# From Goldilocks to Twin Peaks: multiple optimal regimes for quantum
transport in disordered networks

Alexandre R. Coates1, Brendon W. Lovett2 and Erik M. Gauger1<EMAIL_ADDRESS>1 SUPA, Institute of Photonics and Quantum Sciences, Heriot-Watt University,
Edinburgh EH14 4AS, United Kingdom  2 SUPA, School of Physics and Astronomy,
University of St Andrews, St Andrews KY16 9SS, United Kingdom

###### Abstract

Understanding energy transport in quantum systems is crucial for an
understanding of light-harvesting in nature, and for the creation of new
quantum technologies. Open quantum systems theory has been successfully
applied to predict the existence of environmental noise-assisted quantum
transport (ENAQT) as a widespread phenomenon occurring in biological and
artificial systems. That work has been primarily focused on several
‘canonical’ structures, from simple chains, rings and crystals of varying
dimensions, to well-studied light-harvesting complexes. Studying those
particular systems has produced specific assumptions about ENAQT, including
the notion of a single, ideal, range of environmental coupling rates that
improve energy transport. In this paper we show that a consistent subset of
physically modelled transport networks can have at least two ENAQT peaks in
their steady state transport efficiency.

##  I Introduction

Energy transport occurs in many contexts: from circuits and molecular
junctions to processes like photosynthesis and the electron transport chain in
biology [1, 2, 3, 4, 5, 6, 7, 8]. In 2008 the specifics of Environmental
Noise-Assisted Quantum Transport (ENAQT) were first laid out in both
artificial and natural schema [9, 10, 11, 12]. Since then there has been a
proliferation of research into the interplay of coherent quantum transport and
noise in many systems [13, 14, 15], such as the role environmental phonon
coupling can have in overcoming the effects of coherent localisation [16, 17,
18, 19]. From these studies and others, an intuition has been established that
ENAQT produces a single ‘Goldilocks zone’ [20, 21, 22] where dephasing
overcomes limits inherent to fully coherent dynamics, but is not sufficiently
aggressive to spoil its favourable transport characteristics, such as those
brought about by constructive interference.

The study of quantum transport in these various open systems has been
typically carried out on only a few model systems, and some common network
structures; notable exceptions also considering randomly generated networks
include Refs. [15, 23, 24, 25]. In the context of biological photosynthetic
exciton energy transport this is often the Fenna-Matthew-Olson complex (FMO)
[26, 27, 28, 29, 22], and in quantum technologies we see (disordered) chains
and lattices used to simulate many transport scenarios [14, 30, 16, 31, 32,
33, 18, 17, 34, 35, 15]. We see across these contexts that energetic disorder
is common in many systems, with the specifics of these energy landscapes
having a strong effect on quantum transport [25, 23, 16, 36].

Recent work introducing the concept of ‘population uniformisation’ has made
the varying transport behaviour between fully coherent and fully classical
limits explicit [37]. Population uniformisation states that the variance in
on-site populations has a similar qualitative character to the transport
efficiency of open quantum systems, and is minimised when transport efficiency
is maximised, even in the presence of disorder or repulsive interactions [37,
18]. While this framework explicitly frames things in terms of moving from
coherent wavefunctions to classical diffusion and Fick’s Law, we yet again see
the same, singly peaked ENAQT transport efficiency on the standard systems,
including the FMO complex [38].

In this paper we systematically investigate optimal noise rates across
randomly generated transport networks, and show that many have at least two
ENAQT peaks or ‘Goldilocks Zones’ where their transport efficiency is
maximised. Our networks are made of two-level systems, which we model as point
dipoles. We arrange these sites with realistic spacing and effective dipole
moments to ensure the relevance of our results, and consider both uniform and
normally distributed energy landscapes.

##  II Transport Model

###  II.1 Network Setup

We set up each system as follows: first we define a sphere with a fixed radius
of 10 nm, and place two sites at opposite ends of the sphere to act as
injection and extraction sites. We then fill the volume with six additional
sites with random positions and dipole orientations to produce a disordered
but fully connected network, a similar setup and approach has been utilised in
prior works [23, 30, 39]. Figure 1 shows a typical dipole network generated in
this way.

Figure 1: Illustration of a typical dipole system: a fixed volume with
injection and extraction sites in at the poles, and randomly oriented and
positioned sites within the volume. The system is then coupled to phonon
environments with varying strengths ($\Gamma$), and the transport efficiency
($\eta$) measured. On the right we show transport efficiency curves from a
selection of randomly generated networks modelled with phenomenological pure
dephasing that exhibit multiple maxima in their transport efficiency.

Every site is modelled with an identical dipole moment, we use the effective
dipole moments from the FMO complex
$|\boldsymbol{d}|=0.114~{}\text{e}\cdot\text{nm}(\sqrt{30}\text{D})$ [40, 41,
27, 42].

With the structure established we can consider the dipole-dipole interactions,

$V_{i,j}(\boldsymbol{r}_{i,j})=\frac{1}{4\pi\epsilon_{0}}\left(\frac{\boldsymbol{d}_{i}\cdot\boldsymbol{d}_{j}}{|\boldsymbol{r}|^{3}}-3\frac{(\boldsymbol{r}\cdot\boldsymbol{d}_{i})(\boldsymbol{r}\cdot\boldsymbol{d}_{j})}{|\boldsymbol{r}|^{5}}\right)\,,$
(1)

where $\boldsymbol{r}$ is the separation vector between the dipole pair,
$\boldsymbol{d}_{i,j}$ are the respective dipole moments and $\epsilon_{0}$ is
the vacuum permittivity. To ensure the point dipole approximation is remains
appropriate we enforce a minimum separation of 1 nm between every pair of
sites [43]. Throughout this paper we use nanometres, elementary charge and
electron volts, meaning a Coulomb constant
$\left(4\pi\epsilon_{0}\right)^{-1}\approx
1.43996~{}\text{eV}\cdot\text{nm}~{}\text{e}^{-2}$.

To generate the dipole on-site energies we use an average on-site energy
$\bar{\epsilon}=1.5498$ eV (12,500 cm-1), and will later also sample them from
a normal distribution with a standard deviation of 1%, so $\sigma$ = 0.0155eV,
very similar to the energies and disorder found in the FMO complex [40].The
default values we use are summarised in Table 1.

Table 1: Properties used to generate the main network ensemble. The rationale behind our parameter choices is given in the main text. Radius(nm) | N |  $r_{min}$ (nm) |  $|\boldsymbol{d}|$ (e $\cdot$ nm) |  $\bar{\epsilon}$ (eV) |  $\sigma$ (eV)
---|---|---|---|---|---
10 | 8 | 1 | 0.11403 | 1.5498 | 0.0155

With the on-site energies and dipole-dipole interactions defined we can
construct the Hamiltonian. We assume there is only a single excitation in the
system at any time and construct the following excitonic Hamiltonian

$H=\sum^{N}_{i}\epsilon_{i}\ket{i}\bra{i}+\sum_{i\neq
j}^{N}V_{i,j}\ket{i}\bra{j}\,,$ (2)

where $i$ and $j$ are site indices, and the $V_{i,j}$ are evaluated according
to equation 1. We use the single excitation approximation for computational
efficiency as it gives an $N\times N$ Hamiltonian, and is widely used when
modelling open quantum systems with low injection rates, such as light
harvesting complexes [44, 30, 39, 23, 45, 46, 22].

###  II.2 Dynamics

To look at the effect of the environment and finite temperatures on transport
properties, we use the full non-secular Bloch-Redfield master equation [47,
48]

$\begin{split}\dot{\rho_{s}}=&-i[H,\rho_{s}]\\\
&+\gamma_{inj}\mathcal{D}\left[A_{inj}\right]\rho_{s}+\gamma_{ext}\mathcal{D}\left[A_{ext}\right]\rho_{s}\\\
&+\sum_{\omega}\sum_{m,n}S_{m,n}(\omega)(A_{n}(\omega)\rho_{s}A^{\dagger}_{m}(\omega)\\\
&-\frac{1}{2}\\{A^{\dagger}_{m}(\omega)A_{n}(\omega),\rho_{s}\\})\,,\end{split}$
(3)

where $\rho_{s}$ is the system Hamiltonian and $\omega$ are the eigenenergy
splittings [47, 48]. The system-environment interactions, $A_{m}$, are derived
by transforming the relevant site basis operators
$A_{deph,i}=2\ket{i}\bra{i}-\mathbb{I}$ into the Hamiltonian eigenbasis and
$S_{mn}(\omega)$ defines the noise-power spectrum associated with the system-
environment interaction [47, 48, 30]. The noise-power spectrum function is

$S_{m,n}(\omega)=\left(\mathcal{N}_{BE}(\omega,\beta)+\Theta(\omega)\right)\mathcal{J}(\omega),$
(4)

where $\mathcal{N}_{BE}(\omega)$ defines Bose-Einstein statistics at a given
phonon inverse temperature $\beta$, $\Theta(\omega)$ is the Heaviside
function, allowing phonon-assisted transitions from higher to lower
eigenenergies ($\omega>0$), and $\mathcal{J}(\omega)$ is the phonon spectral
density [48]. In this work we use the Drude-Lorentz spectral density, which
has previously been used to model excitonic transfer in light harvesting
complexes [49, 50],

$\mathcal{J}(\omega)=\Gamma\cdot\frac{2}{\pi}\cdot\frac{\omega(1/\tau)}{\omega^{2}+(1/\tau)^{2}},$
(5)

where $\Gamma$ scales the rate of noise in the system from interactions with
the environment, $\tau$ is the correlation time, and $\tau^{-1}$ is the
spectral density peak frequency.

Finally, we have $\mathcal{D}\left[A\right]\rho$ which is the dissipator
superoperator

$\mathcal{D}\left[A\right]\rho=\left(A\rho
A^{\dagger}-\frac{1}{2}\\{A^{\dagger}A,\rho\\}\right)\,.$ (6)

To model extraction and injection, a shelf state is appended to the system.
The extraction operator $A_{ext}$ projects population from the extraction site
to the shelf state, $A_{ext}={\sigma_{shelf}^{+}\sigma_{ext}^{-}}$, and then
that population is re-injected from the shelf state back onto the injection
site with the injection operator
$A_{inj}={\sigma_{inj}^{+}\sigma_{shelf}^{-}}$. Injection and extraction are
matched, $\gamma_{ext,inj}=0.1~{}\text{eV}$, changing this value generally
changes quantitative values but not the qualitative behaviour [37].

