# Generation of entangled waveguided photon pairs by free electrons

Theis P. Rasmussen  These authors contributed equally to this work
POLIMA—Center for Polariton-driven Light–Matter Interactions, University of
Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark  Álvaro Rodríguez
Echarri  These authors contributed equally to this work  ICFO-Institut de
Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860
Castelldefels (Barcelona), Spain  Max-Born-Institut, 12489 Berlin, Germany
Joel D. Cox  POLIMA—Center for Polariton-driven Light–Matter Interactions,
University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark
Danish Institute for Advanced Study, University of Southern Denmark, Campusvej
55, DK-5230 Odense M, Denmark  F. Javier García de Abajo
<EMAIL_ADDRESS>ICFO-Institut de Ciencies Fotoniques,
The Barcelona Institute of Science and Technology, 08860 Castelldefels
(Barcelona), Spain  ICREA-Institució Catalana de Recerca i Estudis Avançats,
Passeig Lluís Companys 23, 08010 Barcelona, Spain

###### Abstract

Entangled photon pairs are a key resource in future quantum-optical
communication and information technologies. While high-power laser light
propagating in bulk nonlinear optical crystals is conventionally used to
generate entangled photons that are routed into optical configurations, such
schemes suffer from low efficiency due to the weak intrinsic nonlinear optical
response of known materials and losses associated with photon in- and out-
coupling. Here, we propose a scheme to generate entangled polariton pairs
directly within optical waveguides using free electrons, whereby the measured
energy loss of undeflected electrons heralds the production of counter-
propagating polaritons pairs that are entangled in energy and direction of
emission. As a paradigmatic example, we study the excitation of plasmon
polaritons in metal strip waveguides that, within specific frequency regimes,
strongly enhance light-matter interactions that lead to two-plasmon generation
in comparison to the probability of single-plasmon excitation. We demonstrate
that, under appropriate conditions, an electron energy loss detected in an
optimal frequency range can reliably signal the generation of a plasmon pair
entangled in energy and momentum. Our proposed scheme can be directly applied
to other types of optical waveguides for in situ generation of entangled
photon pairs in quantum-optics applications.

## Introduction

Entanglement—the nonlocal quantum correlation between particles or
quanta—constitutes a key resource in emerging quantum information and
communication technologies, particularly in the areas of computation Bennett
and DiVincenzo (2000), cryptography Jennewein  _et al._ (2000), and
teleportation Ma  _et al._ (2012). The fragility of quantum states
unfortunately renders entangled particles extremely sensitive to decoherence
introduced by environmental factors. Photons, which interact weakly with their
surrounding environment and propagate at the ultimate speed of light, are
widely used as carriers of quantum information that is typically encoded in
their polarization or momentum states Mair  _et al._ (2001); Fickler  _et al._
(2012); Krutyanskiy  _et al._ (2023). Entangled photon pairs are routinely
generated through spontaneous parametric down-conversion, a second-order
nonlinear process that conserves both the photon polarization and momentum
Kwiat  _et al._ (1995, 1999); Arnaut and Barbosa (2000); Boyd (2020). The low
efficiency of nonlinear optical processes however necessitates the propagation
of intense phase-matched laser light in bulk crystals to generate a
significant response, thus circumscribing the generation of down-converted
photons to macroscopic material platforms from which entangled pairs are
collected and routed in cumbersome optical setups.

Numerous strategies have been proposed to circumvent the undesired decoherence
and losses associated with photon in- and out-coupling, for example, through
the efficient in situ generation of entangled photon pairs within an optical
waveguide via nonlinear optical processes Ilchenko  _et al._ (2004); Yang  _et
al._ (2008); Pomarico  _et al._ (2009); Helt  _et al._ (2015); Luo  _et al._
(2015); Guo  _et al._ (2017). In particular, the counter-propagating guided
photon pairs produced by spontaneous parametric down-conversion under plane
wave illumination impinging normally on an optical waveguide are entangled
through momentum conservation Echarri  _et al._ (2022); Sun  _et al._ (2022).
In addition, the possibility of generating entangled photons directly in
nanophotonic architectures presents appealing prospects for developing
integrated quantum-optical devices Tanzilli  _et al._ (2012). In this context,
polaritons—hybrid light-matter excitations involving polarization
charges—offer the means to confine and manipulate light on nanometer
lengthscales, well below the optical diffraction limit. These excitations can
be guided by engineering low-dimensional materials to form polaritonic
waveguides with enhanced dispersion Fang and Sun (2015); Rasmussen  _et al._
(2021). Notably, the extreme light concentration associated with plasmon
polaritons supported in ultrathin or two-dimensional (2D) materials can
substantially boost the intrinsic optical nonlinearity of their host,
potentially enabling nonlinear light-matter interactions on the few-photon
level Gullans  _et al._ (2013); Sun  _et al._ (2022); Calajó  _et al._ (2023).
Although polariton-based schemes favor mode volume at the expense of quality
factor compared to dielectric waveguides, both platforms rely on relatively
intense optical fields to trigger the generation of entangled photons.

As an alternative to far-field light sources, energetic electrons supply
broadband evanescent electromagnetic fields that can be focused with high
spatial precision to explore light-matter interactions at the nanoscale García
de Abajo (2010). The light emitted by bombarding a photonic structure with
electrons is measured in cathodoluminescence (CL) spectroscopy to spatially
and spectrally map optically active modes Losquin  _et al._ (2015), while
nonradiative excitations in matter can be probed in electron energy-loss
spectroscopy (EELS) by measuring the changes produced in the energy and
momentum distributions of scattered electrons Pettit  _et al._ (1975); Egerton
(1996). The ability of free electrons to generate heralded single photons with
unity-order efficiency was predicted by introducing the concept of phase-
matched propagation in a scheme where free electrons pass tangentially to a
curved waveguide Bendaña  _et al._ (2011), a possibility that has been
realized in recent experiments Feist  _et al._ (2022); Dahan  _et al._ (2020).
Entangled light-electron states are intrinsically generated when a free
electron couples to optical modes, such that each of the created excitations
is associated with a different state of the electron, therefore involving
different energies García de Abajo (2013) and transverse momenta Konečná  _et
al._ (2022). In this context, a strategy to produce distilled entanglement has
been formulated based on pre-shaping the electron wave function Konečná  _et
al._ (2022). In addition, photon entanglement triggered by the interaction
between free electrons and populated optical cavities has been proposed Di
Giulio and García de Abajo (2020); Baranes  _et al._ (2022).

In this work, we introduce a scheme to generate entangled polariton pairs in
the guided modes of an optical waveguide using free electrons. In particular,
we show that free electrons passing close and perpendicularly to a thin metal
waveguide can generate counter-propagating surface-plasmon polaritons that are
heralded in momentum- and energy-resolved measurements of the scattered
electrons. The resulting polariton pairs are entangled in energy and direction
of emission (left and right) in the waveguide. The proposed scheme capitalizes
on the strong near-field enhancement associated with confined plasmons in
ultrathin films to boost the light-matter interaction and excite multiple
guided modes. We anticipate that the entangled plasmon pairs (and photon pairs
after subsequent out-coupling) generated with this procedure under attainable
experimental conditions are advantageous compared to existing photonic devices
Chopin  _et al._ (2023) (e.g., we estimate $\sim 100$ MHz entangled pair
generation rates in our calculations below). In addition, the generation
process guarantees a high degree of synchronization between the two generated
plasmons.

Figure 1: Generation of entangled waveguided plasmons by free electrons. (a)
Illustration of an electron beam that passes close and perpendicularly to a
silver plasmonic waveguide (width $W$, thickness $t$) at a distance $b$ (the
impact parameter) and generates two counter-propagating guided plasmons with
wave vectors $k_{\parallel}$ and $-k_{\parallel}$ along the $y$ direction,
respectively. (b) Dispersion diagram showing the wave-vector- and frequency-
resolved EELS probability $\Gamma^{(1)}(k_{\parallel},\omega)$ calculated for
the silver waveguide depicted in panel (a) with parameters $W=50$ nm, $t=1.18$
nm [5 Ag(111) atomic layers], and $b=1$ nm. The electron velocity (energy) is
$v=0.1\,c$ ($\approx 2.6$ keV). Intense features emerge at the plasmon
dispersion curves. The light cone is indicated by solid black lines, while the
black square inside it shows a $(0.1/W)\times\,0.1$ eV region in
$(k_{\parallel},\hbar\omega)$ space within which electrons can be collected to
herald entangled plasmon-pair emission. (c) Plasmon dispersion relations in
the ribbon of panel (b), where green, purple, and yellow curves correspond to
the first, second, and third plasmon bands, respectively, the gray-shaded area
depicts the light cone, and the regions inside the colored frames denote
different combinations of entangled plasmons that can be generated when an
electron loses energy $\hbar\omega$ (upper black dot) and undergoes no net
momentum change along $\hat{\bf y}$ (undeflected electron). We define even and
odd channels corresponding to the excitation of degenerate (mask pair
$\\{1,1^{\prime}\\}$) and nondegenerate (mask pairs $\\{2,2^{\prime}\\}$ and
$\\{3,3^{\prime}\\}$) plasmon pairs. The sizes of the plasmon masks are
indicated by the black contours in (b) for one of the odd-channel
combinations.

## Results

### Configuration for entangled polariton-pair generation

We explore the generation of entangled waveguided polaritons in the form of
propagating modes supported by optical waveguides with translation invariance
in one spatial direction, taken along $\hat{\bf y}$. As a concrete example, we
consider plasmons supported by a silver ribbon with width $W$ and thickness
$t$ in the $\hat{\bf x}$ and $\hat{\bf z}$ directions, respectively, as
schematically illustrated in Fig. 1a. The ribbon thicknesses are taken to span
a few atomic layers, similar to those studied in recent experiments Abd El-
Fattah  _et al._ (2019). The waveguide supports plasmon bands that are
characterized by the frequency $\omega$ and wave vector $k_{\parallel}$ along
$\hat{\bf y}$. Plasmon modes are confined to the waveguide and lie below the
light cone $\omega=ck_{\parallel}$. In this configuration, a fast electron
passing perpendicularly to the waveguide with velocity vector ${\bf
v}=v\,\hat{\bf x}$ can transfer energy $\hbar\omega$ and momentum $\hbar
k_{\parallel}$ corresponding to the excitation of a plasmon mode that
propagates along $\pm\hat{\bf y}$, but is highly confined in the transverse
$x-z$ plane. The distance $b$ between the electron beam and the ribbon surface
needs to be made smaller than the out-of-plane evanescent decay length of the
plasmon in the surrounding vacuum (typically in the nanoscale). The frequency-
and wave-vector-resolved probability that the electron creates an excitation,
$\Gamma^{(1)}(k_{\parallel},\omega)$ (Fig. 1b, calculated by following the
methods in Refs. García de Abajo and Howie (2002); García de Abajo (2010) and
defined for $\omega>0$ and any real $k_{\parallel}$; see Methods) exhibits
pronounced features below the light cone (solid black line) that follow the
plasmon polariton dispersion relations of the waveguide and can be directly
measured through EELS. The EELS dispersion diagram in Fig. 1b is obtained for
an electron moving with velocity $v=0.1\,c$ ($\approx 2.6$ keV) and impact
parameter $b=1$ nm above a free-standing ultrathin silver strip waveguide with
dimensions $W=50$ nm and $t=1.18$ nm, corresponding to 5 Ag(111) atomic
monolayers (MLs). We use the boundary-element method García de Abajo and Howie
(2002) (BEM) with the metal response described by a background-corrected Drude
model parametrized to fit the experimental data of Ref. Johnson and Christy
(1972): $\epsilon(\omega)=\epsilon_{b}-\omega_{p}^{2}/\omega(\omega+{\rm
i}\gamma)$ with $\epsilon_{b}=4$, $\hbar\omega_{p}=9.17$ eV, and
$\hbar\gamma=21$ meV. We note that only a single waveguide mode can be excited
at low energies, while multiple modes are available at higher energies.

