#  Development of Vertically Integrated Circuits for ILC Vertex Detectors

Ronald Lipton
for the Fermilab Pixel R&D Group
Fermilab
P.O. Box 500  Batavia  Illinois  USA

###### Abstract

We report on studies of vertically interconnected electronics (3D) performed
by the Fermilab pixel group over the past two years. These studies include
exploration of interconnect technology, backside thinning and laser annealing,
the production of the first 3D chip for particle physics, the VIP, and plans
for a commercial two-tier 3D fabrication run. Studies of Direct bond
Interconnect (DBI) oxide bonding and Silicon-on-Insulator based technologies
are presented in other talks in this conference.

##  1 3D electronics

3D electronics is generally defined as consisting of multiple layers of
electronics, thinned, bonded and interconnected to form a monolithic circuit.
This technology has become an area of intense focus in the electronics
industry as a way to improve circuit performance without the expense and
complexity of smaller feature size [2] [3]. 3D electronics provides the
ability to integrate heterogeneous technologies, reduce interconnect lengths,
and expand bus width. These technologies are particularly interesting for
pixel detectors, where they offer new techniques for integrating sensors and
electronics, and provide substantially more processing power per pixel than
conventional technologies.

Fabrication of a 3D stack depends on the development of several techniques
including:

  * •

Bonding between layers including oxide to oxide fusion, copper/tin bonding,
copper/copper bonding, polymer/adhesive bonding

  * •

Wafer thinning using grinding, lapping, etching, and Chemical Mechanical
Polishing (CMP)

  * •

Through wafer via formation and metalization either with isolation using
Through Silicon Vias (TSVs) or without isolation as in Silicon-on-Insulator
(SOI) devices.

  * •

High precision alignment

The technology can also be separated into techniques which form the via before
3D integration is performed (via first) and those which form vias afterwards
(via last).

##  2 VIP Chip

The VIP chip [5] was intended as a demonstration of 3D technology as applied
to an ILC vertex detector. It was fabricated in 0.18 micron SOI CMOS
technology by MIT-Lincoln Labs as part of a DARPA-sponsored multiproject run.
The chip consists of three tiers of electronics interconnected by 3D vias. The
MIT-LL technology utilizes oxide bonding to bond tiers together. This wafer
bonding technology mates activated, planarized silicon oxide surfaces to form
a robust inter-wafer bond [4]. The top ”handle” silicon in the bonded wafer
stack is thinned to $70\mu m$ and the remainder of the silicon is etched away
to expose the oxide surface. Vias are then formed by etching thorough the
insulating oxide to contact internal metalization layers. The process is
repeated to form the required number of tiers.

Figure 1: Schematic of the VIP chip showing the three tiers of electronics.
The drawing on the right shows the metalization for the three layers.

The VIP incorporates an amplifier/disriminator with double correlated sample
and hold in tier three, both a 5-bit digital and an analog time stamp in tier
2, and sparsification and digital logic in tier one, all within a $20\mu m$
square pixel. In ILC operation time stamped hits are stored within the pixel
during the bunch train. The chip utilizes part of the 199 ms period between
trains for readout, while the front end current is reduced to save power. The
front end is designed to consume less than 4 mW/mm${}^{2}\times f$ where $f$
is the front end duty factor. A token passing readout scheme stores addresses
on the periphery, minimizing the logic on the pixel. Figure 1 shows a
schematic of the chip as well as a visualization of the metal layers.

The chip was submitted in October 2006 and received in October 2007. Initial
tests showed that the overall yield was low, with only a few chips showing the
ability to propagate the readout token through the full 64 x 64 matrix. The
single best-performing chip was selected for full testing. Figure 2 shows the
results of the most complete system test, where a pattern of test pulses are
injected into the front-end amplifiers, and a sparse scan is performed,
reading out those channels where the discriminator fired and latched the
readout flag.

Figure 2: Pattern of 119 pixels injected with a test charge (left) and read
out in the subsequent sparse scan.

Our testing has demonstrated the basic functionality of the chip including
propagation of the readout token, threshold scans, input test charge scans,
verification of digital and analog time stamping, full sparsified data
readout, and fixed pattern and temporal noise measurements. No problems could
be found associated with the 3D vias between tiers. Although the chip was
fully functional, we were not fully satisfied with yield and performance.
Performance problems stemmed from large leakage currents in the protection
diodes and transistors, and poor matching of current mirrors. Many of these
issues can be traced to the sensitivity of mixed mode designs in fully
depleted SOI to the transistor environment and process variations [6] [7].
Intrinsic SOI process problems were exacerbated by our aggressive design,
which made extensive use of minimum feature size transistors and dynamic
logic, which is sensitive to transistor leakage current.

An new version of the chip, the VIP2, was submitted to the third DARPA
sponsored 3D multiproject run in October 2008. As a result of useful
interaction with MIT-LL on SOI analog design the overall quality should be
considerably improved. Changes to the chip include:

  * •

Different power and grounding layout

  * •

Larger transistor sizes ($0.18\to 0.45\mu m$) equivalent feature size

  * •

Larger pixels (30 x 30 microns)

  * •

Redundant vias and larger traces in critical paths

  * •

Redesign of current mirrors to reduce thermal effects

  * •

Removal of dynamic logic due to leakage current problems

Changes were also made to improve the overall functionality including
increasing the digital time stamp from 5 to 7 bits. We expect that the changes
will lead to a much more reliable chip which can be bonded to sensors for test
beam studies.

##  3 Commercial 3D Technologies

An R&D process, such as the one provided by MIT-LL has disadvantages of long
turn-around time and process uncertainties. We are now exploring alternative
3D processes implemented as part of a high volume commercial process. Tezzaron
(Naperville Ill) has developed a 3D technology implemented in the high volume
0.13 micron Chartered (Singapore) process [3]. This is a ”via first” process
where through-silicon ”supercontacts” are formed after transistor fabrication
but before any metalization processing. The 6 micron deep by 1 micron diameter
supercontacts are filled with tungsten at the same time as the transistor
contacts are formed. Wafers are finished normally, however there is a top
layer of thin patterned copper which forms both the bond between wafers and
the wafer-to-wafer electrical interconnection. Wafers are then bonded face-to-
face with moderate pressure and temperature. Silicon on the top wafer is
ground down to the supercontacts and the contacts are metalized to form either
external bond pads or to provide connections to the next tier.

Fermilab is organizing a multiproject run in the Tezzaron/Chartered process
with submission expected in Spring of 2009\. The run includes designs from 13
institutes. The Fermilab designs will include a two tier version of the VIP
chip, the VIP2b, as well as test devices for the CMS upgrade and X-ray
imaging. Standard commercial CMOS should provide a reliable process with low
noise, multiple transistor options, better rad hardness, less wasted via area,
faster turn-around, as well as the availability of full wafers for sensor
integration.

##  4 Sensor Integration

The bonding, thinning and lithography process used to build 3D tiers can also
be used for sensor integration with readout. Both Tezzaron and Ziptronix
provide 3D processes, based on copper-copper and oxide bonding respectively,
which can be used to include sensors as a base tier in a 3D stack. 3D
processes offer finer pitch and more robust mechanical interconnection than is
available in solder-based bump bonding. The high planarity and strong
interlayer bonds allow bonded readout ICs to be thinned to 25 $\mu m$ or less,
which can provide access to integrated through-silicon vias.

We have explored two sensor bonding techniques: Cu-Sn and DBI oxide bonding.
The DBI technique is described in another contribution to this conference [8].
We have contracted with RTI to explore Cu-Sn bonding with sparse contacts to
minimize contact mass. These technologies do not have the contact bridging
problems that limit pitch for solder bumps. However they are not self-aligning
which requires special care in aligning the bonded surfaces. In that study
successful Cu and Cu-Sn bump structures were fabricated that were compatible
with $20\mu m$ I/O pitch. Electrical tests indicated that the bonding yield
was $>99$ % for both metallurgies. Samples were also destructively tested to
evaluate bond strength. In both cases the bond strength is considerably higher
than comparable solder bump arrays [9].

##  5 Thinning and Laser Annealing

Wafer thinning is an important part of 3D technology, and the ability to
process and handle thinned silicon is crucial to the goal of constructing a
very low mass vertex detector. In some 3D and SOI technologies it will be
important to thin the devices after topside processing. After thinning a
backside contact ohmic contact must also be formed. The contact is usually
fabricated by a high temperature anneal of an ion implantation. This high
temperature step is unacceptable for fully processed electronics, where the
temperature must be kept below 450 degrees C. to protect the topside
metalization.

We have developed a thinning/implantation/annealing process which limits the
maximum temperature of the topside to below 100 deg C. The wafer is first
bonded to a pyrex carrier using a 3M UV release adhesive designed for wafer
thinning applications. The wafer is then thinned and polished using standard
techniques. The bonded wafer is ion implanted, taking care to ground the
silicon edges and controlling the implantation rate to limit the temperature.
The implantation is then annealed using a eximer laser system which melts the
silicon locally on the backside to a depth of $300$ nm, keeping the topside
close to room temperature.

Figure 3: Secondary Ion Mass Spectrometry (SIMS) phosphorus dose profile of
strip detector before (red) and after (blue) laser annealing.

Initial studies were performed using individual strip sensors, which were
thinned to remove the backside ohmic implant, re-implanted and laser annealed.
All of these devices showed acceptable performance at depletion, with some
variation of leakage current depending on laser dose and annealing
environment. A scan of the dopant concentration before and after the annealing
process is shown in Figure 3. This work was followed by studies using 6” test
wafers donated by Micron Semiconductor. These wafers were thinned to 50
microns on the pyrex handle, implanted and laser annealed at MIT-Lincoln Labs.
Depletion voltage was reduced from 80 to 2.5 volts with acceptable leakage
current. Work is ongoing at Cornell to determine the optimal implantation and
annealing parameters.

