Optical Analogs of Rabi Splitting in Integrated Waveguide‐Coupled Resonators

Realizing optical analogs of quantum phenomena in atomic, molecular, or condensed matter physics has underpinned a range of photonic technologies. Rabi splitting is a quantum phenomenon induced by a strong interaction between two quantum states, and its optical analogs are of fundamental importance for the manipulation of light–matter interactions with wide applications in optoelectronics and nonlinear optics. Here, purely optical analogs of Rabi splitting in integrated waveguide‐coupled resonators formed by two Sagnac interferometers are proposed and theoretically investigated. By tailoring the coherent mode interference, the spectral response of the devices is engineered to achieve optical analogs of Rabi splitting with anti‐crossing behavior in the resonances. Transitions between the Lorentzian, Fano, and Rabi splitting spectral lineshapes are achieved by simply changing the phase shift along the waveguide connecting the two Sagnac interferometers, revealing interesting physical insights about the evolution of different optical analogs of quantum phenomena. The impact of the device's structural parameters is also analyzed to facilitate device design and optimization. These results suggest a new way for realizing optical analogs of Rabi splitting based on integrated waveguide‐coupled resonators, paving the way for many potential applications that manipulate light–matter interactions in the strong coupling regime.


Abstract
Realizing optical analogues of quantum phenomena in atomic, molecular, or condensed matter physics has underpinned a range of photonic technologies.Rabi splitting is a quantum phenomenon induced by a strong interaction between two quantum states, and its optical analogues are of fundamental importance for the manipulation of light-matter interactions with wide applications in optoelectronics and nonlinear optics.Here, we propose and theoretically investigate purely optical analogues of Rabi splitting in integrated waveguide-coupled resonators formed by two Sagnac interferometers.By tailoring the coherent mode interference, the spectral response of the devices is engineered to achieve optical analogues of Rabi splitting with anti-crossing behavior in the resonances.Transitions between the Lorentzian, Fano, and Rabi splitting spectral lineshapes are achieved by simply changing the phase shift along the waveguide connecting the two Sagnac interferometers, revealing interesting physical insights about the evolution of different optical analogues of quantum phenomena.The impact of the device structural parameters is also analyzed to facilitate device design and optimization.These results suggest a new way for realizing optical analogues of Rabi splitting based on integrated waveguide-coupled resonators, paving the way for many potential applications that manipulate light-matter interactions in the strong coupling regime.

I. INTRODUCTION
In atomic, molecular, and condensed matter physics, the response of photon-matter systems is normally characterized via spectroscopic detection of radiation by exploring physical processes like scattering, absorption, or fluorescence, resulting in spectral lineshapes that unveil the nature of the light-matter interactions [1,2].This interaction in a confined electromagnetic environment can be controlled to achieve a coupling regime in which coherent energy exchange occurs between light and matter.Such an exchange, in the strong light-matter coupling regime, results in anti-crossing between the atom-like emitter and the cavity-mode dispersion relations, which is described by the so-called Rabi splitting [2][3][4].
In previous work, optical analogues of Rabi splitting have been realized in PhC and plasmonic cavities [6,39,61].In this paper, we propose and theoretically verify a different way for realizing optical analogues of Rabi splitting based on integrated waveguide-coupled resonators.Similar to the manipulation of the interaction between different quantum states in a multi-level atomic system, the coherent mode interference in the waveguide-coupled resonator formed by two Sagnac interferometers is engineered to achieve optical analogues of Rabi splitting with anti-crossing behavior for the resonances.By changing the phase shift along the waveguide connecting the two Sagnac interferometers, transitions from symmetric Lorentzian spectral lineshape to asymmetric Fano and Rabi splitting spectral lineshapes are also achieved, showing interesting trends for the evolutions of different optical analogues of quantum phenomena.Finally, detailed analysis for the impact of device structural parameters is provided to facilitate device design and optimization.Our results theoretically confirm the effectiveness of realizing optical analogues of Rabi splitting based on integrated waveguide-coupled resonators, which offers new possibilities for many potential applications that manipulate lightmatter interactions in the strong coupling regime.