To complement the Redfield calculations, we also carry out phenomenological
pure dephasing calculations with the Lindblad master equation [47, 51, 52]

$\begin{split}\dot{\rho}=&-i[H,\rho]+\Gamma\sum_{i=1}^{N}\mathcal{D}\left[A_{deph,i}\right]\rho\\\
&+\gamma_{inj}\mathcal{D}\left[A_{inj}\right]\rho+\gamma_{ext}\mathcal{D}\left[A_{ext}\right]\rho\,,\end{split}$
(7)

where $A_{deph,i}$ are Lindblad operators describing the environmental
influence on each site $i$, again these are of the form
$A_{deph,i}=2\ket{i}\bra{i}-\mathbb{I}$ [48]. All other symbols have the same
meaning as in equation 3. This approach is equivalent to the nonsecular Bloch-
Redfield master equation for an infinite temperature and a flat spectral
density [25].

We focus here on the steady state $\rho_{ss}$ which is found by calculating
the null vector of the system evolution Liouvillian. Our figure of merit then
is the excited steady state population on the extraction site

$\eta=\bra{\text{extraction}}\rho_{ss}\ket{\text{extraction}}.$ (8)

This is motivated by the strong correspondence found in prior work between
dynamical and steady state transport properties [18, 53], as well as further
studies suggesting that the steady state is more natural for photosynthetic
systems [54, 55, 2]. The steady state approach also allows us to avoid any
confusion that could arise from the influence of transient effects when
comparing different networks.

For each network considered in this paper, we were interested in how this
transport efficiency $\eta$ changes with $\Gamma$, the noise rate from
coupling to the environment. To do this we considered a large range of noise
rates $\Gamma~{}\mapsto~{}\left[10^{-7}-10~{}\text{eV}\right]$, and for each
value recorded $\eta$ as well as the full steady state population. This range
of $\Gamma$ was chosen as it was broad enough to capture the values where
$\eta$ has maxima for our networks, and additionally show the transport
efficiency decreasing outside these peaks as shown in figure 1.

These results were then filtered to ensure validity, all data presented here
has passed checks on the unity of the steady state trace, non-negativity of
on-site populations and steady state eigenvalues. From that point we could
perform simple peak–finding calculations for each network and spectral density
to directly identify in which cases there was more than one optimal noise rate
or peak in the transport efficiency curves, and how often this occurred.

##  III Results

###  III.1 Environmental Effects

We generated several ensembles of 1,000 dipole networks, each comprising two
fixed and six randomly located sites. For our Redfield calculations the key
spectral density parameters are the temperature, and the peak frequency of the
Drude-Lorentz spectral density ($\tau^{-1}$). Lindbladian pure dephasing acts
as our infinite temperature limit, being equally present at all frequencies.

We start by considering an ensemble of networks with identical splitting
between the two levels on each site, and see how many network have multiple
maxima in their transport efficiency. This approach lets us compare our
results to prior works that have made the same assumption of uniform on-site
energies when modelling FMO and other complexes with dipole interactions [56,
39, 24].

Figure 2: Percentage of dipole networks showing double-ENAQT behaviour in
networks with uniform on-site energies. We see only a slight sensitivity to
the peak frequency in this case, and a slight increase in the behaviour at low
temperatures. This is likely because the eigenenergy splittings are quite
small when the on-site energies are identical, and as such the transport
behaviour depends on the near zero behaviour of the Drude-Lorentz spectral
density, which is a relatively stable region.

Figure 2 shows a surprising result, contrary to prior wisdom we consistently
find a large number of networks with multiple peaks in their transport
efficiency, regardless of the spectral density or temperature. This
illustrates that these fully connected networks can have multiple maxima in
their transport efficiency, but neglects the importance of on-site energies in
transport [25]. We consider the effects of varied on-site energies in figure
3, where the energies are normally distributed as described in table 1.

Figure 3: Percentage of networks with normally distributed energies showing
double-peaked ENAQT behaviour. The lower the temperature and the higher the
environment peak frequency, the more often we observe this phenomenon. There
is a much larger sensitivity to the spectral density than in the energetically
uniform case.

Figure 2 and figure 3 show our main results from this paper. We see that in
every situation we simulate, a sizeable subset of our networks have two peaks
in the their transport efficiency. The Drude-Lorentz peak frequency has a
slight effect on how often we observe this behaviour, but there is a more
pronounced sensitivity to temperature. The lower the temperature, the more
often we see this behaviour. By extension, this is seen least often – but
still clearly represented – in the Lindblad pure dephasing limit. We show what
proportion of these results occur within measured FMO reorganisation energies
in appendix G.

We note that this double peaked phenomenon generally occurs more frequently in
the energetically uniform ensemble. We attribute this to energetic disorder
producing greater localisation. Meaning that not only are the energetic
differences between eigenstates larger, but also those eigenstates are more
tightly confined to specific sites. This greater spatial confinement means
there are fewer pathways from injection site to extraction site. The greater
energetic differences also raise the chance of some eigenstates being so far
detuned that they are effectively inaccessible given the finite range of noise
rates $\Gamma$ we consider.

###  III.2 Structural Effects

As shown on the right of figure 1, different networks have transport
efficiency curves that take a variety of forms. The relative prominence and
position of each transport efficiency peak varies. In some cases peaks are
very well separated with a pronounced dip between them. In other cases the two
are barely distinguishable, almost smearing into each other. This suggests a
strong structural dependence between the form of the curves and the
Hamiltonians of their respective networks.

We closely inspected networks such as the one in Figure 4, which produce
double peaks across a wide range of temperatures and Drude-Lorentz peak
frequencies, to identify what key features might be correlated with having
multiple peaks in the transport efficiency of a system.

Figure 4: A system where we see double-peaked ENAQT for all forms of spectral
density we consider. (a) shows the dipole positions and orientations for this
network, with the $z$-axis pointing upwards, and then out of the page (tilted
by $\pi/2$) in the top and bottom plot, respectively. (b) shows the transport
efficiency in all our cases and (c) shows the network Hamiltonian eigenstates,
with eigenenergies on the $y$-axis and site indices on the $x$-axis (0 and 1
are the injection and extraction sites, respectively). The size of the
diamonds indicate the support of the eigenstate on that site.

We did not find any strong geometric dependence across networks with multiple
maxima in their transport efficiency, but we identified consistent features in
the system eigenstates. Specifically, these systems often have one or more
large gaps in their eigenenergy distributions, as shown in figure 4 (d).

###  III.3 Discussion

An intuitive and tempting interpretation of the multiple optimal transport
regimes would be that two broad sets of energy scales are important: the width
of each band of eigenenergies, and the separation between bands. The intuition
from this being that there is an initial path with an ideal environmental
coupling for ENAQT. However, as phonon couplings become larger, eigenstates
that were previously far–detuned become more accessible. If the new path is
more efficient for exciton transport, then a new peak in the transport
efficiency can be observed. Both the energetically uniform and non-uniform
ensembles consistently have these gaps in their spectra, induced by the
dipole-dipole interactions between sites, and if present, differences in on-
site energies.

Testing this hypothesis in appendix A we do indeed find a positive correlation
with the relative standard deviation of eigenenergy splittings. However the
change in the fraction of multiply peaked networks remains modest, suggesting
there are other factors at play. In the following we briefly summarise how
doubly peaked behaviour correlates with some other aspects.

In appendix B we tested the energetic separation hypothesis in another way,
generating a new independent ensemble of 1000 networks where three of the
dipoles have a fixed offset added to their on-site energies. This approach
encourages more gaps to form in the eigenspectra and we see a modest increase
in the frequency of double peaks.

We also have considered the relative energies of the injection and extraction
sites in these systems. In appendix C we show that double peaks occur more
frequently when injecting at lower energies than the extraction site,
suggesting it may occur more often in less efficient networks (in the sense of
requiring ‘uphill’ energy transport), albeit by no means limited to those.

Another consideration is the number of potential paths in a system from the
injection site to the extraction site. We generated an ensemble of networks
made of paired, disordered, nearest-neighbour chains that only connected at
shared injection and extraction sites, giving only two paths across the
system. In appendix D we show that while this strongly reduces how often a
network has multiple transport efficiency maxima, we do still observe it
against all spectral densities. Lastly, we show in appendix E that double
peaks can be observed against Ohmic and superohmic spectral densities as well.

Overall, our analysis suggests there are a multitude of factors at play which
can positively correlate with and increased occurrence of doubly peaked
network, however, we have not been able to identify specific strongly
predictive factors.

##  IV Conclusion

In this paper, we have shown that transport networks with realistic and
microscopically resolved vibrational interactions frequently feature more than
one optimal regime for transport efficiency. This runs counter to expectations
of there being a single ‘Goldilocks zone’ in any and all cases. We observe
that these multiple optimal transport regimes can occur for energetically
ordered or energetically disordered networks, and occur more often in networks
with less evenly spaced eigenvalues.

## Conflicts of interest

There are no conflicts to declare.

## Acknowledgements

This work was supported by EPSRC Grant Nos. EP/L015110/1, EP/T007214/1,
EP/T01377X/1, and EP/T014032. Calculations carried out in QuTiP [57], data
analysis carried out with pandas DataFrames [58, 59].

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##  Appendix A Role of eigenenergy spacing disorder

Here we present statistical analysis of the role of disorder in the spacing of
eigenenergies, specifically we consider the relative standard deviation
$\frac{\sigma}{\mu}$ where $\sigma$ is the standard deviation of the
eigenenergy differences, and $\mu$ is the average eigenenergy difference. We
find in both our ensembles (energetically uniform and energetically
disordered) that more disorder in eigenenergy spacings is positively
correlated with an increased fraction of networks displaying two optimal
transport regimes.

Figure 5: Histogram of probability density against the relative standard
deviation of network eigenenergy differences for energetically disordered
networks. We see the networks with double peaks occur more often for higher
amounts of disorder. We test this directly in appendix B. Figure 6: Histogram
of probability density against the relative standard deviation of network
eigenenergy differences for 1,000 energetically uniform networks, previously
described in figure 2. Without an energy landscape, we see a wide spread of
relative standard deviations defined by the geometric properties of the
networks. We see a relatively sharp maximum relative standard deviation here
due to the exclusion volume or minimum distance we enforce between dipoles
when generating our systems.

##  Appendix B Networks with offset energies

Based on our intuition of double peaks being related to two effective pathways
with different optimal conditions for ENAQT, we generate a new, independent
ensemble of networks with two different on-site energy scales. We generate the
dipole networks exactly as before, including their random on-site energies.
Then for each network we randomly pick three of the bulk dipoles and shift
their energy upwards by the standard deviation of our on-site energies (+15.5
meV).

Figure 7: The hatched bars show percentage of dipole networks with double-
ENAQT behaviour in our artificially offset networks. The coloured bars show
the prior results from figure 3 for the normally distributed on-site energies.
We see a general increase in double peaked behaviour thanks to this energy
offset.