##  6 Ladder design

3D technologies provide the ability to construct a low mass, dense, tiled
array of chips, which can be used to fabricate ladder and disk planes for the
ILC. Figure 4 shows an example of such a structure. Multi-tier readout ICs are
fabricated utilizing through-silicon vias. These ICs are bonded to an
independently fabricated sensor wafer with a fine pitch technology such as DBI
or cu-cu. Once bonded, the Readout ICs are thinned to reveal the topside TSVs.
Topside interconnections are patterned using standard lithography and wirebond
or other contacts are made. Connections to external power and signal cables
could be made at the ends of the sensor, which could have the appropriate
interconnection patterns. This technique has a number of advantages:

  * •

The sensor can be a fully depleted detector with charge collection by drift
rather than diffusion.

  * •

The sensor wafer serves as a base for the ROICs, obviating the need for
reticle stitching.

  * •

The 3D ROICs can include power control tiers.

  * •

Known good ROIC die can be used.

Final sensor thinning would have to occur after the topside processing. This
could be done by backgrinding a sensor with an imbedded ohmic contact, either
as part of an epitaxial stack or using the SOI technique demonstrated by the
Max Plank Institute [10]. This would avoid the additional implant and laser
annealing steps needed if the backside were not pre-processed.

Figure 4: Conceptual drawing of a thinned ladder(left) based on 3D
interconnects (right). Readout chips could be bonded to the sensor wafer using
an oxide bonding process (DBI) then thinned to $\approx 25\mu m$ to expose
through-silicon vias to provide interconnections. The sensor wafer could be
pre-thinned and mounted on a handle wafer during the DBI processing.

##  7 Conclusions

Fermilab has produced and tested the VIP, the first 3D chip designed for
particle physics applications. The chip demonstrated the required
functionality but suffered from low yield and compromised performance. An
improved version of the chip has been submitted to MIT-LL. The VIP2b, a two-
tier 0.13 micron CMOS chip implemented in the Tezzaron 3D process, will be
submitted this spring. This submission will extend the development of this
technology to applications at super-LHC and in x-ray imaging. We are
continuing to develop wafer thinning, interconnection, and post-processing
technologies aimed at demonstrating the ability to build precise, low mass,
low power vertex detector systems.

## References

  * [1] Presentation:
`http://ilcagenda.linearcollider.org/contributionDisplay.py?contribId=42&sessionId=8&confId=2628`

  * [2] IBM Journal of Research and Development, Volume 52, No. 6, 2008. Issue devoted to 3D technology.
  * [3] Philip Garrou, Christopher Bower, Peter Ramm, Handbook of 3D Integration Technology and Applications of 3D Integrated Circuits, Wiley-VCH, 2008.
  * [4] IEEE Transactions on Electron Devices, Vol. 53, No. 10, October 2006.
  * [5] A Vertically Integrated Pixel Readout Device for the Vertex Detector at the International Linear Collider, FERMILAB-PUB-08-564
  * [6] M. Connell et al., 2007 IEEE/SEMI Advanced Semiconductor Manufacturing Conference.
  * [7] Tenbroek et al., Solid-State Circuits Conference, 1997, pp. 276-279, 16-18 Sept. 1997.
  * [8] `http://ilcagenda.linearcollider.org/contributionDisplay.py?contribId=207&sessionId=21&confId=2628`
  * [9] Allan Huffman, Fabrication, Assembly, and Evaluation of Cu-Cu Bump bonding Arrays for Ultra-fine Pitch Hybridization and 3D Integration, Pixel 2008, Fermilab, Batavia, Illinois, September 22-26, 2008.
  * [10] P. Fischer et al., Nucl. Instrum. Meth. A 582, 843 (2007).


Present address: ]Université Paris-Saclay, CNRS, Centre de Nanosciences et de
Nanotechnologies, 91120, Palaiseau, France

Present address: ]School of Optical and Electronic Information, Huazhong
University of Science and Technology, 1037 Luoyu Road, Wuhan, China

# Nonlinear topological symmetry protection in a dissipative system

Stéphane Coen  Physics Department, The University of Auckland, Private Bag
92019, Auckland 1142, New Zealand  The Dodd-Walls Centre for Photonic and
Quantum Technologies, Dunedin, New Zealand  Bruno Garbin  [  Physics
Department, The University of Auckland, Private Bag 92019, Auckland 1142, New
Zealand  The Dodd-Walls Centre for Photonic and Quantum Technologies, Dunedin,
New Zealand  Gang Xu  [  Physics Department, The University of Auckland,
Private Bag 92019, Auckland 1142, New Zealand  The Dodd-Walls Centre for
Photonic and Quantum Technologies, Dunedin, New Zealand  Liam Quinn  Physics
Department, The University of Auckland, Private Bag 92019, Auckland 1142, New
Zealand  The Dodd-Walls Centre for Photonic and Quantum Technologies, Dunedin,
New Zealand  Nathan Goldman  Center for Nonlinear Phenomena and Complex
Systems, Université Libre de Bruxelles, CP 231, B-1050 Brussels, Belgium
Gian-Luca Oppo  SUPA and Department of Physics, University of Strathclyde,
Glasgow G4 0NG, Scotland, European Union  Miro Erkintalo  Stuart G. Murdoch
Physics Department, The University of Auckland, Private Bag 92019, Auckland
1142, New Zealand  The Dodd-Walls Centre for Photonic and Quantum
Technologies, Dunedin, New Zealand  Julien Fatome  Physics Department, The
University of Auckland, Private Bag 92019, Auckland 1142, New Zealand  The
Dodd-Walls Centre for Photonic and Quantum Technologies, Dunedin, New Zealand
Laboratoire Interdisciplinaire Carnot de Bourgogne (ICB), UMR 6303 CNRS,
Université de Bourgogne Franche-Comté, 9 Avenue Alain Savary, BP 47870,
F-21078 Dijon, France

###### Abstract

We report an experimental and theoretical investigation of a system whose
dynamics is dominated by an intricate interplay between three key concepts of
modern physics: topology, nonlinearity, and spontaneous symmetry breaking. The
experiment is based on a two-mode coherently-driven optical resonator in which
photons interact through the Kerr nonlinearity. In presence of a phase defect
between the modes, a nonlinear attractor develops, which confers a synthetic
Möbius topology to the modal structure of the system. That topology is
associated with an inherently protected exchange symmetry between the modes,
enabling the realization of spontaneous symmetry breaking in ideal, bias-free,
conditions without any fine tuning of parameters. The dynamic manifests itself
by a periodic alternation of the modes from one resonator roundtrip to the
next reminiscent of period-doubling. This extends to a range of localized
structures in the form of domain walls, bright solitons, and breathers, which
have all been observed with remarkable long term stability. A rigorous testing
of the randomness of the symmetry-broken state selection statistics has also
confirmed the robustness of the exchange symmetry in our experiment. Our
results and conclusions are supported by an effective Hamiltonian model
explaining the symmetry protection in our system. The model also shows that
our work has relevance to other systems of interacting bosons and to the
Floquet engineering of quantum matter. Our work could also be beneficial to
the implementation of coherent Ising machines.

##  I Introduction

Topology, nonlinearity, and spontaneous symmetry breaking are key concepts of
modern physics. Topology is concerned with the study of properties that are
invariant under continuous deformations, hence inherently robust against
fluctuations and disorder [1, 2]. Nonlinearity is the source of complexity,
underpinning attractors, chaos, self-organization, and self-localization
(solitons) [3, 4]. _Spontaneous_ symmetry breaking (SSB) underlies much of the
diversity observed in Nature [5, 6, 7], from ferromagnetism [8] to embryo
development [9]. Here we report on an experiment in which these three concepts
are fundamentally intertwined. Nonlinearity and SSB combine to endow our
system with a topological invariant; in return that invariance protects the
underlying symmetry, enabling a robust realization of SSB that is immune to
perturbations.

The discovery of topological states of matter in solid-state materials has
revealed the central role of topology in the classification of quantum matter
[1, 10]. Besides, recent progress in designing and controlling synthetic
lattice systems has allowed for the exploration of topological properties in a
broad class of physical contexts [11, 12], including ultracold gases [11],
photonic crystals [12, 13], mechanics [14], and systems with synthetic
dimensions [15, 13]. Topological robustness has potential wide-ranging
applications, from fault-tolerant quantum computers [16, 17] to lasers capable
of single-mode operation at high power [18]. While topological band structures
concern the properties of single-particle Bloch states, interesting behaviors
were observed by combining these with classical nonlinearities [19]. This
includes, non-exhaustively, edge solitons, nonlinearity-induced topological
phase transitions [20, 21], novel symmetry-protected phases [22], or quantized
Thouless pumping of solitons [23].

Figure 1: (a) Schematic representation of a two-mode coherently driven Kerr
resonator with a $\pi$-phase defect. Inset on the right shows the linear
resonances of the two modes and how only one mode is driven (here the $x$
mode). (b) Equivalent Möbius-like resonator described in terms of the $+$ and
$-$ hybrid modes defined by Eqs. (1). (c) Parametric oscillation of the
$y$-mode, ${|\psi_{y}|^{2}>0}$ (top left) corresponds to an SSB bifurcation
for the occupations of the hybrid modes, ${|\psi_{+}|^{2}\neq|\psi_{-}|^{2}}$
(bottom left). In that regime (P2 SSB), the phase of the $y$-mode changes by
$\pi$ at each resonator roundtrip (top right), corresponding to a periodic
roundtrip-to-roundtrip swapping of the hybrid mode occupations (bottom right).
$t_{\mathrm{R}}$ is the roundtrip time of the resonator.

