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 [5–13]. 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 [14–18], phonon laser [19], unidirectional invisibility [20–22], and breakdown of adiabaticity [23–25].

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 [39–43], to enhance sensing [44–56], and to manipulate the modal content of multimode lasers [57–65].

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 [72–74]. 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 $\frac {d\hat {\rho }}{dt} = \gamma _{N\!L}(2\hat {a}_{1}^{2}\hat {\rho }\hat {a}_{1}^{\dagger } -\hat {a}_{1}^{\dagger }\hat {a}_{1}^{2}\hat {\rho }-\hat {\rho }\hat {a}_{1}^{\dagger }\hat {a}_{1}^{2})$ [77–79], where $\hat {\rho }$ is the corresponding density operator.

 

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.

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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]

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

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:

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

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:

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

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:

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

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

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

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

The corresponding transmittances are

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

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 $\gamma _{N\!L} =0.5$.

 

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 $\gamma _{N\!L} =0.5$.

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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 $\sqrt {p_{in}}=1.0$.

 

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.

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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.

 

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.

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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]

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

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.

 

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 $\gamma _{N\!L} =0.5$.

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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 [81–88].

 

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 $\gamma _{N\!L} =0.5$.

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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 [90–93].

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 [14–25]. 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.