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j}^{N_{M,k}}\beta_{\tau,j^{\prime},k}^{\rm c}.$ (120)

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# Topological superconductivity in quantum wires mediated by helical magnons

Florinda Viñas Boström<EMAIL_ADDRESS>Institut für
Mathematische Physik, Technische Universität Braunschweig, D-38106
Braunschweig, Germany  Division of Solid State Physics and NanoLund, Lund
University, Box 118, S-221 00 Lund, Sweden  Emil Viñas Boström
<EMAIL_ADDRESS>Nano-Bio Spectroscopy Group, Departamento de Física
de Materiales, Universidad del País Vasco, 20018 San Sebastian, Spain  Max
Planck Institute for the Structure and Dynamics of Matter, Luruper Chaussee
149, 22761 Hamburg, Germany

###### Abstract

Triplet superconductivity arises in systems where the effective electronic
attraction favors Cooper pairs with a symmetric spin structure, as may occur
in non-collinear magnetic systems. Here we show that topological triplet
superconductivity can be realized in a one-dimensional quantum wire through
the interaction with magnetic fluctuations of a helical magnet. The magnon-
mediated attraction favors triplet superconductivity over a large sector of
magnetic parameters, and stabilizes a topological phase with a superconducting
gap of the order of $1$ meV over an extended region of chemical potentials.

A promising path to realize fault-tolerant quantum computing is to utilize the
excitations of topologically ordered systems Wen and Niu (1990); Kitaev
(2003); Nayak  _et al._ (2008). These excitations, generically named anyons
Leinaas and Myrheim (1977); Wilczek (1982), are predicted to appear across a
wide range of platforms, such as fractional quantum Hall systems, quantum spin
liquids, and topological superconductors Arovas  _et al._ (1984); Kitaev
(2006); Savary and Balents (2016); Bartolomei  _et al._ (2020); Nakamura  _et
al._ (2020). In particular, the sub-class of excitations called non-abelian
anyons allow for information to be non-locally encoded and processed in the
braiding patterns of the anyon world lines Kitaev (2003); Nayak  _et al._
(2008). In topological superconductors, non-abelian anyons appear at the edges
or in vortex cores of chiral two-dimensional superconductors Fu and Kane
(2008); Nakosai  _et al._ (2013); Sun  _et al._ (2016); Ménard  _et al._
(2017); Palacio-Morales  _et al._ (2019); Wang  _et al._ (2020); Kezilebieke
_et al._ (2020), and at the ends of one-dimensional (1-D) quantum wires
Flensberg (2010); Oreg  _et al._ (2010); Lutchyn  _et al._ (2010, 2018).
However, due to the difficulties associated with realizing such phases,
conclusive evidence of topological superconductivity has not been found in any
experimental platform.

Figure 1: Principles of magnon-mediated superconductivity. a, Experimental
setup with a quantum wire in proximity to a helical magnet. b, The helical
magnetic order (orange arrows) induces an effective spin-orbit interaction and
Zeeman splitting of the electronic bands via the spin-electron coupling $g$.
Magnon fluctuations around the equilibrium magnetic order provide an effective
attractive interaction among the electrons (blue arrows). c, Due to the non-
collinear magnetic structure, magnon fluctuations mediate scattering between
electrons with arbitrary spin projections $\sigma_{i}$.

It has recently been proposed that chiral topological superconductivity can be
realized in two-dimensional heterostructures Yang  _et al._ (2016); Rex  _et
al._ (2019); Bedow  _et al._ (2020); Díaz  _et al._ (2021); Mæland and Sudbø
(2022, 2023); Mæland  _et al._ (2023), where itinerant electrons interact with
a non-collinear magnetic structure, or in chains of magnetic atoms deposited
on a conventional superconductor Nadj-Perge  _et al._ (2014); Neuhaus-
Steinmetz  _et al._ (2022); Schneider  _et al._ (2023); Awoga  _et al._
(2023). In 1-D systems, superconductors with triplet ($p$-wave) pairing are
known to always reside in a topological class that can host non-trivial
topological phases Altland and Zirnbauer (1997); Kitaev (2001); Chiu  _et al._
(2016). To engineer topological superconductivity in 1-D, the dominant
approach has been to combine proximity-induced $s$-wave superconductivity with
strong spin-orbit coupling (SOC) semi-conductors and time-reversal symmetry
breaking magnetic fields Fu and Kane (2008); Law  _et al._ (2009); Stanescu
_et al._ (2011); Alicea  _et al._ (2011); Mourik  _et al._ (2012); Das  _et
al._ (2012); Finck  _et al._ (2013); Albrecht  _et al._ (2016); Deng  _et al._
(2016); Nichele  _et al._ (2017); Zhang  _et al._ (2019); Prada  _et al._
(2020); Aghaee et al. (2023). Although straightforward in principle, this
approach is in practice associated with a number of challenges, the most
prominent being that the time-reversal breaking field must be finely tuned
such that it is large enough to realize an effective spin-polarized regime,
but small enough not to suppresses the effective superconducting gap Fu and
Kane (2008); Oreg  _et al._ (2010); Lutchyn  _et al._ (2010). To avoid this
problem it would be desirable to find a protocol for topological
superconductivity where the spin-polarized regime and the superconducting gap
both are increasing functions of some control parameter.

To address this challenge, we here propose a platform to stabilize triplet
superconductivity by coupling a quantum wire to a helical magnet (see Fig. 1).
The non-collinear magnetic order induces an effective SOC and Zeeman field
among the electrons (see Fig. 2), thereby realizing an effective single-band
regime. The magnon excitations of the helical magnet further mediate an
effective attractive interaction between the electrons, which stabilizes
triplet superconductivity with a superconducting gap of the order of $1$ meV
over a large portion of the magnetic phase space. Within the effective single-
band regime, the system enters a topological phase, with unpaired Majorana
bound states at each end of the wire. Crucially, both the size of the
effective single-band regime and the superconducting gap are increasing
functions of the spin-electron coupling $g$. Our proposal thereby identifies
quantum wires in proximity to helical magnets as a promising platform to
explore topological superconductivity in 1-D.

To investigate magnon-mediated superconductivity in a quantum wire, we
consider an interacting system of spins and electrons described by the lattice
Hamiltonian

$\displaystyle H=$ $\displaystyle-t\sum_{\langle
ij\rangle\sigma}\hat{c}_{i\sigma}^{\dagger}\hat{c}_{j\sigma}-J\sum_{\langle
ij\rangle}\hat{{\bf S}}_{i}\cdot\hat{{\bf S}}_{j}$ (1)
$\displaystyle-\sum_{\langle ij\rangle}{\bf D}_{ij}\cdot(\hat{{\bf
S}}_{i}\times\hat{{\bf S}}_{j})-g\sum_{i}\hat{{\bf s}}_{i}\cdot\hat{{\bf
S}}_{i}.$

Here $\hat{c}_{i\sigma}$ destroys an electron at site $i$ and of spin
projection $\sigma$, and $\hat{{\bf S}}_{i}$ is the spin operator at site $i$
for a spin of magnitude $S$. The parameters $t$, $J$, and ${\bf D}$
respectively determine the nearest-neighbor electronic hopping amplitude, the
exchange interaction, and the antisymmetric Dzyaloshinskii-Moriya interaction
(DMI). The spins and electrons interact via a local spin-electron coupling of
strength $g$, and the electronic spin operator is defined by $\hat{{\bf
s}}_{i}=\sum_{\sigma\sigma^{\prime}}\hat{c}_{i\sigma}^{\dagger}\boldsymbol{\tau}_{\sigma\sigma^{\prime}}\hat{c}_{i\sigma^{\prime}}$,
with $\boldsymbol{\tau}$ the Pauli matrix vector.