The results presented in this Article demonstrate a new facet of the potent
bridging of topology and nonlinearity, and how it can confer robustness to the
realization of SSB. Upon SSB, a system bifurcates to states with a broken
symmetry even though the equations of motion retain that symmetry [5, 6, 7].
Intense interest has been paid to this process in recent years, especially in
optical microresonators where it has been studied, e.g., for optical memories
[24], nonreciprocal propagation in integrated photonic circuits [25, 26],
dynamical control of laser directionality and chirality [27], or sensors with
divergent sensitivities [28, 29]. Experiments on SSB have also been used for
fundamental studies of domain walls [30], of the universal unfolding of the
pitchfork bifurcation [31, 32], or of the Bose-Hubbard dimer model [33]. These
experiments most often rely on an exchange symmetry ($\mathbb{Z}_{2}$) between
two states. Invariably, however, this symmetry is affected by manufacturing
imperfections, experimental biases, or other non-idealities. As a consequence,
instead of a spontaneous, random selection between the two states, one state
is statistically favored over the other [34, 32], hindering the experiments.
Here we show how the combination of nonlinearity and topology can solve that
problem. Specifically, we report on the experimental realization of an optical
resonator with a synthetic modal structure characterized by a Möbius topology
protected by the Kerr nonlinearity. That protection leads to an exact exchange
symmetry, which enables robust, bias-free SSB that does not require any fine
tuning of the system parameters. The generation of truly random binary
sequences based on this platform confirms the robustness of the exchange
symmetry. Additionally, we reveal a new class of topological symmetry-broken
localized structures characterized by a rapid periodic exchange of their
symmetry. Our observations are supported by theory, which reveals in
particular the key role played by degenerate parametric processes. This
connects our work with some recent implementations of so-called coherent Ising
machines [35, 36] that could potentially benefit from our symmetry-protection
scheme. These benefits could extend to experiments on Bose-Einstein
condensates. Indeed, the Kerr nonlinearity stems from interaction processes
among photons, in direct analogy with the nonlinearity inherent to Bose-
Einstein condensates and described by the Gross-Pitaevskii equation [37]. We
also present an effective Hamiltonian approach, similar to that used for the
Floquet engineering of quantum matter [38, 39, 40], to explain aspects of the
emergence of the symmetry in our system. Finally, our work could have
relevance to the symmetry restoration techniques used in mean-field approaches
of quantum many-body systems in nuclear, atomic, and molecular physics [41].

##  II Principle of symmetry protection

Our system consists of a passive nonlinear (Kerr) optical resonator
coherently-driven by a single-frequency field and that presents two distinct
polarization mode families. Hereafter $\psi_{x}$ and $\psi_{y}$ denote the
modes’ classical electric field (complex) amplitudes inside the resonator
scaled such that $|\psi_{x}|^{2}$ and $|\psi_{y}|^{2}$ represent the number of
photons in each mode. In this configuration, the nonlinearity is well known to
induce a polarization SSB bifurcation contingent on the existence of an
exchange symmetry between the two modes [42, 43]. That symmetry is typically
realized by being as close as possible to conditions where the modes are
equally driven and equally detuned from the driving field frequency [26, 25,
32]. Here, we deviate from this paradigm. Specifically, we introduce a
localized $\pi$-phase-shift defect between the two mode families of the
resonator. As a result, their resonance frequencies are separated by half a
free-spectral-range (FSR). In these conditions, the exchange symmetry cannot
be realized as described above: the single-frequency driving field can only be
resonant with one of the modes, assumed here to be the $x$ mode, hence only
that mode is driven and the roundtrip phase detunings differ by $\pi$. These
different aspects are illustrated schematically in Fig. 1(a).

Consider now an alternative description of this resonator in terms of the
hybrid mode amplitudes $\psi_{+}$ and $\psi_{-}$ defined by the unitary
transformation

$\psi_{+}=\frac{1}{\sqrt{2}}\left(\psi_{x}+i\psi_{y}\right),\
\psi_{-}=\frac{1}{\sqrt{2}}\left(\psi_{x}-i\psi_{y}\right).$ (1)

Because of the $\pi$-phase defect, the amplitude of the $y$ mode flips its
sign with respect to the driven mode at each roundtrip in the resonator,
$\psi_{y}\rightarrow\psi_{y}e^{i\pi}=-\psi_{y}$. From the above relations, it
is clear that this is associated with a roundtrip-to-roundtrip _periodic
swapping_ of the $+$ and $-$ hybrid mode amplitudes,
${\psi_{+}\rightleftharpoons\psi_{-}}$ [right side in Fig. 1(c)]. The field
must circulate two roundtrips in the resonator to be restored to its original
state, revealing that the hybrid modes embody the topology of a Möbius strip
[44, 45, 46, 47]. In terms of those hybrid modes, the resonator shown in Fig.
1(a) is thus equivalent to that represented in Fig. 1(b). Note that the Möbius
topology only exists in the synthetic modal dimension rather than in real
physical space [48]. A consequence of the single-sided, non-orientable nature
of the Möbius topology is that the two hybrid mode amplitudes will necessarily
accumulate the _same_ changes over _two_ resonator roundtrips. The resonator
therefore exhibits an exchange symmetry,
${\psi_{+}\rightleftharpoons\psi_{-}}$. That symmetry, coupled with the
nonlinearity, enables SSB between the hybrid modes.

Here, it is important to realize that SSB in our system is fundamentally
intertwined with the generation of the $y$ field. As can be seen from Eqs.
(1), a _symmetric_ state for the hybrid modes, ${\psi_{+}=\psi_{-}}$, is
necessarily associated with a $y$ field of zero amplitude. By contrast, a
_symmetry-broken_ hybrid state, ${\psi_{+}\neq\psi_{-}}$, requires a non-zero
$y$ field. By symmetry, the mirror state,
${\psi_{+}\rightleftharpoons\psi_{-}}$, must also exist, corresponding to a
$y$ field with a phase differing by $\pi$,
${\psi_{y}\rightleftharpoons-\psi_{y}}$. Keeping in mind that the $y$-field is
undriven, it must be generated internally through nonlinear interactions past
a certain threshold of detuning and/or driving strength. What appears as an
SSB bifurcation for the populations $|\psi_{+}|^{2}$, $|\psi_{-}|^{2}$ of the
$+$ and $-$ hybrid modes can therefore be seen as the threshold of parametric
oscillation for the $y$ mode [49] [left side in Fig. 1(c)]. However, our
system is not just a standard parametric oscillator. Above the bifurcation
threshold, the underlying Möbius topology imparted by the $\pi$ phase defect
forces the system to alternate between the two symmetry-broken states one
roundtrip to the next, and to display an overall periodicity of two
roundtrips. As a reminder to these specific characteristics, we will refer to
this regime as P2 SSB. One can also note that, in the P2 SSB regime, the
amplitudes of the two hybrid modes over one resonator roundtrip become
indistinguishable from the amplitude of one of those modes over two subsequent
roundtrips. An alternative interpretation is thus that SSB now occurs between
subsequent resonator roundtrips, in essence breaking the time translation
symmetry, in a way reminiscent of period doubling instabilities [50, 51, 52].

##  III Theoretical analysis

The conclusions of the previous Section are based on the premise that the
phase defect seen by the $y$ mode at each resonator roundtrip is _exactly_
$\pi$. To understand how the Möbius topology and the associated symmetry can
persist in presence of deviations from this condition, we must examine some
aspects in more detail. To that end, we introduce the Hamiltonian
$\hat{H}_{\mathrm{NL}}$ describing the Kerr-mediated interactions taking place
along the resonator roundtrip,

$\frac{1}{\hbar
g}\hat{H}_{\mathrm{NL}}=\frac{A}{2}\left(\hat{a}_{x}^{\dagger}\hat{a}_{x}^{\dagger}\hat{a}_{x}\hat{a}_{x}+\hat{a}_{y}^{\dagger}\hat{a}_{y}^{\dagger}\hat{a}_{y}\hat{a}_{y}\right)+B\left(\hat{a}_{x}^{\dagger}\hat{a}_{y}^{\dagger}\hat{a}_{x}\hat{a}_{y}\right)\\\
+\frac{C}{2}\left(\hat{a}_{x}^{\dagger}\hat{a}_{x}^{\dagger}\hat{a}_{y}\hat{a}_{y}+\hat{a}_{y}^{\dagger}\hat{a}_{y}^{\dagger}\hat{a}_{x}\hat{a}_{x}\right).$
(2)

Here $g$ is the single photon-induced Kerr frequency shift of the resonator
($g>0$) [53] while $\hat{a}_{\sigma}^{\dagger}$ (resp. $\hat{a}_{\sigma}$)
represent creation (annihilation) operators for the mode ${\sigma=x,y}$. These
operators satisfy bosonic commutation relations,
$[\hat{a}_{\sigma},\hat{a}_{\sigma^{\prime}}^{\dagger}]=\delta_{\sigma,\sigma^{\prime}}$.
The three terms on the right-hand side of Eq. (2) describe, respectively, on-
site (Bose-Hubbard) interactions (self-phase modulation), inter-site
interactions (cross-phase modulation), and pair hopping (parametric four-
photon mixing). The latter corresponds to two photons in the $x$ mode being
converted to two photons in the $y$ mode (or vice-versa); since the $y$-mode
is undriven, this term is required as the source of $y$ photons. Without loss
of generality, we will assume ${A=1}$, ${B=2/3}$ and ${C=1/3}$. These are the
values found in silica (which is the Kerr material used in our experimental
demonstration presented below) for modes that are linearly polarized [54, 55].
Generalization to other polarization states will be discussed in Appendix.

