# Generic transverse stability of kink structures in atomic and optical
nonlinear media with competing attractive and repulsive interactions

S. I. Mistakidis  Department of Physics, Missouri University of Science and
Technology, Rolla, MO 65409, USA  G. Bougas  Department of Physics, Missouri
University of Science and Technology, Rolla, MO 65409, USA  G. C. Katsimiga
Department of Physics, Missouri University of Science and Technology, Rolla,
MO 65409, USA  P. G. Kevrekidis  Department of Mathematics and Statistics,
University of Massachusetts Amherst, Amherst, MA 01003-4515, USA

###### Abstract

We demonstrate the existence and stability of one-dimensional (1D) topological
kink configurations immersed in higher-dimensional bosonic gases and nonlinear
optical setups. Our analysis pertains, in particular, to the two- and three-
dimensional extended Gross-Pitaevskii models with quantum fluctuations
describing droplet-bearing environments but also to the two-dimensional cubic-
quintic nonlinear Schrödinger equation containing higher-order corrections to
the nonlinear refractive index. Contrary to the generic dark soliton
transverse instability, the kink structures are generically robust under the
interplay of low-amplitude attractive and high-amplitude repulsive
interactions. A quasi-1D effective potential picture dictates the existence of
these defects, while their stability is obtained through linearization
analysis and direct dynamics in the presence of external fluctuations
showcasing their unprecedented resilience. These generic (across different
models) findings should be detectable in current cold atom and optics
experiments. They also offer insights towards controlling topological
excitations and their usage in topological quantum computers.

### Introduction.

Kink solitons (alias domain walls) are nonlinear excitations encountered in
disparate disciplines ranging from optical Kartashov  _et al._ (2006), and
magnetic media Pérez-Junquera  _et al._ (2008); Buijnsters  _et al._ (2014),
living cellular structures Ivancevic and Ivancevic (2013); Yomosa (1983);
Caspi and Ben-Jacob (1999), folding protein chains Hu  _et al._ (2011a, b) to
atomic gases Tylutki  _et al._ (2020); Katsimiga  _et al._ (2023a) and
cosmology Vilenkin and Shellard (2000). Their topological character has been
recently unveiled in two-dimensional (2D) van der Waals materials Lee  _et
al._ (2023), opening up the possibility of robust computations Wang  _et al._
(2022). Stabilization of multidimensional spatially localized states is a
fundamental challenge of scientific interest in physics and beyond Kartashov
_et al._ (2019). It is well-known that 1D topological (e.g. dark solitons) and
non-topological (bright solitons) defects suffer from the so-called snake-
instability Tikhonenko  _et al._ (1996); Mamaev  _et al._ (1996); Kivshar and
Pelinovsky (2000); Anderson  _et al._ (2001) and infrared catastrophe Bergé
(1998); Fibich (2015), respectively, once embedded in 2D and three-dimensional
(3D) geometries. Mechanisms of suppression of the ensuing instabilities of
defects have also been discussed. These mainly consider nonuniform media,
i.e., incorporating external trapping geometries Proukakis  _et al._ (2004);
Wen  _et al._ (2013); Kevrekidis  _et al._ (2004), or are accompanied by
nonlocal interactions Skupin  _et al._ (2006); Lin  _et al._ (2008); Nath  _et
al._ (2008), or accounting for the active competition of higher-order
nonlinearities Zeng and Zeng (2020).

Cold atoms are ideal quantum many-body simulators, i.e., platforms featuring
remarkable tunability of system parameters, such as e.g. interactions and
external traps Bloch  _et al._ (2008); Lewenstein  _et al._ (2012). In this
context, ultradilute and incompressible self-bound states of matter, referred
to as quantum droplets Luo  _et al._ (2021); Mistakidis  _et al._ (2023) were
recently experimentally detected in Bose mixtures Cheiney  _et al._ (2018);
Semeghini  _et al._ (2018); Cabrera  _et al._ (2018); D’Errico  _et al._
(2019); Burchianti  _et al._ (2020) and dipolar gases Chomaz  _et al._ (2023);
Böttcher  _et al._ (2020). Stabilization of these states is achieved through
the counterbalance of attractive and repulsive interactions modeled by mean-
field nonlinear couplings and quantum fluctuations. The latter are
incorporated perturbatively through the famous Lee-Huang-Yang (LHY) Lee  _et
al._ (1957); Larsen (1963) correction whose sign and form depend on
dimensionality Ilg  _et al._ (2018) leading to an extended Gross-Pitaevskii
equation (eGPE) Petrov (2015); Petrov and Astrakharchik (2016). Such
environments sustain also different kinds of nonlinear excitations such as
dark solitons Edmonds (2023); Katsimiga  _et al._ (2023b); Saqlain  _et al._
(2023), bubbles Katsimiga  _et al._ (2023a); Edmonds (2023), kinks Kartashov
_et al._ (2022); Tylutki  _et al._ (2020); Katsimiga  _et al._ (2023a) and
vortices Li  _et al._ (2018); Yoğurt  _et al._ (2023).

From a complementary nonlinear optics-seeded perspective, the Cubic-Quintic
(CQ) model is an extension of the famous nonlinear Schrödinger model Killip
_et al._ (2017) constituing a universal mathematical toolbox involving higher-
order nonlinearities. In the optical setting, CQ is utilized to study the
propagation of electromagnetic waves Falcão Filho  _et al._ (2013); Reyna  _et
al._ (2020) in photorefractive materials. In this context, the quintic term
accounts for higher-order corrections to the nonlinear refractive index
Akhmediev and Afanasjev (1995), i.e., incorporating susceptibilities up to
fifth-order. Experimentally, it is often also relevant to include the
nontrivial (dissipative) absorption effects. The broad applicability of CQ
manifests by the fact that it has been deployed to describe various phenomena,
for instance, liquid waveguides Falcão Filho  _et al._ (2013), special types
of glasses Boudebs  _et al._ (2003), and colloids containing metallic
nanoparticles Falcão-Filho  _et al._ (2007). In atomic gases, the cubic
(quintic) coupling is related to the two- (three-) body interactions Luckins
and Van Gorder (2018); Cardoso  _et al._ (2011); Bulgac (2002) (see also,
e.g., Kolomeisky  _et al._ (2000)).

In this Letter we explore the dynamical transverse stability of kinks in a
wide range of models from nonlinear optics and atomic physics, such as the 2D
and 3D eGPE settings, as well as the 2D CQ model. A key feature of these
setups is the competition between attractive nonlinearities (dominant at low
density) and repulsive ones (prevailing at high density) whose presence is
proved crucial for the stability of kinks. Specifically, the stability and
robustness of these entities is demonstrated in a twofold manner. Firstly, we
analytically extract and numerically evaluate the relevant for each model
Bogoliubov-de-Gennes (BdG) spectrum. Secondly, we subject the relevant one-
dimensional (1D) kink solutions to external, spatially distributed
fluctuations (customarily present in experimental settings) and subsequently
monitor their remarkable persistence for evolution times of the order of
seconds. This demonstrates the relevance of these 1D topological defects in
current state-of-the-art cold atom and nonlinear optics experiments and their
structural robustness as potential information carriers.

