Abstract

Exceptional points (EPs) have revealed a lot of fundamental physics and promise many important applications. The effect of system nonlinearity on the property of EPs is yet to be well studied. Here, we propose an optical system with nonlinear dissipation to achieve a nonreciprocal EP. Our system consists of a linear whispering-gallery-mode microresonator (WGMR) coupling to a WGMR with nonlinear dissipation. In our system, the condition of EP appearance is dependent on the field intensity in the nonlinear WGMR. Due to the chirality of intracavity field intensity, the EPs and the transmission of the system can be nonreciprocal. Our work may pave the way to exploit nonreciprocal EP for optical information processing.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Dissipation is a natural phenomenon that deviates practical quantum systems from canonical quantum mechanics described by Hermitian Hamiltonians [1,2]. Quantum systems have finite decay times and thus their evolutions are always non-unitary. In this circumstance, a quantum system can be effectively described by a non-Hermitian Hamiltonian with complex eigenvalues and nonorthogonal eigenstates [3,4]. In such a non-Hermitian system, exceptional point (EP) emerges when the coupling strength between two linear modes with the same energy goes beyond half of the dissipation difference [513]. Specifically, the complex eigenvalues and corresponding eigenstates simultaneously coalesce at EPs. Meanwhile, some counter-intuitive features associated with EPs enable a wide range of exciting applications, such as EP-based sensing and measurement [1418], phonon laser [19], unidirectional invisibility [2022], and breakdown of adiabaticity [2325].

Whispering-gallery-mode microresonators (WGMRs) with high quality factors and microscale mode volumes can possess a high intracavity field intensity and a long photon lifetime. Thus, WGMRs provide versatile platforms for fundamental physics studies and practical applications, such as parity-time (PT) symmetry [26], optothermal dynamics [27], unidirectional laser emission [28] and quantum information processing [29]). In 2011, Wiersig predicted that the existence of two or more nanoparticles in the evanescent field of a single WGMR can lead to the appearance of EPs [30], breaking the symmetry of backscattering between two counter-propagating modes [31,32]. In 2014, Peng et al. [33] and Chang et al. [34] experimentally realized a non-Hermitian Hamiltonian and the PT-symmetry in an optical system consisting of two coupled WGMRs with gain and loss, respectively. Going further, these two groups experimentally demonstrated the PT-assisted optical nonreciprocity. Loss-induced suppression and revival of lasing have been observed by Peng et al. using two passively coupled WGMRs [35], and been theoretically discussed in detail by Shu et al. [36]. In this experiment, the pattern of the field intensities in two coupled WGMRs changes from symmetric to extremely asymmetric, namely strongly chiral, when the system transits from the strong-coupling regime to the weak-coupling one [35]. Furthermore, anti-PT symmetry can be realized in three passively coupled WGMRs [37] or a single microcavity [38]. In principle, EPs in WGMRs are very sensitive to the variation of system parameters. They have been exploited to control light flow in a non-reciprocal way [3943], to enhance sensing [4456], and to manipulate the modal content of multimode lasers [5765].

The effect of nonlinearity on systems with PT symmetry also attacks a lot of attention. In 2013, Lumer et al. showed that Kerr nonlinearity can transform a system from broken to full PT symmetry [66]. In 2015, Hassan et al. studied the nonlinear reversal of the PT-symmetric phase transition in two coupled WGMRs with gain and loss by tuning the probe intensity [67]. Subsequently, nonlinearity-induced PT-symmetry was shown in coupled nonlinear waveguides without material gain [68] and higher-order EP in nondissipative non-Hermitian systems with parametric amplification was proposed for sensing [69,70]. Recently, nonreciprocal PT symmetry induced by stimulated Brillouin scattering was realized in two coupled WGMRs [71].

Here we propose an optical system with nonlinear dissipation to obtain nonreciprocal EPs. This system includes two passively coupled WGMRs and two side-coupling waveguides. One WGMR possesses nonlinear dissipation due to two-photon absorption [7274]. The other is a linear WGMR. Our nonlinear dissipative system shows nonreciprocal EPs and the dependence of the EP number on the input field, which are significantly different from those of linear systems.

In a linear optical system, the emergence of EP is independent of the propagation direction of the input field [75,76]. In contrast, the appearance of EP can be nonreciprocal in our nonlinear dissipative system because the different intensities of intracavity fields for opposite-direction inputs can lead to different system dissipation. The transmittance can also be nonreciprocal when input fields are impinged into different WGMRs.

The EPs are the intrinsic property of a linear multimode system. Thus, the input field has no effect on EPs of a linear system [75,76]. However, we find that the input field can modify the property of EPs of a system with nonlinear dissipation. By tuning the input field intensity in a proper range, multiple EPs appear for different frequency detunings between the input field and the cavity mode.

2. System and model

2.1 Coupled-WGMR system

We consider a system consisting of two coupled WGMRs (WGMR1 and WGMR2). The intracavity mode $\hat {a}_{1}$ ($\hat {a}_{2}$) with frequency $\omega _{1}$ ( $\omega _{2}$ ) of WGMR1 (WGMR2) couples to the lower (upper) waveguide with a strength $\kappa _{c1}$ ($\kappa _{c2}$), shown in Fig. 1. The two modes $\hat {a}_{1}$ and $\hat {a}_{2}$ couple to each other with a strength $\kappa _{c}$ and decay linearly with rates $\gamma _{1}$ and $\gamma _{2}$, due to either single-photon absorption or side leakage. Meanwhile, we assume that there is nonlinear dissipation in WGMR1 with rate $\gamma _{N\!L}$ and this dissipation increases with the intracavity photon intensity. For example, nonlinear dissipation with tunable $\gamma _{N\!L}$ can be implemented by two photon absorption of materials consisting of three-level ladder-type atoms [73,77,78]. The two-photon absorption master equation can be described by $\frac {d\hat {\rho }}{dt} = \gamma _{N\!L}(2\hat {a}_{1}^{2}\hat {\rho }\hat {a}_{1}^{\dagger } -\hat {a}_{1}^{\dagger }\hat {a}_{1}^{2}\hat {\rho }-\hat {\rho }\hat {a}_{1}^{\dagger }\hat {a}_{1}^{2})$ [7779], where $\hat {\rho }$ is the corresponding density operator.

 figure: Fig. 1.

Fig. 1. Schematics of coupled WGMRs. Solid (dashed) arrows represent the transmission of an input field impinging in port 1 (port 3). $\kappa _{c}$ is the coupling strength between two WGMRs. $\kappa _{c1}$ and $\kappa _{c2}$ are coupling losses of WGMR1 and WGMR2 that are introduced by lower and upper waveguides, respectively. $\gamma _{1}$ ($\gamma _2$) represents the linear part of losses of WGMR1 (WGMR2). $\gamma _{N\!L}$ represents the nonlinear loss of WGMR1.

Download Full Size | PPT Slide | PDF

Following the quantum jump method used to describe the one-photon loss process presented by Ref. [1], the effective Hamiltonian of the coupled-WGMR system in the rotating frame, with respect to an input field with frequency $\omega _d$, takes the following form ($\hbar =1$) [2,80]

$$\begin{aligned} \hat{H} & =[\Delta_{1}-i(\gamma_{1}+\kappa_{c1})]\hat{a}_{1}^{{\dagger}}\hat{a}_{1} +[\Delta_{2}-i(\gamma_{2}+\kappa_{c2})]\hat{a}_{2}^{{\dagger}}\hat{a}_{2}-i\gamma_{N\!L} (\hat{a}_{1}^{{\dagger}})^{2}\hat{a}_{1}^{2}\\ & \phantom{\qquad}+\kappa_{c}(\hat{a}_{1}^{{\dagger}}\hat{a}_{2}+\hat{a}_{1}\hat{a}_{2}^{{\dagger}})+i\sqrt{2\kappa_{c1}}a_{in}(\hat{a}_{1}-\hat{a}_{1}^{{\dagger}})\;, \end{aligned}$$
where $\Delta _{i}=\omega _{i}-\omega _d$ ($i=1, 2$) is the detuning between the resonance frequency $\omega _{i}$ and the input field frequency $\omega _{d}$. The input field $a_{\textrm {in}}$ probes the WGMR1 mode $\hat {a}_{1}$. The corresponding input field intensity is $p_{in}=|a_{in}|^2$.

The quantum Langevin equations describing the coupled-WGMR system are [2,48]

$$\begin{aligned} \frac{d\hat{a}_{1}}{dt} &={-}i[\Delta_{1}-i(\gamma_{1}+\kappa_{c1})]\hat{a}_{1}-2\gamma_{N\!L} \hat{a}_{1}^{{\dagger}}\hat{a}_{1}^{2}-i\kappa_{c}\hat{a}_{2}-\sqrt{2\kappa_{c1}}a_{in}\\ &-\sqrt{2\kappa_{c1}}\hat{c}_1^{in}-\sqrt{2\gamma_{1}}\hat{c}_2^{in}-2\sqrt{2\gamma_{N\!L}}\hat{c}_3^{in}, \end{aligned}$$
$$\frac{d\hat{a}_{2}}{dt} ={-}i[\Delta_{2}-i(\gamma_{2}+\kappa_{c2})]\hat{a}_{2}-i\kappa_{c}\hat{a}_{1}-\sqrt{2\kappa_{c2}}\hat{c}_4^{in}-\sqrt{2\gamma_{2}}\hat{c}_5^{in}.$$

Here $\hat {c}_1^{in}$ and $\hat {c}_4^{in}$ are noises entering from lower and upper input-output waveguides, whereas $\hat {c}_2^{in}$, $\hat {c}_3^{in}$, and $\hat {c}_5^{in}$ are noises entering from dissipative reservoirs [48]. These noise operators satisfy $\langle \hat {c}_i^{in}(t)\rangle$=0 and $\langle \hat {c}_i^{in\dagger }(t)\hat {c}_i^{in}(t')\rangle =\bar {n}^{th}_i\delta (t-t')$ for reservoirs in thermal equilibrium [2]. Furthermore, at optical frequencies and practical laboratory temperatures, the thermal photon number $\bar {n}^{in}_i=\langle \hat {c}_i^{in\dagger }\hat {c}_i^{in}\rangle$ for (i=1,2,3,4,5) is completely negligible [2]. We can describe the dynamics of $\hat {a}_{1}$ and $\hat {a}_{2}$ by that of the corresponding expectation amplitudes.

Taking the average of Eqs. (2) and (3) and defining the expectation amplitudes $a_{i}=\langle \hat {a}_{i}\rangle (i=1,2)$ [48], we can rewrite Eqs. (2) and (3) describing the dynamics in two coupled WGMRs in the matrix form as follows:

$$\begin{aligned} \frac{d}{dt}\left(\begin{array}{c} a_{1} \\ a_{2} \end{array}\right) & ={-}i\left(\begin{array}{cc} \Delta_{1}-i(\gamma_{1}+\kappa_{c1}+2\gamma_{N\!L}n_1 ) & \kappa_{c} \\ \kappa_{c} & \Delta_{2}-i(\gamma_{2}+\kappa_{c2}) \end{array}\right)\left(\begin{array}{c} a_{1} \\ a_{2} \end{array}\right)-\sqrt{2\kappa_{c1}}\left(\begin{array}{c} a_{in} \\ 0 \end{array}\right)\\ & ={-}iM\left(\begin{array}{c} a_{1} \\ a_{2} \end{array}\right)-\sqrt{2\kappa_{c1}}\left(\begin{array}{c} a_{in} \\ 0 \end{array}\right). \end{aligned}$$

Here we assume the intracavity photon number in WGMR1 is a variable

$$n_1=\langle \hat{a}^{{\dagger}}_1\hat{a}_1\rangle=|a_1|^2,$$
since this can qualitatively show its influence and parameter dependence on the property of this coupled system. The expressions containing the variable $n_1$ below are just formal solutions [66] and their values should be obtained by replacing $n_1$ with $|a_1|^2$ after numerically solving Eq. (4) or its counterpart when probing the linear WGMR2 from port 3.

The characteristic equation and the eigenfrequencies of the coupled-WGMR system can be found from $|\omega I-M|=0$ [75,76]. The eigenfrequencies of two supermodes due to the coupling of the two WGMRs are as follows:

$$\omega_{{\pm}}=\frac{1}{2}[\Delta_{+}-i\Gamma_{+}\pm\sqrt{(\Delta_{-}-i\Gamma_{-})^{2}+4\kappa_{c}^{2}}] \;,$$
with
$$\Gamma_{{\pm}}=(\gamma_{1}+\kappa_{c1}+2\gamma_{N\!L} n_1)\pm(\gamma_{2}+\kappa_{c2}) \;,$$
where $\Delta _{\pm }=\Delta _{1}\pm \Delta _{2}$. $\Gamma _{+}$ and $\Gamma _{-}$ denote the sum of loss and the loss contrast of the two WGMRs. In principle, this method is identical to that adopting a proper ansatz with $\left (a_1,a_2\right )^T=(a'_1,a'_2)^T\exp (-i\omega t)$ and substituting it into Eq. (4) [67]. Furthermore, the term with the square-root represents either the frequency splitting (i.e., in the PT unbroken phase) or the linewidth modification (i.e., in the PT broken phase) due to the coupling between two WGMRs. When the two WGMRs are on resonance that $\Delta _{1}=\Delta _{2}=\Delta _{0}$, the eigenfrequencies of two supermodes are reduced to
$$\omega_{{\pm}}=\Delta_{0}-\frac{i\Gamma_{+}\mp\sqrt{4\kappa_{c}^{2}-\Gamma_{-}^{2}}}{2} \;.$$

Here, the square-root $\sqrt {4\kappa _{c}^{2}-\Gamma _{-}^{2}}$ is real for $2\kappa _{c}-\Gamma _{-}>0$ or imaginary for $2\kappa _{c}-\Gamma _{-}<0$, taking $\Gamma _{-}\geq 0$. At $2\kappa _{c}-\Gamma _{-}=0$, the eigenstates and complex eigenfrequencies ($\omega _{+}$ and $\omega _{-}$) simultaneously coalesce. Therefore, the system is in a non-Hermitian degeneracy and EP appears. According to Eq. (7), the EP is crucially dependent on the intracavity photon number $n_1$ in WGMR1.

To derive the transmission amplitudes, we focus on the steady-state solutions of Eq. (4), i.e. $\frac {da_{1}}{dt} =\frac {da_{2}}{dt}= 0$, and obtain

$$\begin{aligned}a_{1} & ={-}\frac{\sqrt{2\kappa_{c1}}[i\Delta_{2}+(\gamma_{2}+\kappa_{c2})]a_{in}}{[i\Delta_{1}+(\gamma_{1}+\kappa_{c1})+2\gamma_{N\!L} n_1][i\Delta_{2}+(\gamma_{2}+\kappa_{c2})]+\kappa_{c}^{2}} \;,\\ a_{2} & =i\frac{\kappa_{c}\sqrt{2\kappa_{c1}}a_{in}}{[i\Delta_{1}+(\gamma_{1}+\kappa_{c1})+2\gamma_{N\!L} n_1][i\Delta_{2}+(\gamma_{2}+\kappa_{c2})]+\kappa_{c}^{2}} \;. \end{aligned}$$

Furthermore, we use the input-output relations for WGMR1 and WGMR2 and obtain $a_{out2}=a_{in}+\sqrt {2\kappa _{c1}}a_{1}$ and $a_{out3}=\sqrt {2\kappa _{c2}}a_{2}$. We have the transmittance $T_{1\rightarrow 2}$ ($T_{1\rightarrow 3}$) of this system from the input port 1 to the output port 2 (port 3) as follows:

$$T_{1\rightarrow2} =\left|1-\frac{2\kappa_{c1}[i\Delta_{2}+(\gamma_{2}+\kappa_{c2})]}{[i\Delta_{1}+(\gamma_{1}+\kappa_{c1})+2\gamma_{N\!L} n_1][i\Delta_{2}+(\gamma_{2}+\kappa_{c2})]+\kappa_{c}^{2}}\right|^{2},$$
$$T_{1\rightarrow3} =\left|\frac{2\kappa_{c}\sqrt{\kappa_{c1}\kappa_{c2}}}{[i\Delta_{1}+(\gamma_{1}+\kappa_{c1})+2\gamma_{N\!L} n_1][i\Delta_{2}+(\gamma_{2}+\kappa_{c2})]+\kappa_{c}^{2}}\right|^{2}.$$

Here $T_{1\rightarrow 2}$ represents the direct transmittance of the case that the output field and input field are in the lower waveguide, while $T_{1\rightarrow 3}$ represents the cross transmittance of the case that the output field and input field are in different waveguides. These transmittances depend on the linear loss and the coupling strength $\kappa _c$ as in a linear system [35], together with the intracavity photon number $n_1$ in WGMR1 and the two-photon absorption coefficient $\gamma _{N\!L}$. Therefore, in contrast to a linear system, tuning the input field intensity or frequency can significantly change the transmittances of this coupled-WGMR system with nonlinear dissipation even when all system parameters are fixed.

Now we consider the case that the input field is impinged in the upper waveguide and probes WGMR2 ($\hat {a}_{2'}$) other than WGMR1 ($\hat {a}_{1'}$). The effective Hamiltonian describing this coupled-WGMR system takes the form

$$\begin{aligned} \hat{H}' & =[\Delta_{1}-i(\gamma_{1}+\kappa_{c1})]\hat{a}_{1'}^{{\dagger}}\hat{a}_{1'} +[\Delta_{2}-i(\gamma_{2}+\kappa_{c2})]\hat{a}_{2'}^{{\dagger}}\hat{a}_{2'}-i\gamma_{N\!L} (\hat{a}_{1'}^{{\dagger}})^{2}\hat{a}_{1'}^{2}\\ & \phantom{\qquad}+\kappa_{c}(\hat{a}_{1'}^{{\dagger}}\hat{a}_{2'}+\hat{a}_{1'}\hat{a}_{2'}^{{\dagger}})+i\sqrt{2\kappa_{c2}}a_{in}(\hat{a}_{2'}-\hat{a}_{2'}^{{\dagger}}). \end{aligned}$$

The form of $\hat {H}^\prime$ is identical to that of $\hat {H}$ in which the input field probes WGMR1, except the last term representing the probing on WGMR2.

The equations describing the dynamics of the two coupled WGMRs are

$$\frac{d}{dt}\left(\begin{array}{c} a_{1'} \\ a_{2'} \end{array}\right) \!=\!-i\left(\begin{array}{cc} \Delta_{1}-i(\gamma_{1}+\kappa_{c1}+2\gamma_{N\!L}n_{1'} ) & \kappa_{c} \\ \kappa_{c} & \Delta_{2}-i(\gamma_{2}+\kappa_{c2}) \end{array}\right)\left(\begin{array}{c} a_{1'} \\ a_{2'} \end{array}\right)-\sqrt{2\kappa_{c2}}\left(\begin{array}{c} 0 \\ a_{in} \end{array}\right).$$

Here we define the mode amplitudes $a_{i'}=\langle \hat {a}_{i'}\rangle (i=1,2)$ and have the intracavity field intensity in WGMR1 as a variable

$$n_{1'}=\langle \hat{a}^{{\dagger}}_{1'}\hat{a}_{1'}\rangle=|a_{1'}|^2\;.$$

Following a procedure similar to that shown above and taking the input-output relations of $a_{out4}=a_{in}+\sqrt {2\kappa _{c2}}a_{2'}$ and $a_{out1}=\sqrt {2\kappa _{c1}}a_{1'}$, the cavity modes $a_{1'}$ and $a_{2'}$ can be derived in steady state as

$$a_{1'} =i\frac{\kappa_{c}\sqrt{2\kappa_{c2}}a_{in}}{[i\Delta_{1}+(\gamma_{1}+\kappa_{c1})+2\gamma_{N\!L} n_{1'}][i\Delta_{2}+(\gamma_{2}+\kappa_{c2})]+\kappa_{c}^{2}}\;,$$
$$a_{2'} ={-}\frac{\sqrt{2\kappa_{c2}}[i\Delta_{1}+(\gamma_{1}+\kappa_{c1}+2\gamma_{N\!L} n_{1'})]a_{in}} {[i\Delta_{1}+(\gamma_{1}+\kappa_{c1})+2\gamma_{N\!L} n_{1'}][i\Delta_{2}+(\gamma_{2}+\kappa_{c2})]+\kappa_{c}^{2}} \;.$$

The corresponding transmittances are

$$T_{3\rightarrow4} =\left|1-\frac{2\kappa_{c2}[i\Delta_{1}+(\gamma_{1}+\kappa_{c1}+2\gamma_{N\!L} n_{1'})]}{[i\Delta_{1}+(\gamma_{1}+\kappa_{c1})+2\gamma_{N\!L} n_{1'}][i\Delta_{2}+(\gamma_{2}+\kappa_{c2})]+\kappa_{c}^{2}}\right|^{2},$$
$$T_{3\rightarrow1} =\left|\frac{2\kappa_{c}\sqrt{\kappa_{c1}\kappa_{c2}}}{[i\Delta_{1}+(\gamma_{1}+\kappa_{c1})+2\gamma_{N\!L} n_{1'}][i\Delta_{2}+(\gamma_{2}+\kappa_{c2})]+\kappa_{c}^{2}}\right|^{2} \;,$$
where $T_{3\rightarrow 4}$ ($T_{3\rightarrow 1}$) represents the transmittance from the input port 3 to the output port 4 (port 1). Clearly, the direct transmittance $T_{3\rightarrow 4}$ is different from its counterpart $T_{1\rightarrow 2}$ obtained when the input field probes WGMR1. Meanwhile, although the expressions of the cross transmittances $T_{1\rightarrow 3}$ and $T_{3\rightarrow 1}$ are the same, $T_{1\rightarrow 3}$ and $T_{3\rightarrow 1}$ can also be different because the intracavity photon number in WGMR1 for these two cases are different, due to the asymmetric intracavity intensity distribution and the two-photon absorption.

2.2 Nonreciprocal EPs

For studying the influence of the input field on the EPs, one waveguide is decoupled from the corresponding WGMR [35]. In the following section, we specify the case that the upper waveguide was moved away, while the other case can be described in a similar way.

After decoupling the interaction between the upper waveguide and WGMR2, the eigenfrequencies of this simplified coupled-WGMR system can be obtained from Eq. (6) by setting $\kappa _{c2}=0$. Without loss of generality, the resonant frequencies and linear losses of two WGMRs are tuned to be equal with $\Delta _{1}=\Delta _{2}=\Delta _{0}$ and $\gamma _1=\gamma _2=\gamma _0$, respectively. The eigenfrequencies can be described as

$$\omega_{{\pm}}=\Delta_{0}-\frac{i(\kappa_{c1}+2\gamma_0+2\gamma_{N\!L} n_1)\mp\sqrt{4\kappa_{c}^{2}-(\kappa_{c1}+2\gamma_{N\!L} n_1)^{2}}}{2},$$
which is crucially dependent on intracavity photon number $n_1$ in WGMR1.

The influence of the intensity and frequency of the input field on the imaginary parts of $\omega _{\pm }$ and thus the appearance of EP is shown in Fig. 2(a). The corresponding intracavity photon number $n_1$ is shown in Fig. 2(b). Here the coupling between two WGMRs is set to $\kappa _c=1$, the coupling between WGMR1 and the lower waveguide is $\kappa _{c1}=1.8$, the linear loss rates of both WGMRs are set to be equal with $\gamma _1=\gamma _2=1$, and the two-photon absorption rate is $\gamma _{N\!L} =0.5$.

 figure: Fig. 2.

Fig. 2. Imaginary parts of two eigenfrequencies $\omega _{\pm }$ and intracavity photon number in WGMR1 : (a) $\omega ^\textrm {Im}_{\pm }$ as a function of the frequency and intensity of input fields impinging in port 1 ($\kappa _{c1}=1.8$, $\kappa _{c2}=0$); (b) $n_1$ as a function of the frequency and intensity of input fields impinging in port 1 ($\kappa _{c1}=1.8$, $\kappa _{c2}=0$); (c) $\omega ^\textrm {Im}_{\pm }$ as a function of the frequency and intensity of input fields impinging in port 3 ($\kappa _{c1}=0$, $\kappa _{c2}=1.8$); (d) $n_{1'}$ as a function of the frequency and intensity of input fields impinging in port 3 ($\kappa _{c1}=0$, $\kappa _{c2}=1.8$). Here $\kappa _c=1.0$, $\gamma _1=\gamma _2=1.0$, and $\gamma _{N\!L} =0.5$.

Download Full Size | PPT Slide | PDF

Specifically, we focus on the scattering of four input fields when the upper waveguide is moved away. The imaginary parts $\omega ^\textrm {Im}_{\pm }$ of the eigenfrequencies $\omega _{\pm }$ for $\sqrt {p_{in}}\in \{0.7, 0.8, 0.943, 1.0\}$ are shown in the left panel of Fig. 3 with increasing input field intensity from top to bottom. For a weak input field with $\sqrt {p_{in}}=0.7$, there is no EP when changing the input frequency. For a larger input field with $\sqrt {p_{in}}=0.8$, there are four EPs and these EPs symmetrically distributed on the two sides of the resonance frequency. This is because that a local minimum of the photon number in WGMR1 is achieved at resonance and its value $n_{1,\Delta _0=0}$ is smaller than a threshold $n_{1,E\!P}=(\pm 2\kappa _c-\kappa _{c1}-\Gamma _0)/2\gamma _{N\!L} >0$ with $\Gamma _0=\gamma _1-\gamma _2$ (i.e., $n_{1,E\!P}$= 0.2 for the parameters used here). However, when the input field intensity is further increased with $\sqrt {p_{in}}=0.943$, there are only three EPs because $n_{_1,\Delta _0=0}=n_{_1,E\!P}$ is achieved. Furthermore, the number of EP decreases to two when the input field intensity increases further with $\sqrt {p_{in}}=1.0$.

 figure: Fig. 3.

Fig. 3. Imaginary parts $\omega ^\textrm {Im}_{\pm }$ of two eigenfrequencies $\omega _{\pm }$ . Left panel, $\omega ^\textrm {Im}_{\pm }$ as a function of the frequency of input fields impinging in port 1 ($\kappa _{c1}=1.8$, $\kappa _{c2}=0$) with different intensities; Right panel, $\omega ^\textrm {Im}_{\pm }$ as a function of the frequency of input fields impinging in port 3 ($\kappa _{c1}=0$, $\kappa _{c2}=1.8$) with different intensities. Here $\kappa _c=1.0$, $\gamma _1=\gamma _2=1.0$, and $\gamma _{N\!L} =0.5$ are used for both panels.

Download Full Size | PPT Slide | PDF

We include the corresponding functions $\omega ^\textrm {Im}_{\pm }$ and $n_{1'}$ in Figs. 2(c) and 2(d) for the case that the lower waveguide other than the upper waveguide is moved away. All parameters are the same as in Figs. 2(a) and 2(b) except the exchange of $\kappa _{c1}$ and $\kappa _{c2}$ (i.e., $\kappa _{c1}=0$ and $\kappa _{c2}=1.8$). Similarly, we focus on the scattering of four input fields with $\sqrt {p_{in}}\in \{14.5, 14.836, 15.0, 15.5\}$. The imaginary parts $\omega ^\textrm {Im}_{\pm }$ of the eigenfrequencies are shown in the right panel of Fig. 3 with increasing input field intensity from top to bottom. For a weak input field impinged in WGMR2, no EP appears when we use the same system parameters as in Figs. 2(c) and 2(d). However, an EP appears at the resonant frequency when the input field intensity is increased to $\sqrt {p_{in}}=14.836$, and then two EPs appear symmetrically at two detunings with the same magnitude. Therefore, the emergence of EPs and their numbers are dependent on the intensity and frequency of the input field in combination with its direction.

To specify the nonreciprocity of EPs in this system, the imaginary parts $\omega ^\textrm {Im}_{\pm }$ of the eigenfrequencies as a function of the coupling strength $\kappa _c$ and input field intensity $\sqrt {p_{in}}$ are shown in Fig. 4(a) and 4(b) for the cases that the upper and lower waveguide is moved away, respectively. Here we assumed that the input field is resonant with the WGMRs and that all the other parameters in Fig. 4(a) and 4(b) are the same as in Fig. 2(a) and Fig. 2(c), respectively. The condition of EP appearance for the two cases is notably different from each other, leading to the nonreciprocity of EPs. Furthermore, we showed the evolution of $\omega ^\textrm {Im}_{\pm }$ using the resonant input field with critical intensity as $\kappa _c$ was increased in Fig. 4(c) and 4(d). For the case that the upper waveguide is moved away, there is only one EP, shown in Fig. 4(c), when we change the coupling strength $\kappa _c$ using a resonant input field with the critical intensity $\sqrt {p_{in}}=0.943$. Meanwhile, the EP appears at $\kappa _c=1$, which is in coincidence with Fig. 2(a). However, for the case that the lower waveguide is moved away and the input field probes WGMR2, there are three EPs, shown in Fig. 4(d), when changing $\kappa _c$ using a resonant input field with the critical intensity $\sqrt {p_{in}}=14.836$. Note that one EP appears at $\kappa _c=1$, while the other two EPs appear due to the balance of the coupling strength $\kappa _c$ and the nonlinear loss $2\gamma _{N\!L} n_{1'}$ in WGMR1.

 figure: Fig. 4.