To understand specifically the evolution of the phase of the $y$ mode, we
examine the equation of motion for $\hat{a}_{y}$, i.e.,
$d\hat{a}_{y}/dt=i[\hat{H}_{\mathrm{NL}},\hat{a}_{y}]/\hbar$. In the classical
limit, ${\hat{a}_{y}^{\dagger}\rightarrow\psi_{y}}$, we get

$\frac{d\psi_{y}}{dt}=ig\left[\left(A|\psi_{y}|^{2}+B|\psi_{x}|^{2}\right)\psi_{y}+C\psi_{y}^{*}\psi_{x}^{2}\right],$
(3)

where the right-hand side has the form of the interaction terms found in the
Gross-Pitaevskii/nonlinear Schrödinger equation ($*$ denotes complex
conjugation). Separating amplitude and phase,
$\psi_{\sigma}=|\psi_{\sigma}|e^{i\phi_{\sigma}}$, we find that the phase
$\phi_{y}$ of the $y$ mode obeys

$\frac{d\phi_{y}}{dt}=g\left[A|\psi_{y}|^{2}+B|\psi_{x}|^{2}+C|\psi_{x}|^{2}\cos\left(2\phi_{y}-2\phi_{x}\right)\right].$
(4)

The three terms in the above equation represent phase shifts induced by the
Kerr nonlinearity. If we restrict ourselves to high-Q resonators, these can
all be assumed ${\ll 1}$. Indeed, nonlinear effects will only have an
important role when close to resonance. In these conditions, we can use a
single Euler step to approximate the integration of Eq. (4) over one resonator
roundtrip time $t_{\mathrm{R}}$. To obtain the total change in the phase of
the $y$ mode from one roundtrip to the next, we still need to consider the
role of the boundary conditions, which add a linear contribution due to the
resonator detuning (see Appendix for more detail), leading to

$\phi_{y}^{(m+1)}-\phi_{y}^{(m)}=gt_{\mathrm{R}}\left(A|\psi_{y}|^{2}+B|\psi_{x}|^{2}\right)\\\
+gt_{\mathrm{R}}C|\psi_{x}|^{2}\cos\left(2\phi_{y}^{(m)}-2\phi_{x}^{(m)}\right)\underbrace{-\delta_{0}+\pi+\delta_{\pi}}_{-\delta_{y}}.$
(5)

Here $m$ is the roundtrip index, $\delta_{0}=\delta_{x}$ is the roundtrip
phase detuning of the driving field (modulo $2\pi$), also assumed ${\ll 1}$
($\delta_{0}>0$ means red-detuned), and the corresponding detuning for the $y$
mode is defined through $\delta_{x}-\delta_{y}=\pi+\delta_{\pi}$, with
$\delta_{\pi}$ representing deviations from an exact value of $\pi$. We can
observe that all terms above are in fact ${\ll\pi}$, except $\pi$ itself (a
similar argument for the $x$ mode would show that
$\phi_{x}^{(m+1)}-\phi_{x}^{(m)}\ll 1$). Moreover, the pair hopping
(parametric) contribution (${\propto C}$), which is the only phase sensitive
term, is $\pi$ periodic in $\phi_{y}$. This is a well known characteristics of
degenerate parametric oscillators [56]. It follows from Eq. (5) that the only
stable configuration is for the phase of the $y$ mode to step by _exactly_
$\pi$, one roundtrip to the next. That solution acts as an attractor for the
system, irrespective of the value of $\delta_{\pi}$. Effectively, the
intracavity fields will self-adjust through the dissipative nonlinear dynamics
to guarantee a roundtrip-to-roundtrip $y$ mode phase step of $\pi$. In this
way, the Möbius topology of the resonator is intrinsically robust and
protected by the nonlinearity.

As a second step, we examine how the Möbius topology induces an exact exchange
symmetry ${\psi_{+}\rightleftharpoons\psi_{-}}$. To that end, we introduce an
explicit _asymmetry_ and show how it cancels out. Specifically, we consider a
difference in detunings between the $+$ and $-$ hybrid modes. The Hamiltonian
Eq. (2) can be expressed in terms of these modes with creation (annihilation)
operators $\hat{a}_{\pm}^{\dagger}$ ($\hat{a}_{\pm}$); these are related to
the corresponding $x,y$ operators by the same unitary transformations as Eqs.
(1). We find

$\frac{1}{\hbar
g}\hat{H}_{\mathrm{NL}}=\frac{A^{\prime}}{2}\left(\hat{a}_{+}^{\dagger}\hat{a}_{+}^{\dagger}\hat{a}_{+}\hat{a}_{+}+\hat{a}_{-}^{\dagger}\hat{a}_{-}^{\dagger}\hat{a}_{-}\hat{a}_{-}\right)\\\
+{B^{\prime}}\left(\hat{a}_{+}^{\dagger}\hat{a}_{-}^{\dagger}\hat{a}_{+}\hat{a}_{-}\right)$
(6)

where $A^{\prime}=2/3$ and $B^{\prime}=4/3$. For convenience, let us define a
set of angular momentum (Schwinger) operators as

$\displaystyle\hat{J}_{x}$
$\displaystyle=\frac{\hbar}{2}\left(a_{+}^{\dagger}a_{-}+a_{-}^{\dagger}a_{+}\right),$
$\displaystyle\hat{J}_{y}$
$\displaystyle=\frac{\hbar}{2i}\left(a_{-}^{\dagger}a_{+}-a_{+}^{\dagger}a_{-}\right),$
(7) $\displaystyle\hat{J}_{z}$
$\displaystyle=\frac{\hbar}{2}\left(a_{-}^{\dagger}a_{-}-a_{+}^{\dagger}a_{+}\right),$
$\displaystyle\hat{N}$
$\displaystyle=a_{+}^{\dagger}a_{+}+a_{-}^{\dagger}a_{-}.$

These operators satisfy commutation relations
$[\hat{J}_{\mu},\hat{J}_{\nu}]=i\hbar\epsilon_{\mu\nu\lambda}\hat{J}_{\lambda}$
similar to spins ($\epsilon_{\mu\nu\lambda}$ is the Levi-Civita symbol), while
$\hat{N}$ counts the total number of photons. In this part of our analysis, we
assume a lossless resonator (conservative limit) so $\hat{N}$ is constant.
With these notations, a difference in angular frequency detuning
$\delta\Omega$ between the two hybrid modes enter the Hamiltonian as
${\delta\Omega\,\hat{J}_{z}}$, so that the total Hamiltonian of the system
reads

$\displaystyle\hat{H}_{\mathrm{asym}}$
$\displaystyle=\hat{H}_{\mathrm{NL}}+\delta\Omega\,\hat{J}_{z}$ (8)
$\displaystyle=\hbar
g\left[a\,\hat{J}_{z}^{2}/\hbar^{2}+b\,\hat{N}^{2}-c\,\hat{N}\right]+\delta\Omega\,\hat{J}_{z}$

where $a=A^{\prime}-B^{\prime}$, $b=(A^{\prime}+B^{\prime})/4$, and
$c=A^{\prime}/2$.

We now proceed by seeking an effective (Floquet) Hamiltonian
$\hat{H}_{\mathrm{eff}}$ [38, 57] which captures stroboscopically the dynamics
of a full period of the Möbius configuration depicted in Fig. 1(b), i.e.,
every _two_ actual roundtrips of the resonator. The swapping of the two hybrid
modes at each roundtrip is introduced through the discrete application of the
operator
$\hat{U}_{\mathrm{swap}}=\hat{\sigma}_{x}=e^{i(\pi/2)(\hat{\sigma}_{x}-1)}$
(with $\hat{\sigma}_{x}$ a Pauli matrix), leading to

$e^{-i\frac{1}{\hbar}2t_{\mathrm{R}}\hat{H}_{\mathrm{eff}}}=\underbrace{\hat{U}_{\mathrm{swap}}e^{-i\frac{1}{\hbar}t_{\mathrm{R}}\hat{H}_{\mathrm{asym}}}\hat{U}_{\mathrm{swap}}}_{e^{-i\frac{1}{\hbar}t_{\mathrm{R}}\hat{H}_{1}}}e^{-i\frac{1}{\hbar}t_{\mathrm{R}}\hat{H}_{\mathrm{asym}}}.$
(9)

The first three factors on the right hand side of this expression define
$\hat{H}_{1}$ which can be obtained exactly as 111We use the relation
$e^{i\hat{u}}e^{i\hat{v}}e^{-i\hat{u}}=\exp(e^{i\hat{u}}\hat{v}e^{-i\hat{u}})$,
and the fact that $\hat{U}_{\mathrm{swap}}$ is unitary.

