The concept of broken symmetry is used to study bifurcations of equilibria and dynamical instabilities in dynamic model of one-mode laser (nonresonant complex Lorenz model) on the basis of modified Hopf theory. It is shown that an invariant set of stationary points bifurcates into an invariant torus (doubly-periodic branching solution). Influence of the symmetry breaking on stability of branching solutions is investigated as a function of detuning. The invariant torus is found to be stable under the detuning exceeds its critical value, so that dynamically broken symmetry results in the apprearance of low-frequency Goldstone-type mode. If the detuning then goes downward and pumping is kept above the threshold, numerical analysis reveals that after a cascade of period-doublings the strange Lorenz attractor is formed at small values of detuning. It is found that there are three types of the system behavior as pumping increases depending on the detuning. Quantum counterpart of the complex Lorenz model is discussed.
We report a new type of symmetry-breaking bifurcation of solitons in optical systems with parity-time-symmetric potentials. In this bifurcation, the two bifurcated branches of asymmetric solitons exhibit opposite stability, which contrasts all previous symmetry-breaking bifurcations in conservative and non-conservative systems. We show that this novel symmetry-breaking bifurcation can be exploited to achieve unidirectional propagation of high-intensity light beams in parity-time-symmetric potentials.
We report symmetry-breaking and restoring bifurcations of solitons in a fractional Schrödinger equation with the cubic or cubic-quintic (CQ) nonlinearity and a parity-time (PT)-symmetric potential, which may be realized in optical cavities. Solitons are destabilized at the bifurcation point, and, in the case of the CQ nonlinearity, the stability is restored by an inverse bifurcation. Two mutually-conjugate branches of ghost states (GSs), with complex propagation constants, are created by the bifurcation, solely in the case of the fractional diffraction. While GSs are not true solutions, direct…
A laser consisting of two independently-pumped resonators can exhibit mode bifurcations that evolve out of the exceptional points (EPs) of the linear system at threshold. The EPs are non-Hermitian degeneracies occurring at the parity/time-reversal ($\mathcal{PT}$) symmetry breaking points of the threshold system. Above threshold, the EPs become bifurcations of the nonlinear zero-detuned laser modes, which can be most easily observed by making the gain saturation intensities in the two resonators substantially different. Small pump variations can then switch abruptly between different laser beh…
We study the possibility of switching the types of symmetry breaking bifurcation (SBB) in the cylinder shell waveguide with helical double-well potential along propagation direction. This model is described by the one-dimensional nonlinear Schrödinger (NLS) equation. The symmetry- and antisymmetry-breakings can be caused by increasing the applied voltage onto the waveguide in the self-focusing and -defocusing cases, respectively. In the self-focusing case, the type of SBB can be switched from supercritical to subcritical. While in the self-defocusing case, the type of SBB can not be switched b…
Theoretical investigations of dynamical behavior in optical parametric oscillators (OPO) have generally assumed that the cavity detunings of the interacting fields are controllable parameters. However, OPOs are known to experience mode hops, where the system jumps to the mode of lowest cavity detuning. We note that this phenomenon significantly limits the range of accessible detunings and thus may prevent instabilities predicted to occur above a minimum detuning from being evidenced experimentally. As a simple example among a number of instability mechanisms possibly affected by this limitatio…
Symmetry breaking is reported for continuous families of solitons in the nonlinear Schrödinger equation with a two-dimensional complex potential. This symmetry-breaking bifurcation is forbidden in generic complex potentials. However, for a special class of partially parity-time-symmetric potentials, such symmetry breaking is allowed. At the bifurcation point, two branches of asymmetric solitons bifurcate out from the base branch of symmetry-unbroken solitons. Stability of these solitons near the bifurcation point are also studied, and two novel stability properties for the bifurcated asymmetri…
We study symmetry breaking of solitons in the framework of a nonlinear fractional Schrödinger equation (NLFSE), characterized by its Lévy index, with cubic nonlinearity and a symmetric double-well potential. Asymmetric, symmetric, and antisymmetric soliton solutions are found, with stable asymmetric soliton solutions emerging from unstable symmetric and antisymmetric ones by way of symmetry-breaking bifurcations. Two different bifurcation scenarios are possible. First, symmetric soliton solutions undergo a symmetry-breaking bifurcation of the pitchfork type, which gives rise to a branch of asy…