This adjustment to the networks increases the probability of there being a
larger gap in the system eigenenergies, but leaves the geometric properties of
these networks unaffected. As shown in figure 7, we do see some increase in
double-peaked transport efficiency in most cases. Though as expected when
comparing two independent datasets there are fluctuations in the trends.

##  Appendix C Injection and extraction energies

We also consider the importance of where the injection and extraction sites
sit in the eigenenergy landscape. We index the eigenstates from lowest to
highest energy ($\lambda_{1\longrightarrow N}$), and record which eigenstate
is most present on the injection site ($\lambda_{inj}$) and the extraction
site ($\lambda_{ext}$), respectively.

For our networks with no double peaks, there is general symmetry. For the
networks with multiple transport efficiency maxima, we see a preference for
injecting at lower eigenenergies than they extract at
($\lambda_{inj}-\lambda_{ext}<0$). This suggests the effect will be more
prominent in less efficient systems. Though we note that the effect is present
broadly,also clearly occurring in networks where energy transport should be
efficient along a downhill gradient.

Figure 8: Histogram of probability density against the difference in
eigenenergy index of the injection and extraction sites for energetically
disordered systems. Negative values mean the injection is below the
extraction. Where $\lambda_{inj}-\lambda_{ext}=0$, we have a pair of
eigenstates equally present on the injection and extraction sites. This is not
possible in the energetically disordered case due to the extra localisation.
Figure 9: Histogram of probability density against the difference in
eigenenergy index of the injection and extraction sites. Negative values mean
the injection is below the extraction. We again see fewer cases where doubly
peaked networks have the injection far above the extraction, though the
general trend is less clear than for the energetically disordered case (figure
8).

##  Appendix D Double peaks in coupled chains

To further examine the importance of different energy scales, we constructed a
simple system from 8 sites. We define two nearest neighbour chains of three
sites, and connect them to an injection site and extraction site at either
end. Both chains have the same NN coupling (2.5 meV) and average on-site
energy (1.55 eV), but have different amounts of disorder in on-site energies.
We set their standard deviations to $\sigma=15.5\text{ and }1.55$ meV,
respectively, such that each arm will typically have a different amount of
phonon coupling that is optimal for overcoming localisation.

This gives us a scenario with two well-separated paths between injection and
extraction, rather than the many possible traverses in our dipole networks. As
such, there is a general reduction in secondary pathways that have an
opportunity to improve transport efficiency. The results of these calculations
for 1,000 networks are shown in figure 10.

Figure 10: Double-peak rates for 1000 ‘two-armed’ networks. We see a decrease
in double-peak behaviour in every circumstance compared to our main results
shown in figure 3. This suggests that the reduction of available paths or
long-range coupling is limiting how often double peaks occur.

As figure 10 shows, double peaks are still present, though always to a lesser
degree than in our totally random dipole ensembles. A key difference is the
large decrease of double peaks with Lindblad pure dephasing, or at high
temperatures but low peak frequencies. So just having two possible pathways
across a system is not enough to remove the possibility of double peaks, but
does lower the chances of observing it.

##  Appendix E Other spectral densities

The arguments presented in this paper are not unique to the Drude-Lorentz
spectral density or pure dephasing. We can also consider the Ohmic and
superohmic spectral densities. We define these spectral densities with
Gaussian cutoffs as

$\mathcal{J}(\omega)=\Gamma\cdot\left(\frac{\omega}{\omega_{c}}\right)^{S}e^{-\left(\frac{\omega}{\omega_{c}}\right)^{2}},$
(9)

where $\Gamma$ is the system-phonon coupling, and $\omega_{c}$ is the cutoff
frequency. $S=1\text{ and }3$ for our Ohmic and superohmic powers
respectively. The peak frequency $\omega_{peak}$ depends on the cutoff as

$\omega_{peak}=\omega_{c}\sqrt{\frac{S}{2}}.$ (10)

We observe that when secondary peaks appear in these scenarios, they are often
at very high values of $\Gamma$, compared to the range we use to see the same
behaviour in pure dephasing and Drude-Lorentz models. This is slightly
mitigated in the energetically uniform networks where the lack of disorder in
on-site energies has the effect of moving these peaks to lower coupling
strengths where our standard approach can capture them. We present a clear
example in figure 11 of a single energetically uniform network showing double
peaks at all temperatures tested for the Ohmic and superohmic distributions.

Figure 11: Multi-peaked transport efficiency curves for a single network with
uniform on-site energies, showing results from Lindblad pure dephasing, as
well as Ohmic and superohmic spectral densities with $\omega_{peak}=$ 0.1 eV.
We see similar qualitative behaviour across the Lindblad, 300 K and 180 K
curves, with a change in behaviour at 30 K.

As such we can state that these double-peaked effects can and do also occur
for these power law spectral densities. However, they occur over a much
broader range of environmental couplings, and as such, alternative methods
suited to intermediate- and strongly-coupled open quantum systems would be
needed to provide more robust statistics.

##  Appendix F Hamiltonian Details

Here we list out the data for the specific dipole network presented in figure
4.

Table 2: Dipole positions and Energies Dipole | X (nm) | Y (nm) | Z (nm) | Energies (eV)
---|---|---|---|---
0 (inject) | 0.0 | 0.0 | -10.0 | 1.552794
1 (extract) | 0.0 | 0.0 | 10.0 | 1.524548
2 | -5.239018 | -2.013063 | -6.763873 | 1.544986
3 | -2.429034 | 1.463867 | -2.933762 | 1.552236
4 | -1.321062 | 4.226071 | -0.255611 | 1.580151
5 | 3.148822 | -2.374797 | -5.102531 | 1.532472
6 | -1.552033 | 1.351976 | -1.062326 | 1.560427
7 | 1.851469 | 1.995554 | 9.06525 | 1.53166
Table 3: Dipole vector x, y and z components. Dipole |  $\boldsymbol{d}_{x}$ (e $\cdot$ nm) |  $\boldsymbol{d}_{y}$ (e $\cdot$ nm) |  $\boldsymbol{d}_{z}$ (e $\cdot$ nm)
---|---|---|---
0 | 0.0 | 0.0 | 0.114033
1 | 0.0 | 0.0 | 0.114033
2 | -0.054907 | -0.023364 | 0.097174
3 | 0.016014 | -0.040995 | -0.105197
4 | -0.065844 | 0.071077 | -0.060133
5 | 0.02771 | 0.110455 | -0.005934
6 | 0.081605 | -0.079198 | -0.008472
7 | 0.058304 | -0.079757 | 0.056948

##  Appendix G Reorganisation Energies

The goal of this work has been to demonstrate the existence of a new
phenomenon in otherwise very typical transport networks. But answering this
question is separate from demonstrating if this occurs at typical amounts of
coupling to the environment. Also, answering this question cannot be done for
the pure dephasing approach used in many studies, and instead requires
microscopic couplings to be considered. As such we consider our Bloch-Redfield
results in this section, and we calculate the reorganisation energies of our
networks in the standard fashion

$\lambda=\frac{1}{\pi}\int^{\infty}_{0}\frac{\mathcal{J}(\omega)}{\omega}d\omega,$
(11)

where $\lambda$ is the reorganisation energy and the other terms keep their
definitions from equation 4. With this, we can constrain our results to only
those occurring at physically reasonable reorganisation energies, which we
take to be anything below 36 meV (290.4 cm-1), covering previously calculated
reorganisation energies for FMO [60]. Figure 12 shows what percentage of
networks with multiple transport efficiency maxima have both peaks below this
upper bound.

Figure 12: Bar chart showing what percentage of double-peaked networks have
both their peaks below a maximum reorganisation energy of 36 meV. The lower
the Drude-Lorentz peak frequency, the more networks meet this criterion.


# A new lower bound in the $abc$ conjecture

Curtis Bright<EMAIL_ADDRESS>School of Computer Science, University of
Windsor, and Department of Mathematics and Statistics, Carleton University, ,
webpage: www.curtisbright.com

###### Abstract

We prove that there exist infinitely many coprime numbers $a$, $b$, $c$ with
$a+b=c$ and $c>\operatorname{rad}(abc)\exp(6.563\sqrt{\log c}/\log\log c)$.
These are the most extremal examples currently known in the $abc$ conjecture,
thereby providing a new lower bound on the tightest possible form of the
conjecture. Our work builds on that of van Frankenhuysen (1999) who proved the
existence of examples satisfying the above bound with the constant $6.068$ in
place of $6.563$. We show that the constant $6.563$ may be replaced by
$4\sqrt{2\delta/e}$ where $\delta$ is a constant such that all unimodular
lattices of sufficiently large dimension $n$ contain a nonzero vector with
$\ell_{1}$ norm at most $n/\delta$.

###### keywords:

$abc$ conjecture; good $abc$ examples; $abc$ conjecture lower bound

###### keywords:

[2020 Mathematics Subject Classification]11D75, 11H06, 11G50, 11N25

††articletype: RESEARCH ARTICLE††journal: Canadian Mathematical Society

2020 somm#1{ #3:#4 }#2{ #3:#4 #2}

C. Bright

##  1 Introduction

Three natural numbers $a$, $b$, $c$ are said to be an _$abc$ triple_ if they
do not share a common factor and satisfy the equation

$a+b=c.$

Informally, the $abc$ conjecture says that large $abc$ triples cannot be ‘very
composite’, in the sense of $abc$ having a prime factorization containing
large powers of small primes. The _radical_ of $abc$ is defined to be the
product of the primes in the prime factorization of $abc$, i.e.,

$\operatorname{rad}(abc)\coloneqq\prod_{p\mid abc}p.$

The $abc$ conjecture then states that $abc$ triples satisfy

$c=O\big{\lparen}\operatorname{rad}(abc)^{1+\epsilon}\big{\rparen}$ (1.1)

for every $\epsilon>0$, where the implied big-$O$ constant may depend on
$\epsilon$.

Presently, the conjecture is far from being proved; not a single $\epsilon$ is
known for which (1.1) holds.111A proof of the $abc$ conjecture is claimed by
S. Mochizuki, but this has not been accepted by the general mathematical
community. [8] The best known upper bound is due to C. L. Stewart and K. Yu
[10] and says that $abc$ triples satisfy

$c=O\big{\lparen}\exp(\operatorname{rad}(abc)^{1/3}(\log\operatorname{rad}(abc))^{3})\big{\rparen}.$

On the other hand, Stewart and Tijdeman [9] proved in 1986 that there are
infinitly many $abc$ triples with

$c>\operatorname{rad}(abc)\exp\big{\lparen}\kappa\sqrt{\log c}/\log\log
c\big{\rparen}$ (1.2)

for all $\kappa<4$. Such $abc$ triples are exceptional in the sense that their
radical is relatively small in comparison to $c$ and they provide a lower
bound on the best possible form of (1.1). In 1997, van Frankenhuysen [3]
improved this lower bound by showing that (1.2) holds for $\kappa=4\sqrt{2}$,
and in 1999 he improved this to $\kappa=6.068$ using a sphere-packing idea
credited to H. W. Lenstra, Jr. We improve this further by showing that there
are infinitely many $abc$ triples satisfying (1.2) with $\kappa=6.563$.