### Models and Theoretical Analysis.

To highlight the universal features of the existence and stability of kink
solutions in higher spatial dimensions, three different nonlinear models
characterized by competing attractive (at low density) and repulsive (at high
density) interactions are investigated. These correspond to the eGPEs
describing, for instance, self-bound quantum droplets in 2D Petrov and
Astrakharchik (2016); Luo  _et al._ (2021) and 3D Petrov (2015); Luo  _et al._
(2021), and the CQ equation. Experimentally, the droplet bearing systems can
be emulated e.g. by considering two hyperfine states of 39K Semeghini  _et
al._ (2018). In optics the CQ model is typically used to monitor, for
instance, optical beam profiles of topological defects in liquid $\rm{CS}_{2}$
Reyna  _et al._ (2020). In dimensionless form (see also supplemental material
(SM) sup ) these equations read

$\displaystyle
i\partial_{t}\Psi=-\frac{1}{2}\left(\partial^{2}_{x}+\partial^{2}_{y}\right)\Psi+g\left|\Psi\right|^{2}\Psi\ln\left(\left|\Psi\right|^{2}\right),$
(1a) $\displaystyle
i\partial_{t}\Psi=-\frac{1}{2}\left(\partial^{2}_{x}+\partial^{2}_{y}\right)\Psi-\left|\Psi\right|^{2}\Psi+g_{CQ}\left|\Psi\right|^{4}\Psi,$
(1b) $\displaystyle
i\partial_{t}\Psi=-\frac{1}{2}\left(\partial^{2}_{x}+\partial^{2}_{y}+\partial^{2}_{z}\right)\Psi+g_{1}\left|\Psi\right|^{2}\Psi+\left|\Psi\right|^{3}\Psi.$
(1c)

Here, Eqs. (1a) and (1c) refer to the 2D and 3D eGPEs, while Eq. (1b) is the
2D CQ model with $g_{CQ}>0$. In the 2D eGPE, the logarithmic nonlinearity
encompasses both mean-field and first-order quantum fluctuation effects Petrov
and Astrakharchik (2016); Luo  _et al._ (2021), while its 3D counterpart
features a cubic (resp. quartic) attractive with $g_{1}<0$ (resp. repulsive)
mean-field (LHY) contribution Petrov (2015), see also SM sup . Furthermore, in
the CQ model there is an attractive (repulsive) cubic (quintic) nonlinear
coupling commonly accounting for two- (three-) body interactions in the realm
of cold gases Luckins and Van Gorder (2018); Bulgac (2002) or higher order
corrections to the Kerr effect in nonlinear optics Akhmediev and Afanasjev
(1995); Reyna  _et al._ (2020).

Figure 1: Existence of kink configurations. Effective potential for the three
nonlinear models: (a) 2D eGPE and CQ models with $g=g_{CQ}=1$ as well as (b)
the 3D eGPE setup. The effective potentials host the kink configuration, whose
wave function is constrained within $0$ and $u_{*}$. These states occur at
specific chemical potentials (see legends).

To infer the existence of planar kinks (i.e., one dimensional ones embedded in
higher dimensions), the following ansatz is employed,
$\Psi(x,r_{\perp},t)=e^{-i\mu t}u(x)$. Here, $\mu$ denotes the chemical
potential and $r_{\perp}=y$ ($r_{\perp}=(y,z)$) are the transverse coordinates
of the wave function in 2D (3D). A uniform background is assumed along the
transverse directions, and the $u(x)$ waveform is taken to be real without
loss of generality. Introducing this ansatz into Eqs. (1a)-(1c) results in
their reduced effective 1D time-independent standing wave analogues. These can
be cast into Newtonian equations of motion, $d^{2}u(x)/dx^{2}=-dV(u)/du$,
subjected to the relevant for each model effective potential $V(u)$. These
turn out to be

$\displaystyle V_{2D}(u)=\mu
u^{2}-g\frac{u^{4}}{2}\ln\left(\frac{u^{2}}{\sqrt{e}}\right),$ (2a)
$\displaystyle V_{2D}(u)=\mu u^{2}+\frac{u^{4}}{2}-g_{CQ}\frac{u^{6}}{3},$
(2b) $\displaystyle V_{3D}(u)=\mu
u^{2}-\frac{g_{1}}{2}u^{4}-\frac{2}{5}u^{5}.$ (2c)

Integration of the Newtonian equations of motion yields the effective energy,
$E$, of the 1D reduced system Katsimiga  _et al._ (2023b). While the resulting
potentials feature multiple maxima, the frequency parameter $\mu$ can be
generically tuned to render these maxima equi-energetic, thus enabling a
genuine heteroclinic orbit, i.e., a kink, between $u=0$ and $u=u_{*}$ with
$E=0$.

To systematically determine the finite kink backgrounds and chemical
potentials, two conditions need to be satisfied, $V(u_{*})=0$ and
$dV/du|_{u_{*}}=0$, namely, conditions for the maximum at $u=u_{*}$ being
equi-energetic with the one at $u=0$. For the 2D eGPE (CQ) these values lead
analytically to $\mu=-g/(2\sqrt{e})$, $u_{*}=e^{-1/4}$ ($\mu=-3/(16g_{CQ})$
and $u_{*}=\sqrt{3/(4g_{CQ})}$) [Fig. 1(a)]. The $u_{*}$ boundary for the 2D
eGPE separates the parameter region of 2D droplets from that of 2D bubbles Luo
_et al._ (2021); Petrov and Astrakharchik (2016), analogously to what is the
case also in 1D Katsimiga  _et al._ (2023a). For the 3D eGPE, the chemical
potential and extremum $u_{*}$ are $\mu=25g_{1}^{3}/216$ and $u_{*}=-5g_{1}/6$
respectively [Fig. 1(b)]. Notice that while such kink-type states have been
discussed for CQ settings previously (see, e.g., Jin-Liang  _et al._ (2006)),
they are unprecedented, to our knowledge, for the emergent atomic theme of 2D
and 3D quantum droplet settings. Interestingly, the asymmetry of the effective
potential $V_{3D}(u)$ [Fig. 1(b)] suggests the concurrent existence of a
heretofore unexplored quantum droplet (for values of $u<0$) whose analysis is
deferred for future studies.

Within the effective potential framework, unique kink solutions exist for each
coupling strength when competing interactions exist, i.e., when $g_{CQ}\neq
0\neq g_{1}$. In what follows, without loss of generality $g$ and $g_{CQ}$ are
set to unity in order to inspect the stability properties of these topological
defects. On the other hand, the mean-field interaction $g_{1}$ in the 3D eGPE
is routinely tunable in experiments via Feshbach resonances, see Semeghini
_et al._ (2018) and SM sup .

### Spectral stability of kinks in higher dimensions.

According to the aforementioned effective potential description the 1D planar
kink solution occurs for $\mu=-1/(2\sqrt{e})$, $\mu=-3/16$, and e.g.
$\mu=-0.1$ where $g_{1}=-5/6$, within the 2D eGPE, 2D CQ and 3D eGPE model,
respectively. The relevant effective 1D but also the higher dimensional
stationary states are then obtained upon utilizing heteroclinic (tanh-shaped)
initial guesses, bearing the correct asymptotics described above. In the CQ
case, an analytical form exists as
$u=\sqrt{\frac{3}{8}(1+\tanh(\frac{\sqrt{3}x}{2\sqrt{2}}))}$ Jin-Liang  _et
al._ (2006); Birnbaum and Malomed (2008). It is then numerically confirmed
that such 1D planar waveforms satisfy the corresponding 2D and 3D time-
independent version of Eqs. (1a)-(1c).