The competition between exchange and DMI stabilizes a helical magnetic order,
characterized by a propagation vector ${\bf q}$ and a vector $\hat{\bf n}$
defining the plane of polarization. Specifically, the magnitude of the spiral
momentum is given by $\sqrt{2}\tan q=D/J$, while the directions of the
momentum and polarization vectors are determined by the normalized DMI vector
$\hat{\bf D}$. Here we consider a DMI such that the spiral momentum ${\bf q}$
lies along the wire axis $\hat{\bf z}$, and with $\hat{\bf n}$ parallel to
${\bf q}$ (see Fig. 1). The equilibrium spin texture is then a spin helix, and
can be written as ${\bf S}_{i}=\cos({\bf q}\cdot{\bf r}_{i}){\bf
e}_{1}+\sin({\bf q}\cdot{\bf r}_{i}){\bf e}_{2}$, where the vectors ${\bf
e}_{\alpha}$ and $\hat{\bf n}$ define a right-handed orthonormal system Sup .

In the following the magnetic texture is assumed to be commensurate with the
electronic lattice, such that the spin spiral is periodic over a distance
$La$, where $L$ is an integer and $a$ is the electronic lattice parameter. To
describe fluctuations around the equilibrium order, the spin operators
$\hat{\bf S}_{i}$ are expressed in terms of a set of bosonic operators by
performing a Holstein-Primakoff expansion around the local spin axis ${\bf
S}_{i}$ Sup . This results in a diagonal magnon Hamiltonian
$H_{m}=\sum_{np}\Omega_{np}\alpha_{np}^{\dagger}\alpha_{np}$, where
$\Omega_{np}$ is the energy of a magnon with momentum $p$ in band $n$. The
momentum runs over the magnetic Brillouin zone $[-\pi/La,\pi/La]$, and the
number of magnon bands is $L$.

There are two main effects of the spin spiral on the electronic structure:
First, the coupling to the equilibrium magnetic structure ${\bf S}_{i}$
induces an effective SOC and Zeeman splitting of the electronic bands (see
Fig. 2a). Second, the coupling to magnon fluctuations around the helical
configuration generates an effective attractive interaction among the
electrons, which ultimately leads to superconductivity. The effect of the
static spin spiral can be exactly accounted for, by diagonalizing the
electronic sub-system in presence of the spiral ${\bf S}_{i}$ Viñas Boström
_et al._ (2021); Sup , and results in the electron Hamiltonian
$H_{e}=\sum_{k\tau}\epsilon_{k\tau}d_{k\tau}^{\dagger}d_{k\tau}$. Here $\tau$
denotes the electronic bands, and the momentum runs over the electronic
Brillouin zone $[-\pi/a,\pi/a]$. The spiral shifts the minima of the
originally spin-degenerate bands to $\pm q/2$ (Fig. 2a), thereby inducing an
effective Rashba SOC. This shift is independent of the value of $g$, and is
set for any finite $g$ by the momentum of the spin spiral. In addition, the
coupling opens a gap of size $\Delta_{b}=2gS$ at $k=0$ and $k=\pm\pi/a$,
acting like an effective Zeeman field. Together these effects realize an
effective single-band regime for chemical potentials inside the gap.

Figure 2: Parameter dependence of magnon-mediated superconductivity. a,
Electronic band structure for a spin-electron coupling $g=150$ meV and a
spiral momentum $q=\pi/3a$. b, Electronic band structure around $k=0$ for a
spin-electron coupling $g=150$ meV and spiral momenta $q=0.44\pi/a$ (green),
$q=0.5\pi/a$ (gray) and $q=0.57\pi/a$ (blue). The red dashed line indicates
the chemical potential $\mu=-1.4$ eV. c, Symmetry of the dominant
superconducting gap as a function of spin-electron coupling $g$ and spiral
momentum $q$, for a chemical potential $\mu=-1.4$ eV. d, Magnitude of the
dominant superconducting gap as a function of spin-electron coupling $g$ and
spiral momentum $q$, for a chemical potential $\mu=-1.4$ eV. The gray dashed
lines indicate the positions of the van Hove singularities. In all panels the
spin length is $S=1$, the electronic hopping is $t=1$ eV, and the exchange
interaction is $J=10$ meV.

The electron-magnon interaction is found by expressing the spin-electron
coupling (the last term in Eq. 1) in terms of the magnon and band electron
operators. Due to the non-collinear structure of the spin spiral, the coupling
has a non-trivial spin structure (see Fig. 1c), and in general gives rise to
magnon-mediated scattering between electrons with arbitrary spin projections.
An effective electron-electron interaction is obtained by integrating out the
magnons within a finite temperature functional integral formulation Sup . The
effective interaction $U_{\tau_{1}\tau_{2}}^{\tau_{3}\tau_{4}}(k,k^{\prime})$
implicitly depends on the magnetic structure via the spin-electron coupling
$g$, the spiral momentum $q$, and the transformation matrices used to
diagonalize the magnon and electron sub-systems. In the static limit the
superconducting gap $\Delta_{k}^{\tau\tau^{\prime}}$ is determined via the
linearized gap equation

$\displaystyle\Delta_{k}^{\tau_{1}\tau_{2}}$
$\displaystyle=\sum_{k^{\prime}\tau_{3}\tau_{4}}U_{\tau_{1}\tau_{2}}^{\tau_{3}\tau_{4}}(k,k^{\prime})\Big{(}\sum_{\tau}\frac{\tanh(\beta\xi_{k^{\prime}\tau})}{2\xi_{k^{\prime}\tau}}\Big{)}\Delta_{k^{\prime}}^{\tau_{3}\tau_{4}},$
(2)

where $\beta=1/(k_{B}T)$ is the inverse temperature, and
$\xi_{k}=\epsilon_{k}-\mu$. The gap equation can be viewed as an eigenvalue
problem for the susceptibility matrix
$\chi_{\tau_{1}\tau_{2}}^{\tau_{3}\tau_{4}}(k,k^{\prime})$, implicitly defined
by the right hand side of Eq. 2, and superconductivity is signaled by the
largest eigenvalue of this matrix exceeding unity.

