Quasinormal modes of loop quantum black holes near the Planck scale

Douglas M. Gingrich

Department of Physics, University of Alberta, Edmonton, AB T6G 2E1 Canada
TRIUMF, Vancouver, BC V6T 2A3 Canada

e-mail<EMAIL_ADDRESS>
###### Abstract

We compare four loop quantum gravity inspired black hole metrics near the
Planck scale. Spin 0, 1/2, 1, and 2 field perturbations on these backgrounds
are studied. The axial gravitational quasinormal modes are calculated and
compared. The time evolution of the ringdown is examined. We also calculate
the quasinormal modes in the eikonal limit and compare with the predictions
from circular null geodesics.

##  1 Introduction

A number of black hole exterior solutions have been obtained under the loop
quantum gravity (LQG) paradigm [1, 2, 3, 4, 5, 6, 7]. Each is unique, offering
potentially interesting phenomenological implications.

LQG solves the singularity issue of the classical theory by replacing it with
an effective geometry with curvature of the order of the Planck scale. The
theory has a fundamental discreteness referred to as the area gap. The
effective solution is first obtained in a symmetry reduced phase space
framework and then expressed as a quantum corrected spacetime metric.
Polymerization is used in the quantization procedure, where some variables are
replaced by their exponentiated versions.

When comparing different black holes, common geometry parameters need to be
chosen consistently. In addition, a common black hole characteristic depending
on the mass, such as the mass, horizon radius, Hawking temperature, etc. needs
to be chosen in the calculations. We will compare non-rotating and uncharged
black holes, and so choose a common Arnowitt-Deser-Misner (ADM) mass.

Choosing a large mass corresponding to astronomical, primordial, or even small
macroscopic black holes will lead to insignificant different properties from
Schwarzschild black holes for reasonable choices of the parameters. To
emphasize the unique features of the LQG black holes, we would like to choose
the Planck mass $m_{\mathrm{P}}$ as the common mass value. However, the LQG
metrics are an effective solution as they only approximate a full quantum
theory of gravity. The theory does not incorporate the full quantum gravity
effects that are needed near the Planck mass. For macroscopic black hole
masses $M\gtrsim 10^{6}m_{\mathrm{P}}$, the effective theories are valid. LQG
corrections that depend on the mass can become very small for macroscopic
black holes and are usually negligible for solar mass black holes. However, a
full understanding of such quantum corrected black holes requires
investigation of features that are observable in principle if not in practice.
The results presented here will depend on the choice of black hole mass.

For example, the potentials due to spin 0, 1/2, 1, and 2 perturbations on the
fixed black hole backgrounds are indistinguishable for the four different
metrics we consider for $M\gtrsim 1000m_{\mathrm{P}}$. For $M\approx
100m_{\mathrm{P}}$, one of the potentials begins to decrease slightly at the
peak. By $M\approx 10m_{\mathrm{P}}$, a second potential starts to decrease at
the peak. Only when $M\approx 2m_{\mathrm{P}}$, do three of the potentials
deviate from the Schwarzschild case. One of the potentials remain identical to
the Schwarzschild case even when $M=m_{\mathrm{P}}$, as will be explained
later.

When working near the Planck scale it is important to check any approximations
assuming large masses. For calculational purposes, it will occasional be
necessary to make some approximations which will require a mass larger than
the Planck mass, yet can still be significantly below macroscopic black hole
masses.

The theme of this paper is to calculate and compare the eigenvalue solutions
due to perturbations on LQG black hole background metrics. A perturbed black
hole is a dissipative system with a set of eigenmodes called the quasinormal
(QNM) modes. The frequencies depend only on the black hole parameters while
the amplitudes depend on the source causing the excitations. The ringdown
stage of a gravitational wave burst can be modeled by black hole perturbation
theory and is related to the QNMs. The eikonal regime is also examined and
compared with predictions using circular null geodesics. It is important to
confront these prediction with LQG effective solutions [8]. Most previous work
on this topic study a single black hole for different choices of its
parameters. In this work, we fix the parameters and compare the black holes.

There is a large literature on these topics in the context of modified gravity
theory, thus showing their importance. Beside comparing the LQG cases among
themselves, the standard Schwarzschild solution is always presented alongside.
The comparisons with the metric by Modesto [1] will be different. First, the
phenomenology of this metric has been well studied [9, 10, 11, 12, 13, 14].
Secondly, this metric has even been confronted with astronomical data to set
limits on its quantum parameters [15, 16, 17]. We use these constraints and
thus the Modesto metric behaves Schwarzschild-like. It is not our business
here to make value judgments of one black hole solution over another, and we
mean no disrespect to the authors of many recent black hole solutions not
discussed here.

The layout of this paper is as follows. In section 2, we write down the
effective metric functions and discuss their parameters. The critical radii
are discussed and the tortoise coordinate that will be needed in some
subsequent calculations is given. Section 3 discusses the choice for the
parameters used in the studies and the reasons for their choice. In section 4,
we write down the potentials due to different spin-field perturbations on the
background metrics. The potentials will be the basis for subsequent
calculations and indicate the level of difference to anticipate in subsequent
results. QNMs are calculated in section 5. Ringdown and long-time tails are
shown in section 6. The eikonal limit is examined in section 7. At the end of
the paper the most important results are summarized and discussed.

##  2 Black hole metric functions and definitions

In this section, we summarize the metric functions considered and define their
parameters. We state the horizon radii and give the tortoise relationships.
The general static and spherically symmetric metric can be written as

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{g(r)}+h(r)d\Omega^{2}.$ (1)

Throughout, we work in geometric units of $G=c=1$ and express masses in units
the Planck mass and distances in units of the Planck length. Also, when taking
derivatives of metric functions, we use the continuous approximation.

###  2.1 Kelly-Santacruz-Wilson-Erwing (KSW) metric

Kelly, Santacruz, and Wilson-Ewing [6] have obtained a vacuum spherically
symmetric spacetime using an effective framework for the $\bar{\mu}$ scheme of
holonomy corrections by imposing the areal gauge in the classical theory and
then expressing the remaining components of the Ashtekar-Barbero connection in
the Hamiltonian constraint in terms of holonomies of the Planck length. The
resulting metric functions are

$f(r)=g(r)=1-\frac{r_{\mathrm{S}}}{r}\left[1-\frac{\gamma^{2}\Delta
r_{\mathrm{S}}}{r^{3}}\right]\quad\text{and}\quad h(r)=r^{2},$ (2)

where $r_{\mathrm{S}}=2M$ and $M$ is the black hole ADM mass. The parameters
are the Barbero-Immirzi parameter $\gamma$ and the LQG gap parameter $\Delta$
of the full theory. This is the simplest metric we will examine as it involve
only one $1/r^{4}$ quantum correction term to $f(r)$ and $g(r)$, no correction
to $h(r)$, and effectively has only one free parameter, the combination
$\gamma^{2}\Delta$.

Two, one, or zero horizons are possible. The horizon equation is quartic and
we will use the numerical solution. To help aid understanding, expanding about
$\gamma^{2}\Delta/r_{\mathrm{S}}^{2}$ gives the following approximations. The
outer horizon radius is approximately

$r_{+}=\left[1-\frac{\gamma^{2}\Delta}{r_{\mathrm{S}}^{2}}-3\left(\frac{\gamma^{2}\Delta}{r_{\mathrm{S}}^{2}}\right)^{2}+\mathcal{O}\left(\left(\frac{\gamma^{2}\Delta}{r_{\mathrm{S}}^{2}}\right)^{3}\right)\right]r_{\mathrm{S}}\,.$
(3)

The inner horizon radius is approximately

$r_{-}=\left[\left(\frac{\gamma^{2}\Delta}{r_{\mathrm{S}}^{2}}\right)^{1/3}+\frac{1}{3}\left(\frac{\gamma^{2}\Delta}{r_{\mathrm{S}}^{2}}\right)^{2/3}+\frac{1}{3}\left(\frac{\gamma^{2}\Delta}{r_{\mathrm{S}}^{2}}\right)+\mathcal{O}\left(\left(\frac{\gamma^{2}\Delta}{r_{\mathrm{S}}^{2}}\right)^{4/3}\right)\right]r_{\mathrm{S}}\,.$
(4)

A minimum radius exists, given by

$r_{\mathrm{min}}=\left(\frac{\gamma^{2}\Delta}{r_{\mathrm{S}}^{2}}\right)^{1/3}r_{\mathrm{S}}\,.$
(5)

A horizon radius exits, provided

$M\geq\left(\frac{2}{\sqrt{3}}\right)^{3}\sqrt{\gamma^{2}\Delta}.$ (6)

The use of the above metric functions is restricted by the condition on the
minimum mass and radius.

The tortoise relation is

$\frac{dr_{*}}{dr}=\frac{1}{f(r)}.$ (7)

This form of the metric was also obtained more recently using a different
method [18].