Figure 2: Spectral stability of kinks. BdG analysis for the kink waveforms
within (a) the 2D eGPE, (b) 2D CQ and (c) the 3D eGPE models. The absence of a
finite imaginary part, $\Im[\omega]\neq 0$, in the spectrum showcases the
stability of the respective kink states.  Figure 3: Resilience of kink states.
Density snapshots depicting the evolution of perturbed kink solutions within
the (a) 2D eGPE, (b) CQ and (c) 3D eGPE models with $\mu=-0.1$. It can be
readily seen that, in all cases, the kink is robust throughout the evolution
with its core remaining intact and the waveform maintaining its shape despite
being significantly distorted due to external spatial and phase fluctuations.
The box sizes used for the simulations correspond to $L_{x}=L_{y}=400$ and
$L_{z}=6$.

Then, in order to extract the stability properties of these configurations,
small perturbations of the numerically exact (up to a prescribed tolerance)
kink solutions are considered having the form $\Psi=\left(\Psi_{0}+\epsilon
ae^{-i\omega t}+\epsilon b^{\star}e^{i\omega^{\star}t}\right)e^{-i\mu t}$.
Here, $``\star"$ denotes complex conjugation, $a,b$ and $\omega$ refer to the
corresponding eigenvectors, and eigenfrequencies, whilst $\epsilon\ll 1$ is a
small perturbation parameter. Utilizing the above wave function ansatz and
keeping terms of order $\mathcal{O}(\epsilon)$ one arrives at an eigenvalue
problem that, irrespectively of the system under consideration [Eqs.
(1a)-(1c)], can be written as

$\displaystyle\begin{bmatrix}\hat{L}_{11}&\hat{L}_{12}\\\
\hat{L}_{21}&\hat{L}_{22}\end{bmatrix}\begin{bmatrix}a\\\
b\end{bmatrix}=\omega\begin{bmatrix}a\\\ b\end{bmatrix}.$ (3)

The model-dependent diagonal matrix elements are found to be
$\hat{L}_{11}=-\frac{1}{2}\nabla^{2}-\mu+\Psi^{2}_{0}+2\Psi_{0}^{2}\rm{ln}(\Psi^{2}_{0})$,
$\hat{L}_{11}=-\frac{1}{2}\nabla^{2}-\mu-2\Psi^{2}_{0}+3\Psi_{0}^{4}$, and
$\hat{L}_{11}=-\frac{1}{2}\nabla^{2}-\mu+2g_{1}\Psi_{0}^{2}+\frac{5}{2}\Psi_{0}^{3}$,
for the 2D eGPE, CQ, and 3D eGPE respectively. Their relevant off-diagonal
counterparts read
$\hat{L}_{12}=\Psi^{2}_{0}+\Psi^{2}_{0}\rm{ln}(\Psi^{2}_{0})$,
$\hat{L}_{12}=-\Psi^{2}_{0}+2\Psi_{0}^{4}$, and
$\hat{L}_{12}=g_{1}\Psi^{2}_{0}+\frac{3}{2}\Psi_{0}^{3}$, respectively. In all
cases, $\hat{L}_{11}=-\hat{L}_{22}$, $\hat{L}_{12}=-\hat{L}_{21}$ and
$\nabla^{2}$ denotes the $\mathcal{N}$-dimensional Laplace operator.
Additionally, $\Psi_{0}$ designates the stationary, real kink state for each
distinct model.

The stability analysis outcome for all three models investigated herein, is
depicted in Figs. 2(a)-(c). In all scenarios, spectral stability of the kink
state can be inferred from the absence of a finite imaginary part,
$\Im(\omega)\sim 10^{-12}$, in the pertinent BdG spectrum. For all settings,
the first hundred eigenvalues of the continuous quasi-1D BdG spectra are
illustrated, while we note that we verified that their relevant higher
dimensional analogues produce the same stability result. By quasi-1D BdG
spectra, what is meant here is that the transverse direction is decomposed
into Fourier modes (with the “quantization” thereof from a finite transverse
computational domain accounted for) and the stability of (a large number of)
effective 1D spectra for different transverse wavenumbers is confirmed.

### Dynamical confirmation of the absence of transverse instability for the
kinks.

To corroborate our BdG findings we next consider the dynamical evolution of
the stationary kink solution for each of the models under study. Specifically,
we aim not only to mimic unavoidable experimental imperfections but also to
exemplify the robust nature of the kink structure in the presence of
transverse perturbations. To that effect, the kink initial state is
significantly perturbed via a random normal distribution generator,
$\delta(x,r_{\perp})$. The ensuing wave function acquires the form
$\psi(x,r_{\perp},t=0)=\Psi_{0}\left[1+\varepsilon\delta(x,r_{\perp})\right]$.
In this context, $\delta(x,r_{\perp})$ is characterized by zero mean and unit
variance, while $\varepsilon=\varepsilon_{R}+i\varepsilon_{I}$ such that also
phase disturbances are encountered. In particular, the considered amplitudes
of the perturbation range lie in the interval $[10\%,50\%]$ of the solution
amplitude. Typical evolution times (in the dimensionless units adopted herein)
are of the order of $\sim 500$ and $\sim 10^{3}$ for the 3D and 2D settings,
respectively, confirming the longevity of these planar 1D defects upon their
exposure to transverse excitations.

Evidently, despite the significantly excited nature of the initial state the
kink remains intact preserving its shape throughout the evolution, while
“shedding” some of the relevant “radiation” in the form of dispersive
wavepackets, as it can be seen in Figs. 3(a)-(c). Particularly here, distinct
plates correspond to density, $|\Psi(x,y,t)|^{2}$, contours in the $x-y$ plane
taken at different times during the propagation of the perturbed kink
configuration. For the 3D dynamics the $z$-direction is integrated out, while
we note that the robustness of the kink configurations has also been verified
for different values of the chemical potential and hence distinct interaction
strengths.

### Experimental implications.

A binary bosonic mixture in the ultracold regime, where interspecies
attraction is balanced by intraspecies repulsion, can host self-bound states
Petrov (2015); Cheiney  _et al._ (2018); Semeghini  _et al._ (2018). The two
hyperfine levels $\ket{F=1,m_{F}=0}$, $\ket{F=1,m_{F}=-1}$ of 39K in the
vicinity of the $59\,\rm{G}$ intraspecies Feshbach resonance Chin  _et al._
(2010); D’Errico  _et al._ (2007) were proven to be suitable candidates
satisfying the above criteria Cabrera  _et al._ (2018); Semeghini  _et al._
(2018). These two states are experimentally populated using radiofrequency
spectroscopy Cabrera  _et al._ (2018) and their dynamics can be adequately
described by the eGPE models [Eqs. (1a), (1c)] in 2D and 3D. For such a setup
with averaged mean-field scattering length $\delta a=-0.24\,a_{0}$, the
propagation times of the kink defect in 3D ($\sim 500$ in dimensionless units)
refer approximately to $800\,ms$ sup . A corresponding pancake geometry
confining such a 2D mixture can be realized by utilizing a tight harmonic trap
along one spatial dimension Hadzibabic and Dalibard (2011). For an
experimentally relevant trapping frequency $\omega_{z}=2\pi\times 2\,\rm{kHz}$
Cabrera  _et al._ (2018); Saint-Jalm  _et al._ (2019) and $\delta
a=-1.5\,a_{0}$ [see SM sup ], the relevant timescale ($\sim 1000$ in
dimensionless units) for the kink evolution is $2\,$ secs. These long
evolution times guarantee the resilience of the kink structure for its
realization in current state-of-the-art experiments that suffer in addition by
three-body recombination processes Semeghini  _et al._ (2018).

