# Controlled light distribution with coupled microresonator chains
via Kerr symmetry breaking

Alekhya Ghosh1,2,† Arghadeep Pal 1,2,† Lewis Hill 1 Graeme N Campbell 1,3 Toby
Bi 1,2 Yaojing Zhang 1 Abdullah Alabbadi 1,2 Shuangyou Zhang 1 Gian-Luca Oppo
3 and Pascal Del’Haye 1,2,∗ 1Max Planck Institute for the Science of Light,
Staudtstraße 2, D-91058 Erlangen, Germany
2Department of Physics, Friedrich Alexander University Erlangen-Nuremberg,
D-91058 Erlangen, Germany
3Department of Physics, University of Strathclyde, 107 Rottenrow, Glasgow, G4
0NG, UK  †These authors contributed equally to this work
∗Corresponding author<EMAIL_ADDRESS>
###### Abstract

Within optical microresonators, the Kerr interaction of photons can lead to
symmetry breaking of optical modes. In a ring resonator, this leads to the
interesting effect that light preferably circulates in one direction or in one
polarization state. Applications of this effect range from chip-integrated
optical diodes to nonlinear polarization controllers and optical gyroscopes.
In this work, we study Kerr-nonlinearity-induced symmetry breaking of light
states in coupled resonator optical waveguides (CROWs). We discover a new type
of controllable symmetry breaking that leads to emerging patterns of dark and
bright resonators within the chains. Beyond stationary symmetry broken states,
we observe periodic oscillations, switching and chaotic fluctuations of
circulating powers in the resonators. Our findings are of interest for
controlled multiplexing of light in photonic integrated circuits, neuromorphic
computing, topological photonics and soliton frequency combs in coupled
resonators.

##  I Introduction

When a system’s physical or mathematical property remains unchanged under a
certain transformation, it is said to possess symmetry. A sudden collapse of
this symmetry is termed spontaneous symmetry breaking (SSB). SSB has answered
pivotal questions in physics, ranging from the spontaneous breaking of gauge
symmetry [1] to more contemporary models of continuous symmetry breaking in
Rydberg arrays [2], the introduction of entanglement asymmetry [3] and SSB in
quantum phase transitions [4]. The applications of SSB span over a large
spectrum of physics [5, 6, 7, 8].

Kerr-ring resonators – Kerr here referring to cubic nonlinearity
($\chi^{(3)}$) in certain materials – have garnered interest for their
capability to amass high light intensities within minuscule mode volumes,
thereby enhancing the nonlinearity. These resonators have remarkable uses in
optical frequency combs [9], telecommunications [10], spectroscopy [11],
optical clocks [12], and in sub-wavelength distance measurements [13].
Importantly, they also serve as experimental platforms for probing fundamental
physical phenomena like SSB.

Within Kerr-resonators, SSB has been studied extensively between counter-
propagating optical fields during bidirectional pumping [14, 15, 16, 17, 18,
19, 20, 21]. These systems have paved the way for designing optical isolators
[22], circulators [23] and logic gates [24]. A second mechanism for realizing
SSB originates from two co-propagating light fields with mutually orthogonal
polarizations [25, 26, 27, 28, 29], which has led to the creation of
polarization controllers [30], random number generators [31], and vectorial
frequency combs [32]. Recent studies have unveiled SSB of solitons in Fabry-
Pérot Resonators [33, 34]. SSB via optomechanical effects has also been
observed [35]. Recent innovations have expanded the two-field SSB phenomena to
four-field SSB [36, 37].

A myriad of other interesting solutions have been revealed by looking into
slow-time responses, i.e., the evolutions of fields over many resonator round
trips ($t_{\text{r}}$) in coupled cavities [38], photonic dimers [35, 37].
Fast-time (time scale of a single $t_{\text{r}}$) responses of CROW systems
[39] and two-dimensional microresonator arrays [40, 41] have also demonstrated
a rich profusion of soliton dynamics. However, the slow-time response of CROW
systems, rich in potential nonlinear effects, remains largely uncharted.

In this work, we conduct an in-depth study of two distinct CROW systems, and
discussed the occurance of concurrent SSBs amongst different pairs of intra-
resonator circulating intensities. The interplay of linear coupling and
nonlinear interactions in our studied systems offers a vast parameter space to
influence homogeneous responses. We demonstrate that varying input power
levels lead the optical powers in the resonators to shift among different
levels, exhibiting switch-like behaviors. This is promising for controllable
distribution of light in photonic systems and realization of optical digital
memories and computation systems. We also detect oscillations [42, 20, 43]
that cause periodic interchanging of dominant field roles between distinct
resonators and N-level chaotic oscillations in these systems. Precise on-chip
microresonator fabrication methods [44] will make the proposed structures soon
realizable on photonic chips, thus highlighting the pertinence of the work for
guiding experiments in integrated photonics.

Figure 1: CROW configurations. $N$ identical Kerr ring resonators linked in
sequence. Input field directions ensure that each resonator’s circulating
field travels only in a singular direction. Note that for systems with odd and
even numbers of resonators require differing input directions for the end
resonators, see faded resonators of the figure. (a) “Input to end” CROW:
Inputs are provided only to the end-resonators. (b) “Input to all” CROW:
Inputs are connected to all resonators. Input (output) directions are shown by
red (purple) arrows.