$\hat{H}_{1}=e^{i\frac{\pi}{2}\hat{\sigma}_{x}}\hat{H}_{\mathrm{asym}}e^{-i\frac{\pi}{2}\hat{\sigma}_{x}}.$
(10)

Noting that the swapping operation is a single-particle process, we can
substitute $\hat{J}_{x}/\hbar$ for $\hat{\sigma}_{x}/2$ in the exponentials
above and then use the Baker-Campbell-Hausdorff formula. Specifically we have
$e^{i\pi\hat{J}_{x}/\hbar}\,\hat{J}_{z}\,e^{-i\pi\hat{J}_{x}/\hbar}=-\hat{J}_{z}$
while all other terms of $\hat{H}_{\mathrm{asym}}$ are left unchanged, leading
to

$\hat{H}_{1}=\hat{H}_{\mathrm{NL}}-\delta\Omega\,\hat{J}_{z}.$ (11)

Comparing the expressions for $\hat{H}_{\mathrm{asym}}$, Eq. (8), and
$\hat{H}_{1}$, above, we observe that the sign of the asymmetric term is
reversed. It is also clear that $\hat{H}_{\mathrm{asym}}$ commutes with
$\hat{H}_{1}$, which from Eq. (9) leads to
$\hat{H}_{\mathrm{eff}}=(\hat{H}_{1}+\hat{H}_{\mathrm{asym}})/2$. It follows
that the asymmetric term cancels out exactly over two resonator roundtrips,
and we have $\hat{H}_{\mathrm{eff}}=\hat{H}_{\mathrm{NL}}$. Our analysis
therefore shows that the effective Hamiltonian, which describes propagation
every two resonator roundtrips, including two swaps of the hybrid modes, only
contains the original — symmetric — nonlinear interactions; it is immune to
fluctuations and asymmetries in detunings as foreseen in our qualitative
description of the previous Section. This confirms the existence of an exact
exchange symmetry for the hybrid modes,
${\psi_{+}\rightleftharpoons\psi_{-}}$, and the topological robustness of the
P2 SSB regime.

We are now in a position to derive mean-field equations of motion for the
classical fields $\psi_{+}$, $\psi_{-}$ based on the effective Hamiltonian
$\hat{H}_{\mathrm{eff}}$. We have already established that the dissipation
guarantees an exact overall roundtrip phase jump of $\pi$ for the $y$ mode, so
we do not need to consider the influence of $\delta_{\pi}$, the deviation from
an exact $\pi$ phase defect, on the swapping operator
$\hat{U}_{\mathrm{swap}}$. Still, $\delta_{\pi}$ contributes to the linear
detunings through the relation $\delta_{0}-\delta_{y}=\pi+\delta_{\pi}$. This
effect can be taken into account by the following roundtrip phase detuning
operator,

$\hbar\left[\delta_{0}(\hat{a}_{x}^{\dagger}\hat{a}_{x})+(\delta_{0}-\delta_{\pi})(\hat{a}_{y}^{\dagger}\hat{a}_{y})\right]\\\
=\hbar\left(\delta_{0}-\frac{\delta_{\pi}}{2}\right)\hat{N}+\delta_{\pi}\,\hat{J}_{x}.$
(12)

Note that, for the $y$ mode, we have discarded the $\pi$ phase-shift that is
already incorporated in $\hat{H}_{\mathrm{eff}}$. The contribution of the
detunings can be distributed along the resonator roundtrip and combined with
the effective Floquet Hamiltonian, leading to, in the classical limit,

$\displaystyle t_{\mathrm{R}}\frac{d\psi_{+}}{dt}$
$\displaystyle=\biggl{[}-\frac{t_{\mathrm{R}}\Delta\omega}{2}+i\,gt_{\mathrm{R}}\left(A^{\prime}|\psi_{+}|^{2}+B^{\prime}|\psi_{-}|^{2}\right)$
$\displaystyle\qquad-i\left(\delta_{0}-\frac{\delta_{\pi}}{2}\right)\biggr{]}\psi_{+}-i\frac{\delta_{\pi}}{2}\psi_{-}+\sqrt{\theta}\,S,$
(13) $\displaystyle t_{\mathrm{R}}\frac{d\psi_{-}}{dt}$
$\displaystyle=\biggl{[}-\frac{t_{\mathrm{R}}\Delta\omega}{2}+i\,gt_{\mathrm{R}}\left(A^{\prime}|\psi_{-}|^{2}+B^{\prime}|\psi_{+}|^{2}\right)$
$\displaystyle\qquad-i\left(\delta_{0}-\frac{\delta_{\pi}}{2}\right)\biggr{]}\psi_{-}-i\frac{\delta_{\pi}}{2}\psi_{+}+\sqrt{\theta}\,S.$
(14)

To these equations, we have added phenomenologically losses (dissipation),
associated with the linewidth $\Delta\omega$ of the resonances, and driving,
with strength $S$ and external coupling fraction $\theta$. In the conservative
limit applied above, these terms are not included directly. An alternative
classical derivation of the mean-field equations (13)–(14), taking into
account these dissipative effects, is presented in Appendix for completeness.
That derivation confirms in particular that the Möbius topology leads, as for
the detunings, to a robust symmetrization of the driving. Note that the
symmetry of the driving can also be interpreted as stemming from the absence
of driving of the $y$ mode; under these conditions, Eqs. (1) indeed entails
$S_{+}=S_{-}=S=S_{x}/\sqrt{2}$, where $S_{x}$ is the driving strength of the
$x$ mode only. Additionally, we can observe that a deviation $\delta_{\pi}$
from an exact $\pi$ phase defect simply causes a common shift in detuning as
well as some linear coupling between the modes, none of which alters the
exchange symmetry of the equations.

Apart from the absence of kinetic terms, the role of which will be discussed
in Section VI, Eqs. (13)–(14) have the form of coupled, driven, damped, Gross-
Pitaevskii/nonlinear Schrödinger equations. These equations are well known to
exhibit SSB, provided the detunings and the driving strengths of the two modes
are identical [42, 43, 26, 25, 29, 32]. Contrary to previous implementations,
in our system, and thanks to the presence of the $\pi$ phase defect, this
condition is automatically and robustly satisfied. Finally, while the
equations above describe the stroboscopic mean-field evolution of $\psi_{+}$
and $\psi_{-}$ two resonator roundtrips at a time, it is important to keep in
mind that the amplitudes of the two fields swap at each resonator roundtrip
due to the underlying Möbius topology.

##  IV Experimental observation of P2 SSB

Figure 2: Schematic of the experimental setup. FPC: Optical fiber polarization
controller. PBS: Polarizing beam-splitter.

Figure 3: Evidence of P2 SSB. Oscilloscope recordings of the intensity of the
(a) $+$ and (b) $-$ hybrid modes and (c) total intensity over 20 subsequent
resonator roundtrips ($X=50$, $\Delta=10$). The alternation of the hybrid mode
intensities is a key signature of the underlying Möbius topology while the
constant total intensity indicates a high level of symmetry between the hybrid
modes. (d) High and low intensity levels of the $+$ (blue) and $-$ (orange)
hybrid modes versus resonator detuning, revealing a pitchfork bifurcation
($X=50$). The black lines are numerical predictions (dashed lines correspond
to unstable states). (e) Complex plane showing the real (in-phase, $I$) and
imaginary (quadrature, $Q$) components of $\psi_{x}\psi_{y}^{*}$ measured over
1000 resonator roundtrips in the P2 SSB regime. Cyan and purple density plots
correspond, respectively, to odd and even roundtrips and their diametrically
opposite placement signals that the relative phase between the $x$ and $y$
fields changes by $\pi$ at every roundtrip. The spread of each density is
magnified four times for clarity.

To demonstrate experimentally the topologically-protected P2 SSB regime, we
have used, as modes $x$ and $y$, the two polarization modes of a ring
resonator constructed from single-transverse mode silica optical fiber. This
platform enables straightforward implementation of both the $\pi$ phase-shift
defect and the projection of the $x$ and $y$ modes onto the $+$ and $-$ hybrid
modes through manipulations of the polarization state of light with optical
fiber polarization controllers (FPCs) [59]. Note that, for $x$ and $y$ modes
that are orthogonally linearly polarized, the $+$ and $-$ hybrid modes defined
by Eqs. (1) correspond to circular polarization states of opposite handedness.

A schematic of the experimental setup is depicted in Fig. 2. As we used
slightly different resonators with different specifications for different
measurements (in part, to demonstrate the universality of the methodology), we
list here the parameters corresponding to the results presented in this
Section. Variations between setups will be highlighted as appropriate. To
facilitate comparisons, experimental results will be presented in terms of
normalized parameters. Specifically, we will be referring to the normalized
roundtrip phase detuning $\Delta=\delta_{0}/\alpha$ and the normalized driving
power $X=gt_{\mathrm{R}}\theta S_{x}^{2}/\alpha^{3}$, where
$\alpha=t_{\mathrm{R}}\Delta\omega/2$ is the dissipation rate of Eqs.
(13)–(14).

Our first resonator is built around a fiber coupler (beam-splitter) that
recirculates 90 % of the intracavity light. This coupler is also used for the
injection of the coherent driving field ($\theta=0.1$). Another 1 % tap
coupler extracts a small fraction of the intracavity field for analysis. Both
couplers are made up of standard SMF28 (Corning) optical fiber. The rest of
the resonator is made up of a highly nonlinear, low birefringence spun fiber
(iXblue Photonics) [60]. At the 1550 nm driving wavelength, that fiber
exhibits normal group-velocity dispersion (corresponding to repulsive
interactions, or a positive effective mass), which has been selected to avoid
scalar parametric instabilities [55]. Overall, the resonator is 12 m-long,
corresponding to a roundtrip time $t_{\mathrm{R}}$ of 57 ns
($\mathrm{FSR}=17.54$ MHz), and has a resonance linewidth
$\Delta\omega/(2\pi)=650$ kHz. The resonator is driven by a 1550 nm-wavelength
continuous-wave laser (NKT Photonics) with an ultra-narrow linewidth $<1$ kHz,
ensuring coherent driving. The laser frequency can be varied with a
piezoelectric transducer, which provides the ability to scan the detuning.
Alternatively, we can use the transducer to lock the detuning at a set value
via a feedback loop using the technique discussed in Ref. [61]. The driving
laser is intensity-modulated with a Mach-Zehnder amplitude modulator driven by
a pulse pattern generator to generate flat-top $1.1$ ns-long pulses with a
period matching the resonator roundtrip time $t_{\mathrm{R}}$ [62]. Before
injection into the cavity, the driving pulses are amplified with an erbium-
doped fiber amplifier followed by an optical bandpass filter, which reduces
amplified spontaneous emission noise.

