Robust Sensing Based on Exceptional Points in Detuned Non-Hermitian Scattering System

Abstract

Non-Hermitian optics has revealed a series of counterintuitive phenomena with profound implications for sensing, lasing, and light manipulation. While the non-Hermiticity of Hamiltonians has been intensively investigated, recent advancements in the non-Hermitian scattering matrix have given birth to a lot of unique phenomena, such as simultaneous lasing and anti-lasing, reflectionless scattering modes (RSMs), and coherent chaos control. Despite these developments, the investigation has predominantly focused on static and symmetric configurations, leaving the dynamic properties of non-Hermitian scattering in detuned systems, which is essential for applications in sensing and beyond, largely unexplored. Here, we extend the stationary behaviors associated with the RSMs to resonant detuned systems. Contrary to the common belief of exceptional point (EP) sensors as being susceptible to parametric disturbances, we induce an RSM EP in a one-dimensional optical cavity and demonstrate its robustness in displacement sensing against laser frequency drifts up to 10 MHz. Our findings not only contribute to the broader understanding of non-Hermitian scattering phenomena but also pave the way for the next generation of non-Hermitian sensors.

1. Introduction

Owing to the abundant and controllable non-conservative processes in optical systems, non-Hermitian optics has attracted tremendous interest and inspired a host of exotic phenomena [1,2,3,4,5], such as loss-induced transparency [6], laser reviving [7], laser absorbers [8], and unconventional topologies [9]. Beyond the extensively explored physics in non-Hermitian Hamiltonians [3], a class of novel concepts associated with the non-Hermitian scattering matrix, e.g., reflectionless scattering modes (RSMs) [10], transmission peak degeneracies [11], exceptional points (EPs) of scattering matrix zeros [12,13], and coherent perfect absorption states [14], have enabled unique applications such as chiral absorption [12], the suppression of backscattering [15], and coherent control [16]. However, discussions and experiments focused on these topics have primarily been confined to static and geometrically symmetric coupled systems without detuning.

Detuning denotes the resonant mismatch of coupled systems, which is one of the intrinsical sensing targets, particularly in recently emerging non-Hermitian sensors. We emphasize that the detuning used in this papaer refers to the mismatch between the coupled cavities, which is different from the common definition of the frequency deviation in isolated systems or the shift in resonance conditions due to external perturbations. For instance, resonant detuning could come from the displacement-induced assymetric cavity lengths [17], index variation led by temperature drift [18] or the rotation of gyroscopes [19]. The detuning response has been verified to be significantly enhanced by EPs [20]. However, these EP sensors are limited by their susceptibility to fabrication errors and structural instability. Therefore, new strategies for robust EP sensing, such as considering exceptional surfaces [21,22], is quite desired for the practical application of non-Hermitian sensors.

In this work, we first compared two similar and easily confused concepts, bidirectional RSM and RSM EP, in a system without detuning. Then, we discussed their behavior and applications under dynamic detuning conditions. By introducing detuning between two coupled entities, we further explore the generalized RSM existing conditions, the symmetric detuning response and the square-shaped spectrum. For practical applications, we induce an RSM EP within a one-dimensional non-Hermitian coupled cavity, which we propose could significantly enhance the detuning sensing robustness. This protocol ensures operational stability amidst probing laser frequency drifts exceeding 10 MHz, surpassing the traditional cavity-based sensors which require precise frequency stablization equipments.

2. Non-Hermitian Scattering Matrix

A typical binary non-Hermitian system consists of two coupled elements with unbalanced gain or loss, such as the evanescently coupled whispering gallery mode (WGM) cavity systems (Figure 1a) or the directly coupled Fabry–Pérot (FP) cavity system (Figure 1b). According to temporal coupled-mode theory (TCMT), the effective Hamiltonian of a binary coupled system can be written as shown below.