##  II CROW Systems and Model

In our study, we consider CROW systems, as illustrated in Fig. 1. The systems
consist of several identical Kerr ring resonators, forming a coupled resonator
chain. Here, all the light fields supplied via the input ports are assumed to
be identical, i.e., they have equal intensities, frequencies, polarizations,
and phases. To begin with, we consider the situation when inputs are provided
to only the first and last resonators in the chain. This configuration is
depicted in Fig. 1(a). Afterwards, we also study the effects of facilitating
inputs to all of the resonators (as shown in Fig. 1(b)). Due to its more
practical implementability, we first study the system with inputs only to the
two resonators at the ends of the chain. The rich stationary state solutions
and other homogeneous solutions of the system with more inputs are described
later. Finally, in this study, we select the directions of input light fields
that result in a system without counter-propagating fields.
Our modelling begins with normalized coupled Lugiato-Lefever equations (LLEs)
[45, 39]. For a system encompassing $N$ coupled resonators (indexed as
$n=1...N$ for $N\geq 2$), the equations manifest as:

$\begin{split}\frac{\partial\psi_{n}}{\partial\tau}&=-\left(1+i\zeta\right)\psi_{n}+i\\{(1-\delta_{n,1})j\psi_{n-1}\\\
&+(1-\delta_{n,N})j\psi_{n+1}\\}+i|\psi_{n}|^{2}\psi_{n}+\alpha_{n}f,\end{split}$
(1)

where $\psi_{n}=\sqrt{2g_{0}/\kappa}A_{n}$ is the normalized optical field
envelope in the $n^{\text{th}}$ resonator, and within which $A_{n}$ is the
unnormalized field envelope, $g_{0}$ is the Kerr gain, and
$\kappa=\kappa_{l}+\kappa_{e}$ is the total cavity losses, with internal
losses $\kappa_{l}$ and external losses $\kappa_{e}$. The normalized cavity
detuning is given as $\zeta=2\Delta/\kappa$, where the unormalized cavity
detuning is given as $\Delta=\omega_{0}-\omega_{\text{res}}$ (the difference
between the input laser frequency ($\omega_{0}$) and the closest cavity
resonance frequency ($\omega_{\text{res}}$)). The terms within the curly
brackets account for the inter-resonator couplings, where $\delta_{p,q}$ is
the Kronecker Delta function and $j=2J/\kappa$ is the normalized inter-
resonator coupling rate, with $J$ being the unnormalized coupling rate. Input
to the resonators is given as
$f=\sqrt{8\kappa_{e}g_{0}/\kappa^{3}}s_{\text{in}}e^{i\phi_{\text{in}}}$,
where, $s_{\text{in}}$ and $\phi_{\text{in}}$ are respectively the input pump
amplitude and the corresponding phase. In our simulations we assume
$phi_{\text{in}}$ to be constant and set it to $phi_{\text{in}}=0$ for
convenience. The term $\alpha_{n}=1$ when input is provided to the
$n^{\text{th}}$ resonator and $\alpha_{n}=0$ otherwise. The normalized
intracavity intensity in the $n^{\text{th}}$ resonator is given by
$\Psi_{n}=|\psi_{n}|^{2}$ and the input power by $F=|f|^{2}$. We have
neglected dispersion in the systems. The second term (the term within the
curly brackets) on the RHS of Eq. (1) describes the fact that all the
resonators in the chain are coupled to the previous and the next resonator in
the line except for the two end resonators, each of which is just connected to
one adjacent resonator. The third term of Eq. (1) is the self-phase modulation
term, which accounts for the nonlinear effect of a field on itself. The last
term on the RHS of Eq. (1) represents input from outside the system. Since all
the ring resonators in both cases are identical, parameters, such as the
cavity detunings and the Kerr nonlinear gains, $g_{0}$, are the same for all
resonators.

Figure 2: Evolutions of optical intensities in $N=3$ “input to end” CROW
system. Panels (a)-(c) show the evolutions of the field intensities in
different resonators as a function of input power. The field intensities in
the end resonators are depicted in blue and black, whereas, the field
intensity in the middle resonator is depicted in red. For (a), $\zeta=0.5$,
$j=1$. A field intensity crossing point appears in (a). Before the crossing
point, the field intensity in the middle resonator is higher than the fields
in the end resonators and beyond this point, the end resonator fields become
more intense than the middle resonator field. Panel (b) depicts spontaneous
symmetry breaking of the end resonator light field intensities for $\zeta=5$,
$j=2$. Symmetry unbroken and symmetry broken oscillations are depicted in the
upper ($\zeta=3$, $j=2$) and lower ($\zeta=5$, $j=2.5$) panels of (c)
respectively. In this and all successive figures, the dotted lines stand for
the positions of the maxima and the minima of the oscillations, and the shaded
regions in between highlight the span of the oscillations. Upper and lower
panels of (d) are examples of symmetry unbroken and complete symmetry broken
oscillations respectively. Used parameters: $|f|^{2}=22,\zeta=3,j=2$ (d -
upper panel), $|f|^{2}=37.79,\zeta=5,j=2.5$ (d - lower panel). Time step for
integration is $0.005$.

##  III $\mathbf{N=3}$ CROW systems

Our present analysis addresses CROWs with three or more resonators, i.e.
$N\geq 3$. The homogeneous states of the systems are obtained by numerically
evaluating Eq. (1) for a variety of initial conditions and over sufficient
evolution times. The stationary states of the systems, which form a subset of
the homogeneous states, where the circulating fields remain unchanged over
time, can be obtained by setting $\partial\psi_{n}/\partial\tau=0$. For an
$N=3$ system, analytical solutions for the stationary states can be derived
(See Appendix A for more details), but for $N>3$ systems, obtaining analytical
solutions is difficult. Since for $N\geq 3$, the two end resonators are
connected to only one resonator while all other resonators in the CROW system
are connected to two neighbouring resonators, there is an inherent asymmetry
in the system. The circulating field intensity in each resonator $n$ is
asymmetric to that of resonator $m$ for $m\neq N-n+1$. This asymmetry comes
from the linear coupling terms of Eq. (1). Consequently, resonator $n$ is
symmetrical to only resonator $N-n+1$ in terms of coupling arrangements. In
other words, the field intensities are symmetric around the center of the
chain.

