# Control spontaneous symmetry breaking of photonic chirality with
reconfigurable anomalous nonlinearity

Chaohan Cui  James C. Wyant College of Optical Sciences, The University of
Arizona, Tucson, Arizona 85721, USA  Liang Zhang  James C. Wyant College of
Optical Sciences, The University of Arizona, Tucson, Arizona 85721, USA
Linran Fan<EMAIL_ADDRESS>James C. Wyant College of Optical
Sciences, The University of Arizona, Tucson, Arizona 85721, USA

Spontaneous symmetry breaking in nonlinear systems provides a unified method
to understand vastly different phenomena, ranging from Higgs mechanism [1] and
superconductivity [2] to ecological stability [3] and genome generation [4].
Spontaneous symmetry breaking is typically considered as the intrinsic
property of nonlinear systems with fixed occurrence condition and property, as
the form and magnitude of nonlinear interactions cannot be modified [5]. Here,
we report the development of reconfigurable Kerr optical nonlinearity to
control spontaneous symmetry breaking. This is achieved through the
interference between the intrinsic Kerr and cascaded second-order nonlinear
processes [6]. Anomalous Kerr effects including negative self-phase modulation
and strength tuning between competing nonlinear processes have been
demonstrated. With the reconfigurable Kerr nonlinearity, we realize the in-
situ prohibition and facilitation of spontaneous symmetry breaking of photonic
chirality. This work could empower the experimental study of spontaneous
symmetry breaking in unexplored regimes and inspire the development of novel
photonic functions.

The chiral degree of freedom can provide the unique propagation-direction-
dependent property of photons, which is different from conventional photonic
functions based on spatial [7, 8], polarization [9, 10], and spectral-temporal
[11, 12, 13, 14] operations. Photonic chirality is typically introduced by
explicit asymmetries including deformed photonic structures [15, 16], gain-
loss modulation in non-Hermitian systems [17, 18, 19], and interaction with
magnetic materials [20, 21]. Such explicit symmetry breaking of photonic
chirality has been used to build optical isolators [22, 23, 24], realize
topological photonic states [25, 26, 27], and develop unidirectional micro-
lasers [28, 29] in the classical regime, as well as improve single-photon
sources [30, 31, 32] and develop logic operations in the quantum regime [33,
34, 35].

Besides explicit approaches, the symmetry breaking of photonic chirality can
also happen spontaneously with Kerr nonlinearity [36, 37, 38, 39, 40, 41].
This is introduced by the imbalance between self- and cross-phase modulation,
two fundamental Kerr nonlinear effects. In this case, photons can show
propagation-direction-dependent property even though the system configuration
is completely symmetric. Unlike explicit approaches where the symmetry
breaking can be engineered conveniently by photonic structures, the capability
to control the condition of spontaneous symmetry breaking remains exclusive.
The key challenge is the direct modification of Kerr nonlinearity, which
typically has fixed form and magnitude determined by material intrinsic
properties [42].

Here, we report the approach to break the intrinsic restriction of Kerr
nonlinearity, and demonstrate the controlled spontaneous symmetry breaking of
photonic chirality. This is realized through the cascaded second-order
nonlinearity, which is equivalent to reconfigurable artificial Kerr
nonlinearity. In particular, we can selectively control the self-phase
modulation coefficient without affecting the cross-phase modulation. The
capability to tune the relative strength between self- and cross-phase
modulation enables the modification of the spontaneous symmetry breaking
condition for photonic chirality.

Figure 1: Schematics for reconfigurable Kerr nonlinearity. a, Interference
between the intrinsic Kerr and the cascaded second-order nonlinear processes
in the photonic ring cavity. The intrinsic Kerr nonlinearity introduces
resonance shift towards lower frequency (red-shift). This can be modeled as
four-wave mixing with two pump photons (green) annihilated to re-generate two
photons at the same frequency. The cascaded second-order nonlinearity
introduces resonance shift towards higher frequency (blue-shift) with positive
cavity detune. This can be modeled as artificial four-wave mixing, where two
pump photons (green) are first combined to generate one second-harmonic photon
(orange), which drives the parametric down-conversion to re-generate two
photons at the pump frequency. The overall frequency shift is the coherent
combination of the two parallel processes. b, Tuning range of the frequency
shift induced by self-phase modulation with anomalous Kerr nonlinearity. c,
The vector representation of the effective Kerr nonlinear coefficient $g_{\rm
eff}$. By tuning the cavity detune $\Delta$, the effective Kerr nonlinearity
can change along the circle to give frequency shift towards lower (red) and
higher (blue) frequency. Black dashed arrow: intrinsic Kerr nonlinearity;
orange dashed arrow: cascaded second-order nonlinearity; green solid arrow:
effective Kerr nonlinearity.