##  2 Preliminaries

Let $S$ be a set of prime numbers. An $S$-unit is defined to be a rational
number whose numerator and denominator in lowest terms are divisible by only
the primes in $S$. That is, one has

$\text{$S$-units}\coloneqq\set[\bigg{]}{\pm\prod_{p_{i}\in
S}p_{i}^{e_{i}}}{e_{i}\in\mathbb{Z}}.$

This generalizes the notion of units of $\mathbb{Z}$; in particular, the
$\emptyset$-units are $\pm 1$. The _height_ of a rational number $p/q$ in
lowest terms is $h(p/q)\coloneqq\max\\{\lvert p\rvert,\lvert q\rvert\\}$. This
provides a convenient way of measuring the ‘size’ of an $S$-unit. Finally, if
${\vectorsym{x}}=(x_{1},\dotsc,x_{n})$ is a vector in $\mathbb{R}^{n}$, we let

$\lVert{\vectorsym{x}}\rVert_{k}\coloneqq\Big{\lparen}\sum_{i=1}^{n}\lvert
x_{i}\rvert^{k}\Big{\rparen}^{1/k}$

be its standard $\ell_{k}$ norm. The existence of exceptional $abc$ triples
follows from some basic results in the geometry of numbers along with
estimates for prime numbers provided by the prime number theorem. In
particular, we rely on a result of Rankin [6] guaranteeing the existence of a
short nonzero vector in a suitably chosen lattice.

###  2.1 The odd prime number lattice

The result involves in an essential way the _odd prime number lattice_ $L_{n}$
generated by the rows ${\vectorsym{b}}_{1}$, $\dotsc$, ${\vectorsym{b}}_{n}$
of the matrix

$\begin{bmatrix}{\vectorsym{b}}_{1}\\\ {\vectorsym{b}}_{2}\\\
{\vectorsym{b}}_{3}\\\ \vdots\\\
{\vectorsym{b}}_{n}\end{bmatrix}=\begin{bmatrix}\log 3&&&&&\log 3\\\ &\log
5&&&&\log 5\\\ &&\log 7&&&\log 7\\\ &&&\ddots&&\vdots\\\ &&&&\log p_{n}&\log
p_{n}\end{bmatrix}$

where $p_{i}$ denotes the $i$th odd prime number. This lattice has a number of
interesting applications. For example, it is used in Schnorr’s factoring
algorithm [7] and Micciancio’s proof that approximating the shortest vector to
within a constant factor is NP-hard under a randomized reduction [5]. There is
an obvious isomorphism between the points of $L_{n}$ and the positive
$\\{p_{1},\dotsc,p_{n}\\}$-units given by

$\sum_{i=1}^{n}e_{i}{\vectorsym{b}}_{i}\leftrightarrow\prod_{i=1}^{n}p_{i}^{e_{i}}.$

Furthermore, this relationship works well with a natural notion of size, as
shown in the following lemma.

######  Lemma 2.1.

$\lVert{\vectorsym{x}}\rVert_{1}=2\log h\lparen p/q\rparen$ where
${\vectorsym{x}}=\sum_{i=1}^{n}e_{i}{\vectorsym{b}}_{i}$ and
$p/q=\prod_{i=1}^{n}p_{i}^{e_{i}}$ is expressed in lowest terms.

######  Proof 1.

Without loss of generality suppose $p\geq q$. Then

$\lVert{\vectorsym{x}}\rVert_{1}=\sum_{i=1}^{n}\big{\lvert}e_{i}\log
p_{i}\big{\rvert}+\Big{\lvert}\sum_{i=1}^{n}e_{i}\log p_{i}\Big{\rvert}=\log
p+\log q+\log p-\log q=2\log p$

as required, since $h(p/q)=p$ by assumption.

###  2.2 The kernel sublattice

Let $P$ be the set of positive $\\{p_{1},\dotsc,p_{n}\\}$-units, and consider
the map $\phi$ reducing the elements of $P$ modulo $2^{m}$. Since each
$p_{1}$, $\dotsc$, $p_{n}$ is odd, $\phi\colon
P\to(\mathbb{Z}/2^{m}\mathbb{Z})^{*}$ is well-defined. The odd prime number
lattice $L_{n}$ has an important sublattice that we call the _kernel
sublattice_ $L_{n,m}$. It consists of those vectors whose associated
$\\{p_{1},\dotsc,p_{n}\\}$-units lie in the kernel of $\phi$. Formally, we
define

$L_{n,m}\coloneqq\set[\bigg{]}{\sum_{i=1}^{n}e_{i}{\vectorsym{b}}_{i}}{\prod_{i=1}^{n}p_{i}^{e_{i}}\equiv
1\pmod{2^{m}}}.$

Figure 1 plots the first two coordinates of vectors in the kernel sublattice
for varying $m$.

Figure 1: Plots of $\set{(x,y)}{(x,y,z)\in L_{2,m}}$ for $1\leq m\leq 8$.

######  Lemma 2.2.

$L_{n,m}$ is a sublattice of $L_{n}$ of index $2^{m-1}$ when $n\geq 2$.

######  Proof 2.

Note that $L_{n,m}$ is discrete and closed under addition and subtraction.
$L_{n,m}$ also contains the $n$ linearly independent vectors
$\operatorname{ord}_{2^{m}}(p_{i}){\vectorsym{b}}_{i}$ for $1\leq i\leq n$, so
this demonstrates that $L_{n,m}$ is a full-rank sublattice of $L_{n}$.

Since $3$ and $5$ generate $(\mathbb{Z}/2^{m}\mathbb{Z})^{*}$, when $n\geq 2$
we have $\phi(P)=(\mathbb{Z}/2^{m}\mathbb{Z})^{*}$. Since $L_{n}\cong P$ and
$L_{n,m}\cong\ker\phi$ it follows that
$L_{n}/L_{n,m}\cong(\mathbb{Z}/2^{m}\mathbb{Z})^{*}$ by the first isomorphism
theorem. Thus the index of $L_{n,m}$ in $L_{n}$ is
$\lvert(\mathbb{Z}/2^{m}\mathbb{Z})^{*}\rvert=2^{m-1}$.

###  2.3 Hermite’s constant

The _Hermite constant_ $\gamma_{n}$ is defined to be the smallest positive
number such that every lattice of dimension $n$ and volume $\det(L)$ contains
a nonzero vector ${\vectorsym{x}}$ with

$\lVert{\vectorsym{x}}\rVert^{2}_{2}\leq\gamma_{n}\det(L)^{2/n}.$

We are interested in the “Manhattan distance” $\ell_{1}$ norm instead of the
usual Euclidean norm, so we define the related constants $\delta_{n}$ by the
smallest positive number such that every full-rank lattice of dimension $n$
contains a nonzero vector ${\vectorsym{x}}$ with

$\lVert{\vectorsym{x}}\rVert_{1}\leq\delta_{n}\det(L)^{1/n}.$

By Minkowski’s theorem [2] applied to a generalized octahedron (a ‘sphere’ in
the $\ell_{1}$ norm), every full-rank lattice of dimension $n$ contains a
nonzero lattice point ${\vectorsym{x}}$ with
$\lVert{\vectorsym{x}}\rVert_{1}\leq(n!\det(L))^{1/n}$. It follows that
$\delta_{n}\leq(n!)^{1/n}\sim n/e$, but better bounds on $\delta_{n}$ are
known. Blichfeldt [1] showed that

$\delta_{n}\leq\sqrt{\frac{4(n+1)(n+2)}{3\pi(n+3)}}\bigg{\lparen}\frac{2(n+1)}{n+3}\Big{\lparen}\frac{n}{2}+1\Big{\rparen}!\bigg{\rparen}^{1/n}\sim\frac{n}{\sqrt{1.5\pi
e}},$

where $x!\coloneqq\Gamma(x+1)$. Improving this, Rankin [6] showed the
following.

######  Lemma 2.3.

For all integer $n$ and real $x\in[1/2,1]$, we have

$\delta_{n}\leq\Big{\lparen}\frac{2-x}{1-x}\Big{\rparen}^{x-1}\Big{\lparen}\frac{1+xn}{x}(xn)!\Big{\rparen}^{1/n}\frac{n^{1-x}}{x!}\sim\Big{\lparen}\frac{2-x}{1-x}\Big{\rparen}^{x-1}\Big{\lparen}\frac{x}{e}\Big{\rparen}^{x}\frac{n}{x!}.$

######  Corollary 2.4.

Let $\delta$ be a constant such that $\delta_{n}\leq n/\delta+O(\log n)$. Then
a permissible value for $\delta$ is $\max\limits_{1/2\leq x\leq
1}\big{\lparen}\frac{1-x}{2-x}\big{\rparen}^{x-1}\big{\lparen}\frac{e}{x}\big{\rparen}^{x}x!\approx
3.65931$.

######  Proof 3.

Note that $((1+xn)/x)^{1/n}=1+O((\log n)/n)$ and

$(xn)!^{1/n}=\Big{\lparen}\sqrt{2\pi
xn}\mspace{1.5mu}\Big{\lparen}\frac{xn}{e}\Big{\rparen}^{xn}\big{\lparen}1+O(n^{-1})\big{\rparen}\Big{\rparen}^{1/n}=\Big{\lparen}\frac{xn}{e}\Big{\rparen}^{x}\big{\lparen}1+O\big{\lparen}\tfrac{\log
n}{n}\big{\rparen}\big{\rparen}.$

Then by Lemma 2.3 it follows that

$\delta_{n}\leq\Big{\lparen}\frac{2-x}{1-x}\Big{\rparen}^{x-1}\Big{\lparen}\frac{x}{e}\Big{\rparen}^{x}\frac{n}{x!}+O(\log
n),$

and the function
$x\mapsto\big{\lparen}\frac{1-x}{2-x}\big{\rparen}^{x-1}\big{\lparen}\frac{e}{x}\big{\rparen}^{x}x!$
for $1/2\leq x\leq 1$ reaches a maximum of approximately $3.65931$ at
$x\approx 0.645467$.