Although energetic free electrons are typically assumed to undergo weak light-
matter interactions that probe only the linear optical response, actual EELS
measurements contain contributions due to multiple excitation events, which
were observed in the first experimental evidence of the existence of surface
plasmons Powell and Swan (1959), while subsequent theoretical work showed that
coherent states with a Poissonian population of plasmon modes are created by
interaction with the free electrons Šunjić and Lucas (1971); Backes and Ibach
(1983); García de Abajo (2013). In particular, to second-order and neglecting
quantum-coherence effects, the total EELS distribution signal is described by
$\Gamma(k_{\parallel},\omega)=\Gamma^{(1)}(k_{\parallel},\omega)+\Gamma^{(2)}(k_{\parallel},\omega)$
(with units of time $\times$ distance), where the second term, quantifying the
probability of exciting two light quanta (i.e., double loss events), with net
energy and momentum matching those transferred from the electron, is given by
the self-convolution of the linear EELS probability,

$\displaystyle\Gamma^{(2)}(k_{\parallel},\omega)=$ (1)
$\displaystyle\int_{-\infty}^{\infty}dk_{\parallel}^{\prime}\int_{0}^{\omega}d\omega^{\prime}\,\Gamma^{(1)}(k_{\parallel}-k_{\parallel}^{\prime},\omega-\omega^{\prime})\,\Gamma^{(1)}(k_{\parallel}^{\prime},\omega^{\prime}).$

In what follows, we neglect triple and higher-order processes, which
contribute negligible under the conditions here considered.

Assuming that plasmon excitations dominate the electron energy-loss signal,
the leading contribution on the $k_{\parallel}=0$ axis—corresponding to
electrons that lose energy but are undeflected—can be attributed to specific
combinations of the guided modes excited in second-order processes, as
illustrated in Fig. 1c. Detection of an electron for an energy loss
$\hbar\omega$ along the $k_{\parallel}=0$ axis heralds the excitation of
plasmon pairs with momentum and energy $(\hbar
k_{\parallel\ell},\hbar\omega_{\ell})$ and $(\hbar
k_{\parallel\ell^{\prime}},\hbar\omega_{\ell^{\prime}})$ that satisfy
$k_{\parallel\ell}=-k_{\parallel\ell^{\prime}}$ and
$\omega=\omega_{\ell}+\omega_{\ell^{\prime}}$, where $\ell$ and
$\ell^{\prime}$ label plasmon pairs laying either on the same or in different
plasmon bands. The geometrical construction in Fig. 1c (see dashed straight
lines) shows that two different kinds of propagating-plasmon pairs can be
launched, depending on whether they are either degenerate
($\hbar\omega_{\ell}=\hbar\omega_{\ell^{\prime}}=\hbar\omega/2$, orange masks,
which we denote as the even excitation channel) or having different energies
($\hbar\omega_{\ell}\neq\hbar\omega_{\ell^{\prime}}$, red and blue masks, odd
channel). For spectrally narrow plasmons, only the dots labeled
$(k_{\parallel\ell},\omega_{\ell})$ in the figure contribute to losses at
$(k_{\parallel}=0,\omega)$. However, inelastic absorption produces plasmon
broadening, which results in finite-size regions contributing to such losses,
captured in our calculations through integration masks, as we discuss below.

The even channel corresponds to the simultaneous generation of two degenerate
plasmons, which can be regarded as the emission of a single plasmon heralded
by the detection of the other one. In contrast, the two odd-channel
combinations (red and blue masks, respectively) configure an entangled plasmon
pair of the form

$\displaystyle\ket{\text{left},\omega_{2}}\otimes\ket{\text{right},\omega_{3}}+\ket{\text{left},\omega_{3}}\otimes\ket{\text{right},\omega_{2}},$
(2)

formed by plasmons propagating to the left ($k_{\parallel}<0$) and to the
right ($k_{\parallel}>0$) with frequencies $\omega_{2}\neq\omega_{3}$ (see
Fig. 1c). We are thus interested in finding the conditions for which the odd
channel dominates the total energy-loss probability of undeflected electrons
(i.e., with $k_{\parallel}=0$ and $\omega=\omega_{2}+\omega_{3}$) compared
even-channel plasmons as well as non-plasmonic and radiative losses.

Figure 2: Quantifying entangled-plasmon generation. (a) Double-loss
probability obtained from the self-convolution of the EELS probability in Fig.
1b and dominated by two-plasmon events. (b) Calculated single- and two-plasmon
excitation probabilities in a silver waveguide of width $W=10$ nm and
thickness $t=0.236$ nm (1 ML) for an electron with impact parameter $b=1$ nm
and velocity (energy) $v=0.02\,c$ ($\approx 102$ eV) under the configuration
shown in Fig. 1a. Black dotted and solid curves correspond to the total
single-plasmon $\Gamma^{(1)}(k_{\parallel}=0,\omega)$ and two-plasmon
$\Gamma^{(2)}(k_{\parallel}=0,\omega)$ excitation probabilities. Blue and red
curves indicate the probabilities associated with even (orange masks in Fig.
1c) and odd (red and blue masks in Fig. 1c) excitation channels, respectively,
while the purple curve is the sum of both contributions. (c) Probabilities in
(b) plotted in a linear scale within a spectral region of interest (left
vertical scale), along with the fraction of odd-channel events in the total
two-plasmon emission probability (right scale) calculated with either the
numerical BEM (solid orange curve) or the analytical 2D model (dashed orange
curve).

### Plasmon-pair generation with silver ribbons

The probability that the electron undergoes two loss events is given by the
self-convolution of the probability associated with single energy-loss events,
as shown in Eq. (1) and represented in Fig. 2a for a ribbon of width $W=10$ nm
and thickness $t=0.236$ nm (i.e., 1 ML), considering an electron with impact
parameter $b=1$ nm and velocity $v=0.02\,c$. This double-loss probability
distribution features dominant contributions from two-plasmon excitations.
However, other types of events such as non-plasmonic and radiative losses
contribute as well (e.g., through the background signal within the light
cone).

To isolate the contribution from plasmon processes that produce an EELS signal
along the $k_{\parallel}=0$ axis (undeflected electrons), we specify mask
functions $\Theta_{\ell}(k_{\parallel},\omega)$ in momentum–frequency space,
such that $\Theta_{\ell}$ is unity within a curved mask centered around a
specific region within a given plasmon dispersion band $j$ and zero elsewhere
(see Fig. 1c). More precisely, each of the curved mask areas spans a wave
vector range $\pm n_{k_{\parallel}}\Delta k_{\parallel}$ around a central
value $k_{\parallel\ell}$ [determined by the condition that the total wave-
vector–frequency exchange is $(k_{\parallel}=0,\omega)$] as well as a
frequency range $\pm n_{\omega}\Delta\omega$ around the
$k_{\parallel}$-dependent plasmon frequency $\omega_{jk_{\parallel}}$, where
$\Delta\omega=\gamma$ (the Drude damping of the metal), $\Delta
k_{\parallel}=\Delta\omega/v_{jk_{\parallel}}$,
$v_{jk_{\parallel}}=\partial\omega_{jk_{\parallel}}/\partial k_{\parallel}$ is
the plasmon group velocity evaluated at $k_{\parallel}=k_{\parallel\ell}$, and
we set $n_{k_{\parallel}}=10$ and $n_{\omega}=5$ in the calculations as
reasonable scaling factors that allow us to have well-separated plasmons and a
fair representation of the fraction of signal associated with plasmon
launching. As an illustration of these definitions, the two mask contours of
one of the odd-channel combinations are plotted in Fig. 1b (black).

The probababilities for even- and odd-channel plasmon-pair excitation,
$\Gamma^{(2)}_{\text{even}}$ and $\Gamma^{(2)}_{\text{odd}}$, are thus
calculated by restricting the convolution in Eq. (1) to the corresponding mask
areas. We then have

$\displaystyle\Gamma^{(2)}_{\text{even}}(k_{\parallel},\omega)=\int_{-\infty}^{\infty}dk_{\parallel}^{\prime}\int_{0}^{\omega}d\omega^{\prime}\,\Theta_{1^{\prime}}(k_{\parallel}-k_{\parallel}^{\prime},\omega-\omega^{\prime})\Theta_{1}(k_{\parallel}^{\prime},\omega^{\prime})\;\Gamma^{(1)}(k_{\parallel}-k_{\parallel}^{\prime},\omega-\omega^{\prime})\Gamma^{(1)}(k_{\parallel}^{\prime},\omega^{\prime})$
(3a)
$\displaystyle\Gamma^{(2)}_{\text{odd}}(k_{\parallel},\omega)=2\int_{-\infty}^{\infty}dk_{\parallel}^{\prime}\int_{0}^{\omega}d\omega^{\prime}\,\Theta_{2^{\prime}}(k_{\parallel}-k_{\parallel}^{\prime},\omega-\omega^{\prime})\Theta_{2}(k_{\parallel}^{\prime},\omega^{\prime})\;\Gamma^{(1)}(k_{\parallel}-k_{\parallel}^{\prime},\omega-\omega^{\prime})\Gamma^{(1)}(k_{\parallel}^{\prime},\omega^{\prime}).$
(3b)

The factor of two in Eq. (3b) accounts for the two possible odd-channel
combinations indicated by the red and blue masks in Fig. 1c, which have the
same probability (i.e., the same result is obtained when substituting the
$\\{2,2^{\prime}\\}$ subindices by $\\{3,3^{\prime}\\}$).