Our scheme for reconfigurable Kerr nonlinearity can be implemented in an
integrated photonic ring cavity with second-order nonlinearity. We consider
the cavity response to a continuous-wave pump field. The pump field is subject
to two parallel nonlinear processes. The first one is the conventional self-
phase modulation due to the intrinsic Kerr nonlinearity (top path in Fig. 1a).
The cavity instantaneous resonance is shifted by $\delta\omega\propto-
g_{3}\cdot N_{\mathrm{p}}$, with $g_{3}$ the Kerr nonlinear coefficient and
$N_{\mathrm{p}}$ the intra-cavity pump photon number. As all photonic
materials have positive Kerr nonlinear coefficients [42], intrinsic Kerr
nonlinearity causes the resonance shift towards the lower frequency.

Figure 2: Anomalous Kerr nonlinearity. a, Calculated transmission spectrum as
the pump frequency is scanned across the resonance. The transmission minimum
shifts to higher (blue) and lower (red) frequency with negative and positive
Kerr nonlinear coefficients respectively. The transmission spectrum remains
symmetric (grey) if the Kerr nonlinear coefficient is zero. b, Measured
resonance shift dependence on the cavity detune. Gray dashed line: zero
frequency shift. Green dashed line: resonance shift induced by the intrinsic
Kerr nonlinearity. The blue, red, and grey arrows show the condition to
achieve maximum blue shift, maximum red shift, and zero shift, with the
corresponding transmission spectrum shown in (e). c, Measured FWHM dependence
on the cavity detune. d, Measured extinction ratio dependence on the cavity
detune. e, Measured transmission spectrum with maximum blue shift (blue),
maximum red shift (red), and zero shift (grey). f, Measured transmission
spectrum with different cavity detune values. White dashed line: minimum
transmission position; dark dashed lines: half maximum position. Solid lines
in b-e are calculated results. Horizontal and vertical error bars in (b)-(d)
are estimated based on device temperature stability and transmission variance
respectively.

In parallel with the intrinsic Kerr nonlinearity, the pump field also
undergoes the cascaded second-order nonlinear process consisted of second-
harmonic generation and degenerate parametric down-conversion (bottom path in
Fig. 1a). The pump field is first doubled to generate the second-harmonic
field. Then the second-harmonic field drives the degenerate parametric down-
conversion to re-generate the pump field. Therefore, the cascaded second-order
nonlinearity for the pump field is equivalent to an effective Kerr
nonlinearity [6]. In this cascaded processes, the second-order nonlinear
strength $g_{2}$, the second-harmonic resonance linewidth $\gamma$, and the
frequency detune $\Delta$ between the pump and second-harmonic resonances
collectively determine the effective Kerr nonlinear coefficient. The
instantaneous resonance shift is
$\delta\omega\propto-\mathrm{Re}\left[\frac{i|g_{2}|^{2}}{-i\Delta+\gamma/2}\right]\cdot
N_{\mathrm{p}}$ (Supplementary Information I). With a positive detune
$\Delta>0$, the cascaded second-order nonlinearity can lead to the resonance
shift towards higher frequency. This is equivalent to self-phase modulation
with negative Kerr nonlinear coefficients. The overall resonance shift, after
the coherent combination of the intrinsic Kerr and cascaded second-order
nonlinear processes, becomes

$\displaystyle\delta\omega$
$\displaystyle\propto-\mathrm{Re}\left[g_{3}+\frac{i|g_{2}|^{2}}{-i\Delta+\gamma/2}\right]\cdot
N_{\rm p}\stackrel{{\scriptstyle\text{def}}}{{=}}-\mathrm{Re}\left[g_{\rm
eff}\right]\cdot N_{\rm p}$ (1)

with $g_{\rm eff}$ the effective Kerr nonlinear coefficient (Fig. 1c). With
strong second-order nonlinearity $\xi=|g_{2}|^{2}/(g_{3}\gamma)>1$, the
overall resonance shift can be reconfigured towards either lower or higher
frequency depending on the cavity detune $\Delta$ (Fig. 1b). A completely
passive cavity without nonlinear resonance shift can also be obtained at
cavity detune $\Delta/\gamma=\frac{1}{2}(\xi\pm\sqrt{\xi^{2}-1})$. Besides the
reconfigurable resonance shift, the cascaded second-order nonlinearity also
introduces extra loss for the pump field as power is converted into the
second-harmonic field. This is manifested as the imaginary part of the
effective Kerr nonlinearity $g_{\rm eff}$.