II. DEVICE DESIGN AND OPERATION PRINCIPLE
As schematically illustrated in Fig. 1(a), in the strong light-matter coupling regime where the coherent exchange rate of energy between light and matter is higher than the decay rate, a resonance state |r> (e.g., whispering gallery mode and plasmonic resonances) and a matter excitation state |e> (e.g., excitation states of atoms, molecules, plamons, and quantum dots) strongly couple with each other, resulting in the generation of new hybridized eigenstates separated by the Rabi splitting energy ħΩR.In contrast to the original independent eigenstates, the new eigenstates arising from field-induced splitting of energy levels show a clear anticrossing behavior in the spectral response [62], which can be exploited for manipulation of light-matter interaction that has wide applications in low-threshold lasing [63], phase transition modification [39], chemical reactivity tuning [64,65], Bose-Einstein condensation [66][67][68], and optical spin switching [69].

Fig. 1(b)
shows a schematic of the proposed waveguide-coupled resonator, where a bus waveguide is coupled to a closed resonant loop formed by two Sagnac interferometers.The closed resonant loop couples to the bus waveguide to form two directional couplers, and the interference between the waveguides connecting them is similar to that in a Mach-Zehnder interferometer (which is a finite impulse-response (FIR) filter).On the other hand, the interference between the closed resonant loop and the bus waveguide is similar to that in a ring resonator (which is an infinite impulse-response (IIR) filter).As a result, the device consists of both FIR and IIR filter elements.By introducing Sagnac interferometers in the resonant loop, such a device can also be regarded as a hybrid resonator consisting of both traveling-wave (formed by coherent interference between light waves in the closed-loop resonator and the bus waveguide, similar to ring resonators) and standing-wave (formed by coherent interference induced by reflection between the two Sagnac interferometers, similar to Fabry-Perot cavities) resonator elements.The hybrid nature of the device in Fig. 1(b) allows for versatile coherent mode interference that can be tailored for spectral engineering of complex and demanding filtering functions.
The definitions of the device structural parameters are provided in Table I.In our following analysis, we investigate the spectral response of the device in Fig. 1(b) based on the scattering matrix method [33,70], using the values of the transverse electric (TE) mode group index ng = 4.3350 and the propagation loss factor α = 55 m -1 (i.e., 2.4 dB/cm) obtained from the fabricated silicon-on-insulator (SOI) devices in our previous work [70,71].The devices are designed for, but not limited to, the SOI platformthe principles outlined here are universal for all material platforms.
We tailor the spectral response of the device in Fig. 1 states in multi-level atomic systems.Similar Rabi splitting spectra in the visible and midinfrared regions have also been observed for PhC and plasmonic nanocavities [6,39].However, compared to these, the fabrication of waveguide-coupled resonators does not require high lithography resolution and shows higher tolerance to fabrication imperfections such as lithographic smoothing effects and quantization errors due to the finite grid size [70], which makes it much easier to accurately engineer the spectral response and overall device performance.

III. ANTI-CROSSING OF SPLIT RESONANCES
In this section, we investigate the resonance anti-crossing behavior of the device in Fig. 1(b), which is a key feature of Rabi splitting.A typical application of the anti-crossing behavior of the split resonances is to engineer the dispersion of the devices to generate artificial anomalous dispersion in devices that exhibit intrinsic normal dispersion [57,58,72], for applications to nonlinear optics.We also provide an analysis for the influence of the device structural parameters on the Rabi split resonances.To simplify the comparison, we vary only one structural parameter in each figure of this section, while keeping the others the same as in Fig.In Fig. 2 we compare the device spectral response for various ∆L2, i.e., the variation in the length of L2.As ∆L2 changes, both the transmission spectra in Fig. 2(a-i  In practical applications, different values of ΔLi (i = 1, 2, 3) in Figs. 2 -4 can be achieved by varying the physical lengths of the corresponding waveguides in passive devices.By introducing thermo-optic micro-heaters [32,60] or PN junctions [73,74] along the corresponding waveguides to tune their phase shifts, the changes of ΔLi (i = 1, 2, 3) can also be realized via active tuning of passive devices, which allows for tunable Rabi splitting characteristics to meet the requirements of different applications.In addition to phase shifts along the connecting waveguides, we investigate the influence of the coupling strengths of the directional couplers on the Rabi split resonances in Figs. 5 and 6.Since their influence on the center wavelengths of the split resonances is not as significant as varying ΔLi (i = 1, 2, 3) in Figs. 2 -4, we plot the spectra for different coupling strengths in the same figure to highlight the differences.