In Fig. 2b, we analyze single- and two-plasmon excitation probabilities in a
free-standing silver waveguide characterized by a width $W=10$ nm and a
thickness $t=0.236$ nm (1 ML). We take the electron to pass with impact
parameter $b=1$ nm and velocity $v=0.02\,c$ ($\approx 102$ eV) above the
waveguide. Specifically, we plot the probabilities for single-plasmon
excitation $\Gamma^{(1)}$ (dashed black curve) and two-plasmon excitation
$\Gamma^{(2)}$ (solid black curve) together with the decomposition of the
latter into even [$\Gamma^{(2)}_{\text{even}}$] and odd
[$\Gamma^{(2)}_{\text{odd}}$] channels. At low energies, for which the ribbon
only supports a single plasmon mode, the even-channel contribution to the
second-order process dominates the EELS probability. In contrast, above the
single-mode threshold, $\Gamma^{(2)}_{\text{even}}$ decreases rapidly in favor
of the odd-channel contribution $\Gamma^{(2)}_{\text{odd}}$ [nondegenerate
entangled plasmon-pair launching, see Eq. (2)]. Interestingly, the single-
plasmon loss channel $\Gamma^{(1)}$ displays a series of peaks associated with
leaky modes inside the light cone that radiate into free space, thereby
reducing the efficiency of entangled plasmon-pair production. We identify the
frequency ranges for which $\Gamma^{(2)}_{\rm odd}$ is maximized and
$\Gamma^{(1)}$ takes minimum values as ideal for heralding entangled plasmon-
pair generation. In particular, in the $1.6-2$ eV energy-loss region, also
shown in Fig. 2c in linear scale, the sum of even- and odd-channel
probabilities accounts for most of the double-loss processes. Furthermore, the
odd channel accounts for $\sim 80\%$ of the loss probability (see Fig. 2c,
orange curve, and right scale), and consequently, the detection of an electron
under such circumstances signals the emission of an entangled photon pair
given by Eq. (2) with high probability. These results can be improved by
playing with the mask parameters so that the probability associated with
entangled (odd-channel) plasmon-pair emission increases with size. The
contribution of even and odd channels becomes an increasingly smaller fraction
of the total double-loss processes at higher loss energies due to the
excitation of newly available modes, therefore degrading the fidelity of the
entangled pair generation scheme. Indeed, as observed in Fig. 2a at energies
above 1.5 eV, more complex combinations of higher-order modes contribute to
the total EELS signal and introduce further channels to generate multiple
entangled states, thus impeding the ability to confidently distinguish
entangled pairs.

To quantitatively estimate the plasmon pair production efficiency, we
integrate the electron scattering probability density over a region spanning
$\sim 0.1$ eV in energy loss $\hbar\omega$ and $0.1/W$ in lateral wave vector
$k_{\parallel}$, as indicated schematically by the small black square in Fig.
1b. For the 10 nm silver waveguide considered in Fig. 2c, the probability
density $\sim 0.5$ nm eV-1 multiplied by the area of the considered detection
window leads to $5\times 10^{-4}$ entangled plasmon pairs generated per
incident $\sim 100$ eV electron. Considering an electron current of 1 $\mu$A
($6\times 10^{12}$ electrons per second), this generation yield amounts to a
rate of the order of $\sim$3 GHz entangled pairs.

### 2D material limit

We find the enhanced mode confinement offered by smaller waveguides produces
larger yields in the free-electron generation of guided mode pairs, and thus,
motivating a more exhaustive exploration of narrow 2D nanoribbons. The optical
response of metallic nanoribbons with a lateral size well below the free-space
light wavelengths at the frequencies of the supported plasmon resonances is
accurately described in the quasistatic approximation, with the material
response characterized by a 2D conductivity $\sigma(\omega)={\rm i}\omega
t[1-\epsilon(\omega)]/4\pi$, provided the thickness satisfies the condition
$t\ll W$. (Note that we use Gaussian units throughout this work.) In this
limit, we can consider a general Drude response
$\sigma(\omega)=(e^{2}/\hbar)\,{\rm i}\omega_{D}/(\omega+{\rm i}\gamma)$
characterized by a phenomenological damping rate $\gamma$ and a weight
$\omega_{D}$ [e.g., $\omega_{D}={E_{F}}/\pi\hbar$ for graphene doped to a
Fermi energy ${E_{F}}$; or also $\omega_{D}=\hbar\omega_{p}^{2}t/4\pi e^{2}$
for a Drude metal film of permittivity
$\epsilon(\omega)=1-\omega_{p}^{2}/\omega(\omega+{\rm i}\gamma)$].

Following previous works García de Abajo (2013); Mkhitaryan  _et al._ (2021),
we compute the induced field along the electron trajectory by projecting the
external electron field on the eigenmodes associated with the ribbon geometry.
Taking $x$ and $y$ across and along the ribbon-parallel directions (see Fig.
1a), respectively, we obtain the expression

$\displaystyle\Gamma^{(1)}(k_{\parallel},\omega)=\sum_{j}{\rm
Im}\bigg{\\{}\frac{1}{\eta_{\omega}^{-1}-\eta_{jk_{\parallel}}^{-1}}\bigg{\\}}\;\mathcal{I}_{j}(k_{\parallel},\omega)$
(4a) with $\displaystyle\mathcal{I}_{j}(k_{\parallel},\omega)$
$\displaystyle=\frac{2e^{2}}{\hbar
v^{2}}\frac{k_{\parallel}^{2}}{\kappa^{2}}{\rm e}^{-2\kappa b}$ (4b)
$\displaystyle\times\bigg{|}\int_{-W/2}^{W/2}dx\,{\rm e}^{{\rm i}\omega
x/v}\vec{\mathcal{E}}^{*}_{jk_{\parallel}}(x)\cdot\bigg{(}\frac{\omega}{\gamma_{L}^{2}vk_{\parallel}}\hat{\bf
x}+\hat{\bf y}\bigg{)}\bigg{|}^{2}$

for the EELS probability (see Methods for a self-contained derivation), where
$\vec{\mathcal{E}}_{jk_{\parallel}}$ are eigenmode fields labeled by the mode
index $j$ and wave vector $k_{\parallel}$ that satisfy the orthonormality
relation
$\int_{-W/2}^{W/2}dx\;\vec{\mathcal{E}}^{*}_{jk_{\parallel}}(x)\cdot\vec{\mathcal{E}}_{j^{\prime}k_{\parallel}}(x)=W\delta_{jj^{\prime}}$;
the corresponding real eigenvalues $\eta_{jk_{\parallel}}$ determine the
plasmon resonance dispersion through the condition
$\eta_{\omega}=\eta_{jk_{\parallel}}$ (see Methods); and the parameter
$\eta_{\omega}={\rm i}\sigma(\omega)/\omega W$ encapsulates the material-
dependent optical response through the 2D conductivity $\sigma(\omega)$
discussed above. Although the ribbon is treated electrostatically,
relativistic effects in the interaction with the electron are incorporated
through the Lorentz factor $\gamma_{L}=1/\sqrt{1-v^{2}/c^{2}}$, which also
enters the exponential attenuation with impact parameter $b$ through
$\kappa=\sqrt{k_{\parallel}^{2}+(\omega/v\gamma_{L})^{2}}$. By construction,
all the inelastic signal is associated with plasmon launching in this model.

We repeat the analysis in Fig. 2 but use the 2D model [Eqs. (4)] to calculate
the loss probability. The obtained fraction of odd-channel pair generation
(Fig. 2c, dashed orange curve) is similar to the BEM simulation (solid orange
curve). Probabilities for first-order and second-order odd- and even-channel
processes also compare reasonably well, considering that the ribbon aspect
ratio $W/t\sim 40$ is still not too large.

Figure 3: Universal plots of two-plasmon launching in 2D ribbons. (a) Ratio of
two-plasmon to single-photon excitation probabilities
$\Gamma^{(2)}(0,\omega)/\Gamma^{(1)}(0,\omega)$ as a function of electron
velocity $v$ and frequency loss $\omega$ (normalized to speed of light $c$ and
Drude weight $\omega_{D}$, respectively) for undeflected electrons
($k_{\parallel}=0$) after interaction with a 2D ribbon under the conditions of
Fig. 1b. The color scale is saturated at $10^{\pm 2}$. (b,c) Fractions
$\mathcal{F}_{\text{even/odd}}=\Gamma^{(2)}_{\text{even/odd}}\big{/}\big{[}\Gamma^{(1)}+\Gamma^{(2)}\big{]}$
(see upper labels) of undeflected electrons that generate even- and odd-
channel plasmon pairs. All results are obtained from the analytical 2D model
with parameters $\gamma/\omega_{D}=0.01$, $b/W=0.02$, and
$\omega_{D}W/c=0.081$.

The analytical 2D model allows us to explore entangled plasmon-pair generation
over a wide range of material, geometry, and electron parameters. In addition,
upon inspection of Eqs. (4), we find that the scaled probability
$cW^{-2}\Gamma^{(1)}(k_{\parallel},\omega)$ is a function of $k_{\parallel}W$,
$\omega/\omega_{D}$, and $v/c$ that also depends on the parameters
$\omega_{D}W/c$, $\gamma/\omega_{D}$, and $b/W$. The same scaling is inherited
by $cW^{-2}\Gamma^{(2)}(k_{\parallel},\omega)$ and its even- and odd-channel
contributions in virtue of Eqs. (3). Consequently, when probabilities are
integrated over wide masks (see above), the fractions of odd- and even-channel
events for undeflected electrons

$\displaystyle\mathcal{F}_{\text{even/odd}}=\frac{\Gamma^{(2)}_{\text{even/odd}}(0,\omega)}{\Gamma^{(1)}(0,\omega)+\Gamma^{(2)}(0,\omega)}$
(5)

are found to be functions of $\omega/\omega_{D}$, $v/c$, $\omega_{D}W/c$,
$\gamma/\omega_{D}$, and $b/W$. In what follows, we set $b/W=0.02$ (grazing
electron beam) and explore the dependence of $\mathcal{F}_{\rm even/odd}$ on
the rest of the parameters. In addition, Fig. 3a shows the ratio of two-
plasmon to single-plasmon EELS signals calculated from Eqs. (1) and (4) and
plotted as a function of $\omega/\omega_{D}$ and $v/c$ for
$\gamma/\omega_{D}=0.01$ and $\omega_{D}W/c=0.081$. These parameters can be
encountered in polariton-supporting ribbons made of atomically thin materials
such as graphene Ni  _et al._ (2018), hexagonal boron nitride (hBN) Giles  _et
al._ (2018), and MoO3 Ma1  _et al._ (2018) films. We observe that two-plasmon
events dominate over a broad range of electron velocities and energy losses.
Furthermore, although the fractional contribution $\mathcal{F}_{\rm even}$ of
even-channel processes is relatively small (Fig. 3b), we find that entangled
(odd-channel) plasmon pairs reach a broad maximum nearing $\mathcal{F}_{\rm
odd}\approx 80\%$ of the total events associated with the detection of
undeflected electrons (Fig. 3c). These conclusions are maintained for more
lossy ribbons: the maximum fraction of entangled plasmon pairs ($\approx
80\%$) is found to be rather independent of $\gamma$, while the ratio
$\Gamma^{(2)}/\Gamma^{(1)}$ takes large values that scale roughly as $\propto
1/\gamma$. Importantly, to obtain a large fraction of entangled plasmon-pair
generation, the electron energy loss $\hbar\omega$ needs to lie between the
second and third plasmon bands ($1.1\lesssim\omega/\omega_{D}\lesssim 1.7$,
approximately the range plotted in Fig. 3), while higher-order bands produce a
substantial reduction in such fraction at higher energies (not shown). In
addition, the absolute probability of entangled-pair generation
$\Gamma^{(2)}_{\rm odd}$ exhibits a broad maximum piling up near the second
plasmon band ($\omega\gtrsim 1.1\omega_{D}$) and velocities $v\sim 0.02\,c$
(not shown), where the absolute maximum is found (dot in Fig. 3c).