On the mean-field level the electronic Hamiltonian is of the Bogoliubov-de
Gennes (BdG) form, and can be written as
$H=\sum_{k}\Phi_{k}^{\dagger}\mathcal{H}_{k}\Phi_{k}$ with $\mathcal{H}_{k}$
as a $4\times 4$ matrix Sup . The gap $\Delta_{\tau\tau^{\prime}}$ can be
decomposed into four components, given by the singlet gap
$\Delta_{s}=(\Delta_{du}-\Delta_{ud})/2$ and the triplet gaps
$\Delta_{u}=\Delta_{uu}$, $\Delta_{d}=\Delta_{dd}$, and
$\Delta_{p}=(\Delta_{du}+\Delta_{ud})/2$. The subscripts $u$ and $d$ refer to
the band index $\tau$, which can be viewed as an effective spin index, where
$u$ ($d$) denotes the upper (lower) band or effective spin up (down). Due to
the fermionic nature of the electronic operators, the singlet and triplet gaps
are even and odd functions of $k$, respectively.

We begin with analyzing how the symmetry of the dominant superconducting gap
depends on the spiral momentum $q$, the spin-electron coupling $g$, and the
chemical potential $\mu$. For parameters such that the energy difference
$\epsilon_{u}(k_{Fu})-\epsilon_{d}(k_{Fd})\gg\Omega$, with $k_{F\tau}$ the
Fermi momentum in band $\tau$, and $\Omega$ a typical magnon energy, Cooper
pairs will predominantly form from states within the same band. In this
regime, the gap function becomes diagonal in the band indexes. At temperatures
below $10$ K, the pairing interaction is highly restricted to the Fermi
surface, which for a 1-D system consists of the points $\pm k_{F\tau}$.
Restricting the gap equation to the Fermi surface, and neglecting
contributions from inter-band scattering, Eq. 2 reduces to a $2\times 2$
matrix problem and can be solved analytically Sup . The eigenvalues of the
susceptibility matrix are $\lambda_{s}=\chi(k_{F},k_{F})+\chi(k_{F},-k_{F})$
and $\lambda_{p}=\chi(k_{F},k_{F})-\chi(k_{F},-k_{F})$, for singlet and
triplet pairing, respectively. These solutions show that the sign of the term
$\chi(k_{F},-k_{F})$, accounting for scattering between the points $k_{F}$ and
$-k_{F}$, determines the symmetry of the dominant gap. Since only triplet
pairing is consistent with the symmetry of intra-band Cooper pairs, a
consistent superconducting solution requires $\lambda_{p}>\lambda_{s}$. When
$\epsilon_{u}(k_{Fu})-\epsilon_{d}(k_{Fd})\lesssim\Omega$, inter-band
scattering becomes important, and the dominant superconducting gap is expected
to have $s$-wave symmetry.

This qualitative analysis is in good agreement with the numerical solution of
Eq. 2, showing that the dominant superconducting gap has triplet symmetry over
a large fraction of the phase diagram (see Fig. 2b). Only in regions where
$\epsilon_{u}(k_{Fu})-\epsilon_{d}(k_{Fd})\lesssim\Omega$, such as close to
the ferromagnetic limit $q=0$, or when both $k_{F\tau}$ are close to the band
edges (around $k_{F\tau}=0$), does the dominant gap have singlet symmetry. In
the strict ferromagnetic limit, this result can be established analytically
Sup , by noting that ferromagnetic magnons only scatter electrons with
opposite spins. That such a large portion of the phase diagram is dominated by
triplet pairing follows from the fact that singlet pairing requires inter-band
scattering. For $k_{Fd}\neq k_{Fu}$, such scattering results in Cooper pairs
with a finite center-of-mass momentum $k_{Fu}-k_{Fd}\sim q$, which strongly
suppresses singlet pairing at finite $q$ Loder  _et al._ (2010). With
increasing $q$, intra-band magnon scattering therefore quickly becomes the
dominant pairing mechanism, resulting in a gap with triplet symmetry. We note
that in the antiferromagnetic limit $q=\pi$, an effective gap opens between
the electronic bands, and only triplet pairing is possible.

We now analyze how the magnitude of the dominant superconducting gap depends
on the magnetic parameters. In the regime
$\epsilon_{u}(k_{Fu})-\epsilon_{d}(k_{Fd})\gg\Omega$, where inter-band
scattering can be neglected, the zero temperature mean-field gap is given by
$\Delta=2\Omega e^{-1/(\rho_{F}U)}$ Sup . Here $\rho_{F}$ is the electronic
density of states (DOS) at the Fermi level, and $U=U(k,k)-U(k,-k)$ the
effective pairing interaction in the triplet sector. Since $U\sim
g^{2}/\Omega$, the superconducting gap is expected to increase exponentially
with the spin-electron coupling. To estimate $\Delta$, we note that for an
approximately linear dispersion, the electronic DOS tends to the constant
$\rho_{0}=1/(4\pi t)$, while close to a band edge at energy $\epsilon_{0}$ van
Hove singularities of the form
$\rho(\epsilon)=\rho_{0}\sqrt{t/(\epsilon-\epsilon_{0})}$ develop.
Approximating $U$ with its value in the ferromagnetic limit,
$U=4g^{2}S/\Omega$, the dimensionless coupling strength is $\lambda_{\rm
eff}=\rho_{F}U\approx g^{2}S/(\pi t\Omega)$. For the typical values $S=1$,
$t=1$ eV and $\Omega=10$ meV, a spin-electron coupling of $g=100$ meV gives a
gap of $\Delta\sim 1$ meV, with significant enhancements expected at the van
Hove singularities.

The magnitude of the superconducting gap is also obtained from a numerical
solution of Eq. 2, whose largest eigenvalue $\lambda$ is related to the gap by
$\Delta=2\Omega e^{-1/\lambda}$. The magnitude of $\Delta$ is shown in Fig.
2d, and is found to generally increase with $g$ as anticipated above. However,
it also displays a clear non-monotonic behavior with $g$, which can be
attributed to the variation of the electronic DOS with the magnetic
parameters. In particular, a gap between the upper and lowers bands opens at
$k=0$ as a function of $g$, whose position varies with the spiral momentum $q$
(see Fig. 2b). For constant chemical potential, as assumed in Figs. 2c and 2d,
the DOS at the Fermi level therefore changes both with $g$ and $q$, and is
significantly enhanced at the van Hove singularities forming at the band
edges. The diagonal lines in Fig. 2d, emanating from $q\approx\pi/(2a)$, trace
the van Hove singularities as a function of $g$. The overall magnitude of the
gap, of order $\Delta\sim 1$ meV, is in line with the estimate above.

We now investigate the topological phases of the superconducting wire. In the
effective single-band limit, where the chemical potential lies inside the band
gap, the BdG Hamiltonian maps directly onto the Hamiltonian of the Kitaev
chain Kitaev (2001). This system is known to permit a topologically non-
trivial phase with unpaired Majorana bound states appearing at each end of the
quantum wire. Since the effective single-band limit of the model discussed
here can be connected to the Kitaev model by continuously deforming the
dispersion and gap function, its topological phase diagram is identical to the
Kitaev model Lutchyn  _et al._ (2010); Sup . It can be constructed by counting
the number of intersections $N$ between a line at constant chemical potential
and the dispersion, with the topological index given by
$\mathbb{Z}_{2}=(N/2)\mod 2$.

