1. Introduction

Waveguide quantum electrodynamics (QED), where emitters are coupled to a continuum of traveling photons confined in one-dimensional open waveguides, provides a promising platform for enhancing light-matter interactions and mediating interactions between different emitters [1,2]. This field has been sufficiently explored with various candidates such as superconducting circuits [3–5], optical waveguides [6,7], and coupled-resonator (coupled-atom) arrays [8–10]. Up to now, a series of novel phenomena such as chiral photon-atom interactions [11,12], phase transitions [13,14], photonic Anderson localization [15,16], deterministic photon routing [17,18], and single-photon nonreciprocity [19–21] have been achieved based on various techniques and engineered configurations. In particular, waveguide QED is theoretically predicted [22–26] and experimentally verified [27–29] to enable indirect couplings between spatially separated emitters, which have potential applications in large-scale quantum network.

On the other hand, cavity magnonics based on photon-magnon hybridization provides an excellent solid platform for quantum information processing [30–35]. Over the past few years, it has attracted much attention due to the ability to achieve strong couplings in microwave regime and has incited a lot of breakthroughs, for example, gradient memory [36], logic gate [37], magnon blockade [38], and photon-magnon-photon coupling (cavity magnomechanics) [39], just to name a few. Recently, the idea of waveguide QED is introduced to cavity magnonics to achieve level attraction [40], giant nonreciprocity [41], and unconventional singularity [42]. Along this line, an indirect coupling scheme for remote cavity and magnon modes is proposed [43], where the effective coupling can be purely dissipative by tuning the separation distance between them.

In this paper, we begin with revisiting the well-studied model where two separated modes are indirectly coupled via a common waveguide and then generalize it to a single-resonator model. This model is formed by coupling a resonator (microcavity) with a bent waveguide at two different ports such that photons can travel via either external waveguide or intra-resonator path from one port to another, leading to the self interference effect. Such a structure is reminiscent of the "giant atoms", where the atomic size is much larger than the wavelength of the propagating mode [44–50]. Compared with atomic setups, microcavities seem to be a more suitable candidate for realizing quantum hybrid systems and for observing quantum effects at the macroscopic scale. Moreover, the case of traveling-wave cavities may exhibit distinct optical properties compared with that of standing-wave ones (or that of atoms) due to the directionality of the cavity modes. This inspires our research in this paper. It shows that the phase factor induced by the separation between the two ports plays a key role which modifies the resonance frequency, linewidth, and the input term simultaneously. In particular, an optical dark state can be tailored judiciously with which the resonator is effectively decoupled from the waveguide. Moreover, we propose a sensing scheme based on a photon-magnon hybrid model where the self-interference resonator mentioned above is coupled with a ferromagnetic material via the magnetic dipole-dipole interaction. Although self interference has been used for both dispersive and dissipative sensing methods in bare-resonator systems (only an empty resonator is coupled to the waveguide) [51–55], the hybrid model here shows a series of advantages. On one hand, the sensitivity can be markedly improved due to the hybridization induced sharp Fano-like line shapes. The sensing performance can be further optimized by tuning the phase factor and the optimal working region can be changed due to the tunable resonance frequency of the magnon mode. On the other hand, the hybridization in turn enables sensing for the strength of the magnetic field, implying that our scheme can be used as a high-performance magnetometer [56–62].

2. Model and equations

We first briefly revisit a general model in which indirect couplings between remote modes can be achieved. As shown in Fig. 1(a), two spatially separated standing-wave resonators $a$ and $b$ are side-coupled to a common waveguide. The separation distance $L$ is much larger than the wavelengths of the intra-resonator fields, thus there is no direct coupling between the two resonators due to the absence of modal overlap. On the other hand, $L$ is assumed to be much smaller than the coherence length of photons in the waveguide to avoid obvious non-Markovian retarded effects [63–67], which is reasonable for the case of few resonators [68]. It has been shown that the effective dynamic equations of $a$ and $b$ can be written as [22–26]

 

Fig. 1. Schematic diagrams of (a) two separated standing-wave resonators side-coupled with a straight waveguide and (b) a single standing-wave resonator side-coupled with a bent waveguide at two separated ports.

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Here $\omega _{j}$ is the resonance frequency of resonator $j$ ($j=a,b$). $\kappa _{j}=\kappa _{j,i}+\kappa _{j,e}$ is the total loss of resonator $j$, with $\kappa _{j,i}$ the intrinsic loss due to the structure imperfection and $\kappa _{j,e}$ the external loss due to the coupling with the waveguide. Note that the external losses have been doubled here because both resonators are assumed to be two-sided with mirror symmetry [69]. Experimentally, the external loss rate is determined by the resonator-waveguide coupling strength, i.e., $\kappa _{j,e}=2\pi g_{j}^{2}$ with $g_{j}$ the (real) coupling strength between resonator $j$ and the waveguide [25,26,64]. $\phi$ is the phase accumulated by photons traveling in the waveguide from one resonator to another, determined by both the separation distance $L$ and the wave vector of traveling photons in the waveguide [22–26,64–67]. As shown in Fig. 1(a), $a_{\textrm {in}}$ and $a_{\textrm {in}}'$ ($b_{\textrm {in}}$ and $b_{\textrm {in}}'$) are the input fields coming from the left and right sides of resonator $a$ ($b$), respectively. Similarly, we define $a_{\textrm {out}}$ and $a_{\textrm {out}}'$ ($b_{\textrm {out}}$ and $b_{\textrm {out}}'$) as respectively the output fields leaving from the right and left sides of resonator $a$ ($b$).

