# Correlation between normal and superconducting states within the Fermi-
liquid region of the T-p phase diagram of quantum-critical heavy-Fermion
superconductors

M. ElMassalami<EMAIL_ADDRESS>Instituto de Física, Universidade Federal
do Rio de Janeiro, Caixa Postal 68528, 21941-972 Rio de Janeiro RJ, Brazil  P.
B. Castro<EMAIL_ADDRESS>Instituto de Física, Universidade Federal
do Rio de Janeiro, Caixa Postal 68528, 21941-972 Rio de Janeiro RJ, Brazil  M.
B. Silva Neto<EMAIL_ADDRESS>Instituto de Física, Universidade Federal do
Rio de Janeiro, Caixa Postal 68528, 21941-972 Rio de Janeiro RJ, Brazil

###### Abstract

Extensively reported experimental observations indicate that on varying
pressure ($p$) within the T-p phase diagram of most quantum critical heavy
fermion (HF) superconductors, one identifies a cascade of distinct electronic
states which may be magnetic, of Kondo-type, non-conventional superconducting,
Fermi Liquid (FL), or non-FL character. Of particular interest to this work is
the part of the phase diagram lying below a specific phase boundary,
$T^{*}_{\text{{FL}}}(p^{*})$, across which the transport and thermodynamic
properties switch over from non-FL into FL behavior. Remarkably, this
nontrivial manifestation of FL phase is accompanied by (i) the characteristic
$\rho_{o}+\textit{AT}^{2}$ dependence ($\rho_{o}$ = residual resistivity),
(ii) a superconductivity below $T_{c}\leq T^{*}_{\text{{FL}}}(p^{*})$, and
(iii) a universal scaling of $T_{c}$ and $A$: $ln\frac{T_{c}}{\theta}\propto
A^{-\frac{1}{2}}$ ($\theta$ = characteristic energy scale). We consider that
such features are driven by a fluctuation-mediated electron-electron
scattering channel with the mediating quasiparticles being either spin
fluctuations [Mathur et al. Nature 394, 39 (1998)] or valence fluctuations
[Miyake and Watanabe, Phil. Mag.97, 3495–3516 (2017)] depending on the
character of the neighboring instability. On adopting such a scattering
channel and applying standard theories of Migdal-Eliashberg
(superconductivity) and Boltzmann (transport), we derive analytic expressions
that satisfactorily reproduce the aforementioned empirical correlations in
these heavy fermion superconductors.

##  I Introduction

There is ongoing interest in investigating the correlations between the
normal-state and superconducting properties of heavy fermion (HF)
superconduconductors when subjected to a variation in a nonthermal control
parameter such as pressure $p$, concentration of charge carriers $n$,
stoichiometry/doping $x$, disorder, or magnetic field $H$.1; 2; 3; 4; 5 Driven
by ease and convenience, the evolution of these correlations is often
investigated within a T-p phase diagram; there, an extrapolation of the phase
boundaries down to zero temperature often leads to one or two quantum
critical/crossover points.1; 2; 3; 4; 5; 6; 7 One is usually attributed to a
magnetic instability at $p_{m}$; often, it is accompanied by a superconducting
dome wherein the superconductivity is considered to be driven by spin
fluctuations.6 The other point, at $p_{v}$, is usually assigned to a valence
instability7 which, as well, is often accompanied by a superconducting dome
with the superconductivity considered to be driven by valence fluctuations.8;
7

A surge of superconducting dome within the neighborhood of a magnetic quantum
critical point is also manifested in other superconducting family, such as
high-Tc cuprates and Fe-based pnictides and chalcogenides.9; 10 Much of the
understanding of the superconducting and normal-state properties were obtained
from theoretical and experimental investigation of three regions of the dome:
the two neighboring the emergence and disappearance of the superconductivity
(at the left and right $T_{c}\rightarrow 0$) while the third is at the middle
wherein $T_{c}$ is a maximum. Analysis of the character of the left and middle
regions is usually complicated by the additional influence of competing
antiferromagnetic, spin glass, charge and stripes, or pseudogap phases. In
contrast, investigation of the region at the right $T_{c}\rightarrow 0$ limit
is much simpler since only two states are involved, namely the superconducting
and the FL phases.11; 12; 13; 14; 15 This work, in line with Refs. 11; 12; 13;
14; 15, investigate the evolution of (and correlation among) the
superconducting and normal states within the FL region of the HF T-p phase
diagram. As shown below, in contrast to these references, (i) we consider
explicitly the normal-state FL character, (ii) we do not invoke any additional
impurity scattering; rather we consider the spin-fluctuations to be operating
within patches of the sample (the average dimension of a patch is longer than
the mean free path and coherent length while their concentration is reflected
in the excess residual resistivity), and most importantly (iii) we consider
that the very same fluctuation-mediated electron-electron scattering channel
is responsible for the evolution of both the superconductivity and the FL
character.

It is worth recalling that various experiments reveal the presence of phase-
boundary curve, e.g. $T^{*}_{\text{{FL}}}(p^{*})$ within the HF T-p phase
diagram, across which both the transport and thermodynamic properties switch
over from non-Fermi-liquid (NFL) into a Fermi-liquid (FL) behavior. Such a
switch is best illustrated by the electronic contribution to the low-
temperature resistivity, $\rho(T,p)$, of typical HF superconductors: Within
the NFL phase,

$\rho(T>T^{*}_{\text{{FL}}}[p^{*}],p\leq
p^{\text{*}})-\rho_{x}=\rho_{\circ}+A_{\text{NFL}}T^{n},\mathrm{\,\,\,\,\,\,[n<2],}$
(1)

while within the FL phase,

$\rho(T_{c}<T<T^{*}_{\text{FL}}[p^{*}],p>p^{\text{*}})-\rho_{x}=\rho_{\circ}+AT^{2},$
(2)

where $\rho_{x}=\rho_{\circ}^{n.f.}+\rho_{n.f.}(T)$ represents all non-
fluctuation-related (e.g. defects, phonon, magnetic) contributions (see end of
Subsec. II.1). Within the FL region (which starts at the switching of NFL into
FL character and ends when $T_{c}\rightarrow 0$), we denote the excess
contribution of the residual resistivity by $\rho_{\circ}$, that of the
quadratic-in-T coefficient by $A$, and that of the superconductivity
transition by $T_{c}$; all these contributions are attributed to a
fluctuation-mediated electron-electron (e-e) scattering channel (see below).
Generally, these $\rho_{\circ}(p),\,A(p),$ and $T_{c}(p)$ parameters — derived
from experimental $\rho(T,p)$ curves —manifest a dramatic and non-monotonic
variation around each of the critical $p_{m}$ and $p_{v}$ points (see, e.g.,
Figs. 2,3,4).

In this work, we are interested in analyzing and rationalizing the baric
evolution of $\rho_{\circ}(p)$, $A(p)$, and $T_{c}(p)$ within the FL region of
various archetype HF superconductors. Our analysis identified two universal
correlations among these $\rho_{\circ}$, $A$, and $T_{c}$ parameters. We argue
that the fluctuation-mediated electron-electron scattering channel is
responsible for the emergence of the superconductivity, the FL character and
the correlations among them.

The text below is organized as follows. We first recall some helpful insight
regarding the fluctuation-mediated e-e scattering process.6; 7 In Subsec. II.2
we extract, identify, and generalize two main empirical correlations. In
Subsec. II.3 we argue that these correlations are driven by a fluctuation-
mediated e-e interaction channel. Such fluctuations can be either spin
fluctuation6 or valence fluctuation7 (generically referred to as fluctuation).
Although we consider below the spin-fluctuation exchange mechanism 6,
generalization to valence fluctuation7 is implicitly assumed. On adopting such
a mechanism and applying standard Migdal-Eliashberg description of
superconductivity and Boltzmann’s transport theory, we derive analytic
expressions that compare favorably with the empirically-obtained correlations.
Comparison to other fluctuation-bearing/defect-bearing superconductors will be
briefly discussed in Sec. III; there, we demonstrate the generality of our
approach by discussing the Kadowaki-Woods and gap-to-$T_{c}$ ratios of these
HF superconductors.

Figure 1:  A sketch of a fluctuation-mediated e-e interaction channel. (a) An
antiferromagnet structure representing an initial state in the phase diagram
of a typical heavy fermion superconductor. The tiny arrows denote localized
magnetic moments of atomic entities (represented by the small solid circles).
(b) The nonmagnetic state after the quench of localized magnetic moments and,
consequently, the magnetic structure. Within the direct lattice, the two
electrons with ${\bf k}_{1}$ and ${\bf k}_{2}$ scatter into final states ${\bf
k}_{1}^{\prime}$ and ${\bf k}_{2}^{\prime}$. (c) A $\bf{\delta g}=0$ two-
electrons scattering process in the reciprocal space: As that ${\bf q}={\bf
k}_{1}^{\prime}-{\bf k}_{1}-{\bf g}$, the phase space for scattering is quite
limited; a Fermi-liquid sate is possible only at very low temperatures. (d)
The $\bf{\delta g}\neq 0$ two-electrons scattering process in the reciprocal
space: as that ${\bf q}={\bf k}_{1}^{\prime}-{\bf k}_{1}-{\bf g}-\delta{\bf
g}$, the phase space available for scattering is considerably enlarged,
leading to a robust superconducting and Fermi-liquid states. (e) Feynman
diagram of the $\bf{\delta g}=0$ process [as in panel (c)] wherein only
longitudinal modes are involved. (f) The Feynman diagram of the $\bf{\delta
g}\neq 0$ process [as in panel (d)] which is mediated by exchanging all
kinematically unconstrained modes.

##  II Analysis

###  II.1 Some Preliminaries on the fluctuation-mediated e-e interaction
channel

It is recalled that a typical T-p phase diagram of most quantum-critical HF
superconductors exhibits a series of distinct electronic states (see, e.g.,
Figs. 2(a)-3(a) and Refs. [3; 8; 16]). Often, the initial state is an
antiferromagnetic, represented in Fig. 1(a) as simple two-dimension Neel-type
arrangment. Beyond a critical value of the control parameter, $p_{m}$ in the
present cases, the localized moments and the three-dimensional magnetic
structure are quenched leading to a series of "nonmagnetic" states which can
be of Kondo-type, non-conventional superconducting, non-FL, or FL character.
Nevertheless, some remnant magnetic fluctuations persist within various micro-
sized patches which are distributed randomly within an otherwise metallic
matrix.17

These magnetic fluctuations are considered to mediate an e-e scattering
process 6: as shown in Figs. 1(b-f), the two electrons with ${\bf k}_{1}$ and
${\bf k}_{2}$ scatter into final states ${\bf k}_{1}^{\prime}$ and ${\bf
k}_{2}^{\prime}$. In particular, the electron, initially at a state ${\bf
k}_{1}$, goes into a final state ${\bf k}_{1}^{\prime}$ after being scattered
by a mode with wavevector ${\bf q}$; in doing so, it transfers an amount of
energy and quasi-momentum satisfying ${\bf k}_{1}^{\prime}-{\bf k}_{1}-{\bf
q}={\bf g}+\bf{\delta g}$ where ${\bf g}$ is a vector in a reciprocal magnetic
lattice while $\bf{\delta g}$ is an uncertainty in ${\bf g}$ due to the
inherent nature of the process leading to fluctuation (most obvious for the
doping process).

We consider that, within the Fermi-liquid state of these HF superconductors,
an increase in pressure leads to a reduction of the size and density of the
fluctuation-bearing patches,17 till eventually one reaches, at very high
pressure, the fluctuation-free state wherein no modes are available for
mediation. This, as will be detailed below in Figs. 2, 3, and 4, leads to a
complete removal of fluctuation-related features, namely the superconductivity
($T_{c}\rightarrow 0$), the Fermi-liquid state ($A\rightarrow A_{\circ}$), the
residual resistivity ($\rho\rightarrow\rho_{\circ}^{\circ}$), and their
correlations.

Let us now discuss the fluctuation-based mediation within the FL region.
First, we consider the case satisfying $\bf{\delta g}=0$ condition of Figs.
1(c) and 1(e). Here, the quasi-momentum ${\bf q}={\bf k}_{1}^{\prime}-{\bf
k}_{1}-{\bf g}$ is conserved exactly. Accordingly, only longitudinal modes
with a well defined polarization, $\hat{{\bf e}}({\bf q}={\bf
k}_{1}^{\prime}-{\bf k}_{1})$, will be involved. This means that the phase
space available for momentum relaxation will be quite limited leading to, if
any, a very small $T_{c}$ and $A$.