## Discussion

Our proposal to generate entangled photon pairs directly within optical
waveguides using free electrons constitutes an alternative approach to schemes
based on spontaneous parametric down-conversion of intense optical fields in
bulk nonlinear optical materials. Here, the production of photons entangled in
energy and momentum degrees of freedom can be heralded by the detection of
electrons that lose a given amount of energy and are undeflected in the
waveguide direction. To explore the feasibility of the present scheme, we have
simulated the probabilities of generating counter-propagating plasmons
supported by ultrathin silver waveguides that facilitate strong light-matter
interactions with electrons passing along a grazing perpendicular trajectory
at nonrelativistic velocities that can be achieved in integrated electron
sources. Our results indicate that the enhanced field confinement offered by
thin (a few atomic layers) plasmonic waveguides can improve the fidelity of
entangled pair generation, which we quantify by contrasting the probability of
single- and two-plasmon generation. In particular, we identify the production
of nondegenerate counter-propagating plasmons (odd excitation channel) at low
frequencies as the optimal configuration to herald entangled pairs with minor
contamination from undesired single-plasmon emission and non-guided radiated
photons.

In a practical example using silver ribbons, we predict that $\sim
10^{-4}-10^{-5}$ plasmon pairs are generated per incident electron under
attainable conditions by bombarding silver ribbons of $10-100$ nm width and
few-atomic layer thickness with $10^{2}-10^{4}$ eV electrons. Considering an
electron current of 1 $\mu$A, this generation yield amounts to a rate of $\sim
0.1-1$ GHz entangled pairs. To avoid electron-electron Coulomb repulsion for
such high currents, the beam could be spread along the length of the ribbon at
the expense of a small degradation in the temporal synchronization of the two
generated plasmons ($\sim 1$ fs per micron of beam width).

Our exploration of general polaritonic materials based on a rather accurate 2D
model description of thin polariton-supporting ribbons shows that these
results can be extended to long-lived plasmons in graphene Ni  _et al._ (2018)
and hyperbolic phonon-polaritons in hBN Giles  _et al._ (2018) and MoO3 Ma1
_et al._ (2018), which feature large quality factors $\omega/\gamma\sim
10^{2}$ and should thus provide suitable platforms for the implementation of
the proposed electron-driven entangled-plasmon-pair production at comparably
high rates.

As the emission is heralded by the detection of undeflected electrons that
have lost $\lesssim 1$ eV energy (the total generated photon energy), the
signal has to be separated from the background of losses associated with the
direct excitation of a single quantum. We find that 80% of all loss events of
undeflected energy-filtered electrons are associated with the emission of
entangled polariton pairs under a broad range of material, geometry, and
electron parameters. Our proposed mechanism for free-electron-based entangled
pair production, which can be straightforwardly applied to alternative
nanophotonic or optical waveguide geometries of interest, constitutes a
versatile and low-power approach to produce heralded photons directly within
waveguiding architectures for present and future integrated quantum-photonics
technologies.

## Methods

### EELS simulations

The classical EELS probability can be written as García de Abajo (2010)

$\displaystyle\Gamma^{(1)}(\omega)=\frac{e}{\pi\hbar\omega}\int_{-\infty}^{\infty}\\!\\!\\!dt\;{\rm
Re}\big{\\{}{\rm e}^{-{\rm i}\omega t}\;{\bf v}\cdot{\bf E}^{\rm ind}({\bf
r}_{0}+{\bf v}t,\omega)\big{\\}}$ (6a) in terms of the frequency-space self-
induced electric field ${\bf E}^{\rm ind}({\bf r}_{0}+{\bf v}t,\omega)$ acting
back on the electron as it moves along a trajectory ${\bf r}={\bf r}_{0}+{\bf
v}t$ with constant velocity vector ${\bf v}$ (nonrecoil approximation García
de Abajo (2010)) and passing by the position ${\bf r}_{0}$ at time $t=0$.
Given the translational invariance of the ribbon along $y$ (see Fig. 1a), we
can decompose the EELS probability as
$\Gamma^{(1)}(\omega)=\int_{-\infty}^{\infty}dk_{\parallel}\;\Gamma^{(1)}(k_{\parallel},\omega)$,
where
$\displaystyle\Gamma^{(1)}(k_{\parallel},\omega)=\frac{e}{2\pi^{2}\hbar\omega}\int_{-\infty}^{\infty}\\!\\!\\!dx\;{\rm
Re}\big{\\{}{\rm e}^{-{\rm i}\omega x/v}E_{x}^{\rm
ind}(x,k_{\parallel},b,\omega)\big{\\}}$ (6b)

is obtained from Eq. (6a) by writing the induced field as ${\bf E}^{\rm
ind}({\bf r},\omega)=(2\pi)^{-1}\int dk_{\parallel}\,{\rm e}^{{\rm
i}k_{\parallel}y}\,{\bf E}^{\rm ind}(x,k_{\parallel},z,\omega)$ with ${\bf
v}=v\hat{\bf x}$. We use Eq. (6b) to calculate
$\Gamma^{(1)}(k_{\parallel},\omega)$ in Figs. 1 and 2 using an adaptation of
BEM to cope with translationally invariant geometries García de Abajo and
Howie (2002) such as ribbons.

### Zero-thickness 2D limit: derivation of Eqs. (4)

Figure 3 is obtained from Eqs. (4), which is here derived in the limit of a
zero-thickness ribbon, assuming an electrostatic response of the material
through its local, frequency-dependent 2D conductivity $\sigma(\omega)$, but
incorporating relativistic effects in the interaction with the electron.

We start by writing a relation for the total field ${\bf E}({\bf R},\omega)$
acting on each point ${\bf R}$ of a generic 2D structure in response to an
external field ${\bf E}^{\rm ext}({\bf R},\omega)$ in the frequency domain. By
first calculating the 2D charge density from the continuity equation as $({\rm
i}\omega)^{-1}\nabla\cdot{\bf j}^{\rm ind}_{\rm 2D}({\bf R},\omega)$, where
${\bf j}^{\rm ind}_{\rm 2D}({\bf R},\omega)=\sigma(\omega)\,{\bf E}({\bf
R},\omega)$ is the induced current density, we then obtain the induced field
from the Coulomb potential, integrate by parts, and add the external field to
find the self-consistent expression García de Abajo (2013)

$\displaystyle{\bf E}({\bf R},\omega)$ $\displaystyle={\bf E}^{\rm ext}({\bf
R},\omega)$ (7) $\displaystyle+\frac{{\rm i}\sigma(\omega)}{\omega}\int
d^{2}{\bf R}^{\prime}\,M({\bf R}-{\bf R}^{\prime})\cdot{\bf E}({\bf
R}^{\prime},\omega),$

where the $2\times 2$ tensor $M({\bf R})=\nabla\otimes\nabla R^{-1}$ acts on
the plane of the structure, and we note that the 2D coordinates ${\bf R}$ and
${\bf R}^{\prime}$ are restricted to the region occupied by the material
throughout this section. The solution to Eq. (7) can be expressed in terms of
a complete set of eigenmode fields ${\bf E}_{j}({\bf R})$ and real eigenvalues
$\eta_{j}$ of $M$ defined by $\int d^{2}{\bf R}^{\prime}\;M({\bf R}-{\bf
R}^{\prime})\cdot{\bf E}_{j}({\bf R}^{\prime})=(\eta_{j}W)^{-1}{\bf
E}_{j}({\bf R})$ García de Abajo (2013), indexed by $j$, and satisfying the
orthonormality relation $\int d^{2}{\bf R}\;{\bf E}_{j}^{*}({\bf R})\cdot{\bf
E}_{j^{\prime}}({\bf R})=\delta_{jj^{\prime}}$, where $W$ is a characteristic
distance of the structure (e.g., the ribbon width in the present work). This
allows us to project Eq. (7) on the eigenmodes of $M$ and write an analytical
solution for ${\bf E}({\bf r},\omega)$, which, upon multiplication by the
conductivity, yields the 2D induced current density

$\displaystyle{\bf j}^{\rm ind}_{\rm 2D}({\bf R},\omega)=$
$\displaystyle\sigma(\omega)\sum_{j}\frac{1}{1-\eta_{\omega}/\eta_{j}}\;{\bf
E}_{j}({\bf R})$ $\displaystyle\times\int d^{2}{\bf R}^{\prime}\;{\bf
E}_{j}^{*}({\bf R}^{\prime})\cdot{\bf E}^{\rm ext}({\bf R}^{\prime},\omega)$

with $\eta_{\omega}={\rm i}\sigma(\omega)/\omega W$. The induced electric
field at any spatial location ${\bf r}$ is then given by Jackson (1999) ${\bf
E}^{\rm ind}({\bf r},\omega)=({\rm i}/\omega)\int d^{2}{\bf
R}^{\prime}\;\mathcal{G}_{0}({\bf r}-{\bf R}^{\prime},\omega)\cdot{\bf j}^{\rm
ind}_{\rm 2D}({\bf R}^{\prime},\omega)$, where $\mathcal{G}_{0}({\bf
r},\omega)=(k^{2}+\nabla\otimes\nabla){\rm e}^{{\rm i}kr}/r$ is the free-space
electromagnetic Green tensor and $k=\omega/c$. We also write the external
field as ${\bf E}^{\rm ext}({\bf r},\omega)=({\rm i}/\omega)\int d^{3}{\bf
r}^{\prime}\;\mathcal{G}_{0}({\bf r}-{\bf r}^{\prime},\omega)\cdot{\bf j}^{\rm
ext}({\bf r}^{\prime},\omega)$ in terms of the external three-dimensional
current density ${\bf j}^{\rm ext}({\bf r},t)=-e{\bf v}\delta({\bf r}-{\bf
r}_{0}-{\bf v}t)$ associated with the moving electron. More precisely García
de Abajo (2010), ${\bf E}^{\rm ext}({\bf
r},\omega)=(2e\omega/v^{2}\gamma_{L}){\rm e}^{{\rm i}\omega
r_{\parallel}/v}\big{[}{\rm i}\gamma_{L}^{-1}K_{0}(\theta)\hat{{\bf
v}}-K_{1}(\theta)\hat{{\bf r}}_{\perp}\big{]}$, where
$\gamma_{L}=1/\sqrt{1-v^{2}/c^{2}}$ is the Lorentz factor, $K_{m}(\theta)$ are
modified Bessel functions evaluated at $\theta=\omega r_{\perp}/v\gamma_{L}$,
and we use the coordinates $r_{\parallel}=({\bf r}-{\bf r}_{0})\cdot\hat{{\bf
v}}$ and ${\bf r}_{\perp}={\bf r}-{\bf r}_{0}-r_{\parallel}\hat{{\bf v}}$
parallel and perpendicular to the velocity ${\bf v}$, respectively. Inserting
these expressions for the external and the induced fields into Eq. (6a), and
noticing, after some algebra, that the remaining time integral produces a
factor $\big{[}{\bf E}^{\rm ext}({\bf r},\omega)\big{]}^{*}$, we obtain the
result