Fig. 4. Imaginary parts $\omega ^\textrm {Im}_{\pm }$ of two eigenfrequencies $\omega _{\pm }$ . (a) $\omega ^\textrm {Im}_{\pm }$ as a function of the coupling rate $\kappa _c$ and input field intensity for input fields impinging in port 1 ($\kappa _{c1}=1.8$, $\kappa _{c2}=0$); (b) $\omega ^\textrm {Im}_{\pm }$ as a function of the coupling rate $\kappa _c$ and input field intensity for input fields impinging in port 3 ($\kappa _{c1}=0$, $\kappa _{c2}=1.8$); (c and d) $\omega ^\textrm {Im}_{\pm }$ as a function of the coupling rate $\kappa _c$ for input fields impinging in port 1 (c) and port 3 (d). The input field intensities in (c) and (d) are tuned to near the critical values that lead to the emergence of an EP at $\kappa _c=1$ for $\Delta _0=0$, respectively.

Download Full Size | PPT Slide | PDF

2.3 Field-dependent transmission

In practice, it is hard to directly measure the photon number distribution in WGMR1. However, this distribution is correlated to the transmittance of the coupled-WGMR system, because the transmittance is a constant when the coupled system with nonlinear dissipation is in a steady state and it can be obtained from the input-output relation involving the intracavity mode of WGMRs. For instance, $n_1$ is correlated to the transmittance $T_{1\rightarrow 2}$ of this coupled-WGMR system with the input-output relation $a_{out2}=a_{in}+\sqrt {2\kappa _{c1}}a_{1}$, when the upper waveguide is moved away. Specifically, it can be expressed as [35]

$$n_1=\frac{T_{1\rightarrow2}-2\sqrt{T_{1\rightarrow2}}\textrm{cos}\theta+1}{2\kappa_{c1}}p_{in},$$
where $\theta$ represents the phase of the complex coefficient of amplitude transmission $a_{out2}/a_{in}=\sqrt {T_{1\rightarrow 2}}e^{i\theta }$ and $p_{in}=|a_{in}|^2$. In contrast to a constant transmittance of linear systems [35], the transmittance $T_{1\rightarrow 2}$ changes for different input fields and thus depends on $n_1$, whereas $T_{1\rightarrow 2}$ for a given input field can be obtained by a measurement and its formal expression can be derived from Eq. (10) as
$$T_{1\rightarrow2} =\left|1-\frac{2\kappa_{c1}(i\Delta_{0}+\gamma_{0})}{(i\Delta_{0}+\gamma_{0}+\kappa_{c1}+2\gamma_{N\!L} n_1)(i\Delta_{0}+\gamma_{0})+\kappa_{c}^{2}}\right|^{2}.$$

Therefore, the photon number distribution in WGMR1 can also be written as

$$n_{1}=\frac{(\Delta_{0}^{2}-\kappa_{c}^{2}-\gamma_{0}^{2})T_{-}+\kappa_{c1}\gamma_{0}T_{+}-(\kappa_{c1}+2 \gamma_{0})\Delta_{0}\sqrt{T_{1\rightarrow2}}\sin\theta} {2\kappa_{N\!L}(\gamma_{0}T_{-}+\Delta_{0}\sqrt{T_{1\rightarrow2}}\sin\theta)},$$
which is implicitly dependent on the input field intensity and $T_{\pm }=1\pm \sqrt {T_{1\rightarrow 2}}\cos \theta$. From the formal expression of $T_{1\rightarrow 2}$ as shown in Eq. (21), it is easily to show that for a resonant input field with $\Delta _{0}$, an increase of the photon number $n_1$ in WGMR1 increases the transmittance $T_{1\rightarrow 2}$, even though the total loss $\Gamma _+=2\gamma _{0}+\kappa _{c1}+2\gamma _{N\!L} n_1$ of the coupled-WGMR system increases linearly with $n_1$.

Figure 5 shows the transmittance $T_{1\rightarrow 2}$ as a function of the frequency and intensity of an input field. All parameters are the same as in Fig. 2(a). For a resonant input field with $\Delta _0=0$, the transmittance $T_{1\rightarrow 2}$ increases when the intensity $p_{in}$ increases. For a non-resonant input field (e.g., $|\Delta _0|>1.5$), the transmittance $T_{1\rightarrow 2}$ first decreases to a minimum and then increases when $p_{in}$ increases. Meanwhile, the coupled-WGMR system is changed from a strong coupling regime ($p_{in}<p'_{in}$) to a weak coupling regime ($p_{in}>p'_{in}$). Here $p'_{in}$ is the input field intensity required to achieve the EP.

 figure: Fig. 5.

Fig. 5. Transmittance $T_{1\rightarrow 2}$ as a function of the frequency and intensity of input fields. Here $\kappa _c=1.0$, $\kappa _{c1}=1.8$, $\kappa _{c2}=0$, $\gamma _1=\gamma _2=1.0$, and $\gamma _{N\!L} =0.5$.

Download Full Size | PPT Slide | PDF

However, for a given $p_{in}$, the transmittance $T_{1\rightarrow 2}$ first decreases to two local minimums and then increases to a local maximum at the resonant point when continuously decreasing $|\Delta _0|$. This is inverse to the relation between the intracavity number and the detuning, shown in Fig. 2(b). Therefore, the transmittance $T_{1\rightarrow 2}$ performs differently when either the intensity or frequency of an input field is changed for a given coupled-WGMR system. This field-dependent property leads to the nonreciprocal transmission of our coupled-WGMR system.

2.4 Nonreciprocal transmission

In this section, we study the scattering process of a coupled-WGMR system, in which nonreciprocal transmission can be achieved when an input field with the same frequency and intensity is impinged into different WGMRs, shown in Fig. 6. The lower and upper waveguides couple to WGMR1 and WGMR2 with identical coupling rates $\kappa _{c1}=\kappa _{c2}$, respectively. The coupling rate between two WGMRs are set to be unity with $\kappa _c=1$. For simplicity, we study a specified system in which the linear loss of each WGMR is equal to $0.1$ and the nonlinear two-photon absorption rate is $\gamma _{N\!L} =0.5$. We will show that an input field impinging into port 3 will be output into two ports with an equal intensity (i.e., $T_{3\rightarrow 4}=T_{3\rightarrow 1}$) and that an input field impinging into port 1 will be output into the direct transmission mode with vanishing cross transmission (i.e., $T_{1\rightarrow 3}>0$, $T_{1\rightarrow 2}\simeq 0$). This can be referred to as nonreciprocal transmission [8188].

 figure: Fig. 6.

Fig. 6. Transmittances as a function of the frequency and intensity of input fields: (a) $\Delta _T=|T_{3\rightarrow 1}-T_{3\rightarrow 4}|$ represents the transmittance difference when impinging the input field in the port $3$; (b) $T_{3\rightarrow 1}$ represents the transmittance into port $1$ when impinging the input field in the port $3$; (c) $T_{1\rightarrow 2}$ represents the transmittance into port $2$ when impinging the input field in the port $1$; (d) $T_{1\rightarrow 3}$ represents the transmittance into port $3$ when impinging the input field in the port $1$. Here $\kappa _c=1.0$, $\kappa _{c1}=\kappa _{c2}=2.0$, $\gamma _1=\gamma _2=0.1$, and $\gamma _{N\!L} =0.5$.

Download Full Size | PPT Slide | PDF

When the input field probes WGMR2 from port 3, the transmittances $T_{3\rightarrow 1}$ and $T_{3\rightarrow 4}$ of this coupled-WGMR system are described in Eq. (17) and (18). The transmittance difference $\Delta _T=|T_{3\rightarrow 1}-T_{3\rightarrow 4}|$ as a function of the frequency and intensity of the input field is shown in Fig. 6(a). Meanwhile, the transmittance $T_{3\rightarrow 1}$ with the same system parameters is shown in Fig. 6(b). There exist a regime labeled in purple in Fig. 6(a) in which an input field is output into the direct transmission (port 4) and cross transmission (port 1) modes with balanced intensities ($\Delta _T\leq 0.06$); the transmittances $T_{3\rightarrow 1}$ and $T_{3\rightarrow 4}$ are almost equal and can be as large as $0.40$.

When the input field probes WGMR1 from port 1, the corresponding transmittances $T_{1\rightarrow 3}$ and $T_{1\rightarrow 2}$ are described in Eq. (10) and (11). Figures 6(c) and 6(d) show the transmittances $T_{1\rightarrow 3}$ and $T_{1\rightarrow 2}$ as a function of the frequency and intensity of the input field, respectively. For a given input field frequency, the transmittances $T_{1\rightarrow 2}$ and $T_{1\rightarrow 3}$ both continuously decreases when increasing the input field intensity $p_{in}$ with $\sqrt {p_{in}}\in (0,3.0)$. For a given intensity, however, the transmittances $T_{1\rightarrow 2}$ and $T_{1\rightarrow 3}$ change in opposite ways when increasing the frequency detunings $|\Delta _{0}|$, i.e., $T_{1\rightarrow 2}$ increases continuously, while $T_{1\rightarrow 3}$ decreases when $|\Delta _{0}|$ increases. For the regime where both $|\Delta _{T}|$ in Fig. 6(a) and $T_{1\rightarrow 2}$ in Fig. 6(c) approach zero, $T_{1\rightarrow 3}$ can be larger than $0.30$. Therefore, the transmission of this coupled-WGMR system is nonreciprocal and divergent in this regime: An input field is output into the cross transmission mode when it probes WGMR1, while it is equally output into the direct transmission and cross transmission modes when it probes WGMR2.

3. Discussion and summary

So far, we have focused on the nonreciprocal EP and the nonreciprocal transmission of a single probe field. Shi et al. found that the existence of dynamic reciprocity makes optical isolators involving nonlinear processes fail to provide isolation for arbitrary backward-propagating noise coexisting with a forward probe [89]. Although this dynamic reciprocity might present in our coupled-WGMR system involving nonlinear dissipation, nonlinear nonreciprocity presented in the situation with a single probe is still useful for practical applications, such as nonlinear nonreciprocal devices [9093].

The nonlinear dissipation is the main reason leading to nonreciprocal EPs and transmission. The nonlinear loss due to two-photon absorption modifies the total loss of WGMR1 and increases linearly with the increase of the average photon number in WGMR1. Parameters being capable of tuning $n_1$ have influence on the emergence of EP. By decoupling WGMR2 (WGMR1) from the upper (lower) waveguide, i.e. $\kappa _{c2}=0$ ($\kappa _{c1}=0$), we showed that the number of EPs can vary significantly when the frequency and intensity of the probe field change. The EPs appear in a nonreciprocal pattern when probing different WGMRs (see Fig. 3) . This might be useful for studying the interesting EP-based applications [1425]. Furthermore, in our nonlinear dissipative system, a probe field with different intensities or frequencies leads to different intracavity intensities and thus different total losses in WGMR1. As a result, the nonreciprocal transmission is also influenced by the nonlinear two-photon absorption.

Currently, two-photon absorption of natural material is small, while it requires a large two-photon absorption to achieve a high performance of the present protocol for a low intensity input field. In principle, large two-photon absorption nonlinearity can be achieved by using atomic ensembles with two-photon transmission [73]. Furthermore, for a finite two-photon absorption, our protocols can work when high-intensity input fields are impinged into this system, because the extra loss of WGMR1 proportional to $2\gamma _{N\!L} n_1$ or $2\gamma _{N\!L} n_{1'}$ is the main factor influencing the transmittances and the emergence of EP.

In summary, we have shown that nonreciprocal EPs can be induced by nonlinear dissipation in a coupled-WGMR system. Due to the nonlinearity, the condition of nonreciprocal EPs emerging and EP number are dependent on the intensity and frequency of the input field. Meanwhile, the system transmits an input field in a nonreciprocal way, directing the input field probing the linear WGMR into two outputs equally and that probing the nonlinear WGMR primarily into one output. This nonlinear system may be used to study optical nonreciprocity at EPs and to explore EP-based applications.

Funding

National Key Research and Development Program of China (2017YFA0303703, 2019YFA0308700, 2019YFA0308704); Natural Science Foundation of Jiangsu Province (BK20180461); National Natural Science Foundation of China (11874212, 11890704, 11904171).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

1. M. B. Plenio and P. L. Knight, “The quantum-jump approach to dissipative dynamics in quantum optics,” Rev. Mod. Phys. 70(1), 101–144 (1998). [CrossRef]  

2. H. J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations (Springer, 1999).

3. C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70(6), 947–1018 (2007). [CrossRef]  

4. R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14(1), 11–19 (2018). [CrossRef]  

5. W. D. Heiss, “Exceptional points of non-Hermitian operators,” J. Phys. A: Math. Gen. 37(6), 2455–2464 (2004). [CrossRef]  

6. V. Dominguez-Rocha, R. Thevamaran, F. M. Ellis, and T. Kottos, “Environmentally induced exceptional points in elastodynamics,” Phys. Rev. Appl. 13(1), 014060 (2020). [CrossRef]  

7. F. Minganti, A. Miranowicz, R. W. Chhajlany, and F. Nori, “Quantum exceptional points of non-Hermitian Hamiltonians and Liouvillians: The effects of quantum jumps,” Phys. Rev. A 100(6), 062131 (2019). [CrossRef]  

8. S. X. Li, X. Q. Zhang, Q. Xu, M. Liu, M. Kang, J. G. Han, and W. L. Zhang, “Exceptional point in a metal-graphene hybrid metasurface with tunable asymmetric loss,” Opt. Express 28(14), 20083–20094 (2020). [CrossRef]  

9. P. C. Kuo, N. Lambert, A. Miranowicz, H. B. Chen, G. Y. Chen, Y. N. Chen, and F. Nori, “Collectively induced exceptional points of quantum emitters coupled to nanoparticle surface plasmons,” Phys. Rev. A 101(1), 013814 (2020). [CrossRef]  

10. I. Arkhipov, A. Miranowicz, F. Minganti, and F. Nori, “Quantum and semiclassical exceptional points of a linear system of coupled cavities with losses and gain within the Scully-Lamb laser theory,” Phys. Rev. A 101(1), 013812 (2020). [CrossRef]  

11. N. Habler and J. Scheuer, “Higher-order exceptional points: A route for flat-top optical filters,” Phys. Rev. A 101(4), 043828 (2020). [CrossRef]  

12. J. Wen, C. Zheng, X. Kong, S. Wei, T. Xin, and G. Long, “Experimental demonstration of a digital quantum simulation of a general $\mathcal{PT}$-symmetric system,” Phys. Rev. A 99(6), 062122 (2019). [CrossRef]  

13. W.-C. Gao, C. Zheng, L. Liu, T.-J. Wang, and C. Wang, “Experimental simulation of the parity-time symmetric dynamics using photonic qubits,” Opt. Express 29(1), 517–526 (2021). [CrossRef]  

14. J. Wiersig, “Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: Application to microcavity sensors for single-particle detection,” Phys. Rev. Lett. 112(20), 203901 (2014). [CrossRef]  

15. M. P. Hokmabadi, A. Schumer, D. N. Christodoulides, and M. Khajavikhan, “Non-Hermitian ring laser gyroscopes with enhanced Sagnac sensitivity,” Nature 576(7785), 70–74 (2019). [CrossRef]  

16. J. Wiersig, “Prospects and fundamental limits in exceptional point-based sensing,” Nat. Commun. 11(1), 2454 (2020). [CrossRef]  

17. X. Mao, G.-Q. Qin, H. Yang, H. Zhang, M. Wang, and G.-L. Long, “Enhanced sensitivity of optical gyroscope in a mechanical parity-time-symmetric system based on exceptional point,” New J. Phys. 22(9), 093009 (2020). [CrossRef]  

18. G. Q. Zhang, Y. P. Wang, and J. Q. You, “Dispersive readout of a weakly coupled qubit via the parity-time-symmetric phase transition,” Phys. Rev. A 99(5), 052341 (2019). [CrossRef]  

19. H. Jing, S. K. Ozdemir, X. Y. Lu, J. Zhang, L. Yang, and F. Nori, “PT-symmetric phonon laser,” Phys. Rev. Lett. 113(5), 053604 (2014). [CrossRef]  

20. Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011). [CrossRef]  

21. L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2013). [CrossRef]  

22. J.-H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113(12), 123004 (2014). [CrossRef]  

23. C. Dembowski, B. Dietz, H. D. Graf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, “Encircling an exceptional point,” Phys. Rev. E 69(5), 056216 (2004). [CrossRef]  

24. H. Xu, D. Mason, L. Y. Jiang, and J. G. E. Harris, “Topological energy transfer in an optomechanical system with exceptional points,” Nature 537(7618), 80–83 (2016). [CrossRef]  

25. S. Dey, A. Laha, and S. Ghosh, “Nonlinearity-induced anomalous mode collapse and nonchiral asymmetric mode switching around multiple exceptional points,” Phys. Rev. B 101(12), 125432 (2020). [CrossRef]  

26. J. Wen, X. Jiang, L. Jiang, and M. Xiao, “Parity-time symmetry in optical microcavity systems,” J. Phys. B: At., Mol. Opt. Phys. 51(22), 222001 (2018). [CrossRef]  

27. X. Jiang and L. Yang, “Optothermal dynamics in whispering-gallery microresonators,” Light: Sci. Appl. 9(1), 24 (2020). [CrossRef]  

28. X.-F. Jiang, C.-L. Zou, L. Wang, Q. Gong, and Y.-F. Xiao, “Whispering-gallery microcavities with unidirectional laser emission,” Laser Photonics Rev. 10(1), 40–61 (2016). [CrossRef]  

29. P.-B. Li, S.-Y. Gao, and F.-L. Li, “Quantum-information transfer with nitrogen-vacancy centers coupled to a whispering-gallery microresonator,” Phys. Rev. A 83(5), 054306 (2011). [CrossRef]  

30. J. Wiersig, “Structure of whispering-gallery modes in optical microdisks perturbed by nanoparticles,” Phys. Rev. A 84(6), 063828 (2011). [CrossRef]  

31. B. Peng, S. K. Ozdemir, M. Liertzer, W. J. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Natl. Acad. Sci. U. S. A. 113(25), 6845–6850 (2016). [CrossRef]  

32. X.-F. Liu, T.-J. Wang, Y.-P. Gao, C. Cao, and C. Wang, “Chiral microresonator assisted by Rydberg-atom ensembles,” Phys. Rev. A 98(3), 033824 (2018). [CrossRef]  

33. B. Peng, S. K. Ozdemir, F. C. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014). [CrossRef]  

34. L. Chang, X. S. Jiang, S. Y. Hua, C. Yang, J. M. Wen, L. Jiang, G. Y. Li, G. Z. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8(7), 524–529 (2014). [CrossRef]  

35. B. Peng, S. K. Ozdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014). [CrossRef]  

36. F.-J. Shu, C.-L. Zou, X.-B. Zou, and L. Yang, “Chiral symmetry breaking in a microring optical cavity by engineered dissipation,” Phys. Rev. A 94(1), 013848 (2016). [CrossRef]  

37. F. Yang, Y. C. Liu, and L. You, “Anti-PT symmetry in dissipatively coupled optical systems,” Phys. Rev. A 96(5), 053845 (2017). [CrossRef]  

38. F. Zhang, Y. Feng, X. Chen, L. Ge, and W. Wan, “Synthetic anti-PT symmetry in a single microcavity,” Phys. Rev. Lett. 124(5), 053901 (2020). [CrossRef]  

39. A. Calabrese, F. Ramiro-Manzano, H. M. Price, S. Biasi, M. Bernard, M. Ghulinyan, I. Carusotto, and L. Pavesi, “Unidirectional reflection from an integrated “taiji" microresonator,” Photonics Res. 8(8), 1333–1341 (2020). [CrossRef]  

40. B. He, L. Yang, X. Jiang, and M. Xiao, “Transmission nonreciprocity in a mutually coupled circulating structure,” Phys. Rev. Lett. 120(20), 203904 (2018). [CrossRef]  

41. H. Zhang, F. Saif, Y. Jiao, and H. Jing, “Loss-induced transparency in optomechanics,” Opt. Express 26(19), 25199–25210 (2018). [CrossRef]  

42. L. Liu, J. H. Zhang, L. Jin, and L. Zhou, “Transport properties of the non-Hermitian T-shaped quantum router,” Opt. Express 27(10), 13694–13705 (2019). [CrossRef]  

43. T. Liu, G. C. Ma, S. J. Liang, H. Gao, Z. M. Gu, S. W. An, and J. Zhu, “Single-sided acoustic beam splitting based on parity-time symmetry,” Phys. Rev. B 102(1), 014306 (2020). [CrossRef]  

44. Z. P. Liu, J. Zhang, S. K. Ozdemir, B. Peng, H. Jing, X. Y. Lu, C. W. Li, L. Yang, F. Nori, and Y. X. Liu, “Metrology with PT-symmetric cavities: Enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117(11), 110802 (2016). [CrossRef]  

45. W. J. Chen, S. K. Ozdemir, G. M. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548(7666), 192–196 (2017). [CrossRef]  

46. H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548(7666), 187–191 (2017). [CrossRef]  

47. W. J. Chen, J. Zhang, B. Peng, S. K. Ozdemir, X. D. Fan, and L. Yang, “Parity-time-symmetric whispering-gallery mode nanoparticle sensor invited,” Photonics Res. 6(5), A23–A30 (2018). [CrossRef]  

48. H. K. Lau and A. A. Clerk, “Fundamental limits and non-reciprocal approaches in non-Hermitian quantum sensing,” Nat. Commun. 9(1), 4320 (2018). [CrossRef]  

49. M. Zhang, W. Sweeney, C. W. Hsu, L. Yang, A. D. Stone, and L. Jiang, “Quantum noise theory of exceptional point amplifying sensors,” Phys. Rev. Lett. 123(18), 180501 (2019). [CrossRef]  

50. G. Q. Qin, M. Wang, J. W. Wen, D. Ruan, and G. L. Long, “Brillouin cavity optomechanics sensing with enhanced dynamical backaction,” Photonics Res. 7(12), 1440–1446 (2019). [CrossRef]  

51. S. N. Huai, Y. L. Liu, J. Zhang, L. Yang, and Y. X. Liu, “Enhanced sideband responses in a PT-symmetric-like cavity magnomechanical system,” Phys. Rev. A 99(4), 043803 (2019). [CrossRef]  

52. P. Djorwe, Y. Pennec, and B. Djafari-Rouhani, “Exceptional point enhances sensitivity of optomechanical mass sensors,” Phys. Rev. Appl. 12(2), 024002 (2019). [CrossRef]  

53. Q. Zhong, J. Ren, M. Khajavikhan, D. N. Christodoulides, S. K. Özdemir, and R. El-Ganainy, “Sensing with exceptional surfaces in order to combine sensitivity with robustness,” Phys. Rev. Lett. 122(15), 153902 (2019). [CrossRef]  

54. Y. H. Lai, Y. K. Lu, M. G. Suh, Z. Q. Yuan, and K. Vahala, “Observation of the exceptional-point-enhanced Sagnac effect,” Nature 576(7785), 65–69 (2019). [CrossRef]  

55. Z. C. Zhang, Y. P. Wang, and X. G. Wang, “PT-symmetry-breaking-enhanced cavity optomechanical magnetometry,” Phys. Rev. A 102(2), 023512 (2020). [CrossRef]  

56. Y. M. Chu, Y. Liu, H. B. Liu, and J. M. Cai, “Quantum sensing with a single-qubit pseudo-Hermitian system,” Phys. Rev. Lett. 124(2), 020501 (2020). [CrossRef]  

57. L. Feng, Z. J. Wong, R. M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346(6212), 972–975 (2014). [CrossRef]  

58. Z. Lin, A. Pick, M. Loncar, and A. W. Rodriguez, “Enhanced spontaneous emission at third-order Dirac exceptional points in inverse-designed photonic crystals,” Phys. Rev. Lett. 117(10), 107402 (2016). [CrossRef]  

59. H. Hodaei, M. A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346(6212), 975–978 (2014). [CrossRef]  

60. M. Kim, K. Kwon, J. Shim, Y. Jung, and K. Yu, “Partially directional microdisk laser with two Rayleigh scatterers,” Opt. Lett. 39(8), 2423–2426 (2014). [CrossRef]  

61. P. Miao, Z. F. Zhang, J. B. Sun, W. Walasik, S. Longhi, N. M. Litchinitser, and L. Feng, “Orbital angular momentum microlaser,” Science 353(6298), 464–467 (2016). [CrossRef]  

62. J. Perina, A. Luks, J. K. Kalaga, W. Leonski, and A. Miranowicz, “Nonclassical light at exceptional points of a quantum PT-symmetric two-mode system,” Phys. Rev. A 100(5), 053820 (2019). [CrossRef]  

63. E. Lafalce, Q. J. Zeng, C. H. Lin, M. J. Smith, S. T. Malak, J. Jung, Y. J. Yoon, Z. Q. Lin, V. V. Tsukruk, and Z. V. Vardeny, “Robust lasing modes in coupled colloidal quantum dot microdisk pairs using a non-Hermitian exceptional point,” Nat. Commun. 10(1), 561 (2019). [CrossRef]  

64. Q. T. Cao, Y. L. Chen, and Y. F. Xiao, “Chiral emission and Purcell enhancement in a hybrid plasmonic-photonic microresonator,” Light: Sci. Appl. 9(1), 4 (2020). [CrossRef]  

65. T. Wu, W. X. Zhang, H. Z. Zhang, S. S. Hou, G. Y. Chen, R. B. Liu, C. C. Lu, J. F. Li, R. Y. Wang, P. F. Duan, J. J. Li, B. Wang, L. Shi, J. Zi, and X. D. Zhang, “Vector exceptional points with strong superchiral fields,” Phys. Rev. Lett. 124(8), 083901 (2020). [CrossRef]  

66. Y. Lumer, Y. Plotnik, M. C. Rechtsman, and M. Segev, “Nonlinearly induced PT transition in photonic systems,” Phys. Rev. Lett. 111(26), 263901 (2013). [CrossRef]  

67. A. U. Hassan, H. Hodaei, M.-A. Miri, M. Khajavikhan, and D. N. Christodoulides, “Nonlinear reversal of the $\mathcal{PT}$-symmetric phase transition in a system of coupled semiconductor microring resonators,” Phys. Rev. A 92(6), 063807 (2015). [CrossRef]  

68. M.-A. Miri and A. Alù, “Nonlinearity-induced pt-symmetry without material gain,” New J. Phys. 18(6), 065001 (2016). [CrossRef]  

69. Y.-X. Wang and A. A. Clerk, “Non-Hermitian dynamics without dissipation in quantum systems,” Phys. Rev. A 99(6), 063834 (2019). [CrossRef]  

70. A. Roy, S. Jahani, Q. Guo, A. Dutt, S. Fan, M.-A. Miri, and A. Marandi, “Nondissipative non-Hermitian dynamics and exceptional points in coupled optical parametric oscillators,” Optica 8(3), 415–421 (2021). [CrossRef]  

71. J. Ma, J. Wen, S. Ding, S. Li, Y. Hu, X. Jiang, L. Jiang, and M. Xiao, “Chip-based optical isolator and nonreciprocal parity-time symmetry induced by stimulated Brillouin scattering,” Laser Photonics Rev. 14(5), 1900278 (2020). [CrossRef]  

72. N. S. Makarov, M. Drobizhev, and A. Rebane, “Two-photon absorption standards in the 550–1600 nm excitation wavelength range,” Opt. Express 16(6), 4029–4047 (2008). [CrossRef]  

73. H. You, S. M. Hendrickson, and J. D. Franson, “Analysis of enhanced two-photon absorption in tapered optical fibers,” Phys. Rev. A 78(5), 053803 (2008). [CrossRef]  

74. X. Dai, X. Zhang, I. M. Kislyakov, L. Wang, J. Huang, S. Zhang, N. Dong, and J. Wang, “Enhanced two-photon absorption and two-photon luminescence in monolayer MoS2 and WS2 by defect repairing,” Opt. Express 27(10), 13744–13753 (2019). [CrossRef]  

75. M. A. Miri and A. Alù, “Exceptional points in optics and photonics,” Science 363(6422), eaar7709 (2019). [CrossRef]  

76. S. K. Ozdemir, S. Rotter, F. Nori, and L. Yang, “Parity-time symmetry and exceptional points in photonics,” Nat. Mater. 18(8), 783–798 (2019). [CrossRef]  

77. A. Miranowicz, J. Bajer, M. Paprzycka, Y.-X. Liu, A. M. Zagoskin, and F. Nori, “State-dependent photon blockade via quantum-reservoir engineering,” Phys. Rev. A 90(3), 033831 (2014). [CrossRef]  

78. E. S. Guerra, B. M. Garraway, and P. L. Knight, “Two-photon parametric pumping versus two-photon absorption: A quantum jump approach,” Phys. Rev. A 55(5), 3842–3857 (1997). [CrossRef]  

79. Y. R. Shen, “Quantum statistics of nonlinear optics,” Phys. Rev. 155(3), 921–931 (1967). [CrossRef]  

80. T. K. Fryett, C. M. Dodson, and A. Majumdar, “Cavity enhanced nonlinear optics for few photon optical bistability,” Opt. Express 23(12), 16246–16255 (2015). [CrossRef]  

81. K. Xia, G. Lu, G. Lin, Y. Cheng, Y. Niu, S. Gong, and J. Twamley, “Reversible nonmagnetic single-photon isolation using unbalanced quantum coupling,” Phys. Rev. A 90(4), 043802 (2014). [CrossRef]  

82. K. Xia, F. Nori, and M. Xiao, “Cavity-free optical isolators and circulators using a chiral cross-Kerr nonlinearity,” Phys. Rev. Lett. 121(20), 203602 (2018). [CrossRef]  

83. Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9(1), 1797 (2018). [CrossRef]  

84. S. Zhang, Y. Hu, G. Lin, Y. Niu, K. Xia, J. Gong, and S. Gong, “Thermal-motion-induced non-reciprocal quantum optical system,” Nat. Photonics 12(12), 744–748 (2018). [CrossRef]  

85. Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10(10), 657–661 (2016). [CrossRef]  

86. L. Tang, J. Tang, W. Zhang, G. Lu, H. Zhang, Y. Zhang, K. Xia, and M. Xiao, “On-chip chiral single-photon interface: Isolation and unidirectional emission,” Phys. Rev. A 99(4), 043833 (2019). [CrossRef]  

87. S. Zhang, G. Lin, Y. Hu, Y. Qi, Y. Niu, and S. Gong, “Cavity-free circulator with low insertion loss using hot atoms,” Phys. Rev. Appl. 14(2), 024032 (2020). [CrossRef]  

88. C. Liang, B. Liu, A.-N. Xu, X. Wen, C. Lu, K. Xia, M. K. Tey, Y.-C. Liu, and L. You, “Collision-induced broadband optical nonreciprocity,” Phys. Rev. Lett. 125(12), 123901 (2020). [CrossRef]  

89. Y. Shi, Z. Yu, and S. Fan, “Limitations of nonlinear optical isolators due to dynamic reciprocity,” Nat. Photonics 9(6), 388–392 (2015). [CrossRef]  

90. L. D. Bino, J. M. Silver, M. T. M. Woodley, S. L. Stebbings, X. Zhao, and P. Del’Haye, “Microresonator isolators and circulators based on the intrinsic nonreciprocity of the kerr effect,” Optica 5(3), 279–282 (2018). [CrossRef]  

91. D. L. Sounas, J. Soric, and A. Alù, “Broadband passive isolators based on coupled nonlinear resonances,” Nat. Electron. 1(2), 113–119 (2018). [CrossRef]  

92. K. Y. Yang, J. Skarda, M. Cotrufo, A. Dutt, G. H. Ahn, M. Sawaby, D. Vercruysse, A. Arbabian, S. Fan, A. Alù, and J. Vučković, “Inverse-designed non-reciprocal pulse router for chip-based lidar,” Nat. Photonics 14(6), 369–374 (2020). [CrossRef]  

93. L. Tang, J. Tang, H. Wu, J. Zhang, M. Xiao, and K. Xia, “Broad-intensity-range optical nonreciprocity based on feedback-induced Kerr nonlinearity,” Photon. Res., in press (2021).