To obtain the topological phase diagram of the full two-band model, we first
consider the case where $\Delta_{u}\neq 0$ and $\Delta_{d}\neq 0$ but
$\Delta_{s}=\Delta_{p}=0$. In this limit, the model corresponds to two
independent copies of the Kiteav model, and a $\mathbb{Z}_{2}$ invariant can
be defined separately for each band. Denoting these invariants by $Z_{d}$ and
$Z_{u}$, the $\mathbb{Z}_{2}$ index of the full system is given by
$Z_{d}Z_{u}$. Since adding finite inter-band pairings cannot close the
superconducting gap, the full two-band model is continuously connected to the
$\Delta_{s}=\Delta_{p}=0$ limit, and for $\Delta_{u}\neq 0$ and
$\Delta_{d}\neq 0$ the phase diagram can be constructed by counting the number
of times $N$ a line at constant chemical potential crosses the bands
$\epsilon_{k\tau}$ (with $\mathbb{Z}_{2}=(N/2)\mod 2$, see Fig. 3). For $g=0$
the system is always in a trivial regime, while for $g\neq 0$ two regions with
non-trivial topology grow out of the upper and lower band edges. The size of
these regions increase linearly with the magnitude of $g$. We note the in the
antiferromagnetic limit the system is always in the trivial phase, since there
is no effective single-band regime in this case.

Figure 3: Topological phase diagram of magnon-mediated superconductivity as a
function of spin-electron coupling $g$ and chemical potential $\mu$. The
magnetic spiral has a wave length $L=8$ and momentum $q=\pi/(4a)$. The light
regions show the trivial phase, while the dark regions correspond to a
topological phase. The spin length is $S=1$ and the electronic hopping $t=1$
eV.

Our results demonstrate that topological superconductivity is stabilized
whenever the quantum wire is in an effective single-band regime. Such a regime
can be experimentally realized by placing the chemical potential inside the
$k=0$ or $k=\pm\pi/a$ band gap $\Delta_{b}$, induced by the coupling to the
spiral. Magnon scattering further opens a superconducting gap $\Delta$, whose
magnitude is shown in Fig. 2c and is approximately determined by the product
$g^{2}\rho_{F}/\Omega$. To favor topological superconductivity, it is
desirable to find a system with a large $g$ and $\rho_{F}$, but a small
$\Omega$ (however not too small, since $\Omega$ sets the upper limit of
$\Delta$).

To realize the model of Eq. 1, several strategies are possible. The most
straightforward is to consider a quantum wire in proximity to a helical
magnet, as illustrated in Fig. 1. In this case, the wire could consist either
of a simple metal, or a lightly doped or gate tuned semi-conductor (such as
InAs Nilsson  _et al._ (2018); Debbarma  _et al._ (2022)), with a single
active and largely uncorrelated band. For the helical magnet, it is preferable
to use a material with a short spiral wave length ($\lambda\sim 10$ nm), low
magnon energy and large spin length, such as MnGe, MnSi or NiI2 Nagaosa and
Tokura (2013); Song  _et al._ (2022). To maximize $g$, it is desirable to use
a wire geometry where as much as possible of the wire is strongly coupled to
the magnet. Estimating $g$ for a general interface is hard, but based on
earlier work it is expected that $g\sim 10-100$ meV can be achieved Qiao  _et
al._ (2014); Norden  _et al._ (2019); Mæland  _et al._ (2021); Cardoso  _et
al._ (2023).

We have demonstrated that magnon fluctuations of a helical magnet can mediate
triplet superconductivity in a proximate quantum wire. As the induced band gap
and superconducting gap both increase with the spin-electron coupling $g$,
various material engineering techniques such as wire geometry optimization,
microstructuring or cavity-enhanced light-matter couplings can be employed to
maximize its value Moll (2018); Sentef  _et al._ (2018); Nilsson  _et al._
(2018); Barrigón  _et al._ (2019); Viñas Boström  _et al._ (2023). We expect
that, utilizing such strategies, an effective coupling exceeding $g\approx 50$
meV can be achieved, allowing our protocol to surpass present state-of-the-art
experiments with a topological superconducting gap of the order $\Delta\sim
0.1$ meV Aghaee et al. (2023). The present model is a simplified description
of a real quantum wire, and in reality effects coming from repulsive electron-
electron interactions, disorder and a more complex band structure should be
included Das Sarma (2023). While the former is expected to compete with the
magnon-mediated attraction, the later could result in a larger DOS at the
Fermi level, thereby enhancing superconductivity. In conclusion, our results
identify quantum wires in proximity to non-collinear magnets as a promising
platform to explore topological superconductivity in 1-D.

###### Acknowledgements.

We acknowledge fruitful discussions with Oladunjoye A. Awoga. FVB acknowledges
funding from the Swedish Research Council (VR), and EVB acknowledges funding
from the European Union’s Horizon Europe research and innovation programme
under the Marie Skłodowska-Curie grant agreement No 101106809.

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# Real-Time Blind Photonic Interference Cancellation for mmWave MIMO

Joshua C. Lederman1, Weipeng Zhang1, Thomas Ferreira de Lima21, Eric C. Blow1,
Simon Bilodeau1, Bhavin J. Shastri3, Paul R. Prucnal1
1Department of Electrical and Computer Engineering, Princeton University,
Princeton, NJ 08544, USA
2NEC Laboratories America, Princeton, NJ 08540, USA
3Department of Physics, Engineering Physics & Astronomy, Queen’s University,
Kingston, ON K7L 3N6, Canada.

###### Abstract

Multiple-input multiple-output (MIMO) mmWave devices broadcast multiple
spatially-separated data streams simultaneously in order to increase data
transfer rates. Data transfer can, however, be compromised by interference.
Conventional techniques for mitigating interference require additional space
and power not generally available in handheld mobile devices. Here, we propose
a photonic mmWave MIMO receiver architecture capable of interference
cancellation with greatly reduced space and power needs. We demonstrate real-
time photonic interference cancellation with an integrated FPGA-photonic
system that executes a novel zero-calibration micro-ring resonator control
algorithm. The system achieves sub-second cancellation weight determination
latency with sub-Nyquist sampling. We evaluate the impact of canceller design
parameters on performance, establishing that effective photonic cancellation
is possible in handheld devices with less than 30 ms weight determination
latency.

##  I Introduction

Technological development has driven an ever-increasing demand for wireless
communication bandwidth[1, 2, 3]. With sub-7 GHz bands heavily utilized, the
industry has turned to 24-53 GHz (mmWave) bands to meet these growing needs
[4, 5]. At these frequencies, further capacity gains can be realized using
beamforming—the angular steering and filtering of radio-frequency (RF) signals
to optimize wireless communication links. mmWave beamforming may be
implemented using an array of mm-scale antennas in one package suitable for a
handheld device [3, 4, 6, 5]. Hybrid analog-digital beamformers are capable of
transmitting or receiving signals at multiple angles simultaneously, allowing
independent data streams to be transmitted along different spatial paths at
the same time (Fig. 1) [3, 7, 4, 8, 5]. This multiple-input multiple-output
(MIMO) approach multiplies the capacity of an RF link [3, 4, 6, 5].

