## Abstract

Topological insulators (TIs) implemented in synthetic dimensions have recently emerged as an attractive platform to explore higher-dimensional topological phases in compact systems. Here, we present a two-dimensional TI within a single-ring resonator enabled by acousto-optic interactions and electro-optic modulation. In our system, the synthetic dimensions are represented by the range of discrete optical modes supported by the ring resonator and their azimuthal angular order. Gauge fields responsible for the topological order in the synthetic lattice are realized by an array of racetrack couplers coupled to the resonator. We reveal topological bulk and chiral edge bands in time-resolved absorption/transmission spectra, and we show that the proposed system can support reconfigurable and nonreciprocal frequency conversion controlled by the probe frequency detuning. Interestingly, we also show that realistic phase mismatch and disorder in acousto-optic scattering can enable an amorphous TI phase in synthetic space, demonstrating robust nonreciprocal frequency conversion in this regime.

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

## 1. INTRODUCTION

Topological photonics has been intensely investigated over the past decade for its exotic phenomena and inherent degree of robustness of the resulting devices [1–3]. Topologically protected transport of photonic edge states [4–8], the quantum spin Hall effect for light [9–11], synthetic gauge fields for neutral particles [12–14], and topological lasing [15–17] have been demonstrated based on these principles, showcasing a great potential of topological concepts for integrated photonic circuits, as well other relevant applications such as reflective limiters [18,19]. However, conventional topological photonic systems require complex arrays with periodicity in the order of the wavelength, implying large footprints and lack of compatibility with the typical requirements of integrated optoelectronic systems. At the same time, higher-dimensional topological physics has been inspiring a range of exotic phenomena for electronics, photonics, and other platforms; nevertheless, direct implementation of these concepts is hindered by fundamental limitations. For example, topological photonic integrated devices that extend beyond two dimensions are impractical, and fundamental phenomena like the four-dimensional quantum Hall effect [20,21] are not accessible in physical systems relying only on spatial dimensions. Motivated by these challenges, alternative approaches to topological photonics have been recently explored, in particular taking advantage of synthetic dimensions that enable the demonstration of higher-dimensional topological phenomena in lower-dimensional devices [22–36]. This approach utilizes other degrees of freedom (DoFs) of photonic systems as additional dimensions, inducing topological order through gauge fields in this synthetic space. So far, the DoFs exploited in the literature have included the discrete spectrum of modes supported by cavities and waveguide arrays, and spin/orbital angular momenta of optical fields. Based on these principles, two-dimensional (2D) topological insulators (Tis) based on one-dimensional (1D) arrays of coupled resonators [24,26,27], and even zero-dimensional (0D) cavities [36], or three-dimensional (3D) Weyl points in 2D cavity arrays [25], have been demonstrated. However, these synthetic TIs either involve several resonators or face challenges in their experimental realization over small footprints. The recent experimental demonstration of a Hall ladder based on a single-ring resonator has been paving the way to compact topological photonic devices amenable for integration [34]. This synthetic topological lattice was demonstrated in a single resonator supporting clockwise and counterclockwise optical modes coupled by an eight-shape coupler and modulated through electro-optic phenomena, yielding a limited synthetic space, in which one of the synthetic dimensions has just two sites.

Here, we build on the concept of photonic TIs based on synthetic dimensions, revealing the role that optomechanical interactions can play in this platform. Strong light-sound interactions have been drawing significant interest for exploring fundamental mesoscale quantum mechanical phenomena [37,38], enabling microscale nonreciprocal systems [39,40], ultrasensitive mechanical sensors [41,42], quantum information processing [43,44], and topological optomechanical crystals [45,46]. Brillouin scattering, as an example of multimode optomechanical phenomena, occurs via strong cross talk between optical and mechanical modes enabled by nonlinearities in photonic waveguides [47,48]. However, one of the challenges in Brillouin integrated photonics is the implementation of systems with well-confined mechanical (or phononic) modes, which requires challenging technological efforts. To overcome this limitation, optomechanical systems relying on acousto-optic interactions in various piezoelectric platforms have been recently investigated [49–51]. Similar to Brillouin optomechanics, acousto-optic phenomena can be implemented in chip-scale integrated photonics by injecting acoustic pumps through external RF sources. This platform offers interesting opportunities to enable gauge fields in synthetic dimensions, ideal for higher-dimensional topological phenomena.

