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

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

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

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

In this work, we conduct an in-depth study of two distinct CROW systems, and discuss the occurrence of concurrent SSBs among different pairs of intra-resonator circulating intensities. In available literature demonstrating SSB, the symmetries between two opposite directions of propagation (i.e., clockwise and counter-clockwise) [14–18,42] or two mutually orthogonal circular polarizations [25–29] have been broken. However, in CROW systems, we report a different kind of SSB. In a linear CROW system without symmetry breaking, the circulating power in the resonators is mirror symmetric, meaning that the power in the first and last resonator is expected to be the same. The same applies to the second resonator and the second to last resonators as well as the other mirror symmetric resonator pairs in the chain. These equalities of powers or symmetries are broken in nonlinear CROW systems. The interplay of linear coupling and nonlinear interactions in our studied systems offers a vast parameter space to influence homogeneous responses. We demonstrate that varying input power levels leads the optical powers in the resonators to shift among different levels, exhibiting switch-like behaviors. This is promising for controllable distribution of light in photonic systems and realization of optical digital memories and computation systems. We also detect oscillations [20,43,44] that cause periodic interchanging of dominant field roles between distinct resonators and N-level chaotic oscillations in these systems. For silicon nitride resonators with Q-factors of about 108, all the nonlinear phenomena can be observed below input power of 150 mW. Precise on-chip microresonator fabrication methods [45] will make the proposed structures soon realizable on photonic chips, thus highlighting the pertinence of the work for guiding experiments in integrated photonics.

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

Our modelling begins with normalized coupled Lugiato-Lefever equations (LLEs) [39,46]. For a system encompassing N coupled resonators (indexed as n=1,…,N for N≥2), the equations manifest as $$\begin{gathered} ∂ψn∂τ=−(1+iζ)ψn+i[(1−δn,1)jψn−1 \\ +(1−δn,N)jψn+1]+i|ψn|2ψn+αnf, \end{gathered}$$(1)where ψn=2g0/κAn is the normalized optical field envelope in the nth resonator, within which An is the unnormalized field envelope, g0 is the Kerr gain, and κ=κl+κe is the total cavity losses, with internal losses κl and external losses κe. The normalized cavity detuning is given as ζ=2Δ/κ, where the unnormalized cavity detuning is given as Δ=ω0−ωres [the difference between the input laser frequency (ω0) and the closest cavity resonance frequency (ωres)]. The terms within the brackets account for the inter-resonator couplings, where δp,q is the Kronecker delta function and j=2J/κ is the normalized inter-resonator coupling rate, with J being the unnormalized coupling rate. The normalized slow-time τ is defined as τ=2t/κ, where t is the unnormalized slow-time. Input to the resonators is given as f=8κeg0/κ3sineiϕin, where sin and ϕin are respectively the input pump amplitude and the corresponding phase. In our simulations we assume ϕin to be constant and set it to ϕin=0 for convenience. The term αn=1 when input is provided to the nth resonator and αn=0 otherwise. The normalized intracavity intensity in the nth resonator is given by Ψn=|ψn|2 and the input power by F=|f|2. We have neglected dispersion in the systems. The second term (the term within the brackets) on the RHS of Eq. (1) describes the fact that all the resonators in the chain are coupled to the previous and the next resonator in the line except for the two end resonators, each of which is just connected to one adjacent resonator. The third term of Eq. (1) is the self-phase modulation term, which accounts for the nonlinear effect of a field on itself. The last term on the RHS of Eq. (1) represents input from outside the system. Since all the ring resonators in both cases are identical, parameters, such as the cavity detunings and the Kerr nonlinear gains, g0, are the same for all resonators.

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

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

Input power, |f|2, scans for an N=3 CROW, when inputs are provided to only end resonators, are presented in Fig. 3. In Figs. 3(a) and 3(b) the analytical solutions to Eq. (1) are displayed. One may find details on the analytical solutions to Eq. (1) in Appendix A. Figures 3(a) and 3(b) reveal surprisingly rich and interesting dynamics for such a simple system. As mentioned earlier, for nonzero input powers, Ψ2≠Ψ1,3 due to coupling conditions. The difference in coupling causes persistent differences in the evolutions of the fields circulating the resonators, quite evident from the vastly different solution curves of Figs. 3(a) and 3(b). The analytical solutions of the end resonators, Fig. 3(b), reveal not one but two distinct sets of asymmetric solutions occurring for the end resonators. One set (blue) arises in a manner similar to that of previous studies—a pitchfork bifurcation emanating from the (red) symmetric solution line. The more surprising part is the second asymmetric solution set (green), which does not originate from a pitchfork bifurcation of the symmetric line. Indeed, it does not originate from the symmetric solution line at all. The green solution sets form entirely isolated solution “bubbles”, an isolation seen most prominently in Fig. 3(a). Isolated sets of asymmetric solutions have not been seen in any past works, to our knowledge, with the exception of a significantly different setup with unbalanced input conditions [27]. A justified question to ask is the following: as interesting, perhaps, as the discovery of a novel SSB origin is, if these solutions are isolated, does this not imply that they are unreachable under experimental conditions—and hence entirely useless? We report that this is, highly surprisingly, not actually the case. Figures 3(c)–3(f) show, as a counterpart of the discussed analytical results, the results of the numerical integration of Eq. (1) via standard Runge-Kutta methods [Figs. 3(c) and 3(d) are extended versions of Fig. 2(b)]. In Figs. 3(c) and 3(d) the input power is stepwise increased following a suitable system relaxation time, while in Figs. 3(e) and 3(f) the input power is similarly stepwise decreased. Unlike the analytics, which provide the full solution sets, these scans predict the real-world behaviors and evolutions of the circulating fields under experimental conditions. From Fig. 3(d), we see that in the input-increasing-scan, just after |f|2=20, the field intensities of the end resonators suddenly jump away from the red symmetric solution line and begin evolving, instead, along the green, isolated, asymmetric solution line. This is accompanied by a substantial drop in the middle resonator power, as seen in Fig. 3(c). The obvious question is, how. How does the system find the isolated set? The answer lies in the stability of Eq. (1). Performing a linear stability analysis, in Appendix C, we find that, at the point where the isolated asymmetric solutions occur, the symmetric solution line experiences a Hopf bifurcation (Appendix D) leading to system oscillations with wide-ranging intensity changes. These oscillations allow for the system to eventually find, and settle on, attractive and stable, but isolated, asymmetric solutions. This process is shown in Fig. 4. As the increasing-input-scan continues further an optical bistability in the asymmetric states leads to the system losing stability once again and proceeds, this time, to settle on the stable upper branch of the original symmetric solution line. Owing to the strong stability of this upper branch, we find that a reverse scan in this case reveals none of the asymmetric solutions of the forward scan, only bistability jumps. Figure 3 has revealed that even for low values of N, Eq. (1) describes a system capable of extremely intricate dynamics.