Figure 1: mmWave interference scenarios. (a) A spatial channel associated with
one device interferes with another device’s spatial channels. (b) A malicious
source directs interference at a device. Figure 2: mmWave beamforming
receivers. (a) A conventional hybrid beamforming receiver. (b) A photonic-
electronic hybrid beamforming receiver. Each receiver includes a beamforming
phase-shifter array linking the antenna array to six logical antenna ports,
which provide signals ($r_{1},r_{2},\ldots r_{6}$). The conventional receiver
requires six RF chains to digitize the in-phase (I) and quadrature (Q)
components of all sources, with digital interference cancellation allowing the
extraction of target source signals $s_{1}$ and $s_{2}$. The photonic-
electronic receiver performs interference cancellation using RF photonics
prior to downconversion, and only two photonic-RF chains are required for the
two target signals. Sub-Nyquist analog-to-digital converters (ADCs) provide
samples from the target Q signals to drive the PBSS algorithm. DSP: Digital
Signal Processing, PD: Photo-Detector, PBSSE: PBSS Engine, SG: Statistic
Generator.

Beamforming receivers are imperfect, receiving signals from some off-target
angles, particularly those close to the angles of target signals. Interfering
signals arriving at such angles introduce noise that degrades network capacity
[5, 9, 10, 11, 12]. This interference may stem from the other spatial
channels, including those of other devices on the network using the same
spectral resources as in a multi-user MIMO system (Fig. 1a) [4, 5], or from a
malicious actor (Fig. 1b).

Interference may be mitigated by isolating each incoming signal according to
its angle of incidence and subtracting it from the target signals, improving
their signal-to-noise ratios (SNRs) [5, 9, 13, 14, 15]. When implemented
digitally, this technique requires an independent RF chain for every
interference source in order to generate the associated digital reference
signal. mmWave RF chains are costly in space and power, limiting their
quantity in handheld devices, where these resources are constrained [7, 4, 12,
16]. Analog interference cancellation, where the interference is subtracted
prior to signal digitization, addresses this limitation. However, it requires
precise amplitude control, and the variable-gain amplifiers used for analog RF
amplitude control face significant limitations related to circuit size and
complexity, gain resolution, and consistent performance across frequencies and
gain levels [17, 18]. Instead, we propose implementing mmWave MIMO
interference cancellation using RF photonics.

CMOS-compatible integrated silicon photonic devices have proven effective at
broadband, tunable, high-precision scaling of RF signals modulated onto
optical carriers [19, 20]. A single micro-ring resonator (MRR) has achieved 9
bits of amplitude weight precision and effective control of signals from DC to
19.2 GHz using only 1500 ${\mathrm{\SIUnitSymbolMicro m}}^{2}$ of chip area
[21]. Using the broadcast-and-weight architecture [22], a compact photonic
system can perform broadband weighting and summing of RF signals, enabling
photonic self-interference cancellation [23, 24], and photonic blind source
separation (PBSS) [25, 26, 27, 28, 29, 21].

In this work, we present a novel photonic mmWave beamforming receiver
architecture capable of broadband interference cancellation with simpler
hardware and lower power requirements than conventional alternatives. Our
proposed approach relies on _real-time_ photonic weight determination—weight
determination with a latency small compared to the timescale of environmental
change—which has not previously been reported. We demonstrate real-time
photonic interference cancellation, made possible by two innovations:
integration of an MRR weight bank with a field-programmable gate array (FPGA)
and the development of a zero-calibration MRR control scheme. We achieve
reliable sub-second cancellation weight determination latency with sub-Nyquist
sampling, features critical to real-world operation of the proposed system in
a handheld mobile device. We further establish that despite tradeoffs between
hardware complexity, latency, and accuracy, effective photonic interference
cancellation is possible with less than 30 ms weight determination latency.

Figure 3: Experimental blind interference cancellation setup. (a) Setup
diagram. (b) Scale schematic of the experimental MRRs (variation in N-doping
concentration not shown). (c) Packaged photonic chip and controller. (d)
ZCU216 FPGA development board. The experimental setup generates linearly mixed
signals in software and uses an arbitrary waveform generator (AWG) and Mach-
Zehnder Modulators (MZMs) to modulate them onto distinct wavelengths of light,
replacing the beamforming circuitry and input MRR bank of the proposed
receiver. An FPGA performs statistic generation and an integrated central
processing unit (CPU) acts as the PBSS engine. EDFA: erbium-doped fiber
amplifier, PC: polarization controller, I/O: input/output, RFSoC: RF system-
on-chip.

##  II Results

###  II-A A Photonic-RF mmWave MIMO Receiver

In a hybrid digital-analog beamforming receiver, each physical antenna is
connected by an array of splitters, combiners, and phase shifters to a smaller
number of logical antenna ports. Each logical port has an angular sensitivity
that may be independently controlled via the phase-shifter array. As a result,
each port provides a linear mixture of all incident source signals, with
weights dependent on the beamformer tuning, the signals’ angles of incidence,
and the environment. These received signals are connected to a set of RF
chains that perform filtering, downconversion, and digitization (Fig. 2a) [7,
4, 6, 5, 15].

The mixing process, neglecting non-interference noise sources, may be
described as:

$\bm{r}(t)=\bm{M}\bm{s}(t)$ (1)

where $\bm{s}(t)=\left\\{s_{1}(t),s_{2}(t),\ldots,s_{n}(t)\right\\}$
represents the $n$ independent source signals, the independent components
(ICs), which include both target and interference sources;
$\bm{r}(t)=\left\\{r_{1}(t),r_{2}(t),\ldots,r_{l}(t)\right\\}$ represents the
$l$ received signals via the logical antenna ports; and
$\bm{M}\in\mathbb{R}^{l\times n}$ represents the unknown mixing matrix. For
each target source signal $s_{i}(t)$, we seek to identify the cancellation
weight vector $\bm{c}_{i}$ that produces a demixed signal that most closely
approximates the source:

$s_{i}(t)\approx\bm{c}_{i}\cdot\bm{r}(t)=\bm{c}_{i}^{T}\bm{M}\bm{s}(t)$ (2)

We propose the novel photonic receiver architecture shown in Fig. 2b, in which
this vector multiplication is performed using analog photonics. Under our
approach, the received signals are modulated by broadband MRRs onto spaced
wavelengths produced by a co-integrated laser frequency comb [30, 31],
generating a wavelength-division multiplexed optical signal. The multiplexed
optical signal is split between multiple photonic-RF chains, each implementing
vector multiplication with tunable MRRs to perform weighting and balanced
photodetectors to perform summing in a broadcast-and-weight configuration
[22]. Filtering, downconversion, and digitization match that of the
conventional system.

###  II-B Photonic Blind Source Separation

As a consequence of the analog cancellation in the proposed receiver, not all
received signals are digitized. This constraint, in addition to the potential
lack of cooperation of interference sources, prevents the use of many
conventional channel estimation techniques. Instead, we apply PBSS, a proven
statistical approach which does not require full knowledge of the incident
signals or action by transmitters [25, 26, 27, 28, 29, 21]. PBSS consists of
an iterative algorithm that tests different photonic weight vectors $\bm{w}$
and evaluates the resulting mixed signal $m(t)$:

$m(t)=\bm{w}\cdot\bm{r}(t)$ (3)

PBSS requires two statistical measurements of $m(t)$: variance
$\sigma^{2}=\mathbb{E}\left[m^{2}(t)\right]$ and kurtosis
$\kappa=\mathbb{E}\left[m^{4}(t)\right]/\sigma^{4}-3$ (where it is assumed
$\mathbb{E}[m(t)]=0$ for a modulated RF signal).