Equation (1) shows that the indirect coupling between $a$ and $b$ induced by the waveguide is non-Hermitian due to the identical phase accumulation $\phi$ for both directions [see from the identical coefficient $-\sqrt {\kappa _{a,e}\kappa _{b,e}}e^{i\phi }$ in both equations of Eq. (1)]. In particular, purely dissipative couplings can be achieved when $\phi =n\pi$ ($n$ is an arbitrary integer).

Now we consider a single standing-wave resonator $c$ side-coupled with a bent waveguide at two separated ports, as shown in Fig. 1(b). In this case, the Hamiltonian reads $H=\omega _{c}c^{\dagger }c+i\sum _{j=1,2}\sqrt {\kappa _{j,e}}(c_{j,\textrm {in}}+c_{j,\textrm {in}}')(c^{\dagger }-c)$ and the dynamic evolution of $c$ is captured by

Substituting Eq. (3) to Eq. (2), we can obtain the effective dynamic equation of $c$ as

Note that Eq. (1) can be derived via the same method, and the total input-output relations in this case are obtained as

Equations (4) and (5) exhibit a self interference effect which may significantly modify the optical properties of the system. We will discuss this effect in detail in the next two sections. As a supplement, we provide in Appendix A an alternative method based on real-space Schrödinger equation to verify our results at the single-photon level, which leads to essentially the same conclusions. Moreover, we point out that for the case in Fig. 1(a), the effective dynamic equations can be quite different if $a$ and $b$ are traveling-wave resonators, while for the case in Fig. 1(b), the self interference effects are shown to be similar whether $a$ and $b$ are standing-wave or traveling-wave resonators. The related details are shown in Appendix B.

3. Phase-dependent optical response

To study how the separation between the two ports affects the optical response of the single-resonator model in Fig. 1(b), we consider an external input signal injected from the lower side of the waveguide. In this case, $c_{1,\textrm {in}}\rightarrow c_{1,\textrm {in}}+\varepsilon _{s}e^{-i\omega _{s}t}$ comprises both the vacuum input field and external signal, with $\varepsilon _{s}$ and $\omega _{s}$ the amplitude and frequency of the signal, respectively. Under the rotating frame with respect to $\omega _{s}$, the effective dynamic equation of the mean value of $c$ can be written as

Figure 2(a) shows the transmission rate $T$ versus the detuning $\Delta _{c}$ and phase $\phi$. It is clear that the absorption window can be significantly modified by tuning the phase $\phi$ (i.e., the separation distance between the two ports) and the dependence of $T$ on $\phi$ is $2\pi$-periodic. As $\phi$ increases from $0$ to $\pi$ [or from $2n\pi$ to $(2n+1)\pi$ with $n$ an arbitrary integer], the width of the transmission dip decreases gradually, while the position of the dip shows a non-monotonic behavior, i.e., it first moves towards left (the direction towards negative values) and then returns back to the resonance position $\Delta _{c}=0$. Due to the relatively small intrinsic loss, the transmission dip implies in fact a reflection enhancement rather than a strong resonant absorption. To show the details clearly, we plot in Fig. 2(b) the profiles of $T$ versus $\Delta _{c}$ with different values of $\phi$. Indeed, one can find that the transmission dip becomes narrower and narrower as $\phi$ increases from $0$ and its position shift reaches the maximum near $\phi =\pi /2$. This can be understood by the coefficient $-2\sqrt {\kappa _{1,e}\kappa _{2,e}}e^{i\phi }$ of the third term on the right side in Eq. (8), which is complex for $\phi \neq n\pi$. The real part corresponds to a modification of the linewidth (decay rate), while the imaginary part corresponds to a frequency shift. Clearly, the linewidth (frequency shift) of resonator $c$ reaches the minimum (maximum) at $\phi =(2n+1)\pi$ [$\phi =(n+1/2)\pi$]. In particular, the transmission dip disappears completely when $\phi =\pi$, as shown in Fig. 2(b). In this case, the resonator cannot be excited by (or decay to) the waveguide, which demonstrates an optical dark state [72]. One can also understand this result by the term $(\sqrt {\kappa _{1,e}}+\sqrt {\kappa _{2,e}}e^{i\phi })\varepsilon _{s}$ in Eq. (8), which shows that the two input parts completely cancel each other when $\kappa _{1,e}=\kappa _{2,e}$ and $\phi =(2n+1)\pi$.