$\displaystyle\Gamma^{(1)}(\omega)=$
$\displaystyle\frac{W}{\pi\hbar}\sum_{j}{\rm
Im}\bigg{\\{}\frac{1}{\eta_{\omega}^{-1}-\eta_{j}^{-1}}\bigg{\\}}$ (8)
$\displaystyle\times\bigg{|}\int d^{2}{\bf R}^{\prime}\;{\bf E}_{j}^{*}({\bf
R}^{\prime})\cdot{\bf E}^{\rm ext}({\bf R}^{\prime},\omega)\bigg{|}^{2}.$

Equation (8) is general for any 2D structure whose response can be treated in
the electrostatic limit, although relativistic effects are incorporated in the
interaction with the electron through the use of the free-space Green tensor
$\mathcal{G}_{0}$ and the resulting $\gamma_{L}$ factors.

For the ribbon in Fig. 1a, we can write the modes as ${\bf
E}_{jk_{\parallel}}({\bf R})=\vec{\mathcal{E}}_{jk_{\parallel}}(x)\,{\rm
e}^{{\rm i}k_{\parallel}y}/\sqrt{WL}$ with the label $j$ multiplexed into
$\\{j,k_{\parallel}\\}$, $L$ denoting the quantization length along the ribbon
direction $y$, and the mode orthonormality relation now reducing to
$\int_{-W/2}^{W/2}dx\;\vec{\mathcal{E}}^{*}_{jk_{\parallel}}(x)\cdot\vec{\mathcal{E}}_{j^{\prime}k_{\parallel}}(x)=W\delta_{jj^{\prime}}$.
The sum over $j$ is also multiplexed as $\sum_{j}\sum_{k_{\parallel}}$ and we
use the prescription
$\sum_{k_{\parallel}}\rightarrow(L/2\pi)\int_{-\infty}^{\infty}dk_{\parallel}$.
Making use of these elements, setting ${\bf r}_{0}=b\hat{\bf z}$ and ${\bf
v}=v\hat{\bf x}$ (i.e., $r_{\parallel}=x$ and ${\bf r}_{\perp}=y\hat{\bf
y}+b\hat{\bf z}$), inserting the explicit expression for the external field
given above, and retaining only the $k_{\parallel}$ integrand, Eq. (8) readily
leads to

$\displaystyle\Gamma^{(1)}(k_{\parallel},\omega)=\frac{2e^{2}\omega^{2}}{\pi^{2}\hbar\,v^{4}\gamma_{L}^{2}}\sum_{j}{\rm
Im}\bigg{\\{}\frac{1}{\eta_{\omega}^{-1}-\eta_{jk_{\parallel}}^{-1}}\bigg{\\}}$
(9) $\displaystyle\times\bigg{|}\int_{-W/2}^{W/2}\\!\\!dx\,{\rm e}^{{\rm
i}\omega x/v}\;\vec{\mathcal{E}}^{*}_{jk_{\parallel}}(x)\cdot\Big{[}{\rm
i}\gamma_{L}^{-1}I_{0}(b)\,\hat{\bf x}-I_{1}(b)\,\hat{\bf
y}\Big{]}\bigg{|}^{2},$

where $I_{m}(b)=\\!\int_{-\infty}^{\infty}\\!dy\,{\rm e}^{-{\rm
i}k_{\parallel}y}K_{m}\big{(}\omega\sqrt{b^{2}+y^{2}}/v\gamma_{L}\big{)}\,y^{m}/(b^{2}+y^{2})^{m/2}$.
From Eq. (6.677-5) in Ref. Gradshteyn and Ryzhik (2007), we find
$I_{0}(b)=(\pi/\kappa){\rm e}^{-\kappa b}$ with
$\kappa=\sqrt{k_{\parallel}^{2}+(\omega/v\gamma_{L})^{2}}$, while
$I_{1}(b)=(-{\rm i}v\gamma_{L}k_{\parallel}/\omega)I_{0}(b)$ is obtained upon
integration by parts. Inserting these results in Eq. (9) and making some
factor rearrangements, we finally obtain Eqs. (4).

## Acknowledgments

We acknowledge support from ERC (Advanced Grant 789104-eNANO), the European
Commission (Horizon 2020 Grants No. FET-Proactive 101017720-EBEAM and No. FET-
Open 964591-SMART-electron), the Spanish MICINN (PID2020– 112625 GB-I00 and
Severo Ochoa CEX2019-000910-S), the Catalan CERCA Program, and Fundaciós
Cellex and Mir-Puig. J. D. C. is a Sapere Aude research leader supported by
VILLUM FONDEN (grant no. 16498) and Independent Research Fund Denmark (grant
no. 0165-00051B). The Center for Polariton-driven Light–Matter Interactions
(POLIMA) is funded by the Danish National Research Foundation (Project No.
DNRF165).

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$\Leftrightarrow$ | $\langle b^{*}(\rho(i_{1})),...,b^{*}(\rho(i_{k}))\rangle\in\phi(\mathfrak{B})$
$\Leftrightarrow$ | $\langle f(a^{*}(i_{1})),...,f(a^{*}(i_{k}))\rangle\in\phi(\mathfrak{B})$

Hence $f$ is an elementary embedding.

$\Box$
Based on the proof of Theorem 4.4, we have the following corollary:

###### Corollary 4.1

Let $\mathcal{L}$ be a first-order language, $\mathfrak{A},\mathfrak{B}$ be
$\mathcal{L}$-structure, then the following statements are equivalent:

  1. 1.

$\mathfrak{A}\cong\mathfrak{B}$;

  2. 2.

there is an S-homeomorphism
$h:(\mathcal{C}^{\mathfrak{A}\kappa},l_{\mathfrak{A}})\cong(\mathcal{C}^{\mathfrak{B}\kappa},l_{\mathfrak{B}})$
s.t. $h$ maps domain points to domain points.

  3. 3.

For $T\subseteq Th(\mathfrak{A})\cap Th(\mathfrak{B})$, infinite ordinal
$\alpha\geq\|\mathfrak{A}\|\cup\|\mathfrak{B}\|$ and any model points
$a,b\in|\mathcal{C}^{T}_{\alpha}|$ s.t. $a$ represents $\mathfrak{A}$, $b$
represents $\mathfrak{B}$, we have $a\asymp b$.

Proof

  * •

$1\Rightarrow 2$: Since an elementary embedding must be an injection, from the
definition of $h$ in the above proof, it is easy to see that $h$ is injective.
If $f$ is an isomorphism, then for any $b\in B^{\alpha}$, $i\in\alpha$, there
must exist $x_{i}\in A$ s.t. $b(i)=f(x_{i})$. Let $a\in A^{\alpha}$ be
$a(i)=x_{i}$, we know that $h(a)=b$. Therefore, $h$ is a bijection and thus
S-homeomorphism. It is also obvious that if $a$ is a domain point, then $h(a)$
is also a domain point.

  * •

$2\Rightarrow 3$: We know $h(a^{*})$ is domain point, then from the definition
of $\rho$, it is easy to see $B=\\{b^{*}(\rho(i)):i\in\alpha\\}$, then for any
$i\not\in ran(\rho)$, there is an $i^{\prime}\in ran(\rho)$ s.t.
$b^{*}(i)=b^{*}(i^{\prime})$, then $b^{*}\in
l_{\mathfrak{B}}(v_{i}=v_{i^{\prime}})$, then $b\in D_{ii^{\prime}}$. By
definition, $\rho:a\asymp b$.

  * •

$3\Rightarrow 1$: By definittion, for any $i\not\in ran(\rho)$, there is an
$i^{\prime}\in ran(\rho)$ s.t. $b^{*}\in D^{\mathfrak{B}}_{ii^{\prime}}$ i.e.
$b^{*}(i)=b^{*}(i^{\prime})$, then $B=\\{b(i):i\in ran(\rho)\\}=ran(f)$, then
$f$ is a surjection and therefore is an isomorphism.

$\Box$

###### Corollary 4.2

For any first-order language $\mathcal{L}$, any $\mathcal{L}$-theory $T$ and
any infinite cardinal $\kappa$, let $\mathcal{I}(T,\kappa)$ denote the number
of nonisomorphic models of $T$ of cardinality $\kappa$, we have that
$\mathcal{I}(T,\kappa)=\|\\{[a]_{\asymp}:a$ is big model point in
$\mathcal{C}^{T}_{\kappa}\\}\|$.

###### Remark 4.1

Combined with their proofs, Theorem 4.3 and Theorem 4.4 tell us that for any
infinite cardinality $\kappa$, up to isomorphism and relation $\asymp$, all
models of a theory of cardinality$\leq\kappa$ correspond one-to-one with all
model points in the $\kappa$-model space of $T$, and all elementary embeddings
between $T$-models correspond one-to-one with all factor relations between
model points in the $\kappa$-model space of $T$. It is also easy to see that a
subset of a model can be represented as a factor of the model point
representing it; a partial elementary embedding can be represented as a
partial factor relation. In this way, each basic object in semantics of first-
order logic is represented here as a topological object on the FOL space.

An important similarity to the topological representation of semantics of
propositional logic is that, just as the number of points in a Stone space is
independent of the specific theory it corresponds to and the way it
corresponds to it, in an FOL space, the structure consisting of points and
their factor relations is also independent of the specific
$\mathcal{L}$-formation. In other words, through the FOL space, we refine the
general abstract structure of semantics first-order logic independent from the
formal language.

##  5 Application in model theory

In Section 4.4, we showed that the class of FOL spaces is exactly the class of
S-homemorphic spaces of $\alpha$-model spaces. Then, the topological
representation of semantics of first-order logic via $\alpha$-model spaces
naturally leads to a method for discussing model theory using a pure set-
topology language. Introducing our topological representation as such a method
into model theory has at least two foreseeable benefits:

  * •

The topological representation is essentially a homogenization of first-order
structures and the formal language that are heterogeneous from each other.
Under which some distinct objects become the same thing (e.g., both models and
Henkin sets become model points in the FOL space). Thus this treatment can
obviously simplify the discussion and even generate new ideas.

