## Abstract

Photonic gauge potentials are crucial for manipulating charge-neutral photons like their counterpart electrons in the electromagnetic field, allowing the analogous Aharonov–Bohm effect in photonics and paving the way for critical applications such as photonic isolation. Normally, a gauge potential exhibits phase inversion along two opposite propagation paths. Here we experimentally demonstrate phonon-induced anomalous gauge potentials with noninverted gauge phases in a spatial-frequency space, where near-phase-matched nonlinear Brillouin scatterings enable such unique direction-dependent gauge phases. Based on this scheme, we construct photonic isolators in the frequency domain permitting nonreciprocal propagation of light along the frequency axis, where coherent phase control in the photonic isolator allows switching completely the directionality through an Aharonov–Bohm interferometer. Moreover, similar coherent controlled unidirectional frequency conversions are also illustrated. These results may offer a unique platform for a compact, integrated solution to implement synthetic-dimension devices for on-chip optical signal processing.

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

## 1. INTRODUCTION

A gauge potential, which in nature couples only to charged particles, leads to the intriguing Aharonov–Bohm effects even in the absence of an electromagnetic field [1]. Recently, artificial effective gauge fields have been created for neutral particles such as photons in real space [2–8] and also synthetic space [9–14], opening up an emerging field to mold the flow of photons just like their counterpart, electrons. The development of photonic gauge potentials enables significant applications involving unidirectional spatial propagation of light [7], nonreciprocal phase shift in photonic Aharonov–Bohm (AB) effect [2,3,15], robust bulk transport in photonic topological insulator [14], and chiral currents in synthetic dimensions [13]. A key motivation for demonstrating photonic gauge potential is to create effective magnetic fields for photons [10,11,16]. Such an effective magnetic field enables one to break reciprocity for photons and to achieve nonmagnetic photonic isolation [10], which is crucial for on-chip photonic integration.

Previously, the pioneering works of photonic gauge potentials for photonic AB effect [3,15] utilized two standing-wave modulators with different modulation phases. A modulator can convert light between two photonic states with different frequencies. It was noted in [3] that a standing-wave modulator can provide a (gauge-dependent) nonreciprocal phase response. In the Hamiltonian that describes the modulator, the phases associated with the upconversion and downconversion processes are equal in magnitude to the modulation phase and opposite in sign [3]. Such a nonreciprocal phase response corresponds to photonic gauge potential. Cascading two of such modulators with two different modulation phases removes the gauge dependency and allows for nonreciprocal transmission [2]. In another proposal, a traveling-wave spatiotemporal modulation can also induce nonreciprocity through interband/intermode photonic transitions inside waveguide structures [17,18]. Such traveling-wave modulation, in particular, can be created with the use of acoustic phonons [19–22]. In these systems, a single traveling-wave modulator directly creates nonreciprocal intensity dynamics, since the frequency conversion process is phase-matched in the forward direction, and phase-mismatched in the backward direction [23–25].

On the other hand, an emerging field of synthetic dimensions in photonics has attracted broad attention by extending various degrees of freedom of photons [26,27], where physics in synthetic dimensions can exhibit high-dimensional physical phenomena with low-dimensional photonic structures [26]. Moreover, it provides unique opportunities for manipulating these degrees of freedom of photons including optical frequency [9,10,28–32], the orbital angular momentum of light [33,34], and temporal lattice [35]. One major challenge for synthetic dimensions in photonics is to create effective magnetic fields [10,11,16] to manipulate photons so that they become nonreciprocal for some crucial applications such as photonic isolator in frequency space, which is critical for multi-wavelength optical information processing [36]. As in the spatial domain, time reciprocity must be broken to introduce such effective magnetic fields [37]; however, it becomes a nontrivial task for synthetic frequency space.