(1)$H=\left(\begin{array}{cc}{\omega }_1-i{\mathrm{\Gamma }}_1/\phantom{i{\mathrm{\Gamma }}_{1}2}2& \kappa \\ \kappa & {\omega }_2-i{\mathrm{\Gamma }}_2/\phantom{i{\mathrm{\Gamma }}_{2}2}2,\end{array}\right),$

where ${\omega }_{1,2}$ and ${\mathrm{\Gamma }}_{1,2}$ denote the resonant angular frequencies and total loss rates (including intrinsic losses ${\gamma }_{1,2}$ and radiative losses ${\gamma }_{c1,c2}$) of the two subcavities. According to the Lipmann–Schwinger equation [14,23], the scattering matrix of the system can be derived by

(2)$S=1-i{D}^{†}{(\omega -H)}^{-1}D.$

Here, $\omega$ represents the frequency of the excitation source, and D is the diagonal coupling matrix of the system, which is expressed as follows:

(3)$D=\left(\begin{array}{cc}\sqrt{{\gamma }_{c1}}& 0\\ 0& \sqrt{{\gamma }_{c2}}\end{array}\right).$

Therefore, we obtain the $2\times 2$ scattering matrix:

(4)$S\left(\omega \right)=\left(\begin{array}{cc}{r}_1& t\\ t& r_2\end{array}\right)=\left(\begin{array}{cc}1-i{\displaystyle \frac{{\gamma }_{c1}{\mathrm{\Delta }}_2}{{\mathrm{\Delta }}_1{\mathrm{\Delta }}_2-{\kappa }^2}}& -i{\displaystyle \frac{\kappa \sqrt{{\gamma }_{c1}{\gamma }_{c2}}}{{\mathrm{\Delta }}_1{\mathrm{\Delta }}_2-{\kappa }^2}}\\ -i{\displaystyle \frac{\kappa \sqrt{{\gamma }_{c1}{\gamma }_{c2}}}{{\mathrm{\Delta }}_1{\mathrm{\Delta }}_2-{\kappa }^2}}& 1-i{\displaystyle \frac{{\gamma }_{c2}{\mathrm{\Delta }}_1}{{\mathrm{\Delta }}_1{\mathrm{\Delta }}_2-{\kappa }^2}}\end{array}\right),$

where ${\mathrm{\Delta }}_{1,2}={\delta }_{1,2}+i{\mathrm{\Gamma }}_{1,2}/2$ with ${\delta }_{1,2}=\omega -{\omega }_{1,2}$. The difference between ${\omega }_1$ and ${\omega }_2$ is the detuning, which quantifies the extent of resonance mismatch between two sub-cavities. In order to avoid the errors caused by the TCMT approximation, we also adopt the rigorous scattering matrix method to verify the results of the analysis (see Appendix A).

3. Non-Detuned Case

We start from the RSM properties in non-detuned coupled systems. The RSM is used to describe the zero-reflection phenomenon, which is a desirable target as it could avoid unwanted signal echoes in photonic and microwave networks, facilitating secure information transmission and analog computation [24,25]. RSMs has been observed in various platforms [16,25,26,27,28], revealing surprising applications such as thermal mapping [28] and programmable signal routers [24]. RSMs call for purely real reflection zero frequency ${\delta }_{\mathrm{R}0}$. According to Equation (4), for a lossy cavity without detuning (${\gamma }_{1,2,c1,c2}>0$ and ${\delta }_1={\delta }_2=\delta$), RSMs could exist in the following two situations.

The first situation is only a single RSM at $\delta =0$, corresponding to the behavior of ${\delta }_{\mathrm{R}0}$ for port 1 in Figure 1c, which requires

(5)${\kappa }_{\mathrm{RSM},2}=\sqrt{\left({\gamma }_{c2}-{\gamma }_2\right){\mathrm{\Gamma }}_1}/2,{\gamma }_{c2}>{\gamma }_2.$

Similarly, we have ${\kappa }_{\mathrm{RSM},1}=\sqrt{\left({\gamma }_{c1}-{\gamma }_1\right){\mathrm{\Gamma }}_2}/2$ for the port-2 incidence RSM.