PBSS relies on a property stemming from the principles of independent
component analysis (ICA): when $\bm{w}=\bm{c}_{i}$ for some $i$—i.e. when the
mixed output most closely matches one of the independent source
signals—$\kappa$ is at a local minimum. All such local minima can be
determined using principal component analysis (PCA). PCA, which involves
maximization of $\sigma^{2}$, enables the construction of a new weight basis
in which all $\bm{c}_{i}$ are orthogonal, ensuring they can be
deterministically identified.

In the proposed receiver, a low-latency statistic generator calculates
$\sigma^{2}$ and $\kappa$, a PBSS engine runs the $\bm{w}$ optimization
algorithm, and digital-to-analog converters (DACs) control the photonic
weights by tuning the MRRs (Fig. 2b). For the experimental demonstration of
real-time interference cancellation, artificial signals with a 1 GHz carrier
frequency serve as the received signals (see Methods). Critically, the setup
uses an FPGA chip for digital processing, with the FPGA fabric performing
statistic generation and the on-chip CPU running the optimization algorithm
(Fig. 3). A custom DAC board performs micro-ring tuning on the co-packaged
photonic chip.

###  II-C Real-Time PBSS

Figure 4: A typical experimental run of the PBSS algorithm with the
$\bm{M}_{2}$ mixing matrix (Eq. 6). Step 1 identifies the point of lowest
variance, $\bm{i}_{0}$. Step 2 consists of PCA and step 3 of ICA, each
optimizing directional vectors about the circle denoting the linear region. A
0 subscript and dashed lines denote initial positions, while solid lines
denote final positions. Step 4 optimizes the ICs, freeing them from the linear
region to reach higher-amplitude final positions denoted ′. PCs: principal
components.

$\bm{M}$ changes in response to the movement of people and objects in the
environment, but it is assumed to be static during PBSS. PBSS must therefore
run to completion before $\bm{M}$ meaningfully changes, operating in real-time
with low latency. Following PBSS, continuous kurtosis minimization can ensure
the target weight vectors remain accurate, so the latency of the initial PBSS
operation represents the limiting factor. PBSS consists of a series of weight-
set/signal-measure cycles (Fig. 3a), so PBSS latency is the product of cycle
count $N$ and average cycle latency $t$.

No prior work has achieved low-latency, real-time PBSS operation, with
previous PBSS demonstrations facing latencies in the minutes, incompatible
with practical application [21]. Calibration represents a particular
challenge. All previously reported applications of MRR photonic systems rely
on pre-calibration of the MRR weight bank to determine the nonlinear
relationship between the applied weighting currents $\bm{i}$ and the weights
$\bm{w}$ from Eq. 3 [26, 27, 21]. Calibration requires sweeps of current that
add dozens of processing cycles per micro-ring, a significant latency penalty,
and it may be compromised by MRR crosstalk [32]. Furthermore, MRR
characteristics can shift with operating temperature or optical input power,
necessitating frequent recalibration to maintain accuracy. To avoid these
challenges, we propose and demonstrate a novel zero-calibration, error-robust
approach to MRR control:

For properly chosen optical wavelengths, the photonic transfer function
$\bm{f}$, where $\bm{w}=\bm{f}(\bm{i})$, may be approximated as linear about
the zero-weight point $\bm{i}_{0}$ (see Supplementary Notes):

$\bm{f}(\bm{i})\approx\left(\left.\mathrm{D}\bm{f}\right|_{\bm{i}_{0}}\right)(\bm{i}-\bm{i}_{0})\quad\textrm{given}\quad|\bm{i}-\bm{i}_{0}|<i_{max}$
(4)

where $\bm{f}(\bm{i}_{0})=\bm{0}$,
$\left.\mathrm{D}\bm{f}\right|_{\bm{i}_{0}}\in\mathbb{R}^{l\times l}$
represents the matrix of partial derivatives of $\bm{f}(\bm{i})$ evaluated at
$\bm{i}_{0}$, and $i_{max}$ represents the maximum deviation from $\bm{i}_{0}$
where $\bm{f}$ may still be approximated as linear. It follows that the mixed
signal $m(t)$ may be approximated as such:

$m(t)=\bm{w}^{T}\bm{M}\bm{s}(t)\approx\underbrace{(\bm{i}-\bm{i}_{0})}_{\bm{w}^{\prime}}{}^{T}\underbrace{\left(\left.\mathrm{D}\bm{f}\right|_{\bm{i}_{0}}\right)^{T}\bm{M}}_{\bm{M}^{\prime}}\bm{s}(t)$
(5)

Under this approximation, there is an effective weight vector
$\bm{w}^{\prime}$, determined from the applied currents without calibration,
and an unknown effective mixing matrix $\bm{M}^{\prime}$, the product of
$\left.\mathrm{D}\bm{f}\right|_{\bm{i}_{0}}$ and the true mixing matrix
$\bm{M}$. As $\left.\mathrm{D}\bm{f}\right|_{\bm{i}_{0}}$ does not need to be
independently determined, only $\bm{i}_{0}$ must be identified.

Fig. 4 shows a typical experimental PBSS run under this approach with two
source and two received signals. PBSS consists of four steps. First, variance
is minimized to find $\bm{i}_{0}$, establishing the linear region as a circle
centered at $\bm{i}_{0}$ following Eq. 4. PCA and ICA (steps 2 and 3) operate
along the edge of this circle, balancing weighting linearity with mixed signal
SNR, which increases with amplitude. PCA consists of finding an orthogonal
basis in which each successive basis vector points in the direction of highest
variance. ICA performs an analogous process, though it operates in an adjusted
basis derived from PCA and seeks to minimize kurtosis (see Supplementary
Methods).

Estimates of the ICs after step 3 are subject to the cascaded errors of the
previous steps stemming from noise, sampling randomness, drift in $\bm{M}$,
and the linearity approximation. Furthermore, we seek to increase the
amplitude of the post-cancellation signal to maximize SNR, but that requires
pushing $\bm{i}$ out of the linear region. To address both problems, we add a
final step to the algorithm consisting of a kurtosis minimization over the
entire weight field, with each IC estimate serving as the initial position. As
noise raises the kurtosis, this step optimizes both the ICs’ weights and SNRs
without requiring the linearity assumption of Eq. 4. This constitutes step 4.
So long as steps 1-3 place the initial IC estimates within the correct convex
regions, the optimizer will find the correct kurtosis minima. Experimentally,
both the stronger and weaker ICs can be consistently identified and accurately
demodulated under a wide range of $\bm{M}$s.