Secondly, in contrast to above, the surge of fluctuation-based mediation
satisfying ${\bf\delta g}\neq 0$ condition, Figs. 1(d) and 1(f), provides a
source for short wavelength (large ${\bf q}$) phase interference. This
$\bf{\delta g}\neq 0$ condition implies that the quasi-momentum ${\bf q}={\bf
k}_{1}^{\prime}-{\bf k}_{1}-{\bf g}-\delta{\bf g}$ is no longer conserved
since $\bf{\delta g}$ is arbitrary.18; 19; 20 Then, multiple modes
[longitudinal and transverse, of all polarizations $\hat{{\bf e}}({\bf
q}\neq{\bf k}_{1}^{\prime}-{\bf k}_{1}-{\bf g})$] become kinematically
available, Fig. 1(f). As shown in Fig. 1(d), the phase space available for
momentum relaxation is considerably enlarged: this leads to an enhanced
$T_{c}$ and $A$.

Considering the above partitioning of a fluctuation-bearing sample into two
spacial regions, Matthiessen’s rule suggests a sum of two types of
contributions: (i) one type is related to normal (non-fluctuation-bearing)
patches; these are the residual resistivity $\rho_{\circ}^{\circ}$, the
coefficient $A_{\circ}$, the coupling constant $\lambda_{\circ}$, and the mean
free path $\ell_{\circ}$
($\ell_{\circ}\propto\frac{1}{\rho_{\circ}^{\circ}}$). (ii) The other type is
related to fluctuation-bearing patches within which, we consider, a
fluctuation-mediated e-e scattering channel is operating and as such leading
to excess contributions denoted as $A$, $\lambda$, $\ell$, and $\rho_{\circ}$.
Below we consider $\rho_{\circ}$ to be arising from those batches and as such
it measures the strength of the fluctuation-related channel while
$\rho_{\circ}^{\circ}$ as a measure of all normal processes.

The total contribution is denoted as $\rho_{\circ}^{tot}$, $A_{tot}$,
$\lambda_{tot}$, or $\ell_{tot}$. Often, the non-fluctuation-bearing
$\rho_{\circ}^{\circ}$ and $A_{\circ}$ contributions are obtained by
extrapolation. In our case here, these values (shown as large green circle in
Figs. 2, 3, and 4 and tabulated in Table LABEL:Tab-Fit-Values-A-Ro-Theta-F)
were estimated from the very-high-pressure region whereat $T_{c}\rightarrow
0$, indicating, we assume, the quench of the fluctuations. Then each excess,
fluctuation-related contribution $X=X_{tot}-X_{\circ}$, is obtained from 21

$\displaystyle\rho_{\circ}=\rho_{\circ}^{tot}-\rho_{\circ}^{\circ},$ (3a)
$\displaystyle A=A_{tot}-A_{\circ},$ (3b)
$\displaystyle\lambda=\lambda_{tot}-\lambda_{\circ},$ (3c)
$\displaystyle\ell=1/(\frac{1}{\ell_{tot}}-\frac{1}{\ell_{\circ}}).$ (3d)

For theoretical analysis, it is more convenient to measure the strength of the
fluctuation-related scattering channel via the effective mean free path, the
scaling length $\ell\propto 1/\rho_{\circ}$. Then any variation in the control
parameter (e.g. pressure, alloying, or defect incorporation22) would be
reflected in $\ell$ and manifested as a variation in the superconductivity,
the FL character, and the correlation among $\rho_{\circ}(\ell)$,
$T_{c}(\ell)$ and $A(\ell)$; within the FL region of Figs. 2-4, a pressure
increase leads to a reduction of these parameters indicating an increase in
fluctuation-related $\ell$. The functional dependence of $\ell$ on, e.g.,
pressure will not be discussed in this work.

Figure 2:  T-p phase diagram and baric evolution of the parameters of CeCu2Ge2 (data taken from Ref. 3). (a) A semi-log T-p phase diagram showing the evolution of $T_{N}(p)$, $T_{c}(p)$, and the exponent n(p). Vertical dashed arrows represent $p_{m}\approx 9.4\,$ GPa and $p_{v}\approx 16\,$ GPa.3 The dashed lines are visual guides while the curved arrows signal the direction of pressure increase. The red symbols and hatched area mark the FL region. Evolution of (b) $T_{c}(p)$, (c) $A_{tot}(p)$ in a semi-log plot, and (d) $\rho_{\circ}^{tot}(p)$. As discussed in text, for $p\rightarrow$20 GPa, the pressure-induced strong reduction of the fluctuation leads to $T_{c}\rightarrow T_{c}^{\circ}\approx 0$, $A\rightarrow A_{\circ}$ and $\rho_{\circ}^{tot}\rightarrow\rho_{\circ}^{\circ}$: the limits at 20 GPa (large green circles) are shown in Table LABEL:Tab-Fit-Values-A-Ro-Theta-F. (e) Correlation of $[A_{tot}(p)-A_{\circ}]$ with $[\rho_{\circ}^{tot}-\rho_{\circ}^{\circ}]$. The solid line is a best fit to Eq. 9: $A_{0}$ and $A_{2}$ given in Table LABEL:Tab-Fit-Values-A-Ro-Theta-F. (f) $T_{c}$ versus $[\rho_{\circ}-\rho_{\circ}^{\circ}]$: The solid line is calculated based on Eq. (11) of Subsec. II.3.3. (g) $ln(T_{c})\textit{ versus }(A_{tot}-A_{\circ})^{-\frac{1}{2}}$. The solid red line is the best fit to linearized Eq. 10; $\theta$ and $\mathcal{F}$ parameters are given in Table LABEL:Tab-Fit-Values-A-Ro-Theta-F.  Figure 3:  (a) The T-p phase diagram and baric evolution of (b) $T_{c}(p)$, (c) $A_{tot}(p)$, and (d) $\rho_{\circ}^{tot}(p)$ of CeCu2Si2. All curves were taken from the continuous line (the visual guide) of Ref. 8. For a discussion on the critical pressures $p_{m}$ and $p_{v}$ and the related superconductivity regions (SC 1 and SC II) see Ref. 8. The red symbols and hatched area mark the FL region while the dashed lines are visual guides. It is noted that, due to non-stoichiometry, defects, impurities or disorder, the reported parameters of CeCu2Si2 vary considerably from sample to sample; nonetheless, the overall evolution as well as the derived correlations are consistently similar.23; 24; 25; 26 As $p\rightarrow 7$ GPa, $T_{c}\rightarrow 0$, $A\rightarrow A_{\circ}$ and $\rho_{\circ}\rightarrow\rho_{\circ}^{\circ}$ (see text): the extrapolation to 7 GPa (large green circles) are shown in Table LABEL:Tab-Fit-Values-A-Ro-Theta-F. (e) A plot of [$A(p)-A_{\circ}$] versus [$\rho_{\circ}(p)-\rho_{\circ}^{\circ}$]. The best fit to Eq. 9 is shown as solid red curve; the fit parameters $A_{0}$ and $A_{2}$ are given in Table LABEL:Tab-Fit-Values-A-Ro-Theta-F. (f) $T_{c}$ versus [$\rho_{\circ}(p)-\rho_{\circ}^{\circ}$]: For the solid calculated line, see Eq. (11) in Subsec. II.3.3. (g) $ln(T_{c})$ versus $(A-A_{\circ})^{-\frac{1}{2}}$. The straight red line is a best fit to linearization of Eq. (10): $\theta$ and $\mathcal{F}$ are given in Table LABEL:Tab-Fit-Values-A-Ro-Theta-F.  Table 1: Representative fit parameters of CeCu2Ge2, CeCu2Si2 and CeCoIn5. Values of Ao and A2 were obtained from a fit of Eq. 9 to Figs. 2(e),3(e),4(e)): As $A_{1}\approx 0$, it was fixed as $A_{1}\equiv 0$. For all compounds, $A_{2}$, $\theta$ and $\mathcal{F}$ are close to each other, except $A_{2}$ of CeCoIn5 which is orders of magnitude higher than the others: this together with the high value of $T_{c}$ is most probably related to a relatively high strength of the fluctuation-related e-e scattering channel of CeCoIn5. It is worth noting that the strength and evolution of the fit parameters depend critically on the criteria for determining Tc (10-90%, 50%, onset, zero point etc.) and $A$ (the sample geometry, which is often not precisely determined). Additionally, the widely different methods for sample synthesis and thermal treatment adopted by different groups give rise to a corresponding variation in sample-dependent properties. Although these non-fluctuation-related material properties are accounted for by $T_{\circ}$, $A_{\circ}$, and $\rho_{\circ}^{\circ}$, it is expected to lead to a wide scatter in the fit parameters. The scatter in the experimental curves leads to a large error value. $HFs$ | $\rho_{\circ}^{\circ}$ |  Ao |  A2 | $\theta$ | $\mathcal{F}$
---|---|---|---|---|---
- | $\mu\Omega\text{cm}$ | $\mu\Omega\text{cm/K}^{2}$ | $\times 10^{6}\Omega^{-1}\text{cm}^{-1}/\text{K}^{2}$ | K | $\times 10^{-3}(\Omega\text{cm/K}^{2})^{1/2}$
CeCu2Ge2 | 10(1) |  2.9(5)$\times 10^{-3}$ |  2.0(2)$\times 10^{-4}$ | 4.4(2) | 0.36(1)
CeCu2Si2 | 18.5(5) | 0.010(5) |  7.0(5)$\times 10^{-4}$ | 2.6(1) | 0.49(1)
CeCoIn5 | 0.130(5) | 0.082(4) | 6.0(2) | 3.2(1) | 0.18(1)

###  II.2 Empirical analysis: extraction of correlations among $\rho_{\circ}$,
$A$, and Tc

Below, we review the baric evolution of $\rho_{\circ}(p)$, $A(p)$, and
$T_{c}(p)$ of three representative HF superconductors for which detailed
resistivity curves $\rho_{tot}(T,p)$ (essentials for determining these
$\rho_{\circ}$, $A$, and $T_{c}$ parameters) were reported. Special attention
will be directed towards identifying the correlations.

####  II.2.1 CeCu${}_{2}X_{2}$ ($X$=Si, Ge)

The T-p phase diagram and the baric evolution of $T_{c}$, $A_{tot}$, and
$\rho_{\circ}^{tot}$ of CeCu2Ge2 are shown in Figs. 2(a)-2(d) (Refs. 3; 27)
while those of CeCu2Si2 in Figs. 3(a)-3(d) (Refs. 4; 26; 28; 29; 8). As
evident, these plots highlight the dramatic baric evolution in particular
around the critical points.

The phase diagram of each of CeCu2X2 manifests a FL character at sufficiently
higher pressure; this is evidenced as a quadratic-in-T resistivity
contribution, identified in Figs. 2(a) and 3(a) by the hatched area and the
n=2 notation. At 20 GPa for CeCu2Ge2 and 7 GPa for CeCu2Si2, $T_{c}\rightarrow
0$; we identify this as a signal of a pressure-induced quench of the
fluctuation-related FL contributions. Based on Eqs. 3, all parameters
extrapolate to non-fluctuation-related contributions: $A\rightarrow A_{\circ}$
and $\rho_{\circ}\rightarrow\rho_{\circ}^{\circ}$ (see Figs. 2 and 3 and Table
LABEL:Tab-Fit-Values-A-Ro-Theta-F).

Then for the purpose of empirical identification of any possible correlation
within the FL region, we plot
$[A_{\text{tot}}(p)-A_{\circ}]\,\,\it{versus}\,\,[\rho_{\circ}^{tot}(p)-\rho_{\circ}^{\circ}]$
in Figs. 2(e) and 3(e)],
$T_{c}\,\,\it{versus}\,\,[\rho_{\circ}^{tot}(p)-\rho_{\circ}^{\circ}]$ in
Figs. 2(f) and 3(f)], and
$ln[T_{c}(p)]\,\,\it{versus}\,\,1/\sqrt{A_{\text{tot}}(p)-A_{\circ}}$ in Figs.
2(g) and 3(g).

Figures 2(e) and 3(e) show that within the FL state, well above the critical
pressures region, 1; 2; 30; 25; 26 one obtains

$(A-A_{\circ})\approx A_{2}(\rho_{\circ}-\rho_{\circ}^{\circ})^{2},$ (4)

The $A_{\circ}$ and $A_{2}$ parameter of each CeCu2X2 are given in Table
LABEL:Tab-Fit-Values-A-Ro-Theta-F. It is noted that the absence of a linear-
in-$\rho_{\circ}$ term rules out any Koshino-Taylor contribution. On the other
hand, Figs. 2(g) and 3(g) indicate that

$\ln(T_{c})\propto(A-A_{\circ})^{-\frac{1}{2}}.$ (5)

A closer look at Eqs. (4 and 5) suggests that a relation between $T_{c}$ and
$(\rho_{\circ}-\rho_{\circ}^{\circ})$ can be derived; indeed the solid red
curves in Figs. 2(f) and 3(f) were calculated based on Eq. 11 which is a
derived equation (see Subsec. II.3.3).