 

Fig. 2. (a) Pseudo-color map of transmission rate $T$ versus detuning $\Delta _{c}$ and phase $\phi$. (b) Profiles of $T$ versus $\Delta _{c}$ with different values of $\phi$. The other parameters are $\kappa _{c,i}/\kappa _{1,e}=0.1$ and $\kappa _{2,e}/\kappa _{1,e}=1$.

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It is worth pointing out that the external decay rates also play an important role in controlling the optical response, which can be tuned experimentally by changing the separation distance between the resonator and the waveguide. Note that the intrinsic loss of the resonator is viewed to be unchanged as we tune the external decay rates, because it is independent of the resonator-waveguide coupling. In view of this, we plot in Figs. 3(a)–3(c) the transmission rate $T$ versus $\Delta _{c}$ and $\kappa _{2,e}$ for $\phi =0$, $\phi =\pi$, and $\phi =\pi /2$, respectively, and plot in Figs. 3(d)–3(f) the corresponding profiles of $T$ with a set of chosen $\kappa _{2,e}$ to show more details. The transmission rate shows quite different dependence on $\kappa _{2,e}$ for the three cases. In the case of $\phi =0$, the width of the absorption window is positively associated with $\kappa _{2,e}$, while the depth of the window maintains almost invariant, i.e., the reflection can not be suppressed by increasing $\kappa _{2,e}$. In the case of $\phi =\pi$, however, both width and depth of the window strongly depends on $\kappa _{2,e}$. As $\kappa _{2,e}$ increases from $0$ to the value of $\kappa _{1,e}$, the window narrows and shallows rapidly until it disappears completely at $\phi =\pi$ as discussed above. Further increasing $\kappa _{2,e}$ leads to an inverse but much slower process, i.e., the window becomes wider and deeper gradually as $\kappa _{2,e}$ increases. When $\kappa _{2,e}/\kappa _{1,e}=4$, the transmission profile becomes almost identical as that for $\kappa _{2,e}=0$. Such a result can be understood that the coefficient $-2\sqrt {\kappa _{1,e}\kappa _{2,e}}e^{i\phi }$ of the third term in Eq. (8) is purely real for $\phi =n\pi$ and thus there is no frequency shift in this case. However, $\phi =2n\pi$ and $\phi =(2n+1)\pi$ corresponds to constructive and destructive interference between the two external decay channels, respectively. In view of this, one can observe monotonously increasing linewidth (decay rate) for $\phi =2n\pi$ while the linewidth first increases and then reduces monotonously for $\phi =(2n+1)\pi$, as $\kappa _{2,e}$ increases. For the more general case of $\phi \neq n\pi$, such as the case of $\phi =\pi /2$ in Figs. 3(c) and 3(f), increasing $\kappa _{2,e}$ yields a wider transmission dip while the depth of the dip is insensitive to $\kappa _{2,e}$. This is a bit similar to the case of $\phi =2n\pi$. However, one can also observe a position shift that is proportional to $\kappa _{2,e}$, which is distinct from the other two cases. This is because the coefficient mentioned above becomes complex in this case. As $\kappa _{2,e}$ increases, the positive real part results in an enlarged linewidth while the positive imaginary part leads to a monotonously increasing frequency shift.

 

Fig. 3. Pseudo-color maps of transmission rate $T$ versus detuning $\Delta _{c}$ and external decay rate $\kappa _{2,e}$ with (a) $\phi =0$, (b) $\phi =\pi$, (c) $\phi =\pi /2$. Profiles of $T$ versus $\Delta _{c}$ with different values of $\kappa _{2,e}$ and (d) $\phi =0$, (e) $\phi =\pi$, (f) $\phi =\pi /2$. Specifically, we consider $\kappa _{2,e}/\kappa _{1,e}=0$ (blue solid line), $0.5$ (red circles), $1$ (green dots), $2$ (purple dot-dashed line), and $4$ (cyan dashed line) in (d)-(f). The other parameter is $\kappa _{c,i}/\kappa _{1,e}=0.1$.

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4. Tunable sensing with photon-magnon hybridization

Now we consider that resonator $c$ in Fig. 1(b) is a microwave cavity which contains a yttrium iron garnet (YIG) sphere. The YIG sphere is a ferromagnetic crystal whose spins are perfectly ordered in the ground state. By placing the crystal into a microwave resonator, it overlaps with the microwave magnetic field of the resonator mode and the spins are coupled with the photons via the magnetic dipole interaction [30–34,73]. With the low-lying excitation and if the microwave magnetic field is approximately uniform throughout the crystal, the collective spins can be effectively described by a bosonic mode $m$ by using the Holstein-Primakoff transformation [i.e., $S_{-}=(\sqrt {2s-m^{\dagger }m})m$ and $S_{+}=(S_{-})^{\dagger }$ with $S_{\pm }$ the raising and lowering operators of the spins respectively and $s$ the spin quantum number], which is referred to as “magnon mode” [73]. The magnon mode shows a tunable resonance frequency $\omega _{m}$ which is determined by the strength $B$ of the external magnetic field, i.e., $\omega _{m}=\gamma B$ with $\gamma$ the gyromagnetic ratio. With the rotating-wave approximation, the magnon-photon coupling can be described by $J(a^{\dagger }m+m^{\dagger }a)$, where $J=\sqrt {sN/2}g\mu _{B}B$ is the coupling strength with $N$, $g$, and $\mu _{B}$ the total number of the spins, the $g$-factor, and the Bohr magneton, respectively [73]. Using the method developed in Sec. 2 and assuming that an external input signal is injected from the lower side of the waveguide, the effective dynamic equations can be written as