In this work, we explore an on-chip 2D synthetic TI realized within a single-ring resonator. The synthetic lattice is created by coupling optical modes with two DoFs, based on acoustic pumping and electro-optic modulation (EOM). A gauge field is introduced in the synthetic lattice by racetrack couplers coupled to the ring resonator. As a result, we can observe both bulk and topological edge bands by calculating time-resolved absorption and transmittance spectra of the ring resonator. We further construct a finite 2D synthetic lattice supporting reversible one-way, topologically robust frequency conversion corresponding to the nonreciprocal propagation of chiral edge states in synthetic space. Realistic issues such as fabrication imperfections and perturbation from either measurements or environmental effects unavoidably cause deviations from the ideal phase-matching conditions, thereby introducing randomly generated disorder in the synthetic lattice. We show that these seemingly undesirable features allow us to access an amorphous topological phase in synthetic space, and observe amorphous topological transitions induced by disorder [52,53]. Lastly, we present a realistic toroidal ring resonator design that supports the practical requirements to enable our proposal in a practical layout.

## 2. HAMILTONIAN FORMULATION

Two ingredients are needed to generate a 2D synthetic TI in a single-ring resonator. The first consists in coupling the optical modes supported by the resonator with their neighbors in a suitable 2D lattice in synthetic space, where the two synthetic dimensions are denoted by letters $n$ and $m$, as shown in Fig. 1(a). The second ingredient is to create an artificial gauge field in this synthetic space, for example by selecting the phases in the couplings along dimension $n$ to be proportional to index $m$, such that a synthetic phase gradient is established, inducing nontrivial topological order. We consider an optical resonator [Fig. 1(b)] supporting $N$ optical spatial mode families with different resonant frequencies ${\omega _{n,m}}({n = 1,2,..,N})$ and distinct modal profiles and polarizations, where $m$ is the azimuthal number, $m \in {\mathbb Z}$. We assume that the modes share the same group velocity within the frequency range of interest, and we refer to modes with distinct indices $n$ and identical indices $m$ as *intermodes*. Each optical mode family ${a_{n,m}}$ with given $n$ shares the cross-sectional modal profile with a set of longitudinal modes at equally spaced frequencies $E$, commonly known as free spectral range (FSR), but different $m$, and these modes are referred to as *intramodes* of our synthetic lattice. We study a realistic design of the ring resonator in Section 6, showing that its optical modes can indeed meet these requirements over the frequency range of interest.