###  2.2 Gambini-Olmedo-Pullin (GOP) metric

Gambini, Olmedo, and Pullin [5] obtain an interesting black hole metric based
on improved dynamics [19]. They include the ADM mass and its conjugate
momentum in the scalar constraint and represent the scalar constraint as an
operator in the kinematical Hilbert space. By applying group averaging
techniques for both the quantum scalar constraint and the group of finite
spatial diffeomorphisms they obtain the physical states, labeled by the ADM
mass of the spin network. Using some parameterized observables that act as
local operators on each vertex of the spin network they define the physical
observables denoting spacetime metric components. The resulting lowest-order
metric is111We note a missing factor of 2 in the cross term in Eq. 5.3 of [5].

$\displaystyle f(r)$ $\displaystyle=$ $\displaystyle
1-\frac{r_{\mathrm{S}}}{r+r_{0}}+\frac{r_{0}^{3}r_{\mathrm{S}}^{3}}{(r+r_{0})^{6}(1+\frac{r_{s}}{r+r_{0}})^{2}},\quad
g(r)=f(r)\left[1+\frac{\delta r}{2(r+r_{0})}\right]^{-2},$ $\displaystyle
h(r)$ $\displaystyle=$ $\displaystyle
r^{2}\left[1+\frac{r_{0}}{r}\right]^{2}=r^{2}+2r_{0}r+r_{0}^{2},$ (8)

where $r_{\mathrm{S}}=2M_{0}$ and

$r_{0}=\left(\frac{2M_{0}\Delta}{4\pi}\right)^{1/3}.$ (9)

The $M_{0}$ parameter is related to the ADM mass by $M=M_{0}+\delta r/2$,
where $\delta r$ is an independent quantization parameter. The coordinate $r$
is discretized on a lattice with spacing $\delta r$ which typically would be
chosen as the Planck length, $\delta{r}=\ell_{\mathrm{P}}$.

The GOP authors have taken $\gamma=1$ and their definition of the gap
parameter in terms of the KSW gap parameter is
$\Delta_{\mathrm{GOP}}(\gamma=1)\to
4\pi\gamma^{2}\Delta_{\mathrm{KWS}}(\gamma)$. Thus GOP has the same minimum
radius $r_{0}=(\gamma^{2}\Delta r_{\mathrm{S}})^{1/3}$ as KSW, accept
$M_{0}\neq M$ unless $\delta r\to 0$. The parameter $\delta r$ is a choice,
but $r_{0}$ is dictated by $M$ and $\delta r$.

The horizon radius is given by solving a sixth-order polynomial of which there
is only one positive real root for our choice of parameters, which is given
approximately by

$r_{\mathrm{h}}=\left[1-\frac{r_{0}}{r_{\mathrm{S}}}-\frac{1}{4}\left(\frac{r_{0}}{r_{\mathrm{S}}}\right)^{3}+\mathcal{O}\left(\left(\frac{r_{0}}{r_{\mathrm{S}}}\right)^{6}\right)\right]r_{\mathrm{S}}\,.$
(10)

We will use the numerical solution in calculations. The GOP horizon radius
will be significantly smaller than the Schwarzschild radius $2M$.

The tortoise relation is

$\frac{dr_{*}}{dr}=\left[1+\frac{\delta r}{2(r+r_{0})}\right]\frac{1}{f(r)}.$
(11)

###  2.3 Ashtekar-Olmedo-Singh (AOS) metric

In the Ashtekar, Olmedo, and Singh [20, 21, 4] approach, the plaquettes are
chosen to lie on the transition surface, that replaces the classical
singularity and separates the trapped region from the anti-trapped region in
the quantum extension of the Schwarzschild interior. The polymerization scales
$\delta_{b}$ and $\delta_{c}$ are chosen to be appropriate Dirac observables
and a convenient choice of the laps function determines explicit solutions to
the effective dynamics. The mass that characterizes the solution is a constant
of the motion. The effective spacetime structure is then extended to the
exterior region. The metric functions are

$\displaystyle f(r)$ $\displaystyle=$
$\displaystyle\frac{\left[2+\epsilon+\epsilon\left(\frac{r_{\mathrm{S}}}{r}\right)^{1+\epsilon}\right]^{2}}{\left[4(1+\epsilon)^{2}\left(\frac{r_{\mathrm{S}}}{r}\right)^{\epsilon}\right]^{2}}\frac{\left[1-\left(\frac{r_{\mathrm{S}}}{r}\right)^{1+\epsilon}\right]\left[(2+\epsilon)^{2}-\epsilon^{2}\left(\frac{r_{\mathrm{S}}}{r}\right)^{1+\epsilon}\right]}{\left[1+\left(\frac{Lr_{\mathrm{S}}}{r^{2}}\right)^{2}\right]},$
(12) $\displaystyle g(r)$ $\displaystyle=$
$\displaystyle\frac{1}{\left[2+\epsilon+\epsilon\left(\frac{r_{s}}{r}\right)^{1+\epsilon}\right]^{2}}\frac{\left[1-\left(\frac{r_{\mathrm{S}}}{r}\right)^{1+\epsilon}\right]\left[(2+\epsilon)^{2}-\epsilon^{2}\left(\frac{r_{\mathrm{S}}}{r}\right)^{1+\epsilon}\right]}{\left[1+\left(\frac{Lr_{\mathrm{S}}}{r^{2}}\right)^{2}\right]},$
(13) $\displaystyle h(r)$ $\displaystyle=$ $\displaystyle
r^{2}\left[1+\left(\frac{Lr_{\mathrm{S}}}{r^{2}}\right)^{2}\right]=r^{2}+\left(\frac{Lr_{\mathrm{S}}}{r}\right)^{2},$
(14)

where the polymerization scale $\delta_{b}$ is given by

$\epsilon=\sqrt{1+\gamma^{2}\delta_{b}^{2}}-1\qquad\textrm{and}\qquad\delta_{b}=\left(\frac{\sqrt{\Delta}}{\sqrt{2\pi}\gamma^{2}M}\right)^{1/3}\,.$
(15)

The polymerization scale $\delta_{c}$ is given by $L=\gamma
L_{0}\delta_{c}/4$, where

$L_{0}\delta_{c}=\frac{1}{2}\left(\frac{\gamma\Delta^{2}}{4\pi^{2}M}\right)^{1/3},$
(16)

and $L_{0}$ is an infrared regulator. None of the physical results depend on
$L_{0}$ but on combinations with other parameters which are all independent of
the choice of $L_{0}$.

These expressions for the effective metric functions are exact. The effective
metric approaches a flat metric as $1/r$ but the curvature invariants fall-off
slower than that of the Schwarzschild metric. The ADM energy is well-defined
and its value is $M$.

The outer event horizon is $r_{\mathrm{S}}=2M$ and the inner Cauchy horizon is
negligibly small $\sim\epsilon^{2}r_{\mathrm{S}}$. The metric has two
parameters which are fixed by LQG parameters and the mass parameter. The
exponent parameter $\epsilon$ which decreases with mass could have
astrophysical consequences, while the length parameter $L$ is only important
near the Planck scale.

The tortoise relation is

$\frac{dr_{*}}{dr}=\frac{4(1+\epsilon)^{2}\left(\frac{r_{\mathrm{S}}}{r}\right)^{\epsilon}\left[1+\left(\frac{Lr_{\mathrm{S}}}{r^{2}}\right)^{2}\right]}{\left[1-\left(\frac{r_{\mathrm{S}}}{r}\right)^{1+\epsilon}\right]\left[(2+\epsilon)^{2}-\epsilon^{2}\left(\frac{r_{\mathrm{S}}}{r}\right)^{1+\epsilon}\right]}.$
(17)

As $r\to\infty$,

$\frac{dr_{*}}{dr}\to\left[\frac{2(1+\epsilon)}{2+\epsilon}\right]^{2}\left(\frac{r_{\mathrm{S}}}{r}\right)^{\epsilon},$
(18)

which will have important consequences later.

###  2.4 Modesto metric

The earliest loop quantum black hole metric we will consider is by Modesto
[1]. The spherically symmetric Hamiltonian constraint is modified in terms of
holonomies. A large class of semiclassical solutions parameterized by a
generic function of the polymeric parameter are obtained. The function choice
is obtained by picking the one that reproduces the Schwarzschild black hole
solution outside the black hole with the correct asymptotic flat limit. The
metric has been shown to have a $T$-duality and is often referred to as the
self-dual metric. Of the metrics we consider, it has been the most extensively
studied and has a rich phenomenology [9, 10, 11, 12, 13, 14]. The metric
functions are commonly written in the Reissner-Nordström form

$\displaystyle f(r)$ $\displaystyle=$
$\displaystyle\frac{(r-r_{+})(r-r_{-})}{r^{4}+a_{0}^{2}}(r+r_{*})^{2},\quad
g(r)=\frac{(r-r_{+})(r-r_{-})}{r^{4}+a_{0}^{2}}\frac{r^{4}}{(r+r_{*})^{2}},$
(19) $\displaystyle h(r)$ $\displaystyle=$ $\displaystyle
r^{2}\left[1+\left(\frac{a_{0}}{r^{2}}\right)^{2}\right]=r^{2}+\left(\frac{a_{0}}{r}\right)^{2},$
(20)

where $r_{\mathrm{h}}=r_{+}$ is the outer horizon radius, $r_{-}$ is a Cauchy
inner radius, and $r_{*}=\sqrt{r_{+}r_{-}}$; $a_{0}$ is related to the gap
parameter in LQG by $a_{0}=\Delta/(8\pi)$.