References

  • View by:

  1. M. B. Plenio and P. L. Knight, “The quantum-jump approach to dissipative dynamics in quantum optics,” Rev. Mod. Phys. 70(1), 101–144 (1998).
    [Crossref]
  2. H. J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations (Springer, 1999).
  3. C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70(6), 947–1018 (2007).
    [Crossref]
  4. R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14(1), 11–19 (2018).
    [Crossref]
  5. W. D. Heiss, “Exceptional points of non-Hermitian operators,” J. Phys. A: Math. Gen. 37(6), 2455–2464 (2004).
    [Crossref]
  6. V. Dominguez-Rocha, R. Thevamaran, F. M. Ellis, and T. Kottos, “Environmentally induced exceptional points in elastodynamics,” Phys. Rev. Appl. 13(1), 014060 (2020).
    [Crossref]
  7. F. Minganti, A. Miranowicz, R. W. Chhajlany, and F. Nori, “Quantum exceptional points of non-Hermitian Hamiltonians and Liouvillians: The effects of quantum jumps,” Phys. Rev. A 100(6), 062131 (2019).
    [Crossref]
  8. S. X. Li, X. Q. Zhang, Q. Xu, M. Liu, M. Kang, J. G. Han, and W. L. Zhang, “Exceptional point in a metal-graphene hybrid metasurface with tunable asymmetric loss,” Opt. Express 28(14), 20083–20094 (2020).
    [Crossref]
  9. P. C. Kuo, N. Lambert, A. Miranowicz, H. B. Chen, G. Y. Chen, Y. N. Chen, and F. Nori, “Collectively induced exceptional points of quantum emitters coupled to nanoparticle surface plasmons,” Phys. Rev. A 101(1), 013814 (2020).
    [Crossref]
  10. I. Arkhipov, A. Miranowicz, F. Minganti, and F. Nori, “Quantum and semiclassical exceptional points of a linear system of coupled cavities with losses and gain within the Scully-Lamb laser theory,” Phys. Rev. A 101(1), 013812 (2020).
    [Crossref]
  11. N. Habler and J. Scheuer, “Higher-order exceptional points: A route for flat-top optical filters,” Phys. Rev. A 101(4), 043828 (2020).
    [Crossref]
  12. J. Wen, C. Zheng, X. Kong, S. Wei, T. Xin, and G. Long, “Experimental demonstration of a digital quantum simulation of a general $\mathcal{PT}$-symmetric system,” Phys. Rev. A 99(6), 062122 (2019).
    [Crossref]
  13. W.-C. Gao, C. Zheng, L. Liu, T.-J. Wang, and C. Wang, “Experimental simulation of the parity-time symmetric dynamics using photonic qubits,” Opt. Express 29(1), 517–526 (2021).
    [Crossref]
  14. J. Wiersig, “Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: Application to microcavity sensors for single-particle detection,” Phys. Rev. Lett. 112(20), 203901 (2014).
    [Crossref]
  15. M. P. Hokmabadi, A. Schumer, D. N. Christodoulides, and M. Khajavikhan, “Non-Hermitian ring laser gyroscopes with enhanced Sagnac sensitivity,” Nature 576(7785), 70–74 (2019).
    [Crossref]
  16. J. Wiersig, “Prospects and fundamental limits in exceptional point-based sensing,” Nat. Commun. 11(1), 2454 (2020).
    [Crossref]
  17. X. Mao, G.-Q. Qin, H. Yang, H. Zhang, M. Wang, and G.-L. Long, “Enhanced sensitivity of optical gyroscope in a mechanical parity-time-symmetric system based on exceptional point,” New J. Phys. 22(9), 093009 (2020).
    [Crossref]
  18. G. Q. Zhang, Y. P. Wang, and J. Q. You, “Dispersive readout of a weakly coupled qubit via the parity-time-symmetric phase transition,” Phys. Rev. A 99(5), 052341 (2019).
    [Crossref]
  19. H. Jing, S. K. Ozdemir, X. Y. Lu, J. Zhang, L. Yang, and F. Nori, “PT-symmetric phonon laser,” Phys. Rev. Lett. 113(5), 053604 (2014).
    [Crossref]
  20. Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
    [Crossref]
  21. L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2013).
    [Crossref]
  22. J.-H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113(12), 123004 (2014).
    [Crossref]
  23. C. Dembowski, B. Dietz, H. D. Graf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, “Encircling an exceptional point,” Phys. Rev. E 69(5), 056216 (2004).
    [Crossref]
  24. H. Xu, D. Mason, L. Y. Jiang, and J. G. E. Harris, “Topological energy transfer in an optomechanical system with exceptional points,” Nature 537(7618), 80–83 (2016).
    [Crossref]
  25. S. Dey, A. Laha, and S. Ghosh, “Nonlinearity-induced anomalous mode collapse and nonchiral asymmetric mode switching around multiple exceptional points,” Phys. Rev. B 101(12), 125432 (2020).
    [Crossref]
  26. J. Wen, X. Jiang, L. Jiang, and M. Xiao, “Parity-time symmetry in optical microcavity systems,” J. Phys. B: At., Mol. Opt. Phys. 51(22), 222001 (2018).
    [Crossref]
  27. X. Jiang and L. Yang, “Optothermal dynamics in whispering-gallery microresonators,” Light: Sci. Appl. 9(1), 24 (2020).
    [Crossref]
  28. X.-F. Jiang, C.-L. Zou, L. Wang, Q. Gong, and Y.-F. Xiao, “Whispering-gallery microcavities with unidirectional laser emission,” Laser Photonics Rev. 10(1), 40–61 (2016).
    [Crossref]
  29. P.-B. Li, S.-Y. Gao, and F.-L. Li, “Quantum-information transfer with nitrogen-vacancy centers coupled to a whispering-gallery microresonator,” Phys. Rev. A 83(5), 054306 (2011).
    [Crossref]
  30. J. Wiersig, “Structure of whispering-gallery modes in optical microdisks perturbed by nanoparticles,” Phys. Rev. A 84(6), 063828 (2011).
    [Crossref]
  31. B. Peng, S. K. Ozdemir, M. Liertzer, W. J. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Natl. Acad. Sci. U. S. A. 113(25), 6845–6850 (2016).
    [Crossref]
  32. X.-F. Liu, T.-J. Wang, Y.-P. Gao, C. Cao, and C. Wang, “Chiral microresonator assisted by Rydberg-atom ensembles,” Phys. Rev. A 98(3), 033824 (2018).
    [Crossref]
  33. B. Peng, S. K. Ozdemir, F. C. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
    [Crossref]
  34. L. Chang, X. S. Jiang, S. Y. Hua, C. Yang, J. M. Wen, L. Jiang, G. Y. Li, G. Z. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8(7), 524–529 (2014).
    [Crossref]
  35. B. Peng, S. K. Ozdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014).
    [Crossref]
  36. F.-J. Shu, C.-L. Zou, X.-B. Zou, and L. Yang, “Chiral symmetry breaking in a microring optical cavity by engineered dissipation,” Phys. Rev. A 94(1), 013848 (2016).
    [Crossref]
  37. F. Yang, Y. C. Liu, and L. You, “Anti-PT symmetry in dissipatively coupled optical systems,” Phys. Rev. A 96(5), 053845 (2017).
    [Crossref]
  38. F. Zhang, Y. Feng, X. Chen, L. Ge, and W. Wan, “Synthetic anti-PT symmetry in a single microcavity,” Phys. Rev. Lett. 124(5), 053901 (2020).
    [Crossref]
  39. A. Calabrese, F. Ramiro-Manzano, H. M. Price, S. Biasi, M. Bernard, M. Ghulinyan, I. Carusotto, and L. Pavesi, “Unidirectional reflection from an integrated “taiji" microresonator,” Photonics Res. 8(8), 1333–1341 (2020).
    [Crossref]
  40. B. He, L. Yang, X. Jiang, and M. Xiao, “Transmission nonreciprocity in a mutually coupled circulating structure,” Phys. Rev. Lett. 120(20), 203904 (2018).
    [Crossref]
  41. H. Zhang, F. Saif, Y. Jiao, and H. Jing, “Loss-induced transparency in optomechanics,” Opt. Express 26(19), 25199–25210 (2018).
    [Crossref]
  42. L. Liu, J. H. Zhang, L. Jin, and L. Zhou, “Transport properties of the non-Hermitian T-shaped quantum router,” Opt. Express 27(10), 13694–13705 (2019).
    [Crossref]
  43. T. Liu, G. C. Ma, S. J. Liang, H. Gao, Z. M. Gu, S. W. An, and J. Zhu, “Single-sided acoustic beam splitting based on parity-time symmetry,” Phys. Rev. B 102(1), 014306 (2020).
    [Crossref]
  44. Z. P. Liu, J. Zhang, S. K. Ozdemir, B. Peng, H. Jing, X. Y. Lu, C. W. Li, L. Yang, F. Nori, and Y. X. Liu, “Metrology with PT-symmetric cavities: Enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117(11), 110802 (2016).
    [Crossref]
  45. W. J. Chen, S. K. Ozdemir, G. M. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548(7666), 192–196 (2017).
    [Crossref]
  46. H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548(7666), 187–191 (2017).
    [Crossref]
  47. W. J. Chen, J. Zhang, B. Peng, S. K. Ozdemir, X. D. Fan, and L. Yang, “Parity-time-symmetric whispering-gallery mode nanoparticle sensor invited,” Photonics Res. 6(5), A23–A30 (2018).
    [Crossref]
  48. H. K. Lau and A. A. Clerk, “Fundamental limits and non-reciprocal approaches in non-Hermitian quantum sensing,” Nat. Commun. 9(1), 4320 (2018).
    [Crossref]
  49. M. Zhang, W. Sweeney, C. W. Hsu, L. Yang, A. D. Stone, and L. Jiang, “Quantum noise theory of exceptional point amplifying sensors,” Phys. Rev. Lett. 123(18), 180501 (2019).
    [Crossref]
  50. G. Q. Qin, M. Wang, J. W. Wen, D. Ruan, and G. L. Long, “Brillouin cavity optomechanics sensing with enhanced dynamical backaction,” Photonics Res. 7(12), 1440–1446 (2019).
    [Crossref]
  51. S. N. Huai, Y. L. Liu, J. Zhang, L. Yang, and Y. X. Liu, “Enhanced sideband responses in a PT-symmetric-like cavity magnomechanical system,” Phys. Rev. A 99(4), 043803 (2019).
    [Crossref]
  52. P. Djorwe, Y. Pennec, and B. Djafari-Rouhani, “Exceptional point enhances sensitivity of optomechanical mass sensors,” Phys. Rev. Appl. 12(2), 024002 (2019).
    [Crossref]
  53. Q. Zhong, J. Ren, M. Khajavikhan, D. N. Christodoulides, S. K. Özdemir, and R. El-Ganainy, “Sensing with exceptional surfaces in order to combine sensitivity with robustness,” Phys. Rev. Lett. 122(15), 153902 (2019).
    [Crossref]
  54. Y. H. Lai, Y. K. Lu, M. G. Suh, Z. Q. Yuan, and K. Vahala, “Observation of the exceptional-point-enhanced Sagnac effect,” Nature 576(7785), 65–69 (2019).
    [Crossref]
  55. Z. C. Zhang, Y. P. Wang, and X. G. Wang, “PT-symmetry-breaking-enhanced cavity optomechanical magnetometry,” Phys. Rev. A 102(2), 023512 (2020).
    [Crossref]
  56. Y. M. Chu, Y. Liu, H. B. Liu, and J. M. Cai, “Quantum sensing with a single-qubit pseudo-Hermitian system,” Phys. Rev. Lett. 124(2), 020501 (2020).
    [Crossref]
  57. L. Feng, Z. J. Wong, R. M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346(6212), 972–975 (2014).
    [Crossref]
  58. Z. Lin, A. Pick, M. Loncar, and A. W. Rodriguez, “Enhanced spontaneous emission at third-order Dirac exceptional points in inverse-designed photonic crystals,” Phys. Rev. Lett. 117(10), 107402 (2016).
    [Crossref]
  59. H. Hodaei, M. A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346(6212), 975–978 (2014).
    [Crossref]
  60. M. Kim, K. Kwon, J. Shim, Y. Jung, and K. Yu, “Partially directional microdisk laser with two Rayleigh scatterers,” Opt. Lett. 39(8), 2423–2426 (2014).
    [Crossref]
  61. P. Miao, Z. F. Zhang, J. B. Sun, W. Walasik, S. Longhi, N. M. Litchinitser, and L. Feng, “Orbital angular momentum microlaser,” Science 353(6298), 464–467 (2016).
    [Crossref]
  62. J. Perina, A. Luks, J. K. Kalaga, W. Leonski, and A. Miranowicz, “Nonclassical light at exceptional points of a quantum PT-symmetric two-mode system,” Phys. Rev. A 100(5), 053820 (2019).
    [Crossref]
  63. E. Lafalce, Q. J. Zeng, C. H. Lin, M. J. Smith, S. T. Malak, J. Jung, Y. J. Yoon, Z. Q. Lin, V. V. Tsukruk, and Z. V. Vardeny, “Robust lasing modes in coupled colloidal quantum dot microdisk pairs using a non-Hermitian exceptional point,” Nat. Commun. 10(1), 561 (2019).
    [Crossref]
  64. Q. T. Cao, Y. L. Chen, and Y. F. Xiao, “Chiral emission and Purcell enhancement in a hybrid plasmonic-photonic microresonator,” Light: Sci. Appl. 9(1), 4 (2020).
    [Crossref]
  65. T. Wu, W. X. Zhang, H. Z. Zhang, S. S. Hou, G. Y. Chen, R. B. Liu, C. C. Lu, J. F. Li, R. Y. Wang, P. F. Duan, J. J. Li, B. Wang, L. Shi, J. Zi, and X. D. Zhang, “Vector exceptional points with strong superchiral fields,” Phys. Rev. Lett. 124(8), 083901 (2020).
    [Crossref]
  66. Y. Lumer, Y. Plotnik, M. C. Rechtsman, and M. Segev, “Nonlinearly induced PT transition in photonic systems,” Phys. Rev. Lett. 111(26), 263901 (2013).
    [Crossref]
  67. A. U. Hassan, H. Hodaei, M.-A. Miri, M. Khajavikhan, and D. N. Christodoulides, “Nonlinear reversal of the $\mathcal{PT}$-symmetric phase transition in a system of coupled semiconductor microring resonators,” Phys. Rev. A 92(6), 063807 (2015).
    [Crossref]
  68. M.-A. Miri and A. Alù, “Nonlinearity-induced pt-symmetry without material gain,” New J. Phys. 18(6), 065001 (2016).
    [Crossref]
  69. Y.-X. Wang and A. A. Clerk, “Non-Hermitian dynamics without dissipation in quantum systems,” Phys. Rev. A 99(6), 063834 (2019).
    [Crossref]
  70. A. Roy, S. Jahani, Q. Guo, A. Dutt, S. Fan, M.-A. Miri, and A. Marandi, “Nondissipative non-Hermitian dynamics and exceptional points in coupled optical parametric oscillators,” Optica 8(3), 415–421 (2021).
    [Crossref]
  71. J. Ma, J. Wen, S. Ding, S. Li, Y. Hu, X. Jiang, L. Jiang, and M. Xiao, “Chip-based optical isolator and nonreciprocal parity-time symmetry induced by stimulated Brillouin scattering,” Laser Photonics Rev. 14(5), 1900278 (2020).
    [Crossref]
  72. N. S. Makarov, M. Drobizhev, and A. Rebane, “Two-photon absorption standards in the 550–1600 nm excitation wavelength range,” Opt. Express 16(6), 4029–4047 (2008).
    [Crossref]
  73. H. You, S. M. Hendrickson, and J. D. Franson, “Analysis of enhanced two-photon absorption in tapered optical fibers,” Phys. Rev. A 78(5), 053803 (2008).
    [Crossref]
  74. X. Dai, X. Zhang, I. M. Kislyakov, L. Wang, J. Huang, S. Zhang, N. Dong, and J. Wang, “Enhanced two-photon absorption and two-photon luminescence in monolayer MoS2 and WS2 by defect repairing,” Opt. Express 27(10), 13744–13753 (2019).
    [Crossref]
  75. M. A. Miri and A. Alù, “Exceptional points in optics and photonics,” Science 363(6422), eaar7709 (2019).
    [Crossref]
  76. S. K. Ozdemir, S. Rotter, F. Nori, and L. Yang, “Parity-time symmetry and exceptional points in photonics,” Nat. Mater. 18(8), 783–798 (2019).
    [Crossref]
  77. A. Miranowicz, J. Bajer, M. Paprzycka, Y.-X. Liu, A. M. Zagoskin, and F. Nori, “State-dependent photon blockade via quantum-reservoir engineering,” Phys. Rev. A 90(3), 033831 (2014).
    [Crossref]
  78. E. S. Guerra, B. M. Garraway, and P. L. Knight, “Two-photon parametric pumping versus two-photon absorption: A quantum jump approach,” Phys. Rev. A 55(5), 3842–3857 (1997).
    [Crossref]
  79. Y. R. Shen, “Quantum statistics of nonlinear optics,” Phys. Rev. 155(3), 921–931 (1967).
    [Crossref]
  80. T. K. Fryett, C. M. Dodson, and A. Majumdar, “Cavity enhanced nonlinear optics for few photon optical bistability,” Opt. Express 23(12), 16246–16255 (2015).
    [Crossref]
  81. K. Xia, G. Lu, G. Lin, Y. Cheng, Y. Niu, S. Gong, and J. Twamley, “Reversible nonmagnetic single-photon isolation using unbalanced quantum coupling,” Phys. Rev. A 90(4), 043802 (2014).
    [Crossref]
  82. K. Xia, F. Nori, and M. Xiao, “Cavity-free optical isolators and circulators using a chiral cross-Kerr nonlinearity,” Phys. Rev. Lett. 121(20), 203602 (2018).
    [Crossref]
  83. Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9(1), 1797 (2018).
    [Crossref]
  84. S. Zhang, Y. Hu, G. Lin, Y. Niu, K. Xia, J. Gong, and S. Gong, “Thermal-motion-induced non-reciprocal quantum optical system,” Nat. Photonics 12(12), 744–748 (2018).
    [Crossref]
  85. Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10(10), 657–661 (2016).
    [Crossref]
  86. L. Tang, J. Tang, W. Zhang, G. Lu, H. Zhang, Y. Zhang, K. Xia, and M. Xiao, “On-chip chiral single-photon interface: Isolation and unidirectional emission,” Phys. Rev. A 99(4), 043833 (2019).
    [Crossref]
  87. S. Zhang, G. Lin, Y. Hu, Y. Qi, Y. Niu, and S. Gong, “Cavity-free circulator with low insertion loss using hot atoms,” Phys. Rev. Appl. 14(2), 024032 (2020).
    [Crossref]
  88. C. Liang, B. Liu, A.-N. Xu, X. Wen, C. Lu, K. Xia, M. K. Tey, Y.-C. Liu, and L. You, “Collision-induced broadband optical nonreciprocity,” Phys. Rev. Lett. 125(12), 123901 (2020).
    [Crossref]
  89. Y. Shi, Z. Yu, and S. Fan, “Limitations of nonlinear optical isolators due to dynamic reciprocity,” Nat. Photonics 9(6), 388–392 (2015).
    [Crossref]
  90. L. D. Bino, J. M. Silver, M. T. M. Woodley, S. L. Stebbings, X. Zhao, and P. Del’Haye, “Microresonator isolators and circulators based on the intrinsic nonreciprocity of the kerr effect,” Optica 5(3), 279–282 (2018).
    [Crossref]
  91. D. L. Sounas, J. Soric, and A. Alù, “Broadband passive isolators based on coupled nonlinear resonances,” Nat. Electron. 1(2), 113–119 (2018).
    [Crossref]
  92. K. Y. Yang, J. Skarda, M. Cotrufo, A. Dutt, G. H. Ahn, M. Sawaby, D. Vercruysse, A. Arbabian, S. Fan, A. Alù, and J. Vučković, “Inverse-designed non-reciprocal pulse router for chip-based lidar,” Nat. Photonics 14(6), 369–374 (2020).
    [Crossref]
  93. L. Tang, J. Tang, H. Wu, J. Zhang, M. Xiao, and K. Xia, “Broad-intensity-range optical nonreciprocity based on feedback-induced Kerr nonlinearity,” Photon. Res., in press (2021).