To couple the intermodes and introduce a phase gradient in their coupling, we place $N - 1$ tailored racetrack couplers close to the ring resonator, such that each of them shares two successive resonant mode families with the ring, for example the ($n - 1$)th ($n$th) racetrack coupler mainly supports resonant modes ${a_{n - 1,m}}({{a_{n,m}}})$ and ${a_{n,m}}({{a_{n + 1,m}}})$, but it is off-resonant with respect to the other mode families [see Fig. 1(b)]. Next, we add one interdigitated transducer (IDT) in every racetrack coupler, generating an acoustic pump exciting a propagating phonon mode coupled to the optical modes when suitable phase-matching conditions are satisfied, tailored so that the modes can couple only in one direction through forward acousto-optic scattering, while backward scattering is suppressed, since reciprocity is broken by the acoustic traveling wave. Figure 1(c) shows a schematic describing the resulting mode coupling in the frequency-momentum plane, in which the frequency of the phonon mode supported by the ($n - 1$)th racetrack coupler is chosen to satisfy ${{\Omega}_{n - 1}} = {\omega _{n,m}} - {\omega _{n - 1,m}}$, and its wavenumber satisfies ${q_{n - 1}} = {k_{n,m}} - {k_{n - 1,m}}$, where ${k_{n,m}}$ is the wavenumber of the optical mode. In this scheme, a coupling ${G_{n,m}}$ is established between modes ${a_{n - 1,m}}$ and ${a_{n,m}}$ via forward acousto-optic scattering, thus inducing controllable intermodal coupling in the synthetic dimension $n$ in Fig. 1(a). ${G_{n,m}}$ is determined by the overlap integral of the spatial modes, implying that the intermodal couplings are not generally identical to each other because of the difference in modal overlap and group velocity deviations. However, we can acoustically pump them with external piezoelectric sources, so that the effective couplings along one of the synthetic dimensions ($n$), representing different spatial modes, can be made identical by adjusting the acoustic pump strengths and their frequency detuning. The intermodal couplings along the second synthetic dimension ($m$) share the same spatial mode families; hence, they can be assumed identical since the cross-sectional modal profiles are nearly independent of the azimuthal number $m$. Although the group velocity gradually changes with increasing $m$, affecting the phase-matching condition of acousto-optic interaction, in the limited frequency range of interest this deviation is tiny, so that the detrimental effect on the intermodal couplings is negligible. We can include an additional phase ${e^{i{\theta _m}}}$ in the intermodal coupling coefficients through the phase delay due to the optical modes traveling through the coupler, where ${\theta _m}$ is linearly proportional to the azimuthal number $m$, ${\theta _m} = m\theta$, and $\theta$ is determined by the coupler length. A detailed discussion about the relation between $\theta$ and the coupler geometry can be found in Supplement 1.

As shown in Fig. 1(b), we also include an EOM within the ring resonator, whose modulation frequency matches the FSR, so that intramodal couplings can be achieved and controlled along dimension $m$. Every optical mode ${\hat a_{n,m}}$ experiences intermodal couplings controlled by the acoustic pumps and intramodal couplings controlled by the EOM, yielding the desired 2D synthetic lattice in Fig. 1(a). Its Hamiltonian after linearization and the rotating wave approximation (RWA) assumes the form (see details in Supplement 1),

## 3. SYNTHETIC TOPOLOGICAL BANDS

To probe the topology of our synthetic lattice, we couple the ring to a drop-port waveguide [Fig. 1(b)] and develop input-output theory for this system see Supplement 1). We assume that each optical mode ${\hat a_{n,m}}$ is coupled to the waveguide with rate ${\gamma _{n,m}}$. The system is probed with a monochromatic source with input amplitude ${s^ +}$ and frequency ${\omega _s}$ satisfying ${\omega _s} = {\omega _{{n_0},{m_0}}} - {\Delta}$, where ${\Delta}$ is the frequency deviation (detuning) of the driving laser from the targeted mode ${\hat a_{{n_0},{m_0}}}$ at ${\omega _{{n_0},{m_0}}}$, and $| {\Delta} | \ll E,{{\Omega}_n}$. Under RWA, the Hamiltonian coupled to the waveguide is

We numerically examine a strip in synthetic space consisting of $N = 10$ mode families, with periodic boundaries along $m$ and open boundaries along $n$, enforced by the phase-matching conditions imposed by the acousto-optic interactions. The probe frequency ${\omega _s}$ is chosen as ${\omega _s} = {\omega _{{n_0},{m_0}}} - {\Delta}$, where ${\omega _{{n_0},{m_0}}}$ indicates that the source is placed at site $({n_0},{m_0}$) of the synthetic lattice, and ${\Delta}$ determines the excitation of the Floquet eigenmode with eigenfrequency ${\Delta}$, and it does not exceed the bandwidth of the Floquet bands. Our synthetic lattice offers a unique advantage over real space lattices, as we can excite any site in synthetic space and arbitrary eigenstates swiftly by only changing the probe frequency ${\omega _s}$. In addition, the acousto-optic modulation allows us to tune probe and detection frequencies separately, consistent with recent experimental works [49–51]. When probe and detection frequencies are identical, the transmittance spectrum ${{{T}}_{{n_0},{p_0}}}({{k_f},{\Delta}})$ represents the main band window, otherwise ${{{T}}_{n,p}}({{k_f},{\Delta}})$ $({n \ne {n_0},p \ne {p_0}})$ corresponds to a sideband window. Detection at the main band (sideband) corresponds to onsite (offsite) measurements in synthetic space.