Figure 5: PBSS performance and statistic estimator consistency. (a) PBSS
success rate as a function of $S^{2}$ SNR for $\bm{M}_{1}$ and $\bm{M}_{2}$.
The solid lines show linear fits to the data with imperfect success rates, and
dashed lines show the lowest SNR such that all PBSS attempts with higher SNRs
were successful—the SNR floor. (b) Consistency of $S^{2}$ and $K$ as a
function of $f_{s}$. (c) Consistency of $S^{2}$ and $K$ as a function of
$n_{s}$. Dotted lines show the true underlying statistic values, and error
bars show the statistic standard deviations. Estimator SNRs are plotted in
orange. SPS: samples per second.

To achieve real-time operation, we must not just minimize $N$ but also
minimize cycle latency $t$. Cycle latency consists of four non-negligible
components: signal acquisition latency $t_{a}$, statistic calculation latency
$t_{s}$, DAC communication latency $t_{c}$, and photonic weight stabilization
latency $t_{p}$ (Fig. 3a). Previous PBSS demonstrations were limited by a
$t_{s}$ in the 10s of seconds due to the oscilloscope sampling [21]. The novel
implementation of FPGA sampling and statistic generation in the experimental
system makes $t_{s}$ negligible. Instead, the setup is limited by a 2-6 ms
$t_{c}$, but it nevertheless achieves sub-second total latency. Digital
interface optimization, combined with high-frequency PN-junction micro-rings,
render both $t_{c}$ and $t_{p}$ negligible in the proposed system in
comparison to the signal acquisition latency $t_{a}$ [33]. $t_{a}$ represents
the duration required to collect a sufficient number of samples of $m(t)$ to
generate accurate signal statistics, and it sets the latency floor in blind
cancellation systems using ICA.

###  II-D Signal Acquisition and Statistic Generation

We collect $n_{s}$ samples of $m(t)$ at sampling rate $f_{s}$ to generate
estimates of $\sigma^{2}$ and $\kappa$, denoted $S^{2}$ and $K$, respectively.
Variation in these estimates between consecutive measurements, a result of
sampling randomness, may be quantified as an estimator SNR (see Methods). As a
series of statistic estimates drives the PBSS algorithm, low estimator SNRs
have the potential to degrade PBSS performance.

Fig. 5a shows the PBSS success rate as a function of $S^{2}$ SNR for two
representative mixing matrices:

$\bm{M}_{1}=\left(\begin{matrix}0.6&0.4\\\
0.4&0.6\end{matrix}\right)\quad\bm{M}_{2}=\left(\begin{matrix}1&0.5\\\
1&0.2\end{matrix}\right)$ (6)

$\bm{M}_{1}$ represents a symmetrical case in which there is a similarly
powerful interfering signal, corresponding to Fig. 1a. $\bm{M}_{2}$ represents
a case in which powerful jamming interference masks a weaker target signal,
corresponding to Fig. 1b.

PBSS failure typically occurs when an error in the estimate of an IC leaves it
outside the often small convex region with the desired kurtosis minimum,
resulting in a failure to optimize to that minimum. Variance measurements,
which occur earlier in the PBSS process, have a greater ability to cause
cascading errors that result in PBSS failure. Hence, $S^{2}$ SNR is strongly
predictive of PBSS success rate, as shown in Fig. 5a. We find that PBSS is
perfectly successful above a certain $S^{2}$ SNR floor, which depends on the
mixing scenario.

The selection of $n_{s}$ and $f_{s}$ dictates both the estimator SNR for a
given $\bm{M}$ and the interference canceller design. Digitization expense, in
hardware and power requirements, scales with $f_{s}$. With mmWave signals
transferring symbols at hundreds of MBaud or higher, sampling at a super-
Nyquist rate becomes costly in handheld devices [34]. Digitization hardware
and power draw can be greatly reduced by decreasing bit precision, with
precision as low as 1 bit being studied [7, 4, 16, 10]. Such an approach is
expected to become increasingly essential as carrier frequencies continue to
scale. Low-precision analog-to-digital converters (ADCs) can allow effective
signal demodulation, but they degrade the system’s ability to determine
accurate signal statistics to drive the PBSS algorithm.

To address this limitation, we propose the dual-ADC approach for mmWave PBSS
shown in Fig 2b. The demodulation ADCs, operating at a super-Nyquist rate with
low bit precision, take advantage of developments in efficient mmWave front
end design. An additional sub-Nyquist ADC per photonic-RF chain with
sufficient precision to allow accurate statistic generation provides data to
drive the PBSS algorithm. Periodic sub-Nyquist sampling has been successfully
demonstrated for PBSS, but the impact on estimator SNR and thereby PBSS
success rate has not be investigated [35, 29]. There is significant potential
for alignment between frequency components of the measured signal and the
sampling rate, generating data artifacts that reduce performance. This
motivates further analysis.

Fig. 5b shows estimator performance as a function of $f_{s}$, with $n_{s}$
fixed at 2,048. Estimator SNR is flat (with the exception of degradation at
higher sampling rates due to an experimental artifact discussed in
Supplementary Notes). The data indicates that deep sub-Nyquist sampling
represents an effective approach to PBSS.

A low, fixed $f_{s}$ also impacts sample acquisition latency $t_{a}$, the
primary latency component of the proposed system:

$t_{a}=\frac{n_{s}}{f_{s}}$ (7)

We seek to minimize $t_{a}$, and with $f_{s}$ limited by hardware requirements
we must turn to minimizing $n_{s}$. Fig 5c shows the impact of $n_{s}$ on
estimator SNR, with $f_{s}=7.68$ MSPS. The estimator SNRs fall sharply with
$n_{s}$, consistent with theory. By reducing $n_{s}$, we compromise the PBSS
success rate.

$f_{s}$ must be chosen in a tradeoff between hardware costs and latency, while
$n_{s}$ trades off between latency and success rate. Specific values depend on
the use case of the proposed system. One candidate set of sampling parameters,
$n_{s}=2^{14}$ and $f_{s}=122.9$ MSPS, achieves 25.6 dB $S^{2}$ SNR, above the
SNR floor for both mixing matrices tested. With $t_{a}=133\
$\mathrm{\SIUnitSymbolMicro s}$$, total PBSS convergence time for the proposed
system operating with these parameters falls below 30 ms for two sources,
suitable for all but the fastest-changing RF environments.

##  III Discussion

We propose a photonic mmWave MIMO receiver architecture capable of performing
interference cancellation without requiring downconversion and digitization of
the interfering signals, a significant cost in space and power. This system
has particular potential in handheld devices, where space and power are
constrained. By increasing mmWave mobile devices’ tolerance to interference,
greater levels of multi-user spatial multiplexing become feasible,
facilitating network capacity improvements [4, 9].

Our experimental results represent the first demonstration of real-time
applied photonic weight determination, enabled by a novel zero-calibration MRR
control approach and low-latency FPGA-photonic integration. We achieve sub-
second total PBSS latency, consistent with real-time operation even as
$\bm{M}$ shifts due to the movement of people and objects. We further quantify
the performance implications of sub-Nyquist sampling, an approach critical to
practical PBSS at mmWave frequencies. We identify 30 ms as an achievable PBSS
latency in the proposed system, compatible even with fast-changing RF
environments, such as those in moving vehicles.