2021 (2)

2020 (20)

J. Ma, J. Wen, S. Ding, S. Li, Y. Hu, X. Jiang, L. Jiang, and M. Xiao, “Chip-based optical isolator and nonreciprocal parity-time symmetry induced by stimulated Brillouin scattering,” Laser Photonics Rev. 14(5), 1900278 (2020).
[Crossref]

Q. T. Cao, Y. L. Chen, and Y. F. Xiao, “Chiral emission and Purcell enhancement in a hybrid plasmonic-photonic microresonator,” Light: Sci. Appl. 9(1), 4 (2020).
[Crossref]

T. Wu, W. X. Zhang, H. Z. Zhang, S. S. Hou, G. Y. Chen, R. B. Liu, C. C. Lu, J. F. Li, R. Y. Wang, P. F. Duan, J. J. Li, B. Wang, L. Shi, J. Zi, and X. D. Zhang, “Vector exceptional points with strong superchiral fields,” Phys. Rev. Lett. 124(8), 083901 (2020).
[Crossref]

Z. C. Zhang, Y. P. Wang, and X. G. Wang, “PT-symmetry-breaking-enhanced cavity optomechanical magnetometry,” Phys. Rev. A 102(2), 023512 (2020).
[Crossref]

Y. M. Chu, Y. Liu, H. B. Liu, and J. M. Cai, “Quantum sensing with a single-qubit pseudo-Hermitian system,” Phys. Rev. Lett. 124(2), 020501 (2020).
[Crossref]

F. Zhang, Y. Feng, X. Chen, L. Ge, and W. Wan, “Synthetic anti-PT symmetry in a single microcavity,” Phys. Rev. Lett. 124(5), 053901 (2020).
[Crossref]

A. Calabrese, F. Ramiro-Manzano, H. M. Price, S. Biasi, M. Bernard, M. Ghulinyan, I. Carusotto, and L. Pavesi, “Unidirectional reflection from an integrated “taiji" microresonator,” Photonics Res. 8(8), 1333–1341 (2020).
[Crossref]

T. Liu, G. C. Ma, S. J. Liang, H. Gao, Z. M. Gu, S. W. An, and J. Zhu, “Single-sided acoustic beam splitting based on parity-time symmetry,” Phys. Rev. B 102(1), 014306 (2020).
[Crossref]

V. Dominguez-Rocha, R. Thevamaran, F. M. Ellis, and T. Kottos, “Environmentally induced exceptional points in elastodynamics,” Phys. Rev. Appl. 13(1), 014060 (2020).
[Crossref]

J. Wiersig, “Prospects and fundamental limits in exceptional point-based sensing,” Nat. Commun. 11(1), 2454 (2020).
[Crossref]

X. Mao, G.-Q. Qin, H. Yang, H. Zhang, M. Wang, and G.-L. Long, “Enhanced sensitivity of optical gyroscope in a mechanical parity-time-symmetric system based on exceptional point,” New J. Phys. 22(9), 093009 (2020).
[Crossref]

S. X. Li, X. Q. Zhang, Q. Xu, M. Liu, M. Kang, J. G. Han, and W. L. Zhang, “Exceptional point in a metal-graphene hybrid metasurface with tunable asymmetric loss,” Opt. Express 28(14), 20083–20094 (2020).
[Crossref]

P. C. Kuo, N. Lambert, A. Miranowicz, H. B. Chen, G. Y. Chen, Y. N. Chen, and F. Nori, “Collectively induced exceptional points of quantum emitters coupled to nanoparticle surface plasmons,” Phys. Rev. A 101(1), 013814 (2020).
[Crossref]

I. Arkhipov, A. Miranowicz, F. Minganti, and F. Nori, “Quantum and semiclassical exceptional points of a linear system of coupled cavities with losses and gain within the Scully-Lamb laser theory,” Phys. Rev. A 101(1), 013812 (2020).
[Crossref]

N. Habler and J. Scheuer, “Higher-order exceptional points: A route for flat-top optical filters,” Phys. Rev. A 101(4), 043828 (2020).
[Crossref]

X. Jiang and L. Yang, “Optothermal dynamics in whispering-gallery microresonators,” Light: Sci. Appl. 9(1), 24 (2020).
[Crossref]

S. Dey, A. Laha, and S. Ghosh, “Nonlinearity-induced anomalous mode collapse and nonchiral asymmetric mode switching around multiple exceptional points,” Phys. Rev. B 101(12), 125432 (2020).
[Crossref]

S. Zhang, G. Lin, Y. Hu, Y. Qi, Y. Niu, and S. Gong, “Cavity-free circulator with low insertion loss using hot atoms,” Phys. Rev. Appl. 14(2), 024032 (2020).
[Crossref]

C. Liang, B. Liu, A.-N. Xu, X. Wen, C. Lu, K. Xia, M. K. Tey, Y.-C. Liu, and L. You, “Collision-induced broadband optical nonreciprocity,” Phys. Rev. Lett. 125(12), 123901 (2020).
[Crossref]

K. Y. Yang, J. Skarda, M. Cotrufo, A. Dutt, G. H. Ahn, M. Sawaby, D. Vercruysse, A. Arbabian, S. Fan, A. Alù, and J. Vučković, “Inverse-designed non-reciprocal pulse router for chip-based lidar,” Nat. Photonics 14(6), 369–374 (2020).
[Crossref]

2019 (18)

Y.-X. Wang and A. A. Clerk, “Non-Hermitian dynamics without dissipation in quantum systems,” Phys. Rev. A 99(6), 063834 (2019).
[Crossref]

L. Tang, J. Tang, W. Zhang, G. Lu, H. Zhang, Y. Zhang, K. Xia, and M. Xiao, “On-chip chiral single-photon interface: Isolation and unidirectional emission,” Phys. Rev. A 99(4), 043833 (2019).
[Crossref]

X. Dai, X. Zhang, I. M. Kislyakov, L. Wang, J. Huang, S. Zhang, N. Dong, and J. Wang, “Enhanced two-photon absorption and two-photon luminescence in monolayer MoS2 and WS2 by defect repairing,” Opt. Express 27(10), 13744–13753 (2019).
[Crossref]

M. A. Miri and A. Alù, “Exceptional points in optics and photonics,” Science 363(6422), eaar7709 (2019).
[Crossref]

S. K. Ozdemir, S. Rotter, F. Nori, and L. Yang, “Parity-time symmetry and exceptional points in photonics,” Nat. Mater. 18(8), 783–798 (2019).
[Crossref]

M. P. Hokmabadi, A. Schumer, D. N. Christodoulides, and M. Khajavikhan, “Non-Hermitian ring laser gyroscopes with enhanced Sagnac sensitivity,” Nature 576(7785), 70–74 (2019).
[Crossref]

J. Wen, C. Zheng, X. Kong, S. Wei, T. Xin, and G. Long, “Experimental demonstration of a digital quantum simulation of a general $\mathcal{PT}$-symmetric system,” Phys. Rev. A 99(6), 062122 (2019).
[Crossref]

G. Q. Zhang, Y. P. Wang, and J. Q. You, “Dispersive readout of a weakly coupled qubit via the parity-time-symmetric phase transition,” Phys. Rev. A 99(5), 052341 (2019).
[Crossref]

F. Minganti, A. Miranowicz, R. W. Chhajlany, and F. Nori, “Quantum exceptional points of non-Hermitian Hamiltonians and Liouvillians: The effects of quantum jumps,” Phys. Rev. A 100(6), 062131 (2019).
[Crossref]

M. Zhang, W. Sweeney, C. W. Hsu, L. Yang, A. D. Stone, and L. Jiang, “Quantum noise theory of exceptional point amplifying sensors,” Phys. Rev. Lett. 123(18), 180501 (2019).
[Crossref]

G. Q. Qin, M. Wang, J. W. Wen, D. Ruan, and G. L. Long, “Brillouin cavity optomechanics sensing with enhanced dynamical backaction,” Photonics Res. 7(12), 1440–1446 (2019).
[Crossref]

S. N. Huai, Y. L. Liu, J. Zhang, L. Yang, and Y. X. Liu, “Enhanced sideband responses in a PT-symmetric-like cavity magnomechanical system,” Phys. Rev. A 99(4), 043803 (2019).
[Crossref]

P. Djorwe, Y. Pennec, and B. Djafari-Rouhani, “Exceptional point enhances sensitivity of optomechanical mass sensors,” Phys. Rev. Appl. 12(2), 024002 (2019).
[Crossref]

Q. Zhong, J. Ren, M. Khajavikhan, D. N. Christodoulides, S. K. Özdemir, and R. El-Ganainy, “Sensing with exceptional surfaces in order to combine sensitivity with robustness,” Phys. Rev. Lett. 122(15), 153902 (2019).
[Crossref]

Y. H. Lai, Y. K. Lu, M. G. Suh, Z. Q. Yuan, and K. Vahala, “Observation of the exceptional-point-enhanced Sagnac effect,” Nature 576(7785), 65–69 (2019).
[Crossref]

L. Liu, J. H. Zhang, L. Jin, and L. Zhou, “Transport properties of the non-Hermitian T-shaped quantum router,” Opt. Express 27(10), 13694–13705 (2019).
[Crossref]

J. Perina, A. Luks, J. K. Kalaga, W. Leonski, and A. Miranowicz, “Nonclassical light at exceptional points of a quantum PT-symmetric two-mode system,” Phys. Rev. A 100(5), 053820 (2019).
[Crossref]

E. Lafalce, Q. J. Zeng, C. H. Lin, M. J. Smith, S. T. Malak, J. Jung, Y. J. Yoon, Z. Q. Lin, V. V. Tsukruk, and Z. V. Vardeny, “Robust lasing modes in coupled colloidal quantum dot microdisk pairs using a non-Hermitian exceptional point,” Nat. Commun. 10(1), 561 (2019).
[Crossref]

2018 (12)

W. J. Chen, J. Zhang, B. Peng, S. K. Ozdemir, X. D. Fan, and L. Yang, “Parity-time-symmetric whispering-gallery mode nanoparticle sensor invited,” Photonics Res. 6(5), A23–A30 (2018).
[Crossref]

H. K. Lau and A. A. Clerk, “Fundamental limits and non-reciprocal approaches in non-Hermitian quantum sensing,” Nat. Commun. 9(1), 4320 (2018).
[Crossref]

B. He, L. Yang, X. Jiang, and M. Xiao, “Transmission nonreciprocity in a mutually coupled circulating structure,” Phys. Rev. Lett. 120(20), 203904 (2018).
[Crossref]

H. Zhang, F. Saif, Y. Jiao, and H. Jing, “Loss-induced transparency in optomechanics,” Opt. Express 26(19), 25199–25210 (2018).
[Crossref]

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14(1), 11–19 (2018).
[Crossref]

J. Wen, X. Jiang, L. Jiang, and M. Xiao, “Parity-time symmetry in optical microcavity systems,” J. Phys. B: At., Mol. Opt. Phys. 51(22), 222001 (2018).
[Crossref]

X.-F. Liu, T.-J. Wang, Y.-P. Gao, C. Cao, and C. Wang, “Chiral microresonator assisted by Rydberg-atom ensembles,” Phys. Rev. A 98(3), 033824 (2018).
[Crossref]

K. Xia, F. Nori, and M. Xiao, “Cavity-free optical isolators and circulators using a chiral cross-Kerr nonlinearity,” Phys. Rev. Lett. 121(20), 203602 (2018).
[Crossref]

Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9(1), 1797 (2018).
[Crossref]

S. Zhang, Y. Hu, G. Lin, Y. Niu, K. Xia, J. Gong, and S. Gong, “Thermal-motion-induced non-reciprocal quantum optical system,” Nat. Photonics 12(12), 744–748 (2018).
[Crossref]

L. D. Bino, J. M. Silver, M. T. M. Woodley, S. L. Stebbings, X. Zhao, and P. Del’Haye, “Microresonator isolators and circulators based on the intrinsic nonreciprocity of the kerr effect,” Optica 5(3), 279–282 (2018).
[Crossref]

D. L. Sounas, J. Soric, and A. Alù, “Broadband passive isolators based on coupled nonlinear resonances,” Nat. Electron. 1(2), 113–119 (2018).
[Crossref]

2017 (3)

F. Yang, Y. C. Liu, and L. You, “Anti-PT symmetry in dissipatively coupled optical systems,” Phys. Rev. A 96(5), 053845 (2017).
[Crossref]

W. J. Chen, S. K. Ozdemir, G. M. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548(7666), 192–196 (2017).
[Crossref]

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548(7666), 187–191 (2017).
[Crossref]

2016 (9)

F.-J. Shu, C.-L. Zou, X.-B. Zou, and L. Yang, “Chiral symmetry breaking in a microring optical cavity by engineered dissipation,” Phys. Rev. A 94(1), 013848 (2016).
[Crossref]

Z. P. Liu, J. Zhang, S. K. Ozdemir, B. Peng, H. Jing, X. Y. Lu, C. W. Li, L. Yang, F. Nori, and Y. X. Liu, “Metrology with PT-symmetric cavities: Enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117(11), 110802 (2016).
[Crossref]

P. Miao, Z. F. Zhang, J. B. Sun, W. Walasik, S. Longhi, N. M. Litchinitser, and L. Feng, “Orbital angular momentum microlaser,” Science 353(6298), 464–467 (2016).
[Crossref]

Z. Lin, A. Pick, M. Loncar, and A. W. Rodriguez, “Enhanced spontaneous emission at third-order Dirac exceptional points in inverse-designed photonic crystals,” Phys. Rev. Lett. 117(10), 107402 (2016).
[Crossref]

M.-A. Miri and A. Alù, “Nonlinearity-induced pt-symmetry without material gain,” New J. Phys. 18(6), 065001 (2016).
[Crossref]

H. Xu, D. Mason, L. Y. Jiang, and J. G. E. Harris, “Topological energy transfer in an optomechanical system with exceptional points,” Nature 537(7618), 80–83 (2016).
[Crossref]

B. Peng, S. K. Ozdemir, M. Liertzer, W. J. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Natl. Acad. Sci. U. S. A. 113(25), 6845–6850 (2016).
[Crossref]

X.-F. Jiang, C.-L. Zou, L. Wang, Q. Gong, and Y.-F. Xiao, “Whispering-gallery microcavities with unidirectional laser emission,” Laser Photonics Rev. 10(1), 40–61 (2016).
[Crossref]

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10(10), 657–661 (2016).
[Crossref]

2015 (3)

Y. Shi, Z. Yu, and S. Fan, “Limitations of nonlinear optical isolators due to dynamic reciprocity,” Nat. Photonics 9(6), 388–392 (2015).
[Crossref]

T. K. Fryett, C. M. Dodson, and A. Majumdar, “Cavity enhanced nonlinear optics for few photon optical bistability,” Opt. Express 23(12), 16246–16255 (2015).
[Crossref]

A. U. Hassan, H. Hodaei, M.-A. Miri, M. Khajavikhan, and D. N. Christodoulides, “Nonlinear reversal of the $\mathcal{PT}$-symmetric phase transition in a system of coupled semiconductor microring resonators,” Phys. Rev. A 92(6), 063807 (2015).
[Crossref]

2014 (11)

H. Hodaei, M. A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346(6212), 975–978 (2014).
[Crossref]

M. Kim, K. Kwon, J. Shim, Y. Jung, and K. Yu, “Partially directional microdisk laser with two Rayleigh scatterers,” Opt. Lett. 39(8), 2423–2426 (2014).
[Crossref]

L. Feng, Z. J. Wong, R. M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346(6212), 972–975 (2014).
[Crossref]

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113(12), 123004 (2014).
[Crossref]

B. Peng, S. K. Ozdemir, F. C. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

L. Chang, X. S. Jiang, S. Y. Hua, C. Yang, J. M. Wen, L. Jiang, G. Y. Li, G. Z. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8(7), 524–529 (2014).
[Crossref]

B. Peng, S. K. Ozdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014).
[Crossref]

J. Wiersig, “Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: Application to microcavity sensors for single-particle detection,” Phys. Rev. Lett. 112(20), 203901 (2014).
[Crossref]

H. Jing, S. K. Ozdemir, X. Y. Lu, J. Zhang, L. Yang, and F. Nori, “PT-symmetric phonon laser,” Phys. Rev. Lett. 113(5), 053604 (2014).
[Crossref]

K. Xia, G. Lu, G. Lin, Y. Cheng, Y. Niu, S. Gong, and J. Twamley, “Reversible nonmagnetic single-photon isolation using unbalanced quantum coupling,” Phys. Rev. A 90(4), 043802 (2014).
[Crossref]

A. Miranowicz, J. Bajer, M. Paprzycka, Y.-X. Liu, A. M. Zagoskin, and F. Nori, “State-dependent photon blockade via quantum-reservoir engineering,” Phys. Rev. A 90(3), 033831 (2014).
[Crossref]

2013 (2)

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2013).
[Crossref]

Y. Lumer, Y. Plotnik, M. C. Rechtsman, and M. Segev, “Nonlinearly induced PT transition in photonic systems,” Phys. Rev. Lett. 111(26), 263901 (2013).
[Crossref]

2011 (3)

P.-B. Li, S.-Y. Gao, and F.-L. Li, “Quantum-information transfer with nitrogen-vacancy centers coupled to a whispering-gallery microresonator,” Phys. Rev. A 83(5), 054306 (2011).
[Crossref]

J. Wiersig, “Structure of whispering-gallery modes in optical microdisks perturbed by nanoparticles,” Phys. Rev. A 84(6), 063828 (2011).
[Crossref]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[Crossref]

2008 (2)

N. S. Makarov, M. Drobizhev, and A. Rebane, “Two-photon absorption standards in the 550–1600 nm excitation wavelength range,” Opt. Express 16(6), 4029–4047 (2008).
[Crossref]

H. You, S. M. Hendrickson, and J. D. Franson, “Analysis of enhanced two-photon absorption in tapered optical fibers,” Phys. Rev. A 78(5), 053803 (2008).
[Crossref]

2007 (1)

C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70(6), 947–1018 (2007).
[Crossref]

2004 (2)

W. D. Heiss, “Exceptional points of non-Hermitian operators,” J. Phys. A: Math. Gen. 37(6), 2455–2464 (2004).
[Crossref]

C. Dembowski, B. Dietz, H. D. Graf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, “Encircling an exceptional point,” Phys. Rev. E 69(5), 056216 (2004).
[Crossref]

1998 (1)

M. B. Plenio and P. L. Knight, “The quantum-jump approach to dissipative dynamics in quantum optics,” Rev. Mod. Phys. 70(1), 101–144 (1998).
[Crossref]

1997 (1)

E. S. Guerra, B. M. Garraway, and P. L. Knight, “Two-photon parametric pumping versus two-photon absorption: A quantum jump approach,” Phys. Rev. A 55(5), 3842–3857 (1997).
[Crossref]

1967 (1)

Y. R. Shen, “Quantum statistics of nonlinear optics,” Phys. Rev. 155(3), 921–931 (1967).
[Crossref]

Ahn, G. H.

K. Y. Yang, J. Skarda, M. Cotrufo, A. Dutt, G. H. Ahn, M. Sawaby, D. Vercruysse, A. Arbabian, S. Fan, A. Alù, and J. Vučković, “Inverse-designed non-reciprocal pulse router for chip-based lidar,” Nat. Photonics 14(6), 369–374 (2020).
[Crossref]

Almeida, V. R.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2013).
[Crossref]

Alù, A.

K. Y. Yang, J. Skarda, M. Cotrufo, A. Dutt, G. H. Ahn, M. Sawaby, D. Vercruysse, A. Arbabian, S. Fan, A. Alù, and J. Vučković, “Inverse-designed non-reciprocal pulse router for chip-based lidar,” Nat. Photonics 14(6), 369–374 (2020).
[Crossref]

M. A. Miri and A. Alù, “Exceptional points in optics and photonics,” Science 363(6422), eaar7709 (2019).
[Crossref]

D. L. Sounas, J. Soric, and A. Alù, “Broadband passive isolators based on coupled nonlinear resonances,” Nat. Electron. 1(2), 113–119 (2018).
[Crossref]

M.-A. Miri and A. Alù, “Nonlinearity-induced pt-symmetry without material gain,” New J. Phys. 18(6), 065001 (2016).
[Crossref]

An, S. W.

T. Liu, G. C. Ma, S. J. Liang, H. Gao, Z. M. Gu, S. W. An, and J. Zhu, “Single-sided acoustic beam splitting based on parity-time symmetry,” Phys. Rev. B 102(1), 014306 (2020).
[Crossref]

Arbabian, A.

K. Y. Yang, J. Skarda, M. Cotrufo, A. Dutt, G. H. Ahn, M. Sawaby, D. Vercruysse, A. Arbabian, S. Fan, A. Alù, and J. Vučković, “Inverse-designed non-reciprocal pulse router for chip-based lidar,” Nat. Photonics 14(6), 369–374 (2020).
[Crossref]

Arkhipov, I.

I. Arkhipov, A. Miranowicz, F. Minganti, and F. Nori, “Quantum and semiclassical exceptional points of a linear system of coupled cavities with losses and gain within the Scully-Lamb laser theory,” Phys. Rev. A 101(1), 013812 (2020).
[Crossref]

Artoni, M.

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113(12), 123004 (2014).
[Crossref]

Bajer, J.

A. Miranowicz, J. Bajer, M. Paprzycka, Y.-X. Liu, A. M. Zagoskin, and F. Nori, “State-dependent photon blockade via quantum-reservoir engineering,” Phys. Rev. A 90(3), 033831 (2014).
[Crossref]

Bender, C. M.

B. Peng, S. K. Ozdemir, F. C. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

B. Peng, S. K. Ozdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014).
[Crossref]

C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70(6), 947–1018 (2007).
[Crossref]

Bernard, M.

A. Calabrese, F. Ramiro-Manzano, H. M. Price, S. Biasi, M. Bernard, M. Ghulinyan, I. Carusotto, and L. Pavesi, “Unidirectional reflection from an integrated “taiji" microresonator,” Photonics Res. 8(8), 1333–1341 (2020).
[Crossref]

Biasi, S.

A. Calabrese, F. Ramiro-Manzano, H. M. Price, S. Biasi, M. Bernard, M. Ghulinyan, I. Carusotto, and L. Pavesi, “Unidirectional reflection from an integrated “taiji" microresonator,” Photonics Res. 8(8), 1333–1341 (2020).
[Crossref]

Bino, L. D.

Cai, J. M.

Y. M. Chu, Y. Liu, H. B. Liu, and J. M. Cai, “Quantum sensing with a single-qubit pseudo-Hermitian system,” Phys. Rev. Lett. 124(2), 020501 (2020).
[Crossref]

Calabrese, A.

A. Calabrese, F. Ramiro-Manzano, H. M. Price, S. Biasi, M. Bernard, M. Ghulinyan, I. Carusotto, and L. Pavesi, “Unidirectional reflection from an integrated “taiji" microresonator,” Photonics Res. 8(8), 1333–1341 (2020).
[Crossref]

Cao, C.

X.-F. Liu, T.-J. Wang, Y.-P. Gao, C. Cao, and C. Wang, “Chiral microresonator assisted by Rydberg-atom ensembles,” Phys. Rev. A 98(3), 033824 (2018).
[Crossref]

Cao, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[Crossref]

Cao, Q. T.

Q. T. Cao, Y. L. Chen, and Y. F. Xiao, “Chiral emission and Purcell enhancement in a hybrid plasmonic-photonic microresonator,” Light: Sci. Appl. 9(1), 4 (2020).
[Crossref]

Carmichael, H. J.

H. J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations (Springer, 1999).

Carusotto, I.

A. Calabrese, F. Ramiro-Manzano, H. M. Price, S. Biasi, M. Bernard, M. Ghulinyan, I. Carusotto, and L. Pavesi, “Unidirectional reflection from an integrated “taiji" microresonator,” Photonics Res. 8(8), 1333–1341 (2020).
[Crossref]

Chang, L.

L. Chang, X. S. Jiang, S. Y. Hua, C. Yang, J. M. Wen, L. Jiang, G. Y. Li, G. Z. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8(7), 524–529 (2014).
[Crossref]

Chen, G. Y.

P. C. Kuo, N. Lambert, A. Miranowicz, H. B. Chen, G. Y. Chen, Y. N. Chen, and F. Nori, “Collectively induced exceptional points of quantum emitters coupled to nanoparticle surface plasmons,” Phys. Rev. A 101(1), 013814 (2020).
[Crossref]

T. Wu, W. X. Zhang, H. Z. Zhang, S. S. Hou, G. Y. Chen, R. B. Liu, C. C. Lu, J. F. Li, R. Y. Wang, P. F. Duan, J. J. Li, B. Wang, L. Shi, J. Zi, and X. D. Zhang, “Vector exceptional points with strong superchiral fields,” Phys. Rev. Lett. 124(8), 083901 (2020).
[Crossref]

Chen, H. B.

P. C. Kuo, N. Lambert, A. Miranowicz, H. B. Chen, G. Y. Chen, Y. N. Chen, and F. Nori, “Collectively induced exceptional points of quantum emitters coupled to nanoparticle surface plasmons,” Phys. Rev. A 101(1), 013814 (2020).
[Crossref]

Chen, W. J.

W. J. Chen, J. Zhang, B. Peng, S. K. Ozdemir, X. D. Fan, and L. Yang, “Parity-time-symmetric whispering-gallery mode nanoparticle sensor invited,” Photonics Res. 6(5), A23–A30 (2018).
[Crossref]

W. J. Chen, S. K. Ozdemir, G. M. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548(7666), 192–196 (2017).
[Crossref]

B. Peng, S. K. Ozdemir, M. Liertzer, W. J. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Natl. Acad. Sci. U. S. A. 113(25), 6845–6850 (2016).
[Crossref]

Chen, X.

F. Zhang, Y. Feng, X. Chen, L. Ge, and W. Wan, “Synthetic anti-PT symmetry in a single microcavity,” Phys. Rev. Lett. 124(5), 053901 (2020).
[Crossref]

Chen, Y.

Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9(1), 1797 (2018).
[Crossref]

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10(10), 657–661 (2016).
[Crossref]

Chen, Y. F.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2013).
[Crossref]

Chen, Y. L.

Q. T. Cao, Y. L. Chen, and Y. F. Xiao, “Chiral emission and Purcell enhancement in a hybrid plasmonic-photonic microresonator,” Light: Sci. Appl. 9(1), 4 (2020).
[Crossref]

Chen, Y. N.

P. C. Kuo, N. Lambert, A. Miranowicz, H. B. Chen, G. Y. Chen, Y. N. Chen, and F. Nori, “Collectively induced exceptional points of quantum emitters coupled to nanoparticle surface plasmons,” Phys. Rev. A 101(1), 013814 (2020).
[Crossref]

Cheng, Y.

K. Xia, G. Lu, G. Lin, Y. Cheng, Y. Niu, S. Gong, and J. Twamley, “Reversible nonmagnetic single-photon isolation using unbalanced quantum coupling,” Phys. Rev. A 90(4), 043802 (2014).
[Crossref]

Chhajlany, R. W.

F. Minganti, A. Miranowicz, R. W. Chhajlany, and F. Nori, “Quantum exceptional points of non-Hermitian Hamiltonians and Liouvillians: The effects of quantum jumps,” Phys. Rev. A 100(6), 062131 (2019).
[Crossref]

Christodoulides, D. N.

M. P. Hokmabadi, A. Schumer, D. N. Christodoulides, and M. Khajavikhan, “Non-Hermitian ring laser gyroscopes with enhanced Sagnac sensitivity,” Nature 576(7785), 70–74 (2019).
[Crossref]

Q. Zhong, J. Ren, M. Khajavikhan, D. N. Christodoulides, S. K. Özdemir, and R. El-Ganainy, “Sensing with exceptional surfaces in order to combine sensitivity with robustness,” Phys. Rev. Lett. 122(15), 153902 (2019).
[Crossref]

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14(1), 11–19 (2018).
[Crossref]

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548(7666), 187–191 (2017).
[Crossref]

A. U. Hassan, H. Hodaei, M.-A. Miri, M. Khajavikhan, and D. N. Christodoulides, “Nonlinear reversal of the $\mathcal{PT}$-symmetric phase transition in a system of coupled semiconductor microring resonators,” Phys. Rev. A 92(6), 063807 (2015).
[Crossref]

H. Hodaei, M. A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346(6212), 975–978 (2014).
[Crossref]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[Crossref]

Chu, Y. M.

Y. M. Chu, Y. Liu, H. B. Liu, and J. M. Cai, “Quantum sensing with a single-qubit pseudo-Hermitian system,” Phys. Rev. Lett. 124(2), 020501 (2020).
[Crossref]

Clerk, A. A.

Y.-X. Wang and A. A. Clerk, “Non-Hermitian dynamics without dissipation in quantum systems,” Phys. Rev. A 99(6), 063834 (2019).
[Crossref]

H. K. Lau and A. A. Clerk, “Fundamental limits and non-reciprocal approaches in non-Hermitian quantum sensing,” Nat. Commun. 9(1), 4320 (2018).
[Crossref]

Cotrufo, M.

K. Y. Yang, J. Skarda, M. Cotrufo, A. Dutt, G. H. Ahn, M. Sawaby, D. Vercruysse, A. Arbabian, S. Fan, A. Alù, and J. Vučković, “Inverse-designed non-reciprocal pulse router for chip-based lidar,” Nat. Photonics 14(6), 369–374 (2020).
[Crossref]

Dai, X.

Del’Haye, P.

Dembowski, C.

C. Dembowski, B. Dietz, H. D. Graf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, “Encircling an exceptional point,” Phys. Rev. E 69(5), 056216 (2004).
[Crossref]

Dey, S.

S. Dey, A. Laha, and S. Ghosh, “Nonlinearity-induced anomalous mode collapse and nonchiral asymmetric mode switching around multiple exceptional points,” Phys. Rev. B 101(12), 125432 (2020).
[Crossref]

Dietz, B.

C. Dembowski, B. Dietz, H. D. Graf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, “Encircling an exceptional point,” Phys. Rev. E 69(5), 056216 (2004).
[Crossref]

Ding, S.

J. Ma, J. Wen, S. Ding, S. Li, Y. Hu, X. Jiang, L. Jiang, and M. Xiao, “Chip-based optical isolator and nonreciprocal parity-time symmetry induced by stimulated Brillouin scattering,” Laser Photonics Rev. 14(5), 1900278 (2020).
[Crossref]

Djafari-Rouhani, B.

P. Djorwe, Y. Pennec, and B. Djafari-Rouhani, “Exceptional point enhances sensitivity of optomechanical mass sensors,” Phys. Rev. Appl. 12(2), 024002 (2019).
[Crossref]

Djorwe, P.

P. Djorwe, Y. Pennec, and B. Djafari-Rouhani, “Exceptional point enhances sensitivity of optomechanical mass sensors,” Phys. Rev. Appl. 12(2), 024002 (2019).
[Crossref]

Dodson, C. M.

Dominguez-Rocha, V.

V. Dominguez-Rocha, R. Thevamaran, F. M. Ellis, and T. Kottos, “Environmentally induced exceptional points in elastodynamics,” Phys. Rev. Appl. 13(1), 014060 (2020).
[Crossref]

Dong, C.-H.

Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9(1), 1797 (2018).
[Crossref]

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10(10), 657–661 (2016).
[Crossref]

Dong, N.

Drobizhev, M.

Duan, P. F.

T. Wu, W. X. Zhang, H. Z. Zhang, S. S. Hou, G. Y. Chen, R. B. Liu, C. C. Lu, J. F. Li, R. Y. Wang, P. F. Duan, J. J. Li, B. Wang, L. Shi, J. Zi, and X. D. Zhang, “Vector exceptional points with strong superchiral fields,” Phys. Rev. Lett. 124(8), 083901 (2020).
[Crossref]

Dutt, A.

A. Roy, S. Jahani, Q. Guo, A. Dutt, S. Fan, M.-A. Miri, and A. Marandi, “Nondissipative non-Hermitian dynamics and exceptional points in coupled optical parametric oscillators,” Optica 8(3), 415–421 (2021).
[Crossref]

K. Y. Yang, J. Skarda, M. Cotrufo, A. Dutt, G. H. Ahn, M. Sawaby, D. Vercruysse, A. Arbabian, S. Fan, A. Alù, and J. Vučković, “Inverse-designed non-reciprocal pulse router for chip-based lidar,” Nat. Photonics 14(6), 369–374 (2020).
[Crossref]

Eichelkraut, T.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[Crossref]

El-Ganainy, R.

Q. Zhong, J. Ren, M. Khajavikhan, D. N. Christodoulides, S. K. Özdemir, and R. El-Ganainy, “Sensing with exceptional surfaces in order to combine sensitivity with robustness,” Phys. Rev. Lett. 122(15), 153902 (2019).
[Crossref]

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14(1), 11–19 (2018).
[Crossref]

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548(7666), 187–191 (2017).
[Crossref]

Ellis, F. M.