To show the influence of the photon-magnon hybridization on the optical response, we plot in Fig. 4(a) the profiles of $T$ in the case of $\Delta _{m}=\Delta _{c}$ with different values of $\phi$ (the more general case in which $\Delta _{m}\neq \Delta _{c}$ is discussed below). Clearly, a transmission peak emerges in the presence of the YIG sphere, which splits the original dip into two new ones. For $\phi =0$, the separation between the two dips equals to $2J$, which originates from the normal-mode splitting due to the magnon-photon coupling. As $\phi$ increases from $0$, the line shape becomes asymmetric (Fano-like) due to the phase-dependent frequency shift discussed above. During this process, the right dip narrows gradually and the transparency peak tends to be perfect ($T=1$). As a result, the transmission rate can change more drastically near the resonant point $\Delta _{c}=0$ by tuning $\phi$. Note that both dips will shallow rapidly as $\phi$ approaches $\pi$ and disappear completely for $\phi =\pi$ due to the optical dark state. For the purpose of this section, we only consider the phase far away from $(2n+1)\pi$ in the following.

 

Fig. 4. (a) Profiles of transmission rate $T$ versus detuning $\Delta _{c}$ with different values of phase $\phi$. (b) Profiles of $T$ versus $\Delta _{c}$ for the double-ports (blue-solid), single-port (red dashed), and indirectly-coupled (green dotted) models. (c) Maximal sensitivity $S_{\textrm {max}}$ versus $\phi$ for the double-ports (blue solid), single-port (red dashed), and indirectly-coupled (green dotted) models. (d) Pseudo-color map of $T$ versus detunings $\Delta _{c}$ and $\delta$. (e) Profiles of $T$ versus $\Delta _{c}$ with different values of $\delta$. The inset in (e) shows the sensitivity $S$ versus $\Delta _{c}$ with different values of $\delta$. In panel (e) and the inset, the blue solid, red dashed and green dotted lines correspond to $\delta /\kappa _{c,i}=0$, $-0.5$ and $0.5$, respectively. Here we assume $\Delta _{m}=\Delta _{c}$ in (a)-(c) and $\phi =3\pi /4$ in (d) and (e). The other dimensionless parameters are $\{\kappa _{1,e},\kappa _{2,e},\kappa _{c,i},\kappa _{m,i},J\}=\{1,1,0.05,5\times 10^{-3},0.3\}$ for the double-ports model and $\{\kappa _{c,e},\kappa _{c,i},\kappa _{m,e},\kappa _{m,i}\}=\{1,0.05,1.2\times 10^{-4},5\times 10^{-3}\}$ for the indirectly-coupled model. The parameters of the single-port model are the same as those of the double-ports one except for $\kappa _{2,e}=0$.

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We would like to point out that the sharp Fano-like line shapes in Fig. 4(a) have potential applications in sensing parameters associated with $\Delta _{c}$. Within the working region, any small perturbation of the target parameter gives rise to a drastic change in transmission rate. For the case of $\Delta _{c}=\Delta _{m}$ (i.e., $\omega _{c}=\omega _{m}$) shown in Fig. 4(a), one can detect the input frequency fluctuations which are unavoidable in practice and limit the performance of dispersive sensing schemes [54,74,75]. To show the advantage of our model in frequency sensing, we also consider for comparison two general models: the single-port model with $\kappa _{2,e}=0$ and the indirectly-coupled model where the resonator and the YIG sphere couple indirectly with each other via the common waveguide [43]. In fact, the indirectly-coupled model is exactly the one shown in Fig. 1(a), with $a$ and $b$ replacing by modes $c$ and $m$ respectively in this case. According to Eq. (1), the effective dynamic equations can be written as