  * •

In the set-topology language, we do not need to consider the strict
requirement of the formal language in the form of writing, which can simplify
the lines.

In 5.2, we will look at this new topological method through an example. As
mentioned in the introduction, we already have a complete topological proof of
Morley’s theorem. The example in 5.2 is a critical fragment of this entire
proof with some modifications in order to correspond to an intermediate
theorem of the classical proof of Morley’s theorem. We chose this fragment as
the example because, on the one hand, this fragment reflects relatively more
features and advantages of the new method; on the other hand, it follows the
classical proof idea of the theorem, thus making it easier to compare it with
the classical proof.

Before getting into the example, we first explore the connection between our
work and the type space in 5.1. We know that type space is a classical set-
topology notion in model theory. Since many concepts and works in model theory
are constructed based on the type space, not only this connection means that
our new method can directly inherit many set-topology approaches that are
already well established in model theory, but also we need to base on this
discussion when translating concepts in the example.

###  5.1 Connection with type space

Let us start with a basic concept, the formula with parameters. Saying a
formula $\phi(\overline{v},\overline{a})$ with parameters $\overline{a}$ can
be satisfied on a first-order structure $\mathfrak{A}$ is to say there is an
assignment $\sigma$ on $\mathfrak{A}$ containing
$\overline{w}\mapsto\overline{a}$ as a part satisfying
$\phi(\overline{v},\overline{w})$. Then by topological language, this is to
say that the maximal consistent set satisfied by $\sigma$ is in
$S_{\phi(\overline{v},\overline{w})}$. In this way, as with the formula
without parameters, the satisfiability of a formula with parameters can be
expressed as the nonemptyness of a clopen set. Thus, the realizability of a
type with parameters can be expressed as the nonemptyness of the corresponding
closed set. Based on this discussion, the following theorem is more intuitive.

###### Theorem 5.1

Let $\mathcal{L}$ be a language, $T$ be an $\mathfrak{L}$-theory, then for any
$T$-model $\mathfrak{A}$, any nonempty $B\subseteq A$, any
$n\in\mathbb{N}^{+}$ and any $\kappa>\|B\|+\aleph_{0}$, the type space
$S^{\mathfrak{A}}_{n}(B)$ is homeomorphic to a $\kappa\backslash
n$-dimensional complete closed set on $\mathcal{C}_{T}^{\kappa}$. Vice versa.

Proof

  * •

For the type space $S^{\mathfrak{A}}_{n}(B)$, let
$f:\\{v_{i}:i\in\kappa\backslash n\\}\rightarrow B$ be a surjection. We define
$g:S^{\mathfrak{A}}_{n}(B)\rightarrow\mathcal{C}_{T}^{\kappa}$ as
$p\mapsto\\{\phi(\overline{v},\overline{w}):\phi(\overline{v},f(\overline{w}))\in
p\\}$.

    * –

$ran(g)$ is a $\kappa\backslash n$-dimensional complete closed set:

For any $i\in n$, any $x\in ran(g)$ and any $y\sim_{i}x$, we know for any
formula $\phi(\overline{w})\in y$ where $\overline{w}\in\kappa\backslash n$,
$S_{\phi}$ is $\sim_{i}$-saturated and then $\phi\in x$, then for $p\in
g^{-1}[x]$, $\phi(f(\overline{w}))\in p\subseteq\bigcup
S^{\mathfrak{A}}_{n}(B)$ is a sentence. This means
$y^{\prime}:=\\{\phi(\overline{v},f(\overline{w})):\phi(\overline{v},\overline{w})\in
y\\}$ is in $S^{\mathfrak{A}}_{n}(B)$. Clearly $g(y^{\prime})=y$. Hence,
$ran(g)$ is $\sim_{i}$-saturated.

For any formula $\phi(\overline{w})$ s.t. $\Delta(S_{\phi})\cap n=\emptyset$,
and $S_{\phi}\cap ran(g)=\neq\emptyset$, let $x\in S_{\phi}\cap ran(g)$. We
know $\phi(f(\overline{w}))\in p$ for any $p\in g^{-1}[x]$, this means
$\phi(f(\overline{w}))\in p$ for any $p\in S^{\mathfrak{A}}_{n}(B)$. Then for
any $y\in ran(g)$, $\phi\in y$ i.e. $y\in S_{\phi}$.

    * –

$g$ is an injection:

For any $p\neq q\in S^{\mathfrak{A}}_{n}(B)$, we know there is
$q\not\ni\phi(\overline{v},\overline{a})\in p$, then $g(q)\not\in
S_{\phi(\overline{v},\overline{w})}\ni g(p)$ for any $\overline{w}\in
f^{-1}[\overline{a}]$. Then $g(p)\neq g(q)$. Hence $g$ is injective.

    * –

$g$ is continuous:

For any $\phi\in\Lambda_{\mathcal{L}_{\kappa}}$, we denote $\phi$ as
$\phi(\overline{v},\overline{w})$ where $\overline{v}\in\\{v_{i}:i\in
n\\},\overline{w}\in\\{v_{i}:i\not\in n\\}$. $g^{-1}[\phi]=\\{p\in
S^{\mathfrak{A}}_{n}(B):\phi\in g(p)\\}=\\{p\in
S^{\mathfrak{A}}_{n}(B):\phi(\overline{v},f(\overline{w}))\in
p\\}=[\phi(\overline{v},f(\overline{w}))]$ is a clopen set on
$S^{\mathfrak{A}}_{n}(B)$.

    * –

$g^{-1}:ran(g)\rightarrow S^{\mathfrak{A}}_{n}(B)$ is continuous:

For any $\phi(\overline{v},\overline{a})\in\bigcup S^{\mathfrak{A}}_{n}(B)$,
$g[\phi(\overline{v},\overline{a})]=\\{x\in
ran(g):\phi(\overline{v},\overline{w})\in x$ for $\overline{w}\in
f^{-1}[\overline{a}]\\}=ran(g)\cap\bigcap_{\overline{w}\in
f^{-1}[\overline{a}]}S_{\phi(\overline{v},\overline{w})}$ is a closed set on
$ran(g)$. Then for any closed set $s$ on $S^{\mathfrak{A}}_{n}(B)$,
$(g^{-1})^{-1}[s]=g[s]$ is closed. Hence $g^{-1}$ is continuous.

  * •

For any $\kappa\backslash n$-dimensional complete closed set $s$ on
$\mathcal{C}_{T}^{\kappa}$, we know there is a model point $x\in\rho s$ for
some $\rho$. Let $\mathfrak{A}$ be a $T$-model represented by $x$ and
$x^{\prime}$ be a domain point in the inverse image of $x$ in
$\mathcal{C}^{\mathfrak{A}}_{\kappa}$. We define
$B=\\{x^{\prime}(i):i\in\rho[\Delta(s)]\\}$, define
$g:S^{\mathfrak{A}}_{n}(B)\rightarrow\mathcal{C}_{T}^{\kappa}$ as
$p\mapsto\\{\phi(\overline{v},v_{i_{1}},...,v_{i_{m}}):\phi(\overline{v},x^{\prime}(\rho(i_{1})),...,x^{\prime}(\rho(i_{m})))\in
p\\}$. Then for any $y\in\mathcal{C}_{T}^{\kappa}$,  $y\in ran(g)$
$\Leftrightarrow$ for formula $\phi(v_{i_{1}},...,v_{i_{m}})$ in $y$ where
$i_{1},...,i_{m}\in\\{v_{i}:i\not\in n\\}$,
$\mathfrak{A},x^{\prime}\vDash\phi(v_{\rho(i_{1})},...,v_{\rho(i_{m})})$
$\Leftrightarrow$ for formula $\phi(v_{i_{1}},...,v_{i_{m}})$ in $y$ where
$i_{1},...,i_{m}\in\\{v_{i}:i\not\in n\\}$,
$\phi(v_{\rho(i_{1})},...,v_{\rho(i_{m})})\in x$ $\Leftrightarrow$ for formula
$\phi(v_{i_{1}},...,v_{i_{m}})$ in $y$ where
$i_{1},...,i_{m}\in\\{v_{i}:i\not\in n\\}$, $\phi(v_{i_{1}},...,v_{i_{m}})\in
y^{\prime}$ for any $y^{\prime}\in s$ $\Leftrightarrow$ $y\in s$. Hence,
$ran(g)=s$.

Then by a similar way as the first direction, we can prove
$g:S^{\mathfrak{A}}_{n}(B)\cong s$.

$\Box$

###### Remark 5.1

The above theorem corresponds all cofinite-dimensional complete closed sets on
$\kappa$-dimensional spaces to all type spaces over a nonempty set not more
than $\kappa$. In turn, an FOL space can be seen as the merging of a family of
type spaces. Then, the existing type space-based approach in model theory can
be immediately transferred to the FOL space. In addition, Since the FOL space
acts as a merge of type spaces, some connections between different type spaces
can also be formalized as connections on the FOL space. For example, for
models $\mathfrak{A}$, $\mathfrak{A^{\prime}}$, $B\subseteq|\mathfrak{A}|$ and
$B^{\prime}\subseteq|\mathfrak{A^{\prime}}|$ s.t. there is partial elementary
embedding $f:B\rightarrow B^{\prime}$, the relation between
$S^{\mathfrak{A}}_{n}(B)$ and $S^{\mathfrak{A^{\prime}}}_{m}(B^{\prime})$ can
be formalized by a permutation relation on the FOL space. That is, for the
closed sets $s$ representing $S^{\mathfrak{A}}_{n}(B)$ and $t$ representing
$S^{\mathfrak{A^{\prime}}}_{m}(B^{\prime})$, $t\subseteq\rho s$ for some
mapping $\rho$. On the other hand, for a model $\mathfrak{A}$ and a
$B\subseteq|\mathfrak{A}|$, it is easy to see that for different
$m,n\in\mathbb{N}$, the set $s$ representing $S^{\mathfrak{A}}_{m}(B)$ can
become a set representing $S^{\mathfrak{A}}_{n}(B)$ by taking a permutation.
Then, when we need to consider $S^{\mathfrak{A}}_{n}(B)$ for all
$n\in\mathbb{N}$ at the same time, we can combine all these
$S^{\mathfrak{A}}_{n}(B)$ into one subspace of the FOL space, which is a
coinfinite-dimensional complete closed set $s$ s.t. for a closed set $t$
representing $S^{\mathfrak{A}}_{n}(B)$ for an $n$, there is a mapping $\rho$
s.t. $\rho s=t$. The difference is that here an $n$-type is not a point, but
an $\Delta(s)\cup n$-dimensional completely closed set. In particular,
although $n$-type spaces over $\emptyset$ do not have a natural representation
on the FOL space, such a merge space of the $n$-type spaces over $\emptyset$
do has. That is, an $\emptyset$-dimensional completely closed set.

