$ only acts on atoms at the
boundary $x=0$ and at an instantaneous time $t_{1}$. The distribution function
changes from $f_{0}({\bf r}_{1},{\bf k}_{1})$ to $f_{0}({\bf r}_{1},{\bf
k}_{1})+f_{1}({\bf r}_{1},{\bf k}_{1})$ at $t=t_{1}$, and $f_{1}({\bf
r}_{1},{\bf k}_{1})$ is given by

$f_{1}({\bf r}_{1},{\bf k}_{1})=-2\pi\xi\delta(x_{1})\frac{\partial f_{0}({\bf
r}_{1},{\bf k}_{1})}{\partial k_{1x}}.$ (12)

After that, when we measure $N_{x}$ at time $t_{3}$, the distribution function
$f({\bf r},{\bf k})$ at $t=t_{3}$ is given by $f_{0}({\bf r}_{1},{\bf
k}_{1})+f_{1}({\bf r}_{1},{\bf k}_{1})$, where $({\bf r}_{1},{\bf k}_{1})$ and
$({\bf r},{\bf k})$ are related by the mapping Eq. 10-11 with $\delta
t=t_{31}=t_{3}-t_{1}$. Then, it is easy to see that

$N_{x}=\frac{1}{(2\pi)^{2}}\int_{\text{1q}}d^{2}{\bf r}\int d^{2}{\bf
k}f_{1}({\bf r}_{1},{\bf k}_{1}),$ (13)

where “1q” stands for the first quadrant. Similarly, for Step Two, we can
obtain that at time $t=t_{2}$, the distribution function is changed to
$f_{0}({\bf r}_{2},{\bf k}_{2})+f_{2}({\bf r}_{2},{\bf k}_{2})$, with

$f_{2}({\bf r}_{2},{\bf k}_{2})=-2\pi\xi\delta(y_{2})\frac{\partial f_{0}({\bf
r}_{2},{\bf k}_{2})}{\partial k_{2y}}.$ (14)

And we have

$N_{y}=\frac{1}{(2\pi)^{2}}\int_{\text{1q}}d^{2}{\bf r}\int d^{2}{\bf
k}f_{2}({\bf r}_{2},{\bf k}_{2}),$ (15)

where $({\bf r}_{2},{\bf k}_{2})$ is related to $({\bf r},{\bf k})$ by Eq.
10-11 with $\delta t=t_{32}=t_{3}-t_{2}$.

Now we consider Step Three. Following the same discussion, it can be shown
that $f({\bf r},{\bf k})$ at time $t=t_{3}$ contains two parts. The first part
is $f_{0}({\bf r}_{1},{\bf k}_{1})+f_{1}({\bf r}_{1},{\bf k}_{1})$, where
$f_{1}({\bf r}_{1},{\bf k}_{1})$ given by Eq. 12 is due to the force ${\bf
F}_{1}$, and $({\bf r}_{1},{\bf k}_{1})$ and $({\bf r},{\bf k})$ are related
by the mapping Eq. 10-11 with $\delta t=t_{31}$. The second part is
$f_{2}({\bf r}_{2},{\bf k}_{2})+f_{12}({\bf r}_{2},{\bf k}_{2})$ due to the
force ${\bf F}_{2}$, where

$f_{12}=(2\pi)^{2}\xi^{2}\delta(y_{2})\frac{\partial}{\partial
k_{2y}}\left[\delta(x_{1})\frac{\partial f_{0}}{\partial k_{1x}}\right],$ (16)

where $({\bf r}_{2},{\bf k}_{2})$ and $({\bf r},{\bf k})$ are related by the
mapping Eq. 10-11 with $\delta t=t_{32}$, and $({\bf r}_{2},{\bf k}_{2})$ and
$({\bf r}_{1},{\bf k}_{1})$ are related by the same mapping with $\delta
t=t_{21}=t_{2}-t_{1}$. Compared with Eq. 13 and 15, it is easy to see that
$\chi$ defined by Eq. 7 is given by

$\chi=\int_{\text{1q}}d^{2}{\bf r}\int d^{2}{\bf
k}\delta(y_{2})\frac{\partial}{\partial
k_{2y}}\left[\delta(x_{1})\frac{\partial f_{0}}{\partial k_{1x}}\right].$ (17)

This counts for the number of extra atoms cooperatively kicked by the two
pulses to the first quadrant, that is, they are first kicked by the first
pulse to the right side, and then kicked by the second pulse to the first
quadrant. Therefore, this is attributed to the nonlinear conductance.

