Symmetry-breaking-induced dynamics in a nonlinear microresonator

Abstract: Optically induced symmetry-breaking plays a key role in nonlinear photonics. Recently, the experiment has successfully observed the Kerr-nonlinearity-induced chiral symmetry breaking in a single ultrahigh-Q whispering-gallerymicroresonator. Here, we show this symmetrybreaking can generate exotic dynamics between two counter-propagating modes. In particular, we predict two kinds of self-trappings, in which the corresponding relative phase oscillates around π or runs without bound although they have both the nonzero mean energy imbalance. Finally, we also clarify the impacts of the mode loss, finding a dynamical transition from self-trappings to an anharmonic oscillation. The presented scheme offers a new route to understanding the nonlinear dynamics and wave chaos in the microresonator.

measured in the experiment, it is thus interesting to investigate the nonlinear dynamics between CW and CCW modes in the time-domain.
In this paper, we show that this symmetry-breaking can generate exotic dynamics for the single ultrahigh-Q WGM with the modulated Kerr nonlinearity. In particular, when the Kerr nonlinearity parameter goes beyond a critical value, two kinds of self-trappings with both the nonzero mean energy imbalance are found. One is that the relative phase between two modes oscillates around π and is thus called as the π-phase self-trapping. The other is that the relative phase evolution runs without bound and is thus called as the running-phase self-trapping. Finally, we also clarify the impacts of the mode loss, finding a dynamical transition from self-trappings to an anharmonic oscillation (the mean energy imbalance is zero). The presented scheme offers a new route to understand the nonlinear dynamics and wave chaos in the microresonator.

Model and Hamiltonian
We consider a silica toroidal microresonator made of a dispersive medium exhibiting a Kerr nonlinearity, schematically represented in Fig. 1(a). Considering the third-order nonlinear effect and using modal expansion approach [45,46], the Hamiltonian of the system can be written as [37] where a µ (µ = 1, 2) denotes the standing wave mode induced by the intrinsic linear coupling between CW and CCW waves with coupling strength g, ω is the mode frequency. F µηνσ = δ µν δ ησ + δ µσ δ νη + δ µη δ νσ /2, where δ is the Kronecker delta, and repeated indices are summed over. The coefficient Λ is proportional to the scalar Kerr nonlinear susceptibility χ (3) .
In the CW-CCW basis, a cw = (a 1 + ia 2 ) / √ 2 and a ccw = (a 1 − ia 2 ) / √ 2, the dynamics of the Hamiltonian (1) are described by the following coupled equations [37] In Eqs. (2) and (3), the first, second, and third terms represent linear coupling, self-phase modulation, and cross-phase modulation, respectively. It should be noted that the cross-phase modulation induces twice the refractive index change compared to the self-phase modulation. 1   (6) at φ = π. The bifurcation of a single stable fixed point F π splits into two fixed points F ± in the symmetry-breaking regime when γ > γ c = 1.

Stationary solutions
It is worth mentioning that the system possesses chiral symmetry because of the covariant of Eqs. (2) and (3) under the parity transformation, i.e., a cw → a ccw . This chiral symmetry breaks spontaneously, depending on the experimentally tunable parameter γ, i.e., Kerr-nonlinearitymediated interaction between the counter-propagating light. Figure 1(b) shows an example of such symmetry-breaking phenomenon, which has been confirmed experimentally [37,38]. Actually, thanks to the competition between the linear coupling and nonlinearity, two different regimes occurs according to the nonlinearity parameter γ with a critical value γ c (= 1), as shown in Fig. 1(b). Specifically, it becomes obvious by a classical fixed point analysis [ z (t) = 0, φ (t) = 0], which reveals the underlying symmetry breaking. For γ < γ c , two fixed points, F 0 = (z, φ) = (0, 0) and F π = (z, φ) = (0, π), characterize the dynamics. However, for γ > γ c , the steady state point F π becomes unstable and two new stable fixed points are formed, i.e., F π → F ± = [± γ 2 − 1/γ, π]. This implies that a single trajectory around F π splits into to two fixed points F ± .  Figure 2 shows the trajectories of state evolutions in the phase space for the different nonlinearity parameters γ. As shown in Fig. 2(a), in the absence of nonlinearity (γ = 0), the trajectories circulate around the dots at {z = 0, φ = 0} and {z = 0, φ = π}, which correspond to the modes a 1 and a 2 , respectively. For small γ (Rabi regime), the trajectories near the high-frequency mode a 2 are distorted and contracted in the φ direction [see Fig. 2 When γ > γ c (symmetry-breaking regime), the trajectories around F π (red dot) split into two distinct trajectories around the new stable fixed points F ± with the unbalance CW and CCW modes [see the green dots in Fig. 2(c)]. Such pitchfork bifurcation phenomena have also been observed in photonic nanocavities [47] and Bose-Einstein condensates [27]. Noted that by further increasing γ > 2, the partially-closed trajectories around F ± open, as shown by the yellow line in Fig. 2(d). The above discussions show that the system has rich dynamics by controlling γ, which will be discussed in detail later.