The outer and inner horizons, and $r_{*}$ are given by

$r_{+}=\frac{1}{(1+P)^{2}}r_{\mathrm{S}},\quad
r_{-}=\frac{P^{2}}{(1+P)^{2}}r_{\mathrm{S}},\quad
r_{*}=\frac{P}{(1+P)^{2}}r_{\mathrm{S}}.$ (21)

The polymeric function is

$P=\frac{\sqrt{1+\epsilon^{2}}-1}{\sqrt{1+\epsilon^{2}}+1},$ (22)

where $\epsilon=\gamma\delta$, with $\gamma$ is the Barbero-Immirzi parameter
and $\delta$ is the polymeric parameter. The parameter $\delta$ and hence $P$
is a free choice. The metric has two parameters which are independent of black
hole mass. The polymeric parameter could have astrophysical significance,
while the gap parameter is only important near the Planck scale.

The tortoise relation is

$\frac{dr_{*}}{dr}=\frac{r^{4}+a_{0}^{2}}{(r-r_{+})(r-r_{-})r^{2}},$ (23)

which can be analytically integrated to give

$r_{*}=r-\frac{a_{0}^{2}}{r_{+}r_{-}}\left[\frac{1}{r}-\frac{r_{+}+r_{-}}{r_{+}r_{-}}\ln(r)\right]+\frac{1}{r_{+}-r_{-}}\left[\frac{a_{0}^{2}+r_{+}^{4}}{r_{+}^{2}}\ln(r-r_{+})-\frac{a_{0}^{2}+r_{-}^{4}}{r_{-}}\ln(r-r_{-})\right].$
(24)

We now compare the metric functions and tortoise relations. Figure 1(left)
shows the metric functions $f(r)$. Outside the horizon only the GOP metric is
noticeably different from the others. Only GOP has a single horizon. Figure
1(right) shows the radial metric functions $h(r)$. AOS and Modesto show
bounce. GOP has a minimum $h(r)=r_{0}^{2}$ at $r=0$. Except for GOP, $h(r)$ is
negligibly different for $r\gtrsim\ell_{\mathrm{P}}$. GOP $h(r)$ is
significantly different over the $r$ we will consider.

Figure 1: (left) metric functions $f(r)$ and (right) metric functions $h(r)$,
for $M=3m_{\mathrm{P}}$.

Figure 2(left) shows the tortoise coordinate relationships. Three different
$dr_{*}/dr$ behaviors are observed. GOP and AOS are mostly different due to
$f(r)\neq g(r)$.

Analytic forms for the Schwarzschild and Modesto tortoise coordinate $r_{*}$
exist. For the others we use the analytic approximation

$r_{*}=\frac{1}{f^{\prime}(r_{\mathrm{h}})}\ln(r-r_{\mathrm{h}}),+\left.\int\frac{dr_{*}}{dr}\right|_{r\to\infty}dr,$
(25)

where the constant has been set to zero. The first term determines the
behavior as $r\to r_{\mathrm{h}}$ and the second term as $r\to\infty$. The
second term integrates to $r$ for all metrics except AOS which goes as $\sim
r^{1-\epsilon}$. This approximation does well near $r\to r_{\mathrm{h}}$ and
$r\to\infty$. Figure 2(right) shows the tortoise coordinate versus $r$. The
GOP $r_{*}$ is significantly shifted in $r$ due to its lower horizon radius.
The AOS $r_{*}$ is significantly different due to its large $r$ behavior.

Figure 2: (left) tortoise relations and (right) tortoise coordinates, for
$M=3m_{\mathrm{P}}$.

##  3 Choice of parameters

In this section, we discuss our choice of parameters that are subsequently
used in calculations. Some common LQG parameter values that we use are
$\gamma=0.2375$ for the Barbero-Immirzi parameter [22] and
$\Delta=4\pi\sqrt{3}\gamma\ell_{\mathrm{P}}^{2}=5.169\ell_{\mathrm{P}}^{2}$
for the gap parameter. All the metrics contain these parameters so we use
common values.

The KSW parameter combination is
$\gamma^{2}\Delta=4\pi\sqrt{3}\gamma^{3}\ell_{\mathrm{P}}^{2}=0.292\ell_{\mathrm{P}}^{2}$.
The minimum mass to have a horizon radius is $M\geq 0.8m_{\mathrm{P}}$, which
we satisfy.

For the GOP parameters, if $M$ and $\delta r$ are in Planck units,

$r_{0}=\gamma\left[8\pi\sqrt{3}\left(M-\frac{\delta
r}{2}\right)\right]^{1/3}\ell_{\mathrm{P}}=0.8355\left(M-\frac{\delta
r}{2}\right)^{1/3}\ell_{\mathrm{P}}\,.$ (26)

The restriction $\ell_{\mathrm{P}}^{2}/(2r_{0})\leq\delta r\leq r_{0}$ [5]
means $M\gtrsim 2.1m_{\mathrm{P}}$. Assuming $\delta r\geq\ell_{\mathrm{P}}$,
we will take $\delta r=\ell_{\mathrm{P}}$. Hence we pick $M=3m_{\mathrm{P}}$
as our working mass based on the free choice of the discretization parameter
$\delta r$. For $M=3m_{\mathrm{P}}$, $r_{0}=1.134\ell_{\mathrm{P}}$.

For the AOS parameters,

$\gamma^{2}\delta_{b}^{2}=\left(\frac{\gamma^{2}\Delta}{2\pi
M^{2}}\right)^{1/3}=(2\sqrt{3})^{1/3}\gamma\left(\frac{m_{\mathrm{P}}}{M}\right)^{2/3}\,.$
(27)

For $M=3m_{\mathrm{P}}$, $\gamma^{2}\delta_{b}^{2}=(2\sqrt{3})^{1/3}\gamma$
and $\epsilon=0.0829$. Also,

$L=\frac{1}{8}\left(\frac{12}{M}\right)^{1/3}\gamma^{2}\ell_{\mathrm{P}}\,.$
(28)

For $M=3m_{\mathrm{P}}$,
$L=\frac{12^{1/3}}{8}\gamma^{2}\ell_{\mathrm{P}}=0.0112\ell_{\mathrm{P}}$.

For the Modesto polymeric function, we consider constraints from data [15, 16,
17]. Since $P$ is expected to be a universal parameter for all black hole
masses these limits are applicable to our case of a mass near the Planck
scale. The most recent upper limit is $P<6.17\times 10^{-3}$ on the polymeric
function and $|\delta|<0.67$ on the polymeric parameter, at the 95% confidence
level [17]. We take this limit as our parameter value for $P$. The minimum
area parameter is
$a_{0}=\Delta/(8\pi)=\sqrt{3}/2\gamma\ell_{\mathrm{P}}^{2}=0.206\ell_{\mathrm{P}}^{2}$.

All the parameters used in this paper are summarized in Table 1. The
parameters $\gamma$ and $\Delta$ are fixed. $\delta r$ is a choice, while $P$
is based on constraints to data. The remaining parameters are fixed by the
choice of the black hole mass.

The horizon radii using these parameters are shown in Table 2. Also shown in
Table 2 are radii of circular null geodesics $r_{c}$ and shadow radii
$R_{\mathrm{s}}$, which we will discuss late. Figure 3 shows the distance
scales for $M=3m_{\mathrm{P}}$. We notice that $L$ of AOS and $r_{-}$ of
Modesto are negligible. The two GOP parameters $r_{0}$ and $\delta r$ are both
significant. The Modesto parameter $a_{0}$ is not particularly significant.

Table 1: Parameters values. If the parameter depends on mass, $M=3m_{\mathrm{P}}$ has been taken. Parameter | Value
---|---
$\gamma^{2}\Delta$ | $0.292\ell_{\mathrm{P}}^{2}$
$r_{0}$ | $1.134\ell_{\mathrm{P}}$
$\delta r$ | $\ell_{\mathrm{P}}$
$\epsilon$ | $0.0829$
$L$ | $0.0112\ell_{\mathrm{P}}$
$P$ | $0.00617$
$a_{0}$ | $0.206\ell_{\mathrm{P}}^{2}$
Table 2: Characteristic radii in units of $\ell_{\mathrm{P}}$ for $M=3m_{\mathrm{P}}$: $r_{\mathrm{h}}$ is the outer horizon radius, $r_{c}$ is the circular null geodesic radius, and $R_{\mathrm{s}}$ is the shadow radius. | $r_{\mathrm{h}}$ | $r_{c}$ | $R_{\mathrm{s}}$
---|---|---|---
Schwarzschild | 6 | 9 | 15.59
KSW | 5.95 | 8.96 | 15.55
GOP | 3.85 | 6.36 | 12.98
AOS | 6 | 9.12 | 15.39
Modesto | 5.93 | 8.88 | 15.33
Figure 3: Distance scales for $M=3m_{\mathrm{P}}$.