Figure 2 shows the retrieved spectra by exciting the ring resonator with probe frequencies ${\omega _s} = {\omega _{5,3}} + {\Delta}$, ${\omega _{1,2}} + {\Delta}$, ${\omega _{10,2}} + {\Delta}$, i.e., targeting optical modes ${a_{5,3}}$, ${a_{1,2}}$, ${a_{10,2}}$, corresponding to a bulk site (5,3) [Figs. 2(a) and 2(d)], site (1,2) on the left edge [Figs. 2(b) and 2(e)], and site (10,2) on the right edge [Figs. 2(c) and 2(f)] of the synthetic strip, and detecting the signal at frequencies ${\omega _d} = {\omega _{5,3}} + {\Delta}$, ${\omega _{1,1}} + {\Delta}$, ${\omega _{10,3}} + {\Delta}$, i.e., detecting optical modes ${a_{5,3}}$, ${a_{1,1}}$, ${a_{10,3}}$, corresponding to the same bulk site, the neighboring edge site (1,1) and the neighboring edge site (10,3), respectively. The frequency detuning is chosen as $\frac{\Delta}{{{G}_0}} = - 2.608$ to target a bulk mode, and as $\frac{\Delta}{{{G}_0}} = - 1.824$ to target the edge modes, consistent with the eigenvalue spectrum in Supplementary Fig. 2.

In Fig. 2(a) we show the time resolved absorption for ${\omega _s} = {\omega _{5,3}} - 2.608{G}_0$, targeting and detecting the same bulk site (5,3) in the synthetic lattice. The time-dependent response represents the excitation of bulk modes in synthetic space and, as expected, it is symmetric with respect to ${k_f}$ in one period because no chirality is expected for bulk modes. Time-resolved transmission curves in the sideband window are shown in Figs. 2(b) and 2(c) for probe frequencies ${\omega _s} = {\omega _{1,2}} - 1.824{G}_0$ and ${\omega _s} = {\omega _{10,2}} - 1.824{G_0}$ and detected frequencies ${\omega _s} = {\omega _{1,1}} - 1.824{G}_0$ and ${\omega _s} = {\omega _{10,3}} - 1.824{G_0}$, which target sites on the left (1,2) and right (10,2) edges of the synthetic strip, and detect neighboring sites on the left (1,1) and neighboring site on the right (10,3) edges of the synthetic strip. Here the transmission evolves asymmetrically in time within one period due to the expected chiral features of the edge modes in the 2D lattice of Fig. 1(a). The two responses are time-reversed replicas of each other, as expected, since the excitation at edge site (1,2) and detection at edge site (1,1) of the synthetic strip is the time reversal and inversion symmetric of the excitation at edge site (10,2) and detection at edge site (10,3) in synthetic space.

We can reconstruct the complete band structure of these bulk and edge modes performing similar time-resolved calculations by sweeping ${\Delta}$, as shown in Figs. 2(d)–2(f). The spectrum in Fig. 2(d) corresponds to the spectrum of bulk modes, clearly showing the emergence of two bulk bandgaps, in which minimum excitation of bulk modes in synthetic space is observed. We can evaluate the Chern number of these bandgaps (Supplement 1), as indicated in the figure, which have an opposite sign, revealing the topologically nontrivial nature of the synthetic lattice. Therefore, we can expect that these synthetic topological bandgaps support chiral edge states connecting the bulk bands in synthetic space with distinct chiral features, as indeed verified by the transmittance spectra in Figs. 2(e) and 2(f). Specifically, in Fig. 2(e), as we increase the frequency detuning in the lower bulk bandgap, the transmission peak in the time domain shifts from $\pi$ to ${-}\pi$. In the upper bulk bandgap, the shift is in the opposite direction as ${\Delta}$ increases, indicating flipped chirality of the supported edge states. On the other hand, the transmission peak in Fig. 2(f) shifts from ${-}\pi$ to $\pi$ in the lower bandgap and shifts from $\pi$ to ${-}\pi$ in the upper bandgap as ${\Delta}$ increases, exhibiting opposite chirality due to the symmetry constraints mentioned above. We use these chiral features of topological edge states in synthetic space to demonstrate in the next section reversible and nonreciprocal frequency conversion.