V. Dominguez-Rocha, R. Thevamaran, F. M. Ellis, and T. Kottos, “Environmentally induced exceptional points in elastodynamics,” Phys. Rev. Appl. 13(1), 014060 (2020).
[Crossref]

Fan, S.

A. Roy, S. Jahani, Q. Guo, A. Dutt, S. Fan, M.-A. Miri, and A. Marandi, “Nondissipative non-Hermitian dynamics and exceptional points in coupled optical parametric oscillators,” Optica 8(3), 415–421 (2021).
[Crossref]

K. Y. Yang, J. Skarda, M. Cotrufo, A. Dutt, G. H. Ahn, M. Sawaby, D. Vercruysse, A. Arbabian, S. Fan, A. Alù, and J. Vučković, “Inverse-designed non-reciprocal pulse router for chip-based lidar,” Nat. Photonics 14(6), 369–374 (2020).
[Crossref]

Y. Shi, Z. Yu, and S. Fan, “Limitations of nonlinear optical isolators due to dynamic reciprocity,” Nat. Photonics 9(6), 388–392 (2015).
[Crossref]

Fan, S. H.

B. Peng, S. K. Ozdemir, F. C. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

Fan, X. D.

W. J. Chen, J. Zhang, B. Peng, S. K. Ozdemir, X. D. Fan, and L. Yang, “Parity-time-symmetric whispering-gallery mode nanoparticle sensor invited,” Photonics Res. 6(5), A23–A30 (2018).
[Crossref]

Fegadolli, W. S.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2013).
[Crossref]

Feng, L.

P. Miao, Z. F. Zhang, J. B. Sun, W. Walasik, S. Longhi, N. M. Litchinitser, and L. Feng, “Orbital angular momentum microlaser,” Science 353(6298), 464–467 (2016).
[Crossref]

L. Feng, Z. J. Wong, R. M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346(6212), 972–975 (2014).
[Crossref]

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2013).
[Crossref]

Feng, Y.

F. Zhang, Y. Feng, X. Chen, L. Ge, and W. Wan, “Synthetic anti-PT symmetry in a single microcavity,” Phys. Rev. Lett. 124(5), 053901 (2020).
[Crossref]

Franson, J. D.

H. You, S. M. Hendrickson, and J. D. Franson, “Analysis of enhanced two-photon absorption in tapered optical fibers,” Phys. Rev. A 78(5), 053803 (2008).
[Crossref]

Fryett, T. K.

Gao, H.

T. Liu, G. C. Ma, S. J. Liang, H. Gao, Z. M. Gu, S. W. An, and J. Zhu, “Single-sided acoustic beam splitting based on parity-time symmetry,” Phys. Rev. B 102(1), 014306 (2020).
[Crossref]

Gao, S.-Y.

P.-B. Li, S.-Y. Gao, and F.-L. Li, “Quantum-information transfer with nitrogen-vacancy centers coupled to a whispering-gallery microresonator,” Phys. Rev. A 83(5), 054306 (2011).
[Crossref]

Gao, W.-C.

Gao, Y.-P.

X.-F. Liu, T.-J. Wang, Y.-P. Gao, C. Cao, and C. Wang, “Chiral microresonator assisted by Rydberg-atom ensembles,” Phys. Rev. A 98(3), 033824 (2018).
[Crossref]

Garcia-Gracia, H.

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548(7666), 187–191 (2017).
[Crossref]

Garraway, B. M.

E. S. Guerra, B. M. Garraway, and P. L. Knight, “Two-photon parametric pumping versus two-photon absorption: A quantum jump approach,” Phys. Rev. A 55(5), 3842–3857 (1997).
[Crossref]

Ge, L.

F. Zhang, Y. Feng, X. Chen, L. Ge, and W. Wan, “Synthetic anti-PT symmetry in a single microcavity,” Phys. Rev. Lett. 124(5), 053901 (2020).
[Crossref]

Ghosh, S.

S. Dey, A. Laha, and S. Ghosh, “Nonlinearity-induced anomalous mode collapse and nonchiral asymmetric mode switching around multiple exceptional points,” Phys. Rev. B 101(12), 125432 (2020).
[Crossref]

Ghulinyan, M.

A. Calabrese, F. Ramiro-Manzano, H. M. Price, S. Biasi, M. Bernard, M. Ghulinyan, I. Carusotto, and L. Pavesi, “Unidirectional reflection from an integrated “taiji" microresonator,” Photonics Res. 8(8), 1333–1341 (2020).
[Crossref]

Gianfreda, M.

B. Peng, S. K. Ozdemir, F. C. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

Gong, J.

S. Zhang, Y. Hu, G. Lin, Y. Niu, K. Xia, J. Gong, and S. Gong, “Thermal-motion-induced non-reciprocal quantum optical system,” Nat. Photonics 12(12), 744–748 (2018).
[Crossref]

Gong, Q.

X.-F. Jiang, C.-L. Zou, L. Wang, Q. Gong, and Y.-F. Xiao, “Whispering-gallery microcavities with unidirectional laser emission,” Laser Photonics Rev. 10(1), 40–61 (2016).
[Crossref]

Gong, S.

S. Zhang, G. Lin, Y. Hu, Y. Qi, Y. Niu, and S. Gong, “Cavity-free circulator with low insertion loss using hot atoms,” Phys. Rev. Appl. 14(2), 024032 (2020).
[Crossref]

S. Zhang, Y. Hu, G. Lin, Y. Niu, K. Xia, J. Gong, and S. Gong, “Thermal-motion-induced non-reciprocal quantum optical system,” Nat. Photonics 12(12), 744–748 (2018).
[Crossref]

K. Xia, G. Lu, G. Lin, Y. Cheng, Y. Niu, S. Gong, and J. Twamley, “Reversible nonmagnetic single-photon isolation using unbalanced quantum coupling,” Phys. Rev. A 90(4), 043802 (2014).
[Crossref]

Graf, H. D.

C. Dembowski, B. Dietz, H. D. Graf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, “Encircling an exceptional point,” Phys. Rev. E 69(5), 056216 (2004).
[Crossref]

Gu, Z. M.

T. Liu, G. C. Ma, S. J. Liang, H. Gao, Z. M. Gu, S. W. An, and J. Zhu, “Single-sided acoustic beam splitting based on parity-time symmetry,” Phys. Rev. B 102(1), 014306 (2020).
[Crossref]

Guerra, E. S.

E. S. Guerra, B. M. Garraway, and P. L. Knight, “Two-photon parametric pumping versus two-photon absorption: A quantum jump approach,” Phys. Rev. A 55(5), 3842–3857 (1997).
[Crossref]

Guo, G.-C.

Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9(1), 1797 (2018).
[Crossref]

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10(10), 657–661 (2016).
[Crossref]

Guo, Q.

Habler, N.

N. Habler and J. Scheuer, “Higher-order exceptional points: A route for flat-top optical filters,” Phys. Rev. A 101(4), 043828 (2020).
[Crossref]

Han, J. G.

Harney, H. L.

C. Dembowski, B. Dietz, H. D. Graf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, “Encircling an exceptional point,” Phys. Rev. E 69(5), 056216 (2004).
[Crossref]

Harris, J. G. E.

H. Xu, D. Mason, L. Y. Jiang, and J. G. E. Harris, “Topological energy transfer in an optomechanical system with exceptional points,” Nature 537(7618), 80–83 (2016).
[Crossref]

Hassan, A. U.

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548(7666), 187–191 (2017).
[Crossref]

A. U. Hassan, H. Hodaei, M.-A. Miri, M. Khajavikhan, and D. N. Christodoulides, “Nonlinear reversal of the $\mathcal{PT}$-symmetric phase transition in a system of coupled semiconductor microring resonators,” Phys. Rev. A 92(6), 063807 (2015).
[Crossref]

He, B.

B. He, L. Yang, X. Jiang, and M. Xiao, “Transmission nonreciprocity in a mutually coupled circulating structure,” Phys. Rev. Lett. 120(20), 203904 (2018).
[Crossref]

Heine, A.

C. Dembowski, B. Dietz, H. D. Graf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, “Encircling an exceptional point,” Phys. Rev. E 69(5), 056216 (2004).
[Crossref]

Heinrich, M.

H. Hodaei, M. A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346(6212), 975–978 (2014).
[Crossref]

Heiss, W. D.

C. Dembowski, B. Dietz, H. D. Graf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, “Encircling an exceptional point,” Phys. Rev. E 69(5), 056216 (2004).
[Crossref]

W. D. Heiss, “Exceptional points of non-Hermitian operators,” J. Phys. A: Math. Gen. 37(6), 2455–2464 (2004).
[Crossref]

Hendrickson, S. M.

H. You, S. M. Hendrickson, and J. D. Franson, “Analysis of enhanced two-photon absorption in tapered optical fibers,” Phys. Rev. A 78(5), 053803 (2008).
[Crossref]

Hodaei, H.

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548(7666), 187–191 (2017).
[Crossref]

A. U. Hassan, H. Hodaei, M.-A. Miri, M. Khajavikhan, and D. N. Christodoulides, “Nonlinear reversal of the $\mathcal{PT}$-symmetric phase transition in a system of coupled semiconductor microring resonators,” Phys. Rev. A 92(6), 063807 (2015).
[Crossref]

H. Hodaei, M. A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346(6212), 975–978 (2014).
[Crossref]

Hokmabadi, M. P.

M. P. Hokmabadi, A. Schumer, D. N. Christodoulides, and M. Khajavikhan, “Non-Hermitian ring laser gyroscopes with enhanced Sagnac sensitivity,” Nature 576(7785), 70–74 (2019).
[Crossref]

Hou, S. S.

T. Wu, W. X. Zhang, H. Z. Zhang, S. S. Hou, G. Y. Chen, R. B. Liu, C. C. Lu, J. F. Li, R. Y. Wang, P. F. Duan, J. J. Li, B. Wang, L. Shi, J. Zi, and X. D. Zhang, “Vector exceptional points with strong superchiral fields,” Phys. Rev. Lett. 124(8), 083901 (2020).
[Crossref]

Hsu, C. W.

M. Zhang, W. Sweeney, C. W. Hsu, L. Yang, A. D. Stone, and L. Jiang, “Quantum noise theory of exceptional point amplifying sensors,” Phys. Rev. Lett. 123(18), 180501 (2019).
[Crossref]

Hu, Y.

J. Ma, J. Wen, S. Ding, S. Li, Y. Hu, X. Jiang, L. Jiang, and M. Xiao, “Chip-based optical isolator and nonreciprocal parity-time symmetry induced by stimulated Brillouin scattering,” Laser Photonics Rev. 14(5), 1900278 (2020).
[Crossref]

S. Zhang, G. Lin, Y. Hu, Y. Qi, Y. Niu, and S. Gong, “Cavity-free circulator with low insertion loss using hot atoms,” Phys. Rev. Appl. 14(2), 024032 (2020).
[Crossref]

S. Zhang, Y. Hu, G. Lin, Y. Niu, K. Xia, J. Gong, and S. Gong, “Thermal-motion-induced non-reciprocal quantum optical system,” Nat. Photonics 12(12), 744–748 (2018).
[Crossref]

Hua, S. Y.

L. Chang, X. S. Jiang, S. Y. Hua, C. Yang, J. M. Wen, L. Jiang, G. Y. Li, G. Z. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8(7), 524–529 (2014).
[Crossref]

Huai, S. N.

S. N. Huai, Y. L. Liu, J. Zhang, L. Yang, and Y. X. Liu, “Enhanced sideband responses in a PT-symmetric-like cavity magnomechanical system,” Phys. Rev. A 99(4), 043803 (2019).
[Crossref]

Huang, J.

Jahani, S.

Jiang, L.

J. Ma, J. Wen, S. Ding, S. Li, Y. Hu, X. Jiang, L. Jiang, and M. Xiao, “Chip-based optical isolator and nonreciprocal parity-time symmetry induced by stimulated Brillouin scattering,” Laser Photonics Rev. 14(5), 1900278 (2020).
[Crossref]

M. Zhang, W. Sweeney, C. W. Hsu, L. Yang, A. D. Stone, and L. Jiang, “Quantum noise theory of exceptional point amplifying sensors,” Phys. Rev. Lett. 123(18), 180501 (2019).
[Crossref]

J. Wen, X. Jiang, L. Jiang, and M. Xiao, “Parity-time symmetry in optical microcavity systems,” J. Phys. B: At., Mol. Opt. Phys. 51(22), 222001 (2018).
[Crossref]

L. Chang, X. S. Jiang, S. Y. Hua, C. Yang, J. M. Wen, L. Jiang, G. Y. Li, G. Z. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8(7), 524–529 (2014).
[Crossref]

Jiang, L. Y.

H. Xu, D. Mason, L. Y. Jiang, and J. G. E. Harris, “Topological energy transfer in an optomechanical system with exceptional points,” Nature 537(7618), 80–83 (2016).
[Crossref]

Jiang, X.

X. Jiang and L. Yang, “Optothermal dynamics in whispering-gallery microresonators,” Light: Sci. Appl. 9(1), 24 (2020).
[Crossref]

J. Ma, J. Wen, S. Ding, S. Li, Y. Hu, X. Jiang, L. Jiang, and M. Xiao, “Chip-based optical isolator and nonreciprocal parity-time symmetry induced by stimulated Brillouin scattering,” Laser Photonics Rev. 14(5), 1900278 (2020).
[Crossref]

J. Wen, X. Jiang, L. Jiang, and M. Xiao, “Parity-time symmetry in optical microcavity systems,” J. Phys. B: At., Mol. Opt. Phys. 51(22), 222001 (2018).
[Crossref]

B. He, L. Yang, X. Jiang, and M. Xiao, “Transmission nonreciprocity in a mutually coupled circulating structure,” Phys. Rev. Lett. 120(20), 203904 (2018).
[Crossref]

Jiang, X. S.

L. Chang, X. S. Jiang, S. Y. Hua, C. Yang, J. M. Wen, L. Jiang, G. Y. Li, G. Z. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8(7), 524–529 (2014).
[Crossref]

Jiang, X.-F.

X.-F. Jiang, C.-L. Zou, L. Wang, Q. Gong, and Y.-F. Xiao, “Whispering-gallery microcavities with unidirectional laser emission,” Laser Photonics Rev. 10(1), 40–61 (2016).
[Crossref]

Jiao, Y.

Jin, L.

Jing, H.

H. Zhang, F. Saif, Y. Jiao, and H. Jing, “Loss-induced transparency in optomechanics,” Opt. Express 26(19), 25199–25210 (2018).
[Crossref]

Z. P. Liu, J. Zhang, S. K. Ozdemir, B. Peng, H. Jing, X. Y. Lu, C. W. Li, L. Yang, F. Nori, and Y. X. Liu, “Metrology with PT-symmetric cavities: Enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117(11), 110802 (2016).
[Crossref]

H. Jing, S. K. Ozdemir, X. Y. Lu, J. Zhang, L. Yang, and F. Nori, “PT-symmetric phonon laser,” Phys. Rev. Lett. 113(5), 053604 (2014).
[Crossref]

Jung, J.

E. Lafalce, Q. J. Zeng, C. H. Lin, M. J. Smith, S. T. Malak, J. Jung, Y. J. Yoon, Z. Q. Lin, V. V. Tsukruk, and Z. V. Vardeny, “Robust lasing modes in coupled colloidal quantum dot microdisk pairs using a non-Hermitian exceptional point,” Nat. Commun. 10(1), 561 (2019).
[Crossref]

Jung, Y.

Kalaga, J. K.

J. Perina, A. Luks, J. K. Kalaga, W. Leonski, and A. Miranowicz, “Nonclassical light at exceptional points of a quantum PT-symmetric two-mode system,” Phys. Rev. A 100(5), 053820 (2019).
[Crossref]

Kang, M.

Khajavikhan, M.

M. P. Hokmabadi, A. Schumer, D. N. Christodoulides, and M. Khajavikhan, “Non-Hermitian ring laser gyroscopes with enhanced Sagnac sensitivity,” Nature 576(7785), 70–74 (2019).
[Crossref]

Q. Zhong, J. Ren, M. Khajavikhan, D. N. Christodoulides, S. K. Özdemir, and R. El-Ganainy, “Sensing with exceptional surfaces in order to combine sensitivity with robustness,” Phys. Rev. Lett. 122(15), 153902 (2019).
[Crossref]

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14(1), 11–19 (2018).
[Crossref]

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548(7666), 187–191 (2017).
[Crossref]

A. U. Hassan, H. Hodaei, M.-A. Miri, M. Khajavikhan, and D. N. Christodoulides, “Nonlinear reversal of the $\mathcal{PT}$-symmetric phase transition in a system of coupled semiconductor microring resonators,” Phys. Rev. A 92(6), 063807 (2015).
[Crossref]

H. Hodaei, M. A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346(6212), 975–978 (2014).
[Crossref]

Kim, M.

Kislyakov, I. M.

Knight, P. L.

M. B. Plenio and P. L. Knight, “The quantum-jump approach to dissipative dynamics in quantum optics,” Rev. Mod. Phys. 70(1), 101–144 (1998).
[Crossref]

E. S. Guerra, B. M. Garraway, and P. L. Knight, “Two-photon parametric pumping versus two-photon absorption: A quantum jump approach,” Phys. Rev. A 55(5), 3842–3857 (1997).
[Crossref]

Kong, X.

J. Wen, C. Zheng, X. Kong, S. Wei, T. Xin, and G. Long, “Experimental demonstration of a digital quantum simulation of a general $\mathcal{PT}$-symmetric system,” Phys. Rev. A 99(6), 062122 (2019).
[Crossref]

Kottos, T.

V. Dominguez-Rocha, R. Thevamaran, F. M. Ellis, and T. Kottos, “Environmentally induced exceptional points in elastodynamics,” Phys. Rev. Appl. 13(1), 014060 (2020).
[Crossref]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[Crossref]

Kramer, J.

B. Peng, S. K. Ozdemir, M. Liertzer, W. J. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Natl. Acad. Sci. U. S. A. 113(25), 6845–6850 (2016).
[Crossref]

Kuo, P. C.

P. C. Kuo, N. Lambert, A. Miranowicz, H. B. Chen, G. Y. Chen, Y. N. Chen, and F. Nori, “Collectively induced exceptional points of quantum emitters coupled to nanoparticle surface plasmons,” Phys. Rev. A 101(1), 013814 (2020).
[Crossref]

Kwon, K.

La Rocca, G. C.

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113(12), 123004 (2014).
[Crossref]

Lafalce, E.

E. Lafalce, Q. J. Zeng, C. H. Lin, M. J. Smith, S. T. Malak, J. Jung, Y. J. Yoon, Z. Q. Lin, V. V. Tsukruk, and Z. V. Vardeny, “Robust lasing modes in coupled colloidal quantum dot microdisk pairs using a non-Hermitian exceptional point,” Nat. Commun. 10(1), 561 (2019).
[Crossref]

Laha, A.

S. Dey, A. Laha, and S. Ghosh, “Nonlinearity-induced anomalous mode collapse and nonchiral asymmetric mode switching around multiple exceptional points,” Phys. Rev. B 101(12), 125432 (2020).
[Crossref]

Lai, Y. H.

Y. H. Lai, Y. K. Lu, M. G. Suh, Z. Q. Yuan, and K. Vahala, “Observation of the exceptional-point-enhanced Sagnac effect,” Nature 576(7785), 65–69 (2019).
[Crossref]

Lambert, N.

P. C. Kuo, N. Lambert, A. Miranowicz, H. B. Chen, G. Y. Chen, Y. N. Chen, and F. Nori, “Collectively induced exceptional points of quantum emitters coupled to nanoparticle surface plasmons,” Phys. Rev. A 101(1), 013814 (2020).
[Crossref]

Lau, H. K.

H. K. Lau and A. A. Clerk, “Fundamental limits and non-reciprocal approaches in non-Hermitian quantum sensing,” Nat. Commun. 9(1), 4320 (2018).
[Crossref]

Lei, F. C.

B. Peng, S. K. Ozdemir, F. C. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

Leonski, W.

J. Perina, A. Luks, J. K. Kalaga, W. Leonski, and A. Miranowicz, “Nonclassical light at exceptional points of a quantum PT-symmetric two-mode system,” Phys. Rev. A 100(5), 053820 (2019).
[Crossref]

Li, C. W.

Z. P. Liu, J. Zhang, S. K. Ozdemir, B. Peng, H. Jing, X. Y. Lu, C. W. Li, L. Yang, F. Nori, and Y. X. Liu, “Metrology with PT-symmetric cavities: Enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117(11), 110802 (2016).
[Crossref]

Li, F.-L.

P.-B. Li, S.-Y. Gao, and F.-L. Li, “Quantum-information transfer with nitrogen-vacancy centers coupled to a whispering-gallery microresonator,” Phys. Rev. A 83(5), 054306 (2011).
[Crossref]

Li, G. Y.

L. Chang, X. S. Jiang, S. Y. Hua, C. Yang, J. M. Wen, L. Jiang, G. Y. Li, G. Z. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8(7), 524–529 (2014).
[Crossref]

Li, J. F.

T. Wu, W. X. Zhang, H. Z. Zhang, S. S. Hou, G. Y. Chen, R. B. Liu, C. C. Lu, J. F. Li, R. Y. Wang, P. F. Duan, J. J. Li, B. Wang, L. Shi, J. Zi, and X. D. Zhang, “Vector exceptional points with strong superchiral fields,” Phys. Rev. Lett. 124(8), 083901 (2020).
[Crossref]

Li, J. J.

T. Wu, W. X. Zhang, H. Z. Zhang, S. S. Hou, G. Y. Chen, R. B. Liu, C. C. Lu, J. F. Li, R. Y. Wang, P. F. Duan, J. J. Li, B. Wang, L. Shi, J. Zi, and X. D. Zhang, “Vector exceptional points with strong superchiral fields,” Phys. Rev. Lett. 124(8), 083901 (2020).
[Crossref]

Li, P.-B.

P.-B. Li, S.-Y. Gao, and F.-L. Li, “Quantum-information transfer with nitrogen-vacancy centers coupled to a whispering-gallery microresonator,” Phys. Rev. A 83(5), 054306 (2011).
[Crossref]

Li, S.

J. Ma, J. Wen, S. Ding, S. Li, Y. Hu, X. Jiang, L. Jiang, and M. Xiao, “Chip-based optical isolator and nonreciprocal parity-time symmetry induced by stimulated Brillouin scattering,” Laser Photonics Rev. 14(5), 1900278 (2020).
[Crossref]

Li, S. X.

Liang, C.

C. Liang, B. Liu, A.-N. Xu, X. Wen, C. Lu, K. Xia, M. K. Tey, Y.-C. Liu, and L. You, “Collision-induced broadband optical nonreciprocity,” Phys. Rev. Lett. 125(12), 123901 (2020).
[Crossref]

Liang, S. J.

T. Liu, G. C. Ma, S. J. Liang, H. Gao, Z. M. Gu, S. W. An, and J. Zhu, “Single-sided acoustic beam splitting based on parity-time symmetry,” Phys. Rev. B 102(1), 014306 (2020).
[Crossref]

Liertzer, M.

B. Peng, S. K. Ozdemir, M. Liertzer, W. J. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Natl. Acad. Sci. U. S. A. 113(25), 6845–6850 (2016).
[Crossref]

B. Peng, S. K. Ozdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014).
[Crossref]

Lin, C. H.

E. Lafalce, Q. J. Zeng, C. H. Lin, M. J. Smith, S. T. Malak, J. Jung, Y. J. Yoon, Z. Q. Lin, V. V. Tsukruk, and Z. V. Vardeny, “Robust lasing modes in coupled colloidal quantum dot microdisk pairs using a non-Hermitian exceptional point,” Nat. Commun. 10(1), 561 (2019).
[Crossref]

Lin, G.

S. Zhang, G. Lin, Y. Hu, Y. Qi, Y. Niu, and S. Gong, “Cavity-free circulator with low insertion loss using hot atoms,” Phys. Rev. Appl. 14(2), 024032 (2020).
[Crossref]

S. Zhang, Y. Hu, G. Lin, Y. Niu, K. Xia, J. Gong, and S. Gong, “Thermal-motion-induced non-reciprocal quantum optical system,” Nat. Photonics 12(12), 744–748 (2018).
[Crossref]

K. Xia, G. Lu, G. Lin, Y. Cheng, Y. Niu, S. Gong, and J. Twamley, “Reversible nonmagnetic single-photon isolation using unbalanced quantum coupling,” Phys. Rev. A 90(4), 043802 (2014).
[Crossref]

Lin, Z.

Z. Lin, A. Pick, M. Loncar, and A. W. Rodriguez, “Enhanced spontaneous emission at third-order Dirac exceptional points in inverse-designed photonic crystals,” Phys. Rev. Lett. 117(10), 107402 (2016).
[Crossref]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[Crossref]

Lin, Z. Q.

E. Lafalce, Q. J. Zeng, C. H. Lin, M. J. Smith, S. T. Malak, J. Jung, Y. J. Yoon, Z. Q. Lin, V. V. Tsukruk, and Z. V. Vardeny, “Robust lasing modes in coupled colloidal quantum dot microdisk pairs using a non-Hermitian exceptional point,” Nat. Commun. 10(1), 561 (2019).
[Crossref]

Litchinitser, N. M.

P. Miao, Z. F. Zhang, J. B. Sun, W. Walasik, S. Longhi, N. M. Litchinitser, and L. Feng, “Orbital angular momentum microlaser,” Science 353(6298), 464–467 (2016).
[Crossref]

Liu, B.

C. Liang, B. Liu, A.-N. Xu, X. Wen, C. Lu, K. Xia, M. K. Tey, Y.-C. Liu, and L. You, “Collision-induced broadband optical nonreciprocity,” Phys. Rev. Lett. 125(12), 123901 (2020).
[Crossref]

Liu, H. B.

Y. M. Chu, Y. Liu, H. B. Liu, and J. M. Cai, “Quantum sensing with a single-qubit pseudo-Hermitian system,” Phys. Rev. Lett. 124(2), 020501 (2020).
[Crossref]

Liu, L.

Liu, M.

Liu, R. B.

T. Wu, W. X. Zhang, H. Z. Zhang, S. S. Hou, G. Y. Chen, R. B. Liu, C. C. Lu, J. F. Li, R. Y. Wang, P. F. Duan, J. J. Li, B. Wang, L. Shi, J. Zi, and X. D. Zhang, “Vector exceptional points with strong superchiral fields,” Phys. Rev. Lett. 124(8), 083901 (2020).
[Crossref]

Liu, T.

T. Liu, G. C. Ma, S. J. Liang, H. Gao, Z. M. Gu, S. W. An, and J. Zhu, “Single-sided acoustic beam splitting based on parity-time symmetry,” Phys. Rev. B 102(1), 014306 (2020).
[Crossref]

Liu, X.-F.

X.-F. Liu, T.-J. Wang, Y.-P. Gao, C. Cao, and C. Wang, “Chiral microresonator assisted by Rydberg-atom ensembles,” Phys. Rev. A 98(3), 033824 (2018).
[Crossref]

Liu, Y.

Y. M. Chu, Y. Liu, H. B. Liu, and J. M. Cai, “Quantum sensing with a single-qubit pseudo-Hermitian system,” Phys. Rev. Lett. 124(2), 020501 (2020).
[Crossref]

Liu, Y. C.

F. Yang, Y. C. Liu, and L. You, “Anti-PT symmetry in dissipatively coupled optical systems,” Phys. Rev. A 96(5), 053845 (2017).
[Crossref]

Liu, Y. L.

S. N. Huai, Y. L. Liu, J. Zhang, L. Yang, and Y. X. Liu, “Enhanced sideband responses in a PT-symmetric-like cavity magnomechanical system,” Phys. Rev. A 99(4), 043803 (2019).
[Crossref]

Liu, Y. X.

S. N. Huai, Y. L. Liu, J. Zhang, L. Yang, and Y. X. Liu, “Enhanced sideband responses in a PT-symmetric-like cavity magnomechanical system,” Phys. Rev. A 99(4), 043803 (2019).
[Crossref]

Z. P. Liu, J. Zhang, S. K. Ozdemir, B. Peng, H. Jing, X. Y. Lu, C. W. Li, L. Yang, F. Nori, and Y. X. Liu, “Metrology with PT-symmetric cavities: Enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117(11), 110802 (2016).
[Crossref]

Liu, Y.-C.