###  5.2 The example

The example is the following theorem888The formulation of the theorem is
referenced from Theorem 5.2.9 in [11]:

  * •

Let $\mathcal{L}$ be countable and $T$ be an $\mathcal{L}$-theory with
infinite models. For all $\kappa\geq\aleph_{0}$, there is a $T$-model
$\mathfrak{A}$ with $\|\mathfrak{A}\|=\kappa$ such that if
$B\subseteq\mathfrak{A}$ and $n\in\mathbb{N}$, then $\mathfrak{A}$ realizes at
most $\|B\|+\aleph_{0}$ types in $S^{\mathfrak{A}}_{n}(B)$.

First we consider the translation of two concepts: theory in a countable
language and complete theory. Saying a language $\mathcal{L}$ is countable is
equivalent to saying $\Lambda_{\mathcal{L}}$ is countable. Then let $T$ be an
$\mathcal{L}$-theory, which means that on the $\alpha$-model space
$\mathcal{C}_{T}^{\alpha}$ of an $\mathcal{L}$-theory $T$, clopen sets of
dimension $I$ are countable for any finite $I\subseteq\alpha$. We call an FOL
space with this property as a locally-countable FOL space. Saying a theory is
complete is equivalent to saying for any sentence $\phi$, $\phi\in T$ or
$\neg\phi\in T$, which means $l_{T}^{\alpha}(\phi)=\emptyset$ or
$|\mathcal{C}_{T}^{\alpha}|$ for any $\alpha$. Then $\mathcal{C}_{T}^{\alpha}$
has no non-trivial $\emptyset$-dimensional clopen sets. We call an FOL space
with this property as a complete FOL space.

The rest of the discussion on the translation can be organized into the
following proposition:

###### Proposition 5.1

Let $T$ be a theory and $\mathfrak{A}$ be a $T$-model of cardinality $\kappa$.
For any nonempty set $B\subseteq|\mathfrak{A}|$ and $n\in\mathbb{N}$, the
following statements are equivalent:

  1. 1.

$\mathfrak{A}$ realizes $\lambda$ many types in $S^{\mathfrak{A}}_{n}(B)$.

  2. 2.

For $x$ the model point representing $\mathfrak{A}$ in
$\mathcal{C}_{T}^{\kappa}$, let $x^{\prime}$ be defined as in the proof of
Theorem 5.1, we define $I_{B}=\\{i\in\kappa:x^{\prime}(i)\in B\\}$. Then for
any $\kappa\backslash n$-dimensional complete closed set $s$ satisfying there
is a partial mapping $\rho$ defined on $\Delta(s)$ s.t. $\rho s=a|_{I_{B}}$,
there are only $\lambda$ many $b\in s$ s.t. there is mapping
$\rho^{\prime}\supset\rho$, $\rho:b\prec a$.

Proof Let $f$, $g$ be defined as in the direction $2\Rightarrow 1$ of the
proof of Theorem 5.1. For each $I=\\{i_{0},...,i_{n-1}\\}\subseteq\alpha$
($i_{0},...,i_{n-1}$ in ascending order of subscript), defining mapping
$\rho_{I}$ as $k\mapsto i_{k}$ for $k\in n$, $k\mapsto\rho(k)$ for $k\not\in
n$, then for $p\in S^{\mathfrak{A}}_{n}(B)$ we have:

|  $x^{\prime}(i_{0}),...,x^{\prime}(i_{n-1})$ realizes p
---|---
$\Leftrightarrow$ | for $\phi(\overline{v},x^{\prime}(j_{1}),...,x^{\prime}(j_{m}))\in p$, $\mathfrak{A}\vDash\phi(x^{\prime}(i_{0}),...,x^{\prime}(i_{n-1}),x^{\prime}(j_{1}),...,x^{\prime}(j_{m}))$
$\Leftrightarrow$ | for $\phi(\overline{v},,x^{\prime}(j_{1}),...,x^{\prime}(j_{m}))\in p$, $x\in S_{\phi(v_{i_{0}},...,v_{i_{n-1}},v_{j_{1}},...,v_{j_{m}})}$
$\Leftrightarrow$ | for $\phi(\overline{v},,x^{\prime}(j_{1}),...,x^{\prime}(j_{m}))\in p$, $\phi(\overline{v},v_{j^{\prime}_{1}},...,v_{j^{\prime}_{m}})\in\rho_{I}^{-1}x$ for $j^{\prime}_{1},...,j^{\prime}_{m}\in\rho^{-1}[j_{1}],...,\rho^{-1}[j_{m}]$
$\Leftrightarrow$ |  $g(p)=\rho^{-1}_{I}x$ i.e. $\rho_{I}g(p)=a|_{I\cup I_{B}}$

Since $g$ is a bijection between $S^{\mathfrak{A}}_{n}(B)$ and $s$, the
equivalence of statements holds. $\Box$

Obviously, if for any nonempty set $B\subseteq|\mathfrak{A}|$, $\mathfrak{A}$
realizes at most $|B|+\aleph_{0}$ types in $S^{\mathfrak{A}}_{n}(B)$, then
$\mathfrak{A}$ realizes countable types in the type space over $\emptyset$.
Based on the above discussion, we can translate the theorem into the following
proposition.

###### Proposition 5.2

Let $\mathcal{C}_{\alpha}$ be an locally-countable complete FOL space
satisfying $\bigcap_{i,j\in\alpha}-D^{\mathcal{C}}_{ij}\neq\emptyset$, there
is a big model point $a\in\mathcal{C}_{\alpha}$ s.t. for any
$I\subseteq\alpha$, any cofinite-dimensional complete closed set $s$ and any
partial mapping $\rho$ defined on $\Delta(s)$ satisfying $\rho s=a|_{I}$,
there are only $\|I\|+\aleph_{0}$ many $b\in s$ s.t. there is a mapping
$\rho^{\prime}\supset\rho$, $\rho^{\prime}:b\prec a$.

Proof To simplify the discussion, we consider the dimension of $\mathcal{C}$
as a cartesian product $\alpha\times\omega$.

  * •

We say a (partial) mapping
$\rho:\alpha\times\omega\Rightarrow\alpha\times\omega$ is $\alpha$-order-
preserving if for $(i_{1},j_{1}),(i_{2},j_{2})\in
dom(\rho),(i^{\prime}_{1},j^{\prime}_{1})=\rho(i_{1},j_{1}),(i^{\prime}_{2},j^{\prime}_{2})=\rho(i_{2},j_{2})$,
$i_{1}>i_{2}\Leftrightarrow i^{\prime}_{1}>i^{\prime}_{2}$,
$j_{1}=j^{\prime}_{1},j_{2}=j^{\prime}_{2}$.

  * •

For subset $s\subseteq|\mathcal{C}|$, we write $L(s):=\\{\rho s:\rho$ is
$\alpha$-order-preserving$\\}$, $l(s):=\bigcap L(s)$.

Claim: there is a big model point $a$ s.t. $l(\\{a\\})=\\{a\\}$.

Let $a$ be the model point in this Claim, $I,\rho,s,n$ be arbitrary satisfying
the relationship described in the proposition. The only difference is
$\Delta(s)=(\alpha\times\omega)\backslash(1\times n)$ WLOG, we assume that for
any $(i,j)\in I$, $i=\beta+m$ for some limit ordinal $\beta$ and natural
number $m>n$. $I^{\prime}:=\\{i^{\prime}\in\alpha:$ there is $(i,j)\in
I,i^{\prime}=i-n$ for $n\in\mathbb{N}\\}$, we know
$\|I^{\prime}\|+\aleph_{0}=\|I\|+\aleph_{0}$. We define $X:=\\{a|_{t\cup
I}:t\subseteq I^{\prime}\times\omega$ and $\|t\|=n\\}$, $Y:=\\{b\in s:$ there
is $\rho^{\prime}\supseteq\rho,\rho^{\prime}\\{b\\}\in X\\}$. Obviously
$\|X\|=\|Y\|=\|I\|+\aleph_{0}$.

For an arbitrary $b\in s$ and a $\rho^{\prime}\supseteq\rho$ satisfying
$\rho^{\prime}b\prec a$. If $\Delta(\rho^{\prime}\\{b\\})\subseteq
I^{\prime}\times\omega$, then $b\in Y$. If
$\Delta(\rho^{\prime}\\{b\\})\backslash(I^{\prime}\times\omega)\neq\emptyset$,
we assume it contains only one element $(i^{\prime},j^{\prime})$ and for any
$(i,j)\in\rho^{\prime}[1\times n]\backslash\\{(i^{\prime},j^{\prime})\\}$,
$i\neq i^{*}:=min\\{i^{\prime\prime}\in
I^{\prime}:i^{\prime\prime}>i^{\prime}\\}$. Based on the assumed property of
$I$, this is no loss of generality. Then let partial mapping $\eta$ be defined
as $(i^{\prime},j)\mapsto(i^{*},j)$, for $j\in\omega$ and $(i,j)\mapsto(i,j)$
for $i\in I^{\prime}\backslash\\{i^{*}\\},j\in\omega$, we have $\eta$ is
$\alpha$-order-preserving and then $\eta a|_{s\cup I}=a|_{t\cup I}$ for
$t=\\{(i^{*},j^{\prime})\\}\cup\rho^{\prime}[1\times
n]\backslash\\{(i^{\prime},j^{\prime})\\}$. We know $a|_{t\cup I}\in X$ and
$\eta\circ\rho^{\prime}b=a|_{t\cup I}$. Hence, $b\in Y$.

Hence, the proposition holds.

Proof of Claim:

By the following induction procedure, we construct a list of sets converging
to the model point we want.

  * •

BS:  Let $\\{w_{i}:i\in\omega\\}$ list all nonempty clopen sets with dimension
$1\times 1$. $U_{0}:=\\{w_{i}(\frac{(0,2^{i})}{(0,0)}):i\in\omega\\}$,
$x_{0}:=\bigcap\\{-D_{pq}:p,q\in\alpha\times 1\\}\cap l(\bigcap U_{0})$.
Clearly $x_{0}=l(x_{0})$ is non-empty.

  * •

IH:  $x_{m-1}=l(x_{m-1})$ is a nonempty set s.t.
$x_{m-1}|_{\alpha\times(m-1)}$ is an $\alpha\times(m-1)$-dimensional complete
closed set.