The second situation is just the critical coupling scenario (${\gamma }_{c1}={\gamma }_1+{\mathrm{\Gamma }}_2$), at which a pair of RSMs coexist and they coalesce to an RSM EP [Figure 1c, port1] at $\delta =0$ when $\kappa ={\mathrm{\Gamma }}_2/2$ [10]. Both approaches to reaching the RSM exhibit EP bifurcation characteristics from a three-dimensional perspective, as illustrated in Figure 1d, in which the evolution of the real and imaginary parts of ${\delta }_{\mathrm{R}0}$ with $\kappa$ is demonstrated.

Spectrally, these two types of RSM exhibit entirely different characteristics. The first type of RSM, with reflection ${\mathrm{R}}_{\mathrm{L}}$ from left incidence, maintains a standard Lorentzian lineshape. In contrast, at the RSM EP of the second type, the reflection spectrum ${\mathrm{R}}_{\mathrm{R}}$ from the right incidence becomes quartic, as shown in Figure 1e and the enlarged view in Figure 1f. Furthermore, the energy distribution of the optical fields within the two subcavities varies significantly with the direction of incidence, as depicted in Figure 1g,h. It should be noticed that this quartic lineshape does not refer to the squared-Lorentzian spectrum [29,30].

Some previous works also attributed a unidirectional RSM to an EP of a permuted scattering matrix [26,28]. However, here, we clarify that the non-Hermitian RSMs are normally unidirectional except for the unique bidirectional RSM situation $r_1=r_2=0$ which yields ${\kappa }_{\mathrm{RSM},1}={\kappa }_{\mathrm{RSM},2}$; then, it follows that

(6)${\displaystyle \frac{{\gamma }_1}{{\gamma }_2}}={\displaystyle \frac{{\gamma }_{c1}}{{\gamma }_{c2}}}=\eta ,{\gamma }_2,{\gamma }_{c2}\ne 0,$

where $\eta$ denotes the total loss asymmetry of the two subcavities. We term Equation (6) as the loss proportional condition which determines the symmetric reflectivity for two ports. This condition ensures the R-zeros for both ports simultaneously drop to the real plane (Figure 1c), where both the ports become reflectionless at $\kappa =\sqrt{{\gamma }_{c1}{\gamma }_{c2}-{\gamma }_1{\gamma }_2}/2$.

Moreover, if the system has no inner loss (${\gamma }_1={\gamma }_2=0$, which means there is no energy conversion during the scattering processes) and the coupling losses are symmetric (${\gamma }_{c1}={\gamma }_{c2}$), the critical coupling and proportional losses coincide, and thus the bidirectional RSM and RSM EP occur simultaneously (Figure 1c,d).

4. Detuned Case

Now, we turn to the situation of ${\delta }_1\ne {\delta }_2$. For simplicity, we take an antisymmetric detuning ${\omega }_2-{\omega }_0={\omega }_0-{\omega }_1=\mathrm{\Delta }$ which naturally holds in the coupled FP cavities, and it is intimately related to the central mirror displacements d, as shown in the upper panel of Figure 2a. Considering the reflection at port 1 as an example, the bottom panel of Figure 2a maps the $\kappa -{\gamma }_{c1}$ parameter space to delineate the occurrence and the disappearance of the RSMs when $\mathrm{\Delta }$ is involved. In the regime where ${\gamma }_{c1}<{\gamma }_1$, the incident cavity is undercoupled, forbidding the existence of RSMs for any values of $\delta$ and $\kappa$ (see Appendix B). For ${\gamma }_{c1}>{\gamma }_1$, RSMs manifest in the blue region where $\kappa \ge {\kappa }_{\mathrm{RSM}}$, but they vanish in the orange region where $\kappa <{\kappa }_{\mathrm{RSM}}$.