Figure 4: (a) T-p phase diagram and baric evolution of (b) $T_{c}(p)$, (c)
$A_{tot}(p)$, and (d) semilog plot of $\rho_{\circ}^{tot}(p)$ of CeCoIn5. All
curves were taken from the continuous line (the visual guide) of Ref. 16. For
more detail on this phase diagram, in particular, the position of critical
$p_{m}$, see Refs. 16; 31; 32; 33. The red symbols and hatched area identify
the FL region while the dashed lines are visual guides. For $p\rightarrow$4.4
GPa, $T_{c}\rightarrow 0$, $A\rightarrow A_{\circ}$ and
$\rho_{\circ}\rightarrow\rho_{\circ}^{\circ}$ (see text and Table LABEL:Tab-
Fit-Values-A-Ro-Theta-F). (e) A plot of $A_{tot}-A_{\circ}$ versus
$\rho_{\circ}^{tot}-\rho_{\circ}^{\circ}$. The solid curve is the best fit to
Eq. 9 (see Table LABEL:Tab-Fit-Values-A-Ro-Theta-F). (f) $T_{c}$ versus
($\rho_{\circ}^{tot}-\rho_{\circ}^{\circ}$): The solid calculated line is
based on Eq. (11). (g) $ln(T_{c})$ versus $(A-A_{\circ})^{-\frac{1}{2}}$. The
best fit to the linearization of Eq. (10) (see Table LABEL:Tab-Fit-Values-A-
Ro-Theta-F for the fit parameters).

####  II.2.2 CeCoIn5

The T-p phase diagram of CeCoIn5 [Refs. 16; 34; 31; 32] is shown in Fig. 4(a)
while the baric evolution of $T_{c}(p)$, $A_{tot}(p)$ and
$\rho_{\circ}^{tot}(p)$ are shown in Figs. 4(b), 4(c) and 4(d), respectively.

A comparison of the features of CeCoIn5 with those of CeCu${}_{2}X_{2}$
($X$=Si, Ge) suggests an overall similarity in that:

  * •

Figure 4(e) indicates that $A(p)-A_{\circ}$ is large but still quadratic-
in-$[\rho_{\circ}(p)-\rho_{\circ}^{\circ}$],

  * •

Figure 4(f) reveals that the evolution of
$T_{c}[\rho_{\circ}(p)-\rho_{\circ}^{\circ}]$ can be described by Eq. 11 (see
Subsec. II.3.3).

  * •

Figure 4(g) shows that $\ln[T_{c}(p)]$ is linear in
$[A(p)-A_{\circ}]^{-\frac{1}{2}}$ .

Figure 5: (a) The generalized plot of $\frac{T_{c}(p)}{\theta}$versus
$\lambda=\frac{(A-A_{\circ})^{\frac{1}{2}}}{\mathcal{F}}$ of the tabulated HF
superconductors. 3; 8; 25; 16; 35; 36; 37; 38; 39; 40; 41; 42 (b) For
comparison, we include the plot of the same relation in conventional
superconductors (data taken from Ref. [43]). (c) A collapse of data from all
HF and conventional superconductors on the single
$T_{c}/\theta=exp(-1/\lambda)$ curve: an unsurprising result considering
panels (b and c); nevertheless, considering the diversity of the
superconducting materials, this plot is highly nontrivial. Each pair of
sample-dependent $\theta$ and $\mathcal{F}$ was evaluated within the pressure
range wherein Eq. (10) is valid [see Figs. 2(g), 3(g), and 4(g)].
Representative values of $\theta$ and $\mathcal{F}$ are shown in Table
LABEL:Tab-Fit-Values-A-Ro-Theta-F. As evident, the distribution region of the
pairs ($T_{c}/\theta$, $\lambda$) in both HF and strong-coupled
superconductors are similar, with the upper limit being $\approx(0.6,2.1)$
(for $\lambda$ limit of strong superconductors, see Ref. 20 and also Fig. 4 of
Ref. 43). As mentioned for CeCu2X2 and CeCoIn5, $\lambda$ is corrected using
$\frac{(A-A_{\circ})^{\frac{1}{2}}}{\mathcal{F}}$. Similar plot, but with no
$\frac{(A-A_{\circ})^{\frac{1}{2}}}{\mathcal{F}}$ correction, was reported for
both pnictide and chalcognides superconductors.44; 45 Finally, it is
emphasized that the relatively high values of $\frac{T_{c}(p)}{\theta}$ (those
close to 1) should not be substituted in Eq. 14 since this equation is valid
only within the $\frac{T_{c}}{\theta}\ll 1$ condition.46

###  II.3 Theoretical analysis: interpretation of the correlations among
$\rho_{\circ}$, $A$, and Tc

The aforementioned experimental evidences emphasize that in spite of the wide
differences in material properties of these HF superconductors, one observes:
(i) A similarity in the overall evolution of the phase diagrams as well as in
the baric evolution of their $T_{c}(p)$, $A(p)$, and $\rho_{\circ}(p)$. (ii) A
nontrivial manifestation of neighboring superconductivity and FL phases. (iii)
Two correlations (expressed in Eqs. 4 and 5) and a derived one; all manifested
in Figs. 2,3,4. Below we discuss the significance of these features in terms
of a fluctuation-mediated e-e scattering channel which, due to a modification
in the kinematic constraints, is endowed with a significantly enlarged phase
space for scattering, much larger than the traditional Baber or Umklapp e-e
scattering channels.47

Furthermore, the empirical analysis highlights the distinct and nontrivial
contrast between the properties of the FL state of a HF superconductor and
that of conventional weakly-correlated superconductor: It is remarkable that
$A$ is more than five orders of magnitude higher than the contribution
expected for a typical Fermi-liquid metal ($A_{\circ}\simeq
10^{-7}\mu\Omega$cm/K2).48; 49; 50; 51; 52; 53

Another contrast is that a conventional FL superconductor, in contrast to a HF
one, does not manifest a correlation among $\rho_{\circ}$, $T_{c}$ and $A$:
$\rho_{\circ}$ is determined by the electron-impurity scattering,
$\rho_{\circ}\sim|V_{imp}|^{2}$; $T_{c}\sim e^{-1/\lambda}$ is associated with
electron-phonon coupling, $\lambda\sim|V_{ep}|^{2}$; while $A$ depends on e-e
interaction, $A\sim|V_{ee}|^{2}$.

It is worth emphasizing that, although two exceptional cases were reported to
yield a similar BCS-like correlation among $T_{c}$ and $A$ (see Ref. 54 and
Subsec. II.2), none of them manifest a correlation between $A$ and
$\rho_{\circ}$ nor between $T_{c}$ and $\rho_{\circ}$; in fact, Anderson
theorem excludes the latter relation. It is, then, quite puzzling that the
aforementioned empirical analysis, see Figs. 2-4, delineate a distinct FL
phase within which one identifies a surge of superconductivity and pressure-
dependent correlations among $\rho_{\circ}$, $A$ and $T_{c}$. This puzzle can
be resolved if the manifestation of superconductivity and FL state as well as
the correlation of $T_{c}$ (hallmark of superconductivity) and $A$ (hallmark
of normal FL state) are assumed to be driven by a retarded, spin-fluctuation 6
or valence-fluctuation 7 mediated e-e interaction. This is reminiscent of the
defect-induced, phonon-mediated e-e scattering in defectal-bearing systems
(see Ref. 47)

The basic idea is discussed in Subsect.II.1: within a HF superconducting
sample, the fluctuation-related quasiparticles can be created or annihilated
and that their mediation of the e-e interaction, $V^{f}_{ee}$,6; 7 is the
driving mechanism behind the surge of each of these features. We illustrate
this argument by considering the spin-fluctuation case, as in Ref. 6. There,
$I^{2}\chi(\omega)$ [$I$ is an exchange coupling while $\chi(\omega)$ is a
dynamic susceptibility] plays the same role as that of Eliashberg’s electron-
phonon spectral function $\alpha^{2}\mathcal{F}\left(\omega\right)$.
$V^{sf}_{ee}$ can then be obtained from the integral over frequencies of such
an Eliashberg’s type spectral function.

Below, we give a short summary of the main expressions describing
$T_{c}(\ell)$, $A(\ell)$ and their correlations with each other as well as
with $\rho_{\circ}(\ell)$.

####  II.3.1 Expression of $T_{c}(\ell)$

Adopting the traditional strategy for calculating $T_{c}(\ell)$, we solved the
set of coupled linearized Eliashberg’s equations within the BCS limit. In this
limit, $T_{c}(\ell)$ is determined by the superconducting coupling constant
$\lambda(\ell)=N(\epsilon_{F})V_{ee}^{sf}(\ell)=2\int
d\omega\frac{I^{2}\chi(\omega)}{\omega_{opt}^{2}+\omega^{2}}$, where
$\omega_{opt}$ is an optimal frequency at which $\lambda$ is maximal, and can
thus be written as: 47

$T_{c}(\ell)=\theta\,exp\Bigg{(}{\frac{-1-\lambda(\ell)}{\lambda(\ell)-\mu^{*}}}\Bigg{)},$
(6)

where $\mu^{*}$ is the Coulomb pseudopotential and $\theta$ is an energy scale
that depends on the cut-off bandwidth $\omega_{c}^{sf}$.

####  II.3.2 Expression of $A(\ell)$

Applying a variational approach to the linearized version of Boltzmann’s
transport equations within the relaxation time approximation (wherein the
inverse scattering time is calculated by the use of Fermi’s golden rule) we
obtain47 $\rho(\ell)=A(\ell)T^{2}$ and

$A(\ell)=F^{2}_{\ell}\left|\frac{\lambda(\ell)-\mu^{*}}{1+\lambda(\ell)}\right|^{2}.$
(7)

Here, $F_{\ell}$ represents the efficiency of momentum relaxation and the
availability of phase space for scattering.47 Within the FL phase, $\ell$ is
long, $1\ll k_{F}\ell_{p}<\infty$, and $\rho_{\circ}$ is small ($k_{F}$ =
Fermi wave number); as such, Eq. (7) can be expanded around
$\lambda_{\circ}=\lambda(\ell\rightarrow\infty)$ of the normal matrix as

$A(\ell)\simeq
a_{\circ}+a_{1}(\delta\lambda)+a_{2}(\delta\lambda)^{2}+\mathcal{O}[(\delta\lambda)^{3}],$
(8)

wherein
$a_{\circ}=A(\ell\rightarrow\infty)=\frac{|\lambda_{\circ}-\mu^{*}|^{2}}{1+\lambda_{\circ}}$
refer to the negligibly small kinematically-constrained non-fluctuating
contributions; the second and third term [containing
$\delta\lambda=\lambda(\ell)-\lambda_{\circ}$ and the coefficients
$a_{1}=2F^{2}_{\ell}(|\lambda_{\circ}-\mu^{*}|(1+\mu^{*}))/(1+\lambda_{\circ})^{3}$
and
$a_{2}=F^{2}_{\ell}(1+\mu^{*}-2|\lambda_{\circ}-\mu^{*}|)(1+\mu^{*})/(1+\lambda_{\circ})^{4}$]
denote contributions from all kinematically unconstrained relaxation processes
after incorporating the fluctuation-mediated channel.

The second-order polynomial expression of $A(\ell)$, Eq. 8, is reminiscent of
the empirical quadratic-in-$\rho_{\circ}$ of Eq. 4. Below we look for an
analytical description of $A(\rho_{\circ})$. Let us start by recalling that,
in a typical Fermi liquid, the frequency dependence of the imaginary part of
the self energy is given by ${\cal I}m\Sigma(\omega)\sim\omega^{2}$. Then, on
considering the relevant energy scale to be set by $k_{B}T$, one obtains the
characteristic FL quadratic-in-T resistivity. Intuitively, this is related to
the fact that, for a given temperature $T$, $N(E_{F})k_{B}T$ single particles
within the Fermi surface are participating in the fluctuation-mediated two-
particle channel (each single particle can scatter into one another via this
fluctuation-mediated scattering process): this generates the well-known
$AT^{2}$ resistivity contribution. Specifically, our analysis showed that this
channel leads to a FL state with $A$ as in Eq. 7 and superconductivity with
$T_{c}$ as in Eq. 6. Moreover, at $T=0$ limit, the relevant energy scale is
set by the fuzziness of the Fermi surface which is determined by $\delta g$.
Just as the case for the $\omega^{2}$-leading-to-$T^{2}$-contribution, we
expect $\rho_{\circ}\propto\delta\lambda\propto(\delta g)^{2}$.55 This allows
us to establish a correlation among $\rho_{\circ}$ and each of $T_{c}$ and $A$
(each is a function of $\ell$). Guided by these considerations, as well as the
empirical relation of Eq. 4, we consider

$A(\ell)\simeq A_{\circ}+A_{1}(\rho_{\circ})+A_{2}(\rho_{\circ})^{2},$ (9)

where $A_{i}(i=0,1,2$) are functions of $F^{2}_{\ell},~{}\mu^{*},\text{ and
}\lambda_{\circ}$. Eq. 9 suggests three limiting contributions to $A(\ell)$:
(i) The host contribution (dominant $A_{\circ}\neq 0$ when $A_{1}\approx 0$
and $A_{2}\approx 0$); (ii) a single-particle-like Koshino-Taylor-type
$(A-A_{\circ})\propto\rho_{\circ}$, contribution (dominant $A_{1}\neq 0$
contribution); and (iii) an $(A-A_{\circ})\propto(\rho_{\circ})^{2}$
contribution (a dominant $A_{2}$ and a negligible $A_{1}$). Results of Figs.
2(e), 3(e), 4(e) belong to the third case.