The setup includes three FPCs. The first one is placed before the input
coupler ($\mathrm{FPC_{in}}$) to align the polarization state of the driving
field with one of the two principal polarization modes of the resonator,
henceforth defining the $x$ mode. Note that, because of residual
birefringence, these principal modes correspond in practice to polarization
states which evolve along the fiber but map onto themselves at each roundtrip.
The second FPC, placed inside the resonator ($\mathrm{FPC}_{\pi}$), defines
the $\pi$ phase-shift defect. We adjust it for $\delta_{\pi}$ to be as close
as possible to zero, which is controlled by monitoring the linear cavity
resonances while scanning the driving laser frequency. Finally, the third FPC,
placed on the output port of the resonator ($\mathrm{FPC_{out}}$) and followed
by a polarization beam splitter (PBS), can be set so as to split the output
field either in terms of the $x$ and $y$ modes, or in terms of the $+$ and $-$
hybrid modes. The intensity of the two components are then recorded with
individual $12.5$ GHz photodiodes. The total intensity is recorded separately.

To proceed, we record the output intensity levels across our driving pulses
over subsequent resonator roundtrips using a real-time oscilloscope. A typical
measurement is shown as color plots in Figs. 3(a)–(c) in the form of a
vertical concatenation (bottom-to-top) of a 20-roundtrip sequence of real-time
oscilloscope traces plotted against time. Panels (a) and (b) correspond to the
intensities of the $+$ and $-$ hybrid modes while (c) is the total intensity.
The measurements have been obtained with 11 W of peak driving power ($X=50$)
and a roundtrip phase detuning locked to $\delta_{0}=1.16$ rad ($\Delta=10$).
Figures 3(a)–(b) reveal that under these conditions the hybrid mode intensity
levels exhibit a broken symmetry ($|\psi_{+}|^{2}\neq|\psi_{-}|^{2}$) as well
as an anti-phase alternating dynamics: the hybrid modes appear to exchange
their intensities one roundtrip to the next. This alternation is a key
signature of the modal Möbius topology. At the same time, the total intensity
[Fig. 3(c)] appears constant and presents no sign of the periodic alternation
of the hybrid modes. This points to a high level of symmetry between the two
hybrid modes and confirms the realization of the
$\psi_{+}\rightleftharpoons\psi_{-}$ exchange symmetry. We must note that this
remarkable level of symmetry is achieved without any fine tuning of parameters
and can persist for hours.

Further insights can be gained by measuring the high and low intensity levels
of the hybrid modes for a range of resonator detunings. The results of such
measurements are plotted in Fig. 3(d) for $\Delta$ ranging from $-4$ to $10$
and for the same driving power level as Figs. 3(a)–(c). Blue and orange
markers correspond to the $+$ and $-$ hybrid modes, respectively. For low
values of the detuning, the two hybrid modes are found to have the same
intensity and the intracavity field appears symmetric in terms of these modes.
In contrast, beyond $\Delta\simeq 2$, we observe instead two mirror-like,
asymmetric states, whose relative contrast increases with the detuning. The
emergence of these asymmetric states is found to correspond to the start of
the roundtrip-to-roundtrip alternating dynamics described above. These
measurements agree with numerically-calculated steady-state solutions of Eqs.
(13)–(14) and which are shown as black curves in Fig. 3(d) (dashed lines
correspond to unstable solutions). Overall, the data in Fig. 3(d) exhibit the
behavior of a pitchfork bifurcation characteristics of SSB. This result is
remarkable in that it is obtained without having to make any effort to realize
an exchange symmetry. In fact, we observe this behavior as soon as the
$y$-mode phase defect is sufficiently close to $\pi$ (to within about the
resonance linewidth).

To complete these observations, we have also made measurements in terms of the
$x$ and $y$ modes. The emergence of the P2 SSB regime is characterized by the
$y$-mode intensity rising above zero through parametric generation, but the
intensities of the $x$ and $y$ modes do not otherwise reveal the alternating
dynamics characteristics of the underlying Möbius topology. As described
above, the associated dynamics for the $y$-mode should be entirely contained
into a roundtrip-to-round phase step of $\pi$, which calls for a phase
sensitive measurement. To that end, we have used homodyne detection by mixing
the $x$ and $y$ output fields into a single-polarization $90^{\circ}$ optical
hybrid (Kylia). A pair of 40 GHz-bandwidth balanced detectors then provide in
phase, $I\propto\mathrm{Re}(\psi_{x}\psi_{y}^{*})$, and quadrature,
$Q\propto\mathrm{Im}(\psi_{x}\psi_{y}^{*})$, components. Measurements of $I$
and $Q$ over 1000 resonator roundtrips in the P2 SSB regime are reported in
Fig. 3(e), with cyan (respectively, purple) points corresponding to odd (even)
roundtrips. The arrangement of these two groups of points in diametrically
opposite positions in the complex plane reveal unequivocally that the relative
phase between the $x$ and $y$ fields varies in step of $\pi$ at each
roundtrip. The average phase step is found to be $(0.995\pm 0.020)\pi$. Again
this has been obtained without fine tuning of parameters and confirms that the
system is nonlinearly attracted to a roundtrip phase jump of $\pi$ as
predicted in Section III. Note that, in comparison with the rest of the
observations reported in Fig. 3, the data in Fig. 3(e) was obtained with a
slightly shorter but otherwise identical resonator (resonator length of $10.5$
m, roundtrip time $t_{\mathrm{R}}=50.6$ ns, and resonance linewidth
$\Delta\omega/(2\pi)=825$ kHz).

##  V Test of the exchange symmetry

Figure 4: Test results for randomness of the P2 SSB state selection process
using the NIST Statistical Test Suite. The bars indicate the percentage of
sequences (total of $2.4$ million events partitioned into 100 sub-sequences)
that pass the tests with a significance level of $0.01$. The recommended
minimum pass rate of 96 % (black dotted line) is satisfied for all tests. The
normalized driving power was set to $X\simeq 12$.

The results presented in the previous Section provide compelling evidence that
a $\pi$-phase defect combined with the Kerr nonlinearity does confer a robust
Möbius topology and exchange symmetry to a driven resonator, enabling SSB. In
particular, the reproducibility of this behavior and our ability to observe it
over long periods of time without any fine tuning of parameters strongly hint
that the exchange symmetry is exact and protected by the nonlinearity as
predicted by theory. In order to confirm this hypothesis further, we now
present a more stringent and quantitative test of the quality of the exchange
symmetry realized in our system.

Our test is based on the properties of SSB. Specifically, upon crossing the
SSB bifurcation threshold in our resonator, an initially symmetric state will
lose its stability in favor of one of two states with broken symmetry. Under
perfectly symmetrical conditions, this selection should be truly random.
Conversely, a deviation from symmetry will lead to a bias in the state
selection process [34, 63, 32]. The degree of randomness of the state
selection process is therefore a measure of the quality of the exchange
symmetry. To test for such randomness in practice, we have sinusoidally
modulated the resonator detuning at $5.2$ kHz around a mean (locked) value, so
as to cross the SSB threshold repeatedly, and measured the state selection
outcome. This measurement is based on the same resonator as that used to
obtain the results of Fig. 3(e) but with a driving beam modulated so as to
have 19 pulses simultaneously present in the resonator. Each pulse undergoes
SSB independently, providing multiple simultaneous realizations of the
experiment, thus enabling acquisition of a larger amount of data for better
statistical significance. The outcomes of the state selection process are
determined by analyzing the output pulses captured with our real-time
oscilloscope.

Over multiple runs, we have accumulated a sequence of $2.4$ million individual
events and expressed them as zero and one binary values. The randomness of
that sequence was then rigorously assessed using the NIST Statistical Test
Suite for random number generation (NIST STS-2.1.2) which consists of a series
of statistical tests used to detect non-randomness within a given data set
[64]. In each test, a $p$-value is generated which gives the probability of
this result occurring under the null hypothesis. We consider a $p$-value of
$0.01$ or lower to be evidence of non-randomness, or a failure. NIST
recommends that the minimum pass rate for each statistical test should be 96 %
(the whole sequence is partitioned into 100 individual sub-sequences for
testing). The results are presented in Fig. 4 which shows that all tests are
passed with a proportion well over 96 %, highlighting that we have no evidence
of any bias in the SSB state selection process. This confirms the robustness
of the exchange symmetry provided by the Möbius topology in our experiment.

##  VI Topological localized structures

Figure 5: P2 SSB localized structures. (a)–(c), P2 domain walls observed for
repulsive interactions ($\phi_{2}>0$) with $X=50$ and $\Delta=10$. (a)
Temporal intensity profiles of the $+$ (blue) and $-$ (orange) hybrid modes at
a selected odd (left) and even (right) roundtrip. The evolution of these
profiles over 20 subsequent resonator roundtrips is illustrated in (b), (c),
respectively, for the $+$ and $-$ hybrid modes. Inset: close up on one of the
domain walls. (d)–(f), same as (a)–(c) but for P2 bright symmetry-broken
cavity solitons observed for attractive interactions ($\phi_{2}<0$) with
$X=30$ and $\Delta=14.5$. (g)–(k) Breathing dynamics of the P2 bright
symmetry-broken cavity solitons observed with $X=10$ and $\Delta=5.2$. Same
format as the other two examples with the addition of (k) an extra pseudo-
color plot for the evolution of the total intensity and (h) a plot providing
details of the evolution of the intensity at the center of the structure
(blue/orange, $+$/$-$ hybrid mode; green, total intensity).

