Exceptional Point Modulated by Kerr Effect in Anti-Parity-Time Symmetry System

APT symmetry can be implemented in a dissipative system by forming a dissipative coupling between two detuned modes. The dissipative couplings can be obtained by connecting two separated WGMRs with two common waveguides as shown in Fig.1. The WGMRs are of the same material and the renonant frequency $\omega_{0j}(j=1,2)$ differs. As $L$ is much larger than the cavities’ radii, the direct coupling of two WGMRs can be neglected. The modal fields in two WGMRs are governed by the coupled-mode equations

where $a_{j}(j=1,2)$ corresponds to the modal field amplitude in each cavity with $|a_{j}|^{2}$ normalized to the intracavity intensity. $\omega_{j}=\omega_{0j}-\omega$ is the frequency detuning of the j-th WGMR without nonlinear effect, where $\omega$ is the frequency of the input light. $\Gamma_{j}=\gamma_{0j}+2\gamma_{j}+\gamma_{j,NL}-G_{j}$ is the total loss rate of the j-th WGMR, where $\gamma_{0j}$ represents the intrinsic loss rate and $\gamma_{j}$ corresponds to the coupling loss rate. $\gamma_{j,NL}$ is the loss rate lifted by nonlinear effects. $G_{j}$ is the external gain rate of the j-th WGMR. $k=\frac{2\pi\chi^{(3)}}{n^{2}}$ is the Kerr coefficient. $n$ is the linear refractive index and $\chi^{(3)}$ is the third order nonlinear susceptibility, which are considered as a constant in our scheme. The dissipative coupling strength is $g=e^{i\theta_{d}}\sqrt{\gamma_{1}\gamma_{2}}$. The propagating phase factor $\theta_{d}=\beta_{d}(\omega)L$, where $\beta_{d}(\omega)$ is the waveguide dispersion relationship and $L$ denotes waveguides’ length between two WGMRsXiao et al. (2010). $S_{in}$ is the amplitude of input light. Transmission amplitude at Port 2 is defined as $t=\frac{S_{out}}{S_{in}}$ and the transmission rate is $T=|t|^{2}$

The eigenvalues of Eq.(1) reads

where $\Delta=\left(\left(1+k\left|a_{1}\right|^{2}\right)\omega_{1}-{\left(1+k\left|a_{2}\right|^{2}\right)\omega}_{2}+i\left.\left({\Gamma_{1}-\Gamma_{2}}\right)\right)^{2}\right.\\ -4g^{2}$. If $\Delta<0$, two eigenmodes have the same resonant frequency and different linewidth and the system will be in APT symmetric phase; if $\Delta>0$, the two eigenmodes have equal linewidth and divergent resonant frequency, and the system will be in APT symmetry broken phase. $\Delta=0$ is the location of the EP of the system

To analyze the response of this arrangement under linear conditions, we assume the light intensity is small,i.e.,$|a_{1}|^{2},|a_{2}|^{2}\sim 0$. Under these assumptions, Kerr effect can be neglected. Hence this regime could be properly described by linearizing Eq.(1),e.g.,

The eigenvalues of the system in the linear regime read

It is clear that $\Gamma_{1}=\Gamma_{2}$ and $\theta_{d}=\pi$ are necessary to construct APT symmetry in the systemYang, Liu, and You (2017). Hence, the eigenvalues could be further simplified into

Where $\Gamma=\Gamma_{1}=\Gamma_{2}$. $\omega_{0}=\frac{1}{2}(\omega_{1}+\omega_{2})$ and $\delta=\frac{1}{2}(\omega_{1}-\omega_{2})$ define the common- and differential-mode detunings, respectively. In this respect, the two regimes of linear APT symmetry can be identified by whether $g^{2}\gtrless\delta^{2}$. In the first case where $g^{2}>\delta^{2}$, the system remains in APT symmetry unbroken regime. Thus, modal amplitudes in each WGMR are equal and the phase factor $\theta_{l}$ between them is given by $\sin\theta_{l}=-\frac{\delta}{g}$Li et al. (2020); De Carlo et al. (2019); Yang, Liu, and You (2017).

On the other hand, when $\delta^{2}$ increases beyond $g^{2}$, a spontaneous phase transition would occur and drive the system into APT symmetry broken regime. As opposed to the hallmarks in the unbroken phase, the modal field amplitudes in two WGMRs are unequal, and they are phase shifted by $\theta_{l}=\frac{\pi}{2}$ Li et al. (2020); De Carlo et al. (2019); Yang, Liu, and You (2017).

The EP of linear APT symmetric system locates at $\delta^{2}=g^{2}$, which is determined by differential-mode detuning $\delta$ alone (consider $g$ as a constant). As $\delta$ can be easily affected by implementation errors and nonlinear effects, it is difficult to maintain the linear system at its EP precisely.