The reconfigurable Kerr nonlinearity is demonstrated with an aluminum nitride
ring cavity [43]. Phase-matching condition is realized by geometric dispersion
engineering [44]. Due to self-phase modulation, the transmission spectrum of
the pump field deviates from the symmetric Lorentzian shape, and can be used
to quantify the Kerr nonlinear coefficient (Fig. 2a). The speed to scan the
pump frequency is sufficiently fast to avoid the frequency shift induced by
the thermal effect (Supplementary Information III). The pump power is
maintained below the four-wave-mixing threshold to avoid the generation of
optical sidebands. In this low power regime, the frequency difference between
the minimum transmission point and the Lorentzian middle point is proportional
to the real part of the effective Kerr nonlinear coefficient. The imaginary
part can also be obtained through the change in the resonance full-width at
half-maximum (FWHM).

Figure 3: In-situ control of spontaneous symmetry breaking of photonic
chirality. a, Photonic ring cavity with bi-directional pump. b, Calculated
spontaneous symmetry breaking condition with different relative strengths
between self-phase modulation (SPM) and cross-phase modulation (XPM). c,
Cavity detune configuration of the cascaded second-order nonlinearity to
realize different SPM/XPM relative strengths. d, Cavity status under strong
pump with SPM/XPM=1. Symmetric state with equal powers is always stable, as
the resonance shifts are identical in two directions. e, Cavity status under
strong pump with SPM/XPM=1/2. The optical field in one direction dominates, as
cross-phase modulation introduces twice resonance shift than self-phase
modulation. f, Cavity status under strong pump with SPM/XPM=0. The power
difference between different directions become larger as the mismatch between
self- and cross-phase modulation increases.

The cavity detune $\Delta$ is controlled by changing the device temperature,
as the pump and second-harmonic resonances have different thermal-optic
coefficients (Supplementary Information IV). We can clearly observe the change
in the pump transmission spectrum with different cavity detune values (Fig.
2f). The resonance shift follows a Fano shape with respect to the cavity
detune (Fig. 2b). This agrees with the interpretation that the intrinsic Kerr
and cascaded second-order nonlinear processes are coherently combined to give
rise to the effective Kerr nonlinearity. The pump resonance shows maximum
shift towards higher frequency with the cavity detune $\Delta\approx
0.5\gamma$, proving the negative self-phase modulation (blue in Fig. 2e). By
changing the device temperature, we can further reconfigure the pump resonance
to shift towards lower frequency and even exhibit no frequency shift (red and
grey in Fig. 2e). In addition, we also notice that FWHM of the pump resonance
follows the Lorentzian shape with respect to the cavity detune (Fig. 2c). The
largest FWHM is obtained with zero detune $\Delta=0$, in which case maximum
power is converted from the pump field to the second-harmonic field. The power
conversion can be treated as extra intrinsic loss for the pump resonance. As
the cavity is weakly coupled to the bus waveguide with fixed coupling
strength, the increase of the intrinsic loss leads to the drop of the
resonance extinction ratio (Fig. 2d).

Figure 4: Control of spontaneous symmetry breaking condition. a, Relative
power imbalance with fixed cavity detune $\Delta\approx 4\gamma$ and varying
pump power $P_{\rm in}$. The threshold power for intrinsic Kerr nonlinearity
is marked as dashed black line. b, Relative power imbalance with varying
cavity detune $\Delta$ and fixed pump power $P_{\rm in}\approx 1.07P_{0}$.
$P_{0}\approx 51$ mW is the threshold power for spontaneous symmetry breaking
with intrinsic Kerr nonlinearity. c-e, Measured output second-harmonic power
in clockwise (blue) and counterclockwise (red) directions with cavity detune
$\Delta\approx 4\gamma$ and pump power $P_{\rm in}\approx 0.7P_{0}$, $P_{\rm
in}\approx 1.07P_{0}$, $P_{\rm in}\approx 1.07P_{0}$, corresponding to the
purple, blue, and red circle points in (a) respectively. f, Cavity status with
different cavity detune $\Delta$ and pump power $P_{\rm in}$. Blue solid line:
calculated threshold condition for spontaneous symmetry breaking. Shaded blue
area: calculated regime with asymmetric power in two directions. Black dashed
line: threshold power for intrinsic Kerr nonlienarity. Purple points: measured
cavity output with equal power in two directions. Blue/red points: measured
cavity output with optical field in one direction dominant. Orange arrow: the
same condition change as (a). Green arrow: the same condition change as (b).
Power and cavity detune error bars are estimated based on device temperature
stability and transmission variance respectively. Additional data for (f) are
shown in Supplementary Information V.