C. Liang, B. Liu, A.-N. Xu, X. Wen, C. Lu, K. Xia, M. K. Tey, Y.-C. Liu, and L. You, “Collision-induced broadband optical nonreciprocity,” Phys. Rev. Lett. 125(12), 123901 (2020).
[Crossref]

Liu, Y.-X.

A. Miranowicz, J. Bajer, M. Paprzycka, Y.-X. Liu, A. M. Zagoskin, and F. Nori, “State-dependent photon blockade via quantum-reservoir engineering,” Phys. Rev. A 90(3), 033831 (2014).
[Crossref]

Liu, Z. P.

Z. P. Liu, J. Zhang, S. K. Ozdemir, B. Peng, H. Jing, X. Y. Lu, C. W. Li, L. Yang, F. Nori, and Y. X. Liu, “Metrology with PT-symmetric cavities: Enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117(11), 110802 (2016).
[Crossref]

Loncar, M.

Z. Lin, A. Pick, M. Loncar, and A. W. Rodriguez, “Enhanced spontaneous emission at third-order Dirac exceptional points in inverse-designed photonic crystals,” Phys. Rev. Lett. 117(10), 107402 (2016).
[Crossref]

Long, G.

J. Wen, C. Zheng, X. Kong, S. Wei, T. Xin, and G. Long, “Experimental demonstration of a digital quantum simulation of a general $\mathcal{PT}$-symmetric system,” Phys. Rev. A 99(6), 062122 (2019).
[Crossref]

Long, G. L.

G. Q. Qin, M. Wang, J. W. Wen, D. Ruan, and G. L. Long, “Brillouin cavity optomechanics sensing with enhanced dynamical backaction,” Photonics Res. 7(12), 1440–1446 (2019).
[Crossref]

B. Peng, S. K. Ozdemir, F. C. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

Long, G.-L.

X. Mao, G.-Q. Qin, H. Yang, H. Zhang, M. Wang, and G.-L. Long, “Enhanced sensitivity of optical gyroscope in a mechanical parity-time-symmetric system based on exceptional point,” New J. Phys. 22(9), 093009 (2020).
[Crossref]

Longhi, S.

P. Miao, Z. F. Zhang, J. B. Sun, W. Walasik, S. Longhi, N. M. Litchinitser, and L. Feng, “Orbital angular momentum microlaser,” Science 353(6298), 464–467 (2016).
[Crossref]

Lu, C.

C. Liang, B. Liu, A.-N. Xu, X. Wen, C. Lu, K. Xia, M. K. Tey, Y.-C. Liu, and L. You, “Collision-induced broadband optical nonreciprocity,” Phys. Rev. Lett. 125(12), 123901 (2020).
[Crossref]

Lu, C. C.

T. Wu, W. X. Zhang, H. Z. Zhang, S. S. Hou, G. Y. Chen, R. B. Liu, C. C. Lu, J. F. Li, R. Y. Wang, P. F. Duan, J. J. Li, B. Wang, L. Shi, J. Zi, and X. D. Zhang, “Vector exceptional points with strong superchiral fields,” Phys. Rev. Lett. 124(8), 083901 (2020).
[Crossref]

Lu, G.

L. Tang, J. Tang, W. Zhang, G. Lu, H. Zhang, Y. Zhang, K. Xia, and M. Xiao, “On-chip chiral single-photon interface: Isolation and unidirectional emission,” Phys. Rev. A 99(4), 043833 (2019).
[Crossref]

K. Xia, G. Lu, G. Lin, Y. Cheng, Y. Niu, S. Gong, and J. Twamley, “Reversible nonmagnetic single-photon isolation using unbalanced quantum coupling,” Phys. Rev. A 90(4), 043802 (2014).
[Crossref]

Lu, M. H.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2013).
[Crossref]

Lu, X. Y.

Z. P. Liu, J. Zhang, S. K. Ozdemir, B. Peng, H. Jing, X. Y. Lu, C. W. Li, L. Yang, F. Nori, and Y. X. Liu, “Metrology with PT-symmetric cavities: Enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117(11), 110802 (2016).
[Crossref]

H. Jing, S. K. Ozdemir, X. Y. Lu, J. Zhang, L. Yang, and F. Nori, “PT-symmetric phonon laser,” Phys. Rev. Lett. 113(5), 053604 (2014).
[Crossref]

Lu, Y. K.

Y. H. Lai, Y. K. Lu, M. G. Suh, Z. Q. Yuan, and K. Vahala, “Observation of the exceptional-point-enhanced Sagnac effect,” Nature 576(7785), 65–69 (2019).
[Crossref]

Luks, A.

J. Perina, A. Luks, J. K. Kalaga, W. Leonski, and A. Miranowicz, “Nonclassical light at exceptional points of a quantum PT-symmetric two-mode system,” Phys. Rev. A 100(5), 053820 (2019).
[Crossref]

Lumer, Y.

Y. Lumer, Y. Plotnik, M. C. Rechtsman, and M. Segev, “Nonlinearly induced PT transition in photonic systems,” Phys. Rev. Lett. 111(26), 263901 (2013).
[Crossref]

Ma, G. C.

T. Liu, G. C. Ma, S. J. Liang, H. Gao, Z. M. Gu, S. W. An, and J. Zhu, “Single-sided acoustic beam splitting based on parity-time symmetry,” Phys. Rev. B 102(1), 014306 (2020).
[Crossref]

Ma, J.

J. Ma, J. Wen, S. Ding, S. Li, Y. Hu, X. Jiang, L. Jiang, and M. Xiao, “Chip-based optical isolator and nonreciprocal parity-time symmetry induced by stimulated Brillouin scattering,” Laser Photonics Rev. 14(5), 1900278 (2020).
[Crossref]

Ma, R. M.

L. Feng, Z. J. Wong, R. M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346(6212), 972–975 (2014).
[Crossref]

Majumdar, A.

Makarov, N. S.

Makris, K. G.

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14(1), 11–19 (2018).
[Crossref]

Malak, S. T.

E. Lafalce, Q. J. Zeng, C. H. Lin, M. J. Smith, S. T. Malak, J. Jung, Y. J. Yoon, Z. Q. Lin, V. V. Tsukruk, and Z. V. Vardeny, “Robust lasing modes in coupled colloidal quantum dot microdisk pairs using a non-Hermitian exceptional point,” Nat. Commun. 10(1), 561 (2019).
[Crossref]

Mao, X.

X. Mao, G.-Q. Qin, H. Yang, H. Zhang, M. Wang, and G.-L. Long, “Enhanced sensitivity of optical gyroscope in a mechanical parity-time-symmetric system based on exceptional point,” New J. Phys. 22(9), 093009 (2020).
[Crossref]

Marandi, A.

Mason, D.

H. Xu, D. Mason, L. Y. Jiang, and J. G. E. Harris, “Topological energy transfer in an optomechanical system with exceptional points,” Nature 537(7618), 80–83 (2016).
[Crossref]

Miao, P.

P. Miao, Z. F. Zhang, J. B. Sun, W. Walasik, S. Longhi, N. M. Litchinitser, and L. Feng, “Orbital angular momentum microlaser,” Science 353(6298), 464–467 (2016).
[Crossref]

Minganti, F.

I. Arkhipov, A. Miranowicz, F. Minganti, and F. Nori, “Quantum and semiclassical exceptional points of a linear system of coupled cavities with losses and gain within the Scully-Lamb laser theory,” Phys. Rev. A 101(1), 013812 (2020).
[Crossref]

F. Minganti, A. Miranowicz, R. W. Chhajlany, and F. Nori, “Quantum exceptional points of non-Hermitian Hamiltonians and Liouvillians: The effects of quantum jumps,” Phys. Rev. A 100(6), 062131 (2019).
[Crossref]

Miranowicz, A.

I. Arkhipov, A. Miranowicz, F. Minganti, and F. Nori, “Quantum and semiclassical exceptional points of a linear system of coupled cavities with losses and gain within the Scully-Lamb laser theory,” Phys. Rev. A 101(1), 013812 (2020).
[Crossref]

P. C. Kuo, N. Lambert, A. Miranowicz, H. B. Chen, G. Y. Chen, Y. N. Chen, and F. Nori, “Collectively induced exceptional points of quantum emitters coupled to nanoparticle surface plasmons,” Phys. Rev. A 101(1), 013814 (2020).
[Crossref]

F. Minganti, A. Miranowicz, R. W. Chhajlany, and F. Nori, “Quantum exceptional points of non-Hermitian Hamiltonians and Liouvillians: The effects of quantum jumps,” Phys. Rev. A 100(6), 062131 (2019).
[Crossref]

J. Perina, A. Luks, J. K. Kalaga, W. Leonski, and A. Miranowicz, “Nonclassical light at exceptional points of a quantum PT-symmetric two-mode system,” Phys. Rev. A 100(5), 053820 (2019).
[Crossref]

A. Miranowicz, J. Bajer, M. Paprzycka, Y.-X. Liu, A. M. Zagoskin, and F. Nori, “State-dependent photon blockade via quantum-reservoir engineering,” Phys. Rev. A 90(3), 033831 (2014).
[Crossref]

Miri, M. A.

M. A. Miri and A. Alù, “Exceptional points in optics and photonics,” Science 363(6422), eaar7709 (2019).
[Crossref]

H. Hodaei, M. A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346(6212), 975–978 (2014).
[Crossref]

Miri, M.-A.

A. Roy, S. Jahani, Q. Guo, A. Dutt, S. Fan, M.-A. Miri, and A. Marandi, “Nondissipative non-Hermitian dynamics and exceptional points in coupled optical parametric oscillators,” Optica 8(3), 415–421 (2021).
[Crossref]

M.-A. Miri and A. Alù, “Nonlinearity-induced pt-symmetry without material gain,” New J. Phys. 18(6), 065001 (2016).
[Crossref]

A. U. Hassan, H. Hodaei, M.-A. Miri, M. Khajavikhan, and D. N. Christodoulides, “Nonlinear reversal of the $\mathcal{PT}$-symmetric phase transition in a system of coupled semiconductor microring resonators,” Phys. Rev. A 92(6), 063807 (2015).
[Crossref]

Monifi, F.

B. Peng, S. K. Ozdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014).
[Crossref]

B. Peng, S. K. Ozdemir, F. C. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

Musslimani, Z. H.

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14(1), 11–19 (2018).
[Crossref]

Niu, Y.

S. Zhang, G. Lin, Y. Hu, Y. Qi, Y. Niu, and S. Gong, “Cavity-free circulator with low insertion loss using hot atoms,” Phys. Rev. Appl. 14(2), 024032 (2020).
[Crossref]

S. Zhang, Y. Hu, G. Lin, Y. Niu, K. Xia, J. Gong, and S. Gong, “Thermal-motion-induced non-reciprocal quantum optical system,” Nat. Photonics 12(12), 744–748 (2018).
[Crossref]

K. Xia, G. Lu, G. Lin, Y. Cheng, Y. Niu, S. Gong, and J. Twamley, “Reversible nonmagnetic single-photon isolation using unbalanced quantum coupling,” Phys. Rev. A 90(4), 043802 (2014).
[Crossref]

Nori, F.

P. C. Kuo, N. Lambert, A. Miranowicz, H. B. Chen, G. Y. Chen, Y. N. Chen, and F. Nori, “Collectively induced exceptional points of quantum emitters coupled to nanoparticle surface plasmons,” Phys. Rev. A 101(1), 013814 (2020).
[Crossref]

I. Arkhipov, A. Miranowicz, F. Minganti, and F. Nori, “Quantum and semiclassical exceptional points of a linear system of coupled cavities with losses and gain within the Scully-Lamb laser theory,” Phys. Rev. A 101(1), 013812 (2020).
[Crossref]

F. Minganti, A. Miranowicz, R. W. Chhajlany, and F. Nori, “Quantum exceptional points of non-Hermitian Hamiltonians and Liouvillians: The effects of quantum jumps,” Phys. Rev. A 100(6), 062131 (2019).
[Crossref]

S. K. Ozdemir, S. Rotter, F. Nori, and L. Yang, “Parity-time symmetry and exceptional points in photonics,” Nat. Mater. 18(8), 783–798 (2019).
[Crossref]

K. Xia, F. Nori, and M. Xiao, “Cavity-free optical isolators and circulators using a chiral cross-Kerr nonlinearity,” Phys. Rev. Lett. 121(20), 203602 (2018).
[Crossref]

Z. P. Liu, J. Zhang, S. K. Ozdemir, B. Peng, H. Jing, X. Y. Lu, C. W. Li, L. Yang, F. Nori, and Y. X. Liu, “Metrology with PT-symmetric cavities: Enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117(11), 110802 (2016).
[Crossref]

B. Peng, S. K. Ozdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014).
[Crossref]

B. Peng, S. K. Ozdemir, F. C. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

H. Jing, S. K. Ozdemir, X. Y. Lu, J. Zhang, L. Yang, and F. Nori, “PT-symmetric phonon laser,” Phys. Rev. Lett. 113(5), 053604 (2014).
[Crossref]

A. Miranowicz, J. Bajer, M. Paprzycka, Y.-X. Liu, A. M. Zagoskin, and F. Nori, “State-dependent photon blockade via quantum-reservoir engineering,” Phys. Rev. A 90(3), 033831 (2014).
[Crossref]

Oliveira, J. E. B.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2013).
[Crossref]

Ozdemir, S. K.

S. K. Ozdemir, S. Rotter, F. Nori, and L. Yang, “Parity-time symmetry and exceptional points in photonics,” Nat. Mater. 18(8), 783–798 (2019).
[Crossref]

W. J. Chen, J. Zhang, B. Peng, S. K. Ozdemir, X. D. Fan, and L. Yang, “Parity-time-symmetric whispering-gallery mode nanoparticle sensor invited,” Photonics Res. 6(5), A23–A30 (2018).
[Crossref]

W. J. Chen, S. K. Ozdemir, G. M. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548(7666), 192–196 (2017).
[Crossref]

Z. P. Liu, J. Zhang, S. K. Ozdemir, B. Peng, H. Jing, X. Y. Lu, C. W. Li, L. Yang, F. Nori, and Y. X. Liu, “Metrology with PT-symmetric cavities: Enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117(11), 110802 (2016).
[Crossref]

B. Peng, S. K. Ozdemir, M. Liertzer, W. J. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Natl. Acad. Sci. U. S. A. 113(25), 6845–6850 (2016).
[Crossref]

B. Peng, S. K. Ozdemir, F. C. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

H. Jing, S. K. Ozdemir, X. Y. Lu, J. Zhang, L. Yang, and F. Nori, “PT-symmetric phonon laser,” Phys. Rev. Lett. 113(5), 053604 (2014).
[Crossref]

B. Peng, S. K. Ozdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014).
[Crossref]

Özdemir, S. K.

Q. Zhong, J. Ren, M. Khajavikhan, D. N. Christodoulides, S. K. Özdemir, and R. El-Ganainy, “Sensing with exceptional surfaces in order to combine sensitivity with robustness,” Phys. Rev. Lett. 122(15), 153902 (2019).
[Crossref]

Paprzycka, M.

A. Miranowicz, J. Bajer, M. Paprzycka, Y.-X. Liu, A. M. Zagoskin, and F. Nori, “State-dependent photon blockade via quantum-reservoir engineering,” Phys. Rev. A 90(3), 033831 (2014).
[Crossref]

Pavesi, L.

A. Calabrese, F. Ramiro-Manzano, H. M. Price, S. Biasi, M. Bernard, M. Ghulinyan, I. Carusotto, and L. Pavesi, “Unidirectional reflection from an integrated “taiji" microresonator,” Photonics Res. 8(8), 1333–1341 (2020).
[Crossref]

Peng, B.

W. J. Chen, J. Zhang, B. Peng, S. K. Ozdemir, X. D. Fan, and L. Yang, “Parity-time-symmetric whispering-gallery mode nanoparticle sensor invited,” Photonics Res. 6(5), A23–A30 (2018).
[Crossref]

Z. P. Liu, J. Zhang, S. K. Ozdemir, B. Peng, H. Jing, X. Y. Lu, C. W. Li, L. Yang, F. Nori, and Y. X. Liu, “Metrology with PT-symmetric cavities: Enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117(11), 110802 (2016).
[Crossref]

B. Peng, S. K. Ozdemir, M. Liertzer, W. J. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Natl. Acad. Sci. U. S. A. 113(25), 6845–6850 (2016).
[Crossref]

B. Peng, S. K. Ozdemir, F. C. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

B. Peng, S. K. Ozdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014).
[Crossref]

Pennec, Y.

P. Djorwe, Y. Pennec, and B. Djafari-Rouhani, “Exceptional point enhances sensitivity of optomechanical mass sensors,” Phys. Rev. Appl. 12(2), 024002 (2019).
[Crossref]

Perina, J.

J. Perina, A. Luks, J. K. Kalaga, W. Leonski, and A. Miranowicz, “Nonclassical light at exceptional points of a quantum PT-symmetric two-mode system,” Phys. Rev. A 100(5), 053820 (2019).
[Crossref]

Pick, A.

Z. Lin, A. Pick, M. Loncar, and A. W. Rodriguez, “Enhanced spontaneous emission at third-order Dirac exceptional points in inverse-designed photonic crystals,” Phys. Rev. Lett. 117(10), 107402 (2016).
[Crossref]

Plenio, M. B.

M. B. Plenio and P. L. Knight, “The quantum-jump approach to dissipative dynamics in quantum optics,” Rev. Mod. Phys. 70(1), 101–144 (1998).
[Crossref]

Plotnik, Y.

Y. Lumer, Y. Plotnik, M. C. Rechtsman, and M. Segev, “Nonlinearly induced PT transition in photonic systems,” Phys. Rev. Lett. 111(26), 263901 (2013).
[Crossref]

Price, H. M.

A. Calabrese, F. Ramiro-Manzano, H. M. Price, S. Biasi, M. Bernard, M. Ghulinyan, I. Carusotto, and L. Pavesi, “Unidirectional reflection from an integrated “taiji" microresonator,” Photonics Res. 8(8), 1333–1341 (2020).
[Crossref]

Qi, Y.

S. Zhang, G. Lin, Y. Hu, Y. Qi, Y. Niu, and S. Gong, “Cavity-free circulator with low insertion loss using hot atoms,” Phys. Rev. Appl. 14(2), 024032 (2020).
[Crossref]

Qin, G. Q.

G. Q. Qin, M. Wang, J. W. Wen, D. Ruan, and G. L. Long, “Brillouin cavity optomechanics sensing with enhanced dynamical backaction,” Photonics Res. 7(12), 1440–1446 (2019).
[Crossref]

Qin, G.-Q.

X. Mao, G.-Q. Qin, H. Yang, H. Zhang, M. Wang, and G.-L. Long, “Enhanced sensitivity of optical gyroscope in a mechanical parity-time-symmetric system based on exceptional point,” New J. Phys. 22(9), 093009 (2020).
[Crossref]

Ramezani, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[Crossref]

Ramiro-Manzano, F.

A. Calabrese, F. Ramiro-Manzano, H. M. Price, S. Biasi, M. Bernard, M. Ghulinyan, I. Carusotto, and L. Pavesi, “Unidirectional reflection from an integrated “taiji" microresonator,” Photonics Res. 8(8), 1333–1341 (2020).
[Crossref]

Rebane, A.

Rechtsman, M. C.

Y. Lumer, Y. Plotnik, M. C. Rechtsman, and M. Segev, “Nonlinearly induced PT transition in photonic systems,” Phys. Rev. Lett. 111(26), 263901 (2013).
[Crossref]

Ren, J.

Q. Zhong, J. Ren, M. Khajavikhan, D. N. Christodoulides, S. K. Özdemir, and R. El-Ganainy, “Sensing with exceptional surfaces in order to combine sensitivity with robustness,” Phys. Rev. Lett. 122(15), 153902 (2019).
[Crossref]

Richter, A.

C. Dembowski, B. Dietz, H. D. Graf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, “Encircling an exceptional point,” Phys. Rev. E 69(5), 056216 (2004).
[Crossref]

Rodriguez, A. W.

Z. Lin, A. Pick, M. Loncar, and A. W. Rodriguez, “Enhanced spontaneous emission at third-order Dirac exceptional points in inverse-designed photonic crystals,” Phys. Rev. Lett. 117(10), 107402 (2016).
[Crossref]

Rotter, S.

S. K. Ozdemir, S. Rotter, F. Nori, and L. Yang, “Parity-time symmetry and exceptional points in photonics,” Nat. Mater. 18(8), 783–798 (2019).
[Crossref]

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14(1), 11–19 (2018).
[Crossref]

B. Peng, S. K. Ozdemir, M. Liertzer, W. J. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Natl. Acad. Sci. U. S. A. 113(25), 6845–6850 (2016).
[Crossref]

B. Peng, S. K. Ozdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014).
[Crossref]

Roy, A.

Ruan, D.

G. Q. Qin, M. Wang, J. W. Wen, D. Ruan, and G. L. Long, “Brillouin cavity optomechanics sensing with enhanced dynamical backaction,” Photonics Res. 7(12), 1440–1446 (2019).
[Crossref]

Saif, F.

Sawaby, M.

K. Y. Yang, J. Skarda, M. Cotrufo, A. Dutt, G. H. Ahn, M. Sawaby, D. Vercruysse, A. Arbabian, S. Fan, A. Alù, and J. Vučković, “Inverse-designed non-reciprocal pulse router for chip-based lidar,” Nat. Photonics 14(6), 369–374 (2020).
[Crossref]

Scherer, A.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2013).
[Crossref]

Scheuer, J.

N. Habler and J. Scheuer, “Higher-order exceptional points: A route for flat-top optical filters,” Phys. Rev. A 101(4), 043828 (2020).
[Crossref]

Schumer, A.

M. P. Hokmabadi, A. Schumer, D. N. Christodoulides, and M. Khajavikhan, “Non-Hermitian ring laser gyroscopes with enhanced Sagnac sensitivity,” Nature 576(7785), 70–74 (2019).
[Crossref]

Segev, M.

Y. Lumer, Y. Plotnik, M. C. Rechtsman, and M. Segev, “Nonlinearly induced PT transition in photonic systems,” Phys. Rev. Lett. 111(26), 263901 (2013).
[Crossref]

Shen, Y. R.

Y. R. Shen, “Quantum statistics of nonlinear optics,” Phys. Rev. 155(3), 921–931 (1967).
[Crossref]

Shen, Z.

Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9(1), 1797 (2018).
[Crossref]

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10(10), 657–661 (2016).
[Crossref]

Shi, L.

T. Wu, W. X. Zhang, H. Z. Zhang, S. S. Hou, G. Y. Chen, R. B. Liu, C. C. Lu, J. F. Li, R. Y. Wang, P. F. Duan, J. J. Li, B. Wang, L. Shi, J. Zi, and X. D. Zhang, “Vector exceptional points with strong superchiral fields,” Phys. Rev. Lett. 124(8), 083901 (2020).
[Crossref]

Shi, Y.

Y. Shi, Z. Yu, and S. Fan, “Limitations of nonlinear optical isolators due to dynamic reciprocity,” Nat. Photonics 9(6), 388–392 (2015).
[Crossref]

Shim, J.

Shu, F.-J.

F.-J. Shu, C.-L. Zou, X.-B. Zou, and L. Yang, “Chiral symmetry breaking in a microring optical cavity by engineered dissipation,” Phys. Rev. A 94(1), 013848 (2016).
[Crossref]

Silver, J. M.

Skarda, J.

K. Y. Yang, J. Skarda, M. Cotrufo, A. Dutt, G. H. Ahn, M. Sawaby, D. Vercruysse, A. Arbabian, S. Fan, A. Alù, and J. Vučković, “Inverse-designed non-reciprocal pulse router for chip-based lidar,” Nat. Photonics 14(6), 369–374 (2020).
[Crossref]

Smith, M. J.

E. Lafalce, Q. J. Zeng, C. H. Lin, M. J. Smith, S. T. Malak, J. Jung, Y. J. Yoon, Z. Q. Lin, V. V. Tsukruk, and Z. V. Vardeny, “Robust lasing modes in coupled colloidal quantum dot microdisk pairs using a non-Hermitian exceptional point,” Nat. Commun. 10(1), 561 (2019).
[Crossref]

Soric, J.

D. L. Sounas, J. Soric, and A. Alù, “Broadband passive isolators based on coupled nonlinear resonances,” Nat. Electron. 1(2), 113–119 (2018).
[Crossref]

Sounas, D. L.

D. L. Sounas, J. Soric, and A. Alù, “Broadband passive isolators based on coupled nonlinear resonances,” Nat. Electron. 1(2), 113–119 (2018).
[Crossref]

Stebbings, S. L.

Stone, A. D.

M. Zhang, W. Sweeney, C. W. Hsu, L. Yang, A. D. Stone, and L. Jiang, “Quantum noise theory of exceptional point amplifying sensors,” Phys. Rev. Lett. 123(18), 180501 (2019).
[Crossref]

Suh, M. G.

Y. H. Lai, Y. K. Lu, M. G. Suh, Z. Q. Yuan, and K. Vahala, “Observation of the exceptional-point-enhanced Sagnac effect,” Nature 576(7785), 65–69 (2019).
[Crossref]

Sun, F.-W.

Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9(1), 1797 (2018).
[Crossref]

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10(10), 657–661 (2016).
[Crossref]

Sun, J. B.

P. Miao, Z. F. Zhang, J. B. Sun, W. Walasik, S. Longhi, N. M. Litchinitser, and L. Feng, “Orbital angular momentum microlaser,” Science 353(6298), 464–467 (2016).
[Crossref]

Sweeney, W.

M. Zhang, W. Sweeney, C. W. Hsu, L. Yang, A. D. Stone, and L. Jiang, “Quantum noise theory of exceptional point amplifying sensors,” Phys. Rev. Lett. 123(18), 180501 (2019).
[Crossref]

Tang, J.

L. Tang, J. Tang, W. Zhang, G. Lu, H. Zhang, Y. Zhang, K. Xia, and M. Xiao, “On-chip chiral single-photon interface: Isolation and unidirectional emission,” Phys. Rev. A 99(4), 043833 (2019).
[Crossref]

L. Tang, J. Tang, H. Wu, J. Zhang, M. Xiao, and K. Xia, “Broad-intensity-range optical nonreciprocity based on feedback-induced Kerr nonlinearity,” Photon. Res., in press (2021).

Tang, L.

L. Tang, J. Tang, W. Zhang, G. Lu, H. Zhang, Y. Zhang, K. Xia, and M. Xiao, “On-chip chiral single-photon interface: Isolation and unidirectional emission,” Phys. Rev. A 99(4), 043833 (2019).
[Crossref]

L. Tang, J. Tang, H. Wu, J. Zhang, M. Xiao, and K. Xia, “Broad-intensity-range optical nonreciprocity based on feedback-induced Kerr nonlinearity,” Photon. Res., in press (2021).

Tey, M. K.

C. Liang, B. Liu, A.-N. Xu, X. Wen, C. Lu, K. Xia, M. K. Tey, Y.-C. Liu, and L. You, “Collision-induced broadband optical nonreciprocity,” Phys. Rev. Lett. 125(12), 123901 (2020).
[Crossref]

Thevamaran, R.

V. Dominguez-Rocha, R. Thevamaran, F. M. Ellis, and T. Kottos, “Environmentally induced exceptional points in elastodynamics,” Phys. Rev. Appl. 13(1), 014060 (2020).
[Crossref]

Tsukruk, V. V.

E. Lafalce, Q. J. Zeng, C. H. Lin, M. J. Smith, S. T. Malak, J. Jung, Y. J. Yoon, Z. Q. Lin, V. V. Tsukruk, and Z. V. Vardeny, “Robust lasing modes in coupled colloidal quantum dot microdisk pairs using a non-Hermitian exceptional point,” Nat. Commun. 10(1), 561 (2019).
[Crossref]

Twamley, J.

K. Xia, G. Lu, G. Lin, Y. Cheng, Y. Niu, S. Gong, and J. Twamley, “Reversible nonmagnetic single-photon isolation using unbalanced quantum coupling,” Phys. Rev. A 90(4), 043802 (2014).
[Crossref]

Vahala, K.

Y. H. Lai, Y. K. Lu, M. G. Suh, Z. Q. Yuan, and K. Vahala, “Observation of the exceptional-point-enhanced Sagnac effect,” Nature 576(7785), 65–69 (2019).
[Crossref]

Vardeny, Z. V.

E. Lafalce, Q. J. Zeng, C. H. Lin, M. J. Smith, S. T. Malak, J. Jung, Y. J. Yoon, Z. Q. Lin, V. V. Tsukruk, and Z. V. Vardeny, “Robust lasing modes in coupled colloidal quantum dot microdisk pairs using a non-Hermitian exceptional point,” Nat. Commun. 10(1), 561 (2019).
[Crossref]

Vercruysse, D.