Figure 2 shows different kinds of optical intensity distributions that can be
observed in $N=3$ CROW systems. In Fig. 2(a), it can be observed that for
lower input powers, intracavity field intensities in the end resonators behave
symmetrically, but the field intensity in the middle resonator is more than
that of the individual end resonators. However, for a certain value of input
power, the field intensities in the end resonators cross the field intensity
in the middle resonator, and grow steadily after that, with the middle
resonator’s light intensity remaining almost constant. At the crossing point,
there is a momentary occurrence of complete symmetry, where all three
resonators have the same field intensities. Around the crossing point, the
relative distribution of optical intensities between the middle resonator and
the two end resonators can be tuned in a very controlled manner by changing
the input power (discussed in Appendix B). Figure 2(b) reveals a spontaneous
symmetry breaking of the circulating optical intensities in the end
resonators. This adds to the various optical field distribution mechanisms
that can be achieved in the CROW systems. It is important to note that the SSB
depicted in Fig. 2(b) is a novel mechanism, quite different from the usual SSB
observed in Kerr-resonators [16, 18, 26, 37]. The detailed description of the
emergence of this SSB phenomenon is discussed in the next section. Following
the trajectory of the field intensity in the middle resonator (in red) in Fig.
2(b), an interesting characteristic can be observed. The field intensity
remains low for low input powers, jumps to a high value after a certain input
power and finally comes back to a low value in the SSB region. This
effectively allows the system to be used as an all-optical switch with certain
low and high cut-off powers. Apart from the stationary states, numerical
simulations of Eq. (1) also reveal other homogeneous solutions, such as slow-
time oscillations inside the resonators of the chain. Such Kerr induced
oscillations of the field intensities have been observed here with and without
the occurrence of SSB in the system, as depicted in the upper and lower panels
of Fig. 2c respectively. Oscillations of all field intensities are observed in
the upper and lower panels of Fig. 2d respectively. In the upper panel, the
middle resonator field intensity dominates over the end resonator field
intensities, which always oscillate in phase. On the other hand, the lower
panel shows a perfect periodic switching of the field intensities in the end
resonators, where each of the three field intensities becomes dominant over
the other two at certain instances. All of the observed phenomena pave the way
for the $N=3$ system to become an efficient option for optical field routing
in integrated systems.

Figure 3: Circulating field intensities, $\Psi_{n}(=|\psi_{n}|^{2})$, against
input power, $|f|^{2}$, for an $N=3$ “input to end” CROW System. For detuning
$\zeta=5$ and inter-resonator coupling $j=2$, we present in panels (a, b) the
analytical solutions to Eq. (1) for the fields circulating the middle
$\Psi_{2}$ and end resonators $\Psi_{1,3}$ respectively. In panels (c, d) and
panels (e, f), respectively, we display the results of numerical integrations
of Eq. (1) for stepwise increasing, and stepwise decreasing values of the
input field intensity. In all panels, different “relationships” of solutions
are coloured accordingly for visual benefit, by this we mean that when fields
$\Psi_{1,3}$ are on the green solution line, this means that $\Psi_{2}$ is
also on its own respective green solution line. These results are discussed
thoroughly in the main text, but we highlight the possibility of end-
resonator-symmetric solutions (red) and two distinct end-resonator-symmetry-
broken solutions (green and blue).

##  IV Analytical solution for $\mathbf{N=3}$ CROW system

Input power, $|f|^{2}$, scans for an $N=3$ CROW, when inputs are provided to
only end resonators, are presented in Fig. 3. In Fig. 3(a,b) the analytical
solutions to Eq. (1) are displayed. One may find details on the analytical
solutions to Eq. (1) in the Appendix A. Panels (a,b) reveal surprisingly rich
and interesting dynamics for such a simple system. As mentioned earlier, for
nonzero input powers, $\Psi_{2}\neq\Psi_{1,3}$ due to coupling conditions. The
difference in coupling causes persistent differences in the evolutions of the
fields circulating the resonators, quite evident from the vastly different
solution curves of panels (a) and (b).

Figure 4: Occurance of SSB in N=3 CROW system. The left side of the dashed
line corresponds to a scan with an input power of $22.20$, whereas, the right
side of the scan corresponds to a scan with an input power of $22.42$. It can
be seen that starting from a stable “black-blue symmetric” state, the system
after the change in input power, goes to a “black-blue symmetry broken” state
after a small oscillatory evolution. The origin of the oscillations can be
described from the linear stability analysis presented in Appendix C and D.
The blue and black lines represent the circulating field intensities in the
end resonators, whereas, the red line depicts the field in the middle
resonator. The vertical dashed black line marks the point where the input
power is increased. In each case, the detuning is $5$. Time step for
integration is $0.005$.