For the harmonic trap, Eq. 17 can be evaluated exactly and the results
acquires a simple expression as

$\chi=\text{sgn}[\sin(\omega_{x}t_{31})\sin(\omega_{y}t_{32})],$ (18)

where $\text{sgn}[\cdot]$ stands for the sign function. We plot a typical
result in Fig. 2. The extra atoms kicked by the pulses evolve in time under
the force from the harmonic trap. When $t_{31}<\pi/\omega_{x}$ and
$t_{32}<\pi/\omega_{y}$, these extra atoms reside in the first quadrant, which
result in quantized value $\chi=1$, consistent with the Euler characteristic
of the Fermi sea for the quadratic dispersion. For longer evolution time,
$\chi$ is either $+1$ or $-1$. This is because inside a harmonic trap, atoms
oscillate with the same frequency despite their different velocities.

If the dispersion is not the simple quadratic one, such as in the presence of
spin-orbit coupling, or if the trap is not harmonic, the simple expression of
Eq. 18 no longer holds. To this end, we consider the situation that the time
separation between two pulses is short enough such that we can take ${\bf
r}_{1}\approx{\bf r}_{2}$ and ${\bf k}_{1}\approx{\bf k}_{2}$. We also
consider that $t_{31}$ and $t_{32}$ are both sufficiently short. That is to
say, if we require $x>0$ at time $t_{3}$ and $x_{1}=0$ at time $t_{1}$, it
means $v_{1x}>0$. Similarly, if $y>0$ at time $t_{3}$ and $y_{2}=0$ at time
$t_{2}$, it means $v_{2y}>0$. Hence, after the spatial integration we can
effectively replace $\delta(x_{1})$ and $\delta(y_{2})$ by $\Theta(v_{1x})$
and $\Theta(v_{2y})$. Furthermore, by ${\bf v}_{1}\approx{\bf v}_{2}$, Eq. 17
recovers the same expression as the uniform situation discussed in Ref. [25].
Following the same derivation presented in Ref. [25], Eq. 17 is the Euler
characteristic of the two-dimensional Fermi sea. (See Appendix for detailed
derivation.) Thus, it can yield other integers of $\chi$ for nontrivial
dispersions. To be more precise, note that in an external trap, we can
introduce the local Fermi sea determined by the local chemical potential
$\mu({\bf r})=\mu-V({\bf r})$. Hence, for short time separation $t_{21}$ and
$t_{32}$, Eq. 17 probes the Euler characteristic of the local Fermi sea at
${\bf r}\approx 0$, which is the intersection between the interfaces of the
two pulses. Furthermore, it is easy to see that our scheme can be generalized
by varying the areas acted by the two pulses, and we can detect the Euler
characteristic of local Fermi sea at different locations.

Figure 3:  How $\chi$ is affected by the pulse duration $\sigma_{t}$, the
finite width of the edge $\sigma_{r}$, and the temperature $T$. Here we fix
$\omega_{x}=\omega_{y}=\omega$, and $l_{0}$ is the corresponding harmonic
length. (a) $\chi$ as a function of the pulse duration $\sigma_{t}$
(schematically shown in the inset). We fix $\sigma_{r}/l_{0}=2.0$,
$k_{\text{B}}T=0.1\mu$, and $\xi=5$. (b) $\chi$ as a function of the edge
width $\sigma_{r}$ (schematically shown in the inset). We fix
$\omega\sigma_{t}=0.2$, $k_{\text{B}}T=0.1\mu$, and $\xi=5$. (c) $\chi$ as a
function of the temperature $T$. We fix $\omega\sigma_{t}=0.2$ and
$\sigma_{r}/l_{0}=2.0$, and $\xi=5$. (d) $\chi$ for different pulse strength
$\xi$. We fix $\omega\sigma_{t}=0.2$, $\sigma_{r}/l_{0}=2.0$, and
$k_{\text{B}}T=0.1\mu$. The red squares only measure the first quadrant and
the blue dots average over the first and the third quadrants.