####  II.3.3  Correlation between $T_{c}(\ell)$ and $A(\ell)$

Combining the expressions for $T_{c}(\ell)$ in Eq. 6 and $A(\ell)$ in Eq. 7,
we arrive at:

$T_{c}(\ell)=\theta\,e^{-\mathcal{F}/\sqrt{A(\ell)}}.$ (10)

This universal and exact kinematic scaling relation (valid for long $\ell$,
low $\rho_{\circ}$: the Fermi-liquid region) is the essence of the relations
of Eq. (5) and Figs. 2(g), 3(g), and 4(g). Eq. (10) is most remarkably
manifested in Fig. 5: an increase (decrease) of pressure leads to a downwards
(upwards) flow of $T_{c}(\ell)$ towards weaker (stronger) couplings, without
ever leaving the curve given by Eq. (10).

Finally, a simplified but approximate relation between $T_{c}$ and
$(\rho_{\circ}-\rho_{\circ}^{\circ})$ can be obtained in the specific case
wherein the quadratic-in-$\rho_{\circ}$ term in Eq. 9 is dominant.
Substituting Eqs. (4 and 7) into Eq. 10, we obtain

$T_{c}(\ell)\approx\theta\,exp\big{(}\frac{-\mathcal{F}}{\sqrt{A_{2}}\big{[}\rho_{\circ}(\ell)-\rho_{\circ}^{\circ}\big{]}}\big{.}$
(11)

On substituting $\theta$, $\mathcal{F}$, $A_{2}$, and $\rho_{\circ}^{\circ}$
of Table LABEL:Tab-Fit-Values-A-Ro-Theta-F into this equation, one obtains the
solid curves of Figs. 2(f), 3(f), 4(f); this excellent description of
experimental data with no fitting parameters is no surprise since, as
mentioned above, Eq. (11) is a derived one.

##  III Discussion and Conclusions

The similarity of the phase diagrams shown in Figs. 2,3,4, as well as those of
other HF superconductors,1; 2 suggests a generalized T-p phase diagram that
highlights the similarity in the cascade of distinct electronic states and in
the overall evolution of $T_{c}(\ell)$, $A(\ell)$, and $\rho_{\circ}(\ell)$.
Of particular interest to this work is the nontrivial manifestation, in all
phase diagrams, of a superconductivity, a FL character and their correlations
(see Figs. 2, 3, 4, and 5). We argue above that the surge of all these
features and correlations is driven by the spin-fluctuation/valence-
fluctuation mediated e-e channel operating within the FL range of the studied
HF superconductors.56

It happened that for these fluctuations to be well defined quasiparticles and
for the Migdal-Eliashberg theoretical framework to be applicable, one needs to
ensure that $\ell$ is long which for the spin-fluctuation case translates into
$\chi(\omega)$ and $V^{sf}_{ee}$ being weak and, as a consequence, reduced
$\rho_{\circ}$, $A$, and Tc. In fact this argument, turned around, can be used
to define the FL range which can be reached by an application of higher
pressure (see Figs. 2\- 4), strong magnetic field,31; 32; 33 or incorporation
of nonmagnetic impurities: 57; 58; 59; 60 All promote the weakening of the
fluctuation-mediated scattering process.

Based on these arguments, the following inferences can be drawn: First, Figs.
2\- 4 indicate that on moving out of the FL region but towards the quantum
critical points, one observes a continuous enhancement in $\rho_{\circ}$, $A$,
and $T_{c}$. This continuity is suggestive of a similarity in the scattering
interaction, the strength of which is enhanced on reducing the pressure.
However, Figs. 2-4 also indicate that below the FL region our analytical
expressions do not reproduce the observed baric evolution of $A$ or $T_{c}$.
This shortcoming is related to the breakdown of the long-$\ell$ condition.
Nevertheless, this does not invalidate our analysis within the FL region. It
is worth repeating that the availability of a FL state is not a precondition
for the applicability of our approach. Rather, the surge of the e-e scattering
channel gives rise to the FL state, the superconductivity and their
correlation.

Second, it was reported that an application of a magnetic field ($H>H_{2}$)
within the superconducting dome of CeCoIn5 leads to a quench of
superconductivity in favor of a FL normal-state.31; 32; 33 We attribute this
field-induced FL character within the NFL region to a field-induced weakening
of the strength of the spin-fluctuation and as such to an increase in $\ell$,
in reminiscence of the aforementioned increase in applied pressure or in the
incorporation of nonmagnetic impurities. Further analysis is underway.

Third, we argued above that within the long-$\ell$ region, the spin-
fluctuation/valence-fluctuation modes are mediating the e-e interaction and
this in turn leads to the scaling between $\frac{T_{c}}{\theta}$ and
$(A_{tot}-A_{\circ})$ (see Fig. 5(a)). This is similar to the emergent phonon-
mediated scaling reported for other non-HF superconductors: Fig. 5(b)
demonstrates the scaling in conventional superconductors.47; 43 Similar
scaling was reported for the Fe-based pnictides44 and chalcogenides:45
Although the superconductivity in these series are considered to be driven by
spin-fluctuation mediated pairing10 however we did not include these Fe-based
materials in Fig.5 since these reports did not include a correction for the
non-fluctuation-related contribution: $A_{tot}$ was used instead of
$(A_{tot}-A_{\circ})$.

We extend our present analysis by including a discussion47 of two specific
parameters of the HF superconductors: (i) the Kadowaki-Woods ratio and (ii)
the gap-to-$T_{c}$ ratio. We show that the influence of the fluctuation
processes on both ratios can be accounted for by including an additional
factor which embodies the material properties of these HF superconductors.

(i) Jacko et al.61 accounted for the wide difference among the Kadowaki-Woods
ratio of a variety of strongly coupled systems by demonstrating that

$\frac{A}{\gamma^{2}}=\Big{(}\frac{81}{4\pi\hbar
k_{B}^{2}e^{2}}\Big{)}\Big{(}\frac{1}{d^{2}nN^{2}(\epsilon_{F})\langle
v_{0x}^{2}\rangle}\Big{)}=\Big{(}f_{con}\Big{)}\Big{(}\frac{1}{f_{mat}}\Big{)},$
(12)

wherein $\langle v_{0x}^{2}\rangle$ is an average of the carrier velocity
squared (a measure of the anisotropies), $n$ is the carrier density,
$N(\epsilon_{F})$ is the density of states at the Fermi level, and $d\sim 1$
is a dimensionless number. Apparently, due to the material-dependent second
factor, $\frac{A}{\gamma^{2}}$ of Eq. 12 is nonuniversal. This shortcoming
would be most evident for the ratio of the fluctuation-bearing HF
superconductors within their Fermi-liquid region (see Figs. 2-4). Our
evaluation of the Kadowaki-Woods ratio (Eq. 12) of these HF superconductors
gives

$\frac{A}{\gamma^{2}}=\Big{(}\frac{81}{4\pi\hbar
k_{B}^{2}e^{2}}\Big{)}\Big{(}\frac{F_{\ell}}{d^{2}nN^{2}(\epsilon_{F})\langle
v_{0x}^{2}\rangle}\Big{)}=\Big{(}f_{con}\Big{)}\Big{(}\frac{F_{\ell}}{f_{mat}}\Big{)},$
(13)

wherein the material-dependent $F_{\ell}$ factor is as described in Eq. (7).
It is not close to 1 as assumed during the derivation of Eq. (12).61 Rather
$F_{\ell}>1$: a larger apparent ratio because of the easing of the kinematic
constraints in these fluctuation-bearing systems. As in Ref. 61, the universal
character is restored only when expressed as
$\frac{A}{\gamma^{2}}\Big{(}\frac{f_{mat}}{F_{\ell}}\Big{)}=\frac{A.f_{mat}^{{}^{\prime}}}{\gamma^{2}}=f_{con}$.

(ii) Carbotte46 derived an approximate expression for the gap-to-$T_{c}$ ratio
of strong-coupled superconductor. Modified to the case of a fluctuation-
bearing superconductor within $\frac{T_{c}}{\theta}\ll 1$ region, this
expression reads as

$\displaystyle\frac{2\Delta(\ell)}{k_{B}T_{c}(\ell)}\approx
3.53\left\\{1+12.5\left[\frac{T_{c}(\ell)}{\theta}\right]^{2}\ln{\left[\frac{\theta}{2T_{c}(\ell)}\right]}\right\\},$
(14a) $\displaystyle\approx
3.53\left\\{1+12.5\left[e^{-\mathcal{F}/\sqrt{A(\ell)}}\right]^{2}\ln{\left[\frac{1}{2e^{-\mathcal{F}/\sqrt{A(\ell)}}}\right]}\right\\}$
(14b)

Thus, within $\frac{T_{c}}{\theta}\ll 1$ region, the gap-to-$T_{c}$ ratio can
be fine-tuned: From
$2\Delta(\ell\rightarrow\infty)/k_{B}T_{c}(\ell\rightarrow\infty)=3.53$, the
universal BCS value, up to a higher nonuniversal value by varying the control-
parameter that modifies $\ell$.

Finally, it is worth adding that our internally-consistent empirical and
theoretical analyses are based on a clear identification of the difference
between the fluctuation-related and normal (non-fluctuation-related)
contributions within the fluctuation-related FL region (which starts when the
resistivity manifests the $T^{2}$ contribution and ends when $T_{c}\rightarrow
0$).

In summary, we investigated the superconductivity, the FL transport, and their
correlations within the FL region of the T-p phase diagram of representative
quantum-critical HF superconductors. Empirically, on varying the control
parameters, (i) the normal-state resistivities manifest the characteristic FL,
$\rho_{o}+\textit{AT}^{2}$, character with $A\propto\rho_{\circ}^{2}$ and (ii)
the superconducting state manifests a correlation of $T_{c}$ with $A$
($ln\frac{T_{c}}{\theta}\propto A^{-\frac{1}{2}}$). We attribute the surge of
these superconducting and FL transport features and their correlations to a
fluctuation-mediated e-e scattering channel. Theoretically, on adopting many-
body techniques, we derive analytic expressions for $T_{c}(\rho_{\circ})$ and
$A(\rho_{\circ})$ and their correlations that reproduce satisfactorily the
aforementioned empirical correlations.

## Methods

We analyzed the extensively reported pressure-dependent resistivities of
various HF superconductors. Within the FL region, we looked for any
correlation among the superconductivity (as measured by $T_{c}$), the Fermi-
liquid character (as measured by $A$), and the excess in the residual
resistivity ($\rho_{\circ}$ as a measure of the strength of the scattering
channel) when the control parameter is varied. Using graphical and analytical
procedures, we managed to identify the empirical expression discussed in
Subsec. II.2. More importantly, we managed to identify the spin-
fluctuation/valence-fluctuation mediated e-e scattering channel as being the
driving mechanism behind these correlations. Guided by the inferences drawn
from the empirical expressions and the identified channel, we formulated the
theoretical framework outlined in Subsec. II.2 and Sec. II.3: Basically, we
started with the spin-fluctuation/valence-fluctuation exchange mechanism which
is different from the traditional mechanisms in that it takes into
consideration the modification in the kinematic constraints which, in turn,
lead to a significant enlargement in the phase space available for scattering.
Then, after applying the standard theories of Migdal-Eliashberg
(superconductivity) and Boltzmann (transport), we managed to derive the
analytic expressions of Subsec. II.2 which satisfactorily explain the
empirical observations.

## Acknowledgments

We acknowledge partial financial support from Brazilian agency CNPq.