The analytical solutions of the end resonators, Fig. 3(b), reveal not one but
two distinct sets of asymmetric solutions occurring for the end resonators.
One set (blue) arises in a manner similar to that of previous studies – a
pitchfork bifurcation emanating from the (red) symmetric solution line. The
more surprising part is the second asymmetric solution set (green), which does
not originate from a pitchfork bifurcation of the symmetric line. Indeed, it
does not originate from the symmetric solution line at all. The green solution
sets form entirely isolated solution “bubbles”, an isolation seen most
prominently in Fig. 3(a). Isolated sets of asymmetric solutions have not been
seen in any past works, to our knowledge, with the exception of a
significantly different setup with unbalanced input conditions [27]. A
justified question to ask is the following; as interesting, perhaps, as the
discovery of a novel SSB origin is, if these solutions are isolated, does this
not imply that they are unreachable under experimental conditions – and hence
entirely useless? We report that this is, highly surprisingly, not actually
the case. Panels (c-f) of Fig. 3 show, as a counterpart of the discussed
analytical results, the results of the numerical integration of Eq.(1) via
standard Runge-Kutta methods (Fig. 3(c) and (d) are extended versions of Fig.
2(b)). In panels (c,d) the input power is stepwise increased following a
suitable system relaxation time, while in panels (e,f) the input power is
similarly stepwise decreased. Unlike the analytics, which provide the full
solution sets, these scans predict the real-world behaviors and evolutions of
the circulating fields under experimental conditions. From panel (d), we see
that in the input-increasing-scan, just after $|f|^{2}=20$, the field
intensities of the end resonators suddenly jump away from the red symmetric
solution line and begin evolving, instead, along the green, isolated,
asymmetric solution line. This is accompanied by a substantial drop in the
middle resonator power, as seen in panel (c). The obvious question is how? How
does the system find the isolated set? The answer lies in the stability of Eq.
(1). Performing a linear stability analysis, in Appendix C, we find that, at
the point that the isolated asymmetric solutions occur, the symmetric solution
line experiences a Hopf bifurcation (Appendix D) leading to system
oscillations with wide-ranging intensity changes. These oscillations allow for
the system to eventually find, and settle on, attractive and stable, but
isolated, asymmetric solutions. This process is shown in Fig. 4. As the
increasing-input-scan continues further an optical bistability in the
asymmetric states leads to the system losing stability once again and
proceeds, this time, to settle on the stable upper branch of the original
symmetric solution line. Owed to the strong stability of this upper branch, we
find that a reverse scan in this case reveals none of the asymmetric solutions
of the forward scan, only bistability jumps. Fig. 3 has revealed that even for
low values of $N$, Eq. (1) describes a system capable of extremely intricate
dynamics.

Figure 5: Evolutions of optical intensities in $N=5$ (a-c) and $N=10$ (d-f)
“input to end” CROW systems. Panels (a),(b - upper panel) show the evolutions
of the field intensities in different resonators as a function of input power.
The end resonator field intensities are depicted in cyan and blue. In (a), the
end resonator field intensities and the neighboring resonator’s field
intensities (depicted in red and green) display spontaneous symmetry
breakings. In (b), the field intensities within coupling-wise symmetric
resonators always remain symmetric and they display oscillations. Overlapping
(lower-left panel) and non-overlapping (lower-right panel) oscillations are
observed. In (c), we demonstrate three of the possible light intensity
distribution conditions in the resonators with bright-red (dark-red) referring
to bright (dark) resonators (left configuration:
$\zeta=3.62,J=1,|f|^{2}=59.18$, middle configuration:
$\zeta=3.62,J=1,|f|^{2}=37.88$ and right configuration:
$\zeta=3,J=2,|f|^{2}=112.99$). Input power scans for $N=10$ CROW systems are
depicted in (d) and (e-upper panel). All the symmetric field pairs undergo
SSBs (in intensity) in (d) and upper panel of (e), followed by bistability
jumps and symmetry broken oscillations. Moreover, symmetry unbroken
oscillations are observed in the upper panel of (e). In the lower panel of
(e), three examples of oscillations of field intensities in time are depicted
from the different oscillatory regions as shown in the corresponding upper
panel. Panel (f) presents three possible light distribution conditions (left
configuration: $\zeta=1.5,J=4,|f|^{2}=37.88$, middle configuration:
$\zeta=1,J=6,|f|^{2}=270.67$ and right configuration:
$\zeta=1,J=6,|f|^{2}=88.38$). Used parameters: (a) $\zeta=3.62$ and $j=1$, (b)
$\zeta=3$ and $j=2$, (d) $\zeta=1.5$ and $j=4$, and (e) $\zeta=1$ and $j=6$.
Time step for integration is $0.005$.

##  V CROW systems with $\mathbf{N>3}$

To reveal the full potential of the CROW systems, we continue to study the
homogeneous solutions in $N>3$ systems. Figure 5 shows SSB phenomena occurring
in the CROW systems with $N=5$ and $N=10$, with inputs provided only to the
end resonators. In both systems, SSB bifurcations can be observed between
different symmetric field pairs.

For lower input powers in the $N=5$ system,
$|\psi_{1}|^{2}=|\psi_{5}|^{2}\neq|\psi_{2}|^{2}=|\psi_{4}|^{2}\neq|\psi_{3}|^{2}$
& $|\psi_{3}|^{2}\neq|\psi_{1}|^{2}$. In Fig. 5(a), where $\zeta=3.5$ and
$j=1$, the field intensity in the middle resonator (in black, index $3$) is
suppressed greatly for all input power values. The intensities of the fields
within the end resonators (index $1$ and $5$ \- depicted in blue and cyan) and
the set of resonators coupled to the end resonators (index $2$ and $4$ \-
depicted in red and green) display spontaneous symmetry breakings. The system
also shows bistability jumps. In the upper panel of Fig. 5(b), we can observe
two isolated regions with symmetry unbroken oscillations in the system. The
lower panel shows oscillations of circulating field intensities in time
corresponding to the two different regions. Fig. 5(c) depicts some of the
possible steady state light field intensity distributions among different
resonators for the $N=5$ CROW system. Here, the state of a resonator is
assumed to be bright if the circulating field intensity within the resonator
is more than the average of all the resonator’s field intensities in the
respective CROW arrangement for a particular combination of system parameters
and input powers.

Fig. 5(d-e) depicts the input power scan of the $N=10$ system for two
different sets of parameters. In Fig. 5(d), independent SSBs of all field
intensities within resonators with symmetric coupling conditions are observed,
with SSB bubbles crossing each other. The initial SSB region is followed by a
symmetry restored region, which then is followed by regions of oscillating
full asymmetric solutions, and subsequently non-oscillating full asymmetric
solutions. Fig. 5(e) depicts SSB of all coupling-wise symmetric pairs,
symmetry restored regions with and without oscillations, and bistability jumps
of the circulating field intensities leading to a second region of full
asymmetry. This full-asymmetric region also displays oscillations for certain
ranges of input powers with and without overlaps. The lower panels of Fig.
5(e) show examples of field intensity oscillations in time corresponding to
three different oscillatory regions.