In this paper, we experimentally demonstrate an anomalous gauge potential for photons in frequency space. For this purpose, we consider an optomechanical system supporting both optical and acoustic resonance. Thus, an optical photon will also experience upconversion and downconversion through the interaction with acoustic phonons. However, in our system, the phases associated with the two time-reversed upconversion and downconversion processes are not equal in magnitude, which is caused by a near-phase-matching condition in the nonlinear Stokes/anti-Stokes scattering processes involving a traveling-wave phonon. In contrast with [3], therefore, we refer to such a scenario of having unequal upconversion and downconversion phases as having an anomalous gauge potential for photons. And we refer to such an optomechanical system as a gauge-field modulator in the frequency space. Based on this anomalous gauge potential, we demonstrate a photonic isolator in the frequency space, constructed as an interferometer with a parallel-connected pair of an electro-optic modulator and a gauge-field modulator (GM), which can achieve unidirectional frequency conversion. Moreover, we also demonstrate the direction of the frequency conversion can be coherently controlled by controlling these upconversion and downconversion phases. This compact direction-dependent GM enables nonreciprocal light transmission in the synthetic frequency domain, opening up new avenues for the synthetic dimension toward practical applications in all-optical signal processing and on-chip photonic integration.

## 2. RESULTS

#### A. Photonic Isolator in the Frequency Space

A photonic isolator in the frequency space allows nonreciprocal transmission between two signal ports (waveguides or scattering channels) at different carrier frequencies, i.e., ${{{\omega}}_0}$, ${{{\omega}}_1}$, as shown in Fig. 1(b). In contrast, a photonic isolator in the spatial domain allows light in one direction but blocks light in the opposite direction *at the same frequency* by breaking time reciprocity in an asymmetric scattering matrix [37]; and a unidirectional frequency converter only permits frequency conversion in one direction, while its reciprocal conversion in the other direction is not guaranteed in prior works [19] (more theoretical analysis using scattering matrix among the three cases has been included in Supplement 1, Note 1). Here the photonic isolator permits light frequency conversion in one direction (${\omega _0} \to {\omega _1}$) while forbidding the opposite one (${\omega _0} \nleftarrow {\omega _1}$) between the two ports at different frequency states. This is crucial for the emerging field in photonic synthetic dimensions [26,28–30,33–35], as well as optical information processing where the ever-growing hunger for signal bandwidths has pushed the limit of it into multi-frequency domains, e.g., dense wavelength-division multiplexing (DWDM). Inspired by recent works of spatial photonic isolators based on photonic gauge potentials and effective magnetic flux [3,4], we would like to construct similar gauge potentials for photonic isolation, but in the frequency space.

In the spatial case, it is straightforward to create effective gauge potentials by imprinting the modulation phase during the frequency conversion processes in traditional electro-optical modulators (EOMs) [2,3]. For example, photonic AB effects [3,15] have been constructed with gauge potentials using two serial-connected EOMs, which are individually phase-controlled, and a filtered single-frequency channel has to be implemented to enable photonic isolation in the spatial domain. Similar approaches using two tandem phase modulators have also been demonstrated to achieve spatial photonic isolation [38,39]. It is important to note that during these processes with frequency conversion, a traditional EOM exhibits a reciprocal conversion phase in the frequency axis, i.e., ${\varphi _ \pm} = - {\varphi _ \mp}$ (${+}/ -$ for anti-Stokes and Stokes processes, respectively), which is associated with a normal gauge potential in EOMs [3]. Instead, to realize such a photonic isolator in the frequency space, we consider an interferometer with two frequency modulators/converters including an EOM and a GM in parallel connection [Fig. 1(a)], where the forward transmission (${\omega _0} \to {\omega _1}$) is determined by the interference phases, i.e., ${\varphi _0} + {\varphi _{\rm{Forward}}}$ between the EOM and the GM, while the backward one (${\omega _0} \leftarrow {\omega _1}$) contains a nonreciprocal phase ${-}{\varphi _0} + {\varphi _{\rm{Backward}}}$, leading to unequal transmission rates between the two signal ports. Obviously, this requires the proposed GM to contain path-depended, unequal conversion phases, i.e., ${\varphi _{\rm{Forward}}} \ne - {\varphi _{\rm{Backward}}}$, which is *anomalous* as compared to the normal gauge potential in EOM, i.e., ${\varphi _ \pm} = - {\varphi _ \mp}$. In contrast, if the GM is replaced by another EOM, these two parallel-connected EOMs [3] will exhibit reciprocal phases for the above process, resulting in equal transmissions.