## References

  * Gegenwart  _et al._ (2008) P. Gegenwart, Q. Si, and F. Steglich, Nat. Phys. 4, 186 (2008).
  * Coleman (2007) P. Coleman, Handbook of Magnetism and Advanced Magnetic Materials (2007).
  * Jaccard  _et al._ (1999) D. Jaccard, H. Wilhelm, K. Alami-Yadri, and E. Vargoz, Physica B 259-261, 1 (1999).
  * Yuan  _et al._ (2003) H. Q. Yuan, F. M. Grosche, M. Deppe, C. Geibel, G. Sparn, and F. Steglich, Science 302, 2104 (2003).
  * Löhneysen  _et al._ (2007) H. v. Löhneysen, A. Rosch, M. Vojta, and P. Wölfle, Rev. Mod. Phys. 79, 1015 (2007).
  * Mathur  _et al._ (1998) N. D. Mathur, F. M. Grosche, S. R. Julian, I. R. Walker, D. M. Freye, R. K. W. Haselwimmer, and G. G. Lonzarich, Nature 394, 39 (1998).
  * Miyake and Watanabe (2017) K. Miyake and S. Watanabe, Philos. Mag. 97, 3495 (2017).
  * Holmes  _et al._ (2004) A. T. Holmes, D. Jaccard, and K. Miyake, Phys. Rev. B 69, 024508 (2004).
  * Pines (2013) D. Pines, J. Phys. Chem. B 117, 13145 (2013).
  * Scalapino (2012) D. J. Scalapino, Reviews of Modern Physics 84, 1383 (2012).
  * Maier  _et al._ (2020) T. A. Maier, S. Karakuzu, and D. J. Scalapino, Phys. Rev. Research 2, 033132 (2020).
  * Li  _et al._ (2021) Z. Li, S. Kivelson, and D. Lee, npj Quantum Mater 6, 36 (2021).
  * Dean  _et al._ (2013) M. Dean, G. Dellea, R. S. Springell, F. Yakhou-Harris, K. Kummer, N. Brookes, X. Liu, Y. Sun, J. Strle, T. Schmitt, _et al._ , Nature materials 12, 1019 (2013).
  * Lee-Hone  _et al._ (2020) N. Lee-Hone, H. Özdemir, V. Mishra, D. Broun, and P. Hirschfeld, Phys. Rev. Res. 2, 013228 (2020).
  * Lee-Hone  _et al._ (2017) N. Lee-Hone, J. Dodge, and D. Broun, Phys. Rev, B 96, 024501 (2017).
  * Sidorov  _et al._ (2002) V. Sidorov, M. Nicklas, P. Pagliuso, J. Sarrao, Y. Bang, A. V. Balatsky, and J. D. Thompson, Phys. Rev. Lett. 89, 157004 (2002).
  * (17) The assumption of a formation of a patch within which quantum critical fluctuations emerge does not contradict the requirement that quantum critical fluctuations are scale invariant and extend over a large-sized region.This is because the length scale of each patch is taken to be longer than the mean free path and coherence length. This assumption is is invoked for taking care of the excess residual resistivity and its usefulness in reconciling the influence from each of disorder, pressure or magnetic field. Within each patch, $\bf{\delta g}\neq 0$.
  * Bergmann (1971) G. Bergmann, Phys. Rev. B 3, 3797 (1971).
  * Bergmann (1976) G. Bergmann, Phys. Rep. 27, 159 (1976).
  * Gurvitch (1986) M. Gurvitch, Phys. Rev. Lett. 56, 647 (1986).
  * (21) In cases where $X_{\circ}^{\circ}\approx 0$, one considers $X\approx X_{tot}$.
  * (22) An incorporation of nonmagnetic impurities 57; 58; 59; 60 leads to a reduction in $\chi(\omega)$ and $V^{sf}_{ee}$ and, as a consequence, to a drop in both A and Tc.
  * Rosch (1999) A. Rosch, Phys. Rev. Lett. 82, 4280 (1999).
  * Kambe and Flouquet (1997) S. Kambe and J. Flouquet, Solid State Commun. 103, 551 (1997).
  * Sheikin  _et al._ (2000) I. Sheikin, D. Braithwaite, J.-P. Brison, W. Assmus, and J. Flouquet, J. Low Temp. Phys. 118, 113 (2000).
  * Gegenwart  _et al._ (1998) P. Gegenwart, C. Langhammer, C. Geibel, R. Helfrich, M. Lang, G. Sparn, F. Steglich, R. Horn, L. Donnevert, A. Link, and W. Assmus, Phys. Rev. Lett. 81, 1501 (1998).
  * Honda  _et al._ (2013) F. Honda, T. Maeta, Y. Hirose, Y. Ōnuki, A. Miyake, and R. Settai, J. Korean Phys. Soc. 63, 345 (2013).
  * Bellarbi  _et al._ (1984) B. Bellarbi, A. Benoit, D. Jaccard, J. M. Mignot, and H. F. Braun, Phys. Rev. B 30, 1182 (1984).
  * Rueff  _et al._ (2011) J.-P. Rueff, S. Raymond, M. Taguchi, M. Sikora, J.-P. Itié, F. Baudelet, D. Braithwaite, G. Knebel, and D. Jaccard, Phys. Rev. Lett. 106, 186405 (2011).
  * Miranda and Dobrosavljević (2005) E. Miranda and V. Dobrosavljević, Rep. on Prog. in Phys. 68, 2337 (2005).
  * Paglione  _et al._ (2003) J. Paglione, M. A. Tanatar, D. G. Hawthorn, E. Boaknin, R. W. Hill, F. Ronning, M. Sutherland, L. Taillefer, C. Petrovic, and P. C. Canfield, Phys. Rev. Lett. 91, 246405 (2003).
  * Bianchi  _et al._ (2003) A. Bianchi, R. Movshovich, I. Vekhter, P. G. Pagliuso, and J. L. Sarrao, Phys. Rev. Lett. 91, 257001 (2003).
  * Bauer  _et al._ (2005) E. D. Bauer, C. Capan, F. Ronning, R. Movshovich, J. D. Thompson, and J. L. Sarrao, Phys. Rev. Lett. 94, 047001 (2005).
  * Ronning  _et al._ (2006) F. Ronning, C. Capan, E. D. Bauer, J. D. Thompson, J. L. Sarrao, and R. Movshovich, Phys. Rev. B 73, 064519 (2006).
  * Kotegawa  _et al._ (2006) H. Kotegawa, K. Takeda, T. Miyoshi, S. Fukushima, H. Hidaka, T. C. Kobayashi, T. Akazawa, Y. Ohishi, M. Nakashima, A. Thamizhavel, _et al._ , J. Phys. Soc. Jpn. 75, 044713 (2006).
  * Araki  _et al._ (2002) S. Araki, M. Nakashima, R. Settai, T. C. Kobayashi, and Y. Onuki, J. Phys.: Condens. Matter 14, L377 (2002).
  * Nakashima  _et al._ (2004) M. Nakashima, K. Tabata, A. Thamizhavel, T. C. Kobayashi, M. Hedo, Y. Uwatoko, K. Shimizu, R. Settai, and Y. Onuki, J. Phys.: Condens. Matter 16, L255 (2004).
  * Nakashima  _et al._ (2006) M. Nakashima, H. Kohara, A. Thamizhavel, T. D. Matsuda, Y. Haga, M. Hedo, Y. Uwatoko, R. Settai, and Y. Ōnuki, Physica B: Cond Matter 378, 402 (2006).
  * Onuki  _et al._ (2008) Y. Onuki, Y. Miyauchi, M. Tsujino, Y. Ida, R. Settai, T. Takeuchi, N. Tateiwa, T. D. Matsuda, Y. Haga, and H. Harima, J. Phys. Soc. Jpn. 77, 37 (2008).
  * Hassinger  _et al._ (2008) E. Hassinger, G. Knebel, K. Izawa, P. Lejay, B. Salce, and J. Flouquet, Phys. Rev. B 77, 115117 (2008).
  * de Visser  _et al._ (1984) A. de Visser, J. Franse, and A. Menovsky, J. Magn. Magn. Mater. 43, 43 (1984).
  * Ott  _et al._ (1983) H. R. Ott, H. Rudigier, Z. Fisk, and J. L. Smith, Phys. Rev. Lett. 50, 1595 (1983).
  * Nunez-Regueiro  _et al._ (2012) M. Nunez-Regueiro, G. Garbarino, and M. D. Nunez-Regueiro, J Phys: Conf. Series 400, 022085 (2012).
  * Castro  _et al._ (2018) P. B. Castro, J. L. Ferreira, M. B. S. Neto, and M. ElMassalami, J. Phys.: Conf. Ser. 969, 012050 (2018).
  * Soares  _et al._ (2018) C. Soares, M. ElMassalami, Y. Yanagisawa, M. Tanaka, H. Takeya, and Y. Takano, Sci. Rep. 8, 7041 (2018).
  * Carbotte (1990) J. P. Carbotte, Rev. Mod. Phys. 62, 1027 (1990).
  * ElMassalami and Neto (2021) M. ElMassalami and M. B. S. Neto, Phys. Rev. B 104, 014520 (2021).
  * Lawrence and Wilkins (1973) W. E. Lawrence and J. W. Wilkins, Phys. Rev. B 7, 2317 (1973).
  * Lawrence (1976) W. E. Lawrence, Phys. Rev. B 13, 5316 (1976).
  * MacDonald (1980) A. H. MacDonald, Phys. Rev. Lett. 44, 489 (1980).
  * MacDonald  _et al._ (1981) A. H. MacDonald, R. Taylor, and D. J. W. Geldart, Phys. Rev. B 23, 2718 (1981).
  * Patton and Zaringhalam (1975) B. Patton and A. Zaringhalam, Phys. Lett. A 55, 95 (1975).
  * Pethick  _et al._ (1986) C. J. Pethick, D. Pines, K. F. Quader, K. S. Bedell, and G. E. Brown, Phys. Rev. Lett. 57, 1955 (1986).
  * not (a) (a), see Ref. 47 for a discussion on the two reported exceptions.
  * not (b) (b), based on Eq. (5) of Ref. 62, for low defects, one obtains $\delta\lambda_{\ell}\propto\frac{1}{\ell}\propto\rho_{\circ}$.
  * not (c) (c), such an argument, supported by the aforementioned agreement between theory and experiments, is not in contradiction with the fact that there are difference in the details of the corresponding phase diagrams (e.g. the number and type of $T_{c}$-domes, the number and type of the critical/crossover points $p_{m}$, $p_{v}$, etc.): In fact, irrespective of these details, all studied phase diagrams manifest quantum critical instabilities which are accompanied by critical spin-fluctuations6 or valence-fluctuations8; 7.
  * Smith  _et al._ (1984) J. L. Smith, Z. Fisk, J. O. Willis, B. Batlogg, and H. R. Ott, J Appl Phys 55, 1996 (1984).
  * Adrian and Saemann-Ischenko (1988) G. Adrian and G. Saemann-Ischenko, Zeitschrift für Physik B Condensed Matter 72, 235 (1988).
  * Adrian and Adrian (1988) G. Adrian and H. Adrian, Physica C: Superconductivity 153-155, 435 (1988).
  * Adrian and Adrian (1989) G. Adrian and H. Adrian, J Less Common Met 149, 313 (1989).
  * Jacko  _et al._ (2009) A. C. Jacko, J. O. Fjærestad, and B. J. Powell, Nat Phys 5, 422 (2009).
  * Bergmann (1969) G. Bergmann, Z. Physik 228, 25 (1969).


# Flexible Coherent Optical Access: Architectures, Algorithms, and
Demonstrations

Ji Zhou, Zhenping Xing, Haide Wang, Kuo Zhang, Xi Chen, Qiguang Feng, Keshuang
Zheng, Yijia Zhao, Zhen Dong, Tao Gui, Zhicheng Ye, and Liangchuan Li
Manuscript received xxxx; revised xxxx. (Corresponding authors: Liangchuan
Li).J. Zhou and H. Wang are with the Department of Electronic Engineering,
College of Information Science and Technology, Jinan University, Guangzhou
510632, China.Z. Xing, K. Zhang, X. Chen, Q. Feng, K. Zheng, Y. Zhao, Z. Dong,
T. Gui, Z. Ye, and L. Li are with Optical Research Department, Huawei
Technologies Co Ltd, Dongguan, 523808, China.

###### Abstract

To cope with the explosive bandwidth demand, significant progress has been
made in the ITU-T standardization sector to define a higher-speed passive
optical network (PON) with a 50Gb/s line rate. Recently, 50G PON becomes
mature gradually, which means it is time to discuss beyond 50G PON. For
ensuring an acceptable optical power budget, beyond 50G PON will potentially
use coherent technologies, which can simultaneously promote the applications
of flexible multiple access such as time/frequency-domain multiple access
(TFDMA). In this paper, we will introduce the architectures, algorithms, and
demonstrations for TFDMA-based coherent PON. The system architectures based on
an ultra-simple coherent transceiver and specific signal spectra are designed
to greatly reduce the cost of ONUs. Meanwhile, fast and low-complexity digital
signal processing (DSP) algorithms are proposed for dealing with upstream and
downstream signals. Based on the architectures and algorithms, we
experimentally demonstrate the first real-time TFDMA-based coherent PON, which
can support at most 256 end users, and peak line rates of 100Gb/s and 200Gb/s
in the upstream and downstream scenarios, respectively. In conclusion, the
proposed technologies for the coherent PON make it more possible to be applied
in the future beyond 50G PON.

###### Index Terms:

Time/frequency-domain multiple access, ultra-simple coherent transceiver,
beyond 50G, passive optical network.