Specifically, at an RSM EP (Figure 2b, point 1), two R-zeros, initially coalesced at the origin, diverge from the real axis with $\mathrm{\Delta }$, leading to the disappearance of RSMs. Similarly, point 2 shown in Figure 2c represents a scenario where critical coupling is maintained, resulting in the simultaneous vanishing of two separated RSMs. However, at a more general position such as point 3, although there was no RSM at the beginning, detuning can lead an R-zero to the real axis, giving rise to an new RSM at $\delta \ne 0$, as shown in Figure 2d. This means that in the detuned situation, the critical coupling ceases to be a necessary condition for the existence of RSMs at non-zero $\delta$. Finally, for point 4, Figure 2e elucidates that two RSMs originally on opposite sides of the real axis cannot be brought to it through detuning, precluding the emergence of new RSMs.

It can be observed that the four quadrants of the parameter space have different impacts on the detuning response of R-zero and the reflection spectrum. However, it is important to note that their near-symmetry about the central point of an RSM EP (e.g., points 3 and 4 are nearly symmetric to point 1) does not manifest as symmetry in the reflection spectra or their R-zero.

In addition to inducing new RSMs, we find that detuning can also expand the quartic spectrum at the exact RSM EP into a square-shaped reflection spectrum map, as shown in Figure 3a. It should be noted that this does not always happen, as it requires the loss of the incident cavity to be close to 0 (see Appendix C). As shown in Figure 3b, while ensuring the critical coupling condition for port 1 and merging of two RSMs before the detuning, a slight introduction of ${\gamma }_1$ will destroy the symmetry of the map. Figure 3c,d show the reflection spectra before and after the detuning in two scenarios. Before the detuning, both are symmetrical. After the detuning, only the former retains its symmetry, while the latter loses it.

5. Robust RSM EP Sensing

Metrological applications utilizing a single cavity critically depend on a movable mirror to transduce displacement and vibration into detectable variations in the light wave phase or intensity [31]. Nonetheless, the constancy of probing laser frequency, indispensable for high-precision measurements, is compromised by mechanical instabilities and thermal perturbations. To counteract these fluctuations, complex frequency stabilization mechanisms, including injection locking and active feedback loops [32], are mandated.

Now, we introduce a method that can get rid of these supplementary stabilization apparatus, by leveraging the detuning response of an RSM EP (Figure 1b). The detuning is originated from the asymmetric subcavity lengths, which could be directly led by the displacements or vibrations by the central mirror movements. Classically, it could serve as an effective platform to achieve quantum Fock state readout [33]. For simplicity, we take the case of ${\gamma }_1={\gamma }_2=0$ and ${\gamma }_{c1}={\gamma }_{c2}$, where the transmission spectrum shares the same quartic properties to expound our robust metrology protocol.

Figure 4a illustrates the relationship among the transmission, incident frequency and central mirror displacement, for a linear single cavity widely applied in fundamental detection devices such as displacement and acceleration detection. Typically, the detection approach involves irradiating the cavity with a narrow-band laser near one peak, such as at point A, whereupon displacement causes a reduction in light intensity to point B. Points A and A′ (as well as points B and B′) denote the frequency fluctuations $\omega$, which can be attributed to the inherent instability of the laser source, described by the application of a random perturbing function. Since both sets of points experience the same displacement perturbation at their respective jumping frequencies, these jumps are linearly reflected in the signal for linear cavities, rendering the displacement readout severely unstable.

In contrast, employing the RSM EP, as depicted in Figure 4b, the intensity at points A and B is almost indistinguishable from that at A′ and B′. This ensures a stable light intensity response to displacement within a range of frequency drift. For visualization, applying a sinusoidal vibration signal $d\left(t\right)\propto \sin \mathrm{c}\left(t\right)$ to the central mirror, both detection methods can replicate it through variations in light intensity $T\left(t\right)$ yet exhibit marked differences under varying degrees of frequency disturbance, as shown in Figure 4c–e for ${\sigma }_f$ equal to 1 MHz, 5 MHz, and 10 MHz. It is evident that with increasing frequency perturbation, the response of the Hermitian cavity quickly becomes overwhelmed by noise, whereas the non-Hermitian cavity demonstrates robust resistance to frequency noise. These results highlight the self-stabilizing capability of RSM EP-based displacement or vibration detection, potentially reducing the footprint required for frequency stabilization equipment.