1(c).
We first investigate the influence of the coupling strengths of the directional couplers in the two Sagnac interferometers (i.e., ts1 and ts2), which determines the reflectivity of the Sagnac interferometers.Fig. 5(a) shows the transmission and reflection spectra for various ts1.As can be seen, the extinction ratios of the Rabi split resonances (defined as the difference between the maximum and minimum transmission) remain almost unchanged, while the spectral interval between the split resonances increases with ts1.Similar trends are also observed in Fig. 5(b) that plot the transmission and reflection spectra versus ts2.These result from the fact that changes in ts1 or ts2 alter the mutual coupling between the light waves propagating in two opposite directions, which varies the degree of Rabi mode splitting that have various spectral intervals between the split resonances.In Fig. 6, we investigate the influence of varying the strengths of the directional couplers between the bus waveguide and Sagnac interferometers (i.e., tb1 and tb2), i.e., the energy coupling strength between the closed resonant loop and bus waveguide.and tb2 on the transmission and reflection spectra.Although tb1 and tb2 have more influence on the extinction ratios of the Rabi split resonances than ts1 and ts2, their influence on the spectral interval between the split resonances is not as significant.This is mainly because the spectral interval between the split resonances, which reflects the degree of Rabi mode splitting, is determined by the coupling strength between the bidirectional light waves within the closed resonant loop.On the other hand, the extinction ratios of the split resonances are determined by the energy exchange between the resonant loop and the bus waveguide, which is similar to the different coupling regimes in ring resonators [75].
In practical applications, dynamic tuning of the coupling strength of the directional couplers in Figs. 5 and 6 can be realized by using Mach-Zehnder interferometric couplers to replace the directional couplers and adjusting the phase difference between the two arms.
Recently, compact tunable directional couplers have also been demonstrated by directly integrating thermo-optic micro-heaters above the coupling regions [76,77], which induces thermal gradients that lead to phase velocity mismatch between the coupled modes of the waveguides, enabling dynamic tuning of the coupling strength.

IV. TRANSITIONS BETWEEN LORENTZIAN, FANO, AND RABI SPLITTING SPECTRAL LINESHAPES
The spectroscopic detection of radiation via physical processes such as scattering, absorption, or fluorescence has been widely used for characterizing the response of photon-matter systems, resulting in featured spectral lineshapes that unveil the fundamentals of light-matter interactions [1,2].In addition to Rabi splitting, there are other typical resonance spectral lineshapes.For example, the symmetric Lorentzian spectral lineshape, which describes the finite radiative lifetimes of excited states, and is commonly seen in the response spectra of ring resonators and Fabry-Perot cavities [71,73,[78][79][80].In addition, Fano resonances, which feature an asymmetric spectral lineshape induced by interference between a discrete quantum state and a continuum band of states [81,82], have underpinned many applications such as switching [83], sensing [84,85], lasing [86], and directional scattering [87,88].Despite originating from different underlying physics, the different spectral lineshapes are related to each other and sometimes can transit from one to another.In this section, we engineer the spectral response of the device in Fig. 1(b) to achieve transitions between Lorentzian, Fano, and Rabi splitting resonance spectral lineshapes.The versatility and degree of control achieved using these advanced design techniques based on Sagnac interferometers [89][90][91] will likely have wide applications to many different areas beyond linear optical filters, including nonlinear optics and microcomb based devices  as well as quantum optical photonic chips [149][150][151][152][153][154][155][156][157][158][159][160][161] and photonic integrated chips based on novel 2D materials and structures .