##  I Introduction

Driven by the sustainably growing traffic demand, the line rate of passive
optical network (PON) is rising steadily. Fig. 1 shows the roadmap for IEEE
and ITU-T standards of PON. Significant progress has been made in the ITU-T
standardization sector to define higher-speed (HS) PON with a line rate of
50Gb/s [1, 2]. Recently, 50G PON becomes mature gradually, which means it is
time to discuss beyond 50G PON [3, 4, 5]. For ensuring an acceptable optical
power budget, beyond 50G PON will potentially use coherent technologies [6, 7,
8]. Among all the potential solutions, a coherent PON based on flexible time-
domain/frequency-domain multiple access (TFDMA) is one of the most appealing
options, which can combine the statistical multiplexing capability of TDMA and
the dedicated frequency allocation capability of FDMA [9, 10, 11]. Based on
digital subcarrier multiplexing (DSCM), FDMA-based coherent PON allows each
optical network unit (ONU) to transmit and receive only a subset of
subcarriers, which significantly reduces the bandwidth of its transceiver[12,
13, 14]. Other benefits of coherent PON include: 1) it improves the receiver
sensitivity due to the use of a local oscillator (LO), and 2) it can use
C-band wavelength resources that have not been used in previous PONs since the
dispersion can be effectively compensated by digital signal processing (DSP)
[15, 16, 17].

Figure 1: The roadmap for IEEE and ITU-T standards of PONs.

Unfortunately, the use of a full coherent transceiver at each ONU is still
cost-prohibitive for the PON scenario. To further reduce the cost of an ONU,
it has been proposed to use a single-polarization heterodyne receiver based on
Alamouti coding rather than a fully coherent receiver in the downstream
scenario [18, 19, 20]. In the upstream scenario, The ONU can also use a single
Mach-Zehnder modulator (MZM) instead of a dual-polarization in-phase and
quadrature MZM. Such an ultra-simple coherent transceiver contains one
digital-to-analog converter (DAC), one single MZM, one optical coupler, one
balanced photo-detector (BPD), one analog-to-digital converter (ADC), and two
lasers. Two lasers become the major cost in the ultra-simple coherent
transceiver [21]. The high-cost external cavity laser used in traditional
coherent transceivers cannot meet the requirement of the ultra-simple coherent
transceiver.

In this paper, we experimentally demonstrate the first real-time TFDMA-based
coherent PON using an ultra-simple transceiver. The proposed PON can support a
splitting ratio up to 1:256, and peak line rates of 100Gb/s and 200Gb/s in the
upstream and downstream scenarios, respectively. In addition, we prove that
high-precision DSP-aided frequency locking makes the cost-effective
distributed feedback (DFB) laser feasible for the ultra-simple coherent
transceiver. This paper is an extended version of our published post-deadline
paper in OFC 2023 [22]. More detailed information about the DSP algorithms is
added in this extended version.

The main contributions of this paper are as follows:

  * •

The system architectures based on an ultra-simple coherent transceiver and
specific signal spectra are designed to greatly reduce the cost of ONUs.

  * •

We propose fast and low-complexity DSP algorithms for effectively processing
upstream and downstream signals in TFDMA-based coherent PON.

  * •

We demonstrate the first real-time TFDMA-based coherent PON, which can support
at most 256 end users, and peak line rates of 100Gb/s and 200Gb/s in the
upstream and downstream scenarios, respectively.

The remainder of this paper is organized as follows. The system architectures
and specific signal spectra are shown in Section II. In Section III, the DSP
algorithms for TFDMA-based coherent PON are introduced in detail. In Section
IV, the experimental setups and results are given. Finally, the paper is
concluded in Section V.

Figure 2: (a) The architecture diagram of the coherent PON with 256 splitter
ratio. (b) Transmitter (Tx) and receiver (Rx) devices of the ultra-simple
coherent transceiver at the ONU. (c) Tx and Rx devices of the full coherent
transceiver at the OLT.

##  II System Architecture and Spectra Design

In this section, we will introduce the system architectures with an ultra-
simple coherent transceiver at the ONU and a full coherent transceiver at the
OLT. Meanwhile, the specific signal spectra are designed for the upstream and
downstream scenarios in the TFDMA-based coherent PON.

The system architecture of the coherent PON is shown in Fig. 2(a), which is
designed to support at most 256 ONUs by using 1:4, 1:8, and 1:8 passive
optical splitters. Without loss of generality, we test 4 ONUs on the edge of
the network. The transmitter and receiver devices of the ultra-simple coherent
transceiver at the ONU are depicted in Fig. 2(b). The ultra-simple coherent
transceiver consists of one DAC, one single MZM, one optical coupler, one ADC,
one single BPD, and two DFB lasers. Two DFB lasers are used as the LO and
optical carrier for the downstream and upstream scenarios, respectively.
Therefore, Single-polarization heterodyne detection and unidimensional signal
generation are implemented by the ultra-simple coherent transceiver. For the
downstream scenario, an Alamouti-coding signal should be received at the ONU
to avoid the state-of-polarization (SOP)-caused signal disappearance. For the
upstream scenario, the transmitted signal at the ONU should be a real-valued
carrier-less amplitude phase (CAP) signal to tolerate direct-current (DC)
leakage. For generating the Alamouti-coding signal and detecting the CAP
signal at the OLT, a full coherent transceiver can be deployed, as shown in
Fig. 2 (c). The full coherent transceiver at the OLT makes it possible to
gradually evolve the line rate by updating the transceiver at the ONU.

Figure 3: (a) Designed spectra with a bandwidth granularity of 12.5GHz for the
downstream scenario. (b) Designed spectra with a bandwidth granularity of
6.25GHz for the upstream scenario. SC: Subcarrier. BW: Bandwidth. DS:
Downstream. US: Upstream.

Figure 3 (a) shows the designed signal spectra with a bandwidth granularity of
12.5GHz for the downstream scenario. In the downstream scenario,
5$\times$12.5Gbaud digital subcarriers with a granularity of 12.5GHz are
filled in the whole bandwidth. Only 4 digital subcarriers carry data, and the
central subcarrier is blank for implementing the heterodyne detection. Then,
Alamouti coding with a rate of 1/2 is used to implement the single-
polarization detection. However, the capacity of the Alamouti-based dual-
polarization optical system is equivalent to that of a single-polarization
optical system with the same bandwidth. Therefore, the peak line rate of the
downstream scenario is only 200Gb/s when 16QAM is modulated on 50GHz
bandwidth. We use probabilistic constellation shaping 16QAM (PCS-16QAM) to
achieve flexible-rate adaption from 100Gb/s to 200Gb/s for making full use of
optical power budget [23, 24, 25]

In the upstream scenario, each ONU transmits a real-valued CAP signal, as
shown in Fig. 3 (b). The reasons are that 1) a single MZM can only modulate a
real-valued signal, and 2) a guard band around DC frequency is required. The
MZM at the ONU is biased at the null point. To achieve steep rising and
falling edges, the burst-mode signals are generated by switching on/off the
DAC rather than the laser. However, due to the limited extinction ratio of the
MZM, the DC leakage of laser power is not negligible when the DAC is off,
hence the guard band is necessary. We further break each 12.5Gbaud subcarrier
into 2$\times$6.25Gbaud subcarriers for providing fine-granularity
transmission in the upstream scenario. More frequency resources can be
provided for dedicated usage in the upstream scenario. Dedicated subcarriers
allow high-end users to get free from rogue ONUs, which only exist in the
upstream scenario. Each ONU can transmit either the inner (Type-1) or the
outer (Type-2) subcarrier in CAP with two subcarriers. In our work, only
quadrature phase shift keying (QPSK) was modulated for the upstream scenario
to obtain a peak line rate of 25Gb/s per ONU and a total peak line rate of 100
Gb/s from the OLT’s perspective.

In conventional FDMA-based PON, four subcarriers for the downstream scenario
and eight subcarriers for the upstream scenario cannot support the bandwidth
allocation for 256 ONUs. However, if the subcarrier number is increased, it is
hard to compensate for the phase noise by the DSP algorithm, and the guard
band increases to decrease the spectral efficiency. The TDMA is added to the
subcarriers to implement TFDMA, which is a feasible method for increasing the
number of ONUs. In the TFDMA, the subcarriers can be individually allocated to
low-latency and bandwidth-hungry ONUs. Meanwhile, for the common ONUs, the
subcarrier can be divided into some time slots to provide flexible bandwidth
allocation. In conclusion, the system architectures, specific signal spectra,
and TFDMA can be used to support low-latency or low-cost ONUs.

##  III DSP Algorithms for Coherent PON

Traditional coherent DSP algorithms are not suitable for TFDMA-based coherent
PON. In this section, we will introduce the specially designed DSP algorithms
for the TFDMA-based coherent PON. The DSP algorithms for the upstream and
downstream scenarios work in the burst mode and continuous mode [26, 27, 28],
respectively. For the upstream scenario, burst-mode DSP algorithms based on
training sequences are used to achieve fast convergence for reducing the
overhead and improving spectral efficiency. For the downstream scenario, the
continuous-mode DSP algorithms at the ONU are sensitive to computational
complexity and power consumption. Only a part of the DSP algorithms should be
always turned on to track the dynamical distortions, such as frequency-offset
estimation (FOE), timing error detection, coefficient estimation of the
equalizer, and carrier-phase estimation (CPE). At the time slots of other
ONUs, the DSP algorithms processing the payloads and forward error correction
(FEC) can be turned off to reduce power consumption. In addition, frame
synchronization is required only once when the ONU is initially registered.

###  III-A Frame Detection and Coarse FOE

In this subsection, periodic sequences are designed to simultaneously
implement frame detection and coarse FOE.

####  III-A1 Frame Detection

For the upstream scenario, frame detection is required to recognize two
symmetric frequency tones for confirming the arrival of burst fame. The
symmetric frequency tones are generated by a periodic sequence. Frame
detection is not necessary for the continuous-mode downstream scenario. The
frame detection can be implemented by the following steps. Firstly, a sliding
window operation is applied to the received signal. Then, the extracted signal
is down-sampled and transferred to the frequency domain by a fast Fourier
transform (FFT). Finally, the average power of the non-zero points is
calculated. The frequency points with a power less than average power are
filtered out. These mentioned operations are repeated three times to find the
accurate frequency tones.

####  III-A2 Coarse FOE

After the frame detection, the frequency offset should be compensated to
ensure the signal spectrum within the frequency-domain range of the matched
filter. Fig. 4 shows the frequency tones with and without frequency offset.
The detected frequency tones after the frame detection can be used to estimate
the frequency offset by

$\Delta f_{\text{Coarse}}=\frac{1}{2}\times\left(f_{1}+f_{2}\right)$ (1)

where $f_{1}$ and $f_{2}$ are the frequencies of the $-f_{0}+\Delta f$ and
$f_{0}+\Delta f$, respectively. $\Delta f$ is the actual frequency offset. The
FFT size confines the accuracy of the FOE, which is limited by the parallelism
of real-time implementation. For example, when the FFT size is 32 and the
signal bandwidth is 8GHz, the resolution of one frequency point is only
250MHz, which may lead to $\pm 125$MHz deviation for the FOE. Therefore, the
FOE based on the detected frequency tones is coarse, and more fine FOE is
required.

Figure 4: The frequency tones with and without frequency offset (FO).

###  III-B Burst- and Continuous-Mode Timing Recovery

For the upstream scenario of the TFDMA-based coherent PON, burst-mode timing
recovery should be used to accelerate convergence to reduce the overhead. For
the downstream scenario, the timing recovery works in continuous mode.

Fig. 5 shows the burst-mode timing recovery with sampling phase initialization
for the upstream scenario, and continuous-mode timing recovery for the
downstream scenario. The common structure of burst-mode and continuous-mode
timing recovery is the frequency-domain Godard algorithm. For the upstream
scenario, an appropriate sampling phase initialization plays a significant
role in reducing the convergence time of timing recovery. Fortunately, the
initialized sampling phase offset $\tau_{0}$ can be estimated by using two
frequency tones in Subsection III-A, which can be calculated by

$\tau_{0}=\frac{1}{4\pi f_{0}}\text{arg}[R(f_{0})\times R^{*}(-f_{0})]$ (2)

where $(.)^{*}$ denotes the conjugate operation. $R(f)$ is the received
frequency tones with the initialized sampling phase offset such as
$\delta(f-f_{0})e^{j2\pi f_{0}\tau+\phi_{0}}$ and $\delta(f+f_{0})e^{-j2\pi
f_{0}\tau+\phi_{0}}$. $\phi_{0}$ is the phase noise, which does not influence
the estimation of sampling phase offset. The initialized sampling phase offset
is injected into the frequency-domain Godard algorithm to reduce the
convergence time.

The frequency-domain Godard algorithm is implemented using the signal spectrum
$\mathbf{S}$ after match filtering with a roll-off factor $\beta$. The timing
error is estimated as

$e=\sum_{k=\frac{(1-\beta)}{2sps}N}^{\frac{(1+\beta)}{2sps}N-1}\operatorname{Im}\left[S_{k}\cdot
S_{k+(1-1/sps)N}^{*}\right]$ (3)

where $\operatorname{Im}(\cdot)$ denotes the imaginary part of a complex
value. $sps$ is the samples per symbol. $N$ is the number of the frequency
points after the $N$-FFT, which should choose the value that allows the upper
and lower bounds to be integers. Based on Eq. (3), only $K$ frequency points
are used to estimate timing error with relatively low complexity where $K$ is
equal to $\beta N/sps-1$. Finally, the estimated phase is updated after every
iteration.