6. Discussion

It should be noted that while the theoretical assumption of ${\gamma }_1={\gamma }_2=0$ is employed to demonstrate ideal symmetry of the system, in practical systems, a small inherent loss, such as ${\gamma }_1={\gamma }_2=0.01$, is feasible and maintains the essential symmetrical properties of the transmission spectra near the RSM EP. Morevover, Figure 4c–e show that the transmission response amplitude for both the linear cavity and the RSM EP falls within the range of 0.1–0.2 for the same displacement signal. This indicates that the RSM EP maintains comparable sensitivity to the linear cavity while enhancing sensing robustness. The underlying reason for this robustness is the separation of the eigenfrequency and spectral peak (or dip) frequency due to the introduction of non-Hermiticity. Nonetheless, efforts to further enhance sensitivity near the RSM EP would narrow the robust region, reflecting the inherent trade-off of robustness and sensitivity.

7. Conclusions

This study expands the scope of non-Hermitian scattering explorations to include detuning (resonant mismatch). By properly engineering the loss and coupling, we investigated several unique non-Hermitian scattering phenomena both with and without detuning. We clarified that detuning could give rise to RSMs in static scenarios where RSMs were previously absent. Near an RSM EP, the detuning response can be designed to be symmetric, resulting in a square-shaped spectrum map. To demonstrate the potential of a detuned non-Hermitian scattering system, we introduced an application of enhancing the robustness of displacement sensing against environmental instabilities. This capability represents a substantial improvement over previous fragile non-Hermitian sensors and even the traditional linear sensing methods. We believe that the present work fills the theoretical gap in non-Hermitian scattering investigations and establishes a foundation for creating advanced and reliable non-Hermitian optical sensors.

Author Contributions

Conceptualization, J.X.; methodology, J.X.; software, J.X.; validation, J.X., Y.M.; formal analysis, K.L.; investigation, J.X.; writing—original draft preparation, J.X.; writing—review and editing, Y.M.; visualization, J.X.; supervision, Z.Z.; funding acquisition, K.L. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Dataset available on request from the authors.

Acknowledgments

We thank Shi Chen, Xingqiao Chen and Hanyu Wang for discussions.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

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Figures

Figure 1. RSM EP and bidirectional RSMs in non-detuned coupled systems. Typical platforms such as (a) coupled WGM microcavities and (b) coupled FP cavities. (c) Evolution of the R-zeros with [Forumla omitted. See PDF.] on the complex plane. The dark red sphere represents the simultaneous appearance of an RSM EP and a bidirectional RSM. (d) Three-dimensional evolution of R-zeros with [Forumla omitted. See PDF.] upon illumination from the two ports under the condition of loss proportional. The bidirectional RSM occurs at [Forumla omitted. See PDF.], which is highlighted by an orange sphere. (e) Corresponding reflection spectrum of c. A bidirectional zero reflection emerges, and the zoom-in panel (f) shows an RSM EP of the left incidence and is featured by a flat-bottom lineshape. (g,h) gives the optical field distribution within the two subcavities under left and right incidence, respectively.

Figure 2. RSM distribution on the parametric space with and without detuning. (a) Top: coupled FP cavities with antisymmetric detuning induced by the central mirror displacements. Illustration of [Forumla omitted. See PDF.] 2-dimensional parameter space. The blue region allows the existence of RSMs when detuning is involved. The gray shadowed region and the orange region forbid any RSMs. (b–e) Top: the motion of R-zeros with detuning for points 1–4 in (a). Bottom: reflection spectra before (blue lines) and after (gray lines) detuning for points 1–4. The number of the RSMs is labeled in each panel, and a detuning-induced RSM is marked in (d).