V. CONCLUSIONS
We theoretically investigate optical analogues of Rabi splitting in integrated waveguidecoupled resonators formed by two Sagnac interferometers.Coherent mode interference in the proposed device is tailored to achieve optical analogues of Rabi splitting with a clear avoided behavior for the resonances.Moreover, transitions between the Lorentzian, Fano, and Rabi splitting resonance lineshapes are achieved by changing the phase shift along the connecting waveguide between the two Sagnac interferometers.A detailed analysis for the influence of the structural parameters on the device performance is also provided.Our work offers new possibilities for realizing optical analogues of Rabi splitting based on integrated waveguidecoupled resonators and provides new prospects for realizing optical analogues of quantum physics by exploiting Sagnac interference in integrated photonic devices.This will allow for attractive benefits in providing devices with compact footprint, high scalability, high fabrication tolerance, and mass producibility for practical applications.
(b) with bidirectional light propagation to realize optical analogues of Rabi splitting.The power transmission and reflection spectra with input from Port 1 are depicted in Fig. 1(c).Unless elsewhere specified, the spectral response of the device is assumed to have an input from Port 1 in Fig. 1(b), with the transmitted light at Port 2 and the reflected light back to Port 1.In Fig. 1(c), the device structural parameters are tb1 = 0.98, ts1 = 0.87, tb2 = 0.90, ts2 = 0.71, Ls1 = Ls2 = 100 µm, and L1 = L2 = L3 =100 µm.As can be seen, both the transmission spectra at Port 2 and the reflection spectra at Port 1 show Rabi-like splitting that features asymmetric split resonances, which result from coherent mode interference, analogous with the interaction between different quantum

Fig. 2 .
Fig. 2. Influence of the variation in the length of L2 (ΔL2) on the device spectral response.(a) Rabi split resonances for various ΔL2 at the transmission port.(b) Rabi split resonances for various ΔL2 at the reflection port.In (a) and (b), (i) shows the power transmission spectra, (ii) shows the center wavelengths of the two resonances in (i) as functions of ΔL2, and (iii) shows the Q factors of the two resonances in (i) as functions of ΔL2.The structural parameters are kept the same as those in Fig. 1(c) except for the variation in the length of L2.

) and the reflection spectra in Fig. 2 (Fig. 3 2 ,
Fig.3shows the device spectral response for various ΔL3, i.e., the variation in the length

Fig. 3 .In Fig. 4 ,
Fig. 3. Influence of the variation in the length of L3 (ΔL3) on the device spectral response.(a) Rabi split resonances for various ΔL3 at the transmission port.(b) Rabi split resonances for various ΔL3 at the reflection port.In (a) and (b), (i) shows the power transmission spectra, (ii) shows the center wavelengths of the two resonances in (i) as functions of ΔL3, and (iii) shows the Q factors of the two resonances in (i) as functions of ΔL3.The structural parameters are kept the same as those in Fig. 1(c) except for the variation in the length of L3.

Fig. 4 .
Fig. 4. Influence of the variation in the length of L1 (ΔL1) on the device spectral response.(a) Rabi split resonances for various ΔL1 at the transmission port.(b) Rabi split resonances for various ΔL1 at the reflection port.In (a) and (b), (i) shows the power transmission spectra, (ii) shows the center wavelengths of the two resonances in (i) as functions of ΔL1, and (iii) shows the Q factors of the two resonances in (i) as functions of ΔL1.The structural parameters are kept the same as those in Fig. 1(c) except for the variation in the length of L1.

Fig. 5 .
Fig. 5. (a) Influence of ts1 on Rabi split resonances at (i) the transmission port and (ii) the reflection port.(b) Influence of ts2 on Rabi split resonances at (i) the transmission port and (ii) the reflection port.In (a) and (b), the structural parameters are kept the same as those in Fig. 1(c) except for ts1 and ts2, respectively.

Fig. 6 (
a) shows the transmission and reflection spectra versus tb1, where the left resonance extinction ratio decreases with tb1, while the right extinction ratio shows the opposite trend, reflecting their trade-off.The transmission and reflection spectra versus tb2 are shown in Fig. 6(b).Unlike the trade-off between the extinction ratios of the two resonances in Fig. 6(a), both resonance extinction ratios in Fig. 6(b) decrease with tb2, highlighting the difference in the influence of tb1

Fig. 6 .
Fig. 6.(a) Influence of tb1 on Rabi split resonances at (i) the transmission port and (ii) the reflection port.(b) Influence of tb2 on Rabi split resonances at (i) the transmission port and (ii) the reflection port.In (a) and (b), the structural parameters are kept the same as those in 1(c) except for tb1 and tb2, respectively.

Fig. 7
Fig.7shows the power transmission and reflection spectra for different values of L3.The