Exceptional, but Separate: Precursors to Spontaneous Symmetry Breaking

TL;DR

This work clarifies the nuanced relationship between spontaneous symmetry breaking (SSB) and exceptional points (EPs) in nonlinear Kerr resonators. By deriving the Jacobian of stationary two-field LLE systems and introducing invariants $ ext{η}$ and $ ext{ν}$, the authors show that Jacobian EPs (including dual and single EP$2$) structure the parameter space and mark symmetry-phase boundaries, yet SSB does not generally coincide with these EPs. Crucially, crossing a Jacobian EP is shown to be a necessary precursor to SSB, establishing a general principle that EPs are not universally predictive of SSB but are essential for its onset. The findings, supported by three real optical platforms and analytical expressions for stability and EP conditions, have practical implications for predicting and controlling nonlinear optical behavior in photonic devices.

Abstract

Spontaneous symmetry breaking (SSB) and exceptional points (EPs) are often assumed to be inherently linked. Here we investigate the intricate relationship between SSB and specific classes of EPs across three distinct, real-world scenarios in nonlinear optics. In these systems, the two phenomena do not coincide for all classes of EPs; they can occur at dislocated points in parameter space. This recurring behavior across disparate platforms implies that such decoupling is not unique to these optical systems, but likely reflects a more general principle. Our results highlight the need for careful analysis of assumed correlations between SSB and EPs in both theoretical and applied contexts. They deepen our understanding of nonlinear dynamics in optical systems and prompt a broader reconsideration of contexts where EPs and SSB are thought to be interdependent.

Figures (4)

• Figure 1: Illustrative example systems in which an exceptional point (EP) and spontaneous symmetry breaking (SSB) either coincide (left) or are dislocated (right). Panels (a,b) show the SSB of two system properties, $P_1$ and $P_2$, occurring where the relation $P_1=P_2$ abruptly breaks as a system parameter, $\alpha$, is varied. Panels (c,d) depict EPs along $\alpha$, where two eigenvalues, $\lambda_1$ and $\lambda_2$ become degenerate and their associated eigenvectors coalesce. While SSB is often assumed to coincide with an EP, as in panels (a,c), this is not guaranteed for all types of EPs: panels (b,d) illustrate that the SSB and an EP -- such as one derived from the system Jacobian -- can occur dislocated in parameter space, as we exemplify in this work.
• Figure 2: Kerr resonator schematics. (a) Two co-propagating light field components, $E_{1,2}$ -- with left- and right-circular polarizations, respectively -- are coupled into a Kerr ring resonator using a single linearly polarized input pump. (b) Two identical input pumps introduce counter-propagating light fields, $E_{1,2}$, into the Kerr ring resonator. (c) In a Fabry-Pérot cavity, reflections at the cavity boundaries cause rebounded fields to coexist, still true when multiple polarizations $E_{1,2}$ are present. In all cases, $\tau$ denotes the fast-time axis used for modelling, which ranges from $0$ to $\tau_\mathrm{R}$, the round-trip time.
• Figure 3: Solutions to Eq. \ref{['Eq:GenLLE']} are shown in panels (a) and (b) as functions of cavity detuning $\theta$ and input intensity $|E_{\mathrm{in}}|^2$, respectively. Panels (a,c,e) correspond to an input intensity of $|E_{\mathrm{in}}|^2 = 1.75$, while (b,d,f) use a detuning of $\theta = 2.5$. The solutions from (a) and (b) are inserted into Eq. \ref{['Eq:EVs']} to compute the eigenvalues $\lambda_{\pm_1\pm_2}$. The real and imaginary parts of these eigenvalues are plotted in panels (c,d) and (e,f), respectively. Red dots mark SSB bifurcations, purple dots denote single EP$2$s, blue dots indicate dual EP$2$s, and green dots show the limits of optical bistability. The stability of the solutions in (a) and (b) can be inferred from the real parts of the corresponding eigenvalues in (c) and (d): when $\Re(\lambda) > 0$ for any $\lambda \equiv \lambda_{\pm_1\pm_2}$, the associated solution is unstable, indicated by dashed lines across the figure. Eigenvalue branches that deviate from $\Re(\lambda) = -1$ in (c,d) and from $\Im(\lambda) = 0$ in (e,f) are labeled. All lines associated with symmetry-broken solutions are shown in blue. Symmetric solutions are colored black for $\Re(\lambda_{\pm+})$ and gray for $\Re(\lambda_{\pm-})$.
• Figure 4: Structure of the Jacobian. (a) Full view of the Jacobian's structure in the $\eta$-$\nu$-plane, showing its EP configuration and stability regions. (b,c) Zoom-ins on the origin and the region near the SSB bifurcation, respectively. Green region: Jacobian is quasi-chiral symmetry (qCS) unbroken and $\mathcal{PT}$-broken. Orange region: $\mathcal{PT}$-unbroken and qCS-broken. White regions: both symmetries broken. The regions are separated by a curve of single EP$2$s (purple) and a curve of dual EP$2$s (dark blue). The gray line marks the stability boundary of Eq. \ref{['Eq:GenLLE']}; the gray dashed area denotes instability. Overlaid are the trajectories of the detuning scan corresponding to Fig. \ref{['fig:EigenPlot']}(a,c,e). The black line represents the symmetric solution, with dashed segments indicating instability. When the trajectory enters the unstable region, the system either exhibits optical bistability (dark green dots) or undergoes SSB (red dots). The asymmetric solution is shown as the light blue curve. As $\theta \to \pm \infty$, the Jacobian asymptotically approaches the dual EP$2$ curve from within the qCS-unbroken region (green).