The best possible value $\delta$ can achieve in Corollary 2.4 is unknown, but
the Minkowski–Hlawka theorem [2] applied to an generalized octahedron shows
that in any dimension $n$ there is always a full-rank lattice $L$ with all of
its nonzero lattice points ${\vectorsym{x}}$ having
$\lVert{\vectorsym{x}}\rVert_{1}>(\zeta(n)\mspace{1.5mu}n!\det(L))^{1/n}\mspace{-1.5mu}/2$;
here $\zeta$ is the Riemann zeta function. It follows that
$\delta_{n}>(\zeta(n)\mspace{1.5mu}n!)^{1/n}\mspace{-1.5mu}/2\sim n/(2e)$, so
we must have $\delta\leq 2e$.

###  2.4 A full-rank kernel sublattice

Since $L_{n,m}\in\mathbb{R}^{n+1}$ is of dimension $n$ (i.e., not full-rank)
it is awkward to use Rankin’s result on $L_{n,m}$ directly. The basis matrix
of $L_{n,m}$ cannot simply be rotated to embed it in $\mathbb{R}^{n}$, since
rotation does not preserve the $\ell_{1}$ norm. To circumvent this and work
with a full-rank lattice we adjoin the new basis vector
${\vectorsym{b}}_{n+1}=[0,\dotsc,0,n^{3}]$ to $L_{n}$ to form a full-rank
lattice $\overline{L}_{n}$ (and similarly a full-rank lattice
$\overline{L}_{n,m}$).

######  Lemma 2.5.

The volume of $\overline{L}_{n,m}$ is $2^{m-1}n^{3}\prod_{i=1}^{n}\log p_{i}$
when $n\geq 2$.

######  Proof 4.

The basis matrix of $L_{n}$ adjoined with ${\vectorsym{b}}_{n+1}$ is an upper-
triangular matrix, so $\det(\overline{L}_{n})=n^{3}\prod_{i=1}^{n}\log p_{i}$.
The index of $\overline{L}_{n,m}$ in $\overline{L}_{n}$ is $2^{m-1}$ when
$n\geq 2$ by the same argument as in Lemma 2.2, so
$\det(\overline{L}_{n,m})=2^{m-1}\det(\overline{L}_{n})$.

Our choice of $m$ will ultimately be asymptotic to $n\log_{2}n$, and in this
case $\det(\overline{L}_{n,m})^{1/(n+1)}$ grows slightly more than linearly in
$n$.

######  Lemma 2.6.

If $m\sim n\log_{2}n$ then
$\det(\overline{L}_{n,m})^{1/(n+1)}=O(n^{1+\epsilon})$ for all $\epsilon>0$.

######  Proof 5.

Lemma 2.5 implies
$\det(\overline{L}_{n,m})^{1/(n+1)}<2^{m/n}n^{3/n}\big{\lparen}\prod_{i=1}^{n}\log
p_{i}\big{\rparen}^{1/n}$. Note that
$m/n=\log_{2}n+o(\log_{2}n)<(1+\epsilon)\log_{2}n$ for all $\epsilon>0$ and
sufficiently large $n$. Thus $2^{m/n}<n^{1+\epsilon}$ for sufficiently large
$n$, and the remaining factors are $O(n^{\epsilon})$ since $n^{3/n}=O(1)$ and
$\big{\lparen}\prod_{i=1}^{n}\log p_{i}\big{\rparen}^{1/n}<\log p_{n}=O(\log
n)$.

Finally, we will require the fact that any vector in $\overline{L}_{n}$
including a nontrivial coefficient on ${\vectorsym{b}}_{n+1}$ must be
sufficiently large (have length at least $n^{3}$ in the $\ell_{1}$ norm).

######  Lemma 2.7.

If ${\vectorsym{x}}=\sum_{i=1}^{n+1}e_{i}{\vectorsym{b}}_{i}$ then
$\lVert{\vectorsym{x}}\rVert_{1}\geq n^{3}\lvert e_{n+1}\rvert$.

######  Proof 6.

We have $\lVert{\vectorsym{x}}\rVert_{1}=\sum_{i=1}^{n}\lvert e_{i}\rvert\log
p_{i}+\big{\lvert}\sum_{i=1}^{n}e_{i}\log p_{i}+e_{n+1}n^{3}\big{\rvert}$.

Without loss of generality suppose that $e_{n+1}>0$ and for contradiction
suppose $\lVert{\vectorsym{x}}\rVert_{1}<n^{3}e_{n+1}$. Then

$\sum_{i=1}^{n}e_{i}\log
p_{i}+e_{n+1}n^{3}\leq\Big{\lvert}\sum_{i=1}^{n}e_{i}\log
p_{i}+e_{n+1}n^{3}\Big{\rvert}<n^{3}e_{n+1}-\sum_{i=1}^{n}\lvert
e_{i}\rvert\log p_{i}$

implies $\sum_{i=1}^{n}(e_{i}+\lvert e_{i}\rvert)\log p_{i}<0$, and this is
nonsensical since the left-hand side is nonnegative.

###  2.5 Asymptotic formulae

Let $x\coloneqq p_{n}$ and let $\pi(x)$ be the prime counting function, so
that $n=\pi(x)-1$. The prime number theorem [4] states that
$\pi(x)\sim\operatorname{li}(x)$ where $\operatorname{li}(x)$ is the
logarithmic integral $\int_{0}^{x}\frac{\,\mathrm{d}t}{\log t}$ with
asymptotic expansion

$\operatorname{li}(x)=\frac{x}{\log
x}+\frac{x}{\log^{2}x}+\frac{2x}{\log^{3}x}+O\Big{\lparen}\frac{x}{\log^{4}x}\Big{\rparen}.$
(2.1)

In fact, the error term $\pi(x)-\operatorname{li}(x)$ is
$O(x/\exp(C\log^{1/2}x))$ for some constant $C>0$. The following estimates are
consequences of this (cf. [9, Lemma 2]). For the convenience of the reader,
proofs are given in the appendix.

######  Lemma 2.8.

$\sum_{i=1}^{n}\log p_{i}=n\log
p_{n}-n-p_{n}/\log^{2}p_{n}+O(p_{n}/\log^{3}p_{n})$.

######  Lemma 2.9.

$\sum_{i=1}^{n}\log\log p_{i}=n\log\log
p_{n}-p_{n}/\log^{2}p_{n}+O(p_{n}/\log^{3}p_{n})$.

##  3 Exceptional $abc$ triples

For our purposes the importance of the kernel sublattice is that it lets us
show the existence of $abc$ triples in which $c$ is large relative to
$\operatorname{rad}(abc)$. The following lemma shows how this may be done.

######  Lemma 3.1.

For all $m\lesssim n\log_{2}n$ and sufficiently large $n$, there exists an
$abc$ triple satisfying

$\frac{2^{m-1}}{\prod_{i=1}^{n}p_{i}}\operatorname{rad}(abc)\leq
c\quad\text{and}\quad 2\log c\leq\frac{n+O(\log
n)}{\delta}\Big{\lparen}2^{m-1}n^{3}\prod_{i=1}^{n}\log
p_{i}\Big{\rparen}^{1/(n+1)}.$

######  Proof 7.

By the definition of $\delta$ from Corollary 2.4, for all sufficiently large
$n$ there exists a nonzero ${\vectorsym{x}}\in\overline{L}_{n,m}$ with

$\lVert{\vectorsym{x}}\rVert_{1}\leq\Big{\lparen}\frac{n+1}{\delta}+O(\log
n)\Big{\rparen}\det\lparen\overline{L}_{n,m}\rparen^{1/(n+1)}.$ (3.1)

Say ${\vectorsym{x}}=\sum_{i=1}^{n+1}e_{i}{\vectorsym{b}}_{i}$. For
sufficiently large $n$ we must have $e_{n+1}=0$, since by Lemma 2.7 if
$e_{n+1}\neq 0$ then $\lVert{\vectorsym{x}}\rVert_{1}\geq n^{3}$. This would
contradict (3.1) since by Lemma 2.6 the right-hand side is
$O(n^{2+\epsilon})$.

Let $\prod_{i=1}^{n}p_{i}^{e_{i}}=p/q$ be expressed in lowest terms. By
construction of the kernel sublattice, we have that $p/q\equiv 1\pmod{2^{m}}$.
Let $c\coloneqq h(p/q)=\max\\{p,q\\}$, $b\coloneqq\min\\{p,q\\}$, and
$a\coloneqq c-b$, so that $a$, $b$, $c$ form an $abc$ triple. Furthermore, we
see that

$c\equiv b\pmod{2^{m}}$

so that $c=b+k2^{m}$ for some positive integer $k\leq c/2^{m}$. Note $a$ is
divisible by $2$ and any other prime that divides it also divides $k$, so that
$\operatorname{rad}(a)\leq 2k\leq c/2^{m-1}$. Furthermore, by construction of
$b$ and $c$, $\operatorname{rad}(bc)\leq\prod_{i=1}^{n}p_{i}$ and the first
bound follows. The second bound follows from (3.1) and Lemmas 2.1 and 2.5.

###  3.1 Optimal choice of $m$

The first bound in Lemma 3.1 allows us to show the existence of infinitely
many $abc$ triples whose ratio of $c$ to $\operatorname{rad}(abc)$ grows
arbitrarily large. Using the second bound, we can even show that this ratio
grows faster than a function of $c$. It is not immediately clear how to choose
$m$ optimally, i.e., to maximize the ratio $c/\operatorname{rad}(abc)$.

For convenience, let $R$ denote the right-hand side of the second inequality
in Lemma 3.1 with $l_{n}\coloneqq O(\log n)$. Then
$2^{m-1}=\big{\lparen}\frac{\delta
R}{n+l_{n}}\big{\rparen}^{n+1}/(n^{3}\prod_{i=1}^{n}\log p_{i})$, so the
bounds of Lemma 3.1 can be rewritten in terms of $R$:

$\frac{(\delta R/(n+l_{n}))^{n+1}}{n^{3}\prod_{i=1}^{n}p_{i}\log
p_{i}}\operatorname{rad}(abc)\leq c\quad\text{and}\quad 2\log c\leq R.$ (3.2)

The question now becomes how to choose $R$ in terms of $n$ so that
$c/\operatorname{rad}(abc)$ is maximized.