In the experimental results discussed so far, we have only considered cases
where the intracavity field is homogeneous across the nanosecond driving
pulses. The spatial extension of the driving provides however the opportunity
to support more complex, inhomogeneous field configurations. The P2 SSB regime
naturally extends to such structures because the mechanism giving rise to the
Möbius topology of our resonator and to the two-roundtrip alternating dynamics
is purely local. We now discuss this aspect in more detail.

Inhomogeneous field structures can be accounted for by introducing a kinetic
term into the Hamiltonian. This corresponds to adding terms of the form
$-i(\phi_{2}/2)\partial^{2}\psi_{\pm}/\partial\tau^{2}$ to, respectively, the
mean-field equation for the $+$ [Eq. (13)] and $-$ [Eq. (14)] hybrid modes.
Here $\tau$ is a fast time variable describing variations of the fields along
the resonator. In the optical context relevant to our experiments, $\phi_{2}$
represents chromatic dispersion integrated along the roundtrip of the
resonator. With these kinetic terms, Eqs. (13)–(14) take the form of coupled
driven-dissipative Gross-Pitaevskii equations. Such equations have been used,
e.g., to describe exciton-polariton condensates [65, 66, 37]. In optics, these
equations are known as Lugiato-Lefever equations (LLEs). The LLE was initially
introduced as a paradigmatic model of pattern formation in dissipative optical
systems [67]. It is also known to support localized structures referred to as
cavity solitons [68, 69]. In the temporal domain [70], these solitons underlie
the new key technology of microresonator broadband optical frequency combs
[71, 72]. In their coupled form, Eqs. (13)–(14) exhibit an even richer
dynamics [66, 61, 73]. Of particular relevance here are the recent
experimental observations of two-component localized structures emerging
through SSB, namely domain walls between homogeneous symmetry-broken states
[30] and bright symmetry-broken vector cavity solitons [74, 75]. Given the
formal equivalence of the equations describing these structures and our
system, it should be clear that these structures have a P2 counterpart; in
fact the only difference with what was previously reported should be the
roundtrip-to-roundtrip alternation of the two components.

We start by considering P2 domain walls, which exist in presence of repulsive
interactions ($\phi_{2}>0$). Experimental evidence are presented in Figs.
5(a)–(c). Panel (a) show intensity profiles of the $+$ (blue) and $-$ (orange)
hybrid modes at a selected odd (left) and even (right) roundtrip, while panels
(b) and (c) present the roundtrip-to-roundtrip evolution of those intensities
over 20 roundtrips as pseudo-color plots similar to those used in Fig. 3.
These plots reveal a field structure whose components are perfectly anti-
correlated and, at the same time, alternate roundtrip-to-roundtrip. The
driving pulses are subdivided into domains realizing, at any one time, one or
the other of the two homogenous symmetry-broken states. By symmetry, the other
component has the opposite domain structure. The domains are segregated by
sharp kink-like temporal transitions, which correspond to dissipative
(polarization) domain walls or PDWs [30, 76, 77]. A close up of a domain wall
obtained with a 70 GHz-bandwidth photodiode and _sampling_ oscilloscope is
shown in inset (this measurement is bandwidth limited with simulations
predicting a rise/fall time of 4 ps). We must also point out that the total
intensity (not shown) does not reveal any structure, neither the domain walls,
nor the roundtrip-to-roundtrip alternation. These observations were performed
with the same resonator and in the same conditions ($X=50$, $\Delta=10$) as
that used for the results of Figs. 3(a)–(c) (for which we have
$\phi_{2}\simeq+0.5\ \mathrm{ps/THz}$). We have only added a shallow 10 GHz
sinusoidal phase modulation imprinted onto the driving so as to align and trap
the PDWs onto a controlled temporal reference grid [78]. The domain walls were
excited spontaneously by ramping up the detuning through the SSB bifurcation
threshold.

Next we consider P2 bright symmetry-broken temporal cavity solitons. Because
these require attractive interactions, i.e., a negative effective mass or,
equivalently, anomalous chromatic dispersion ($\phi_{2}<0$), we have used here
a different resonator than for all the other observations reported so far in
this Article. The resonator was 86 m-long and entirely made up of standard
SMF28 (Corning) optical fiber with an input coupler that recirculates 95 % of
the light ($\theta=0.05$). The roundtrip time $t_{\mathrm{R}}=420$ ns, the
resonance linewidth $\Delta\omega/(2\pi)=57$ kHz, and $\phi_{2}=-1.7\
\mathrm{ps/THz}$. The driving and detection schemes were the same as before.
Experimental results are presented in Figs. 5(d)–(f) using the same layout as
for the P2 domain walls. Here we can observe a sequence of five bright
solitons, which are aligned on a $2.87$ GHz driving phase modulation grid
[78]. Spectral measurements (not shown) indicate a pulse duration of $1.6$ ps
(not resolved by the oscilloscope). These solitons are symmetry-broken with
one hybrid mode component dominating over the other. Due to the high driving
level ($X=30$, corresponding to $2.5$ W peak driving power, and with
$\Delta=14.5$), the contrast between the components is almost 100 %. Again,
the symmetry implies the existence of two mirror-like states, i.e., solitons
exhibiting a different dominant component, and our data confirms that these
simultaneously coexist in the resonator. Finally, we observe a regular
alternation of the components, with $+$ mode-dominant solitons becoming $-$
dominant and vice versa, each roundtrip to the next, as expected from the
Möbius topology.

Bright symmetry-broken cavity solitons are also known to go through a Hopf
bifurcation for lower detuning values, leading to a breathing dynamics. This
dynamics is characterized by slow in-phase oscillations of the intensities of
the two components of the soliton [75]. In Figs. 5(g)–(k), we provide evidence
of this dynamic in the P2 SSB regime. As for the other structures discussed
above, panels (i) and (j) show color plots of the evolution of the $+$ and $-$
hybrid mode temporal intensity profiles. Additionally, panel (k) displays the
corresponding evolution for the total intensity. The latter clearly reveals
the breathing dynamics, occurring here with a period of about 9 resonator
roundtrips, while the roundtrip-to-roundtrip alternation characteristics of
the P2 SSB regime is only detected in the individual components. The
superposition of breathing and alternation is better seen in panel (h), with
curves highlighting the evolution at the center of the soliton in terms of $+$
(blue), $-$ (orange), and total (green) intensities; these correspond to
cross-sections through the center of panels (i)–(k), respectively. Note that
we have selected here a lower driving level ($X=10$) than in Figs. 5(d)–(f),
with the detuning set to $\Delta=5.2$. This choice leads to less contrast
between the soliton components [compare Figs. 5(d) and (g)], and provides a
clearer illustration of the overall complexity of the P2 symmetry-broken
breather dynamics.

Finally, we must point out that all the structures reported in Fig. 5 could
consistently be maintained and observed for long periods of time, from 30 to
60 minutes. This confirms that the robustness of the Möbius topology extends
to complex inhomogeneous structures.

##  VII Discussion

We have described theoretically and experimentally a system characterized by a
complex interplay between topology, nonlinearity, and SSB. Specifically, we
have shown that a $\pi$ phase-defect can force a driven Kerr resonator with
two modes $x$ and $y$ towards an attractor arising through nonlinear
degenerate parametric interactions, so that the system displays a modal Möbius
topology. That topology is associated with an exchange symmetry, which
together with the nonlinearity, enables SSB. Because of the Möbius topology,
the two symmetry-broken states of our resonator alternate roundtrip-to-
roundtrip, which we refer to as P2 SSB. In the experiment, this alternation is
directly seen in the intensity of the $+$ and $-$ hybrid modes. Homodyne
measurements have also enabled us to detect the corresponding phase steps in
the evolution of the $x$ and $y$ modes of the resonator. Because the Möbius
topology arises through a purely local mechanism, it extends to inhomogeneous
field distributions, giving rise to a range of P2 localized structures which
we have also been able to observe. Thanks to the robustness associated with
the topology, SSB is realized without any bias or fine tuning of parameters.
This fact has been theoretically confirmed by an effective Floquet Hamiltonian
model and, experimentally, through rigorous testing of the randomness of the
symmetry-broken state selection statistics. The inherent robustness is also
seen in the fact that all the different behaviors we have reported can be
obtained without any sensitive adjustments of parameters, and are consistently
observed over long period of times, easily exceeding 30 minutes (typically
only limited by our detuning locking feedback loop). As the realization of SSB
is often hampered by biases and non-idealities, we believe that our findings
can have interest for a number of applications, in photonics, but more broadly
in other bosonic systems.