K. Y. Yang, J. Skarda, M. Cotrufo, A. Dutt, G. H. Ahn, M. Sawaby, D. Vercruysse, A. Arbabian, S. Fan, A. Alù, and J. Vučković, “Inverse-designed non-reciprocal pulse router for chip-based lidar,” Nat. Photonics 14(6), 369–374 (2020).
[Crossref]

Vuckovic, J.

K. Y. Yang, J. Skarda, M. Cotrufo, A. Dutt, G. H. Ahn, M. Sawaby, D. Vercruysse, A. Arbabian, S. Fan, A. Alù, and J. Vučković, “Inverse-designed non-reciprocal pulse router for chip-based lidar,” Nat. Photonics 14(6), 369–374 (2020).
[Crossref]

Walasik, W.

P. Miao, Z. F. Zhang, J. B. Sun, W. Walasik, S. Longhi, N. M. Litchinitser, and L. Feng, “Orbital angular momentum microlaser,” Science 353(6298), 464–467 (2016).
[Crossref]

Wan, W.

F. Zhang, Y. Feng, X. Chen, L. Ge, and W. Wan, “Synthetic anti-PT symmetry in a single microcavity,” Phys. Rev. Lett. 124(5), 053901 (2020).
[Crossref]

Wang, B.

T. Wu, W. X. Zhang, H. Z. Zhang, S. S. Hou, G. Y. Chen, R. B. Liu, C. C. Lu, J. F. Li, R. Y. Wang, P. F. Duan, J. J. Li, B. Wang, L. Shi, J. Zi, and X. D. Zhang, “Vector exceptional points with strong superchiral fields,” Phys. Rev. Lett. 124(8), 083901 (2020).
[Crossref]

Wang, C.

W.-C. Gao, C. Zheng, L. Liu, T.-J. Wang, and C. Wang, “Experimental simulation of the parity-time symmetric dynamics using photonic qubits,” Opt. Express 29(1), 517–526 (2021).
[Crossref]

X.-F. Liu, T.-J. Wang, Y.-P. Gao, C. Cao, and C. Wang, “Chiral microresonator assisted by Rydberg-atom ensembles,” Phys. Rev. A 98(3), 033824 (2018).
[Crossref]

Wang, G. Z.

L. Chang, X. S. Jiang, S. Y. Hua, C. Yang, J. M. Wen, L. Jiang, G. Y. Li, G. Z. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8(7), 524–529 (2014).
[Crossref]

Wang, J.

Wang, L.

X. Dai, X. Zhang, I. M. Kislyakov, L. Wang, J. Huang, S. Zhang, N. Dong, and J. Wang, “Enhanced two-photon absorption and two-photon luminescence in monolayer MoS2 and WS2 by defect repairing,” Opt. Express 27(10), 13744–13753 (2019).
[Crossref]

X.-F. Jiang, C.-L. Zou, L. Wang, Q. Gong, and Y.-F. Xiao, “Whispering-gallery microcavities with unidirectional laser emission,” Laser Photonics Rev. 10(1), 40–61 (2016).
[Crossref]

Wang, M.

X. Mao, G.-Q. Qin, H. Yang, H. Zhang, M. Wang, and G.-L. Long, “Enhanced sensitivity of optical gyroscope in a mechanical parity-time-symmetric system based on exceptional point,” New J. Phys. 22(9), 093009 (2020).
[Crossref]

G. Q. Qin, M. Wang, J. W. Wen, D. Ruan, and G. L. Long, “Brillouin cavity optomechanics sensing with enhanced dynamical backaction,” Photonics Res. 7(12), 1440–1446 (2019).
[Crossref]

Wang, R. Y.

T. Wu, W. X. Zhang, H. Z. Zhang, S. S. Hou, G. Y. Chen, R. B. Liu, C. C. Lu, J. F. Li, R. Y. Wang, P. F. Duan, J. J. Li, B. Wang, L. Shi, J. Zi, and X. D. Zhang, “Vector exceptional points with strong superchiral fields,” Phys. Rev. Lett. 124(8), 083901 (2020).
[Crossref]

Wang, T.-J.

W.-C. Gao, C. Zheng, L. Liu, T.-J. Wang, and C. Wang, “Experimental simulation of the parity-time symmetric dynamics using photonic qubits,” Opt. Express 29(1), 517–526 (2021).
[Crossref]

X.-F. Liu, T.-J. Wang, Y.-P. Gao, C. Cao, and C. Wang, “Chiral microresonator assisted by Rydberg-atom ensembles,” Phys. Rev. A 98(3), 033824 (2018).
[Crossref]

Wang, X. G.

Z. C. Zhang, Y. P. Wang, and X. G. Wang, “PT-symmetry-breaking-enhanced cavity optomechanical magnetometry,” Phys. Rev. A 102(2), 023512 (2020).
[Crossref]

Wang, Y.

L. Feng, Z. J. Wong, R. M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346(6212), 972–975 (2014).
[Crossref]

Wang, Y. P.

Z. C. Zhang, Y. P. Wang, and X. G. Wang, “PT-symmetry-breaking-enhanced cavity optomechanical magnetometry,” Phys. Rev. A 102(2), 023512 (2020).
[Crossref]

G. Q. Zhang, Y. P. Wang, and J. Q. You, “Dispersive readout of a weakly coupled qubit via the parity-time-symmetric phase transition,” Phys. Rev. A 99(5), 052341 (2019).
[Crossref]

Wang, Y.-X.

Y.-X. Wang and A. A. Clerk, “Non-Hermitian dynamics without dissipation in quantum systems,” Phys. Rev. A 99(6), 063834 (2019).
[Crossref]

Wei, S.

J. Wen, C. Zheng, X. Kong, S. Wei, T. Xin, and G. Long, “Experimental demonstration of a digital quantum simulation of a general $\mathcal{PT}$-symmetric system,” Phys. Rev. A 99(6), 062122 (2019).
[Crossref]

Wen, J.

J. Ma, J. Wen, S. Ding, S. Li, Y. Hu, X. Jiang, L. Jiang, and M. Xiao, “Chip-based optical isolator and nonreciprocal parity-time symmetry induced by stimulated Brillouin scattering,” Laser Photonics Rev. 14(5), 1900278 (2020).
[Crossref]

J. Wen, C. Zheng, X. Kong, S. Wei, T. Xin, and G. Long, “Experimental demonstration of a digital quantum simulation of a general $\mathcal{PT}$-symmetric system,” Phys. Rev. A 99(6), 062122 (2019).
[Crossref]

J. Wen, X. Jiang, L. Jiang, and M. Xiao, “Parity-time symmetry in optical microcavity systems,” J. Phys. B: At., Mol. Opt. Phys. 51(22), 222001 (2018).
[Crossref]

Wen, J. M.

L. Chang, X. S. Jiang, S. Y. Hua, C. Yang, J. M. Wen, L. Jiang, G. Y. Li, G. Z. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8(7), 524–529 (2014).
[Crossref]

Wen, J. W.

G. Q. Qin, M. Wang, J. W. Wen, D. Ruan, and G. L. Long, “Brillouin cavity optomechanics sensing with enhanced dynamical backaction,” Photonics Res. 7(12), 1440–1446 (2019).
[Crossref]

Wen, X.

C. Liang, B. Liu, A.-N. Xu, X. Wen, C. Lu, K. Xia, M. K. Tey, Y.-C. Liu, and L. You, “Collision-induced broadband optical nonreciprocity,” Phys. Rev. Lett. 125(12), 123901 (2020).
[Crossref]

Wiersig, J.

J. Wiersig, “Prospects and fundamental limits in exceptional point-based sensing,” Nat. Commun. 11(1), 2454 (2020).
[Crossref]

W. J. Chen, S. K. Ozdemir, G. M. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548(7666), 192–196 (2017).
[Crossref]

B. Peng, S. K. Ozdemir, M. Liertzer, W. J. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Natl. Acad. Sci. U. S. A. 113(25), 6845–6850 (2016).
[Crossref]

J. Wiersig, “Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: Application to microcavity sensors for single-particle detection,” Phys. Rev. Lett. 112(20), 203901 (2014).
[Crossref]

J. Wiersig, “Structure of whispering-gallery modes in optical microdisks perturbed by nanoparticles,” Phys. Rev. A 84(6), 063828 (2011).
[Crossref]

Wittek, S.

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548(7666), 187–191 (2017).
[Crossref]

Wong, Z. J.

L. Feng, Z. J. Wong, R. M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346(6212), 972–975 (2014).
[Crossref]

Woodley, M. T. M.

Wu, H.

L. Tang, J. Tang, H. Wu, J. Zhang, M. Xiao, and K. Xia, “Broad-intensity-range optical nonreciprocity based on feedback-induced Kerr nonlinearity,” Photon. Res., in press (2021).

Wu, J.-H.

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113(12), 123004 (2014).
[Crossref]

Wu, T.

T. Wu, W. X. Zhang, H. Z. Zhang, S. S. Hou, G. Y. Chen, R. B. Liu, C. C. Lu, J. F. Li, R. Y. Wang, P. F. Duan, J. J. Li, B. Wang, L. Shi, J. Zi, and X. D. Zhang, “Vector exceptional points with strong superchiral fields,” Phys. Rev. Lett. 124(8), 083901 (2020).
[Crossref]

Xia, K.

C. Liang, B. Liu, A.-N. Xu, X. Wen, C. Lu, K. Xia, M. K. Tey, Y.-C. Liu, and L. You, “Collision-induced broadband optical nonreciprocity,” Phys. Rev. Lett. 125(12), 123901 (2020).
[Crossref]

L. Tang, J. Tang, W. Zhang, G. Lu, H. Zhang, Y. Zhang, K. Xia, and M. Xiao, “On-chip chiral single-photon interface: Isolation and unidirectional emission,” Phys. Rev. A 99(4), 043833 (2019).
[Crossref]

K. Xia, F. Nori, and M. Xiao, “Cavity-free optical isolators and circulators using a chiral cross-Kerr nonlinearity,” Phys. Rev. Lett. 121(20), 203602 (2018).
[Crossref]

S. Zhang, Y. Hu, G. Lin, Y. Niu, K. Xia, J. Gong, and S. Gong, “Thermal-motion-induced non-reciprocal quantum optical system,” Nat. Photonics 12(12), 744–748 (2018).
[Crossref]

K. Xia, G. Lu, G. Lin, Y. Cheng, Y. Niu, S. Gong, and J. Twamley, “Reversible nonmagnetic single-photon isolation using unbalanced quantum coupling,” Phys. Rev. A 90(4), 043802 (2014).
[Crossref]

L. Tang, J. Tang, H. Wu, J. Zhang, M. Xiao, and K. Xia, “Broad-intensity-range optical nonreciprocity based on feedback-induced Kerr nonlinearity,” Photon. Res., in press (2021).

Xiao, M.

J. Ma, J. Wen, S. Ding, S. Li, Y. Hu, X. Jiang, L. Jiang, and M. Xiao, “Chip-based optical isolator and nonreciprocal parity-time symmetry induced by stimulated Brillouin scattering,” Laser Photonics Rev. 14(5), 1900278 (2020).
[Crossref]

L. Tang, J. Tang, W. Zhang, G. Lu, H. Zhang, Y. Zhang, K. Xia, and M. Xiao, “On-chip chiral single-photon interface: Isolation and unidirectional emission,” Phys. Rev. A 99(4), 043833 (2019).
[Crossref]

K. Xia, F. Nori, and M. Xiao, “Cavity-free optical isolators and circulators using a chiral cross-Kerr nonlinearity,” Phys. Rev. Lett. 121(20), 203602 (2018).
[Crossref]

J. Wen, X. Jiang, L. Jiang, and M. Xiao, “Parity-time symmetry in optical microcavity systems,” J. Phys. B: At., Mol. Opt. Phys. 51(22), 222001 (2018).
[Crossref]

B. He, L. Yang, X. Jiang, and M. Xiao, “Transmission nonreciprocity in a mutually coupled circulating structure,” Phys. Rev. Lett. 120(20), 203904 (2018).
[Crossref]

L. Chang, X. S. Jiang, S. Y. Hua, C. Yang, J. M. Wen, L. Jiang, G. Y. Li, G. Z. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8(7), 524–529 (2014).
[Crossref]

L. Tang, J. Tang, H. Wu, J. Zhang, M. Xiao, and K. Xia, “Broad-intensity-range optical nonreciprocity based on feedback-induced Kerr nonlinearity,” Photon. Res., in press (2021).

Xiao, Y. F.

Q. T. Cao, Y. L. Chen, and Y. F. Xiao, “Chiral emission and Purcell enhancement in a hybrid plasmonic-photonic microresonator,” Light: Sci. Appl. 9(1), 4 (2020).
[Crossref]

Xiao, Y.-F.

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10(10), 657–661 (2016).
[Crossref]

X.-F. Jiang, C.-L. Zou, L. Wang, Q. Gong, and Y.-F. Xiao, “Whispering-gallery microcavities with unidirectional laser emission,” Laser Photonics Rev. 10(1), 40–61 (2016).
[Crossref]

Xin, T.

J. Wen, C. Zheng, X. Kong, S. Wei, T. Xin, and G. Long, “Experimental demonstration of a digital quantum simulation of a general $\mathcal{PT}$-symmetric system,” Phys. Rev. A 99(6), 062122 (2019).
[Crossref]

Xu, A.-N.

C. Liang, B. Liu, A.-N. Xu, X. Wen, C. Lu, K. Xia, M. K. Tey, Y.-C. Liu, and L. You, “Collision-induced broadband optical nonreciprocity,” Phys. Rev. Lett. 125(12), 123901 (2020).
[Crossref]

Xu, H.

H. Xu, D. Mason, L. Y. Jiang, and J. G. E. Harris, “Topological energy transfer in an optomechanical system with exceptional points,” Nature 537(7618), 80–83 (2016).
[Crossref]

Xu, Q.

Xu, Y. L.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2013).
[Crossref]

Yang, C.

L. Chang, X. S. Jiang, S. Y. Hua, C. Yang, J. M. Wen, L. Jiang, G. Y. Li, G. Z. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8(7), 524–529 (2014).
[Crossref]

Yang, F.

F. Yang, Y. C. Liu, and L. You, “Anti-PT symmetry in dissipatively coupled optical systems,” Phys. Rev. A 96(5), 053845 (2017).
[Crossref]

Yang, H.

X. Mao, G.-Q. Qin, H. Yang, H. Zhang, M. Wang, and G.-L. Long, “Enhanced sensitivity of optical gyroscope in a mechanical parity-time-symmetric system based on exceptional point,” New J. Phys. 22(9), 093009 (2020).
[Crossref]

Yang, K. Y.

K. Y. Yang, J. Skarda, M. Cotrufo, A. Dutt, G. H. Ahn, M. Sawaby, D. Vercruysse, A. Arbabian, S. Fan, A. Alù, and J. Vučković, “Inverse-designed non-reciprocal pulse router for chip-based lidar,” Nat. Photonics 14(6), 369–374 (2020).
[Crossref]

Yang, L.

X. Jiang and L. Yang, “Optothermal dynamics in whispering-gallery microresonators,” Light: Sci. Appl. 9(1), 24 (2020).
[Crossref]

S. N. Huai, Y. L. Liu, J. Zhang, L. Yang, and Y. X. Liu, “Enhanced sideband responses in a PT-symmetric-like cavity magnomechanical system,” Phys. Rev. A 99(4), 043803 (2019).
[Crossref]

M. Zhang, W. Sweeney, C. W. Hsu, L. Yang, A. D. Stone, and L. Jiang, “Quantum noise theory of exceptional point amplifying sensors,” Phys. Rev. Lett. 123(18), 180501 (2019).
[Crossref]

S. K. Ozdemir, S. Rotter, F. Nori, and L. Yang, “Parity-time symmetry and exceptional points in photonics,” Nat. Mater. 18(8), 783–798 (2019).
[Crossref]

W. J. Chen, J. Zhang, B. Peng, S. K. Ozdemir, X. D. Fan, and L. Yang, “Parity-time-symmetric whispering-gallery mode nanoparticle sensor invited,” Photonics Res. 6(5), A23–A30 (2018).
[Crossref]

B. He, L. Yang, X. Jiang, and M. Xiao, “Transmission nonreciprocity in a mutually coupled circulating structure,” Phys. Rev. Lett. 120(20), 203904 (2018).
[Crossref]

W. J. Chen, S. K. Ozdemir, G. M. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548(7666), 192–196 (2017).
[Crossref]

Z. P. Liu, J. Zhang, S. K. Ozdemir, B. Peng, H. Jing, X. Y. Lu, C. W. Li, L. Yang, F. Nori, and Y. X. Liu, “Metrology with PT-symmetric cavities: Enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117(11), 110802 (2016).
[Crossref]

F.-J. Shu, C.-L. Zou, X.-B. Zou, and L. Yang, “Chiral symmetry breaking in a microring optical cavity by engineered dissipation,” Phys. Rev. A 94(1), 013848 (2016).
[Crossref]

B. Peng, S. K. Ozdemir, M. Liertzer, W. J. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Natl. Acad. Sci. U. S. A. 113(25), 6845–6850 (2016).
[Crossref]

H. Jing, S. K. Ozdemir, X. Y. Lu, J. Zhang, L. Yang, and F. Nori, “PT-symmetric phonon laser,” Phys. Rev. Lett. 113(5), 053604 (2014).
[Crossref]

B. Peng, S. K. Ozdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014).
[Crossref]

B. Peng, S. K. Ozdemir, F. C. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

Yilmaz, H.

B. Peng, S. K. Ozdemir, M. Liertzer, W. J. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Natl. Acad. Sci. U. S. A. 113(25), 6845–6850 (2016).
[Crossref]

B. Peng, S. K. Ozdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014).
[Crossref]

Yoon, Y. J.

E. Lafalce, Q. J. Zeng, C. H. Lin, M. J. Smith, S. T. Malak, J. Jung, Y. J. Yoon, Z. Q. Lin, V. V. Tsukruk, and Z. V. Vardeny, “Robust lasing modes in coupled colloidal quantum dot microdisk pairs using a non-Hermitian exceptional point,” Nat. Commun. 10(1), 561 (2019).
[Crossref]

You, H.

H. You, S. M. Hendrickson, and J. D. Franson, “Analysis of enhanced two-photon absorption in tapered optical fibers,” Phys. Rev. A 78(5), 053803 (2008).
[Crossref]

You, J. Q.

G. Q. Zhang, Y. P. Wang, and J. Q. You, “Dispersive readout of a weakly coupled qubit via the parity-time-symmetric phase transition,” Phys. Rev. A 99(5), 052341 (2019).
[Crossref]

You, L.

C. Liang, B. Liu, A.-N. Xu, X. Wen, C. Lu, K. Xia, M. K. Tey, Y.-C. Liu, and L. You, “Collision-induced broadband optical nonreciprocity,” Phys. Rev. Lett. 125(12), 123901 (2020).
[Crossref]

F. Yang, Y. C. Liu, and L. You, “Anti-PT symmetry in dissipatively coupled optical systems,” Phys. Rev. A 96(5), 053845 (2017).
[Crossref]

Yu, K.

Yu, Z.

Y. Shi, Z. Yu, and S. Fan, “Limitations of nonlinear optical isolators due to dynamic reciprocity,” Nat. Photonics 9(6), 388–392 (2015).
[Crossref]

Yuan, Z. Q.

Y. H. Lai, Y. K. Lu, M. G. Suh, Z. Q. Yuan, and K. Vahala, “Observation of the exceptional-point-enhanced Sagnac effect,” Nature 576(7785), 65–69 (2019).
[Crossref]

Zagoskin, A. M.

A. Miranowicz, J. Bajer, M. Paprzycka, Y.-X. Liu, A. M. Zagoskin, and F. Nori, “State-dependent photon blockade via quantum-reservoir engineering,” Phys. Rev. A 90(3), 033831 (2014).
[Crossref]

Zeng, Q. J.

E. Lafalce, Q. J. Zeng, C. H. Lin, M. J. Smith, S. T. Malak, J. Jung, Y. J. Yoon, Z. Q. Lin, V. V. Tsukruk, and Z. V. Vardeny, “Robust lasing modes in coupled colloidal quantum dot microdisk pairs using a non-Hermitian exceptional point,” Nat. Commun. 10(1), 561 (2019).
[Crossref]

Zhang, F.

F. Zhang, Y. Feng, X. Chen, L. Ge, and W. Wan, “Synthetic anti-PT symmetry in a single microcavity,” Phys. Rev. Lett. 124(5), 053901 (2020).
[Crossref]

Zhang, G. Q.

G. Q. Zhang, Y. P. Wang, and J. Q. You, “Dispersive readout of a weakly coupled qubit via the parity-time-symmetric phase transition,” Phys. Rev. A 99(5), 052341 (2019).
[Crossref]

Zhang, H.

X. Mao, G.-Q. Qin, H. Yang, H. Zhang, M. Wang, and G.-L. Long, “Enhanced sensitivity of optical gyroscope in a mechanical parity-time-symmetric system based on exceptional point,” New J. Phys. 22(9), 093009 (2020).
[Crossref]

L. Tang, J. Tang, W. Zhang, G. Lu, H. Zhang, Y. Zhang, K. Xia, and M. Xiao, “On-chip chiral single-photon interface: Isolation and unidirectional emission,” Phys. Rev. A 99(4), 043833 (2019).
[Crossref]

H. Zhang, F. Saif, Y. Jiao, and H. Jing, “Loss-induced transparency in optomechanics,” Opt. Express 26(19), 25199–25210 (2018).
[Crossref]

Zhang, H. Z.

T. Wu, W. X. Zhang, H. Z. Zhang, S. S. Hou, G. Y. Chen, R. B. Liu, C. C. Lu, J. F. Li, R. Y. Wang, P. F. Duan, J. J. Li, B. Wang, L. Shi, J. Zi, and X. D. Zhang, “Vector exceptional points with strong superchiral fields,” Phys. Rev. Lett. 124(8), 083901 (2020).
[Crossref]

Zhang, J.

S. N. Huai, Y. L. Liu, J. Zhang, L. Yang, and Y. X. Liu, “Enhanced sideband responses in a PT-symmetric-like cavity magnomechanical system,” Phys. Rev. A 99(4), 043803 (2019).
[Crossref]

W. J. Chen, J. Zhang, B. Peng, S. K. Ozdemir, X. D. Fan, and L. Yang, “Parity-time-symmetric whispering-gallery mode nanoparticle sensor invited,” Photonics Res. 6(5), A23–A30 (2018).
[Crossref]

Z. P. Liu, J. Zhang, S. K. Ozdemir, B. Peng, H. Jing, X. Y. Lu, C. W. Li, L. Yang, F. Nori, and Y. X. Liu, “Metrology with PT-symmetric cavities: Enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117(11), 110802 (2016).
[Crossref]

H. Jing, S. K. Ozdemir, X. Y. Lu, J. Zhang, L. Yang, and F. Nori, “PT-symmetric phonon laser,” Phys. Rev. Lett. 113(5), 053604 (2014).
[Crossref]

L. Tang, J. Tang, H. Wu, J. Zhang, M. Xiao, and K. Xia, “Broad-intensity-range optical nonreciprocity based on feedback-induced Kerr nonlinearity,” Photon. Res., in press (2021).

Zhang, J. H.

Zhang, M.

M. Zhang, W. Sweeney, C. W. Hsu, L. Yang, A. D. Stone, and L. Jiang, “Quantum noise theory of exceptional point amplifying sensors,” Phys. Rev. Lett. 123(18), 180501 (2019).
[Crossref]

Zhang, S.

S. Zhang, G. Lin, Y. Hu, Y. Qi, Y. Niu, and S. Gong, “Cavity-free circulator with low insertion loss using hot atoms,” Phys. Rev. Appl. 14(2), 024032 (2020).
[Crossref]

X. Dai, X. Zhang, I. M. Kislyakov, L. Wang, J. Huang, S. Zhang, N. Dong, and J. Wang, “Enhanced two-photon absorption and two-photon luminescence in monolayer MoS2 and WS2 by defect repairing,” Opt. Express 27(10), 13744–13753 (2019).
[Crossref]

S. Zhang, Y. Hu, G. Lin, Y. Niu, K. Xia, J. Gong, and S. Gong, “Thermal-motion-induced non-reciprocal quantum optical system,” Nat. Photonics 12(12), 744–748 (2018).
[Crossref]

Zhang, W.

L. Tang, J. Tang, W. Zhang, G. Lu, H. Zhang, Y. Zhang, K. Xia, and M. Xiao, “On-chip chiral single-photon interface: Isolation and unidirectional emission,” Phys. Rev. A 99(4), 043833 (2019).
[Crossref]

Zhang, W. L.

Zhang, W. X.

T. Wu, W. X. Zhang, H. Z. Zhang, S. S. Hou, G. Y. Chen, R. B. Liu, C. C. Lu, J. F. Li, R. Y. Wang, P. F. Duan, J. J. Li, B. Wang, L. Shi, J. Zi, and X. D. Zhang, “Vector exceptional points with strong superchiral fields,” Phys. Rev. Lett. 124(8), 083901 (2020).
[Crossref]

Zhang, X.

Zhang, X. D.

T. Wu, W. X. Zhang, H. Z. Zhang, S. S. Hou, G. Y. Chen, R. B. Liu, C. C. Lu, J. F. Li, R. Y. Wang, P. F. Duan, J. J. Li, B. Wang, L. Shi, J. Zi, and X. D. Zhang, “Vector exceptional points with strong superchiral fields,” Phys. Rev. Lett. 124(8), 083901 (2020).
[Crossref]

Zhang, X. Q.

Zhang, Y.

L. Tang, J. Tang, W. Zhang, G. Lu, H. Zhang, Y. Zhang, K. Xia, and M. Xiao, “On-chip chiral single-photon interface: Isolation and unidirectional emission,” Phys. Rev. A 99(4), 043833 (2019).
[Crossref]

Zhang, Y.-L.

Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9(1), 1797 (2018).
[Crossref]

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10(10), 657–661 (2016).
[Crossref]

Zhang, Z. C.

Z. C. Zhang, Y. P. Wang, and X. G. Wang, “PT-symmetry-breaking-enhanced cavity optomechanical magnetometry,” Phys. Rev. A 102(2), 023512 (2020).
[Crossref]

Zhang, Z. F.

P. Miao, Z. F. Zhang, J. B. Sun, W. Walasik, S. Longhi, N. M. Litchinitser, and L. Feng, “Orbital angular momentum microlaser,” Science 353(6298), 464–467 (2016).
[Crossref]

Zhao, G. M.

W. J. Chen, S. K. Ozdemir, G. M. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548(7666), 192–196 (2017).
[Crossref]

Zhao, X.

Zheng, C.

W.-C. Gao, C. Zheng, L. Liu, T.-J. Wang, and C. Wang, “Experimental simulation of the parity-time symmetric dynamics using photonic qubits,” Opt. Express 29(1), 517–526 (2021).
[Crossref]

J. Wen, C. Zheng, X. Kong, S. Wei, T. Xin, and G. Long, “Experimental demonstration of a digital quantum simulation of a general $\mathcal{PT}$-symmetric system,” Phys. Rev. A 99(6), 062122 (2019).
[Crossref]

Zhong, Q.

Q. Zhong, J. Ren, M. Khajavikhan, D. N. Christodoulides, S. K. Özdemir, and R. El-Ganainy, “Sensing with exceptional surfaces in order to combine sensitivity with robustness,” Phys. Rev. Lett. 122(15), 153902 (2019).
[Crossref]

Zhou, L.

Zhu, J.

T. Liu, G. C. Ma, S. J. Liang, H. Gao, Z. M. Gu, S. W. An, and J. Zhu, “Single-sided acoustic beam splitting based on parity-time symmetry,” Phys. Rev. B 102(1), 014306 (2020).
[Crossref]

Zi, J.

T. Wu, W. X. Zhang, H. Z. Zhang, S. S. Hou, G. Y. Chen, R. B. Liu, C. C. Lu, J. F. Li, R. Y. Wang, P. F. Duan, J. J. Li, B. Wang, L. Shi, J. Zi, and X. D. Zhang, “Vector exceptional points with strong superchiral fields,” Phys. Rev. Lett. 124(8), 083901 (2020).
[Crossref]

Zou, C.-L.

Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9(1), 1797 (2018).
[Crossref]

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10(10), 657–661 (2016).
[Crossref]

F.-J. Shu, C.-L. Zou, X.-B. Zou, and L. Yang, “Chiral symmetry breaking in a microring optical cavity by engineered dissipation,” Phys. Rev. A 94(1), 013848 (2016).
[Crossref]

X.-F. Jiang, C.-L. Zou, L. Wang, Q. Gong, and Y.-F. Xiao, “Whispering-gallery microcavities with unidirectional laser emission,” Laser Photonics Rev. 10(1), 40–61 (2016).
[Crossref]

Zou, X.-B.

Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9(1), 1797 (2018).
[Crossref]

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10(10), 657–661 (2016).
[Crossref]

F.-J. Shu, C.-L. Zou, X.-B. Zou, and L. Yang, “Chiral symmetry breaking in a microring optical cavity by engineered dissipation,” Phys. Rev. A 94(1), 013848 (2016).
[Crossref]

J. Phys. A: Math. Gen. (1)

W. D. Heiss, “Exceptional points of non-Hermitian operators,” J. Phys. A: Math. Gen. 37(6), 2455–2464 (2004).
[Crossref]

J. Phys. B: At., Mol. Opt. Phys. (1)

J. Wen, X. Jiang, L. Jiang, and M. Xiao, “Parity-time symmetry in optical microcavity systems,” J. Phys. B: At., Mol. Opt. Phys. 51(22), 222001 (2018).
[Crossref]

Laser Photonics Rev. (2)

X.-F. Jiang, C.-L. Zou, L. Wang, Q. Gong, and Y.-F. Xiao, “Whispering-gallery microcavities with unidirectional laser emission,” Laser Photonics Rev. 10(1), 40–61 (2016).
[Crossref]

J. Ma, J. Wen, S. Ding, S. Li, Y. Hu, X. Jiang, L. Jiang, and M. Xiao, “Chip-based optical isolator and nonreciprocal parity-time symmetry induced by stimulated Brillouin scattering,” Laser Photonics Rev. 14(5), 1900278 (2020).
[Crossref]

Light: Sci. Appl. (2)

X. Jiang and L. Yang, “Optothermal dynamics in whispering-gallery microresonators,” Light: Sci. Appl. 9(1), 24 (2020).
[Crossref]

Q. T. Cao, Y. L. Chen, and Y. F. Xiao, “Chiral emission and Purcell enhancement in a hybrid plasmonic-photonic microresonator,” Light: Sci. Appl. 9(1), 4 (2020).
[Crossref]

Nat. Commun. (4)

E. Lafalce, Q. J. Zeng, C. H. Lin, M. J. Smith, S. T. Malak, J. Jung, Y. J. Yoon, Z. Q. Lin, V. V. Tsukruk, and Z. V. Vardeny, “Robust lasing modes in coupled colloidal quantum dot microdisk pairs using a non-Hermitian exceptional point,” Nat. Commun. 10(1), 561 (2019).
[Crossref]

H. K. Lau and A. A. Clerk, “Fundamental limits and non-reciprocal approaches in non-Hermitian quantum sensing,” Nat. Commun. 9(1), 4320 (2018).
[Crossref]

J. Wiersig, “Prospects and fundamental limits in exceptional point-based sensing,” Nat. Commun. 11(1), 2454 (2020).
[Crossref]

Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9(1), 1797 (2018).
[Crossref]

Nat. Electron. (1)

D. L. Sounas, J. Soric, and A. Alù, “Broadband passive isolators based on coupled nonlinear resonances,” Nat. Electron. 1(2), 113–119 (2018).
[Crossref]

Nat. Mater. (2)

S. K. Ozdemir, S. Rotter, F. Nori, and L. Yang, “Parity-time symmetry and exceptional points in photonics,” Nat. Mater. 18(8), 783–798 (2019).
[Crossref]

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2013).
[Crossref]

Nat. Photonics (5)

L. Chang, X. S. Jiang, S. Y. Hua, C. Yang, J. M. Wen, L. Jiang, G. Y. Li, G. Z. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8(7), 524–529 (2014).
[Crossref]

S. Zhang, Y. Hu, G. Lin, Y. Niu, K. Xia, J. Gong, and S. Gong, “Thermal-motion-induced non-reciprocal quantum optical system,” Nat. Photonics 12(12), 744–748 (2018).
[Crossref]

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10(10), 657–661 (2016).
[Crossref]

Y. Shi, Z. Yu, and S. Fan, “Limitations of nonlinear optical isolators due to dynamic reciprocity,” Nat. Photonics 9(6), 388–392 (2015).
[Crossref]

K. Y. Yang, J. Skarda, M. Cotrufo, A. Dutt, G. H. Ahn, M. Sawaby, D. Vercruysse, A. Arbabian, S. Fan, A. Alù, and J. Vučković, “Inverse-designed non-reciprocal pulse router for chip-based lidar,” Nat. Photonics 14(6), 369–374 (2020).
[Crossref]

Nat. Phys. (2)

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14(1), 11–19 (2018).
[Crossref]

B. Peng, S. K. Ozdemir, F. C. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

Nature (5)

Y. H. Lai, Y. K. Lu, M. G. Suh, Z. Q. Yuan, and K. Vahala, “Observation of the exceptional-point-enhanced Sagnac effect,” Nature 576(7785), 65–69 (2019).
[Crossref]

W. J. Chen, S. K. Ozdemir, G. M. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548(7666), 192–196 (2017).
[Crossref]

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548(7666), 187–191 (2017).
[Crossref]

M. P. Hokmabadi, A. Schumer, D. N. Christodoulides, and M. Khajavikhan, “Non-Hermitian ring laser gyroscopes with enhanced Sagnac sensitivity,” Nature 576(7785), 70–74 (2019).
[Crossref]

H. Xu, D. Mason, L. Y. Jiang, and J. G. E. Harris, “Topological energy transfer in an optomechanical system with exceptional points,” Nature 537(7618), 80–83 (2016).
[Crossref]

New J. Phys. (2)

X. Mao, G.-Q. Qin, H. Yang, H. Zhang, M. Wang, and G.-L. Long, “Enhanced sensitivity of optical gyroscope in a mechanical parity-time-symmetric system based on exceptional point,” New J. Phys. 22(9), 093009 (2020).
[Crossref]

M.-A. Miri and A. Alù, “Nonlinearity-induced pt-symmetry without material gain,” New J. Phys. 18(6), 065001 (2016).
[Crossref]

Opt. Express (7)

Opt. Lett. (1)

Optica (2)

Photonics Res. (3)

A. Calabrese, F. Ramiro-Manzano, H. M. Price, S. Biasi, M. Bernard, M. Ghulinyan, I. Carusotto, and L. Pavesi, “Unidirectional reflection from an integrated “taiji" microresonator,” Photonics Res. 8(8), 1333–1341 (2020).
[Crossref]

W. J. Chen, J. Zhang, B. Peng, S. K. Ozdemir, X. D. Fan, and L. Yang, “Parity-time-symmetric whispering-gallery mode nanoparticle sensor invited,” Photonics Res. 6(5), A23–A30 (2018).
[Crossref]

G. Q. Qin, M. Wang, J. W. Wen, D. Ruan, and G. L. Long, “Brillouin cavity optomechanics sensing with enhanced dynamical backaction,” Photonics Res. 7(12), 1440–1446 (2019).
[Crossref]

Phys. Rev. (1)

Y. R. Shen, “Quantum statistics of nonlinear optics,” Phys. Rev. 155(3), 921–931 (1967).
[Crossref]

Phys. Rev. A (21)

A. Miranowicz, J. Bajer, M. Paprzycka, Y.-X. Liu, A. M. Zagoskin, and F. Nori, “State-dependent photon blockade via quantum-reservoir engineering,” Phys. Rev. A 90(3), 033831 (2014).
[Crossref]

E. S. Guerra, B. M. Garraway, and P. L. Knight, “Two-photon parametric pumping versus two-photon absorption: A quantum jump approach,” Phys. Rev. A 55(5), 3842–3857 (1997).
[Crossref]

H. You, S. M. Hendrickson, and J. D. Franson, “Analysis of enhanced two-photon absorption in tapered optical fibers,” Phys. Rev. A 78(5), 053803 (2008).
[Crossref]

K. Xia, G. Lu, G. Lin, Y. Cheng, Y. Niu, S. Gong, and J. Twamley, “Reversible nonmagnetic single-photon isolation using unbalanced quantum coupling,” Phys. Rev. A 90(4), 043802 (2014).
[Crossref]

L. Tang, J. Tang, W. Zhang, G. Lu, H. Zhang, Y. Zhang, K. Xia, and M. Xiao, “On-chip chiral single-photon interface: Isolation and unidirectional emission,” Phys. Rev. A 99(4), 043833 (2019).
[Crossref]

S. N. Huai, Y. L. Liu, J. Zhang, L. Yang, and Y. X. Liu, “Enhanced sideband responses in a PT-symmetric-like cavity magnomechanical system,” Phys. Rev. A 99(4), 043803 (2019).
[Crossref]

F.-J. Shu, C.-L. Zou, X.-B. Zou, and L. Yang, “Chiral symmetry breaking in a microring optical cavity by engineered dissipation,” Phys. Rev. A 94(1), 013848 (2016).
[Crossref]

F. Yang, Y. C. Liu, and L. You, “Anti-PT symmetry in dissipatively coupled optical systems,” Phys. Rev. A 96(5), 053845 (2017).
[Crossref]

Z. C. Zhang, Y. P. Wang, and X. G. Wang, “PT-symmetry-breaking-enhanced cavity optomechanical magnetometry,” Phys. Rev. A 102(2), 023512 (2020).
[Crossref]

Y.-X. Wang and A. A. Clerk, “Non-Hermitian dynamics without dissipation in quantum systems,” Phys. Rev. A 99(6), 063834 (2019).
[Crossref]

A. U. Hassan, H. Hodaei, M.-A. Miri, M. Khajavikhan, and D. N. Christodoulides, “Nonlinear reversal of the $\mathcal{PT}$-symmetric phase transition in a system of coupled semiconductor microring resonators,” Phys. Rev. A 92(6), 063807 (2015).
[Crossref]

J. Perina, A. Luks, J. K. Kalaga, W. Leonski, and A. Miranowicz, “Nonclassical light at exceptional points of a quantum PT-symmetric two-mode system,” Phys. Rev. A 100(5), 053820 (2019).
[Crossref]

P. C. Kuo, N. Lambert, A. Miranowicz, H. B. Chen, G. Y. Chen, Y. N. Chen, and F. Nori, “Collectively induced exceptional points of quantum emitters coupled to nanoparticle surface plasmons,” Phys. Rev. A 101(1), 013814 (2020).
[Crossref]

I. Arkhipov, A. Miranowicz, F. Minganti, and F. Nori, “Quantum and semiclassical exceptional points of a linear system of coupled cavities with losses and gain within the Scully-Lamb laser theory,” Phys. Rev. A 101(1), 013812 (2020).
[Crossref]

N. Habler and J. Scheuer, “Higher-order exceptional points: A route for flat-top optical filters,” Phys. Rev. A 101(4), 043828 (2020).
[Crossref]

J. Wen, C. Zheng, X. Kong, S. Wei, T. Xin, and G. Long, “Experimental demonstration of a digital quantum simulation of a general $\mathcal{PT}$-symmetric system,” Phys. Rev. A 99(6), 062122 (2019).
[Crossref]

G. Q. Zhang, Y. P. Wang, and J. Q. You, “Dispersive readout of a weakly coupled qubit via the parity-time-symmetric phase transition,” Phys. Rev. A 99(5), 052341 (2019).
[Crossref]

F. Minganti, A. Miranowicz, R. W. Chhajlany, and F. Nori, “Quantum exceptional points of non-Hermitian Hamiltonians and Liouvillians: The effects of quantum jumps,” Phys. Rev. A 100(6), 062131 (2019).
[Crossref]

X.-F. Liu, T.-J. Wang, Y.-P. Gao, C. Cao, and C. Wang, “Chiral microresonator assisted by Rydberg-atom ensembles,” Phys. Rev. A 98(3), 033824 (2018).
[Crossref]

P.-B. Li, S.-Y. Gao, and F.-L. Li, “Quantum-information transfer with nitrogen-vacancy centers coupled to a whispering-gallery microresonator,” Phys. Rev. A 83(5), 054306 (2011).
[Crossref]

J. Wiersig, “Structure of whispering-gallery modes in optical microdisks perturbed by nanoparticles,” Phys. Rev. A 84(6), 063828 (2011).
[Crossref]

Phys. Rev. Appl. (3)

V. Dominguez-Rocha, R. Thevamaran, F. M. Ellis, and T. Kottos, “Environmentally induced exceptional points in elastodynamics,” Phys. Rev. Appl. 13(1), 014060 (2020).
[Crossref]

P. Djorwe, Y. Pennec, and B. Djafari-Rouhani, “Exceptional point enhances sensitivity of optomechanical mass sensors,” Phys. Rev. Appl. 12(2), 024002 (2019).
[Crossref]

S. Zhang, G. Lin, Y. Hu, Y. Qi, Y. Niu, and S. Gong, “Cavity-free circulator with low insertion loss using hot atoms,” Phys. Rev. Appl. 14(2), 024032 (2020).
[Crossref]

Phys. Rev. B (2)

T. Liu, G. C. Ma, S. J. Liang, H. Gao, Z. M. Gu, S. W. An, and J. Zhu, “Single-sided acoustic beam splitting based on parity-time symmetry,” Phys. Rev. B 102(1), 014306 (2020).
[Crossref]

S. Dey, A. Laha, and S. Ghosh, “Nonlinearity-induced anomalous mode collapse and nonchiral asymmetric mode switching around multiple exceptional points,” Phys. Rev. B 101(12), 125432 (2020).
[Crossref]

Phys. Rev. E (1)

C. Dembowski, B. Dietz, H. D. Graf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, “Encircling an exceptional point,” Phys. Rev. E 69(5), 056216 (2004).
[Crossref]

Phys. Rev. Lett. (15)

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113(12), 123004 (2014).
[Crossref]

J. Wiersig, “Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: Application to microcavity sensors for single-particle detection,” Phys. Rev. Lett. 112(20), 203901 (2014).
[Crossref]

H. Jing, S. K. Ozdemir, X. Y. Lu, J. Zhang, L. Yang, and F. Nori, “PT-symmetric phonon laser,” Phys. Rev. Lett. 113(5), 053604 (2014).
[Crossref]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[Crossref]

Z. P. Liu, J. Zhang, S. K. Ozdemir, B. Peng, H. Jing, X. Y. Lu, C. W. Li, L. Yang, F. Nori, and Y. X. Liu, “Metrology with PT-symmetric cavities: Enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117(11), 110802 (2016).
[Crossref]

F. Zhang, Y. Feng, X. Chen, L. Ge, and W. Wan, “Synthetic anti-PT symmetry in a single microcavity,” Phys. Rev. Lett. 124(5), 053901 (2020).
[Crossref]

B. He, L. Yang, X. Jiang, and M. Xiao, “Transmission nonreciprocity in a mutually coupled circulating structure,” Phys. Rev. Lett. 120(20), 203904 (2018).
[Crossref]

Q. Zhong, J. Ren, M. Khajavikhan, D. N. Christodoulides, S. K. Özdemir, and R. El-Ganainy, “Sensing with exceptional surfaces in order to combine sensitivity with robustness,” Phys. Rev. Lett. 122(15), 153902 (2019).
[Crossref]

M. Zhang, W. Sweeney, C. W. Hsu, L. Yang, A. D. Stone, and L. Jiang, “Quantum noise theory of exceptional point amplifying sensors,” Phys. Rev. Lett. 123(18), 180501 (2019).
[Crossref]

T. Wu, W. X. Zhang, H. Z. Zhang, S. S. Hou, G. Y. Chen, R. B. Liu, C. C. Lu, J. F. Li, R. Y. Wang, P. F. Duan, J. J. Li, B. Wang, L. Shi, J. Zi, and X. D. Zhang, “Vector exceptional points with strong superchiral fields,” Phys. Rev. Lett. 124(8), 083901 (2020).
[Crossref]

Y. Lumer, Y. Plotnik, M. C. Rechtsman, and M. Segev, “Nonlinearly induced PT transition in photonic systems,” Phys. Rev. Lett. 111(26), 263901 (2013).
[Crossref]

Y. M. Chu, Y. Liu, H. B. Liu, and J. M. Cai, “Quantum sensing with a single-qubit pseudo-Hermitian system,” Phys. Rev. Lett. 124(2), 020501 (2020).
[Crossref]

Z. Lin, A. Pick, M. Loncar, and A. W. Rodriguez, “Enhanced spontaneous emission at third-order Dirac exceptional points in inverse-designed photonic crystals,” Phys. Rev. Lett. 117(10), 107402 (2016).
[Crossref]

C. Liang, B. Liu, A.-N. Xu, X. Wen, C. Lu, K. Xia, M. K. Tey, Y.-C. Liu, and L. You, “Collision-induced broadband optical nonreciprocity,” Phys. Rev. Lett. 125(12), 123901 (2020).
[Crossref]

K. Xia, F. Nori, and M. Xiao, “Cavity-free optical isolators and circulators using a chiral cross-Kerr nonlinearity,” Phys. Rev. Lett. 121(20), 203602 (2018).
[Crossref]

Proc. Natl. Acad. Sci. U. S. A. (1)

B. Peng, S. K. Ozdemir, M. Liertzer, W. J. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Natl. Acad. Sci. U. S. A. 113(25), 6845–6850 (2016).
[Crossref]

Rep. Prog. Phys. (1)

C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70(6), 947–1018 (2007).
[Crossref]

Rev. Mod. Phys. (1)

M. B. Plenio and P. L. Knight, “The quantum-jump approach to dissipative dynamics in quantum optics,” Rev. Mod. Phys. 70(1), 101–144 (1998).
[Crossref]

Science (5)

B. Peng, S. K. Ozdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014).
[Crossref]

H. Hodaei, M. A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346(6212), 975–978 (2014).
[Crossref]

P. Miao, Z. F. Zhang, J. B. Sun, W. Walasik, S. Longhi, N. M. Litchinitser, and L. Feng, “Orbital angular momentum microlaser,” Science 353(6298), 464–467 (2016).
[Crossref]

L. Feng, Z. J. Wong, R. M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346(6212), 972–975 (2014).
[Crossref]

M. A. Miri and A. Alù, “Exceptional points in optics and photonics,” Science 363(6422), eaar7709 (2019).
[Crossref]

Other (2)

L. Tang, J. Tang, H. Wu, J. Zhang, M. Xiao, and K. Xia, “Broad-intensity-range optical nonreciprocity based on feedback-induced Kerr nonlinearity,” Photon. Res., in press (2021).

H. J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations (Springer, 1999).

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1.
Fig. 1. Schematics of coupled WGMRs. Solid (dashed) arrows represent the transmission of an input field impinging in port 1 (port 3). $\kappa _{c}$ is the coupling strength between two WGMRs. $\kappa _{c1}$ and $\kappa _{c2}$ are coupling losses of WGMR1 and WGMR2 that are introduced by lower and upper waveguides, respectively. $\gamma _{1}$ ( $\gamma _2$ ) represents the linear part of losses of WGMR1 (WGMR2). $\gamma _{N\!L}$ represents the nonlinear loss of WGMR1.
Fig. 2.
Fig. 2. Imaginary parts of two eigenfrequencies $\omega _{\pm }$ and intracavity photon number in WGMR1 : (a) $\omega ^\textrm {Im}_{\pm }$ as a function of the frequency and intensity of input fields impinging in port 1 ( $\kappa _{c1}=1.8$ , $\kappa _{c2}=0$ ); (b) $n_1$ as a function of the frequency and intensity of input fields impinging in port 1 ( $\kappa _{c1}=1.8$ , $\kappa _{c2}=0$ ); (c) $\omega ^\textrm {Im}_{\pm }$ as a function of the frequency and intensity of input fields impinging in port 3 ( $\kappa _{c1}=0$ , $\kappa _{c2}=1.8$ ); (d) $n_{1'}$ as a function of the frequency and intensity of input fields impinging in port 3 ( $\kappa _{c1}=0$ , $\kappa _{c2}=1.8$ ). Here $\kappa _c=1.0$ , $\gamma _1=\gamma _2=1.0$ , and $\gamma _{N\!L} =0.5$ .
Fig. 3.
Fig. 3. Imaginary parts $\omega ^\textrm {Im}_{\pm }$ of two eigenfrequencies $\omega _{\pm }$ . Left panel, $\omega ^\textrm {Im}_{\pm }$ as a function of the frequency of input fields impinging in port 1 ( $\kappa _{c1}=1.8$ , $\kappa _{c2}=0$ ) with different intensities; Right panel, $\omega ^\textrm {Im}_{\pm }$ as a function of the frequency of input fields impinging in port 3 ( $\kappa _{c1}=0$ , $\kappa _{c2}=1.8$ ) with different intensities. Here $\kappa _c=1.0$ , $\gamma _1=\gamma _2=1.0$ , and $\gamma _{N\!L} =0.5$ are used for both panels.
Fig. 4.
Fig. 4. Imaginary parts $\omega ^\textrm {Im}_{\pm }$ of two eigenfrequencies $\omega _{\pm }$ . (a) $\omega ^\textrm {Im}_{\pm }$ as a function of the coupling rate $\kappa _c$ and input field intensity for input fields impinging in port 1 ( $\kappa _{c1}=1.8$ , $\kappa _{c2}=0$ ); (b) $\omega ^\textrm {Im}_{\pm }$ as a function of the coupling rate $\kappa _c$ and input field intensity for input fields impinging in port 3 ( $\kappa _{c1}=0$ , $\kappa _{c2}=1.8$ ); (c and d) $\omega ^\textrm {Im}_{\pm }$ as a function of the coupling rate $\kappa _c$ for input fields impinging in port 1 (c) and port 3 (d). The input field intensities in (c) and (d) are tuned to near the critical values that lead to the emergence of an EP at $\kappa _c=1$ for $\Delta _0=0$ , respectively.
Fig. 5.
Fig. 5. Transmittance $T_{1\rightarrow 2}$ as a function of the frequency and intensity of input fields. Here $\kappa _c=1.0$ , $\kappa _{c1}=1.8$ , $\kappa _{c2}=0$ , $\gamma _1=\gamma _2=1.0$ , and $\gamma _{N\!L} =0.5$ .
Fig. 6.
Fig. 6. Transmittances as a function of the frequency and intensity of input fields: (a) $\Delta _T=|T_{3\rightarrow 1}-T_{3\rightarrow 4}|$ represents the transmittance difference when impinging the input field in the port $3$ ; (b) $T_{3\rightarrow 1}$ represents the transmittance into port $1$ when impinging the input field in the port $3$ ; (c) $T_{1\rightarrow 2}$ represents the transmittance into port $2$ when impinging the input field in the port $1$ ; (d) $T_{1\rightarrow 3}$ represents the transmittance into port $3$ when impinging the input field in the port $1$ . Here $\kappa _c=1.0$ , $\kappa _{c1}=\kappa _{c2}=2.0$ , $\gamma _1=\gamma _2=0.1$ , and $\gamma _{N\!L} =0.5$ .

Equations (22)

Equations on this page are rendered with MathJax. Learn more.

H ^ = [ Δ 1 i ( γ 1 + κ c 1 ) ] a ^ 1 a ^ 1 + [ Δ 2 i ( γ 2 + κ c 2 ) ] a ^ 2 a ^ 2 i γ N L ( a ^ 1 ) 2 a ^ 1 2 + κ c ( a ^ 1 a ^ 2 + a ^ 1 a ^ 2 ) + i 2 κ c 1 a i n ( a ^ 1 a ^ 1 ) ,
d a ^ 1 d t = i [ Δ 1 i ( γ 1 + κ c 1 ) ] a ^ 1 2 γ N L a ^ 1 a ^ 1 2 i κ c a ^ 2 2 κ c 1 a i n 2 κ c 1 c ^ 1 i n 2 γ 1 c ^ 2 i n 2 2 γ N L c ^ 3 i n ,
d a ^ 2 d t = i [ Δ 2 i ( γ 2 + κ c 2 ) ] a ^ 2 i κ c a ^ 1 2 κ c 2 c ^ 4 i n 2 γ 2 c ^ 5 i n .
d d t ( a 1 a 2 ) = i ( Δ 1 i ( γ 1 + κ c 1 + 2 γ N L n 1 ) κ c κ c Δ 2 i ( γ 2 + κ c 2 ) ) ( a 1 a 2 ) 2 κ c 1 ( a i n 0 ) = i M ( a 1 a 2 ) 2 κ c 1 ( a i n 0 ) .
n 1 = a ^ 1 a ^ 1 = | a 1 | 2 ,
ω ± = 1 2 [ Δ + i Γ + ± ( Δ i Γ ) 2 + 4 κ c 2 ] ,
Γ ± = ( γ 1 + κ c 1 + 2 γ N L n 1 ) ± ( γ 2 + κ c 2 ) ,
ω ± = Δ 0 i Γ + 4 κ c 2 Γ 2 2 .
a 1 = 2 κ c 1 [ i Δ 2 + ( γ 2 + κ c 2 ) ] a i n [ i Δ 1 + ( γ 1 + κ c 1 ) + 2 γ N L n 1 ] [ i Δ 2 + ( γ 2 + κ c 2 ) ] + κ c 2 , a 2 = i κ c 2 κ c 1 a i n [ i Δ 1 + ( γ 1 + κ c 1 ) + 2 γ N L n 1 ] [ i Δ 2 + ( γ 2 + κ c 2 ) ] + κ c 2 .
T 1 2 = | 1 2 κ c 1 [ i Δ 2 + ( γ 2 + κ c 2 ) ] [ i Δ 1 + ( γ 1 + κ c 1 ) + 2 γ N L n 1 ] [ i Δ 2 + ( γ 2 + κ c 2 ) ] + κ c 2 | 2 ,
T 1 3 = | 2 κ c κ c 1 κ c 2 [ i Δ 1 + ( γ 1 + κ c 1 ) + 2 γ N L n 1 ] [ i Δ 2 + ( γ 2 + κ c 2 ) ] + κ c 2 | 2 .
H ^ = [ Δ 1 i ( γ 1 + κ c 1 ) ] a ^ 1 a ^ 1 + [ Δ 2 i ( γ 2 + κ c 2 ) ] a ^ 2 a ^ 2 i γ N L ( a ^ 1 ) 2 a ^ 1 2 + κ c ( a ^ 1 a ^ 2 + a ^ 1 a ^ 2 ) + i 2 κ c 2 a i n ( a ^ 2 a ^ 2 ) .
d d t ( a 1 a 2 ) = i ( Δ 1 i ( γ 1 + κ c 1 + 2 γ N L n 1 ) κ c κ c Δ 2 i ( γ 2 + κ c 2 ) ) ( a 1 a 2 ) 2 κ c 2 ( 0 a i n ) .
n 1 = a ^ 1 a ^ 1 = | a 1 | 2 .
a 1 = i κ c 2 κ c 2 a i n [ i Δ 1 + ( γ 1 + κ c 1 ) + 2 γ N L n 1 ] [ i Δ 2 + ( γ 2 + κ c 2 ) ] + κ c 2 ,
a 2 = 2 κ c 2 [ i Δ 1 + ( γ 1 + κ c 1 + 2 γ N L n 1 ) ] a i n [ i Δ 1 + ( γ 1 + κ c 1 ) + 2 γ N L n 1 ] [ i Δ 2 + ( γ 2 + κ c 2 ) ] + κ c 2 .
T 3 4 = | 1 2 κ c 2 [ i Δ 1 + ( γ 1 + κ c 1 + 2 γ N L n 1 ) ] [ i Δ 1 + ( γ 1 + κ c 1 ) + 2 γ N L n 1 ] [ i Δ 2 + ( γ 2 + κ c 2 ) ] + κ c 2 | 2 ,
T 3 1 = | 2 κ c κ c 1 κ c 2 [ i Δ 1 + ( γ 1 + κ c 1 ) + 2 γ N L n 1 ] [ i Δ 2 + ( γ 2 + κ c 2 ) ] + κ c 2 | 2 ,
ω ± = Δ 0 i ( κ c 1 + 2 γ 0 + 2 γ N L n 1 ) 4 κ c 2 ( κ c 1 + 2 γ N L n 1 ) 2 2 ,
n 1 = T 1 2 2 T 1 2 cos θ + 1 2 κ c 1 p i n ,
T 1 2 = | 1 2 κ c 1 ( i Δ 0 + γ 0 ) ( i Δ 0 + γ 0 + κ c 1 + 2 γ N L n 1 ) ( i Δ 0 + γ 0 ) + κ c 2 | 2 .
n 1 = ( Δ 0 2 κ c 2 γ 0 2 ) T + κ c 1 γ 0 T + ( κ c 1 + 2 γ 0 ) Δ 0 T 1 2 sin θ 2 κ N L ( γ 0 T + Δ 0 T 1 2 sin θ ) ,

Metrics