On the other hand, the 2D CQ equation is used to describe, for instance, the
propagation of electromagnetic waves in optical media incorporating higher-
order nonlinear refractive indices Sutherland (2003). Characteristic materials
featuring such properties are chalcogenide glasses Boudebs  _et al._ (2003),
such as liquid $\rm{CS}_{2}$ Falcão Filho  _et al._ (2013); Reyna  _et al._
(2020). In such setups (where optical vortices and their destabilization has
been monitored Desyatnikov  _et al._ (2005)), the time-evolution can be probed
by measuring the transmission of the kink along the axial direction of the
nonlinear medium. Considering a light beam of $900\,nm$ Falcão Filho  _et al._
(2013), a propagation distance of $10^{3}$ corresponds to $0.14\,mm$ axial
depth for the kink transmission. In the above-discussed setups, namely either
cold bosonic mixtures or nonlinear optical media, kink defects can be
imprinted in the bulk by means of the well-established technique of phase
masking Denschlag  _et al._ (2000); Pedaci  _et al._ (2006) or utilizing
digital micromirror device patterned optical traps Saint-Jalm  _et al._
(2019); Navon  _et al._ (2021); Gauthier  _et al._ (2016).

### Conclusions and future challenges.

The existence and corresponding spectral and dynamical stability of 1D kink
solutions embedded in higher-dimensional gases or nonlinear optical media
featuring competing attractive lower order and repulsive higher order
interactions is unveiled. This is exemplarily demonstrated for three distinct,
representative models referring to the 2D and 3D eGPE droplet bearing systems
and the 2D CQ setting relevant in nonlinear optics, in order to establish the
breadth and generality of results. We explicated the existence of these 1D
topological defects upon analytically extracting, in each case, a quasi-1D
effective potential picture, subsequently supplemented by their spectral
stability inferred through linearization analysis. Additionally, monitoring
their dynamical evolution in the presence of external fluctuations, with
amplitudes up to 50% of the actual solution, revealed their profound
robustness and resilience, for times up to few seconds.

These results suggest the use of kink defects as promising candidates for
topological quantum computing. Indeed, contrary to the generically (in such
nonlinear Schrödinger models) transversely unstable dark soliton states, which
have been studied and reviewed extensively Kivshar and Pelinovsky (2000),
kinks appear to defy the proneness to transverse modulations. Importantly,
also, these results pave the way for significant further both physical and
mathematical studies. On the latter side, a rigorous proof of such stability
is a key open direction. On the other hand, from a physical perspective, our
effective potential analysis suggests the existence of further lower-
dimensional nonlinear excitations (including droplets, bubbles, etc.), whose
transverse dynamics and potential dynamical features may be of further
theoretical and experimental interest in its own right. Such studies are
currently in progress and will be reported in future publications.

### Acknowledgments.

We acknowledge fruitful discussions with B. Malomed. This material is based
upon work supported by the U.S. National Science Foundation under the awards
PHY-2110030 and DMS-2204702 (PGK).

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Supplementary Material:
Generic transverse stability of kink structures in atomic and optical
nonlinear media with competing repulsive and attractive interactions

##  I Three-dimensional attractive mixture

The three-dimensional setting discussed in the main text corresponds to an
attractive bosonic mixture, where the intercomponent attraction slightly
exceeds the intracomponent repulsion. Namely,
$a_{12}+\sqrt{a_{11}a_{22}}\lesssim 0$, with $a_{ij}$ denoting the relevant 3D
scattering lengths that are tunable with the Feshbach resonance technique
Semeghini  _et al._ (2018). In this interaction regime, it is theoretically
predicted Petrov (2015); Luo  _et al._ (2021) and experimentally observed
Cabrera  _et al._ (2018); Semeghini  _et al._ (2018) that the bosonic mixture
hosts many-body self-bound states, known as quantum droplets. The beyond mean-
field first-order LHY correction is proven to be crucial for sustaining such
structures, minimizing the underlying energy functional of the mixture. The
resulting extended mean-field equations (eGPEs) then feature competing
nonlinearities, expressed through a cubic and a quartic term Petrov (2015).

When the ratio between the density of the components satisfies
$n_{1}/n_{2}=\sqrt{a_{22}/a_{11}}$, the two coupled extended mean-field
equations reduce to a single effective one Semeghini  _et al._ (2018),
describing an attractive bosonic mixture. The dimensional form of the eGPE
reads Semeghini  _et al._ (2018)

$\displaystyle i\hbar\frac{\partial\Psi}{\partial
t}=-\frac{\hbar^{2}}{2m}\nabla^{2}\Psi+\frac{8\pi\hbar^{2}\delta
a}{m}\frac{\sqrt{a_{22}/a_{11}}}{(1+\sqrt{a_{22}/a_{11}})^{2}}\left|\Psi\right|^{2}\Psi$
$\displaystyle+\frac{128\sqrt{\pi}\hbar^{2}}{3m}(a_{11}a_{22})^{5/4}\left|\Psi\right|^{3}\Psi.$
(S1)

Here, $m$ is the mass of the atoms, $\Psi$ is the 3D wave function describing
the reduced single component, and the averaged mean-field interactions $\delta
a=a_{12}+\sqrt{a_{11}a_{22}}$.

To render Eq. (S1) dimensionless, the following transformations are employed,

$t=\tilde{t}\frac{mL^{2}}{\hbar},\quad
r=\tilde{r}L,\quad\Psi=\tilde{\Psi}L^{-3/2},$ (S2)

where e.g. $L\simeq 1.6\,\mu m$ is the length of the box along the $z$
direction. With the above units, the 3D eGPE becomes dimensionless,

$i\frac{\partial\Psi}{\partial
t}=-\frac{1}{2}\nabla^{2}\Psi+g_{1}\left|\Psi\right|^{2}\Psi+g_{2}\left|\Psi\right|^{3}\Psi.$
(S3)

The $\sim$ notation has been dropped for simplicity. Moreover, the
dimensionless interaction strengths $g_{1},g_{2}$ in these units read

$\displaystyle g_{1}=\frac{8\pi\delta
a\sqrt{a_{22}/a_{11}}}{L(1+\sqrt{a_{22}/a_{11}})^{2}},$ (S4a) $\displaystyle
g_{2}=\frac{128\sqrt{\pi}}{3}\left(a_{11}a_{22}\right)^{5/4}L^{-5/2}.$ (S4b)

Rescaling the wavefunction as $\Psi=\Psi^{\prime}g_{2}^{-1/3}$, we absorb the
prefactor in front of the quartic term and we recover Eq. (1c) in the main
text. For the two hyperfine states of 39K Semeghini  _et al._ (2018); Cabrera
_et al._ (2018); Cheiney  _et al._ (2018), it was shown that droplets can form
when $\delta a$ is tuned to small negative values by means of Fano-Feshbach
resonances Chin  _et al._ (2010). Relevant timescales correspond to
$mL^{2}/\hbar\approx 1.6\,ms$.