Figure 3. Dependence of the port-1 reflectivity on displacement and incident frequency near an RSM EP when (a) [Forumla omitted. See PDF.] and (b) [Forumla omitted. See PDF.]. The square-shaped spectrum map given in (a) corresponds to a symmetric reflection spectrum (c) before and after detuning. As a comparison, when [Forumla omitted. See PDF.], the reflection spectrum (d) loses symmetry after detuning.

Figure 4. Comparison of displacement sensing via linear and square-shaped spectra. The three-dimensional plots illustrate the dependence of transmission on displacement and incident frequency for (a) a single FP cavity and (b) a non-Hermitian coupled cavity, respectively. Left insets depict the corresponding schematic. The right inset of (a) gives a sinusoidal vibration signal. In (a,b), points A and B represent the light intensities at two displacement positions under unperturbed frequency conditions, while A′ and B′ denote the responses at an alternate frequency position following perturbation. Panels (c–e) display the detuning responses of both systems to an input displacement signal subject to random frequency disturbances of magnitudes (c) 1 MHz, (d) 5 MHz, and (e) 10 MHz, respectively.

Appendix A. Numerical Model of the One-Dimensional Cavity

In the coupled FP cavity, the phase accumulation during a round trip in the left or right subcavity is given by the following:(A1)$$\begin{array}{c}{\phi }_l(\omega ,d)=exp[2i(\omega +i{\gamma }_1)(l+d)/c]\hfill \\ {\phi }_r(\omega ,d)=exp[2i(\omega +i{\gamma }_2)(l-d)/c]\hfill \end{array}$$ where $\omega$, d, l and c denote the angular frequency of the incident wave, displacement of the central mirror, half of the whole cavity length, and vacuum speed of light (assuming that the real part of the index inside the cavities is 1). Treating the middle and right mirrors as a whole, its effective reflection and transmission coefficients could be derived as (A2)$$\begin{array}{c}{r}_{\mathrm{eff}}=r_c+{\displaystyle \frac{{\displaystyle r_r{t_c}^2{\phi }_r}}{{\displaystyle 1-r_r{r}_c{\phi }_r}}}\hfill \\ \\ t_{\mathrm{eff}}={\displaystyle \frac{{\displaystyle t_r{t}_c{\phi }_r}}{{\displaystyle 1-r_r{r}_c{\phi }_r}}}\hfill \end{array}$$

Then, the total reflection and transmission spectra under left incident waves take the following form:(A3)$$\begin{array}{c}{r}_1=r_l+{\displaystyle \frac{{\displaystyle r_{\mathrm{eff}}{t_l}^2{\phi }_l}}{{\displaystyle 1-r_l{r}_{\mathrm{eff}}{\phi }_l}}},R_1={\left|r_1\right|}^2\hfill \\ \\ t_1={\displaystyle \frac{{\displaystyle t_{\mathrm{eff}}{t}_l{\phi }_l}}{{\displaystyle 1-r_l{r}_{\mathrm{eff}}{\phi }_l}}},T_1={\left|t_L\right|}^2\hfill \end{array}$$

This direct numerical model could help us check the efficiency of TCMT for the one-dimensional cavity.