#### B. Photonic Gauge-Field Modulator

Unlike the normal gauge potential in traditional EOMs, the proposed GM with an anomalous gauge potential should possess path-depended, unequal conversion phases. To realize such anomalous gauge potential, a conceptual GM is illustrated in Fig. 1(b), where a photon undergoes a frequency conversion propagating through the GM with an internal and directional temporal dynamic modulation induced by acoustic phonons through electrostriction [22]. As a result, photons propagating from the left and the right both experience frequency conversion through the structure, but with different phases depending on the propagation direction, which is caused by the traveling-wave modulation according to a unique near-phase-matching condition for frequency conversion (more details in the next section). For example, in the forward Stokes process (${\omega _0} \to {\omega _1} = {\omega _0} - \Omega$), the photon acquires the phase ${\varphi _{\rm{Forward}}} = \int_0^1 {A_F} \cdot {\rm d}\vec l{_F}$, where ${A_F}$ is the effective gauge potential in the spatial-frequency space for the forward Stokes process (1,0 refer to ${\omega _1}$, ${\omega _0}$ states). Meanwhile, in the backward direction, the photon acquires the phase ${\varphi _{\rm{Backward}}} = \int_1^0 {A_B} \cdot {\rm d}\vec l{_B}$ for the backward anti-Stokes process (${\omega _0} \leftarrow {\omega _1} = {\omega _0} - \Omega$), where ${A_B}$ is the corresponding gauge potential. This direction-dependent *anomalous gauge potential* in the GM can be defined as ${A_F} \ne {A_B}$, such that ${\varphi _{\rm{Backward}}} \ne - {\varphi _{\rm{Forward}}}$. In contrast, within traditional EOMs [3,4], such photon conversion phases are exactly opposite due to the fact of standing-wave modulation, i.e., ${\varphi _{\rm{Forward}}} = \int_0^1 A \cdot {\rm d}\vec l{_F} = \int_1^0 A \cdot {\rm d}\vec l{_B} = - {\varphi _{\rm{Backward}}}$, which indicates the same *normal gauge potential* ${{A}}$ in the spatial-frequency space. As shown above, anomalous gauge potential in the GM becomes the key ingredient to construct photonic isolators in the frequency space, which prevents unwanted optical signals from transmitting between different frequency channels. But how to experimentally realize this anomalous gauge potential for the GM remains a big technical challenge.

#### C. GM Modulator Based on Cavity-Enhanced Brillouin Scattering

Experimentally, we can construct such GMs with anomalous gauge potential through phonon-induced dynamic modulation based on the Brillouin scattering process in Fig. 2. Particularly, we consider a tapered fiber-coupled microsphere resonator of diameter around 100 µm with ${Q_{\rm{optical}}} \gt {10^7}$ in the ambient environment [Supplement 1, Note 2]. Such a microsphere cavity can support both optical and acoustic whispering-gallery-mode (WGM) resonances, where a strong optical pump beam is launched into one optical WGM, simultaneously generating a forward Stokes photon in a second optical WGM and an extra acoustic forward-traveling phonon through an electrostriction-induced stimulated Brillouin scattering (SBS) process [22,40]. When pumped above the threshold, an SBS laser can be excited to populate strong acoustic phonons in the microcavity [41]. Here the frequency difference between the two optical WGMs has to exactly match with the acoustic waves’ frequency [Figs. 2(b) and 2(d)], which has been verified by comparing the frequency difference of the two optical WGMs and the frequency beating note of Stokes waves in Supplement 1, Note 2. When the SBS starts lasing in the microcavity, its Stokes wave’s frequency beating reveals a linewidth around 6.2 kHz in the spectrum, corresponding to a mechanical $Q$ factor ${\sim}{Q_{\rm{mech}}}\sim2.6 \times {10^4}$, which can resonantly enhance the acoustic field in the microcavity. In this manner, the resonant-enhanced acoustic phonons can induce a strong traveling dynamic modulation inside the microcavity, which is essential for building a GM. Note that, the high $Q$ factor is essential to reduce the device’s input power; such acoustic phonons can also be generated through other methods [18,19,21]

Once the cavity-enhanced phonons are excited through the forward SBS laser in a free-running manner, we launch a secondary probe beam to sense the dynamic modulation. In Fig. 2(a), the original pump is frequency-locked to the optical modes through thermal locking [42], and its power is maintained at a constant level to ensure steady acoustic waves inside the cavity. Under this condition, the probe laser can experience the dynamic modulations when scanning through the frequency spectrum, exhibiting a Stokes shift with a frequency equaling to the acoustic wave’s frequency [Fig. 2(c)], which have been verified by comparing the frequency beating notes of two pairs, i.e., the pump and its Stokes, the probe and its Stokes, in Figs. 2(b)–2(d). Effectively, this setup also composes a scheme of Brillouin enhanced four-wave mixing involving the pump, the probe, and their Stokes waves [43–45] [Fig. 2(b)]. More surprisingly, anti-Stokes scattering can also occur simultaneously in the same apparatus, thanks to the near-phase-matching conditions which shall be discussed in the following section.