Fig. 5(f) shows some of the possible bright-dark conditions achievable in
$N=10$ CROW systems. These demonstrate the power distribution capabilities of
the CROW systems. Different arrangements of bright-dark resonators, achievable
via tuning the input power can be used as different binary states. Therefore,
the system can be used as an optical analog-to-digital converter where an
analog optical input is transformed into a digital binary bit-string and
multi-bit logical operations can be performed on them. Moreover, in Ref. [39],
the authors have discussed the possibilities of having different dynamical
fast-time solutions in the systems. These solutions, e.g., solitons, depend on
the interplay of the dispersion profiles and nonlinear gains in the systems.
The power redistribution that can be achieved in CROW systems can
significantly change the intensity-dependent nonlinear gains in the
resonators, affecting the fast-time dynamics. Therefore, by exploiting the
interactions of the Kerr effect and coupling between resonators one can gain
control over the fast-time dynamics in these systems.

Figure 6: Colormap of different symmetry breaking conditions as function of
input power and detuning for $N=3$ (a), $N=5$ (b), $N=10$ (c) and $N=20$ (d)
“input to end” CROW systems. Purple regions stand for symmetry unbroken case
and the yellow stands for completely asymmetric case. The colors in between
(different shades of green) stand for different levels of symmetry breakings
in the systems. The watershed portions with stripes in each panel display the
regions of oscillations. In all cases, the scans are done from lower to higher
input powers for certain detuning values. In other words, Eq. (1) is scanned
for all detuning values starting from zero input power. For each input power,
the initial values of the field amplitudes are selected to be the steady state
value of the last step (smaller input power). The scans for all detunings are
done in parallel. For the scans $j=2\text{(a), }1\text{(b), }3.5\text{(c), and
}3\text{(d).}$

Motivated by the fact that a rich variety of SSBs can be observed in the CROW
system, we performed input power-detuning scans for $N=3,5,10,20$ and the
corresponding results are depicted in Fig. 6. All the scans are performed for
increasing input power. The scans demonstrate different thresholded symmetry
breaking conditions and oscillations of the circulating field intensities
through different colored regions. Two fields are considered to be symmetric
if the difference of their normalized intensities lies within an upper limit
(here we choose 0.05). These scans can be used to allocate different amounts
of homogeneous power in different resonators.

Figure 7: Evolutions of optical intensities in “input to all” CROW systems for
$N=4$ (b), $N=5$ (c), $N=10$ (e) and $N=50$ (f). Panel (a) depicts the
schematics for a system with $N=4$. Panel (b) presents the evolutions of the
field intensities in an $N=4$ CROW system with increasing input power for
$\zeta=1.5$ and $j=1$. Oscillations, bistability jumps and SSBs are observed
sequentially in the system with increasing input power. Panel (c) shows the
evolutions of the field intensities in an $N=5$ CROW system with increasing
input power for $\zeta=2.5$ and $j=2$. Panel (d) displays the Poincaré section
plot of (c). In the region where dots of all colors look randomly scattered,
chaos is observed. Panel (e) and the upper panel of (f) present evolutions of
the field intensities for $N=10$ ($\zeta=1.5$, $j=1$) and $N=50$ ($\zeta=1.5$,
$j=1$) CROW systems respectively. In both cases, the fields within coupling-
wise symmetric resonators break their symmetry with increasing input power,
afterwards, they form two bunches, one at high power and one at low power.
Thereafter in panel (e), by a bistability jump, the field intensities form two
inverse bifurcation structures. In the upper panel of (f), several bistability
jumps can be observed, where different field intensities group and regroup at
different jumps. Finally, two inverse bifurcation structures form. The lower
panel of (f) demonstrates the bright (yellow)-dark (purple) conditions of
different resonators for different input powers.

##  VI “input to all” CROW systems

In this section of the paper, we aim to study the CROW configurations where
all the resonators are provided with input fields. Since in these systems, all
the resonators in the chain have access to input power (as shown in Fig.
7(a)), richer nonlinear effects are expected to be observed. The high degree
of controllability of fields, due to inputs to all the resonators, makes this
section important, especially for the experimentalists working on integrated
coupled resonator systems.

For $N=4$, the full asymmetry of circulating field intensities in different
resonators is depicted in Fig. 7(b) for a wide range of input power values.
The end resonator field intensities (in blue and green) initially remain
symmetric and have less power than the middle resonators (index $2$ and $3$,
shown in black and red). With increasing input power, before the SSB region, a
region of oscillation appears. The near-switching behaviors of the field
intensities are depicted in the Appendix E. It is important to note that in
the full asymmetric region, the crossing of the various fields’ intensities
leads to two very localized regions of three-level asymmetry. These particular
crossings are intriguing since despite the different coupling conditions for
the end and middle resonators, at some distinct input power levels one of the
middle resonators becomes symmetric to one of the end resonators. The two SSB
bubbles close by forming inverse bifurcation structures after a certain input
power leading to two separated pairs of circulating field intensities. Here,
each pair contains the fields within resonators that have symmetric coupling
conditions, as observed for lower input powers. It can be observed that the
dominance order of intensities of the pairs switches between before and after
the SSB region.