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

For lower input powers in the N=5 system, |ψ1|2=|ψ5|2≠|ψ2|2=|ψ4|2≠|ψ3|2 and |ψ3|2≠|ψ1|2. In Fig. 5(a), where ζ=3.5 and j=1, the field intensity in the middle resonator (in black, index 3) is suppressed greatly for all input power values. The intensities of the fields within the end resonators (indices 1 and 5—depicted in blue and cyan) and the set of resonators coupled to the end resonators (indices 2 and 4—depicted in red and green) display spontaneous symmetry breakings. The system also shows bistability jumps. In the upper panel of Fig. 5(b), we can observe two isolated regions with symmetry unbroken oscillations in the system. The lower panel shows oscillations of circulating field intensities in time corresponding to the two different regions. Figure 5(c) depicts some of the possible steady state light field intensity distributions among different resonators for the N=5 CROW system. Here, the state of a resonator is assumed to be bright if the circulating field intensity within the resonator is more than the average of all the resonator’s field intensities in the respective CROW arrangement for a particular combination of system parameters and input powers.

Figures 5(d) and 5(e) depict the input power scans of the N=10 system for two different sets of parameters. In Fig. 5(d), independent SSBs of all field intensities within resonators with symmetric coupling conditions are observed, with SSB bubbles crossing each other. The initial SSB region is followed by a symmetry restored region, which then is followed by regions of oscillating full asymmetric solutions, and subsequently non-oscillating full asymmetric solutions. Figure 5(e) depicts SSB of all coupling-wise symmetric pairs, symmetry restored regions with and without oscillations, and bistability jumps of the circulating field intensities leading to a second region of full asymmetry. This full asymmetric region also displays oscillations for certain ranges of input powers with and without overlaps. The lower panels of Fig. 5(e) show examples of field intensity oscillations in time corresponding to three different oscillatory regions. For CROW systems with an odd number of resonators, we observe (N−1)/2 symmetry broken field pairs, since the middle resonator is not symmetric to any other resonator. However, for CROW systems with an even number of resonators, there are N/2 symmetric pairs at low power; therefore, we can observe up to N/2 symmetry broken field pairs.

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

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

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

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

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

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

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

It can be noted that due to the randomness associated with SSB, the dominance of circulating field amplitudes following symmetry breaking of each symmetric pair in a CROW is chosen randomly. For different simulation runs, the intracavity power plots can swap among the two resonators undergoing symmetry breaking. Similarly, in experiments, in a perfect symmetric case, after symmetry breaking, the dominance will be randomly assigned each time the system is pumped. This can be avoided, in order to achieve repeatability, by applying small input biases to the resonators. The resonators with the bias fields will always dominate in terms of circulating powers over their counterpart. However, this solution is not applicable for “input to end” CROW systems, where we cannot apply bias to all the resonators. In this case, relative detunings can be introduced between different resonators to achieve similar effects.

From an experimental outlook, we can consider silicon nitride resonators with radii 100 μm. If we assume the internal losses to be κl=50MHz and under the condition of critical coupling, i.e., κe=50MHz, one unit of normalized input power (|f|2=1) corresponds to 0.109 mW.

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

Future research will address the dynamic behavior of optical fields in the resonators and the effects of asymmetry in different parameters on the homogeneous states of the systems. SSBs in other complex arrangements of microring resonators will also be addressed. The controllable distribution of light field intensities among different resonators could be a key feature for large-scale optical computing and light-field steering in integrated photonics. The combination of linear coupling between different resonators and optical nonlinearities in high-Q microresonators makes the CROW systems a promising candidate for integrated optical neural networks. CROW systems are also promising candidates for observing symmetry broken vector solitons with N different values of circulating intensities [32]. These can be useful for generating N distinct interconnected frequency combs, which would be very useful in neuromorphic computing, telecommunications, and especially in space technologies due to compactness. Together with the latest concepts of dispersion engineering [47–49] the studied nonlinear effects in this work will lead to a lot more interesting soliton dynamics.

Acknowledgment. AG and AP acknowledge the support from the Max Planck School of Photonics. LH acknowledges funding provided by the SALTO funding scheme from the MPG and Centre National de la Recherche Scientifique (CNRS). The authors thank Prof. Gian-Luca Oppo for the fruitful discussions.

Author Contributions.PDH, AG, and LH defined the research project. AG performed the theoretical analysis with support from LH and AP. AP, AG, and LH completed the numerical simulations. AG, AP, LH, and PDH wrote the manuscript with the help of all other authors. PDH and LH supervised the project.