The key to realize reconfigurable Kerr nonlinearity is the cascaded second-
order nonlinear process, which requires the phase matching condition. It
provides the unique capability that we can selectively modify the target Kerr
nonlinear process without affecting the others. This is achieved by designing
the phase matching condition only for the optical fields involved in the
target Kerr nonlinear process. For example, we consider the same photonic ring
cavity with second-order nonlinearity, but pumped in both clockwise and
counter-clockwise directions (Fig. 3a). As phase matching condition is only
valid for optical fields along one direction, only the self-phase modulation
coefficient is modified by the cascaded second-order nonlinearity. To modify
the cross-phase modulation, we need sum-frequency generation between counter-
propagating fields. This requires zero wave-vector for the second-harmonic
field, which cannot be realized. Therefore, cross-phase modulation coefficient
remains unchanged.

The bi-directional configuration is of critical importance for numerous
photonic applications including optical gyroscope [45], single-particle
detection [46], and soliton referencing [47]. It is recently realized that
spontaneous symmetry breaking of photonic chirality can also be introduced in
this bi-directional configuration due to the imbalance between self- and
cross-phase modulation [37, 36, 38, 39, 40]. For intrinsic Kerr nonlinearity,
the ratio between self- and cross-phase modulation has a fixed value of $1/2$
in solid-state materials [48]. The optical field along one direction causes
twice resonance shift for the opposite direction. Therefore, the optical field
with smaller power will experience larger resonance shift, which further
suppresses its amplitude due to the larger mismatch between the input
frequency and cavity resonance. With this positive feedback mechanism, the
symmetric state with equal power in both directions becomes unstable even the
input pumps have the same power (Supplementary Information II). Then the
optical field in one direction becomes stronger, showing spontaneous chiral
behavior (Fig. 3e).

Obviously, the condition for spontaneous symmetry breaking is directly
determined by the relative strength between self- and cross-phase modulation.
With the capability to selectively control self-phase modulation without
affecting cross-phase modulation, we can engineer a single cavity to exhibit
different behaviors for spontaneous symmetry breaking (Fig. 3b-f). If we
enhance the self-phase modulation to the same strength of cross-phase
modulation, spontaneous symmetry breaking can be prohibited as optical fields
in both directions will experience the same resonance shift regardless of the
power distribution (Fig. 3d). If we suppress the self-phase modulation,
spontaneous symmetry breaking can be realized more easily as the resonance
shift difference is larger with lower pump power (Fig. 3f).

To observe spontaneous symmetry breaking, optical fields with identical
frequency and power are coupled into the cavity in both directions
(Supplementary Information III). When the optical frequency is scanned across
the pump resonance, we monitor the output second-harmonic power, which is
proportional to the square of the intra-cavity pump photon number. The
photonic chirality $C$ is defined as the relative intra-cavity power imbalance

$\displaystyle
C=\frac{(N_{\mathrm{p}}^{\mathrm{CCW}}-N_{\mathrm{p}}^{\mathrm{CW}})}{(N_{\mathrm{p}}^{\mathrm{CCW}}+N_{\mathrm{p}}^{\mathrm{CW}})}$
(2)

with $N_{\mathrm{p}}^{\mathrm{CCW}}$ and $N_{\mathrm{p}}^{\mathrm{CW}}$ the
intra-cavity pump photon number along clockwise and counter-clockwise
directions respectively. Larger power imbalance indicates stronger spontaneous
symmetry breaking of photonic chirality.