With increasing $N$, more and more interesting nonlinear phenomena are
observed. For an $N=5$ system, Fig. 7(c) shows the input power-dependent
redistribution of relative optical intensities among different resonators. The
intensity of the middle resonator field (index = $3$, in black) gradually
decreases from being higher than any other fields at small input powers to
being extremely suppressed after the complete symmetry breaking of all the
fields. Figure 7(d) gives an insight into the oscillations in the system which
is portrayed via the Poincaré section plot, where the maxima and minima of the
temporal oscillations of the respective field intensities are presented by
dots for each input power. At the beginning of the oscillatory regime, the
maxima and minima of the initially symmetric field intensities overlap
completely. Therefore three regions of oscillations appear, the overlap
regions of which gradually increase with input power. After the region of
periodic oscillations, the system drives into the region of chaos with
increasing input power, where all the field intensities oscillate
asymmetrically. This symmetry broken chaotic oscillation region ends in
symmetry broken stationary states. The full asymmetry closes with $2$ inverse
bifurcation structures. The SSB enforces high suppression of circulating power
in the middle resonator, whereas, the outer resonators always have higher
circulating field intensities. A detailed overview of the oscillations for the
$N=5$ system is given in Appendix E.

For $N=10$, as shown in Fig. 7(e), we again observe an SSB region, however,
the SSB bubbles here twist and cross each other to provide different levels of
symmetry breakings. At the end, following bistability jumps, the field
intensities in the resonators form two inverse bifurcation structures, one by
the fields within the end resonators, and one by all other fields. It is
noteworthy that all the inner resonators at this point behave almost
symmetrically. For $N=50$, depicted in the upper panel of Fig. 6(f), SSB
bubble crossings generate a much more complex scenario with many possible
levels of SSB. However, a noticeable phenomenon in this case is the group
formation of the field intensities within the SSB bubble, where two pairs of
fields with little differences in intensities emerge. These intermediate pairs
split up with multiple bistability jumps and a series of regroupings occur.
Finally, the field intensities merge into two symmetric pairs through two
inverse bifurcation structures. For the first time, it has been observed that
the field intensities jump between different levels of circulating power,
forming a unique cage-like diagram in the regrouping section. Even if various
resonators in the system have coupling-wise asymmetry, it is seen in both Fig.
7(e) and (f - upper panel), after the inverse bifurcations only two bunches
sustain, one by the fields within the end resonators, and one by all other
fields. All the inner resonators at this point behave almost symmetrically.
These SSB-induced intra-cavity field distributions in coupled resonator
systems are not only intriguing for generating multi-logic levels with higher
functionalities in all-optical devices but also give an idea for observing
various soliton dynamics at different input power levels.

In the lower panel of Fig. 7(f), we demonstrate the corresponding dark-bright
conditions of different resonators as a function of input power. As mentioned
earlier, a resonator is considered to be bright (shown in yellow) if the field
intensity within that resonator is more than the average of all the field
intensities of all the resonators. Otherwise, it is considered to be dark
(shown in purple). The dark-bright condition plot shows that these
configurations of CROW systems have great potential in all-optical computing.
The input-output access to all the resonators makes it ideal for loading and
unloading a bit-stream of data in a digital computing platform, with the
ability to perform logical operations via the control of optical field
distributions within the resonators.

##  VII Discussions and outlook

To summarise, a theoretical framework has been developed to examine different
states of light in CROW systems. In CROW systems with inputs to the end
resonators, we can observe a plethora of different symmetry breaking
phenomena. The spontaneous symmetry breaking causes different light field
intensities in different resonators. At low input powers, mirror symmetric
pairs of resonators within the chain experience equal circulating powers. When
the input power is increased, more complex light distributions within the
resonator chain emerge, corresponding to multiple concurrent spontaneous
symmetry breaking events. Symmetry broken oscillations are observed in
$N=3,5,10\text{ and }20$ CROW systems for higher input powers and detunings.
Periodic switchings between different pairs of field intensities are also
observed. Due to the access of higher input powers to all resonators, richer
nonlinear phenomena are observed in CROW systems with all resonators coupled
to input waveguides. Extended regions of $N$-level SSBs and chaotic
oscillations are observed in this type of CROW configuration.

Future research will address the dynamic behavior of optical fields in the
resonators and the effects of asymmetry in different parameters on the
homogeneous states of the systems. SSBs in other complex arrangements of
microring-resonators will also be addressed. The controllable distribution of
light field intensities amongst different resonators could be a key feature
for large-scale optical computing and light-field steering in integrated
photonics. The combination of linear coupling between different resonators and
optical nonlinearities in high-Q microresonators makes the CROW systems a
promising candidate for integrated optical neural networks. CROW systems are
also promising candidates for observing symmetry broken vector solitons with
$N$ different values of circulating intensities [32]. These can be useful for
generating $N$ distinct interconnected frequency combs which would be very
useful in neuromorphic computing, telecommunications and especially in space
technologies due to compactness. Together with the latest concepts of
dispersion engineering [46, 47, 48] the studied nonlinear effects in this work
will lead to a lot more interesting soliton dynamics.

###### Acknowledgements.

This work was supported by the European Union’s H2020 ERC Starting Grant
”CounterLight” 756966 and the Max Planck Society. AG and AP acknowledges the
support from Max Planck School of Photonics. LH acknowledges funding provided
by the SALTO funding scheme from the Max-Planck-Gesellschaft (MPG) and Centre
national de la recherche scientifique (CNRS).
AG and AP contributed equally to this work. PDH, AG and LH defined the
research project. AG performed the theoretical analysis with support from LH
and AP. AP, AG and LH completed the numerical simulations. AG, AP, LH and PDH
wrote the manuscript with the help of all other authors. PDH and LH supervised
the project.