##  II Two-dimensional droplet setting

Here, we elaborate on the 2D eGPE supporting quantum droplets, its
dimensionless form, and the connection of the interaction strengths with the
experimentally tunable 3D scattering lengths. Quantum droplets can also form
in two dimensions Petrov and Astrakharchik (2016), where the logarithmic
nonlinearity incorporates both the LHY term and the mean-field contributions.
Similar to 3D, the corresponding coupled system of eGPEs can be reduced to an
effective single-component one Otajonov  _et al._ (2020), which reads

$i\hbar\frac{\partial\Psi}{\partial
t}=-\frac{\hbar^{2}}{2m}\nabla^{2}\Psi+\frac{\hbar^{2}}{m}g|\Psi|^{2}\Psi\ln\left(\frac{|\Psi|^{2}}{n_{0}\sqrt{e}}\right).$
(S5)

The dimensionless interaction strength $g$ is related to the 2D scattering
lengths through the expression Petrov and Astrakharchik (2016); Otajonov  _et
al._ (2020),

$\displaystyle
g=\frac{4\pi}{\ln\left(\frac{a^{(2D)}_{12}\sqrt{a^{(2D)}_{11}a^{(2D)}_{22}}}{[a^{(2D)}_{11}]^{2}\Delta}\right)\ln\left(\frac{a^{(2D)}_{12}\sqrt{a^{(2D)}_{11}a^{(2D)}_{22}}}{[a^{(2D)}_{22}]^{2}\Delta}\right)},$
(S6a)
$\displaystyle\rm{where}~{}~{}\Delta=\exp\left\\{-\frac{\ln^{2}\left(\frac{a^{(2D)}_{22}}{a^{(2D)}_{11}}\right)}{2\ln\left(\frac{[a^{(2D)}_{12}]^{2}}{a^{(2D)}_{11}a^{(2D)}_{22}}\right)}\right\\}.$
(S6b)

In these expressions, $a^{(2D)}_{\sigma\sigma^{\prime}}$ refer to the 2D
scattering lengths within ($\sigma=\sigma^{\prime}$) or between
($\sigma\neq\sigma^{\prime}$) the components. Moreover,
$n_{0}=\frac{e^{-2\gamma-3/2}}{\pi
a^{(2D)}_{12}\sqrt{a^{(2D)}_{11}a^{(2D)}_{22}}}\Delta\sqrt{\frac{4\pi}{g}}$ is
the droplet equilibrium density in the thermodynamic limit Petrov and
Astrakharchik (2016) and $\gamma\approx 0.577$ is the Euler-Mascheroni
constant Abramowitz  _et al._ (1988).

In the case of a tightly confined mixture in the $z$-direction due to the
presence of a harmonic trap with oscillator length
$a_{\perp}=\sqrt{\hbar/(m\omega_{\perp})}$, the scattering lengths are
connected to their 3D counterparts, $a_{\sigma\sigma^{\prime}}$ by Petrov and
Shlyapnikov (2001); Petrov  _et al._ (2000)

$a^{(2D)}_{\sigma\sigma^{\prime}}=2e^{-\gamma}a_{\perp}\sqrt{\frac{\pi}{0.9}}\exp\left\\{-\frac{a_{\perp}\sqrt{\pi}}{a_{\sigma\sigma^{\prime}}\sqrt{2}}\right\\}.$
(S7)

Below, we measure time and length with respect to $m/(gn_{0}\hbar\sqrt{e})$
and $\sqrt{gn_{0}\sqrt{e}}^{-1}$ respectively Saint-Jalm  _et al._ (2019) and
normalize the 2D wave function in terms of the total particle number,
$\int|\Psi|^{2}~{}dxdy=gN=g(N_{1}+N_{2})$. In this sense the corresponding
dimensionless eGPE is retrieved,

$i\frac{\partial\Psi}{\partial
t}=-\frac{1}{2}\nabla^{2}\Psi+|\Psi|^{2}\Psi\ln\left(|\Psi|^{2}\right).$ (S8)

##  III Cubic-quintic model

Below we elaborate on the dimensionalization of the cubic-quintic equation in
the context of non-linear optical media discussed in the main text [Eq. (1b)].
Particularly, the relevant model considers light beams passing through
nonlinear media, being characterized by high-order refractive index prefactors
(stemming from an expansion of the refractive index as a function of the light
intensity). In this context, the propagation of the electric field envelope
$E=E(x,y,z)$ along the axial direction $z$ can be described by the following
cubic-quintic equation Reyna  _et al._ (2020); Falcão Filho  _et al._ (2013),

$i\frac{\partial E}{\partial
z}=-\frac{1}{2k}\nabla^{2}E-2kn_{2}c\epsilon_{0}\left|E\right|^{2}E-4kn_{4}c^{2}\epsilon_{0}^{2}\left|E\right|^{4}E.$
(S9)

Here, $k$ is the wavenumber, $n_{0}$ refers to the linear refractive index,
and $n_{2}$, $n_{4}$ are the higher-order nonlinear refractive indices.
Moreover, $\epsilon_{0}$ and $c$ are the vacuum permittivity and speed of
light respectively. The nabla operator describes the dispersion of the beam
along the transversal directions, $x$ and $y$.

To cast the above equation in dimensionless form, we employ the coordinate
transformations, $\\{x,y,z\\}=r_{0}\\{\tilde{x},\tilde{y},\tilde{z}\\}$, and
$E=E_{0}\tilde{E}$ where,

$r_{0}=k^{-1},\quad E_{0}=\frac{1}{\sqrt{2n_{2}c\epsilon_{0}}}.$ (S10)

The CQ subsequently reads,

$i\frac{\partial E}{\partial
z}=-\frac{1}{2}\nabla^{2}E-\left|E\right|^{2}E+g_{CQ}\left|E\right|^{4}E,$
(S11)

where $g_{CQ}=-n_{4}/n_{2}^{2}$. For liquid carbon disulfide $\rm{CS}_{2}$
employed in Refs. Reyna  _et al._ (2020); Falcão Filho  _et al._ (2013),
$g_{CQ}=450$.


# Enhancements for 5G NR PRACH Reception: An AI/ML Approach

Rohit Singh1, Anil Kumar Yerrapragada2, Jeeva Keshav S3, Radha Krishna Ganti4
Department of Electrical Engineering
Indian Institute of Technology Madras
Chennai, India 600036
Email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Random Access is an important step in enabling the initial attachment of a
User Equipment (UE) to a Base Station (gNB). The UE identifies itself by
embedding a Preamble Index (RAPID) in the phase rotation of a known base
sequence, which it transmits on the Physical Random Access Channel (PRACH).
The signal on the PRACH also enables the estimation of propagation delay,
often known as Timing Advance (TA), which is induced by virtue of the UE’s
position. Traditional receivers estimate the RAPID and TA using correlation-
based techniques. This paper presents an alternative receiver approach that
uses AI/ML models, wherein two neural networks are proposed, one for the RAPID
and one for the TA. Different from other works, these two models can run in
parallel as opposed to sequentially. Experiments with both simulated data and
over-the-air hardware captures highlight the improved performance of the
proposed AI/ML-based techniques compared to conventional correlation methods.

###### Index Terms:

PRACH, 5G, AI/ML, DNN, RAPID, Timing Advance, 3GPP

##  I Introduction

In any cellular system, the UEs have to first identify and attach themselves
to a base station. This procedure is known as Random Access (RA). In order to
facilitate this procedure, the 5G standards have a provision for a dedicated
channel known as the Physical Random Access Channel (PRACH). On the PRACH, the
UE sends a randomly rotated version of a known base sequence. The identity of
the UE, known as Random Access Preamble ID (RAPID), is embedded in this
rotation. Typically the base sequence is a Zadoff-Chu sequence [1], well known
for its Constant Amplitude Zero Auto Correlation (CAZAC) properties. The
receiver can then leverage these properties and extract the RAPID by
correlation with the known base sequence. In addition, the correlation also
helps in the estimation of the propagation delay, which the signal incurs as
it traverses the distance between the UE and the gNB. The delay value is
reported back to the UE and is then used by the UE to advance its future
transmissions. In this paper, we refer to the propagation delay value as the
Timing Advance (TA). Though the primary interest lies in estimating the TA,
the propagation delay estimate also captures any fine drift in time
synchronization between the UE and gNB.