## 4. TUNABLE NONRECIPROCAL FREQUENCY CONVERSION

In the previous section, we assumed that the synthetic dimension $m$ extends to infinity, and therefore we imposed periodic boundary conditions. In reality, the group index gradually varies as a function of frequency; thereby, the phase-matching condition cannot be satisfied over a very large range of $m$. This feature appears to be a drawback, but it effectively introduces a second boundary to the synthetic lattice, thus offering the chance to explore routing of topological edge states in synthetic space. We introduce sharply terminated boundaries in synthetic dimension $m$ at sites $\pm{m_0}$ by placing additional resonators near the ring [36], off-resonant with respect to the modes of interest except for modes ${\hat a_{n, \pm {m_0}}}$. Assuming that ${\hat a_{n, \pm {m_0}}}$ majorly couple to optical modes ${\hat d_{n, \pm}}$ supported by the external resonators with coupling coefficient ${\lambda}_{n, \pm {m_0}}^ \pm$, and the probing source targets optical mode ${\hat a_{{n_0},{m_0}}}$, the Hamiltonian becomes

If the coupling between the external resonators and the ring resonator is large enough so that effective potential walls are created at synthetic sites $\pm{m_0}$, the signal is prevented from propagating beyond modes $\pm{m_0}$. The steady-state transmittance for optical mode ${\hat a_{n,m}}$ is then calculated in this geometry, with details shown in Supplement 1.

Propagation of topological states in our 2D synthetic space implies that the optical mode excited by the probe signal is converted to other modes as time evolves, as confirmed by the results in Fig. 2, and the total outgoing signal will consist of a superposition of optical modes. The conversion can be observed by focusing the detector on either the main band window or the sideband windows, as shown in Figs. 3(a) and 3(b), respectively, as we scan the frequency detuning. When the probe detuning is within the bulk bandgap and the detector frequency is in the main band window, the transmission at mode ${a_{1,0}}$, corresponding to an edge site in the synthetic lattice [upper red arrow in Fig. 3(c)], is smaller compared to probing mode ${a_{4,0}}$ that lies in the synthetic bulk [red curve versus blue curve in Fig. 3(a)]. This confirms that mode conversion occurs at edge sites in synthetic space when ${\Delta}$ is in the bulk bandgap. The sideband window detection [Fig. 3(b)] reveals one-way frequency conversion of optical modes: we can observe signal isolation between ${a_{1,0}}$ and ${a_{1, - 16}}$, where ${a_{1, - 16}}$ represents a site at the lower boundary of the synthetic lattice [lower red arrow in Fig. 3(c)], defined as ${s_{{\rm iso}}} = {T_{1, - 16 \to 1,0}}/{T_{1,0 \to 1, - 16}}$. This quantity is significantly larger when ${\Delta}$ is in the upper bandgap (indicated by the right shaded region) than when ${\Delta}$ is in the bulk band or in the lower bandgap (left shaded region) in Fig. 3(b). This confirms again that, when ${\Delta}$ is in the upper (lower) bandgap, the excited modes convert toward modes with increasing (decreasing) indices $m$ and fixed index $n = 1$, stemming from the excitation of chiral edge modes propagating clockwise (counterclockwise) in synthetic space.