We first vary the pump power while fixing the cavity detune at which the self-
phase modulation is suppressed (Fig. 4a). With low pump power, the optical
fields have the same amplitude in both directions (Fig. 4c). By increasing the
pump power, we observe the spontaneous symmetry breaking with the optical
field in one direction significantly stronger than the other (Fig. 4d&e). Due
to the suppression of the self-phase modulation, the threshold power for
spontaneous symmetry breaking is significantly lower compared to the intrinsic
Kerr nonlinearity case (Fig. 4a). Next, we fix the pump power, and vary the
cavity detune $\Delta$ for the cascaded second-order nonlinearity (Fig. 4b).
With negative cavity detune, the self-phase modulation is enhanced. Therefore,
the optical fields along both directions remain the same. By increasing the
cavity detune, the imbalance between self- and cross-phase modulation becomes
larger, leading to the spontaneous symmetry breaking. As shown in Fig. 4f, the
spontaneous symmetry breaking condition is collectively determined by the pump
power and cavity detune. The threshold pump power can be continuously tuned by
the cavity detune. Therefore, we can either prevent or facilitate the
occurrence of spontaneous symmetry breaking by increasing or decreasing the
threshold pump power.

In summary, we have demonstrated the capability to control the spontaneous
symmetry breaking of photonic chirality. This is achieved by directly
modifying Kerr nonlinear interactions with the cascaded second-order
nonlinearity. Beyond the fixed intrinsic Kerr nonlinearity, we can realize
reconfigurable anomalous Kerr effects, including negative self-phase
modulation and strength tuning between competing Kerr nonlinear processes.
This flexible mechanism can find broad applications in photonic technologies
based on Kerr nonlinearity, such as frequency comb generation, quantum state
generation, and frequency conversion. Moreover, the control of spontaneous
symmetry breaking condition can provide novel insights to the study of
nonlinear dynamics in broad fields ranging from condensed-matter physics to
nanomechanics.

##  .1 Acknowledgments

CC, LZ, and LF acknowledge the support from U.S. Department of Energy, Office
of Advanced Scientific Computing Research (Field Work Proposal ERKJ355);
Office of Naval Research (N00014-19-1-2190); National Science Foundation
(ECCS-1842559). Device fabrication is performed in the OSC cleanrooms at the
University of Arizona and the cleanroom of Arizona State University.

##  .2 Author Contributions Statement

CC and LF conceived the idea, designed the experiment, and wrote the
manuscript. LZ fabricated the device. CC derived the theoretical model and
operated the experiment. LF supervised the work.

##  .3 Competing Interests Statement

The authors declare no competing interests.