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##  Appendix A 3 resonator CROW - Analytical Solutions

For three resonator “input to all” CROW system the equations of motion takes
the form:

$\displaystyle\frac{\partial\psi_{1}}{\partial\tau}$ $\displaystyle=$
$\displaystyle-\left(1+i\zeta_{1}\right)\psi_{1}+ij\psi_{2}+i|\psi_{1}|^{2}\psi_{1}+f,$
(2a) $\displaystyle\frac{\partial\psi_{2}}{\partial\tau}$ $\displaystyle=$
$\displaystyle-\left(1+i\zeta_{2}\right)\psi_{2}+i(j\psi_{1}+j\psi_{3})+i|\psi_{2}|^{2}\psi_{2}+f$
(2b) $\displaystyle\frac{\partial\psi_{3}}{\partial\tau}$ $\displaystyle=$
$\displaystyle-\left(1+i\zeta_{3}\right)\psi_{3}+ij\psi_{2}+i|\psi_{3}|^{2}\psi_{3}+f.$
(2c)

Here $\zeta_{i}$ represents the normalized detuning of the $i^{\text{th}}$
resonator. In steady state, all the equations become equal to zero. Solving
the steady state equations and considering equal input to all the resonators,
one can obtain

$\displaystyle\Psi_{1}^{3}-\Psi_{3}^{3}+2\zeta_{3}\Psi_{3}^{2}-2\zeta_{1}\Psi_{1}^{2}+(1+\zeta_{1}^{2})\Psi_{1}-(1+\zeta_{3}^{2})\Psi_{3}=0,$
(3a)
$\displaystyle\Psi_{2}^{3}(C_{3}^{2}+1)+\Psi_{2}^{2}(2C_{2}C_{3}+2C_{4})+\Psi_{2}(C_{2}^{2}+C_{4}^{2})=\Psi_{1}C_{5},$
(3b)
$\displaystyle\left|-\left(1-\frac{j^{2}}{D_{1}}-\frac{j^{2}}{D_{3}}\right)+i(\Psi_{2}-\zeta_{2})\right|^{2}\Psi_{2}=\left|1-i\left(\frac{j}{D_{1}}+\frac{j}{D_{3}}\right)\right|^{2}F,$
(3c)

where $\Psi_{n}=|\psi_{n}|^{2}$, $D_{n}=-(1+i\zeta_{n})+i\Psi_{n}$ (for
$n=1,2$ or $3$), $C_{1}=1-\zeta_{2}\zeta_{3}-j\zeta_{3}$,
$C_{2}=C_{1}+\zeta_{2}\Psi_{3}+j\Psi_{3}$, $C_{3}=\zeta_{3}-\Psi_{3}$,
$C_{4}=-C_{3}-\zeta_{2}-j$,
$C_{5}=\\{1+C_{3}(\Psi_{1}-\zeta_{1}-j)-j\zeta_{1}+j\Psi_{1}\\}^{2}+\\{\zeta_{1}-\Psi_{1}+C_{3}+2j\\}^{2}$,
and $F=|f|^{2}$.

For “input to end” CROW system, $f_{2}=0$, therefore Eqs. (3) takes the form

$\displaystyle\Psi_{1}^{3}-\Psi_{3}^{3}+2\zeta_{3}\Psi_{3}^{2}-2\zeta_{1}\Psi_{1}^{2}+(1+\zeta_{1}^{2})\Psi_{1}-(1+\zeta_{3}^{2})\Psi_{3}=0,$
(4a)
$\displaystyle\Psi_{2}^{3}-2\zeta_{2}\Psi_{2}^{2}+\Psi_{2}(1+\zeta_{2}^{2})=\Psi_{1}\left|j\left(\frac{D_{3}}{D_{1}}+1\right)\right|^{2},$
(4b)
$\displaystyle\left|D_{2}+j^{2}\left(\frac{1}{D_{1}}+\frac{1}{D_{3}}\right)\right|^{2}\Psi_{2}=\left|\left(\frac{j}{D_{1}}+\frac{j}{D_{3}}\right)\right|^{2}F.$
(4c)

##  Appendix B Field intensity crossings for $\mathbf{N=3}$ “input to end”
CROW systems

For symmetric CROW system, Eq. (4b) takes the form,

$\displaystyle\Psi_{2}^{3}-2\zeta\Psi_{2}^{2}+\Psi_{2}(1+\zeta^{2})-4j^{2}\Psi_{1}=0,$
(5)

where we have considered $\zeta_{1}=\zeta_{2}=\zeta_{3}=\zeta$ and
$D_{1}=D_{2}=D_{3}$. At the crossing points, $\Psi_{1}=\Psi_{2}$,

$\displaystyle\Psi_{2}=\zeta\pm\sqrt{4j^{2}-1},$ (6)

Further using Eq. (4c), at the crossing points, one can obtain,

$\displaystyle F=|f|^{2}=\left|\frac{-3\pm
i\sqrt{4j^{2}-1}}{2}\right|^{2}\left(\zeta\pm\sqrt{4j^{2}-1}\right),$ (7)

The conditions for having different numbers of crossing are mentioned in the
following table.

Number of crossing points |  Conditions
---|---
$0$ | $4j^{2}<1$
$1$ |  $4j^{2}>1$ and $\sqrt{4j^{2}-1}>\zeta$
$2$ |  $4j^{2}>1$ and $\sqrt{4j^{2}-1}<\zeta$
Table 1: Number of field intensity crossing points for $N=3$ Figure 8: Field
intensity crossings for $N=3$ “input to end” CROW systems. Panels (a-c) show
input power scan of resonator light intensities for CROW system with $N=3$. In
panel (a) the end-resonator field intensities (shown in black) do not cross
the field intensity in the middle resonator (shown in red). The middle
resonator field intensity crosses end resonator field intensities once in
panel (b) and twice in panel (c). Panel (d) shows the ratio of field
intensities circulating in end resonators and middle resonator. Brown line
shows the ratio for (b), and green line shows the ratio for (c). Panel (d)
demonstrates the potential for relative power distributions among the
resonators in $N=3$ CROW systems. Used parameters: (a) $\zeta=0.5$, $j=0.1$,
(b) $\zeta=0.5$, $j=1$, and (c) $\zeta=1.66$, $j=0.6$.