Fig. 4(b) shows that the transmission profiles are quite different for the three cases. In the single-port model, one can still observe a transmission peak which is induced by the photon-magnon coupling. However, the working window (transmission peak) is much wider than that (right sharp dip) of our double-ports model, which should correspond to a worse sensing performance. Note that the optical response in this case is physically independent of $\phi$. In the indirectly-coupled model, the transmission profile maintains the Lorentz shape. This is because the external decay rate of the magnon mode (the coupling strength between the YIG sphere and the waveguide) is typically much smaller than that of the resonator ($\kappa _{m,e}/\kappa _{c,e}\sim 10^{-4}$) in experiments [41–43]. Instead, the direct photon-magnon coupling induced by the magnetic dipole interaction can be larger than the decay rates of the two modes (corresponding to the strong coupling regime). This implies that the photon-magnon hybridization is much weaker in the indirectly-coupled model. Such a working window with large linewidth is apparently not suitable for sensing. We have also checked that the transmission profile of the indirectly-coupled model is also quite insensitive to $\phi$ due to the small coefficients of the $\phi$-dependent terms in Eq. (14). For quantitative estimation, we introduce the sensitivity $S=|dT/d\Delta _{c}|$ [24] and plot in Fig. 4(c) the maximal sensitivity $S_{\textrm {max}}$ over the whole frequency range as a function of $\phi$ in the case of $\Delta _{c}=\Delta _{m}$. It shows that the maximal sensitivity is markedly enhanced in the presence of the self interference. In addition, the performance of our sensing scheme can be further optimized by tuning $\phi$, with the maximum of $S_{\textrm {max}}$ obtained at $\phi \approx 0.59\pi$. As predicted in Fig. 4(b), the wide Lorentz-shape window of the indirectly-coupled model corresponds to a quite low sensitivity. Although the transmission rate is not a monotonic function of $\Delta _{c}$ near the working region, one can further determine the target parameter with different values of $\phi$. For different $\phi$ [see the uppermost and lowermost panels in Fig. 4(a) for example], the values of $T$ locating at the right side of the sharp dip changes much more drastically than those locating at the left side.

As mentioned above, one major feature of our scheme based on cavity magnonics is the controllable frequency of the magnon mode so that the working region can be tuned by changing the strength of the external magnetic field. As shown in Fig. 4(d), the position of the sharp right dip is determined by the detuning $\delta =\Delta _{c}-\Delta _{m}=\omega _{c}-\omega _{m}$ between the photon and magnon modes, while the left transmission dip is nearly unaffected by the change in $\delta$ (its position is in fact determined by the self interference induced frequency shift, as discussed above). This can be seen clearly in Fig. 4(e), where we do find that the left transmission dip is quite insensitive to $\delta$. The sharp right dip, which is approximately located at $\Delta _{c}=\delta$, shows slightly changed width for different $\delta$, i.e., the width of the right dip increases mildly as $\delta$ decreases from positive to negative. The inset in Fig. 4(e) depicts the sensitivity $S$ versus $\Delta _{c}$ with the three chosen values of $\delta$ in Fig. 4(e). We can find that a wider transmission dip corresponds to a lower sensitivity peak. In view of this, our scheme shows better performance within the blue-detuned region ($\Delta _{c}>0$). Moreover, Figs. 4(d) and 4(e) suggest that our sensing scheme maintains high performance even for $\omega _{m}\neq \omega _{c}$. Therefore our scheme can also be used for sensing fluctuations related to $\omega _{c}$ such as frequency shifts induced by the thermorefractive and thermoelastic effects [76,77].

On one hand, it has been shown above that the photon-magnon hybridization can markedly enhance the sensitivity due to the sharp Fano-like line shapes and the sensitivity can be further improved by tuning the phase $\phi$. On the other hand, the transmission rate is also dependent on $\Delta _{m}$ according to Eq. (12), implying that our hybrid model can be used in turn as a solid-state magnetometer [59–62]. In other words, one can detect the resonance frequency of the magnon mode and thereby the strength of the magnetic field via the transmission rate. As shown in Fig. 5(a), there is always a transmission peak near the resonant point $\Delta _{m}=\Delta _{c}=0$ (we assume $\Delta _{c}=0$ in Fig. 5 for simplicity). In the case of $\phi \neq 0$, the transmission profile becomes Fano-like similar to that in Fig. 4(a), giving rise to both a peak and a dip near the resonant point. As $\phi$ increases from $0$, the transmission peak increases obviously while the depth of the transmission dip only changes slightly. Meanwhile, the off-resonant transmission rate increases markedly with $\phi$, implying that the transmission rate changes more and more drastically near the resonant point by increasing $\phi$ suitably. We also plot in Fig. 5(b) the maximal sensitivity $S_{\textrm {max}}'$ of our model to find out the optimal phase $\phi$ for magnetometry and those of the single-port and indirectly-coupled models for comparison, with the definition of the sensitivity becoming $S'=|dT/d\Delta _{m}|$ in this case. Once again, our scheme shows much better performance than the other two models. The $\phi$ dependence of the maximal sensitivity of our model is quite similar with that in Fig. 4(b), with the optimal performance obtained near $\phi =0.59\pi$ as well. Although it is challenging in experiments to control the phase $\phi$ precisely, our scheme shows much higher sensitivity than those of the other two general models within almost the whole range of $\phi$. In addition, one can control $\phi$ more precisely in the microwave regime considered here: an obvious fluctuation of $\phi$ requires a considerably large change in $L$ in this case due to the much smaller wave vector than that in the optical regime.