In ideal conditions, the correlation leads to a single peak that contains
information about the RAPID and the Timing Advance. Traditional receivers use
thresholds in order to identify the correct peak. However, setting up these
thresholds requires tedious experiments and simulations [2]. Improper
thresholds result in false alarms and/or missed detections. Furthermore, since
the RAPID selection is random, it is possible for multiple UEs to use the same
RAPID, possibly leading to collisions. Collisions lead to re-transmissions,
which in turn increases the initial attach latency and also consumes more
energy for every re-transmission.

In this paper, we direct our attention toward the RAPID detection and TA
estimation of a single user, both of which we recast as AI/ML classification
problems. We make the following contributions.

###  I-A Our Contributions

  * •

Prior AI/ML implementations of the PRACH receiver rely upon input data from
multiple correlation zones [3]. This paper presents an approach that avoids
correlations altogether. Both the RAPID and TA classification models require
no pre-processing of the input and take frequency domain data directly from
the OFDM grid.

  * •

Several existing works implement RAPID detection and TA estimation in a
sequential manner, in which incorrect detection of RAPID could result in TA
estimation errors. In the proposed receiver, the RAPID detection and TA
estimation models can work in parallel using the same received signal input
and are independent of each other.

  * •

In this paper, we study the effects of training and testing with different
channel models. In particular, we study the generalization performance of the
models under various configurations of delay spread values (TDLC150 and
TDLC300).

  * •

Most works in AI/ML for wireless communications deal with simulated data and
environments. Hardware impairments are rarely incorporated. In this paper, we
analyze the performance of the proposed AI/ML models using Over-the-air (OTA)
captures obtained from the 5G Testbed at IIT Madras [4].

##  II Background on 5G NR Physical Random Access Channel

This section offers background information on Random Access Procedures. We
also describe, compare, and contrast existing prior works on both conventional
and AI/ML-based receivers for PRACH.

###  II-A 5G Random Access Procedures

The 5G specification supports two types of Random Access (RA) procedures. The
first is contention-based in which a UE is free to select any preamble
randomly from a pool of preambles. The second is contention-free in which the
gNB assigns preamble IDs to the UE. In this paper, we focus on the Contention
Based Random Access (CBRA) procedure which is a 4-step process described in
Fig. 1. The RA procedure begins with PRACH (Msg 1) signaling. In this step,
the UE selects a RAPID from a pool of possible values and applies the
corresponding cyclic shift to the base sequence as specified in [5]. We note
that at this point, the UE has already ascertained the PRACH time-frequency
locations in the resource grid from prior messages in the Synchronization
Signal Block (SSB) and System Information Block #1 (SIB1), both of which are
periodically broadcast by the gNB.

The gNB acknowledges decoding of the PRACH signal through a Random Access
Response (RAR, Msg 2) on the Physical Downlink Shared Channel (PDSCH). This is
immediately followed by a scheduled uplink transmission (Msg 3) on the
Physical Uplink Shared Channel (PUSCH). The fourth step includes scheduled
downlink transmissions (Msg 4) on the PDSCH for Contention Resolution (CR) and
for further initial attachment processing that ends in the UE latching on to
the gNB. If the PRACH detection is incorrect, it will be evident at Msg 3, and
the RA process restarts, causing significant latency in the initial attach
process. In order to reduce this initial attach latency, PRACH receivers
should be robust enough to reduce incorrect detections even in multipath and
low SNR environments.

Figure 1: Contention Based Random Access (CBRA) Procedure

###  II-B PRACH Sequence Generation and Transmission

In this paper, we focus on the PRACH signaling in Msg1 of the RA process.
PRACH sequence generation starts with the selection of a base sequence which
is given by,

$x_{u}(n)=e^{\frac{-j\pi un(n+1)}{L_{RA}}},n=0,1,.....,L_{RA}-1.$ (1)

The length of the sequence $L_{RA}$, depends on the PRACH preamble format
given by Table 6.3.3.1-2 in TS 38.211 [5]. The initial sequence number $u$ is
selected from Table 6.3.3.1-4 in TS 38.211. A cyclic rotation is applied to
$x_{u}(n)$ resulting in $x_{u,v}(n)=x_{u}((n+C_{v})\text{ mod }L_{RA})$. The
parameter $C_{v}$ is a function of the parameter $N_{CS}$ which is given in
Table 6.3.3.1-7 in TS 38.211. The table indices are configured by higher
layers.

Without loss of generality, we use commonly configured parameters for the
purpose of illustration and experiments. We use the following values:
$L_{RA}=139$ and $N_{CS}=13$ resulting in $C_{v}=vN_{CS}$ where
$v=0,1,.....,\left\lfloor\frac{L_{RA}}{N_{CS}}\right\rfloor-1$.

The 5G standards have a provision for 64 Random Access Preamble Indices
(RAPID). For the configured values,
$\left\lfloor\frac{L_{RA}}{N_{CS}}\right\rfloor=10$, and hence each base
sequence can accommodate $10$ RAPID values. In order to encompass $64$ RAPID
values, $7$ base sequences are required. In this paper, we develop neural
network models for a single base sequence and $10$ RAPID values. Other base
sequences can be incorporated similarly.

The sequences $x_{u}(n)$ and $x_{u,v}(n)$ are generated in the time domain. In
order to map $x_{u,v}(n)$ to the OFDM resource grid, it is first converted to
the frequency domain as follows,

$\begin{split}y_{u,v}(k)&=\sum_{n=1}^{L_{RA}-1}x_{u,v}(n)e^{\frac{-j2\pi
nk}{L_{RA}}}=X_{u}(k)e^{\frac{-j2\pi kC_{v}}{L_{RA}}},\end{split}$ (2)

where,

$X_{u}(k)=\sum_{n=1}^{L_{RA}-1}x_{u}(n)e^{\frac{-j2\pi nk}{L_{RA}}}.$ (3)

Then, $y_{u,v}(k)$ can be mapped to the frequency resources using the tables
6.3.3.2-2 to 6.3.3.2-4 given in TS 38.211 [5]. After $y_{u,v}(k)$ is mapped to
the resource grid, the signal is OFDM modulated, up-converted to the carrier
frequency, and transmitted over the channel. At the receiver, after down-
conversion and OFDM demodulation, the frequency domain signal is given by,

$\tilde{y}_{u,v}(k)=h(k)X_{u}(k)e^{\frac{-j2\pi
k}{L_{RA}}\left[{C_{v}-\frac{TA\cdot L_{RA}}{N}}\right]}+w(k),$ (4)

where $h(k)$ represents the fading channel, $N$ is the OFDM FFT size, and
$w(k)$ is the Additive White Gaussian Noise (AWGN). Note that Eq.(4) includes
one rotation that uniquely identifies the RAPID (through $C_{v}$ and $u$) and
a second rotation that is induced by the propagation delay. In this paper, we
assume that $L_{RA}=139$, $N=4096$, and a sub-carrier spacing of $30$ kHz.
Hence a propagation delay of $\frac{4096}{139}=29.4$ samples (caused by a
distance of $71$m) causes the correlation peak at the receiver to shift by one
sample. In this paper, we evaluate the performance of a small cell with a
radius of $850$ meters, which corresponds to $12$ discrete values of TA.