##  Appendix C Eigen value analysis for $\mathbf{N=3}$ CROW system

For $N=3$ CROW system, the normalized coupled LLE equations take the form:

$\displaystyle\frac{\partial\psi_{1}}{\partial\tau}$
$\displaystyle=-\left(1+i\zeta\right)\psi_{1}+ij\psi_{2}+i|\psi_{1}|^{2}\psi_{1}+f,$
(8a) $\displaystyle\frac{\partial\psi_{2}}{\partial\tau}$
$\displaystyle=-\left(1+i\zeta\right)\psi_{2}+i\left(j\psi_{1}+j\psi_{3}\right)+i|\psi_{2}|^{2}\psi_{2},$
(8b) $\displaystyle\frac{\partial\psi_{3}}{\partial\tau}$
$\displaystyle=-\left(1+i\zeta\right)\psi_{3}+ij\psi_{2}+i|\psi_{3}|^{2}\psi_{3}+f,$
(8c)

where $\zeta$ is the normalized detuning, $j$ is the normalized coupling,
$\psi_{n}$ is the normalized field envelop in the $i^{\text{th}}$ resonator
and $f$ is the input field amplitude. If we define
$\psi_{n}=\bar{\psi_{n}}+\delta\psi_{n}$, where $\bar{\psi_{n}}$ is the steady
state value and $\delta\psi_{n}$ is the infinitesimal perturbation.
Considering the complex conjugates of each field envelopes, the evaluation
equations for the perturbations can be written as $\dot{A}=\mathcal{J}A$,
where
$A=\left[\delta\psi_{1},\delta\psi_{1}^{*},\delta\psi_{2},\delta\psi_{2}^{*},\delta\psi_{3},\delta\psi_{3}^{*}\right]^{T}$.
The Jacobian matrix $\mathcal{J}$ can be written as

$\displaystyle\mathcal{J}=\begin{bmatrix}\Delta_{1}&i|\bar{\psi_{1}}|^{2}&ij&0&0&0\\\
-i|\bar{\psi_{1}}|^{2}&\Delta_{1}^{*}&0&-ij&0&0\\\
ij&0&\Delta_{2}&i|\bar{\psi_{2}}|^{2}&ij&0\\\
0&-ij&-i|\bar{\psi_{2}}|^{2}&\Delta_{2}^{*}&0&-ij\\\
0&0&ij&0&\Delta_{3}&i|\bar{\psi_{3}}|^{2}\\\
0&0&0&-ij&-i|\bar{\psi_{3}}|^{2}&\Delta_{3}^{*},\end{bmatrix}$ (9)

where $\Delta_{n}=-\left(1+i\zeta-i2|\bar{\psi_{n}}|^{2}\right)$. When the
real part of any of the eigenvalues of the matrix becomes positive, the system
becomes unstable to perturbations.

##  Appendix D Origin of the novel symmetry breaking mechanism in
$\mathbf{N=3}$ CROW system

In the main text, we have discussed about the occurrence of a novel type of
SSB in $N=3$ CROW system, where the system jumps from an initial symmetric
state to an isolated set of asymmetric solutions via oscillations. The system
starts to oscillate when the real part of any of its’ eigenvalues goes above
zero. This is depicted in Fig. 9.

Figure 9: SSB mechanism in $N=3$ CROW system. Panel (a) and (b) show real and
imaginary parts of eigenvalues of the Jacobian matrix of the system as given
by (9). The shaded region in panel (a) depicts the region where y-axis is
above $0$. When the real part of any of the eigenvalues enters the shaded
region, the system becomes unstable. The dashed line (which is at same place
as the dashed line in Fig. 4) depicts the input power value for which real
part of one eigenvalue of the system goes above zero. The SSB between the
circulating intensities in the end resonators occurs at this point.

##  Appendix E Oscillations in CROW systems

In the main paper, we have observed different types of oscillations in “input
to end” CROW systems. We have also observed the perfect periodic switching in
$N=3$ CROW systems. In this section, we will observe different types of
oscillations present in the $N>3$ CROW systems with different input
conditions.

Figure 10: Switchings in “input to end” CROW systems. Periodic switchings of
the fields within coupling-wise symmetric resonators for $N=5$ (a-b), $N=10$
(c-d), $N=20$ (e-f,g-h) are depicted. Perfect sinusoidal switchings (a,c,e,g)
are confirmed by the complete overlaps of the phase space plots (b,d,f,h).
Used parameters: (a) $|f|^{2}=108.34$, $\zeta=3$, $j=1$, (a) $|f|^{2}=39.18$,
$\zeta=4.2$, $j=3.5$, (a) $|f|^{2}=67.53$, $\zeta=4$, $j=3$, (a)
$|f|^{2}=120.05$, $\zeta=4$, $j=3$. Time step for integration is $0.005$.
Figure 11: Oscillations in “input to all” CROW systems. (a) Input power scan
of resonator light intensities for CROW system with $N=4$, $\zeta=1.5$ and
$j=1$. Evolutions of field intensities over time in the oscillatory region
(shaded region in (a)) $|f|^{2}=5.68$. (c) Input power scan of resonator light
intensities for CROW system with $N=5$, $\zeta=2,5$ and $j=2$. (d) and (e)
shows oscillations with maintained symmetry between fields within resonators
with symmetric coupling conditions, whereas (f) shows switching oscillations
between them. (g) depicts 5-field chaos. (h) shows near-switching
oscillations. (i)-(j) shows completely asymmetric periodic oscillations.
$|f|^{2}=12.93\text{(d), }20.20\text{(e), }24.44\text{(f), }29.09\text{(g),
}34.14\text{(h)}$. Time step for integration is $0.005$.