More physical insights on the GM can be understood by considering the internal phase-matching conditions in Fig. 3. Still, under a free-running SBS laser to induce phonons, we consider that a probe photon at the frequency ${\omega _0}$ injects from the left undergoing a Brillouin scattering process and converts into ${\omega _1} = {\omega _0} \pm {{\Omega}}$, where ${\pm}{{\Omega}}$ depends on anti-Stokes/Stokes processes involving phonon frequency ${{\Omega}}$. Normally, the phase-matching condition among two-photon and one-phonon modes has to be fulfilled to ensure the conversion efficiency; for example, SBS Stokes waves have to reverse their propagation directions to satisfy the phase-matching due to the large wave vector of high-frequency phonons (${\sim}\;{{11}}\;{\rm{GHz}}$) inside optical fibers [46]. However, in the current configuration, optical waves and the acoustic wave both travel in the forward direction with a phonon frequency in the range from 50 MHz to 500 MHz [47] (depend on different trials due to experimental stability issues, but it is a fixed frequency for each experiment). For these low-frequency phonons, the wave vector of phonons $G$ is much smaller than that of photons, i.e., $k \gg G$, which can be estimated as $G\sim0.02k$ for 100 MHz phonons and 1550 nm photons. As a result, a crucial near-phase-matching condition can be obtained from the mismatching vector:

where ${k_i}$ denotes the wave vector corresponding to the probe and Stokes/anti-Stokes photons. Here a forward-type SBS [22,40] in the cavity induces a forward-traveling phonon around 100 MHz, and its wave vector is much smaller than that of the photons centered at 1550 nm wavelength. Due to the small frequency difference, the wave vectors of the probe ${k_0}$ and of its anti-Stokes/Stokes ${k_1}$ are with similar magnitudes, i.e., ${k_0} \approx {k_1}$. As a result, the phase mismatch ${{\Delta}}k \approx 0$ effectively vanishes, creating a near-phase-matching condition.In this near-phase-matching regime, GMs can support anti-Stokes and Stokes processes in both propagation directions due to the small phonon wave vector $G$, as shown in Fig. 3. Take Stokes processes as an example: in the forward direction (${\omega _0} \to {\omega _1} = {\omega _0} - \Omega$), both the probe and the Stokes waves are aligned along the internal phonon’s direction, and the momentum mismatching gives ${{\Delta}}{k_{\rm{FSS}}} = {k_0} - ({k_1} + G)$. As a comparison, the backward Stokes (${\omega _0} - \Omega = {\omega _1} \leftarrow {\omega _0})$ in the same GM gives ${{\Delta}}{k_{\rm{BSS}}} = {k_0} - ({k_1} - G)$. One notices that ${{\Delta}}{k_{\rm{FSS}}}$, ${{\Delta}}{k_{\rm{BSS}}} \ll {k_i}$ according to the above argument, enabling the possibility for Stokes scattering generations in both directions, which is also verified experimentally later in Fig. 4. In contrast, prior works using dynamical modulation by AOMs and phonons in waveguides for spatial optical isolators or nonreciprocal modulation [17–19] have to rely on phase-matching conditions to achieve nonreciprocal light flow. Here we emphasize this near-phase-matching condition enables anti-Stokes and Stokes processes in both propagation directions, which is hard to archive in the phase-matching scenarios.

Experimentally, we exam anti-Stokes and Stokes frequency conversions for both propagation directions over a wide range (1539–1561 nm) within the GM, as shown in Fig. 3. First, acoustic phonons are excited with a strong pump beam through forward SBS to form a GM in a microcavity. A secondary probe beam is launched through the same tapered fiber for frequency conversions; in the meantime, an optical switch is added to control the light propagation directions [Fig. 3(b)]. While keeping the pump fixed in the forward direction, we illustrate the four cases of anti-Stokes/Stokes processes in both forward and backward directions with their beating note frequency spectrum (horizontal) while scanning the probe wavelength (vertical). The continuous beating note during the probe wavelength scanning in the four cases clearly indicates nonlinear Brillouin scattering can occur in our enhanced-phonon microcavity in Fig. 3(c). The experimental results show that these conversions are continuous over a wide frequency spectrum when tuning the wavelength of the probe, verifying the nature of the near-phase-matching condition in Eq. (1) due to a small mismatching vector ${\rm{\Delta k}}$ [Fig. 3(a)]. Moreover, this condition can be applied to both anti-Stokes/Stokes processes, as well as forward and backward directions, greatly contrasting with the phase-matching cases [19], where only one phase-matched path is preferred while prohibiting propagation/conversions in the other direction.