Taking the logarithm of the first inequality in (3.2) gives

$(n+1)\log\Big{\lparen}\frac{\delta R}{n+l_{n}}\Big{\rparen}-3\log
n-\sum_{i=1}^{n}\log p_{i}-\sum_{i=1}^{n}\log\log
p_{i}+\log\operatorname{rad}(abc)\leq\log c.$

Using the asymptotic formulae in Lemmas 2.8 and 2.9 with $\log(n+l_{n})=\log
n+O(l_{n}/n)$, this becomes

$n\log\Big{\lparen}\frac{e\delta R}{np_{n}\log
p_{n}}\Big{\rparen}+\frac{2p_{n}}{\log^{2}p_{n}}+O\Big{\lparen}\frac{p_{n}}{\log^{3}p_{n}}\Big{\rparen}+\log\operatorname{rad}(abc)\leq\log
c.$ (3.3)

By the prime number theorem
$n=\operatorname{li}(p_{n})+O(p_{n}/\log^{2}p_{n})$ and (2.1) the leftmost
term becomes

$n\log\bigg{\lparen}\frac{e\delta R}{p_{n}^{2}\big{\lparen}1+1/\log
p_{n}+O(1/\log^{2}p_{n})\big{\rparen}}\bigg{\rparen},$

and with $\log(1+1/x)=1/x+O(1/x^{2})$ as $x\to\infty$, this is

$n\log\Big{\lparen}\frac{e\delta R}{p_{n}^{2}}\Big{\rparen}-\frac{n}{\log
p_{n}}+O\Big{\lparen}\frac{n}{\log^{2}p_{n}}\Big{\rparen}.$

Using (2.1) again on the last two terms and putting this back into (3.3), we
get

$n\log\Big{\lparen}\frac{e\delta
R}{p_{n}^{2}}\Big{\rparen}+\frac{p_{n}}{\log^{2}p_{n}}+O\Big{\lparen}\frac{p_{n}}{\log^{3}p_{n}}\Big{\rparen}+\log\operatorname{rad}(abc)\leq\log
c,$ (3.4)

and our goal becomes to choose $R$ as a function of $n$ to maximize
$n\log(e\delta R/p_{n}^{2})$. Choosing $R$ as asymptotically slow-growing as
possible in terms of $n$ will maximize this in terms of $R$. We must take
$R>p_{n}^{2}/(e\delta)$ for the logarithm to be positive, so we take
$R\coloneqq kp_{n}^{2}$ for some constant $k$. Note that with this choice
$m\sim n\log_{2}n$, so Lemma 3.1 applies. We have that $n\log(e\delta
R/p_{n}^{2})$ simplifies to

$n\log(e\delta k)\sim\frac{p_{n}}{\log p_{n}}\log(e\delta
k)=\frac{\sqrt{R/k}}{\log\sqrt{R/k}}\log(e\delta k)\sim\frac{2\sqrt{R/k}}{\log
R}\log(e\delta k).$

For fixed $R$ this is maximized when $k\coloneqq e/\delta$. Using
$R=ep_{n}^{2}/\delta$ in (3.4),

$2n+\frac{p_{n}}{\log^{2}p_{n}}+O\Big{\lparen}\frac{p_{n}}{\log^{3}p_{n}}\Big{\rparen}+\log\operatorname{rad}(abc)\leq\log
c.$

By the prime number theorem and (2.1) again,

$\frac{2p_{n}}{\log
p_{n}}+\frac{3p_{n}}{\log^{2}p_{n}}+O\Big{\lparen}\frac{p_{n}}{\log^{3}p_{n}}\Big{\rparen}+\log\operatorname{rad}(abc)\leq\log
c.$

Rewriting in terms of $R$,

$\frac{2\sqrt{\delta R/e}}{\log\sqrt{\delta R/e}}+\frac{3\sqrt{\delta
R/e}}{\log^{2}\sqrt{\delta
R/e}}+O\Big{\lparen}\frac{\sqrt{R}}{\log^{3}R}\Big{\rparen}+\log\operatorname{rad}(abc)\leq\log
c.$

Simplifying,

$\frac{4\sqrt{\delta R/e}}{\log(\delta R/e)}+\frac{12\sqrt{\delta
R/e}}{\log^{2}(\delta
R/e)}+O\Big{\lparen}\frac{\sqrt{R}}{\log^{3}R}\Big{\rparen}+\log\operatorname{rad}(abc)\leq\log
c.$

Using $1/(x+y)=1/x-y/x^{2}+O(x^{-3})$ as $x\to\infty$ this gives

$\frac{4\sqrt{\delta R/e}}{\log(R/2)}+\frac{(12-4\log(2\delta/e))\sqrt{\delta
R/e}}{\log^{2}R}+O\Big{\lparen}\frac{\sqrt{R}}{\log^{3}R}\Big{\rparen}+\log\operatorname{rad}(abc)\leq\log
c.$

Using that $2\delta<e^{4}$ the second term on the left is positive, and so for
sufficiently large $R$ the middle two terms are necessarily positive.
Therefore for sufficiently large $R$ this can be simplified to

$\frac{4\sqrt{\delta R/e}}{\log(R/2)}+\log\operatorname{rad}(abc)\leq\log c.$

Using that $2\log c\leq R$ from (3.2) and the increasing monotonicity of
$\sqrt{R}/\log(R/2)$ for sufficiently large $R$, we finally achieve that

$\frac{4\sqrt{2(\delta/e)\log c}}{\log\log
c}+\log\operatorname{rad}(abc)\leq\log c.$

Taking the exponential, this proves the following theorem.

######  Theorem 3.1.

There are infinitely many $abc$ triples satisfying

$\exp\bigg{\lparen}\frac{4\sqrt{2(\delta/e)\log c}}{\log\log
c}\bigg{\rparen}\operatorname{rad}(abc)\leq c.$

Using the permissible value for $\delta$ derived by Rankin’s bound in
Corollary 2.4, the constant in the exponent becomes approximately $6.56338$.
As mentioned in Section 2.3, the best known upper bound on $\delta$ is $2e$,
meaning that the constant in the exponent would become $8$ if this upper bound
was shown to be tight.

## Acknowledgments

The author would like to thank the reviewer for their detailed review and
useful feedback they provided on the first draft of this paper.

## References

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  * [5] D. Micciancio, The shortest vector in a lattice is hard to approximate to within some constant, in Proceedings 39th Annual Symposium on Foundations of Computer Science, IEEE Comput. Soc, 1998, https://doi.org/10.1109/sfcs.1998.743432.
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  * [7] C. P. Schnorr, Factoring integers and computing discrete logarithms via diophantine approximation, in Advances in Cryptology — EUROCRYPT ’91, Springer Berlin Heidelberg, 1991, pp. 281–293, https://doi.org/10.1007/3-540-46416-6_24.
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## Appendix

######  Lemma 2.8.

$\sum_{i=1}^{n}\log p_{i}=n\log
p_{n}-n-p_{n}/\log^{2}p_{n}+O(p_{n}/\log^{3}p_{n})$.

######  Proof 8.

Let $x\coloneqq p_{n}$, so the prime number theorem (with error term) gives
$n=\operatorname{li}(x)+O(x/\log^{4}x)$. Rearranging the asymptotic expansion
of the logarithmic integral (2.1) gives

$\displaystyle x$ $\displaystyle=n\log x-\frac{x}{\log
x}-\frac{2x}{\log^{2}x}+O\Big{\lparen}\frac{x}{\log^{3}x}\Big{\rparen}$
$\displaystyle=n\log
x-n-\frac{x}{\log^{2}x}+O\Big{\lparen}\frac{x}{\log^{3}x}\Big{\rparen}.$

An alternate form of the prime number theorem is $x=\sum_{p\leq x}\log
p+O(x/\log^{3}x)$, so the left-hand side may be replaced by
$\sum_{i=1}^{n}\log p_{i}$ from which the result follows.

######  Lemma 2.9.

$\sum_{i=1}^{n}\log\log p_{i}=n\log\log
p_{n}-p_{n}/\log^{2}p_{n}+O(p_{n}/\log^{3}p_{n})$.

######  Proof 9.

By Abel’s summation formula with $f(k)\coloneqq\log\log k$ and

$a_{k}\coloneqq\begin{cases}1&\text{if $k$ is an odd prime}\\\
0&\text{otherwise}\end{cases}$

for $k$ up to $x\coloneqq p_{n}$, we have

$\sum_{i=1}^{n}\log\log p_{i}=n\log\log x-\int_{2}^{x}\frac{\pi(t)-1}{t\log
t}\,\mathrm{d}t.$

We have $\pi(t)-1=t/\log t+O(t/\log^{2}t)$ by the prime number theorem, so
that

$\int_{2}^{x}\frac{\pi(t)-1}{t\log
t}\,\mathrm{d}t=\int_{2}^{x}\frac{\,\mathrm{d}t}{\log^{2}t}+O\bigg{\lparen}\int_{2}^{x}\frac{\,\mathrm{d}t}{\log^{3}t}\bigg{\rparen}.$

The first integral on the right works out to

$\int_{2}^{x}\frac{\,\mathrm{d}t}{\log^{2}t}=\operatorname{li}(x)-\frac{x}{\log
x}+O(1)=\frac{x}{\log^{2}x}+O\Big{\lparen}\frac{x}{\log^{3}x}\Big{\rparen}$

by the asymptotic expansion of the logarithmic integral. The second integral
on the right can split in two (around $\sqrt{x}$) and then estimated by

$\int_{2}^{\sqrt{x}}\frac{\,\mathrm{d}t}{\log^{3}t}+\int_{\sqrt{x}}^{x}\frac{\,\mathrm{d}t}{\log^{3}t}\leq\frac{\sqrt{x}}{\log^{3}2}+\frac{x-\sqrt{x}}{\log^{3}\sqrt{x}}=O\Big{\lparen}\frac{x}{\log^{3}x}\Big{\rparen}.$

Putting everything together gives

$\sum_{i=1}^{n}\log\log p_{i}=n\log\log
x-\frac{x}{\log^{2}x}+O\Big{\lparen}\frac{x}{\log^{3}x}\Big{\rparen}.$


# Wafer-Scale Fabrication of InGaP-on-Insulator for Nonlinear and Quantum
Photonic Applications

Lillian Thiel<EMAIL_ADDRESS><EMAIL_ADDRESS>Electrical and Computer
Engineering Department, University of California, Santa Barbara, CA 93106
Joshua E. Castro  Electrical and Computer Engineering Department, University
of California, Santa Barbara, CA 93106  Trevor J. Steiner  Materials
Department, University of California, Santa Barbara, CA 93106  Catherine L.
Nguyen  Thorlabs Crystalline Solutions, Santa Barbara, California 93101
Audrey Pechilis  Physics Department, University of California, Santa Barbara,
CA 93106  Liao Duan  Physics Department, University of California, Santa
Barbara, CA 93106  Nicholas Lewis  Electrical and Computer Engineering
Department, University of California, Santa Barbara, CA 93106  Garrett D. Cole
Thorlabs Crystalline Solutions, Santa Barbara, California 93101  John E.
Bowers  Electrical and Computer Engineering Department, University of
California, Santa Barbara, CA 93106  Materials Department, University of
California, Santa Barbara, CA 93106  Galan Moody  Electrical and Computer
Engineering Department, University of California, Santa Barbara, CA 93106

###### Abstract

The development of manufacturable and scalable integrated nonlinear photonic
materials is driving key technologies in diverse areas such as high-speed
communications, signal processing, sensing, and quantum information. Here, we
demonstrate a novel nonlinear platform—InGaP-on-insulator—optimized for
visible-to-telecommunication wavelength $\chi^{\left(2\right)}$ nonlinear
optical processes. In this work, we detail our 100-mm wafer-scale InGaP-on-
insulator fabrication process realized via wafer bonding, optical lithography,
and dry-etching techniques. The resulting wafers yield 1000s of components in
each fabrication cycle, with initial designs that include chip-to-fiber
couplers, 12.5-cm-long nested spiral waveguides, and arrays of microring
resonators with free-spectral ranges spanning 400-900 GHz. We demonstrate
intrinsic resonator quality factors as high as 324,000 (440,000) for single-
resonance (split-resonance) modes near 1550 nm corresponding to 1.56 dB/cm
(1.22 dB/cm) propagation loss. We analyze the loss versus waveguide width and
resonator radius to establish the operating regime for optimal 775 nm – 1550
nm phase matching. By combining the high $\chi^{\left(2\right)}$ and
$\chi^{\left(3\right)}$ optical nonlinearity of InGaP with wafer-scale
fabrication and low propagation loss, these results open promising
possibilities for entangled-photon, multi-photon, and squeezed light
generation.