 

Fig. 5. (a) Profiles of transmission rate $T$ versus detuning $\Delta _{m}$ with $\phi =0$ (blue solid), $\phi =\pi /4$ (red dashed), $\phi =\pi /2$ (green dotted), and $\phi =3\pi /4$ (purple dot-dashed). (b) Maximal sensitivity $S_{\textrm {max}}'$ versus $\phi$ for the double-ports (blue solid), single-port (red dashed), and indirectly-coupled (green dotted) models. Here we assume $\Delta _{c}=0$. The other dimensionless parameters are $\{\kappa _{1,e},\kappa _{2,e},\kappa _{c,i},\kappa _{m,i},J\}=\{1,1,0.05,5\times 10^{-3},0.3\}$ for the double-ports model and $\{\kappa _{c,e},\kappa _{c,i},\kappa _{m,e},\kappa _{m,i}\}=\{1,0.05,1.2\times 10^{-4},5\times 10^{-3}\}$ for the indirectly-coupled model. The parameters of the single-port model are the same as those of the double-ports one except for $\kappa _{2,e}=0$.

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Finally, we point out that the photon-magnon interaction is formally identical with that of the Jaynes-Cummings model, implying that our scheme in this section can be generalized to other models with such interactions. However, there are three reasons for which we choose the photon-magnon model: (i) One major advantage of the magnon mode is its tunable resonance frequency controlled by the external magnetic field, which enables the tunable working window. (ii) Magnetometers are of significant values within a wide range of practical applications, such as geology, navigation, and archaeology. Cavity magnonics provide a promising platform for realizing highly-sensitive solid-state magnetometers, which allows optical readout of the magnetic field strength. (iii) The magnon mode of the YIG sphere shows extremely weak dissipation [78], which enables the ultra-narrow working window and thereby promotes the sensing performance.

5. Conclusions

In summary, we have systematically studied the self interference effect of a resonator side-coupled with a bent waveguide at two separated ports. While the well-studied model of two indirectly coupled resonators shows quite different effective couplings for standing-wave and traveling-wave resonators, the present model supports similar self interference effects regardless of the resonator configuration. The theory is also verified at the single-photon level with the method based on real-space Schrödinger equation. It shows that controllable optical response can be achieved by tuning the separation distance between the two ports. Besides the resonance frequency and linewidth, there is also an interference effect between input fields at the two ports, which may lead to an optical dark state under specific conditions. Moreover, the controllable Fano-like line shapes in the photon-magnon hybrid model are proved to be useful in frequency sensing. We show that the sensitivity can be markedly enhanced in the hybrid system and the working region can be changed flexibly due to the tunable resonance frequency of the magnon mode. On the other hand, our scheme can also be used for magnetometry which allows for optical readout of the magnetic field strength. The results in this paper are expected to inspire more potential applications of the self interference effect, such as microcavity soliton generation [79] and the control of bound states in the continuum [45,49].

Appendix

A. Verification in real space

It is known that the waveguide can be described by a bath of harmonic oscillators [69], therefore the total Hamiltonian of the model in Fig. 1(b) can be given by

(17)$$H=H_{\textrm{c}}+H_{\textrm{w}}+H_{\textrm{int}},$$

where $H_{\textrm {c}}=\omega _{c}c^{\dagger }c$ and $H_{\textrm {w}}=\int \omega _{k}a_{k}^{\dagger }a_{k}dk$ are the free Hamiltonians of resonator $c$ and the waveguide, respectively, where $a_{k}$ ($a_{k}^{\dagger }$) is the annihilation (creation) operator of a traveling photon in the waveguide with frequency $\omega _{k}$ and wave vector $k$. In addition,

(18)$$H_{\textrm{int}}=\int (g_{1}c^{\dagger}a_{k}+g_{2}e^{ikx_{0}}c^{\dagger}a_{k}+h.c.)dk,$$

is the interaction Hamiltonian between the resonator and the waveguide under the rotating-wave approximation (RWA). The locations of ports $1$ and $2$ are assumed to be $x=0$ and $x=x_{0}$, respectively. According to Refs. [1,71], if the resonance frequency $\omega _{c}$ of the resonator is far away from the cut off frequency of the waveguide dispersion, the whole system can be described conveniently in the real space, with

(19)$$H_{\textrm{w}}=\int dx[{-}iv_{g}a_{F}^{\dagger}(x)\frac{d}{dx}a_{F}(x)+iv_{g}a_{B}^{\dagger}(x)\frac{d}{dx}a_{B}(x)]$$

and

(20)$$\begin{aligned} H_{\textrm{int}}=&\int dx\{g_{1}\delta(x)[c^{\dagger}a_{F}(x)+c^{\dagger}a_{B}(x)+h.c.]\\ &+g_{2}\delta(x-x_{0})[c^{\dagger}a_{F}(x)+c^{\dagger}a_{B}(x)+h.c.]\} \end{aligned}$$

in this case. Here, $v_{g}$ is the group velocity of the traveling photons in the waveguide. $a_{F}^{\dagger }(x)$ [$a_{B}^{\dagger }(x)$] is the bosonic operator creating a forward (backward) traveling photon at position $x$. Note that “forward” (“backward”) here refers to the direction from the lower side to the upper side (from the upper side to the lower side) of the waveguide. Assuming that the whole system is initially prepared in the single-excitation manifold and considering that the RWA preserves the number of excitations, the wave function can be given by