The 1 GHz experimental mixed signals, with 200 MBaud symbol rates, more
accurately reflect a mmWave signal post-downconversion, rather than pre-
downconversion. While downconversion prior to photonic modulation is feasible,
the greatest hardware reduction may be achieved by performing cancellation
directly on the received signals. MRRs have demonstrated performance up to
19.2 GHz [21], and recent advancements in novel integrated silicon photonic
components show promise toward fully extending integrated photonics to the
mmWave domain [19, 36, 37].

We anticipate order-of-magnitude improvements in PBSS latency with the
implementation of available low-latency digital communication protocols (e.g.
serial peripheral interface) and high-frequency PN junction micro-rings [33].
Furthermore, the number of cycles per PBSS optimization step was fixed, and
dynamic adjustment of the cycles per optimization has the potential to both
reduce latency by terminating optimization early and increase performance by
extending or restarting optimization when convergence has not been achieved.
Better characterization of performance with multiple emitters and a variety of
mixing matrices is essential to validate the expected broad applicability of
the proposed approach. Synergies and conflicts between the photonic linear
mixer and the beamforming apparatus should also be explored.

##  IV Methods

###  IV-A Experimental Setup

The experimental source signals are two binary phase-shift keyed (BPSK)
signals consisting of distinct 1,137-bit repeating random sequences. Each has
a 200 MBaud data rate and a carrier frequency offset from 1 GHz by 176 kHz in
opposite directions to prevent artifacts generated from a perfect alignment
(the frequency offset is not used for signal discrimination). The signals are
mixed in software and generated by a Keysight N8196A 92 GSPS AWG, which
modulates the signals using MZMs onto distinct C-band laser frequencies
generated by two Pure Photonics PPCL500 lasers. The light is polarization-
controlled, amplified, and coupled onto a photonic chip, which performs linear
mixing. The output intensities are received by a Discovery Semiconductor
DSC-R405ER balanced photodetector, and the resulting signal is split between a
Tektronix DPO73304SX 100 GSPS oscilloscope and an analog input to the Xilinx
ZCU216 FPGA development board. The on-board CPU directs the system and
communicates with a separate custom weight control board, which contains DACs
to tune the micro-ring weights. The setup largely matches that of Zhang et.
al. [21].

###  IV-B Optimization Approach

TABLE I: Steps of the Proposed PBSS Algorithm. Step | Direction | Parameter | Domain | Repeats
---|---|---|---|---
1 | Minimize | Variance | Field | 1
2 | Maximize | Variance | Hypersphere | $a-1$
3 | Minimize | Kurtosis | Hypersphere | $a-1$
4 | Minimize | Kurtosis | Field | $a$

The PBSS algorithm consists of the optimizations listed in Table I, performed
using the Nelder-Mead derivative-free algorithm with a fixed forty iterations
per optimization [38]. $a$ represents the number of distinguishable
independent signals. The optimization domain for PCA and ICA consists of a
hypersphere centered at $\bm{i}_{0}$ with a 0.6 mA radius, corresponding to
the boundary of the linear region. Each component identified constrains the
optimization domain for later components under the requirement that later
components are orthogonal to earlier components in the original (PCA) or
whitened (ICA) basis. Hence the hypersphere dimensionality reduces by one
dimension for each optimization.

###  IV-C Statistic Generation

$n_{s}$ samples of $m(t)$, denoted $m_{1},m_{2},\ldots,m_{n_{s}}$, are used to
generate estimators $S^{2}$ and $K$ of the variance and kurtosis,
respectively:

$S^{2}=\frac{1}{n_{s}}\sum_{i=1}^{n_{s}}m_{i}^{2}\quad
K=\frac{1}{S^{4}}\frac{1}{n_{s}}\sum_{i=1}^{n_{s}}m_{i}^{4}-3$ (8)

The quality of these estimators for the purpose of PBSS is quantified by their
SNR. The SNR of an estimator $A$ is defined as the ratio of the estimator mean
$\mu(A)=\mathbb{E}[A]$ to the estimator standard deviation
$\sigma(A)=\sqrt{\mathbb{E}\left[\left(A-\mu(A)\right)^{2}\right]}$.

###  IV-D Data Collection

Estimator SNR and PBSS success rate data, shown in Fig. 5, were collected
under a fixed set of sampling rates $f_{s}$ and sample counts $n_{s}$. $f_{s}$
varied from 960 kHz to 1.966 GHz by power-of-2 scaling factors, and $n_{s}$
varied from $2^{8}$ to $2^{16}$ by powers of 2. PBSS success rate is defined
as the percentage of PBSS attempts, out of 32, which enable the successful
demodulation of both source signals with no bit errors over the full bit
sequence. Tests on all combinations of allowed $f_{s}$ and $n_{s}$ values are
shown in Fig. 5a. $S^{2}$ and $K$ SNR data shown in Fig. 5b-c was collected
with the $\bm{M}_{1}$ mixing matrix at the high-variance $\bm{i}=(0\
$\mathrm{mA}$,\ 3\ $\mathrm{mA}$)$ point, with estimator mean and standard
deviations calculated from 32 statistic measurements.

##  V Data Availability

The data generated for this work have been deposited in the Figshare database
and are are available at https://doi.org/10.6084/m9.figshare.22773689.

##  VI Code Availability

All code used in this study is available from the corresponding author upon
request.

##  VII Acknowledgements

This research is supported by the National Science Foundation (NSF)
(ECCS-2128616 to P. Prucnal and ECCS-1642962 to P. Prucnal), the Office of
Naval Research (ONR) (N00014-18-1-2297 to P. Prucnal and N00014-20-1-2664 to
P. Prucnal), and the Defense Advanced Research Projects Agency (HR00111990049
to P. Prucnal). The devices were fabricated at the Advanced Micro Foundry
(AMF) in Singapore through the support of CMC Micro-systems. J. C. Lederman
acknowledges support from the Department of Defense (DoD) through the National
Defense Science & Engineering (NDSEG) Fellowship Program. B. J. Shastri
acknowledges support from the Natural Sciences and Engineering Research
Council of Canada (NSERC). S. Bilodeau acknowledges funding from the Fonds de
recherche du Québec - Nature et technologies.

##  VIII Author Contributions

J.C.L., W.Z., and T.F.L. conceived the idea for the experiment. J.C.L.
conceived the application of PBSS to MIMO interference cancellation and
developed the proposed design and the zero-calibration MRR control scheme.
W.Z. developed the experimental photonic setup, including the DAC control
board and the associated control software. J.C.L. integrated the ZCU216 and
wrote the associated software, including the FPGA fabric design and the
implementation of the PBSS algorithm, with support from T.F.L. and W.Z.
T.F.L., W.Z., S.B., and E.C.B provided theoretical and experimental support.
J.C.L wrote the manuscript with support from T.F.L., B.J.S., and W.Z. P.P.
supervised the research and contributed to the vision and execution of the
experiment.

##  IX Competing Interests

The authors declare no competing interests.

##  X Materials & Correspondence

Correspondence and material requests should be addressed to J. C. Lederman.

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