††preprint: AIP/123-QED

Chip-scale nonlinear quantum light sources that can efficiently generate bi-
photon pairs at high rates and quality are a key component for optical quantum
computing Alexander  _et al._ (2024); Arrazola et al. (2021), entanglement-
based quantum key distribution Steiner  _et al._ (2023); Liu  _et al._ (2022),
quantum time transfer Lafler and Lanning (2023), and quantum sensing Moody et
al. (2022). Integrated photonic microring resonators comprised of nonlinear
optical materials can generate entangled bi-photon pairs via $\chi^{(2)}$
spontaneous parametric down conversion (SPDC) or $\chi^{(3)}$ spontaneous
four-wave mixing (SFWM) processes Kues  _et al._ (2019); Dutt  _et al._
(2024), and they can be readily integrated with scalable chip-based photonic
integrated circuits (PICs). Such sources have been demonstrated on many
different material platforms, all of which face tradeoffs between optimal
material properties and the constraints of existing fabrication
technologiesBaboux et al. (2023), as shown in Fig. 1(a).

Figure 1: a) Comparison of material properties and capabilities for potential
nonlinear quantum light source platforms. b) 100 mm InGaP-on-insulator
(InGaPOI) wafer after full fabrication process. Figure 2: a) InGaPOI process
flow diagram. Step 1: Direct wafer-to-wafer bonding of InGaP to a thermally
oxidized silicon base wafer. Step 2: Removal of GaAs growth substrate via wet
etch. Step 3: Definition of waveguides in InGaP layer via deep-UV optical
lithography and ICP etch. Step 4: ALD and PECVD deposition of oxide cladding
layers. Step 5: Definition of Ti/Pt resistive heaters. b) Atomic Force
Microscopy scan showing rms surface roughness of 0.22 nm. c) Optical
microscope image of ring resonators. d) Scanning electron microscope image of
waveguide cross-section. e) Optical microscope image of nested spirals.

Silicon and silicon nitride photonics have already successfully scaled to mass
production of low-loss PICs Shekhar  _et al._ (2024); Tran et al. (2022), but
they are at a disadvantage compared to other materials when it comes to
optical nonlinearity Moody  _et al._ (2020). Neither material has an intrinsic
$\chi^{(2)}$ response; they also exhibit a weak $\chi^{(3)}$ nonlinearity
(nitride) or suffer from two-photon and free-carrier absorption (silicon) at
1550 nm, which puts a fundamental limit on the performance of quantum light
sources based on these platforms. Many different III-V materials including
Gallium Arsenide (GaAs), Aluminum Gallium Arsenide (AlGaAs), Aluminum Nitride
(AlN), and Indium Phosphide (InP) are attractive for this purpose because of
their strong $\chi^{(2)}$ and $\chi^{(3)}$ responses, their large index
contrast, and, with advances in fabrication techniques, they have enabled
bright entangled-photon pair sources Mobini  _et al._ (2022); Chang  _et al._
(2022); Guo  _et al._ (2017); Kumar  _et al._ (2019). SFWM from AlGaAs-on-
insulator resonators, for example, has enabled $>$ 20 GHz/mW2 pair generation
rates, coincidence-to-accidental (CAR) ratio $>$ 3,000, up to $10^{6}$
detected coincidences per second, better than $99\%$ heralded single-photon
purity, and entanglement visibility $>99\%$ Steiner et al. (2021); Castro  _et
al._ (2022). These chip-scale sources now produce entangled-photon pairs with
rates that are competitive with the best solid-state single-photon emitters
and bulk nonlinear optical systems, but with the inherent scalability,
compactness, and efficiency enabled by room-temperature PIC devices Moody  _et
al._ (2020).

Compared to SFWM, for materials with broken inversion symmetry, the stronger
$\chi^{(2)}$ response can more efficiently generate photon pairs through
spontaneous parametric down-conversion (SPDC) Appas  _et al._ (2022); Placke
_et al._ (2024); Appas  _et al._ (2023). One promising $\chi^{(2)}$ material
is indium gallium phosphide (In0.49Ga0.51P) lattice-matched to GaAs—a III-V
semiconductor with underdeveloped potential. In addition to its high
$\chi^{(2)}$ and $\chi^{(3)}$ nonlinearities ($\chi^{(2)}$ $\approx$ 220 pm/V,
1.5 times larger than AlGaAs and 10 times larger than lithium niobate), InGaP
has a relatively wide bandgap of 1.9 eV (645 nm) compared to many other III-V
materials developed for quantum PICsUeno et al. (1997). As a result, InGaP
devices have already demonstrated efficient pair generation using SFWM and
SPDC in suspended microcavities fabricated with electron-beam lithography Zhao
and Fang (2022); Akin  _et al._ (2024); Chopin  _et al._ (2023). If the
challenge of scalable fabrication of high quality devices can be overcome,
InGaP has the potential to become a leading platform for integrated quantum
photonics. This challenge is nontrivial, as shown by years of work on more
established platforms including silicon, silicon nitride, and AlGaAs. To that
end, we have developed an InGaP-on-insulator (InGaPOI) process that can meet
the requirements of scale and compatibility for integration. We present wafer-
scale fabrication of InGaPOI microring resonators and 12.5-cm-long spirals
with propagation losses as low as 1.22 dB/cm and intrinsic quality factors
greater than 4 $\times$ 105 at 1550 nm, demonstrated for devices across a 100
mm wafer shown in Fig. 1(b). To the best of our knowledge, these results are
the first 100 mm wafer-scale fabrication process demonstrated for InGaP-on-
insulator which can be immediately scaled to 200 mm production with our
existing wafer bonding capabilities, opening exciting prospects for
manufacturing highly nonlinear quantum photonic wafers.

The InGaPOI fabrication process begins with a wafer-scale, low-temperature
plasma-activated bonding process which leverages expertise in substrate-
transferred epitaxial structures for the development of compound semiconductor
on insulator devices Chang  _et al._ (2022) and high-performance precision
laser optics Cole  _et al._ (2023). The InGaP epitaxial wafer (MOCVD-grown by
Twenty-One Semiconductors) was inspected to ensure its suitability for
bonding. Atomic force and scanning optical microscopy were used to confirm RMS
surface microroughness $<$ 1 nm and micrometer-scale defect density $<<1000$
cm-2. The wafer was then thoroughly rinsed in a spin-cleaning system with a
combination of solvents (acetone and isopropanol), a nonionic surfactant, and
deionized water. After cleaning, the wafer was re-inspected to ensure no
particles or remnant organic contaminants remain on its surface.

Directly before bonding, an oxygen plasma activation process was carried out
on both the InGaP and thermal-oxide-on-silicon substrate wafers. After a final
inspection, the two wafers were loaded into a commercial bonding tool (EVG 520
IS semi-automated wafer bonding system). The bond chamber was evacuated to a
pressure $<1\times 10^{-4}$ mbar and the wafers were mechanically pressed
together at room temperature with 9 kN of force. A 12-hour-long anneal at 150
°C was then performed external to the bonder to enhance the interfacial energy
of the contacted wafers.

Figure 3: a) Modal cross-section for the phase-matched SFWM 1550 nm TE00 mode.
b) Modal cross-section for the phase-matched SPDC TM00 mode. c) Modal cross-
section for optimized phase-matched SPDC 1550 nm TE00. d) Waveguide dispersion
near 1550 nm for SFWM designs of InGaP with 400nm thickness. e) Frequency
mismatch versus waveguide width for 102-nm thick InGaP 775 nm – 1550 nm SPDC
designs.

Removal of the GaAs substrate was performed using a NH4OH:H2O2 wet etch
without any prior mechanical lapping of the GaAs wafer. The etch terminates on
the Al0.8Ga0.2As etch stop layer, which is then selectively removed with a
dilute HF-based wet chemical etch. The surface of the 102-nm-thick InGaP
waveguiding layer was thoroughly cleaned and inspected to verify surface
quality and defect density before further processing. For waveguide feature
definition, a 90-nm thick SiO2 hard mask was deposited using atomic layer
deposition (ALD). An anti-reflection coating and photoresist (DUV42P6 and
UV0.8-6) were spun-on and then patterned via deep-UV optical lithography with
an ASML stepper tool. After development, a thermal reflow process was used to
smooth the sidewall roughness in the photoresist that may transfer to the hard
mask and then waveguide profile during etching Xie  _et al._ (2020). The
reflow process must be carefully calibrated to maximize improvement in
sidewall roughness while preserving feature size and uniformity across the
wafer.

After the reflow process, an inductively coupled plasma (ICP) etch was used to
define the features in the hard mask, followed by thorough cleaning to remove
resist and AR coating residue. A BCl3/Cl2/N2 ICP etch was then used to define
the waveguides in the InGaP. Next 30 nm of ALD SiO2 and 1.5 $\mu$m of PECVD
SiO2 were deposited as waveguide cladding, and Ti/Pt resistive heaters were
patterned on top of the cladding above specific regions of devices for thermo-
optic phase tuning. Finally, the wafer underwent a facet-etching process and
was diced into 45 separate die for screening and characterization.

Both SFWM and SPDC processes are possible with InGaP-on-insulator devices;
however, optimal designs for each process require very different waveguide
cross-sections. For any nonlinear process, conservation of energy, phase
matching, high confinement of the optical mode within the waveguide, and
substantial mode overlap are important for optimizing the efficiency. For SFWM
of the fundamental transverse electric (TE) mode, a near-zero dispersion
waveguide design allows for generation of entangled pairs spanning a larger
bandwidth for waveguides and a larger range of frequency bins for a single
resonator. Using the dispersion curves from Tanaka et al.Tanaka  _et al._
(1986), we calculate the waveguide dispersion for the fundamental TE mode at
1550 nm for various SFWM designs and find an optimal cross-section of 400 nm
$\times$ 650 nm (Fig. 3(a) and Fig. 3(d)).