Figure 5: The burst-mode timing recovery with sampling phase initialization
for the upstream scenario, and continuous-mode timing recovery for the
downstream scenario. Interp.: Interpolation. NCO: Numerically controlled
oscillator. Est.: estimation.

###  III-C Frame Synchronization and Fine FOE

In this subsection, training sequences are designed to simultaneously
implement frame synchronization and fine FOE.

####  III-C1 Frame Synchronization

Frame synchronization is implemented based on the specially designed sequences
$[\mathbf{s_{1}}~{}\mathbf{s_{2}}]$ and $[\mathbf{s_{3}}~{}\mathbf{s_{4}}]$
for X polarization and Y polarization, respectively. The $\mathbf{s_{1}}$ and
$\mathbf{s_{3}}$ consist of QPSK symbols with different pseudorandom binary
sequences. The $\mathbf{s_{2}}$ and $\mathbf{s_{4}}$ are separately generated
by

$s_{2/4}(i)=s_{1/3}(i)\times pn(i)$ (4)

where $i$ is from 1 to $L$. $\mathbf{pn}$ is a pseudo-random noise (PN)
sequence with the length of $L$. Finally, the specially designed sequences
repeat three times to generate the successive sequences
$\left[\mathbf{s_{1/3}}~{}\mathbf{s_{2/4}},~{}\mathbf{s_{1/3}}~{}\mathbf{s_{2/4}},~{}\mathbf{s_{1/3}}~{}\mathbf{s_{2/4}}\right]$.

Frame synchronization is realized by using a sliding window and the cross-
correlation on each polarization, which is expressed as

$P(d)=\sum_{i=0}^{L-1}r(d+i)\times\left[pn(i)\times r^{*}(d+i+L)\right]$ (5)

where $\mathbf{r}$ is the received signal of $\mathbf{s}$. The frame
synchronization is based on the timing metric, which is calculated by

$M(d)=\frac{|P(d)|^{2}}{P_{r}^{2}(d)}.$ (6)

where the half-symbol energy $P_{r}(d)$ is defined as

$P_{r}(d)=\frac{1}{2}\sum_{i=0}^{2L-1}|r(d+i)|^{2}.$ (7)

There are five sharp peaks in the $M(d)$. To enhance the tolerance for noise,
$M(d)$ with $0$, $L$, $2L$, $3L$ and $4L$ delay are stacked over, which can be
expressed as

$M^{\prime}(d)=\sum_{i=0}^{4}M(d+i\times L),$ (8)

The highest-peak position of $M^{\prime}(d)$ is the accurate frame
synchronization position.

####  III-C2 Fine FOE

The successive sequences for frame synchronization can also be used to
implement fine FOE. For convenient analysis, we define
$\left[\mathbf{s_{1/3}}~{}\mathbf{s_{2/4}},~{}\mathbf{s_{1/3}}~{}\mathbf{s_{2/4}},~{}\mathbf{s_{1/3}}~{}\mathbf{s_{2/4}}\right]$
as [$\mathbf{s_{B1}}$, $\mathbf{s_{B2}}$, $\mathbf{s_{B3}}$]. The received
signal of $\mathbf{s_{B1}}$, $\mathbf{s_{B2}}$ and $\mathbf{s_{B3}}$ can be
defined as $\mathbf{r_{B1}}$, $\mathbf{r_{B2}}$ and $\mathbf{r_{B3}}$,
respectively. When the fine frequency offset is considered, the received
signal can be defined as

$r_{\text{B}i}(n)=s_{\text{B}i}(n)\cdot\exp({j\frac{2\pi n}{R_{s}}\cdot\Delta
f_{\text{Fine}}})$ (9)

where $i$ is from 1 to 3. $\Delta f_{\text{Fine}}$ is the fine frequency
offset. $R_{s}$ is the baud rate. The fine FOE estimates the frequency offset
by [29]

$\displaystyle\Delta f_{\text{Fine}}$ $\displaystyle=\frac{R_{s}}{4\pi
L}\cdot\arg{\left(\mathbf{r_{B2}}\times\mathbf{r_{B1}^{H}}+\mathbf{r_{B3}}\times\mathbf{r_{B2}^{H}}\right)}$
(10) $\displaystyle=\frac{R_{s}}{4\pi
L}\cdot\arg{\left[2|\mathbf{s_{B1}}|^{2}\cdot\exp\left({j\frac{4\pi
L}{R_{s}}\cdot\Delta f_{\text{Fine}}}\right)\right]}$

where $\arg(\cdot)$ represents the operation of taking the angle of a complex
value and $(\cdot)^{\mathbf{H}}$ denotes the conjugate transpose operation.
Due to the average effect, the fine FOE is more accurate with the increase of
$L$. However, the range of fine FOE is from $-R_{s}/4L$ to $R_{s}/4L$, which
decreases with the increase of $L$. Therefore, the $L$ should be set to a
suitable value for balancing the accuracy and range. When the $L$ is set to
$10$ and $R_{s}$ is set to 8Gbaud, the range of fine FOE is from $-200$MHz to
$200$MHz. If the frequency offset of the laser is locked within $\pm 100$MHz,
the coarse FOE is not required.

Figure 6: The block diagram for the burst- and continuous-mode multiple-input-
multiple-output (MIMO) equalizer.

###  III-D Burst- and Continuous-Mode MIMO Equalizer

For the upstream scenario, the burst-mode multiple-input-multiple-output
(MIMO) equalizer is used to accelerate convergence for reducing the overhead.
For the downstream scenario, the MIMO equalizer works in continuous mode. Fig.
6 shows the block diagram for the burst- and continuous-mode MIMO equalizer.
Different from the continuous-mode MIMO equalizer, the burst-mode MIMO
equalizer has a coefficient initialization. For fast converging the
coefficients, LS-based coefficient estimation is implemented using a designed
const amplitude zero auto-correlation training sequence [30]. After the
coefficient initialization, it switches to decision-directed-LMS (DD-LMS)
algorithm to track the coefficients. For the continuous-mode MIMO equalizer,
the DD-LMS algorithm tracks the coefficients all the time. Before the DD-LMS
algorithm, the carrier-phase noise should be estimated and compensated based
on a pilot-based CPE. In our work, the MIMO equalizer at the ONU should be
specially designed for dealing with Alamouti-coding received signals [31, 32].
Meanwhile, the MIMO equalizer at the OLT can be simplified to a multiple-
input-single-output equalizer due to the transmitted single-polarization real-
valued signal.

Figure 7: The structure of a pilot-based CPE integrated with the maximum
likelihood (ML) algorithm.

###  III-E Pilot-Based CPE

As Fig. 7 shows, a pilot-based CPE integrated with the maximum likelihood
algorithm is used to estimate the carrier phase noise with low computational
complexity. One pilot symbol is periodically inserted into every $M$ payload
symbol for the pilot-based CPE, which estimates the phase noise of the pilot
symbol as

$\phi_{\text{Pilot}}(n)=\arg\left[r_{p}^{*}(n)\times p(n)\right]$ (11)

where $r_{p}(j)$ is the $n$-th received pilot symbol after the MIMO equalizer
and $p(n)$ denotes the $n$-th pilot symbol. The phase noise of the $M-1$
symbols between the $n$-th and the $(n+1)$-th pilot symbols is initialized as
$\phi_{\text{Pilot}}(n)$. Then, the following maximum likelihood algorithm is
employed to estimate the residual phase noise as

$\phi_{\text{ML}}(i)=\frac{1}{2Q+1}\sum_{l=i-Q}^{i+Q}\arg\left[q(l)\times\hat{q}^{*}(l)\right]$
(12)

where $q(l)$ denotes the received signal after the carrier phase compensation.
$\hat{q}(l)$ is its decision. $Q$ is the half-length of the average filter.
After the pilot-based CPE and compensation, the QAM can be decided to
regenerate the bit sequence, which can finally be sent into FEC to correct the
error bits.

Figure 8: (a) The linecards including field-programmable gate array (FPGA)
integrated with 31GSa/s DACs and ADCs for upstream and downstream scenarios.
(b) Schematic diagrams of the full coherent transceiver and the simplified
coherent transceiver. AWG: arbitrary waveform generator. CDM: coherent driver
modulator. ICR: integrated coherent receiver. SP: single polarization. Het.:
heterodyne.

##  IV Experimental Setups and Results

As Fig. 8 (a) and Fig. 8 (b) show, we developed the linecards including field-
programmable gate array (FPGA) integrated with 31GSa/s DACs and ADCs for
upstream and downstream scenarios. At the OLT, a commercial coherent driver
modulator (CDM) and integrated coherent receiver (ICR) were used to implement
the full coherent transceiver. One FPGA integrated with four DACs and One FPGA
integrated with four ADCs were used to generate the transmitted signal and
process the received signal, respectively. At the ONU, single-polarization MZM
and single-polarization heterodyne ICR were used to implement the ultra-simple
coherent transceiver. Only one FPGA integrated with one DAC and one ADC was
employed to generate the transmitted signal and process the received signal.
The proposed ultra-simple coherent transceiver for ONU has low cost and low
power consumption.

Figure 9: (a) Measured spectra of downstream and upstream scenarios. (b)
Measured waveforms of TDM downstream and TDMA upstream signals. (c) Rising and
falling edges of the burst-mode signals. Figure 10: Time and frequency source
allocations for (a) downstream test with AWG at the transmitter and FPGA at
the receiver, (b) downstream test with FPGA, and (c) upstream test with FPGA.
Figure 11: The BER versus ROP for (a) downstream test with AWG at the
transmitter and FPGA at the receiver, (b) downstream test with FPGA at both
transmitter and receiver, and (c) upstream test with FPGA at both transmitter
and receiver.

The measured downstream and upstream spectra are shown in Fig. 9 (a). Due to
the limited sampling rate of the DACs, two experimental cases were done for
the downstream scenario: 1) using a linecard to generate one subcarrier with
data, and 2) using an AWG to generate four subcarriers with data and one blank
subcarrier. In both cases, we used a linecard for implementing the real-time
DSP to process the downstream signals at the ONU, and the same card was also
used to generate an upstream CAP signal with two subcarriers. Specifically, we
generated a CAP signal with either the inner subcarrier or the outer
subcarrier at each ONU, and then we used another linecard for the burst-mode
DSP to process the upstream signal at the OLT. Fig. 9 (b) depicts the
waveforms of the downstream TDM signals and the upstream TDMA signals. It is
worth noting that the downstream TDM signal is a continuous signal and the
upstream TDMA signal is a burst signal. Fig. 9 (c) zooms in on the rising
edges and the falling edges of the burst signals. The double-sided arrows
denote the dimension of 10ns. Obviously, the rising and falling edges are much
less than 1ns. Simultaneously, the receiver DSPs can provide real-time
estimations of laser frequency offsets. The estimated laser frequency offsets
were sent to the microcontroller units (MCUs), which then control the
temperature and the driving injection currents of the DFB lasers at the ONUs
to achieve real-time frequency locking.

Figure 10 shows the time and frequency source allocations for the downstream
and upstream tests. As Fig. 10 (a) shows, when the AWG was used at the
transmitter for the downstream scenario, ONU1 and ONU2 were assigned with the
dedicated subcarriers on the frequency of f4 and f3, respectively. ONU3 and
ONU4 shared the subcarrier on the frequency of f2, which occupied half of the
time slots. Note that although an ONU only detects one subcarrier, it can
receive the power of all four data subcarriers. As Fig. 10 (b) depicts, when
the FPGA linecard was used at the transmitter of OLT, we only tested the
dedicated subcarrier cases with different spectral efficiency. Fig. 10 (c)
depicts that ONU1 and ONU2 were assigned to different time slots of the
subcarrier on the frequency of f5 for the upstream scenario. ONU3 and ONU4
shared the subcarrier on the frequency of f6.

As Fig. 11 (a) depicts, to achieve total line rates of 100Gb/s, 150Gb/s, and
200Gb/s in the downstream scenario (assuming sending the same modulation
formats to all ONUs), the required ROPs are $-36.8$dBm, $-32$dBm, and $-25$dBm
at the BER threshold of $2\times 10^{-2}$, respectively. $15\%$ overhead soft-
decision FEC with a BER threshold of $2\times 10^{-2}$ is considered. An EDFA
was employed after the CDM at the OLT to amplify optical output power to 7dBm.
Therefore, the optical power budgets are 43.8dB, 39dB, and 32dB for line rates
of 100Gb/s, 150Gb/s, and 200Gb/s, respectively. It is worth noting that the
EDFA can be replaced by a cost-effective booster SOA for commercial
deployment. As Fig. 11 (b) shows, the required ROPs for 25Gb/s QPSK, 37.5Gb/s
PCS-16QAM, and 50Gb/s 16QAM flexible-rate transmission are $-41.8$dBm,
$-37$dBm, and $-31$dBm, respectively. The power of one subcarrier was measured
as the ROP in the test cases.