TABLE I: Features of various AI/ML approaches.

###  II-C Existing receivers for 5G NR PRACH

The primary goal of a PRACH receiver is to accurately determine the RAPID and
the TA. Here, we summarize both the conventional and AI/ML approaches taken by
other works in the design of PRACH receivers.

####  II-C1 Conventional approaches

Most existing PRACH receivers are correlation-based [6, 7] where the received
time domain samples are correlated with all the possible base sequences, and
the one that results in the maximum correlation value is identified as the
transmitted base sequence. The shift in the correlation peak maps to the RAPID
and the TA. If a Zadoff Chu Sequence is correlated with the rotated version of
the same sequence, it will result in a strong peak at the shift. If it is
correlated with any other sequence, the correlation will be ideally zero (or
below a threshold [2, 8]).

A common implementation (such as the one used at the IIT Madras 5G testbed
[4]) of a correlation-based PRACH receiver involves dividing the $139$ length
correlation output into multiple preamble windows. Each window corresponds to
a particular RAPID value and the shift of the peak from the boundary of the
window indicates the TA. Fig. 2, shows three example peaks corresponding to
three different cases with the same transmitted RAPID value of 6. In the first
example, if the transmitted sequence is correlated with the base sequence, the
peak occurs at the $78$th position ($C_{v}=6\times N_{CS}=78$) as shown in
Fig. 2. In the second example, due to the propagation delay of $236$ samples,
the peak shifts to the $70$th position which is still contained within the
same preamble window (Estimated RAPID=6, TA=8 in Fig. 2). In the third
example, adverse channel effects along with the propagation delay result in
the peak shifting into the next preamble window causing errors in both RAPID
detection and TA estimation (Estimated RAPID=5, TA=2 in Fig. 2).

Figure 2: Example correlation plot, showing division into preamble windows and
shifting of the peaks due to propagation delay and fading channel effects.

####  II-C2 AI/ML approaches

Figure 3: The proposed AI/ML architecture replaces correlation operations with
2 neural networks that can run in parallel.

Any PRACH receiver should be designed for RAPID detection, TA estimation,
Collision Detection and Contention Resolution (CD/CR), and false detection. As
summarized in Table I, prior work is focussed on one or more of the above
outcomes. For ease of understanding the AI/ML landscape for PRACH, we compile
and summarize some of the key similarities and differences between our work
and some of the prior works in Table I.

CD and CR are crucial in massively connected IoT scenarios. For such a
scenario, [3] develops a framework for CD and CR using multiple neural network
classifiers. First, classifier models are trained to predict the number of UEs
likely to collide. Separate classifiers are designed for TA estimation. AI/ML
models such as k-Nearest Neighbors (kNN) [9, 10], Naive Bayes [9, 10], and
Decision Tree Classification (DTC) [10] have shown to improve the True/False
classification performance. A further gain is achieved through ensemble
learning methods [10]. An initial attempt at RAPID detection and TA estimation
is made by [11]. However, this work does not consider any impairments such as
fading or noise. We note that most of the above methods (1) rely on
correlation data for their inputs (2) consider the longer 839-length PRACH
sequences and (3) focus on the 4-step RA.

The 2-step RA process, which was introduced later in 5G Release 16, has seen
some interest. For 2-step RA, AI/ML-based TA estimation using RSRP values
collected at an edge RAN controller is shown in [12]. The trained model is
then deployed to different UEs and updated periodically or as needed. A method
for collision avoidance for 2-step RA is shown in [13]. Recurrent Neural
Networks (RNN) and Long Short Term Memory (LSTM) models are trained with a
publicly available dataset containing time instances at which UEs connect to
the network. Using the model to predict whether a UE is likely to transmit on
the PRACH, the base station preemptively allocates resources to the UE, thus
eliminating the need for the UE to even transmit and reducing collisions.

Since most of the deployed 5G systems around the world use the 4-step RA
process, in this paper we focus on RAPID and TA detection for the 4-step RA.
We focus on the 5G PRACH A1 format, where a 139-length Zadoff Chu sequence is
transmitted by the UE. We observe that while the 139-length sequences are not
studied in detail in the current literature there is a significant advantage
of the 139-length sequence over the 839-length sequence in that it consumes
fewer radio resources. A further distinction of our work is that the input to
our proposed AI/ML-based PRACH receiver is directly extracted from the
frequency domain resources after performing OFDM-FFT. This is in contrast to
other works, which require a correlation of the received samples with the base
sequence. Furthermore, we focus on model generalization, another aspect, which
to the best of our knowledge, is the first such attempt. The first level of
model generalization is with respect to the different simulated channel
scenarios (TDLC10, TDLC150, and TDLC300), and the second, is with respect to
the hardware OTA data captured at the IIT Madras 5G Testbed [4].

##  III Proposed AI/ML model for the PRACH receiver

As described in the previous section, both the phase rotations in Eq. 4, come
from discrete sets. Hence RAPID and TA detection can be considered as a
classification of one element out of the respective discrete sets.

###  III-A Neural Network Architectures

In this work, we use a Residual Network (ResNet)-like architecture with four
convolutional layers to classify both RAPID and TA. ResNets are popular
because they use skip connections to overcome vanishing gradients. The input
to the RAPID detection model is directly taken from the received frequency
domain resources after the OFDM FFT. The input to the TA estimation model is
the $139$-point IFFT of the extracted frequency domain data. The input
dimension to both networks is $139\times 2\times 2$ ($139$ resource elements,
$2$ OFDM symbols, and real and imaginary parts of the samples). The input is
first passed through a Batch normalization (BN) block followed by a Leaky
Rectified Linear Unit (LReLU) activation function. In both models, this is
followed by three convolutional layers, each with $64$ filters. Each of these
layers is followed by a BN block and a LReLU activation function. The output
is passed through another convolutional layer with $2$ filters. The output of
the final convolutional layer is added to the output of the initial BN + LReLU
output. The sum is then flattened and passed to the output via a dense layer
of size $128$, with ReLU activation. After the flatten operation, we use
dropout as a regularizer with a probability of $0.4$. The RAPID output layer
is of size $10$ and the TA output layer is of size $12$. Both output layers
use the softmax activation function. Stochastic Gradient Descent (SGD)
optimizer with a learning rate of $10^{-2}$ is used in case of RAPID detection
and an Adam optimizer with a learning rate of $10^{-3}$ is used for TA
estimation. The detailed architecture of both the neural networks is shown in
Fig. 4.

###  III-B Dataset Generation

Two types of data are considered in this paper - simulated data and Over-the-
Air (OTA) data captured on testbed hardware.

####  III-B1 Simulated datasets

These datasets are generated using the MATLAB 5G Toolbox. We consider an SNR
range of $-20$ dB to $+20$ dB in the steps of $5$ dB. For each SNR, $100000$
data instances are generated. Each data instance is generated by randomly
choosing a RAPID and applying it to the base sequence. The corresponding OFDM-
modulated signal is transmitted over a TDLC channel and the received samples
are extracted. These are the inputs to the models. The labels are the chosen
RAPID values. To simulate a propagation delay (timing advance), a random
number of zero samples are added before transmitting the signal over the
channel. This translates to a phase shift in the frequency domain at the
receiver. In this case, the ground truth TA label is calculated as the floor
of the ratio of the propagation delay to 29.4.