##  4 Potentials

The perturbation due to a field of spin $s$ on a black hole background
spacetime can be written in the Schrödinger-like form as

$\left[\frac{d^{2}}{dr_{*}^{2}}+\omega^{2}-V_{s}(r)\right]\psi(r_{*})=0,$ (29)

where $\omega$ is the quasinormal mode frequency and $r_{*}$ is the tortoise
coordinate. The potentials for spin 0, 1/2, 1, and 2 can be written as [12]

$\displaystyle V_{0}(r)$ $\displaystyle=$
$\displaystyle\nu_{0}\frac{f}{h}+\frac{\partial_{*}^{2}\sqrt{h}}{\sqrt{h}},$
(30) $\displaystyle V_{1/2}(r)$ $\displaystyle=$
$\displaystyle\nu_{1/2}\frac{f}{h}\pm\sqrt{\nu_{12}}\partial_{*}\sqrt{\frac{f}{h}},$
(31) $\displaystyle V_{1}(r)$ $\displaystyle=$
$\displaystyle\nu_{1}\frac{f}{h},$ (32) $\displaystyle V_{2}(r)$
$\displaystyle=$
$\displaystyle\nu_{2}\frac{f}{h}+\frac{(\partial_{*}h)^{2}}{2h^{2}}-\frac{\partial_{*}^{2}\sqrt{h}}{\sqrt{h}},$
(33)

where $\nu_{0}=\nu_{1}=\ell(\ell+1)$, $\nu_{1/2}=\ell(\ell+1)+1/4$, and
$\nu_{2}=\ell(\ell+1)-2$. The metrics functions $f$ and $h$ are functions of
$r$, and the derivatives are with respect to $r_{*}$. The first term in the
different potentials is the same to within an $\ell$-dependent constant. The
$\ell(\ell+1)$ piece of the first term is the “centrifugal term”.

Figure 4 and 5 show the potentials versus $r$ and versus $r_{*}$,
respectively. The potentials in Fig. 4 do not depend on the metric function
$g(r)$, while the potentials in Fig. 5 only depend on the $g(r)$ through the
tortoise coordinate. The different behavior of the GOP potentials are
noticeable. The AOS potentials also noticeably approach the Schwarzschild
potentials slower with increasing $r$. The greatest difference is for $s=0$
and the smallest difference is for $s=2$. For $s=\ell=0$, the first term in
the potential vanishes and only depends on $h(r)$.

Figure 4: Potentials versus $r$ for spin 0, 1/2, 1, and 2 perturbations on the
loop quantum gravity black hole background metrics. $M=3m_{\mathrm{P}}$ has
been used. Figure 5: Potentials versus $r_{*}$ for spin 0, 1/2, 1, and 2
perturbations on the loop quantum gravity black hole background metrics.
$M=3m_{\mathrm{P}}$ has been used.

The gravitational perturbations of a spherically symmetric black hole can be
divided into odd (axial) parity and even (polar) parity. In what follows, we
only consider the axial symmetric gravitational perturbations.

##  5 Quasinormal modes

In this section, we compute the axial gravitational QNMs. There are many
methods that can be used to solve for the eigenvalues due to the
perturbations. We employ the Wentzel-Kramers-Brillouin (WKB) approximation
method [23, 24] and use the following generic formula to order $p$:

$\omega^{2}=V_{0}-i\sqrt{-2V_{0}^{\prime\prime}}\left[n+\frac{1}{2}+\sum_{i=2}^{p}\Lambda_{i}\right],\quad
n=0,1,\ldots\,,$ (34)

where $V_{0}^{\prime\prime}$ is the second derivative of $V(r_{*})$ calculated
at the maximum of the potential, and $n$ is the overtone number. The method
depends only on $\ell$ through the potential. The correction terms
$\Lambda_{i}$ represent the higher orders of approximation and depend on
higher-order derivatives of the potential.

The WKB method is valid only for asymptotically flat spacetime, or for
spacetime admitting wavelike solutions at spatial infinity and is not
applicable for $\ell<n$. The WKB formula converges only asymptotically and
does not guarantee convergence in each order. The method is not particularly
accurate for low $\ell$ and high $n$. We employ a third-order WKB method
beyond the eikonal approximation [25], which uses up to sixth-order
derivatives in the correction terms. Higher-order methods exist [26, 27].
Although many QNM studies use multiple methods, that is not our intent here.
For qualitative comparison purposes, the third-order WKB approximation is
adequate.

For the GOP, AOS, and Modesto potentials, the number of terms in the sixth
derivative is beyond our computational abilities. For the GOP case, we make a
change of variable $r+r_{0}\to r$, approximate $f(r)$ to order $1/r^{4}$, and
expand the Jacobian $dr/dr_{*}$ to first order in $\delta r/r$, which is good
to better than 1%. For the AOS case, we use the improved small $\epsilon$
approximation of Ref. [28] which is good to order $\epsilon$. For the Modesto
case, we approximate $r_{-}/r\to 0$. These are the only analytical
approximations made in this paper and are only made when calculating the
derivatives for the WKB method.

The QNMs are shown in Fig. 6. The blocks of points are $\ell=2$, $\ell=3$, and
$\ell=4$ when going from left to right, while $n$ increases from bottom to top
as $0\leq n<\ell$. We reproduce the Schwarzschild QNMs for all modes examined,
and agree well with previous results for the other cases where available. The
GOP values agree with Ref. [29]. That paper looks at massless scalar,
electromagnetic, axial gravitational perturbations. The paper uses $M_{0}=10$
and $\delta r$ very small, which makes quantitative comparisons difficult. The
AOS values agree with Ref. [28], which considers scalar perturbations. GOP and
AOS black holes oscillate with higher frequency and less damping than
Schwarzschild black holes. We found no modes with positive damping, which
indicates these LQG black holes are stable against massless axial
gravitational perturbations.

Figure 6: Axial gravity quasinormal modes for loop quantum gravity black holes
for $\ell=2,3,4$ and $0\leq n<\ell$. The points for a given metric have $\ell$
values that increase from left to right and $n$ values that increase from
bottom to top.

##  6 Ringdown waveforms

In this section, we calculate the time-domain profile of an axial-
gravitational perturbation on the background given by the black hole metrics.
We follow the method used by Gundlach, et al [30].

In a finite time domain, the Schrödinger-like Eq. 29 can be written as

$\left[\frac{\partial^{2}}{\partial r_{*}^{2}}-\frac{\partial^{2}}{\partial
t^{2}}-V(r_{*})\right]\Psi(t,r_{*})=0.$ (35)

Using light-cone coordinates, $u=t-r_{*}$ (retarded time) and $v=t+r_{*}$
(advanced time), we obtain

$\left[4\frac{\partial^{2}}{\partial u\partial v}-V(u,v)\right]\Psi(u,v)=0.$
(36)

We take the perturbation as a Gaussian wavepacket centered at $v_{c}=10$ with
a width $\sigma=3$. Without loss of generality, we take the initial conditions
as

$\Psi(0,v)=\exp\left[-\frac{(v-v_{c})^{2}}{2\sigma^{2}}\right]\quad\text{and}\quad\Psi(u,0)=\Psi(0,0).$
(37)

The observer is placed at future timelike infinity, which we approximate as
$r=10r_{\mathrm{h}}$.

The partial differential equation is solved numerically using forward
differences. We take $\Delta v=\Delta u=0.1$, and note that taking $0.05$
provides no visible benefit. For the purpose of brevity, we choose units in
which $r_{\mathrm{h}}=1$. This means $r,r_{*},t$ are expressed in units of
$r_{\mathrm{h}}$. The QNM frequency $\omega$ is in units of
$r_{\mathrm{h}}^{-1}$ and units for the QNM potential $V$ are
$r_{\mathrm{h}}^{-2}$. We have scaled the LQG black hole parameters
accordingly. This is allowed since the wave equation is linear and we are not
interested in the amplitude.

To validate the procedure, we first consider a Schwarzschild black hole and
obtain the ringdown waveform. We observe the usual power-law tail at late
time. The result is in agreement with Ref. [14], and others. Figure 7 shows
the ringdown waveforms for the LQG black holes. The KSW, GOP, and Modesto
curves are similar to Schwarzschild case (not shown). The AOS ringdown curve
is significantly different.

Figure 7: (top-left) KSW, (top-right) GOP, (bottom-left) AOS, (bottom-right)
Modesto ringdown waveforms of a Gaussian wavepacket at $r=10r_{\mathrm{h}}$.