## References

  * Bernstein [1974] J. Bernstein, Reviews of Modern Physics 46, 7 (1974).
  * Baskaran and Anderson [1988] G. Baskaran and P. W. Anderson, Physical Review B 37, 580 (1988).
  * Borile  _et al._ [2012] C. Borile, M. A. Muñoz, S. Azaele, J. R. Banavar, and A. Maritan, Physical Review Letters 109, 038102 (2012).
  * Takeuchi  _et al._ [2017] N. Takeuchi, P. Hogeweg, and K. Kaneko, Nature Communications 8, 250 (2017).
  * Malomed [2013] B. A. Malomed,  _Spontaneous symmetry breaking, self-trapping, and Josephson oscillations_ (Springer, 2013).
  * Cui  _et al._ [2022] C. Cui, L. Zhang, and L. Fan, Nature Physics 18, 497 (2022).
  * Efron [1994] U. Efron,  _Spatial light modulator technology: materials, devices, and applications_ , Vol. 47 (CRC press, 1994).
  * Zhao  _et al._ [2015] N. Zhao, X. Li, G. Li, and J. M. Kahn, Nature Photonics 9, 822 (2015).
  * Han and Li [2005] Y. Han and G. Li, Optics Express 13, 7527 (2005).
  * Crespi  _et al._ [2011] A. Crespi, R. Ramponi, R. Osellame, L. Sansoni, I. Bongioanni, F. Sciarrino, G. Vallone, and P. Mataloni, Nature Communications 2, 1 (2011).
  * Fan  _et al._ [2016] L. Fan, C.-L. Zou, M. Poot, R. Cheng, X. Guo, X. Han, and H. X. Tang, Nature Photonics 10, 766 (2016).
  * Lukens and Lougovski [2017] J. M. Lukens and P. Lougovski, Optica 4, 8 (2017).
  * Fan  _et al._ [2019] L. Fan, C.-L. Zou, N. Zhu, and H. X. Tang, Nature Photonics 13, 323 (2019).
  * Raymer and Walmsley [2020] M. G. Raymer and I. A. Walmsley, Physica Scripta 95, 064002 (2020).
  * Redding  _et al._ [2012] B. Redding, L. Ge, Q. Song, J. Wiersig, G. S. Solomon, and H. Cao, Physical Review Letters 108, 253902 (2012).
  * Cao and Wiersig [2015] H. Cao and J. Wiersig, Reviews of Modern Physics 87, 61 (2015).
  * Miao  _et al._ [2016] P. Miao, Z. Zhang, J. Sun, W. Walasik, S. Longhi, N. M. Litchinitser, and L. Feng, Science 353, 464 (2016).
  * Feng  _et al._ [2017] L. Feng, R. El-Ganainy, and L. Ge, Nature Photonics 11, 752 (2017).
  * El-Ganainy  _et al._ [2018] R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, Nature Physics 14, 11 (2018).
  * Decker  _et al._ [2007] M. Decker, M. Klein, M. Wegener, and S. Linden, Optics Letters 32, 856 (2007).
  * Liu  _et al._ [2014] M. Liu, D. A. Powell, I. V. Shadrivov, M. Lapine, and Y. S. Kivshar, Nature Communications 5, 4441 (2014).
  * Sayrin  _et al._ [2015] C. Sayrin, C. Junge, R. Mitsch, B. Albrecht, D. O’Shea, P. Schneeweiss, J. Volz, and A. Rauschenbeutel, Physical Review X 5, 041036 (2015).
  * Xia  _et al._ [2018] K. Xia, F. Nori, and M. Xiao, Physical Review Letters 121, 203602 (2018).
  * Sohn  _et al._ [2021] D. B. Sohn, O. E. Örsel, and G. Bahl, Nature Photonics 15, 822 (2021).
  * Raghu and Haldane [2008] S. Raghu and F. D. M. Haldane, Physical Review A 78, 033834 (2008).
  * Hafezi  _et al._ [2011] M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, Nature Physics 7, 907 (2011).
  * Zhao  _et al._ [2019] H. Zhao, X. Qiao, T. Wu, B. Midya, S. Longhi, and L. Feng, Science 365, 1163 (2019).
  * Peng  _et al._ [2016] B. Peng, Ş. K. Özdemir, M. Liertzer, W. Chen, J. Kramer, H. Yılmaz, J. Wiersig, S. Rotter, and L. Yang, Proceedings of the National Academy of Sciences 113, 6845 (2016).
  * Carlon Zambon  _et al._ [2019] N. Carlon Zambon, P. St-Jean, M. Milićević, A. Lemaître, A. Harouri, L. Le Gratiet, O. Bleu, D. Solnyshkov, G. Malpuech, I. Sagnes, _et al._ , Nature Photonics 13, 283 (2019).
  * Mitsch  _et al._ [2014] R. Mitsch, C. Sayrin, B. Albrecht, P. Schneeweiss, and A. Rauschenbeutel, Nature Communications 5, 1 (2014).
  * Söllner  _et al._ [2015] I. Söllner, S. Mahmoodian, S. L. Hansen, L. Midolo, A. Javadi, G. Kiršanskė, T. Pregnolato, H. El-Ella, E. H. Lee, J. D. Song, _et al._ , Nature Nanotechnology 10, 775 (2015).
  * Cui  _et al._ [2021] C. Cui, L. Zhang, and L. Fan, Optics Letters 46, 4753 (2021).
  * Junge  _et al._ [2013] C. Junge, D. O’shea, J. Volz, and A. Rauschenbeutel, Physical Review Letters 110, 213604 (2013).
  * Shomroni  _et al._ [2014] I. Shomroni, S. Rosenblum, Y. Lovsky, O. Bechler, G. Guendelman, and B. Dayan, Science 345, 903 (2014).
  * Lodahl  _et al._ [2017] P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller, Nature 541, 473 (2017).
  * Cao  _et al._ [2017] Q.-T. Cao, H. Wang, C.-H. Dong, H. Jing, R.-S. Liu, X. Chen, L. Ge, Q. Gong, and Y.-F. Xiao, Physical Review Letters 118, 033901 (2017).
  * Del Bino  _et al._ [2017] L. Del Bino, J. M. Silver, S. L. Stebbings, and P. Del’Haye, Scientific Reports 7, 1 (2017).
  * Garbin  _et al._ [2020] B. Garbin, J. Fatome, G.-L. Oppo, M. Erkintalo, S. G. Murdoch, and S. Coen, Physical Review Research 2, 023244 (2020).
  * Cao  _et al._ [2020] Q.-T. Cao, R. Liu, H. Wang, Y.-K. Lu, C.-W. Qiu, S. Rotter, Q. Gong, and Y.-F. Xiao, Nature Communications 11, 1 (2020).
  * Silver  _et al._ [2021] J. M. Silver, L. Del Bino, M. T. Woodley, G. N. Ghalanos, A. Ø. Svela, N. Moroney, S. Zhang, K. T. Grattan, and P. Del’Haye, Optica 8, 1219 (2021).
  * Xu  _et al._ [2021] G. Xu, A. U. Nielsen, B. Garbin, L. Hill, G.-L. Oppo, J. Fatome, S. G. Murdoch, S. Coen, and M. Erkintalo, Nature Communications 12, 1 (2021).
  * Gaeta  _et al._ [2019] A. L. Gaeta, M. Lipson, and T. J. Kippenberg, Nature Photonics 13, 158 (2019).
  * Xiong  _et al._ [2012] C. Xiong, W. H. Pernice, X. Sun, C. Schuck, K. Y. Fong, and H. X. Tang, New Journal of Physics 14, 095014 (2012).
  * Zhang  _et al._ [2021] L. Zhang, C. Cui, J. Yan, Y. Guo, J. Wang, and L. Fan, arXiv preprint arXiv:2111.12784 (2021).
  * Lai  _et al._ [2019] Y.-H. Lai, Y.-K. Lu, M.-G. Suh, Z. Yuan, and K. Vahala, Nature 576, 65 (2019).
  * Chen  _et al._ [2017] W. Chen, Ş. Kaya Özdemir, G. Zhao, J. Wiersig, and L. Yang, Nature 548, 192 (2017).
  * Yang  _et al._ [2017] Q.-F. Yang, X. Yi, K. Y. Yang, and K. Vahala, Nature Photonics 11, 560 (2017).
  * Boyd [2020] R. W. Boyd,  _Nonlinear optics_ (Academic press, 2020).