(21)$$|\psi\rangle=\int dx[\phi_{F}(x)a_{F}^{\dagger}(x)+\phi_{B}(x)a_{B}^{\dagger}(x)]|G\rangle+u_{c}c^{\dagger}|G\rangle,$$

where $|G\rangle$ is the ground state of the whole system with no photon in the resonator and the waveguide. $u_{c}$ and $\phi _{j}(x)$ ($j=F,\,B$) are the excitation amplitudes of resonator $c$ and waveguide mode $a_{j}(x)$, respectively. The equations of the excitation amplitudes can be obtained by solving the Schrödinger equation, which reads

(22)$$\begin{aligned} &E\phi_{F}(x)={-}iv_{g}\frac{d}{dx}\phi_{F}(x)+[g_{1}\delta(x)+g_{2}\delta(x-x_{0})]u_{c},\\ &E\phi_{B}(x)=iv_{g}\frac{d}{dx}\phi_{B}(x)+[g_{1}\delta(x)+g_{2}\delta(x-x_{0})]u_{c},\\ &Eu_{c}=\omega_{c}u_{c}+g_{1}[\phi_{F}(0)+\phi_{B}(0)]+g_{2}[\phi_{F}(x_{0})+\phi_{B}(x_{0})]. \end{aligned}$$

Now we consider that a single photon is incident from the lower side of the waveguide. In this case $\phi _{F}(x)$ and $\phi _{B}(x)$ can be written as [21]

(23)$$\begin{aligned} &\phi_{F}(x)=e^{ikx}\{\theta({-}x)+A[\theta(x)-\theta(x-x_{0})]+t\theta(x-x_{0})\},\\ &\phi_{B}(x)=e^{{-}ikx}\{r\theta({-}x)+B[\theta(x)-\theta(x-x_{0})]\}, \end{aligned}$$

where $\theta (x)$ is the Heaviside step function satisfying $\partial \theta (bx-a)/\partial x=b\delta (bx-a)$. Here, $A$ ($B$) denotes the amplitude of the forward (backward) wave in the region of $0<x<x_{0}$, while $r$ ($t$) denotes the reflection (transmission) amplitude at $x=0$ ($x=x_{0}$). Substituting Eq. (23) into Eq. (22)), one can obtain

(24)$$\begin{aligned} 0=&-iv_{g}(A-1)+g_{1}u_{c},\\ 0=&-iv_{g}(t-A)e^{ikx_{0}}+g_{2}u_{c},\\ 0=&-iv_{g}(r-B)+g_{1}u_{c},\\ 0=&-iv_{g}Be^{{-}ikx_{0}}+g_{2}u_{c},\\ Eu_{c}=&\omega_{c}u_{c}+\frac{g_{2}}{2}(te^{ikx_{0}}+Ae^{ikx_{0}}+Be^{{-}ikx_{0}})\\ &+\frac{g_{1}}{2}(A+B+1+r). \end{aligned}$$

where $E=v_{g}k$ for $x\neq 0$ and $x\neq x_{0}$. In this way, we can obtain the effective equations of $u_{c}$, i.e.,

(25)$$\begin{aligned} i\frac{du_{c}}{dt}=Eu_{c}=&(\omega_{c}-i\gamma_{1}-i\gamma_{2})u_{c}+g_{1}+g_{2}e^{ikx_{0}}\\ &-2i\sqrt{\gamma_{1}\gamma_{2}}e^{ikx_{0}}u_{c} \end{aligned}$$

with $\gamma _{1}=g_{1}^{2}/v_{g}$ and $\gamma _{2}=g_{2}^{2}/v_{g}$ [1]. Clearly, Eq. (25) shows essentially the same self interference effect as that in Eq. (4). The input terms, however, have also the same form if we only consider the input field coming from the lower side of the waveguide ($c_{2,\textrm {in}}'=0$) in Eq. (4).

One can also verify the effective dynamic equation Eq. (1) of the model in Fig. 1(a). With similar procedures, we have

(26)$$\begin{aligned} &i\frac{du_{a}}{dt}=Eu_{a}=(\omega_{a}-i\gamma_{a})u_{a}-i\sqrt{\gamma_{a}\gamma_{b}}e^{ikx_{0}}u_{b}+g_{a},\\ &i\frac{du_{b}}{dt}=Eu_{b}=(\omega_{b}-i\gamma_{b})u_{b}-i\sqrt{\gamma_{a}\gamma_{b}}e^{ikx_{0}}u_{a}+g_{b}e^{ikx_{0}} \end{aligned}$$

with $\gamma _{a}=g_{a}^{2}/v_{g}$ and $\gamma _{b}=g_{b}^{2}/v_{g}$. Here $u_{a}$ and $u_{b}$ are the single-photon excitation amplitudes of resonators $a$ and $b$, respectively. Once again, the effective coupling and input terms in Eq. (1) (in the case of $b_{\textrm {in}}'=0$) have the same forms as these in Eq. (26), which verifies our conclusions in Sec. 2.