Appendix B. RSM Existence Conditions with Cavity Detuning

Figure 2a illustrates where the RSMs are allowed in the parametric space when detuning is involved. Here, we analyze the number of RSMs with different parameters. First, we analyze the reflection zeros for the port-1 incidence $r_1=0$:(A4)$$\begin{array}{cc}(\delta -\mathrm{\Delta })(\delta +\mathrm{\Delta })+i{\displaystyle \frac{{\mathrm{\Gamma }}_2}{2}}(\delta -\mathrm{\Delta })+i({\displaystyle \frac{{\mathrm{\Gamma }}_1}{2}}-{\gamma }_{c1})(\delta +\mathrm{\Delta })\hfill & \\ & \hfill -{\displaystyle \frac{1}{4}}{\mathrm{\Gamma }}_1{\mathrm{\Gamma }}_2-{\kappa }^2+{\displaystyle \frac{{\mathrm{\Gamma }}_2}{2}}{\gamma }_{c1}=0\end{array}$$ The reflection zeros are (A5)$$\begin{array}{cc}{\delta }_{\mathrm{R}0}=-{\displaystyle \frac{1}{4}}i({\gamma }_1-{\gamma }_{c1}+{\mathrm{\Gamma }}_2)\hfill & \pm {\displaystyle \frac{1}{2}}[-{\displaystyle \frac{1}{4}}{({\gamma }_1-{\gamma }_{c1}+{\mathrm{\Gamma }}_2)}^2\hfill \\ \hfill & +({\gamma }_1-{\gamma }_{c1}){\mathrm{\Gamma }}_2+4{\kappa }^2+4{\mathrm{\Delta }}^2-2i({\gamma }_1-{\gamma }_{c1}-{\mathrm{\Gamma }}_2){\mathrm{\Delta }]}^{-1/2}\hfill \end{array}$$ RSMs call for purely real ${\delta }_{\mathrm{R}0}$, which requires that (A6)$${{\mathrm{\Delta }}_{\mathrm{RSM}}}^2={\displaystyle \frac{{({\gamma }_1-{\gamma }_{c1}+{\mathrm{\Gamma }}_2)}^2({\kappa }^2-{{\kappa }_{\mathrm{RSM},1}}^2)}{4{\mathrm{\Gamma }}_2({\gamma }_{c1}-{\gamma }_1)}}$$ All terms in the equation are real numbers; therefore, in order for the equation to have a solution, it necessitates (A7)$${\gamma }_{c1}>{\gamma }_1,\kappa \ge {\kappa }_1$$ The first condition of Equation (A7) is the overcoupling requirement inherited from the non-detuned situation. When the second conditon is further satisfied, it is possible to reproduce one or more RSMs by inducing detuning at (as marked in Figure 2d):(A8)$${\delta }_{\mathrm{RSM}}={\displaystyle \frac{2{\mathrm{\Delta }}_{\mathrm{RSM}}({\mathrm{\Gamma }}_2+{\gamma }_{c1}-{\gamma }_1)}{{\mathrm{\Gamma }}_2-({\gamma }_{c1}-{\gamma }_1)}}$$ Conversely, for $\kappa <{\kappa }_1$, RSM is no longer achievable. This result indicates that there is only one RSM for a given $\mathrm{\Delta }$ except for the critical coupling scenario (2 RSMs at $\mathrm{\Delta }=0$).

Appendix C. Derivation of the Detuning Response Symmetry Condition

According to Equation (4), we derive the reflection detuning symmetry condition by directly solving $R_1=R_2$, which yields (A9)$${\left|1-i{\displaystyle \frac{{\gamma }_{c1}{\mathrm{\Delta }}_2}{{\mathrm{\Delta }}_1{\mathrm{\Delta }}_2-{\kappa }^2}}\right|}^2={\left|1-i{\displaystyle \frac{{\gamma }_{c1}({\mathrm{\Delta }}_2-2\mathrm{\Delta })}{(-2\mathrm{\Delta }-{\mathrm{\Delta }}_1)(2\mathrm{\Delta }-{\mathrm{\Delta }}_2)-{\kappa }^2}}\right|}^2$$ It requires that (A10)$${\gamma }_{c1}={\gamma }_2+{\gamma }_{c2},{\gamma }_1=0.$$ This result implies that to obtain the symmetric reflection maps shown in Figure 3a, the critical coupling condition of the whole cavity should be satisfied, and the incident cavity must be lossless.

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