Exotic dynamics
We first consider the simplest case without the Kerr nonlinearity (γ = 0). The energy oscillation of the system is entirely dominated by the linear coupling. The dynamical equation for the energy imbalance is obtained as Equation (8) shows that the energy imbalance exhibits a standard Rabi oscillation with the frequency ω = 2g [48]. This Rabi oscillation corresponds to the trajectories in Fig. 2(a).
If there exists the Kerr nonlinearity with γ < 1 (Rabi regime), the nonlinear effect alters the dynamics, resulting in an anharmonic oscillation. This is the situation shown in Fig. 2(b). Note that the dynamics is characterized by two fixed points, however, the corresponding trajectories around these are differently distorted. The motion around F 0 is known as the zero-phase modes with z = 0 and φ = 0, while the trajectories around F π is known as the π-phase modes with z = 0 and φ = π [49]. To see the difference between them, in Figs. 3(a) and 3(b) we numerically plot the dynamics of the energy imbalance z (t) and the relative phase φ (t) for these two types of oscillations, respectively. It can be seen that the difference between the zero-and π-phase oscillations manifests itself most pronouncedly in the modified oscillation frequency. Obviously, for the small-amplitude oscillation with |z| 1, Eq. (5) is linearized around the fixed point F 0 and F π , and the dynamical equation is given by where "±" denote the zero-and π-phase modes, respectively. We obtain sinusoidal oscillations with the periods (in unscale units) independent of the initial conditions z (0) and φ (0). While for the large-amplitude oscillation, no exact solution can be found and the period of the anharmonic oscillation is dependent of the initial conditions z (0) and φ (0), as well as γ. Figure 3(c) shows the period τ versus γ, which implies that the oscillation period is reduced (enhanced) in the case of the zero-phase (π-phase) oscillation.
In addition to the anharmonic oscillation, other striking effects occur in our system by further increasing γ. For instance, for a fixed value of the initial energy imbalance, if the nonlinearity parameter γ exceeds a critical value, the energy becomes self-trapped. This describes the physical fact that the temporal mean energy imbalance is nonzero, i.e., z 0, as shown in Figs. 2(c) and 2(d). There are different ways in which this state can be achieved, and all of them correspond to the condition that H 0 = H (z (0) , φ (0)) > 1. For a given value of the initial condition {z (0) , φ (0)}, the critical parameter for self-trapping is given bỹ On the other hand, when γ (> 1) remains constant and the initial value φ (0) mπ (m is an integer), there is a critical energy imbalance z c by requiring H 0 = 1, that is, γz c /2− 1 − z 2 c cos [φ (0)] = 1. Intriguingly, there are two type self-trapping according to the dynamics of the relative phase. As shown in Fig. 2(c), in the symmetry-breaking regime, but close to the critical value (i.e., 1 < γ < 2), all self-trapping modes have an oscillating phase difference (green line), which is called the π-phase self-trapping. When γ > 2, the partially-closed trajectories open, as shown by the yellow line in Fig. 2(d). This implies that the relative phase evolution runs without bound, which is thus called the running-phase self-trapping. Figure 4 plots the dynamics of the energy imbalance z(t) and relative phase φ(t), showing the typical features of the πand running-phase self-trappings, respectively. In order to see these different kinds of self-trappings, we map the dynamics of the system onto the dynamical behaviors of a classical particle moving in an effective potential W (z). By combining Eqs. (5) and (7), we have with the effective potential Here, W 0 = 1 − H 2 0 is the total energy (potential barrier). Figures 5(a) and 5(b) display the potential W (z) − W(0) for different z(0). In Fig. 5(a), for z (0) = 0.3, the potential is parabolic and the (small-amplitude) oscillations are sinusoidal. For z (0) = 0.8, the trajectory of z (t) becomes markedly nonsinusoidal, given the double-well structure. For z (0) > 0.8, the total energy is smaller than the potential barrier (horizontal black lines), forcing the particle to become localized in one of the two wells. This corresponds to the running-phase self-trapping. In Fig. 5(b), for γ = 1.5 and φ (0) = π, this potential always has a double-well structure for a wide range of z (0) (< z c ). This corresponds to the π-phase self-trapping. Noted that when γ > 2 and φ (0) = π, the system is self-trapping for all values of z (0), which consists of the runningand π-phase self-trappings. Specifically, for small z (0), the phase φ (t) is unbounded and the system exhibits the running-phase self-trapping. However, for large z (0), the relative phase φ (t) becomes localized around π and remains bounded. A clear observation feature of the above dynamics behaviors is the time period of oscillation. In Figs. 5(c) and 5(d), we plot the inverse period 1/τ as a function of the ratio between the initial energy imbalance z (0) and the critical energy imbalance z c . Figure 5(c) shows the case for φ (0) = 0 and γ = 5 (z c = 0.8). The initial parts of the graph for z (0) z c mark the sinusoidal small-amplitude oscillations. With increasing z (0), the oscillations become highly anharmonic with the inverse period that first increases and then decreases, displaying a critical slowing down. The system enters into the running-phase self-trapping if z (0) z c . For φ (0) = π [ Fig. 5(d)], the self-trapping occurs when z (0) < z c . At z (0) = z c , the time period diverges since the system reaches the fixed point F π [Fig. 2(c)], and for z (0) > z c , the self-trapping disappears.