B. Traveling-wave-resonator scheme

If we consider two separated traveling-wave resonators (such as WGM resonators) coupled with a common waveguide, as shown in Fig. 6(a). The dynamic equations of the two clockwise (CW) resonator modes $a$ and $b$ can be written as

(27)$$\begin{aligned} &\frac{da}{dt}={-}(i\omega_{a}+\kappa_{a})a+\sqrt{\kappa_{a,e}}a_{\textrm{in}},\\ &\frac{db}{dt}={-}(i\omega_{b}+\kappa_{b})b+\sqrt{\kappa_{b,e}}b_{\textrm{in}}, \end{aligned}$$

where $\kappa _{j}=\kappa _{j,i}+\kappa _{j,e}/2$ ($j=a,\,b$) in this case. Here $\omega _{j}$, $\kappa _{j,i}$ and $\kappa _{j,e}$ have the same meaning as those in Eq. (1). Due to the directionality of the traveling-wave resonator modes, the external decay rates are not doubled in this case even if the resonators are two-sided. For the same reason, the CW modes can only be excited by the forward (right-going) input field $a_{\textrm {in}}$ coming from the left side of the waveguide. As shown in Fig. 6(a), the input-output relations of each CW mode can be written as

(28)$$\begin{aligned} &a_{\textrm{out}}=a_{\textrm{in}}-\sqrt{\kappa_{a,e}}a,\\ &b_{\textrm{in}}=a_{\textrm{out}}e^{i\phi},\\ &b_{\textrm{out}}=b_{\textrm{in}}-\sqrt{\kappa_{b,e}}b \end{aligned}$$

in this case, which leads to

(29)$$\begin{aligned} \frac{da}{dt}=&-(i\omega_{a}+\kappa_{a})a+\sqrt{\kappa_{a,e}}a_{\textrm{in}},\\ \frac{db}{dt}=&-(i\omega_{b}+\kappa_{b})b-\sqrt{\kappa_{a,e}\kappa_{b,e}}e^{i\phi}a+\sqrt{\kappa_{b,e}}e^{i\phi}a_{\textrm{in}}. \end{aligned}$$

Equation (29) shows that the effective coupling between $a$ and $b$ is unidirectional in this case, i.e., the dynamics of $b$ depends on $a$ but not vice versa. This type of interactions have been used to achieve the chiral exceptional point (EP) in an indirectly coupled WGM resonator system where backscattering induced couplings are also considered [80].

 

Fig. 6. Schematic diagrams of (a) two separated traveling-wave resonators side-coupled with a straight waveguide and (b) a single traveling-wave resonator side-coupled with a bent waveguide at two different ports.

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However, if we consider a single traveling-wave resonator coupled with a bent waveguide at two different ports, as shown in Fig. 6(b), the dynamic equation of mode $c$ [which is a counter-clockwise (CCW) mode in this configuration] is given by

(30)$$\frac{dc}{dt}={-}(i\omega_{c}+\kappa_{c})c+\sqrt{\kappa_{1,e}}c_{1,\textrm{in}}+\sqrt{\kappa_{2,e}}c_{2,\textrm{in}},$$

where $\kappa _{c}=\kappa _{c,i}+(\kappa _{1,e}+\kappa _{2,e})/2$. $\omega _{c}$, $\kappa _{c,i}$ and $\kappa _{j,e}$ ($j=1,\,2$) have the same meanings as those in Eq. (2). As discussed above, we only consider two forward input fields $c_{1,\textrm {in}}$ and $c_{2,\textrm {in}}$ at the two ports in this case. Similarly, with the local input-output relations, one can obtain the effective dynamic equation

(31)$$\begin{aligned} \frac{dc}{dt}=&-(i\Delta_{c}+\kappa_{c})c-\sqrt{\kappa_{1,e}\kappa_{2,e}}e^{i\phi}c\\ &+(\sqrt{\kappa_{1,e}}+\sqrt{\kappa_{2,e}}e^{i\phi})c_{1,\textrm{in}} \end{aligned}$$

and the total input-output relation

(32)$$c_{\textrm{out}}=(c_{1,\textrm{in}}-\sqrt{\kappa_{1,e}}c)e^{i\phi}-\sqrt{\kappa_{2,e}}c.$$

Now we can find that although the dynamic equations show significant difference for the cases of Fig. 1(a) and Fig. 6(a), the self-interference terms appear in a similar form for a single standing-wave or traveling-wave resonator coupled with a waveguide at separated ports. Compared with the standing-wave scheme in Fig. 1(b), the single-traveling-wave-resonator scheme here shows the advantage in preventing reflections.

Funding

National Natural Science Foundation of China (11774024, 11875011, 12074030, U1930402); National Key Research and Development Program of China (2016YFA0301200); Science Challenge Project (TZ2018003).

Acknowledgments

L. Du thanks Y.-T. Chen and Q.-S. Zhang for helpful discussions.

Disclosures

The authors declare no conflicts of interest.

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