Figure 4: NN Architecture

####  III-B2 Over-the-Air (OTA) Hardware captured datasets

Validation of AI/ML models with data collected from deployed base station
provides insights into the robustness of the model with respect to the effects
of hardware elements like DACs, ADCs, up/down converters, amplifiers, etc. We
captured the data on the hardware deployed at the state-of-the-art 5G testbed
at IIT Madras [4] to analyze model performance. The captures help incorporate
real-time channel conditions and hardware impairments that may/may not be
included in the training dataset. The setup (as shown in Fig. 5) consists of
an N5182B Vector Signal Generator (VSG) for transmitting the 5G signal at a
center frequency of $3.64962$ GHz (sub-6 GHz raster in the n78 band). The
commercial omnidirectional wideband monopole antenna transmits signals
originating from the VSG.

A multi-channel receiver front end connected to a dual-polarized antenna at
the base station receives the signals over the air (for this paper, we utilize
only one antenna and one transceiver chain). Other receiver components include
an in-house Low Noise Amplifier (LNA) with 60dB gain at the receiver front end
and an ADRV 9009 RF transceiver. We place the transmitter about one meter away
from the receiver to emulate one of the real-time wireless scenarios. The
PRACH signal is transmitted from the VSG through an antenna, over the air, and
then received at the base station’s antenna, amplified at the LNA, followed by
the transceiver. The signal out of the 16-bit ADC is then collected and used
for testing. A total of $56000$ OTA data instances are captured with the
hardware.

Figure 5: IIT-Madras 5G testbed setup with the Remote Radio Head used as a
receiver and the VSG used as a transmitter.

###  III-C Training and testing

In this paper, we consider $3$ cases for training and testing, as described
below. In all cases, we use a $75-25$% train-test split. Of the training
dataset, $30$% of the data is for validation. In all cases, training is done
with a combination of data corresponding to $8$ SNRs ($-15$dB to $20$dB in
steps of $5$dB).

####  III-C1 Training and testing on same channel scenario (Simulated
Datasets)

We train both the RAPID and TA classifier models with datasets that include
signals received over TDLC150 channels (Training dataset size = $600000$).
Testing is performed on data of the same channel scenario. We repeat the
process for datasets that include signals received over TDLC300 channels
(Training dataset size = $600000$). These experiments highlight the best-case
performance for the simulated datasets.

####  III-C2 Training and testing on different channel scenarios (Simulated
Datasets)

We generate datasets with signals received over TDLC10, TDLC150, and TDLC300
channels. First, we train the model with a combined dataset (Training dataset
size = $1800000$) containing signals received over all 3 channel conditions.
We then compare this training with signals received over only TDLC300 channels
(Training dataset size = 600000). Testing is done with TDLC150 data. The
experiments help analyze the generalization performance of the model across
channels of different delay spread values.

####  III-C3 Training and testing on OTA Hardware datasets

In the case of OTA hardware captures, we look at three scenarios. First, the
OTA test data is fed to the model trained on a mixed dataset containing
signals received on TDLC10, TDLC150, and TDLC300 channels. Second, the OTA
test data is fed to a model trained on a combination of the mixed simulation
data as well as some OTA data (Training dataset size = $1800000+42000$).
Lastly, the OTA test data is fed to a model trained purely with OTA data
(Training dataset size = $42000$). These experiments help determine the
generalization capability of the model across simulation and hardware data.

##  IV Results

This section shows the performance analysis of the proposed neural network
models and compares it with that of conventional correlation-based approaches.
Fig. 6 and Fig. 7 show the accuracy of RAPID detection and TA Estimation
respectively. We can see that NN clearly outperforms the correlation-based
approach in both cases. In the lower SNR range, the correlation-based approach
is impaired by both fading and noise. In the higher SNR range, the noise
reduces but the fading remains, leading to a saturation in the accuracy. This
is because in channels such as TDLC150 and TDLC300, the number of taps is high
and each tap, other than the first tap, causes a different shift in the
correlation peak. If the power of the first tap is significant compared to
other taps, the other correlation peaks will be insignificant. In the case of
TDLC150 and TDLC300 channels, we observed that the other taps have significant
power and the combined influence of those taps on the correlation peak leads
to incorrect detections. One shift in the detected peak from the correct
location leads to a shift in the estimated TA by one. For $N_{CS}$= 13, it
takes 13 shifts of the TA to make the detected RAPID incorrect by 1. Hence
RAPID detection is more robust than TA estimation.

Figure 6: RAPID Detection Accuracy of the NN compared with correlation-based
detection. Testing is performed on the same channel scenario as training
Figure 7: Timing Advance Estimation Accuracy of the NN compared with
correlation-based estimation. Testing is performed on the same channel
scenario as training

Fig. 8 and Fig. 9 focus on the generalization of the model in RAPID detection
and TA estimation. We observe that if we train the NN with a mixed dataset of
signals received over various channel scenarios, testing with one subset of
the dataset does not result in a drop in performance. Training with mixed
datasets, however, can be memory and time-prohibitive. Consequently, we train
the model with signals received over a more complex TDLC300 channel and test
with data from a less complex channel such as TDLC150. We note that there is
no significant drop in performance compared to the mixed training case.

Lastly, Fig. 10 shows the RAPID detection accuracy vs SNR for the OTA captured
data. When the neural network is trained with simulation data alone, it is
unable to capture the impairments that are caused by the hardware elements on
the receiver path such as ADC, filters, Low Noise Amplifiers (LNA), etc. To
include these impairments, we first train with a pure hardware dataset. This
is the best-case scenario, but it is impractical due to the complex nature of
the capturing process. A more practical training approach is to combine
hardware data with simulated data. Currently, we are limited by the amount of
captures that can be taken from the testbed. With a sufficiently large data
collection, the performance of combined training will approach pure hardware
training. Note that these hardware captures are taken in a lab environment
where the transmitter and receiver are one meter apart, resulting in a Line-
of-sight scenario. Hence, correlation-based decoding also performs well at
SNRs greater than -10 dB.

Figure 8: Generalization performance of the NN-based RAPID detection across
channel scenarios Figure 9: Generalization performance of the NN-based Timing
Advance estimation across channel scenarios. Figure 10: RAPID detection
accuracy vs SNR for OTA hardware datasets

##  V Conclusion

A robust PRACH receiver helps users connect to the network seamlessly. In this
work, we have built one such receiver that is implemented using two parallel
and independent Neural Networks, one for RAPID detection and another for TA
estimation. The NNs take frequency domain samples directly extracted from the
OFDM resource grid as input rather than the output of operations such as
correlation. Results show that the NN-based PRACH receiver outperforms
traditional correlation-based receivers for a single base sequence. This
indicates that our AI/ML-based receiver decodes the PRACH signal with a higher
probability, which will lead to a reduction in PRACH retransmissions. We have
also shown initial promising indications that the NN generalizes well across
channel scenarios of various delay spreads and Over-the-Air data captured on a
deployed base station. Future work will incorporate more base sequences, more
variations in channel scenarios (simulated and OTA), and multi-user collision
detection.

## Acknowledgment

The authors would like to thank the Department of Telecommunications (DOT),
India, for funding the 5G Testbed project and the Ministry of Electronics and
Information Technology (MeitY) for funding this work through the project ”Next
Generation Wireless Research and Standardization on 5G and Beyond”.

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