Lossy resonator
In real experiments, the dissipation usually exists [37,38]. In this section, we consider this effect on the dynamical behavior of our system. In such case, the coupled-mode equation becomes where κ µ is the decay rate for the mode a µ and ∆ is the detuning of input light. In the CW-CCW basis, we have Due to the existence of the dissipation, the total energy is no longer conserved and we have to add a new dynamical variable where A 0 is the total energy of these two modes at time t = 0. The evolution equations for z (t) = [A 1 (t) − A 2 (t)] /A 0 , φ (t) and η (t) are given respectively by where χ = (κ 1 + κ 2 ) /4g, ξ = (κ 2 − κ 1 ) /4g, and γ = ΛA 0 /2. In Fig. 6, we show the zero-phase [Figs. 6(a) and 6(c)] and π-phase oscillations [Figs. 6(b) and 6(d)] under different dissipations in the Rabi regime (γ = 0.5). It can be seen that the oscillation amplitude of z(t) decays continuously for all cases due to the presence of the dissipative term. While the dynamics of the φ (t) are quite different. For the zero-phase oscillation, φ(t) decays with the time. For the π-phase oscillation, the relative phase oscillates around π with gradually increased amplitude in small dissipation [ Fig. 6(b)], and in large dissipation, there is a transition from the π-phase oscillation to the zero-phase oscillation at a critical time [ Fig. 6(d)].
In Fig. 7, we show the dynamics of two-type self-trappings in the presence of the dissipative terms. For the π-phase self-trapping [Figs. 7(a) and 7(c)], the system works in the self-trapping regime at the beginning and then moves into the anharmonic oscillation. For the running-phase self-trapping [Figs. 7(b) and 7(d)], small losses induce a dynamical transition from the selftrapping to the π-phase oscillation [ Fig. 7(b)], large losses induce a dynamical transition of φ(t) from an π-phase oscillation to an oscillation regime around 2π for long enough time [ Fig. 7(d)].

Discussion and conclusion
We first give a brief estimation on the possible experimental parameters to show the feasibility of our model. In general microresonator experiments (Q∼ 1.2 × 10 8 ) [4,37], the linear coupling strength is g = 2π × 16 MHz, and the decay rates are κ 1 = 2π × 0.5 MHz, κ 2 = 2π MHz. Then we get χ ≈ 0.023 and γ ≈ 0.008. These parameters show that our predicted dynamics can be observed in current experiments. We should note that the emergence of various exotic dynamics depends heavily on the nonlinear parameter γ. In realistic experiments, we can modulate it by adjusting the input pump power. According to Ref. [37], the effect of four-wave mixing will occur when the input power is higher enough. To achieve the regime of running-phase self-trapping (γ > 2), we can decrease the fiber-cavity coupling rate or increase the value of χ (3) . In addition, the dynamics of WGM in the time-domain can be observed by reflection measurement and drop-port measurement technique [48]. We believe that our work will motivate experimental studies of nonlinear optical properties of microresonator. In summary, motivated by recent experimental development, we have investigated the dynamics between counter-propagating modes in a nonlinear microresonator. When the chiral symmetry of system is broken spontaneously by increasing the Kerr-nonlinearity parameter, we have found two kinds of self-trappings, in which the corresponding relative phase oscillates around π or runs without bound although they have both the nonzero mean energy imbalance. We have also clarified the impacts of the mode loss, finding a dynamical transition from self-trappings to an anharmonic oscillation. The presented scheme offers a new route to understanding the nonlinear dynamics and wave chaos in the microresonator.