  * •

IS:  List $\\{u:u$ is a clopen set s.t.
$\omega\times(m-1)\subseteq\Delta(u)\subseteq\omega\times m\\}$ as
$\\{u_{i}:i\in\omega\\}$. Use recursion on $\omega$, we define a list of non-
empty closed sets $x_{m-1}=y_{0}\supseteq y_{1}\supseteq...$ where
$y_{i}=l(y_{i})$ for each $i$ . The $n$-th inductive step is:

    * –

We consider the clopen set $u_{n}$. For each $\alpha$-order-preserving mapping
$\rho$, $K^{\rho}:=\\{i\in\alpha:(i,j)\in\Delta(\rho u_{n})$ for some j$\\}$
and $k^{\rho}:=\|K^{\rho}\|$. Clearly $k^{\rho}=k^{\rho^{\prime}}$ for any
$\alpha$-order-preserving $\rho,\rho^{\prime}$, so we write it as $k$. Let
$H=(\alpha,[\alpha]^{k})$ be the complete $k$-hypergraph on $\alpha$. Choosing
an arbitrary point $a\in y_{n-1}$, let hyperedge coloring
$f:[\alpha]^{k}\rightarrow 2$ be defined as $f(I)=1\Leftrightarrow
a\in\rho_{I}u_{n}$ where $\rho_{I}$ is $\alpha$-order-preserving and
$K^{\rho_{I}}=I$. WLOG assume $a\not\in l(u_{n})$, then there is finite
$X\subseteq L(u_{n})$ s.t. $a\not\in\bigcap X$. Since $y_{n-1}=l(y_{n-1})$,
there is $h\in\mathbb{N}^{+}$ s.t. for any $I\subseteq\kappa$ with $\|I\|=h$,
$a\not\in\bigcap\\{\rho u_{n}:\rho$ is $\alpha$-order-preserving and
$K^{\rho}\subseteq I\\}$. This means there is no monochromatic clique of order
$h$ colored $1$. Then by Ramsey’s theorem for hypergraph, we know that for any
$h^{\prime}\in\alpha$, there is a monochromatic clique of order $h^{\prime}$
colored $0$. Since $y_{n-1}=l(y_{n-1})$, this means for any
$h^{\prime}\in\alpha$, $a\in\bigcap\\{\rho-u_{n}:\rho$ is $\alpha$-order-
preserving and $K^{\rho}\subseteq h^{\prime}\\}$, and then, $a\in l(-u_{n})$.
Let $y_{i}:=y_{i-1}\cap l(-u_{n})$.

$x^{\prime}_{m}:=\bigcap_{i\in\omega}y_{i}$. By compactness,
$x^{\prime}_{m}=l(x^{\prime}_{m})\neq\emptyset$. Clearly
$x^{\prime}_{m}|_{\alpha\times m}$ is a complete closed set of dimension
$\alpha\times m$.

List $\\{u:u$ is a clopen set s.t.
$(0,m)\in\Delta(v)\subseteq\\{(0,m)\\}\cup(\omega\times m)$ and
$x^{\prime}_{m}\subseteq[u]_{(0,m)}\\}$ as $\\{w_{i}:i\in\mathbb{N}^{+}\\}$.
Let $U_{m}:=\\{w_{i}(\frac{(r_{i},p^{i}_{m+1})}{(0,m)}):p_{m+1}$ is the
$(m+1)$-th prime$,i\in\mathbb{N}^{+}\\}$ where $r_{i}=min\\{i^{\prime}:$ there
is a $j^{\prime}$ s.t. $(i^{\prime},j^{\prime})\in\Delta([w_{i}]_{(0,m)})\\}$.
Let $x_{m}:=l(\bigcap U_{m})\cap x^{\prime}_{m-1}$, by compactness, we know
$x_{m}=l(x_{m})\neq\emptyset$.

Let $x:=\bigcap_{i<\omega}x_{i}$, we know $x$ is a singleton by compactness
and $T_{2}$ property, know the point in it is a model point by the
construction of $U_{m}$, know $x=l(x)$ by IH. $\Box$

###### Remark 5.2

The classical proof of this theorem involves many concepts, like the Skolem
function, order-indiscernible, Ehrenfeucht–Mostowski model, etc., and while we
do a rough count of the proof length, it can be found that the length of the
complete proof (counting the proofs of all the antecedent propositions) is not
less than five pages in many model theory textbooks. In contrast, our proof is
slightly more than one page long in a similar typographic format and avoids
defining many concepts. Even if all the translation processes are involved,
the length is no more than four pages in this format. Moreover, since
translation is only needed when the problem is introduced, the proportion of
translation length is much lower when we use this method for more significant
problems. This crude comparison of proof lengths at least illustrates our
proof’s lower complexity. And this is not because we have adopted a completely
different proof idea. In fact, it is easy to see by comparison that the
$\alpha$-order-preserving mapping plays a similar role to order-indiscernible
in the proof, while the model point in the claim is a topological analog of
the E–M model. And, as in the classical proof, we also use the S-theorem. The
above facts highlight the distinct advantage of the new method as a
homogeneous treatment.

##  6 Conclusion

The Topological Representation. In the first part, Sections 2, 3, and 4, which
are also the main body of this paper, a topological representation of
semantics of first-order is presented. For a first-order theory $T$, all
$T$-models, subsets of $T$-models, elementary embeddings, and partial
elementary embeddings between $T$-models, are represented as points and purely
topologically defined point-to-point relations in the model space or
$\alpha$-model space of $T$, respectively. Moreover, isomorphic topological
structures of these model spaces are depicted as FOL spaces. This way, the
construction of semantics of first-order logic is transformed into the
construction of abstract topological structures. We have discussed this
representation fully in the previous sections. Here we add one more
perspective on the topologization space.

By Theorems 4.3 and 4.4, for a theory $T$, $T$-models can also be represented
as basis-preserving C-mappings from topologization spaces to the model space
of $T$. Elementary embeddings between $T$-models can also be represented as
basis-preserving C-mappings between these topologization spaces. From the
perspective of category theory, this representation can also be regarded as a
generalization of the Stone duality. For the dual space of a Boolean algebra,
a point $x$ in it can also be viewed as a continuous mapping from a single
point topological space to $x$ (this is the dual morphism in the topological
category of a homomorphism from Boolean space to 0-1 algebra). As a
generalization of this observation, in category theory, a general element of
an object is a morphism directed at it, and an object can be determined by its
general elements of a certain type.[2] In this paper, a basis-preserving
mapping from the topologization space of a first-order structure to an FOL
space $\mathcal{C}$ can be regarded as a general element of $\mathcal{C}$.

The application in model theory. As a discipline, model theory must deal not
only with syntactic objects like formulas and theories but also with concrete
mathematical structures like first-order structures. The objects it deals with
are not homogeneous. In this paper, through the topological representation, we
homogenize the heterogeneous objects treated by the model theory into
components of the same kind of objects. This treatment brings with it the
great advantage that we can express complex relations between heterogeneous
objects in model theory by simple topological nations. This feature is amply
demonstrated in the in our example. We know that many in-depth discussions in
abstract model theory, including the classical proof of Morley’s theorem,
requires a series of complex definitions. This is evident in our example. the
complexity of these definitions makes it difficult for a newcomer to have a
precise and natural intuition of the connections between them. Here, all these
concepts are reduced to primitive objects such as closed sets, points, and
dimensions with certain properties, and their relationships are thus
intuitively represented. In general, many of the not-so-fundamental,
’auxiliary line’ constructions of first-order logic are transformed into some
basic and natural intrinsic structures in FOL space.

In the application example, it is easy to see that our proof method is very
syntactic. It is actually a Henkin construction process if translated into the
language of model theory. This phenomenon is common throughout the topological
proof of Morley’s theorem. What is the reason for this? It is easy to see that
there is a very natural connection between cylindric space and cylindric
algebra, so the link between cylindric space and first-order syntax is more
direct and closed than for specific first-order structures. Topological
treatment can therefore be seen to some extent as the ’syntacticization’ of
first-order semantic objects. Based on this, the inherent structure of FOL
spaces brings an advantage: It significantly extends the range of applications
of the syntactic approach in model theory. Furthermore, in the topological
treatment, we are freed from the constraints of formal languages, which
simplifies the complexity of expression when performing syntactic
constructions in proofs.

While simplifying the syntactic concepts, the topological treatment does make
some purely semantic concepts, such as models, subsets, elements of models,
etc., less intuitive. However, these problems are solved by tools introduced
in the first part of this paper.

The proof we give in the application is considerably simpler than its
conventional counterpart. This simplification is a direct result of these
features and tools mentioned earlier. And I think these features and tools
work in essence because model theory, to a large extent, is a discipline that
only studies the first-order syntactic properties of a mathematical structure
and therefore does not rely as much on purely semantic concepts.

Related works. We have already presented other works on the topological
representation of semantics of first-order in Introduction. Here, we only
discuss work related to topological methods or homogenization methods in model
theory.

As mentioned above, our topological method can be seen as a homogenization
approach in model theory. Of course, there are many ways of homogenization,
but topological structures have unique advantages. For example, Section II.6.2
of [13] introduces the pointless topology method to transform a topological
space into a complete lattice. Can we use this class of complete lattices as a
means of homogenization? An obvious problem is that this structure lacks
simple, effective means to distinguish open sets, closed sets, and points,
which will cause many difficulties in the basic discussion. For example, the
equivalence relations corresponding to quantifiers will be difficult to
define.

Cylindric algebra is also a homogeneous tool that comes easily to mind.
However, if one tries to translate the representation of semantics of first-
order by the model space into the language of cylindric algebra, one finds
that these concepts become very complex. Moreover, the language of algebra
lacks the flexibility of the language of topology. For example, the concept of
point permutation will be difficult to define.

However, some people have indeed developed an algebraic approach to model
theory. The paper [12] proposed a set of methods to deal with model theoretic
problems using cylindric set algebra. But its discussion is primarily based on
the non-algebraic details of cylindric set algebra. For example, isomorphisms
between first-order structures only correspond to base-isomorphisms involving
specific $\alpha$-sequences. Then, this approach does not do anything about
homogenization but simply transposes the heterogeneity in the classical
approach into the algebraic approach.

As mentioned in Introduction, the main idea of this paper is inspired by
Pinter’s work in [14]. Pinter further developed a topological version of model
theory based on ultrafilter space of $\omega$-dimensional local-finite
dimensional cylindric algebra in [15]. Different from us, the main work in
[14] and [15] is algebraic, and the conclusion on the topological side is
mainly the consequence of the algebraic side. Therefore, linking cylindric
algebras of different dimensions is difficult, and relations like permutation
are difficult to define. So the representation of uncountable models can only
be specially constructed by a complex and unnatural method. However, some
basic intuitions are similar. For example, the concept of “model point" in
[15] can be regarded as a particular form of the concept of model point in
this article.

Future works. As has been mentioned many times, in a forthcoming paper, we
will prove a duality between the category of cylindric spaces with C-mappings
between them and the category of cylindric algebras as a natural
generalization of Stone duality.

By modifying the definition of cylindric space, we may construct duality or
even topological model theory for variants of first-order logic. It will be
exciting and may bring unexpected results.

Our work on the topological method in model theory is preliminary, but this
method has shown considerable application potential in abstract model theory.
Furthermore, it also has application potential when dealing with specific
structures, for example, in algebraic model theory. This requires us to build
on existing work by introducing distinguished clopen sets to represent basic
predicates and functions.

## Acknowledgments

I am very grateful to Dr. Shengyang Zhong for his valuable and specific
advice, assistance, and guidance in writing this paper.

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