##  .4 Methods

Device fabrication and parameters. Devices were fabricated from 1-$\mu$m
aluminium nitride grown on sapphire substrates using metalorganic vapour-phase
epitaxy (MOCVD). FOx-16 resist was used for patterning photonic circuits with
electron-beam lithography. After development with tetramethylammonium
hydroxide (TMAH), plasma etching with Cl2/BCl3/Ar was used to transfer the
pattern to the aluminium nitride layer. Finally, SiO2 cladding was deposited
by plasma-enhanced chemical vapour deposition (PECVD).

The ring cavity has a radius of 60 $\mu$m and width of 1.1 $\mu$m. The
fundamental and second-harmonic resonances have quality factors of $5.82\times
10^{5}$ near 1560 nm and $1.83\times 10^{5}$ near 780 nm respectively. The
intrinsic Kerr nonlinear coefficient is estimated as $g_{3}\approx 2\pi\times
2.0$ Hz. This gives the threshold power of spontaneous symmetry breaking with
intrinsic Kerr nonlinearity $P_{\rm 0}\approx 51$ mW. The second-order
nonlinearity strength is estimated as $g_{2}\approx 2\pi\times 122$ kHz. This
leads to relative strength between cascaded and intrinsic Kerr nonlinearity
$\xi=|g_{2}|^{2}/(g_{3}\gamma)\approx 3.7$ (Supplementary Information IV).

Measurement setup. Self-phase modulation is measured with 10 mW on-chip pump
power, which is strong enough to introduce significant frequency shift without
generating optical sidebands by four-wave mixing. Fast frequency scan is
realized with single-sideband electro-optic modulation. The time to scan
across the resonance is less than 20 ns (scanning rate 36 GHz/$\mu$s). The
transmitted light is detected with a 1 GHz IR photodetector. Therefore,
thermal-optic effect (millisecond timescale) is excluded in the measurement.

The spontaneous symmetry breaking is observed by measuring the output second-
harmonic field under bi-directional pump. We use a lower scanning rate
(scanning rate 6 GHz/$\mu$s), as the visible photodetector has a smaller
bandwidth (125 MHz). The complete measurement setup is shown in Supplementary
Information III.

##  .5 Data availability

All the data supporting the plots within this paper and the code are available
from the corresponding author upon reasonable request.