To visually demonstrate this one-way routing, the transmittance magnitude of all optical modes is plotted in the synthetic lattice in Figs. 3(c) and 3(d), in which the probed mode is fixed at ${a_{1,0}}$. Since topological edge states in the lower and upper bandgaps manifest opposite chirality, due to the opposite Chern number of the corresponding bandgaps, nonreciprocal frequency conversion can be reversed by changing the sign of the frequency detuning. For instance, in Fig. 3(c), the frequency detuning is selected at a frequency in the upper bulk bandgap, and one-way frequency conversion is revealed by the excitation of modes ${a_{1,m}}$ with $m \ge 0$, corresponding to the sites at the left boundary above the probed site, and of modes ${a_{n,16}}$, corresponding to the sites at the top boundary of the synthetic lattice. If the frequency detuning is shifted to the lower bandgap, one-way frequency conversion occurs in the opposite direction, corresponding to excited modes ${a_{1,m}}$ with negative $ m $ and modes ${a_{n, - 16}}$ [Fig. 3(d)]. The *ad hoc* tunability of nonreciprocal frequency conversion based on our platform enables exciting opportunities for applications such as tunable directional frequency comb generation.

## 5. AMORPHOUS TIs IN SYNTHETIC SPACE

The topological nature of edge propagation along the synthetic boundary implies that one-way frequency conversion is expected to persist when phase-matching conditions are relaxed. In other words, the strength of the acousto-optic interactions and the EOM frequency can deviate from their optimal values without affecting the previous findings, as long as the topological bandgap does not close. In reality, many independent factors, such as geometric dispersion due to structural bends and imperfections, group velocity deviations due to the material dispersion, the imprecise alignment of multiple acoustic sources, can cause a deviation of intramodal and intermodal couplings from their ideal values. These nonidealities cannot be precisely controlled in an experiment; thus, it is reasonable to assume that they can be described with random noise introduced in the intramodal and intermodal couplings. In turn, this noise breaks the translational symmetry of the synthetic lattice, while its onsite potential is unaffected by these factors under the RWA. The occurrence of disorder in the couplings not only enables us to test the topological robustness of our synthetic TI, but it also offers the opportunity to access an amorphous topological phase in synthetic space.

To study the emergence of such amorphous phase, we consider a 2D synthetic lattice with periodic boundary conditions enforced in both directions. Random disorder is introduced in both intermodal and intramodal couplings, characterized by noise level $x$ (see the definition of $x$ in Supplement 1). The eigenvalue variation of the synthetic amorphous TI for different noise levels indicates a topological phase transition in their eigenenergies at the critical noise level ${x_t} \approx 2$ [blue colored dots in Fig. 4(a)]. When $x \gt {x_t}$, the topological bandgap closes, and the topological phase of the synthetic amorphous lattice becomes trivial due to the large amount of disorder. We calculated the Bott index, a defining topological invariant in disordered TIs (Supplement 1), to study the persistent topological nature of the randomly perturbed synthetic lattice as the strength of the disorder changes. The Bott index is numerically evaluated for synthetic lattices of various sizes, and it converges to unity (topological phase) when $x \lt {x_t}$ and to zero (trivial phase) when $x \gg {x_t}$ as the system size becomes infinite [Fig. 4(b)]. In order to demonstrate the robust nature of nonreciprocal frequency conversion in our ring resonator and the amorphous topological phase transition caused by increasing disorder, the transmittance magnitude of all optical modes, evaluated in the amorphous synthetic TI, are mapped to the corresponding synthetic lattice [Fig. 4(c)]. When the disorder intensity is moderate (for example, $x = 0.1$), nonreciprocal frequency conversion is clearly preserved, as revealed by the mapped field distribution majorly localized at the upper boundary of the synthetic lattice in the left panel of Fig. 4(c), indicating the topological robustness of topological edge propagation and nonreciprocal frequency conversion. When the disorder is large enough such that the amorphous synthetic lattice transitions to the trivial phase, one-way frequency conversion is no longer preserved, as shown by the randomly scattered field distributions in the right panel of Fig. 4(c). These simulations verify our speculation that the strict phase-matching conditions of the acousto-optic and EOM interactions are not strictly required to realize nonreciprocal frequency conversion, as long as the disorder of the amorphous TI is not large enough to close the bandgap, significantly enhancing the possibility of an experimental implementation of our proposal.