There are at least three features worth considering in these ringdown curves.
The time at which the ringing stops is visually the same for all black holes
except AOS which rings considerably longer. This is due to the significantly
different tortoise relationship. A second feature is the oscillation
frequency. It is not our purpose in presenting these curves to have an
alternative means of calculating the QNMs. However, we simply fit the region
$100\leq t/M\leq 200$, to the function $Ae^{\omega_{I}t}\sin(\omega_{R}+B)$
and obtain consistent results to the WKB determination, both for real and
imaginary parts of the frequency; thus partly validating the ringing behavior
in the curves. The last feature we will comment on is the power-law tails. The
long-time tails are linear on a log-log plot. By performing power-law fits in
the region $500\leq t/M\leq 580$ we obtain a power of $-7.25$ for all ringdown
curves except AOS for which we obtain $-4.4$, with significantly more
uncertainty. Since our potential is for $s=2$, we can not compare the power-
law behavior exactly with Ref. [30] but point out that we obtain a consistent
$1/4$ extra contribution to their case of $\ell=2$ with no initial static
field. The exceptional power-law behavior of AOS is due to the unique
asymptotic behavior at large $r$. The AOS tail appears less smooth. As a
check, we used the metric approximation of Ref. [28] but obtained the same
behavior.

##  7 Eikonal limit and circular null geodesics

In this section, we calculate the quasinormal modes in the eikonal, or
geometric optics, limit using the WKB approximation and compare with the
quasinormal modes predicted by circular null geodesics. The real part of the
complex QNM frequency can be determined by the angular velocity at the
unstable null geodesic and the imaginary part can be related to the
instability time scale of the orbit. It is important to test these
relationships for LQG black holes [31].

Based on the conditions of a stationary, spherically symmetric, and
asymptotically flat metric, Ref [32], and others, have derived the QNM
frequency from the circular null geodesic:

$\omega=\left(\ell+\frac{1}{2}\right)\Omega_{c}-i\left(n+\frac{1}{2}\right)|\lambda_{c}|,$
(38)

where the subscript $c$ means that the functions are evaluated at the radius
of the circular null geodesic, and $\Omega_{c}$ and $\lambda_{c}$ are the
coordinate angular velocity and the Lyapunov exponent of the circular null
geodesic, respectively. The principal Lyapunov exponent is the inverse
instability time scale associated with the geodesic motion. It is a measure of
the average rate at which nearby trajectories converge or diverge in phase
space. A positive Lyapunov exponent indicates a divergence between nearby
trajectories. The WKB results are accurate in the eikonal regime. Equation
(38) has been shown to be identical to the first-order WKB approximation [32].

The radius of the null geodesic $r_{c}$, is obtained by solving
$f_{c}h_{c}^{\prime}=f_{c}^{\prime}h_{c}$, where the prime means
differentiation with respect to $r$. The values of $r_{c}$ can be found in
Table 2. For the Schwarzschild case, $r_{c}=3M$.

The coordinate angular velocity is

$\Omega_{c}=\sqrt{\frac{f_{c}}{h_{c}}},$ (39)

and the positive Lyapunov exponent is

$\lambda_{c}=\sqrt{\frac{g_{c}}{2h_{c}}(f_{c}h_{c}^{\prime\prime}-f_{c}^{\prime\prime}h_{c})}.$
(40)

In the eikonal regime ($\ell\gg 0$), the potentials for all spin-field
perturbations is approximately

$V(r)\approx\ell^{2}\frac{f}{h}.$ (41)

Thus the results obtained by solving the Schrödinger-like wave Eq. 29 are
independent of the spin-field perturbation and so we only consider the
gravitational case. For GOP, this potential does not depend on $\delta r$.

Setting $\ell=1000$ (approximating infinity) and $n=0$ to $4$, we calculate
the QNM frequencies in the eikonal limit using the WKB approximation method
and using Eq. (38). The results are shown in Fig. 8 and the agreement between
the two methods is good for all metrics. The GOP eikonal results are
significantly different due to the smaller horizon radius and the others are
marginally different.

Figure 8: QNMs of the axial gravitational perturbation for $\ell=1000$ and
$n=0$ to 4 calculated by (solid markers) the WKB approximation method and
(dashed lines) the QNM circular null geodesic (CNG) relations for $M=3m_{p}$.
For CNG, the real frequencies are independent of $n$ and the lines between the
markers are simply to aid the eye.

For free, we calculate the shadow radius

$R_{s}=\frac{1}{\Omega_{c}}$ (42)

of the black holes, which corresponds to the angular semi-diameter of the
shadow around a black hole as seen by an observer at spatial infinity. The
results are shown in Table 2. For the Schwarzschild case,
$R_{\mathrm{s}}=3\sqrt{3}M$. The GOP is significantly different due to the
smaller CNG radius.

##  8 Discussion

We have compared four LQG black hole metrics at $M=3m_{\mathrm{P}}$ using the
LQG parameters. The KSW metric adds a maximum contribution to the metric
function $f(r)$ of 0.8%. KSW appears identical to Schwarzschild in terms of
potentials, QNMs, and ringdown. Differences of 0.2% and 0.5% in the eikonal
limit are observed for the real and imaginary parts, respectively. To observe
the quantum effects of the KSW metric, one needs to consider black hole masses
closer to the Planck scale.

The GOP black hole is the most different compared to the other black holes. It
is the only metric we examined with an arbitrary free parameter $\delta r$
which we choose to be the Planck length. This causes the ADM mass, and hence
horizon radius, to be about a factor of 0.6 small than the other black holes.
This is a main source of difference. The validity of the solution forces the
ADM mass to be greater than $2.1m_{\mathrm{P}}$. The QNMs are 20% and $-12$%
different in the eikonal limit for real and imaginary parts, respectively.

The main difference for AOS black holes is due to the asymptotic behavior at
large $r$. However, QNMs are mostly independent of shifts in $r$. The QNMs in
the eikonal limit differ by 1.1% and 0.8% in the real and imaginary part,
respectively. The main difference is in the ringdown. The ring lasts about 1.6
times longer, with a smaller long-time power-law slope.

At $M=3m_{\mathrm{P}}$ and using a data-constrained polymeric function value,
the Modesto metric adds a maximum contribution to the metric function $f(r)$
of 2.5% and the potential is marginally higher at the peak. Modesto appears
similar to Schwarzschild in terms of QNMs and ringdown. Differences of 1.7%
and $-1.4$% in the eikonal limit are observed for the real and imaginary
parts, respectively. The ability to observe the quantum effects of the Modesto
metric is unlikely, and one needs to consider black hole masses closer to the
Planck scale.

We have calculate and compare the eigenvalue solutions due to perturbations on
LQG black hole background metrics. Not all possible LQG black hole metrics
have been considered, and others are worthy of investigation. The GOP and AOS
metrics are deserving of further study. The parameters of these metrics could
be constraint by data in the way the Modesto polymeric function is
constrained. This is the first QNM comparison between LQG black holes.

## Acknowledgments

I thank Saeed Rastgoo for pointing out some of the LQG metrics and helpful
discussions. We acknowledge the support of the Natural Sciences and
Engineering Research Council of Canada (NSERC). Nous remercions le Conseil de
recherches en sciences naturelles et en génie du Canada (CRSNG) de son
soutien.

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# Asymmetrical temporal dynamics of topological edge modes in Su-Schrieffer-
Heeger lattice with Kerr nonlinearity

Ghada Alharbi  School of Physics and Astronomy, Cardiff University, Cardiff
CF24 3AA, United Kingdom  Department of Physics, University of Tabuk, Tabuk
741, Kingdom of Saudi Arabia  Stephan Wong  School of Physics and Astronomy,
Cardiff University, Cardiff CF24 3AA, United Kingdom  Center for Integrated
Nanotechnologies, Sandia National Laboratories, Albuquerque, New Mexico 87185,
USA  Yongkang Gong  School of Physics and Astronomy, Cardiff University,
Cardiff CF24 3AA, United Kingdom  Sang Soon Oh<EMAIL_ADDRESS>School of
Physics and Astronomy, Cardiff University, Cardiff CF24 3AA, United Kingdom

###### Abstract

Optical bistability and oscillating phases exist in a Sagnac interferometer
and a single ring resonator made of $\chi^{(3)}$ nonlinear medium where the
refractive indices are modulated by the light intensity due to the Kerr
nonlinearity. An array of coupled nonlinear ring resonators behave similarly
but with more complexity due to the presence of the additional couplings.
Here, we theoretically demonstrate the bifurcation of topological edge modes
which leads to optical bistability in the Su-Schrieffer-Heeger lattice with
the Kerr nonlinearity. Additionally, we demonstrate periodic and chaotic
switching behaviors in an oscillating phase resulting from the coupling
between the topological edge mode and bulk modes with different chiralities,
i.e., clockwise and counter-clockwise circulations.

bi-stability, Kerr effect, topological photonics

##  I Introduction

Asymmetrical states emerge when a system lose the balance and thus a symmetry
between different components is broken. This can lead to bistability, where
the system has two stable states for a single excitation. In some cases,
asymmetrical dynamic states can emerge with periodic or chaotic oscillatory
behaviors. In photonic systems, optical bistability can appear when the light
transmits through a cavity with a nonlinear medium leading to two different
optical states, where one mode is dominant (switched on) and the other is
quenched (switched off) [1]. For instance, the stable symmetry breaking has
been studied for counter-propagating light beams in a Sagnac interferometer
[2] and micro-resonator with Kerr nonlinearity [3, 4, 5, 6, 7]. More
interestingly, nonlinear optical ring resonators can present rich temporal
dynamics with oscillatory behaviors, displaying various types of mode
switching, such as chaotic, periodic, and self-switching dynamics [8]. These
various dynamics are the result of the nonlinear interaction between the
counter-propagating modes and they manifest the symmetry breaking, i.e.,
unequal intensities of the two counter-propagating modes.