By symmetry, if we measure the atom number in the third quadrant, it is
equivalent to measuring atom number in the first quadrant but with $\xi$
replaced by $-\xi$ for both pulses. For ideal situations, since $\chi$ is a
nonlinear response purely proportional to $\xi^{2}$, we can conclude that
$\chi$ takes the same value for the first and the third quadrant. If we
measure the atom number in the second or the fourth quadrants, it equals to
the atom number measured in the first quadrant but with $\xi$ replaced by
$-\xi$ for one of the two pulses. Thus, $\chi$ takes the opposite value if
measured in the second or the fourth quadrants. This is also evident in Fig.
2. Fig. 2 shows that $\chi$ is either $+1$ or $-1$. In Fig. 2(a), when
$\omega_{x}=\omega_{y}$ and when $t_{2}\rightarrow t_{1}$, the dynamical
trajectory of atoms always lie in the first and the third quadrants,
therefore, $\chi$ is always $+1$.

Practical Considerations. Now we turn to more realistic effects in
experiments. We will take into account the effects of finite temperature and
finite edge width and duration of pulses. For example, we consider a cloud of
a total of $N\sim 10^{4}$ atoms confined to a two-dimensional isotropic trap
with frequency $\omega_{x}=\omega_{y}=\omega=2\pi\times 100$Hz, which
corresponds to a harmonic length $l_{0}=\sqrt{\hbar/(m\omega)}\approx 1.6\mu$m
for 40K. We set the chemical potential $\mu=100\hbar\omega$, which gives a
cloud radius $\approx 14l_{0}$. We approximate the $\delta$-functions with
Gaussian function, namely

$\delta(t-t_{i})\rightarrow\frac{1}{\sqrt{2\pi}\sigma_{t}}e^{-(t-t_{i})^{2}/2\sigma_{t}^{2}},$
(19)
$\displaystyle\delta(x)\rightarrow\frac{1}{\sqrt{2\pi}\sigma_{r}}e^{-\frac{x^{2}}{2\sigma_{r}^{2}}},\
\
\delta(y)\rightarrow\frac{1}{\sqrt{2\pi}\sigma_{r}}e^{-\frac{y^{2}}{2\sigma_{r}^{2}}}.$
(20)

We numerically compute $\chi$ defined by Eq. 7 by solving the collisionless
Boltzmann equation [32]. We consider the quadratic dispersion and take $\omega
t_{1}=0.4\pi$, $\omega t_{2}=0.8\pi$, and $\omega t_{3}=1.2\pi$. In the ideal
situation, this yields $\chi=1$. Considering the practical effects, $\chi$
deviates from unity as shown by red squares in Fig. 3.