Figure 4: Typical ring resonator transmission spectrum over 1550-1600 nm.
Free-spectral range analysis was used to identified the 1550 nm TE00 modes as
the higher extinction mode family. Inset) Lorentzian fit for resonance with
$Q_{L}$= 213,000, $Q_{i}$= 342,000, $\alpha$ = 1.56 dB/cm Figure 5: a)
Propagation loss versus ring waveguide width for rings with radius = 32
$\mu$m. b) Propagation loss versus radius for rings with 1.45 $\mu$m width.
For a) and b), values are each averaged across identical device designs from
at least three die, looking at device designs with closest to critical
coupling. c) Highest intrinsic quality factors for two devices from each of
nine die across the wafer. On left of each die panel, identical device design
with radius = 32 $\mu$m and width = 1.45 $\mu$m. On right, device with any
design parameters with highest intrinsic quality factor. Devices included have
radii of 32-40 $\mu$m and widths of 1.35-1.55 $\mu$m. Resonances are fit using
the split resonance transfer function described in the text with a sinusoidal
background correction to account for minor facet reflections.

Because of the strong dispersion of InGaP (refractive index $n$ = 3.41 at 775
nm and $n$ = 3.12 at 1550 nm), quasi-phase matching for SPDC can only be
obtained by using high-aspect-ratio cross sections as shown in Fig. 3(b) and
Fig. 3(c) for the fundamental TM mode at 775 nm and fundamental TE mode as
1550 nm, respectively. Considering conservation of energy
($2\omega_{TE,1550}=\omega_{TM,775}$) and the phase-matching condition
$|2m_{TE,1550}-m_{TM,775}|=2$, where $\omega$ is the angular frequency of the
indicated mode and $m$ is the azimuthal number of the mode in a resonator, we
show that in the resonator, the phase-matched waveguide width varies sharply
with waveguide height, presenting additional considerations for designing
resonators for SPDC. For a waveguide height of 102 nm, the ideal width is
$\sim$ 1.2 $\mu$m, although a 2 nm change in waveguide height requires a
$\sim$100 nm change in the waveguide width to achieve phase matching.

With these considerations, we fabricated a full wafer (Fig. 1(b)) that
includes nine nominally identical dies (with five sub-dies on each for a total
of 45 chips) of ring resonator and spiral waveguide designs with various
parameter sweeps. After die singulation, individual chips are tested in a
custom-built PIC testbed with fiber-to-chip end coupling using lensed
polarization-maintaining fibers, chip temperature stabilization at 20 °C,
electrical probes for thermo-optic tuning, and a microscope for probe and
fiber alignment. For each device, light is coupled on and off chip via
waveguide tapers. For resonator devices, pulley coupler designs are used for
waveguide bus-to-ring power transfer. The parameters swept include ring
radius, ring waveguide width, and pulley coupler bus waveguide width and
angle. For the pulley coupler designs, the coupling gap was fixed at 300 nm
and the angle is swept to control the coupling. Transmission spectra for each
device were measured using a continuously tunable laser scanning across
1530-1600 nm. Light coupled off the chip was measured with a fast photodiode
and oscilloscope with a separate fiber-based Mach Zehnder interferometer for
wavelength calibration. In this work, we focus on characterization of the
performance of devices near 1550 nm.

A representative linear transmission spectrum is shown in Fig. 4 for a
resonator with 32 $\mu$m radius and 1.35 $\mu$m width. For SPDC ring designs,
the dimensions required for quasi-phase-matching result in resonators that can
support higher-order TE modes in the telecom wavelength range. For devices
with multiple mode families, the target fundamental TE mode was identified and
isolated by analyzing the free spectral range of each mode family. Resonances
associated with the fundamental TE mode are highlighted in dark blue in the
spectrum shown in Fig. 4. Each resonance is fit using an analytical model
developed by Moresco et al. that takes into account the interplay between
standing-wave modes due to back-scattering that leads to resonance splitting
in some of the devices, and the traveling ring modes in all of the devices
Moresco  _et al._ (2013). From the fits, we can determine the loaded quality
factor $Q_{L}$, intrinsic quality factor $Q_{i}$ that is inversely
proportional to the waveguide propagation loss $\alpha$, and the reflection
coefficient $\rho$ that determines the correlation length and root-mean-square
value of the surface roughness that leads to the back-scatter. For modes
without resonance splitting, fits from the model agree with a Lorentzian fit
to within $\sim 1\%$. The inset to Fig. 4 shows that $Q_{i}$ up to 324,000 is
measured from this particular device corresponding to a propagation loss of
1.56 dB/cm.

Parameter sweeps of ring designs can be used to extract trends in propagation
loss as a function of ring radius and ring waveguide width, both of which help
assess the potential performance of InGaPOI devices for nonlinear frequency
conversion and quantum light generation. Maintaining low propagation loss at
small ring radii enables smaller mode volumes and therefore higher conversion
efficiency. Figure 5(a) shows the average propagation loss measured from three
die across the wafer for increasing microring resonator radius from 20 to 40
$\mu$m. Nominally identical resonators from each die with 1.45 $\mu$m
waveguide widths are chosen for device designs closest to critical coupling to
the bus waveguide. With increasing radius the average propagation loss
decreases from $\sim 5.4$ dB/cm to $\sim 2.4$ dB/cm. From this curve, we
estimate that for radii below $\sim 20-25$ $\mu$m, scattering from the
resonator due to greater mode overlap with the sidewall begins to dominate
over material absorption loss and governs the loaded quality factor $Q_{L}$.

We next examined the average propagation loss for different resonator
waveguide widths, shown in Fig. 5(b). For increasing width, the loss decreases
down to $\sim 2.5$ dB/cm as a result of less mode overlap with the etched
waveguide sidewalls. For both data from the radius and width sweeps in Fig. 5,
error bars are shown as the standard deviation from the mean of at least three
different nominally identical devices. For smaller radius and width, larger
device-to-device variation is observed due to the larger mode interaction with
the sidewalls. In these designs, fabrication process variations have a greater
impact on scattering and thus the measured $Q$. Figure 5(b) illustrates that
low propagation loss over a range of ring waveguide widths ensures that the
high device performance demonstrated here reflects that of device designs that
also meet the phase-matching requirements for this waveguide thickness.

Finally, we show in Fig. 5(c) linear transmission plots from the highest-$Q$
devices from each die across the wafer, ranging from 194,000 to 440,000 (not
necessarily similar resonator FSR or waveguide widths). We see that for some
of the highest-$Q$ resonators, the resonances are split due to backscattering,
which is more apparent with increasing $Q$ as expected from the model by
Moresco et al. discussed previously. In Table 1, we compare the linear
propagation loss determined from these measurements to those reported for
other $\chi^{\left(2\right)}$ photonic platforms that have been developed
specifically for nonlinear optical conversion and pair generation processes.

Notably, while the loss from many other platforms have reached sub-dB/cm,
excluding AlGaAsOI they are all fabricated using electron-beam lithography
(EBL), compared to 100-mm wafer scale fabrication using deep-UV
photolithography in our study. The 1.22 dB/cm propagation loss at 1550 nm
reported here is the lowest amongst the most recently proposed heterogeneously
integrated nonlinear platforms (ScAlNOI and GaNOI), with strong outlook for
improvements in device performance. Surface passivation of InGaP with an Al2O3
layer has been shown to provide a $3\times$ improvement in the intrinsic
quality factor at telecom wavelengthsAkin  _et al._ (2024). Use of deuterated
SiO2 has been shown to reduce material absorption from the cladding layer by
$\sim 7\times$ at 1550 nm and has the potential to provide performance
improvement across the visible wavelength range Jin  _et al._ (2020).

Table 1: A comparison of the 1550 nm linear propagation loss for different $\chi^{\left(2\right)}$ integrated photonic platforms developed for nonlinear frequency conversion and quantum light generation. The GaNOI loss is estimated from the published data; the other data are reported from the listed citations. DUV-PL: deep-UV photolithography. EBL: electron-beam lithography. Nonlinear | Propagation Loss | Fabrication
---|---|---
Material | @ 1550 nm | Method
InGaPOI (this work) | 1.22 dB/cm | DUV-PL, 100-mm wafer
Suspended InGaP Akin  _et al._ (2024) | 0.8 dB/cm | EBL
AlGaAsOI Xie  _et al._ (2020) | 0.2 dB/cm | DUV-PL, 100-mm wafer
LNOI Luo  _et al._ (2018) | 0.5 dB/cm | EBL
GaNOI Zeng  _et al._ (2024) |  $\sim$2 dB/cm | EBL
ScAlNOI Friedman  _et al._ (2024) | 1.6 dB/cm | EBL
SiC Guidry  _et al._ (2020) | 0.38 dB/cm | EBL

In conclusion, we presented a new InGaP-on-insulator 100-mm wafer-scale
fabrication process with 200 nm feature sizes using direct wafer bonding and
substrate removal, deep-UV photolithography, and dry etching methods. Each
wafer produces 1000s of photonic components with waveguide propagation loss
down to 1.22 dB/cm and 1550 nm resonator quality factors up to 440,000.
Numerical and experimental characterization of the loss and dispersion with
waveguide cross section establishes the optimal regime for SFWM and SPDC
nonlinear processes. Considering the large $\chi^{\left(2\right)}$ and
$\chi^{\left(3\right)}$ nonlinearities, wide bandgap (1.9 eV) and high
refractive index ($>3$), these results establish InGaP as a promising
candidate for a wide range of nonlinear photonic applications, including
second harmonic generation, quantum frequency conversion, broadband entangled-
pair generation, and squeezed light generation.

###### Acknowledgements.

A portion of this work was performed in the UCSB Nanofabrication Facility, an
open access laboratory. We thank Ifeanyi Achu, Navin Lingaraju, Joseph
Christesen, and Cale Gentry from SRI International for technical input and
assistance in device modeling and designs. Wafers were obtained from Twenty-
One Semiconductors GmbH. This work was supported by the NSF Quantum Foundry
through the Q-AMASE-i Program (Grant No. DMR-1906325), the NSF CAREER Program
(Grant No. 2045246), and the Air Force Office of Scientific Research (Grant
No. FA9550-23-1-0525). L.T. acknowledges support from the NSF Graduate
Research Fellowship Program.

## Data Availability Statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

## Author Declarations

The authors have no conflicts to disclose.

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