Recently, topologically protected modes have been widely studied in photonic
systems due to their intriguing properties, such as unidirectional light
propagation and robustness to defects and disorders [9, 10]. In particular,
topological edge modes in a one-dimensional (1D) Su-Schrieffer-Heeger (SSH)
configuration and two-dimensional (2D) photonic quantum spin-Hall or quantum
valley-Hall structures have been employed in an array of coupled lasers,
namely topological lasers [11, 12, 13, 14, 15, 16]. Moreover, nonlinear
topological photonics has been studied in various platforms such as waveguide
arrays [17, 18, 19, 20], microcavity polariton systems [21] and optical
resonators [22, 23, 24, 23, 25, 26, 27, 28]. The phase diagrams of a nonlinear
SSH model and nonlinear breathing kagome model were drawn for the nonlinear
parameters and coupling coefficients between sites [26]. Also, the edge
solitons have shown to be stable at any energy when the ratio between the weak
and strong couplings falls below a critical value [29]. Up until now, however,
no research has demonstrated spontaneous symmetry breaking coming from the
nonlinear response for the edge modes in photonic topological insulators.

In this paper, we theoretically show that we can observe asymmetrical temporal
dynamics, including optical bistabilities and oscillation phases for
topological edge modes in a nonlinear 1D SSH model. The system consists of an
array of coupled ring resonators with the Kerr nonlinearity. Using the
Lugiato-Lefever equation [30] with additional nearest neighbor couplings, we
demonstrate the optical bistability of the topological edge mode in the
nonlinear SSH lattice. Finally, we use Poincaré section plots, composed of the
maxima of the oscillating intensities, to display the oscillation phases
featuring periodic and chaotic switching.

##  II Linear SSH model with two counter-propagating modes

We start by considering a linear SSH chain which does not have any resonance
frequency shift coming from a nonlinearity. As shown in Fig. 1(a), the one-
dimensional chain has $(N+1)$ unit cells and every unit cell hosts two ring
resonators; one on the sublattice A, and the other on the sublattice B. The
$(N+1)$-th unit cell has only one ring resonator that belongs to the
sublattice A, resulting in $M=2N+1$ ring resonators in total. In photonics,
this SSH model can be implemented by alternating the gap size between the ring
resonators, resulting in different intra- and inter-cell coupling coefficients
$v$ and $w$ (Fig. 1(b)). The coupled ring resonators are excited by two
optical pumps with the same intensity, both of which are coupled into the
first ring resonator but in the opposite directions exciting clockwise (CW)
and counter-clockwise (CCW) modes, respectively. Then, the optical waves
propagate back and forth through all the resonators via the couplings between
the ring resonators.

Figure 1: (a) A 1D array of coupled ring resonators with alternating gap
sizes. $s_{in,\pm}$ is the amplitude of input beams. (b) A schematic of the
SSH model with resonators with Kerr nonlinearity. $B$ is the XPM strength, $v$
and $w$ are the nearest neighbor coupling coefficients between ring
resonators, and $\gamma_{c}$ is the coupling coefficient between the waveguide
and the first ring resonator.

To calculate the intensities of circulating optical waves at all ring
resonators, we describe the time evolution of the field amplitudes $a_{n}(t)$
and $b_{n}(t)$ in the $n$-th unit cell (Fig. 1(b)) for which we use the
temporal coupled mode theory [31, 32]. Then, the coupled mode equations are
written as:

$\displaystyle\frac{da_{n,\pm}}{dt}$ $\displaystyle=$ $\displaystyle
i\Big{(}\omega_{0}+i\gamma_{n}\Big{)}a_{n,\pm}+ivb_{n,\mp}+iwb_{n-1,\mp}$
$\displaystyle+\delta_{n,1}\gamma_{c}s_{in},$
$\displaystyle\frac{db_{n,\pm}}{dt}$ $\displaystyle=$ $\displaystyle
i\Big{(}\omega_{0}+i\gamma^{\prime}_{n}\Big{)}b_{n,\pm}+iva_{n,\mp}+iwa_{n+1,\mp},$
(1)

where

$\displaystyle\gamma_{n}$ $\displaystyle=$
$\displaystyle\gamma_{0}+\delta_{n,1}\gamma_{c},$
$\displaystyle\gamma^{\prime}_{n}$ $\displaystyle=$ $\displaystyle\gamma_{0}.$
(2)

Here, the subscript $\pm$ denotes the mode propagation directions, CW and CCW,
respectively. $\delta_{n,1}$ is the Kronecker delta and $\omega_{0}$ is the
resonance frequency of the uncoupled ring resonators. The two input beams with
the same amplitude $s_{in}$, which is given as $\sqrt{I_{s}}e^{i\omega t}$ for
pump intensity $I_{s}$, are coupled to the CW and CCW modes in the first ring
($a_{1,\pm}$) with the waveguide-to-ring coupling coefficient $\gamma_{c}$.
Note that only $a_{n}$’s and $b_{n}$’s are time-dependent functions and we
have omitted the symbol $(t)$ for brevity.

To be more compact, we express the coupled mode equations (Eq. (1)) in a
matrix form by using the Hamiltonian $\mathbf{H}$ as

$\displaystyle\frac{d\mathbf{x}}{dt}=\mathbf{H}\mathbf{x}+\mathbf{S}$ (3)

where

$\mathbf{x}=(a_{1,+},b_{1,-},a_{2,+},b_{2,-},\ldots,a_{1,-},b_{1,+},a_{2,-},b_{2,+},\ldots)^{\mathsf{T}}.$

Note that the Hamiltonian $\mathbf{H}$ can be differently defined after
multiplying $i$ in both sides, which makes the equation look like the
Schrödinger equation and makes its eigenvalues correspond to the real parts of
the frequencies. However, we have chosen this notation to make Eq. (3) similar
to Lugiato-Lefever equation which we will explain in the following section.
Then, the Hamiltonian $\mathbf{H}$ can be split into two terms like:

$\mathbf{H}=\mathbf{H}_{0}+\mathbf{H}_{c}$ (4)

where

$\mathbf{H}_{0}=i(\omega_{0}+i\gamma_{0})\mathbb{I}_{M}$ (5)

with $\mathbb{I}_{M}$ the $(M\times M)$ identity matrix. The ring-to-ring
coupling is expressed as:

$\mathbf{H}_{c}=i\begin{pmatrix}0&v&0&0&\cdots\\\ v&0&w&0&\cdots\\\
0&w&0&v&\cdots\\\ 0&0&v&0&\cdots\\\
\vdots&\vdots&\vdots&\vdots&\ddots\end{pmatrix}.$ (6)

Finally, the source term $\mathbf{S}$ is expressed as
$[s_{in,+},0,0,\cdots,s_{in,-},0,0,\cdots]^{\mathsf{T}}$. For the remainder of
this paper, we assume the symmetric pumping by setting $s_{in,+}=s_{in,-}$.

The coupled mode equations (Eq. (3)) can be solved in both frequency and time
domains. For example, in the frequency domain, by assuming
$\mathbf{x}=\mathbf{\tilde{x}}\exp(i\omega t)$ and $\mathbf{S}=0$, we obtain
an eigenvalue equation

$i\omega\mathbf{\tilde{x}}=\mathbf{H}\mathbf{\tilde{x}}.$ (7)

Solving the eigenvalue equation gives an frequency spectrum with so-called
zero-energy modes that are topologically protected and localized on one of the
edges of the SSH chain with the smaller coupling coefficient among $v$ and
$w$. In this work, we will use the term _edge modes_ because their frequencies
deviate from the resonance frequency $\omega_{0}$ and thus they are not any
more zero-energy modes for nonlinear cases. We call the rest of the modes
_bulk modes_ as the mode fields are delocalized over the entire SSH lattice.

##  III Lugiato-Lefever equation for nonlinear SSH model

Now we introduce the Kerr nonlinearity in the linear SSH model. In optics, the
Kerr nonlinearity induces various nonlinear effects, for instance, self-phase
modulation (SPM), cross-phase modulation (XPM), four-wave mixing, and two-
photon absorption [33]. Here, we only consider the SPM and XPM for the
counter-propating rotating modes in the ring resonators, both of which lead to
a shift of the resonance frequencies of the CW or CCW modes. Although only the
couplings due to the XPM are shown in Fig. 1(b), the frequency shift
$\Delta\omega$ is expressed by $(AI_{n,+}+BI_{n,-})$ where $I_{n,+}$ and
$I_{n,-}$ are the intensities of the CW and CCW modes in the $n$-th ring
resonator, respectively. $A$ and $B$ are the SPM and XPM nonlinear
coefficients, respectively.