Switching from O-band wavelength to C-band wavelength can reduce fiber loss.
The measured total loss of 1:256 splitter and 20 km fiber is 31 dB in the lab.
The situation in a real-world network may be worse, and it might be
challenging to achieve 1:256 splitting ratio when 16QAM is modulated.
Fortunately, the optical power budgets of PCS-16QAM and QPSK are less
stringent. With the development of high-bandwidth transceiver devices, it is
possible to further increase the symbol rate and reduce the entropy of
PCS-16QAM. Thus, it is promising to simultaneously achieve 1:256 splitting
ratio and 200 Gb/s downstream line rate in the real-world network.

Figure 11 (c) shows that the required ROP is approximately $-45$dBm for the
upstream scenario with 12.5Gb/s per ONU. The ONUs have the ability to transmit
data on both the inner and the outer CAP with two subcarriers for achieving
the maximum rate of 25Gb/s per ONU. The required ROP should be approximately
$-42$dBm for the upstream scenario with 25Gb/s per ONU. We used one single MZM
at the transmitter of each ONU, thus the insertion loss of the ONU transmitter
is much less than that of the OLT transmitter. The launch optical power was
approximately 1dBm without any optical amplifier at the ONUs. Therefore, the
optical power budget of the upstream scenario can achieve 43dB for a peak line
rate of 100Gb/s.

Fig. 12 shows the real-time frequency-locking results for the DFB laser when
the environment temperature was varied between 10${}^{\text{o}}$C and
60${}^{\text{o}}$C. The topic subfigure shows the estimated frequency offset
of the DFB laser using the DSP. The middle and bottom subfigures show the
injection current and target temperature of TEC (Thermo-Electric Cooler) set
by the MCU, respectively. When the environment temperature was changed, the
injection current and target temperature of TEC transitorily fluctuated and
immediately settled down. By controlling both the current and the temperature,
the frequency offset is strictly confined within ($-100$, $100$) MHz, which
can be well compensated by the receiver DSP. The experimental results verify
the cost-effective DFB laser can meet the requirement for the coherent PON.

Figure 12: The real-time frequency-locking results for the DFB laser when the
environment temperature was varied between 10 ${}^{\text{o}}$C and
60${}^{\text{o}}$C.

##  V Conclusions

In this paper, we introduce the architectures, algorithms, and demonstrations
for the TFDMA-based coherent PONs. The system architectures consist of a fully
coherent transceiver at the OLT and an ultra-simple coherent transceiver at
the ONU, which can greatly reduce the cost and power consumption of the ONU.
Meanwhile, the Almouti coding and specific spectra are designed to implement
the proposed system architectures. Fast and low-complexity DSP algorithms are
used for processing upstream and downstream signals. Based on the system
architectures and the DSP algorithms, the first real-time TFDMA-based coherent
PON was experimentally demonstrated to support at most 256 end users, and the
peak line rates of 100Gb/s and 200Gb/s in the upstream and downstream
scenarios, respectively. Meanwhile, the real-time frequency locking within a
frequency offset of ($-100$, $100$) MHz verifies the feasibility of a cost-
effective DFB laser. In conclusion, the system architectures, DSP algorithms,
and real-time demonstrations enable TFDMA-based coherent PON in the future
beyond 50G PON.

## References

  * [1] “50-Gigabit-capable passive optical networks (50G-PON): Physical media dependent (PMD) layer specification,” ITU-T G.9804.3, 2021.
  * [2] D. Zhang, D. Liu, X. Wu, and D. Nesset, “Progress of ITU-T higher speed passive optical network (50G-PON) standardization,” _Journal of Optical Communications and Networking_ , vol. 12, no. 10, pp. D99–D108, 2020.
  * [3] “PON transmission technologies above 50 Gb/s per wavelength,” ITU-T work programme G. suppl. VHSP, 2023.
  * [4] J. Zhang and Z. Jia, “Coherent Passive Optical Networks for 100G/$\lambda$-and-Beyond Fiber Access: Recent Progress and Outlook,” _IEEE Network_ , vol. 36, no. 2, pp. 116–123, 2022.
  * [5] N. Suzuki, H. Miura, K. Mochizuki, and K. Matsuda, “Digital Coherent based PON Technologies and Beyond-100G Optical Access Systems,” in _2022 27th OptoElectronics and Communications Conference (OECC) and 2022 International Conference on Photonics in Switching and Computing (PSC)_. IEEE, 2022, pp. 1–3.
  * [6] A. Teixeira, A. Shahpari, R. Ferreira, F. P. Guiomar, and J. D. Reis, “Coherent access,” in _Optical Fiber Communication Conference_. Optica Publishing Group, 2016, pp. M3C–5.
  * [7] D. Lavery, S. Erkılınç, P. Bayvel, and R. I. Killey, “Recent Progress and Outlook for Coherent PON,” in _Optical Fiber Communication Conference_. Optica Publishing Group, 2018, pp. M3B–1.
  * [8] M. S. Faruk and S. J. Savory, “Coherent access: Status and opportunities,” in _2020 IEEE Photonics Society Summer Topicals Meeting Series (SUM)_. IEEE, 2020, pp. 1–2.
  * [9] J. Zhang, Z. Jia, M. Xu, H. Zhang, and L. A. Campos, “Efficient preamble design and digital signal processing in upstream burst-mode detection of 100G TDM coherent-PON,” _Journal of Optical Communications and Networking_ , vol. 13, no. 2, pp. A135–A143, 2021.
  * [10] M. Xu, Z. Jia, H. Zhang, L. A. Campos, and C. Knittle, “Intelligent Burst Receiving Control in 100G Coherent PON with 4$\times$ 25G TFDM Upstream Transmission,” in _Optical Fiber Communication Conference_. Optica Publishing Group, 2022, pp. Th3E–2.
  * [11] H. Zhang, Z. Jia, L. A. Campos, and C. Knittle, “Low-Cost 100G Coherent PON Enabled by TFDM Digital Subchannels and Optical Injection Locking,” in _Optical Fiber Communication Conference_. Optica Publishing Group, 2023, pp. W1I–4.
  * [12] D. Welch, A. Napoli, J. Bäck, N. Swenson, W. Sande, J. Pedro, F. Masoud, A. Chase, C. Fludger, H. Sun _et al._ , “Digital Subcarriers: A Universal Technology for Next Generation Optical Networks,” in _Optical Fiber Communication Conference_. Optica Publishing Group, 2022, pp. Tu3H–1.
  * [13] D. Welch, A. Napoli, J. Bäck, S. Buggaveeti, C. Castro, A. Chase, X. Chen, V. Dominic, T. Duthel, T. A. Eriksson _et al._ , “Digital subcarrier multiplexing: enabling software-configurable optical networks,” _Journal of Lightwave Technology_ , vol. 41, no. 4, pp. 1175–1191, 2022.
  * [14] M. M. Hosseini, J. Pedro, N. Costa, A. Napoli, J. E. Prilepsky, and S. K. Turitsyn, “Multi-period Planning in Metro-Aggregation Networks using Point-to-Multipoint Transceivers,” in _IEEE Global Communications Conference_. IEEE, 2022, pp. 2921–2926.
  * [15] S. Xing, G. Li, J. Chen, J. Zhang, N. Chi, Z. He, and S. Yu, “First Demonstration of PS-QAM based Flexible Coherent PON in Burst-Mode with 300G Peak-Rate and Record Dynamic-Range and Net-Rate Product up to 7,104 dB· Gbps,” in _Optical Fiber Communications Conference_. IEEE, 2022, p. Th4A.4.
  * [16] Z. Wei, J. Zhang, W. Li, and D. V. Plant, “Time-variant entropy regulated multiple access for flexible coherent PON,” _Optics Letters_ , vol. 47, no. 19, pp. 5148–5151, 2022.
  * [17] J. Zhang, G. Li, S. Xing, and N. Chi, “Flexible and adaptive coherent PON for next-generation optical access network,” _Optical Fiber Technology_ , vol. 75, p. 103190, 2023.
  * [18] M. S. Faruk, X. Li, and S. J. Savory, “Experimental demonstration of 100/200-Gb/s/$\lambda$ PON downstream transmission using simplified coherent receivers,” in _Optical Fiber Communication Conference_. Optica Publishing Group, 2022, pp. Th3E–5.
  * [19] A. Hraghi, G. Rizzelli, A. Pagano, V. Ferrero, and R. Gaudino, “Analysis and experiments on C band 200G coherent PON based on Alamouti polarization-insensitive receivers,” _Optics Express_ , vol. 30, no. 26, pp. 46 782–46 797, 2022.
  * [20] X. Li, M. S. Faruk, and S. J. Savory, “Bidirectional symmetrical 100 Gb/s/$\lambda$ coherent PON using a simplified ONU transceiver,” _IEEE Photonics Technology Letters_ , vol. 34, no. 16, pp. 838–841, 2022.
  * [21] R. Gaudino, V. Curri, G. Bosco, G. Rizzelli, A. Nespola, D. Zeolla, S. Straullu, S. Capriata, and P. Solina, “On the use of DFB lasers for coherent PON,” in _Optical Fiber Communication Conference_. Optica Publishing Group, 2012, pp. OTh4G–1.
  * [22] Z. Xing, K. Zhang, X. Chen, Q. Feng, K. Zheng, Y. Zhao, Z. Dong, J. Zhou, T. Gui, Z. Ye, and L. Li, “First Real-time Demonstration of 200G TFDMA Coherent PON using Ultra-simple ONUs,” in _Optical Fiber Communication Conference_. Optica Publishing Group, 2023, p. Th4C.4.
  * [23] G. Liu, J. Zhou, Y. Huang, G. Wang, Y. Bo, Y. Wu, Y. Lu, J. He, M. Li, Z. Ye _et al._ , “Flexible transceiver for an access network: a multicarrier entropy loading approach,” _Journal of Optical Communications and Networking_ , vol. 15, no. 7, pp. 442–448, 2023.
  * [24] J. Zhou, J. He, X. Lu, G. Wang, Y. Bo, G. Liu, Y. Huang, L. Li, C. Yang, H. Wang _et al._ , “100G fine-granularity flexible-rate passive optical networks based on discrete multi-tone with PAPR optimization,” _Journal of Optical Communications and Networking_ , vol. 14, no. 11, pp. 944–950, 2022.
  * [25] W. Shen, S. Xing, G. Li, Z. Li, A. Yan, J. Wang, J. Zhang, and N. Chi, “Demonstration of Beyond 100G Three-Dimensional Flexible Coherent PON in Downstream with Time, Frequency and Power Resource Allocation Capability,” in _Optical Fiber Communication Conference_. Optica Publishing Group, 2023, pp. W1I–5.
  * [26] H. Wang, J. Zhou, Z. Xing, Q. Feng, K. Zhang, K. Zheng, X. Chen, T. Gui, L. Li, J. Zeng, J. Yang, W. Liu, C. Yu, and Z. Li, “Fast-Convergence Digital Signal Processing for Coherent PON Using Digital SCM,” _Journal of Lightwave Technology_ , vol. 41, no. 14, pp. 4635–4643, 2023.
  * [27] S. J. Savory, “Digital coherent optical receivers: Algorithms and subsystems,” _IEEE Journal of selected topics in quantum electronics_ , vol. 16, no. 5, pp. 1164–1179, 2010.
  * [28] J. Zhang, Z. Jia, M. Xu, H. Zhang, and L. A. Campos, “Efficient preamble design and digital signal processing in upstream burst-mode detection of 100G TDM coherent-PON,” _Journal of Optical Communications and Networking_ , vol. 13, no. 2, pp. A135–A143, 2020.
  * [29] M. Morelli and U. Mengali, “An improved frequency offset estimator for OFDM applications,” in _1999 IEEE Communications Theory Mini-Conference (Cat. No. 99EX352)_. IEEE, 1999, pp. 106–109.
  * [30] F. Pittala, I. Slim, A. Mezghani, and J. A. Nossek, “Training-aided frequency-domain channel estimation and equalization for single-carrier coherent optical transmission systems,” _Journal of Lightwave Technology_ , vol. 32, no. 24, pp. 4849–4863, 2014.
  * [31] D. Lavery and S. J. Savory, “Digital Coherent Technology for Long-Reach Optical Access,” in _Optical Fiber Communication Conference_. Optica Publishing Group, 2014, pp. Tu2F–1.
  * [32] M. Erkılınç, R. Emmerich, K. Habel, V. Jungnickel, C. Schmidt-Langhorst, C. Schubert, and R. Freund, “PON transceiver technologies for $\geq$ 50 Gbits/s per $\lambda$: Alamouti coding and heterodyne detection,” _Journal of Optical Communications and Networking_ , vol. 12, no. 2, pp. A162–A170, 2020.