We study the spontaneous temporal symmetry breaking instability in a coherently-driven passive optical Kerr resonator observed experimentally by Xu and Coen in Opt.~Lett.~{\bf 39}, 3492 (2014). We perform a detailed stability analysis of the Lugiato-Lefever model for the optical Kerr resonators and analyze the temporal bifurcation structure of stationary symmetric and the emerging asymmetric states as a function of the pump power. For intermediate pump powers a pitchfork loop is responsible for the destabilization of symmetric states towards stationary asymmetric ones while at large pump power…
Bifurcation structure of soliton self-injection locking in microresonators
Emergent Complexity in Physical Systems Laboratory (ECPS), Swiss Federal Institute of Technology Lausanne (EPFL)（复杂物理系统实验室，洛桑联邦理工学院）；Laboratoire Temps-Fréquence, Université de Neuchâtel（时间频率实验室，纳沙泰尔大学）
We introduce discrete systems in the form of straight (infinite) and ring-shaped chains, with two symmetrically placed nonlinear sites. The systems can be implemented in nonlinear optics (as waveguiding arrays) and BEC (by means of an optical lattice). A full set of exact analytical solutions for symmetric, asymmetric, and antisymmetric localized modes is found, and their stability is investigated in a numerical form. The symmetry-breaking bifurcation (SBB), through which the asymmetric modes emerge from the symmetric ones, is found to be of the subcritical type. It is transformed into a super…
It is analytically shown that symmetry breaking, in dissipative systems, affects the nature of the bifurcation at onset of instability resulting in transitions from super to subcritical bifurcations. In the case of a nonlinear fiber cavity, we have derived an amplitude equation to describe the nonlinear dynamics above threshold. An analytical expression of the critical transition curve is obtained and the predictions are in excellent agreement with the numerical solutions of the full dynamical model.
We explore the dynamics of spontaneous breakdown of mirror symmetry in a pair of identical optomechanical cavities symmetrically coupled to a waveguide. Large optical intensities enable optomechanically-induced nonlinear detuning of the optical resonators, resulting in a pitchfork bifurcation. We investigate the stability of this regime and explore the possibility of inducing multistability. By injecting proper trigger pulses, the proposed structure can toggle between two asymmetric stable states, thus serving as a low-noise nanophotonic all-optical switch or memory element.
The combination of linear and nonlinear potentials, both shaped as a single well, enables competition between the confinement and expulsion induced by the former and latter potentials, respectively. We demonstrate that this setting leads to spontaneous symmetry breaking (SSB) of the ground state in the respective generalized nonlinear Schrödinger (Gross-Pitaevskii) equation, through a spontaneous off-center shift of the trapped mode. Two different SSB bifurcation scenarios are possible, depending on the shape of the nonlinearity-modulation profile, which determines the nonlinear potential. If …
We report results of the analysis of the spontaneous symmetry breaking (SSB) in the basic (actually, simplest) model which is capable to produce the SSB phenomenology in the one-dimensional setting. It is based on the Gross-Pitaevskii - nonlinear Schroedinger equation with the cubic self-attractive term and a double-well-potential built as an infinitely deep potential box split by a narrow (delta-functional) barrier. The barrier's strength, epsilon, is the single free parameter of the scaled form of the model. It may be implemented in atomic Bose-Einstein condensates and nonlinear optics. The …
We analyze nonlinear guided waves in a planar waveguide made of a left-handed material surrounded by a Kerr-like nonlinear dielectric. We predict that such a waveguide can support fast and slow symmetric and antisymmetric nonlinear modes. We study the symmetry breaking bifurcation and asymmetric modes in such a symmetric structure. We analyze nonlinear dispersion properties of the guided waves, and show that the modes can be both forward and backward.
We perform a theoretical study of nonlinear optical isolator devices based on coupled microcavities with gain and loss. Using coupled-mode theory, we derive a correspondence between the boundary of asymptotic stability in the nonlinear regime, where gain saturation is present, and the PT-breaking transition in the underlying linear system. For zero detuning and weak input intensity, the onset of optical isolation can be rigorously derived, and corresponds precisely to the PT transition point. When the couplings to the external ports are unequal, the isolation ratio exhibits an abrupt jump at t…
The realization of spontaneous symmetry breaking (SSB) requires a system that exhibits a near perfect symmetry. SSB manifests itself through a pitchfork bifurcation, but that bifurcation is fragile, and perturbed by any asymmetry or imperfections. Consequently, exploiting SSB for real-world applications is challenging and often requires cumbersome stabilization techniques. Here, we reveal a novel method that automatically leads to symmetric conditions, and demonstrate its practical application in coherently-driven, two-mode, passive Kerr resonators. More specifically, we show that introducing …