For simplicity, we use the normalized Lugiato-Lefever equation to describe the
field amplitudes in our nonlinear SSH model [30]. Notably, the equation is
equivalent to the one derived from the temporal coupled-mode theory (see the
Appendix A for more details). With the time-varying envelope amplitudes
$\tilde{a}(t)$, $\tilde{b}(t)$, defined as $a(t)=\tilde{a}(t)e^{i\omega t}$,
$b(t)=\tilde{b}(t)e^{i\omega t}$ respectively, the Lugiato-Lefever equation
for a 1D SSH array of nonlinear ring resonators can be written as:

$\displaystyle\frac{d\tilde{a}_{n,\pm}}{d\bar{t}}$ $\displaystyle=$
$\displaystyle-\tilde{a}_{n,\pm}-i\eta\Delta\tilde{a}_{n,\pm}+i\eta(A|\tilde{a}_{n,\pm}|^{2}+B|\tilde{a}_{n,\mp}|^{2})\tilde{a}_{n,\pm}$
$\displaystyle+iv\tilde{b}_{n\mp}+iw\tilde{b}_{n-1,\mp}+\delta_{n,1}s_{in},$
$\displaystyle\frac{d\tilde{b}_{n,\pm}}{d\bar{t}}$ $\displaystyle=$
$\displaystyle-\tilde{b}_{n,\pm}-i\eta\Delta\tilde{b}_{n,\pm}+i\eta(A|\tilde{b}_{n,\pm}|^{2}+B|\tilde{b}_{n,\mp}|^{2})\tilde{b}_{n,\pm}$
(8) $\displaystyle+iv\tilde{a}_{n\mp}+iw\tilde{a}_{n+1,\mp},$

where $\bar{t}=t\gamma_{0}$ is the dimensionless time. The first term on the
right-hand side represents damping, while the second term stands for detuning
($\Delta=(\omega-\omega_{0})/\gamma_{0}$), which is the difference between the
frequency of the continuous wave input beams and the resonance frequency of a
single ring resonator. The third and fourth terms correspond to the SPM and
XPM, respectively, with the normalized nonlinear coefficients $A$ and $B$, and
$\eta=+1$ for a self-focusing medium or $\eta=-1$ for a self-defocusing
medium. The terms with $v$ and $w$ refer to the intra- and inter- couplings
between ring resonators as in the linear case. Finally, we add a nonlinear
term to Eq. (4) to have the Hamiltonian for our nonlinear SSH model:

$\mathbf{H}=\mathbf{H}_{0}+\mathbf{H}_{c}+\mathbf{H}_{NL},$ (9)

where

$\mathbf{H}_{0}=-(1+i\eta\Delta)\mathbb{I}_{M},$ (10)

and

$\mathbf{H}_{NL}=i\eta\mathbb{I}_{M}\times\begin{pmatrix}A|\tilde{a}_{1,+}|^{2}+B|\tilde{a}_{1,-}|^{2}\\\
A|\tilde{b}_{1,-}|^{2}+B|\tilde{b}_{1,+}|^{2}\\\
A|\tilde{a}_{2,+}|^{2}+B|\tilde{a}_{2,-}|^{2}\\\
A|\tilde{b}_{2,-}|^{2}+B|\tilde{b}_{2,+}|^{2}\\\ \vdots\\\
A|\tilde{a}_{1,-}|^{2}+B|\tilde{a}_{1,+}|^{2}\\\
A|\tilde{b}_{1,+}|^{2}+B|\tilde{b}_{1,-}|^{2}\\\
A|\tilde{a}_{2,-}|^{2}+B|\tilde{a}_{2,+}|^{2}\\\
A|\tilde{b}_{2,+}|^{2}+B|\tilde{b}_{2,-}|^{2}\\\ \vdots\end{pmatrix}.$ (11)

To obtain the temporal evolution of the amplitudes for nonlinear SSH lattice,
we can solve the time-dependent equation (Eq.(3)) with this Hamiltonian.
However, we cannot solve the equation in the frequency domain by simply
solviing an eigenvalue equation because it is a system of nonlinear equations.

Figure 2: (a) Optical bistability for seven ring resonators ($M=7$) with
detuning $\Delta=1.85$, $v=3$ and $w=7$. (b) Optical bistability with the Kerr
nonlinearity in the first ring resonator only. (c), (d) The distributions of
intensity for the input intensities corresponding to the dashed vertical lines
in (a)

##  IV Optical bistability of topological edge modes

Optical bistability in a single ring resonator is a result of the XPM between
two counter-propagating modes [34]. This means that the Kerr nonlinearity
leads to a shift in the resonance frequency of the two counter-propagating
modes due to both SPM and XPM, and the coupling via XPM between them leads to
spontaneous symmetry breaking above a certain threshold pump intensity [3].
Here, we want to address the question whether we can observe the optical
bistability using an edge mode in a nonlinear SSH lattice model.

To theoretically observe the optical bistability in the nonlinear SSH lattice,
we consider a SSH array of seven ring resonators ($M=7$) with the nonlinear
parameter and the detuning in Ref. [34] and the alternating coupling
coefficients ($v=3$ and $w=7$). In our simulations, we scan the pump intensity
$I_{s}$ for a certain interval with random initial conditions. Indeed, as
shown in Fig. 2(a), we observe the optical bistability in the range between
$\log(I_{s}+1)=1.9$ and $\log(I_{s}+1)=2.9$ of pump intensity. Note that the
field amplitudes are relatively large for odd sites only (sublattice A), and
the intensity decreases exponentially along the right direction for both
single stable (Fig. 2(c)) and bistable cases (Fig. 2 (d)), which is the
reminiscence of the zero-energy edge modes.

Figure 3: Poincaré section of the maxima of the CW and CCW intensity time
series for the first ring resonator in 1D SSH lattice composed seven ring
resonators with alternating coupling, $v=3$ and $w=7$. (a) Maximum intensity
curves for low input intensities $I_{s}=0.05$ (cyan), $0.1$ (green), $0.2$
(blue), $0.4$ (red), $0.6$ (yellow), $0.8$ (brown). (b) For high input
intensities $I_{s}=2$ (red), $6$ (green), $10$ (blue), $14$ (cyan), $16$
(yellow), $20$ (brown), both optical bistability and oscillation regions
appear. The vertical dashed lines indicate frequencies determined by the
eigenvalue equation for the linear case (Eq.(7)).

To explain the origin of the observed optical bistability, we hypothesize that
the optical bistability comes from the symmetry breaking in the first ring
only. First, optical bistability in a single ring resonator can occur when the
pump intensity is above a certain threshold, called a bifurcation point. This
means the first ring will show the optical bistability first as we increase
the pump intensity under an excitation close to the zero-energy frequency.
Indeed, the detuning $\Delta=1.85$ is smaller than the topological band gap
($2|v-w|=8$) meaning the zero-energy edge mode is dominantly excited even
though it is off-resonance. This is supported by the field intensity
distribution in Fig. 2(d). As the intensities in the rest of rings are much
smaller than the first ring (Fig. 2(d)), only the first ring introduces
bistability and the modes in the first ring couple to the other rings
successively instead of having additional optical bistability from the rest of
the rings. Second, to confirm this propagation, we consider the Kerr effect
only in the first ring resonator but keep all other parameters the same. This
is equivalent to switching off the Kerr effect in the $2N$ ring resonators
except the first ring resonator in our original setting. As shown in Fig.
2(b), the intensity-intensity curve has almost identical shape as the original
one except slight reduction in the range of $I_{s}$ and slight change in the
difference between two counter-propagating mode intensities.

Figure 4: Poincaré sections of the maxima of oscillating coupled intensity as
a function of detuning for input intensity $I_{s}=20$, for the first ring
resonator from the 1D SSH array. The Poincaré sections for coupling
coefficients ($v=3$, $w=7$) and XPM strength ($B$) of 4 and 7 in (a) and (b),
respectively. The red shading indicates a symmetric case, the yellow indicates
optical bistability, and the cyan indicates oscillations. The vertical dashed
lines refer to resonance frequencies for the linear 1D SSH lattice. Figure 5:
Snapshots of intensity distribution in the nonlinear 1D SSH lattice for the CW
modes in the left column and the CCW modes in the right column for different
values of detuning with $I_{s}=20$, $B=4$. Figure 6: Time series of intensity
and their phase space trajectories for $A=1$, $B=4$ and $I_{s}=20$ at
different values of detuning, for the first ring resonators from 1 D SSH array
of 7 rings. (a), (b) Optical bistability phase with $\Delta=5.51$. (c), (d)
Oscillations without overlapping trajectories. (e), (f) Periodic switching for
$\Delta=5.93$. (g), (h) Chaotic switching with $\Delta=7.73$

##  V Asymmetrical temporal dynamics

Now, let us look at the temporal evolution of the optical intensities of a 1D
SSH array that contains seven ring resonators. To visualize oscillation and
chaotic phases in our nonlinear system, we will use the Poincaré section
obtained by plotting all the local maxima in a time series of oscillating
intensities [8]. Since the intensity of each ring in the 1D SSH array follows
the same pattern as the intensity of the first ring (see Fig. A1 in Appendix
B), we plot the Poincaré sections for the first ring resonator only.