As a final remark, we would like to mention that phase-defect-induced
topological protection can be generalized to systems with more than two modes,
potentially leading to P3, P4, $\ldots$ regimes of SSB. In what follows, we
present brief arguments to illustrate the general idea, restricting our
attention to three modes for simplicity. Consider a resonator with modes $x$,
$y$, and $z$, where the $y$ and $z$ modes present phase defects of about,
respectively, $2\pi/3$ and $4\pi/3$ with respect to the $x$ mode. This could
be implemented, e.g., using etched waveguide-based adiabatic mode converters
or photonic crystals [48, 79]. The resonance frequencies of the three mode
families would be shifted in steps of a third of the FSR. With these modes, it
is possible to define hybrid modes $\alpha$, $\beta$, $\gamma$, such that the
hybrid mode amplitudes would be converted at each roundtrip
$\alpha\rightarrow\beta\rightarrow\gamma\rightarrow\alpha$, and display a
generalized Möbius topology over a cycle of three resonator roundtrips. This
will realize a cyclic $\mathbb{Z}_{3}$ symmetry. The nonlinear four-wave
interactions enabled by energy conservation are of the form $x+x\rightarrow
y+z$ (and permutations). Using the same approach as in Section III, it is easy
to show that these interactions make the $2\pi/3$ and $4\pi/3$ roundtrip phase
steps robust attractors of the system, making the $\mathbb{Z}_{3}$ symmetry
exact, and protecting the topology. We have further confirmed that parameters
exist where this system exhibits three symmetry-broken states, leading to P3
SSB, where these states cycle over three roundtrips.

As a particular application, we must note that the symmetry-broken states
described in our Article could potentially be used to realize artificial
spins. In the P2 SSB regime, these would correspond to spin-$1/2$ and could be
exploited to implement an Ising machine. Such a machine aims to find the
ground state of interacting spins described by the Ising Hamiltonian with an
appropriate physical system. The interest rests on the fact that many complex
combinatorial optimization problems relevant to modern society, such as drug
discovery [80] or the analysis of social networks [81], can be mapped onto an
Ising Hamiltonian [82, 83]. We must note that coherent optical Ising machines
based on degenerate parametric oscillation have been particularly successful
in recent years [35, 36]. A P2 SSB Ising machine would work along the same
principle, but the inherent topological robustness of the Möbius topology
could provide additional benefits. In this context, using more than two modes
would offer even more tantalizing possibilities. For example, P3 symmetry-
broken states would correspond to spin-$1$ particles, and open the way to the
realization of a Potts machine [84].

###### Acknowledgements.

We acknowledge financial support from The Royal Society of New Zealand, in the
form of Marsden Funding (18-UOA-310) as well as M.E.’s Rutherford Discovery
Fellowship (RDF-15-UOA-015). J.F. thanks the Conseil régional de Bourgogne
Franche-Comté, mobility (2019-7-10614), financial support from the CNRS, the
IRP Wall-IN project, and the FEDER, Solstice project. N.G. is supported by the
FRS-FNRS (Belgium), the EOS program (CHEQS project), and the ERC (TopoCold and
LATIS projects).

*

## Appendix A

We present here a classical derivation of the mean-field equations (13)–(14).
This derivation complements the Hamiltonian description of Section III, which
was derived in the conservative limit, by including dissipation and driving
explicitly. For completeness, we also consider modes with arbitrary
polarization states, i.e., arbitrary values of the $A$, $B$, and $C$
coefficients [see Eq. (2)]. We start from the classical equations of motion
for the two modes, i.e., Eq. (3) and a similar equation for $\psi_{x}$. These
equations are complemented by boundary conditions,

$\begin{pmatrix}\psi_{x}\\\
\psi_{y}\end{pmatrix}^{(m+1)}=e^{-\alpha}M\begin{pmatrix}\psi_{x}\\\
\psi_{y}\end{pmatrix}^{(m)}+\sqrt{\theta}\,S_{\mathrm{in}}\begin{pmatrix}\cos\chi\\\
\sin\chi\end{pmatrix}.$ (15)

Here, $\alpha=t_{\mathrm{R}}\Delta\omega/2$ is the dissipation rate of the
resonator, as introduced before, $S_{\mathrm{in}}$ is the total driving
amplitude, with the angle $\chi$ describing a split of the driving between the
two modes ($\chi=0$ corresponds to only driving the $x$ mode as previously
considered, and $S_{x}=S_{\mathrm{in}}\cos\chi$), and the matrix $M$
represents the action of the detunings,

$M=\begin{pmatrix}e^{-i\delta_{0}}&0\\\
0&e^{i\pi}e^{-i(\delta_{0}-\delta_{\pi})}\end{pmatrix}.$ (16)

We proceed by restricting ourselves to high-Q resonators operated close to
resonance and by following an approach similar to that of Refs. [85, 86]. In
these conditions, $\alpha$, $\theta$, $\delta$, and $\delta_{\pi}$ can be
assumed $\ll 1$. Similarly, all the nonlinear terms in the equations of motion
are small and these equations can be integrated over one roundtrip time
$t_{\mathrm{R}}$ by a single Euler step, as we have done with Eq. (4).
Combining the result with the boundary conditions, Eq. (15), and expanding all
small terms at first order, we can express the fields at roundtrip $m+1$ in
terms of those at roundtrip $m$ as

$\displaystyle\psi_{x}^{(m+1)}$
$\displaystyle=\phantom{-}\left[1-\alpha-i\delta_{0}+i\,gt_{\mathrm{R}}\left(A|\psi_{x}^{(m)}|^{2}+B|\psi_{y}^{(m)}|^{2}\right)\right]\psi_{x}^{(m)}+i\,gt_{\mathrm{R}}C\,{\psi_{x}^{(m)}}^{*}{\psi_{y}^{(m)}}^{2}+\sqrt{\theta}\,S_{\mathrm{in}}\cos\chi,$
(17) $\displaystyle\psi_{y}^{(m+1)}$
$\displaystyle=-\left[1-\alpha-i\left(\delta_{0}-\delta_{\pi}\right)+i\,gt_{\mathrm{R}}\left(A|\psi_{y}^{(m)}|^{2}+B|\psi_{x}^{(m)}|^{2}\right)\right]\psi_{y}^{(m)}-i\,gt_{\mathrm{R}}C\,{\psi_{y}^{(m)}}^{*}{\psi_{x}^{(m)}}^{2}+\sqrt{\theta}\,S_{\mathrm{in}}\sin\chi.$
(18)

Above, we can observe that the leading contribution for the $y$ mode is a
change of sign, as expected from the $\pi$ phase defect.

The next step is to iterate Eqs. (17)–(18) and seek expressions for
$\psi_{x,y}^{(m+2)}$ in terms of the $m$th roundtrip fields. At first order,
we can assume that, in the nonlinear terms that result,
$\psi_{x}^{(m+1)}\approx\psi_{x}^{(m)}$ and
$\psi_{y}^{(m+1)}\approx-\psi_{y}^{(m)}$. We can also observe that the $y$
component of the driving will cancel out. Because of the double action of the
$\pi$ phase defect over _two_ roundtrips, we then find that the changes in the
fields over two roundtrips, $\psi_{x,y}^{(m+2)}-\psi_{x,y}^{(m)}$, are first
order quantities. This enables us to introduce a slow-time derivative over two
resonator roundtrips, defined as
$d./dt=\left[.^{(m+2)}-.^{(m)}\right]/(2t_{\mathrm{R}})$, which transforms the
difference equations we get into the following differential equations (this
latter step is analogous to the stroboscopic Floquet approach used in Section
III),

$\displaystyle t_{\mathrm{R}}\frac{d\psi_{x}}{dt}$
$\displaystyle=\Bigl{[}-\alpha-i\delta_{0}+i\,gt_{\mathrm{R}}\left(A|\psi_{x}|^{2}+B|\psi_{y}|^{2}\right)\Bigr{]}\psi_{x}$
$\displaystyle\qquad\quad+i\,gt_{\mathrm{R}}C\,\psi_{x}^{*}\psi_{y}^{2}+\sqrt{\theta}\,S_{\mathrm{in}}\cos\chi,$
(19) $\displaystyle t_{\mathrm{R}}\frac{d\psi_{y}}{dt}$
$\displaystyle=\Bigl{[}-\alpha-i(\delta_{0}-\delta_{\pi})+i\,gt_{\mathrm{R}}\left(A|\psi_{y}|^{2}+B|\psi_{x}|^{2}\right)\Bigr{]}\psi_{y}$
$\displaystyle\qquad\quad+i\,gt_{\mathrm{R}}C\,\psi_{y}^{*}\psi_{x}^{2}.$ (20)

where we have dropped the roundtrip index $m$.

The last step is to express the equations above in terms of the $+$ and $-$
hybrid mode amplitudes defined by Eqs. (1). This leads to Eqs. (13)–(14), with

$\displaystyle A^{\prime}$ $\displaystyle=\frac{1+B-C}{2}A,$ $\displaystyle
B^{\prime}$ $\displaystyle=(1+C)A,$ (21)

but with additional parametric terms,
$i\,gt_{\mathrm{R}}C^{\prime}\psi_{+}^{*}\psi_{-}^{2}$ for Eq. (13) and
$i\,gt_{\mathrm{R}}C^{\prime}\psi_{-}^{*}\psi_{+}^{2}$ for Eq. (14), where

$C^{\prime}=\frac{1-B-C}{2}A.$ (22)

We can observe that these terms do not break the
$\psi_{+}\rightleftharpoons\psi_{-}$ exchange symmetry of the equations, and
overall do not change the phenomenology. When the $x$ and $y$ modes correspond
to linear polarization states, $C^{\prime}=0$ as in Eq. (6).

Note that the above analysis assumes a $\pi$ phase defect that is purely
localized. Additional considerations show that distributing the phase defect
over a certain fraction of the resonator length amounts to a phase-mismatch
for the parametric interaction (non conservation of momentum). This leads to
less efficient generation of the $y$ mode but P2 SSB remains possible. When
parametric generation cannot overcome the losses of the resonator anymore, the
P2 SSB regime eventually disappears.

Kinetic terms (chromatic dispersion) can also be included in our derivation in
a straightforward manner.

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