In ideal situations, ${\bf F}_{i}$ ($i=1,2$) is a $\delta$-function pulse.
Therefore, ${\bf F}_{1}$ only acts on $f_{0}$, giving rise to $\ f_{1}$. ${\bf
F}_{2}$ acts on $f_{0}$ and $f_{1}$, giving rise to $f_{2}$ and $f_{12}$,
respectively. In practical situations, because ${\bf F}_{i}$ has finite
duration, ${\bf F}_{1}$ also influences on $f_{1}$, and ${\bf F}_{2}$ also
influences on $f_{2}$ and $f_{12}$. That is to say, $f_{12}$ contains
contribution beyond $\xi^{2}$ order. The high-order corrections is dominated
by $\sim\xi^{3}$ order contribution for small $\sigma_{t}$ and small $\xi$ and
it contributes to a significant part of the total deviation. Due to the
symmetry argument presented above, for the $\sim\xi^{n}$ order contribution
with odd $n$, the measurements performed in the first and the third quadrants
take opposite values. Hence, we propose a method to reduce the deviation. This
method is to measure $\chi$ and $\chi^{\prime}$ in the first and the third
quadrants, respectively, and then average over these two results, as described
by Step Four. The averaged results are shown by blue dots in Fig. 3. As one
can see, after the average, $\chi$ can be $\sim 0.95$ for very practical
experimental conditions, such as $\omega\sigma_{t}=0.2$,
$\sigma_{r}/l_{0}=2.0$, $k_{\text{B}}T=0.1\mu$, and $\xi=5$. These conditions
can be routinely reached within current ultracold atom experimental
capability. Moreover, since the applied pulses are a far-detuned weak
potential, we expect the atom loss to be negligible in this process. On the
other hand, here, we choose the total number of atoms to be $\sim 10^{4}$,
which requires a higher accuracy in measuring atom number. One can also reduce
the total atom number, but this will require more stringent requirements on
experimental parameters, since a smaller Fermi surface is much more fragile.
In a real experiment, one can further optimize these parameters to reach the
best balance.

Summary and Outlook. Ultracold atomic gas is an ideal platform to
experimentally observe the recently proposed quantized nonlinear conductance
in Ref. [25]. The key message of this work is to show that this effect can be
observed with practical experimental conditions, calling for immediate
experimental implementations. The experimental observation will enable the
study of the interaction effect on quantized nonlinear transport in a highly
controllable way. The Euler characteristic of Fermi sea is also related to
topological multipartite entanglement entropy of Fermi liquids [33]. Further
exploration of Euler characteristic can advance our understanding of the role
of topology in high-dimensional Fermi liquids and enrich the topological
physics.

Acknowledgment. We thank Zhe-Yu Shi, Chengshu Li, Yanting Cheng, and Qi Gu for
helpful discussions. The project is supported by Beijing Outstanding Young
Scholar Program, Innovation Program for Quantum Science and Technology
2021ZD0302005, NSFC Grant No. 11734010 and the XPLORER Prize. F.Y. is also
supported by Chinese International Postdoctoral Exchange Fellowship Program
(Talent-introduction Program) and Shuimu Tsinghua Scholar Program at Tsinghua
University.

## Appendix

Here we elaborate on the short-time response for an arbitrary dispersion and
trap profile. We show that in the short-time limit, one recovers the
expression for Euler characteristics derived in Ref. [25].

In the limit $t_{31},t_{32}\to 0$, we can linearize the trajectory of motion

$\displaystyle x_{1}\approx x-t_{31}v_{1x},$ (21) $\displaystyle y_{2}\approx
y-t_{32}v_{2y},$ (22)

and use ${\bf k_{1}}\approx{\bf k_{2}}\approx{\bf k}$ in the derivatives. In
this way, we obtain an asymptotic form of $\chi$,

$\chi=\int_{\text{1q}}d^{2}{\bf r}\int d^{2}{\bf k}\delta(y-t_{32}v_{2y})\\\
\frac{\partial}{\partial k_{y}}\left[\delta(x-t_{31}v_{1x})\frac{\partial
f_{0}}{\partial k_{x}}\right].$ (23)

Next, we perform the integration over the 1st quadrant. The $\delta$-functions
put a constraint on the velocity and generate two step functions
$\Theta(v_{2y})$ and $\Theta(v_{1x})$, giving

$\chi=\int d^{2}{\bf k}\Theta(v_{2y})\frac{\partial}{\partial
k_{y}}\Bigg{[}\Theta(v_{1x})\frac{\partial f_{0}}{\partial k_{x}}\Bigg{]}$
(24)

Further utilizing ${\bf v_{1}}\approx{\bf v_{2}}\approx{\bf v}$ and performing
integration by parts, we arrive at

$\chi=-\int d^{2}{\bf k}\Theta(v_{x})\frac{\partial\Theta(v_{y})}{\partial
k_{y}}\frac{\partial f_{0}}{\partial k_{x}}.$ (25)

The above equation, which is obtained in the short time limit
$t_{31},t_{32}\to 0$ for arbitrary dispersion and trap profile, is the same as
the expression for the uniform case in Ref. [25].