## 6. PRACTICAL IMPLEMENTATION

In this section, we provide a concrete example of a toroidal ring resonator geometry in which the optical modes meet the requirements of our proposal, and thus provide a platform to implement the synthetic 2D TI in an experiment. The ring resonator has a torus geometry, and it is made of a material that supports both piezoelectric and electro-optic responses (for example, silicon). The structure geometry embedded in air is shown in Fig. 5(a) and described in the figure caption. We aim at an operation across a range of frequencies for which the azimuthal number of the supported modes is large enough so that the deviation in FSR among different mode families is limited. For example, Fig. 5(b) shows the cross-sectional field profiles of six optical modes, whose dispersions are plotted in the frequency range of interest in Fig. 5(c). We find that their dispersion curves are approximately parallel to each other, since the average FSR for these modes is $\bar E \cong 37.6\; {\rm GHz}$, and the standard deviation is ${\delta _{{\rm FSR}}} \cong 0.48\;{\rm GHz} $, less than 1.3% of $\bar E$. These features ensure that EOM of the selected modes is experimentally implementable in an integrated platform. In addition, we can scale up the design to decrease the average FSR and further ease the experimental effort. The phonon modes can be chosen accordingly to achieve maximum intermodal couplings induced by acoustic pumps. The blue-shaded region in Fig. 5(c) is shown in more detail in Fig. 5(d), to highlight the frequency difference between neighbors. The phase-matching conditions denoted by gray shaded regions in this panel are given as illustrating examples, and other choices of phase-matching conditions and of the order of indices $n$ assigned to the different mode families can be made, as long as the constraint $\mathop \sum \limits_{n = 1}^5 {{\Omega}_n} \le E$ is met, where ${{\Omega}_{n - 1}}(n \gt 1)$ is the frequency difference between ${a_{n - 1}}$ and ${a_n}$, which equals the frequency of the propagating phonon mode excited by the acoustic pump. Therefore, the required average acoustic modulation frequency ${\bar \Omega}$ is less than 7.52 GHz, which is realistically achievable in an experiment. The effective couplings ${G_n}$ and ${\kappa _n}$ between optical modes can assume values in the range from 1 MHz up to 1 GHz, and the frequency detuning of the laser ${\Delta}$ is approximately in the same range as the effective couplings, thus meeting the condition ${\Delta} \ll {\bar \Omega}$, $\bar E$, which is necessary to remain within the RWA.

## 7. CONCLUSION

In this paper, we have introduced a scheme to realize synthetic 2D TIs within a single-ring resonator using the combination of acoustic pumps and an EOM imparting gauge fields in a synthetic lattice formed by the coupled modes of the ring resonator. The described synthetic topological platform can be realized in various realistic experimental setups. Such an implementation requires the fabrication of a ring resonator supporting several optical mode families, well separated in frequency space, with approximately the same FSR, and similar group velocities in the frequency range of interest. A specific design for such a ring resonator is provided and meets these requirements. The racetrack couplers are required to support traveling phonon modes inducing tailored intramodal couplings with the respective optical modes, while the coupling to other optical modes should be negligible, since the phase-matching conditions are not satisfied. Their cross-sectional overlap integral should at the same time be minimized by judiciously choosing their profile. These requirements are addressable with appropriate design of the couplers.

We have used this platform to show both bulk and chiral edge bands of synthetic lattice, clearly reflected in time-resolved absorption/transmittance spectra. Based on these principles, we have also shown the possibility of inducing topologically protected and highly reconfigurable nonreciprocal frequency conversion. Such a unidirectional conversion is controllable with the detuning of the probe signal and pump parameters, thus providing exciting opportunities for achieving robust and highly flexible nanophotonic spectrum management, frequency comb generation. The inherent robustness to disorder stemming from these topological features also enables access to amorphous topological phases in this synthetic space, with exciting opportunities for classical and quantum optics. Several other topological phenomena may be revealed based on the same platform, such as the emergence of Hofstadter butterfly spectra [54], topological charge pumping [55], and Bloch oscillations [56].

## Funding

Office of Naval Research (N00014-19-1-2011); Air Force Office of Scientific Research MURI (FA9550-18-1-0379); Simons Foundation.

## Disclosures

The authors declare no conflicts of interest.

## Data availability

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

## Supplemental document

See Supplement 1 for supporting content.

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