Another feature of such EPs is their tremendous response towards external perturbations. Then, we will analyze the response to perturbations of the linear EPs. Assuming the linear system is at its EP initially, and a perturbation $\varepsilon$ influences resonant frequency of WGMRs. The frequency detuning of WGMR1 becomes $\omega_{1}(\varepsilon)=\omega_{1}+m\varepsilon,m\in R$ and that of WGMR2 is $\omega_{2}(\varepsilon)=\omega_{2}+n\varepsilon,n\in R$. Assuming $m-n=2$, $\delta$ changes into $\delta+\varepsilon$. Transmission spectrum in the proximity of the linear EP is depicted in Fig.(2). In the APT symmetry broken phase, a real splitting appears in the spectrum and the distance between transmission peaks is determined by the scale of perturbations. The parameters of stimulation in Fig.(2) are $\delta=g=-10GHz,\Gamma=150MHz$.

We introduce the frequency splitting in APT symmetry broken regime, defined as $\Delta\omega=\omega_{+}-\omega_{-}$, as a criterion to appraise the responding ability of the system. A larger frequency splitting towards perutrbation indicates better responding ability of the system. The frequency splitting in linear condition is

When $\varepsilon\ll 1$, $\Delta\omega_{L}\approx 2\sqrt{2|g|\varepsilon}$, which is proportional to $\sqrt{\varepsilon}$. Therefore, the responding ability of such EP system is greatly enhanced compared with that of conventional system, whose response is proportional to $\varepsilon$.

As fields in APT-coupled configuration start to grow, Kerr effect comes into play, as described by Eq.(1). Yet, Kerr effect’s influence is beyond the common-mode detuning of the system, it might be able to trigger the spontaneous phase transition of APT symmetry, as we will show later.

Suppose that $a_{2}=\rho e^{i\theta_{k}}a_{1}$, with $\rho>0,\theta_{k}\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$ and solve Eq.(1) in the stationary conditon $\frac{d}{dt}a_{1}=\frac{d}{dt}a_{2}=0$, which reads

After dividing Eq.(7a) by $a_{1}$ and Eq(7b) by $a_{2}$, upon subtraction comes that

Eq.(8) can be solved for the real part and imaginary parts. To vanish the real part of Eq.(8), $\rho-\frac{1}{\rho}=0$ or $\cos\theta_{k}=0$ must be satisfied. If $\rho-\frac{1}{\rho}=0$, the system is in APT symmetric regime, while $\cos\theta_{k}=0$ corresponds to the nonlinear system in APT symmetry broken regime, and the EP locates at where the two conditions are satisfied simutaneously, as we will prove later.

Nonlinear effect could not only shift EPs of APT symmetry, but also affect the responding ability of the system. In this section, analyses on how nonlinear effct influence the transmission spectrum and frequency splitting have been made.

High pump intensity increases light absorption of the cavity, causing significant thermal effect as well as Kerr effect. In this section, we will investigate the influence of thermal effect induced by high intensity and other mechanisms causing the variance of temperature will be neglected.

When taking thermal effect into consideration, the dynamic evolution of the system under XPM can be described by

$\beta=\frac{1}{n}\frac{dn}{dT}+\alpha$ is a temperature coefficient which contains both thermal index change and thermal expansion. $\theta_{j}(j=1,2)$ is the temperature difference between the mode volume of the j-th WGMR and the ambient. $\delta_{\theta}$ is determined by thermal diffusion coefficient and half thickness of the mode in radial directionGrudinin and Vahala (2009). $\nu=\frac{\beta\alpha_{A}nc}{4\pi C_{h}\rho\delta_{\theta}}$ defines the frequency shift coefficient due to thermal effect, where $c$ is light speed in vacuum, $\alpha_{A}$ is the absorption coefficient of the WGMR, $C_{h}$ is the heat capacity of the WGMR and $\rho$ is the density of the WGMR.

Set all time derivatives in Eq.(16) to zero and its static solution comes that

As $\theta_{j}$ is proportional to pump light intensity $|a_{p}^{j}|^{2}$, the frequency shift induced by thermal effect is also proportional to the intensity of pump light in each cavity. The total frequency shift lifted by nonlinear effect in the j-th WGMR is

where $\mu=k+\nu$ is the total nonlinear frequency shift coefficient. As the frequency shift caused by thermal effect is usually in the same direction with Kerr effect, thermal effect raised by high pump intensity could be incorporated with Kerr nonlinearity to manipulate the EP more effectively.

Silica is taken as an example here to demonstrate the influence of thermal effect, as seen in Fig.(7). $k$ for silica is only $2.36\times 10^{-20}m^{2}/W$Zhu et al. (2019), while $\nu$ for silica could be up to $2.37\times 10^{-14}m^{2}/W$Grudinin and Vahala (2009), which is $6$ orders larger than its Kerr coefficient. In Fig.(7), EP shift is almost $6$ orders larger with thermal effect. The EP shift $g-\delta$ is greatly enhanced with the help of thermal effect and manipulation of EPs could be accomplished with less pump intensity.

We have presented a scheme to modulate EPs in APT symmetry configuration by Kerr effect. The position of EP is manipulated by light intensity, which can also trigger the spontaneous phase transition of APT symmetry. Modulating EPs by SPM will hamper the responding ability of the APT symmetric system. The XPM scheme is then introduced to realize the enhancement of the system’s response to detectable perturbations. Moreover, thermal effect induced by high pump intensity can reduce the pump intensity required to manipulate the EP. Our research would be helpful to experimental realization of such APT structure and pave the way for appications of APT symmetry in integrated photonics devices.

The data that support the findings of this study are available from the corresponding author upon reasonable request.