In the rest of the Appendix, we follow Ref. [25] and show that Eq. 25 is
indeed the Euler characteristic of the Fermi sea. First one can add a null
term to Eq. 25

$\int d^{2}{\bf k}\Theta(v_{x})\frac{\partial\Theta(v_{y})}{\partial
k_{x}}\frac{\partial f_{0}}{\partial k_{y}}.$ (26)

This term can be rewritten using the chain rule as

$\begin{split}&\int d^{2}{\bf
k}\Theta(v_{x})\frac{\partial\Theta(v_{y})}{\partial v_{y}}\frac{\partial
v_{y}}{\partial k_{x}}\frac{\partial f_{0}}{\partial E}\frac{\partial
E}{\partial k_{y}}\\\ &=\int d^{2}{\bf k}\Theta(v_{x})\frac{\partial
v_{y}}{\partial k_{x}}\frac{\partial f_{0}}{\partial E}\hbar
v_{y}\delta(v_{y}).\end{split}$ (27)

The term $v_{y}\delta(v_{y})$ in the integrand ensures the null term to be
zero.

After adding the null term and performing integration by parts, one gets

$\begin{split}\chi&=\int d^{2}{\bf
k}f_{0}\Bigg{[}\frac{\partial\Theta(v_{x})}{\partial
k_{x}}\frac{\partial\Theta(v_{y})}{\partial
k_{y}}-\frac{\partial\Theta(v_{x})}{\partial
k_{y}}\frac{\partial\Theta(v_{y})}{\partial k_{x}}\Bigg{]}\\\ &=\int d^{2}{\bf
k}f_{0}\delta({\bf v})\Bigg{[}\frac{\partial v_{x}}{\partial
k_{x}}\frac{\partial v_{y}}{\partial k_{y}}-\frac{\partial v_{x}}{\partial
k_{y}}\frac{\partial v_{y}}{\partial k_{x}}\Bigg{]}.\end{split}$ (28)

Because of the factor $f_{0}$ and $\delta({\bf v})$, only points below the
Fermi surface with ${\bf v}=0$ contribute to the integration. One can then
perform the change of variables by replacing the momentum integration with a
velocity integration, which introduces a Jacobian determinant

$J=\det\Bigg{[}\frac{\partial(k_{x},k_{y})}{\partial(v_{x},v_{y})}\Bigg{]},$
(29)

giving

$\chi=\int d^{2}{\bf v}f_{0}\delta({\bf v})|J|\Bigg{[}\frac{\partial
v_{x}}{\partial k_{x}}\frac{\partial v_{y}}{\partial k_{y}}-\frac{\partial
v_{x}}{\partial k_{y}}\frac{\partial v_{y}}{\partial k_{x}}\Bigg{]},$ (30)

with $|J|$ the absolute value of $J$.

Noticing that the terms inside the bracket is simply the inverse of the
Jacobian determinant, we get

$\chi=\sum\text{sgn}\det\Bigg{[}{\frac{\partial(v_{x},v_{y})}{\partial(k_{x},k_{y})}}\Bigg{]}\Bigg{\lvert}_{{\bf
v}=0},$ (31)

where the summation runs over all critical points with ${\bf v}=0$ below the
Fermi surface. Finally, utilizing ${\bf v}=\nabla_{\bf k}E/\hbar$, we arrive
at the expression

$\chi=\sum\text{sgn}\det\Bigg{[}\frac{\partial^{2}E}{\partial k_{i}\partial
k_{j}}\Bigg{]}\Bigg{\lvert}_{{\bf v}=0}.$ (32)

This is exactly the formula for the Euler characteristics in the Morse theory
[25, 27].

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