As shown in Fig. 3 (a)(b), the nonlinear SSH lattice exhibits both bistability
and oscillation phases in the range of detuning corresponding to the edge mode
and two bulk modes with positive detuning for the linear SSH lattice (denoted
as the vertical dashed lines). Here, we set $A=1$ and $B=4$ and change the
input intensity from 0.05 to 20 denoted with different colors. As we can see
in the zoomed view of the plots, these spectra show seven resonance modes; one
edge mode with the largest intensity in the middle and six bulk modes on both
sides of the edge mode having three on each side. For low input intensities
(Fig. 3(a)), the edge mode shifts dramatically and its intensity increases
significantly, whereas the bulk modes shift less and their intensities
increase slightly. This is due to the localization of intensity at the first
ring resonator. For high input intensities (Fig. 3(b)), the CW and CCW modes
for edge mode undergo an interaction between them via XPM, leading to an
optical bistability. Remarkably, the high input intensity leads to the
interaction between the edge mode and the bulk mode near $\Delta=5.2$,
resulting in a series of oscillation phases occurring for both CW and CCW
modes.

The range of detuning of asymmetrical phases also depends on the XPM strength.
Figure 4 (a) and (b) compare the Poincaré sections for two different values of
XPM strength $B$. One can observe that increasing $B$ leads to a larger range
of detuning for asymmetrical phases including optical bistability (yellow) and
oscillation phases (blue). The Poincaré section as a function of XPM strength
can be found in Fig. A2 in Appendix C.

To better understand the asymmetrical dynamic modes, we show the spatial
distributions and temporal changes of the excited mode intensity for different
detuning in Fig. 5 and Fig. 6. Here, we focus on the case of $I_{s}=20$ and
$B=4$ as an example of high input intensity. For the optical bistability
($\Delta=4.77$) shown in Fig. 5(a),(b), both CW and CCW modes have contrasting
intensity values, whereas their profiles are similar to the zero-energy edge
mode’s profile in a linear SSH model with exponentially decaying non-zero odd-
site intensities and zero even-site intensities. The deviations can be
attributed to the off-resonance excitation and the interaction between CW and
CCW modes via the nonlinear process (XPM). Note that the excited mode is
stable as they have constant intensities and appear as two separate points its
phase space (Fig. 6 (a),(b)). When we increase the detuning further to
$\Delta=5.51$, both CW and CCW mode profile deviates further away from the
zero-energy edge mode but the CCW mode profile deviates less still having low
intensities at even sites (Fig. 5(c),(d)). Here, the largest value at the
first site is related to the zero-energy edge mode and also due to the fact we
are exciting the ring resonators from the waveguide on the left side. The
dynamics for this detuning (Fig. 6 (c)(d)) is periodically oscillatory,
resulting in two distinct regions in the phase space meaning the CW mode
intensity is always larger than the CCW mode intensity (the trajectory for CW
is further away from the origin). For slightly larger detuning of
$\Delta=5.93$, the two trajectories are merged into one meaning that the
intensities between the two modes alternates. In the phase space, they are in
different two points with the $\pi$ phase difference in the same trajectory.
For a large detuning of $\Delta=7.73$, we see chaotic oscillations showing two
separate trajectories covering a similar region in the phase space (Fig.
6(g),(h)).

In contrast to the optical bistability coming from the coupling between two
counter-propagating modes via nonlinearity, the emergence of the periodic and
chaotic oscillations come from the coupling between the edge mode and the bulk
modes. The reasoning is below. First, the resonance frequency shift of the
edge mode when increasing the intensity is much larger than the ones for the
bulk modes as shown in Fig. 4. Second, although the intensity distributions
for CW and CCW are close to the bulk modes, there are clear signatures of the
edge modes, i.e., an exponentially decaying odd-site intensities and nearly
zero even-site intensities (Fig. 5 (d)(f)(g)). Thus, our numerical simulations
confirm that the edge mode overlap with bulk mode due to the Kerr nonlinearity
results in the periodic and chaotic oscillations.

##  VI Conclusions

In summary, we have numerically demonstrated the optical bistability and
various types of oscillations in a 1D SSH model composed of ring resonators
with Kerr nonlinearity. When the nonlinear terms are introduced in the
Lugiato-Lefever equation, the first ring’s CW and CCW mode intensities are
symmetric until the pump intensity reaches a bifurcation point. Above the
bifurcation point, the symmetry is spontaneously broken due to the splitting
of the resonance frequencies of the two CW and CCW modes in the first ring
resonator. For the high input intensity regime, we have observed oscillating
phases including periodic and chaotic oscillations. The periodic oscillation
phases can also be classified into two different phases where the trajectories
are separate or identical in the phase space of the mode intensities. This
emergence of the oscillating phases can be attributed to the coupling between
the edge mode and bulk mode due to the large shift of resonance peaks of the
edge mode.

We believe that our theoretical models and numerical results will provide
valueable insight in understanding the complex dynamics in coupled nonlinear
resonator systems with two chiral modes. Additionally, the various spatio-
temporal dynamics could be applied to optical switching devices as well as the
stability analysis of coupled lasers.

###### Acknowledgements.

We are grateful to Daniel Leykam for fruitful discussions. The work is part-
funded by the European Regional Development Fund through the Welsh Government
(80762-CU145 (East)).

##  Appendix A Derivation of the Lugiato-Lefever equation using the coupled
mode theory

The coupled mode theory [31] has been used to describe the field amplitude $a$
propagating in an optical ring resonator, which can be written as

$\frac{da}{dt}=i\omega_{0}a-\gamma a+\gamma_{c}s.$ (12)

Here $\omega_{0}$ refers to the resonance frequency, $\gamma$ and $\gamma_{c}$
are the damping and coupling with the source coefficient, respectively. we can
express field amplitude $a$ in terms of envelope amplitude $\tilde{a}$ as:

$a=\tilde{a}e^{i\omega t},$ (13)

by substituting in Eq.(12) :

$\frac{d\tilde{a}}{dt}=[i(\omega_{0}-\omega)-\gamma]\tilde{a}+\gamma_{c}\tilde{s},$
(14)

where $-\tilde{\Delta}=\omega_{0}-\omega$, then we can rewrite this equation
in terms of detuning as :

$\frac{d\tilde{a}}{dt}=[-\gamma-i\tilde{\Delta}]\tilde{a}+\gamma_{c}\tilde{s}.$
(15)

This equation is equivalent to the Lugiato-Lefever equation without
nonlinearity terms; the terms in RHS correspond to damping, detuning, and
source terms, respectively.

##  Appendix B Site-dependence of Poincaré sections for nonlinear SSH model

Figure A1 (a), (b) displays Poincaré sections of maxima of oscillating in
coupled intensities $I_{max,\pm}$ for odd (sublattice A) and even (sublattice
B) sites in the 1D SSH lattice, respectively. The odd-site intensities of the
CW and CCW modes follow the same pattern as the ones for the first ring
resonator, while the even-site intensities follow the same patter as the one
the second ring resonator. The symmetry is broken, i.e., the CW and CCW mode
intensities are notequal for the optical bistability and oscillation phases.

Figure A1: Poincaré sections of the maxima of oscillating coupled intensity as
a function of detuning for a 1D SSH lattice ($M=7$) with the same parameters
in Fig. 4. (a) For odd sites in the main text ($I_{s}=20,v=3,w=7,A=1,B=4$)
with the cyan, green, blue, and red colors corresponding to the 1st, 3rd, 5th,
and 7th ring resonators respectively. (b) For even sites with the red, green
and blue colors corresponding to the 2nd, 4th, and 6th ring resonators,
respectively.

##  Appendix C Poincaré sections of the maxima of oscillating coupled
intensity as a function of B

Figure A2: Poincaré section of the maxima of oscillating coupled intensity as
a function of $B$ for input intensity $I_{s}=20$, for the first ring resonator
from the 1D SSH array. The Poincaré sections for coupling coefficients ($v=3$,
$w=7$) and $\Delta=5.5$. Cyan shading indicates oscillations and yellow refers
to optical bistability.

In Fig. A2 we scan $B$ from $1$ to $7$ to observe oscillations in coupled
intensity for $A=1$, $I_{s}=20$, $v=3$ and $w=7$. Here, the Poincaré section
can be divided into three regions: symmetric stable, asymmetric stable and
asymmetric unstable states. We observe a series of bifurcations and
oscillation windows for $B\geq 1.9$.

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