We stress that near-phase-matching conditions along both forward and backward directions are the key ingredients enabling nonreciprocal and noninverted phases during frequency modulation with similar conversion efficiency in our GM. The unequal beating notes in Fig. 3(a) indicate the unequal conversion efficiencies among four scenarios resulted from these nonconstant phase-mismatching vectors during wavelength scanning. However, these conversions are companying with low conversion efficiency, estimated to be less than 0.01%, due to the limited length and nonresonant propagation. But we do observe some resonant spikes in all the cases, where either the optical resonances near the probe or the Stokes/anti-Stokes can enhance these conversions resulting in spike patterns on the beating spectra in Fig. 3(c), which is verified experimentally by comparing it with the linear transmission spectrum in Supplement 1, Note 3. Also, further studies are required to improve this resonant-enhanced conversion efficiency.

#### D. Nonreciprocal Conversion Phase in an Aharonov–Bohm Interferometer

Importantly, the anomalous gauge potential in the GM leads to nonreciprocal conversion phases depending on the direction. As a result, the difference between these two mismatching vectors reveals a hidden feature of *nonreciprocal phase* during the frequency conversions: for the forward (left to right) Stokes process, the Stokes wave gains an additional conversion phase as ${e^{i{\varphi _{\rm{FSS}}}}}$, where ${\varphi _{\rm{FSS}}} = {{\Delta}}{k_{\rm{FSS}}}L = \int_0^1 {A_F} \cdot {\rm d}\vec l{_F}$, L is the GM’s path length, and $A$ is the effective gauge field [Supplement 1, Note 4]. Moreover, for the forward anti-Stokes process, the additional phase is ${e^{i{\varphi _{\rm{FAS}}}}}$, where ${\varphi _{\rm{FAS}}} = {{\Delta}}{k_{\rm{FAS}}}L = \int_1^0 {A_F} \cdot {\rm d}\vec l{_F}$. We notice that ${{\Delta}}{k_{\rm{FSS}}} = - {{\Delta}}{k_{\rm{FAS}}}$, which leads to ${\varphi _{\rm{FSS}}} = - {\varphi _{\rm{FAS}}}$. Therefore, one can safely define the effective gauge potential ${A_F}$ along the frequency axis of light for the forward propagation. Meanwhile, we can obtain the conversion phase from a backward Stokes process (${\omega _0} - \Omega = {\omega _1} \leftarrow {\omega _0})$ as ${\varphi _{\rm{BSS}}} = {{\Delta}}{k_{\rm{BSS}}}L = \int_0^1 {A_B} \cdot {\rm d}\vec l{_B}$. Furthermore, the backward anti-Stokes process (${\omega _0} \leftarrow {\omega _1} = {\omega _0} - \Omega$) associated with the photonic isolator mentioned above poses a conversion phase of ${\varphi _{\rm{BAS}}} = {{\Delta}}{k_{\rm{BAS}}}L = \int_1^0 {A_B} \cdot {\rm d}\vec l{_B} = - {\varphi _{\rm{BSS}}}$ due to ${{\Delta}}{k_{\rm{BAS}}} = ({{k_1} - {\rm{G}}}) - {k_0} = - {{\Delta}}{k_{\rm{BSS}}}$, which indicates a crucial conclusion that the backward Stokes and anti-Stokes processes support the same effective gauge potential ${A_B}$ along the frequency axis of light for the backward propagation.

Moreover, there clearly exists a nonzero nonreciprocal phase difference between the forward and backward Stokes processes (to be verified experimentally in Fig. 4):

Therefore, our GM supports the path-dependent effective gauge potentials, i.e., ${A_F} \ne {A_B}$. In contrast, normal EOMs pose the same phase for the Stokes process from either direction, which cannot provide the necessary nonreciprocal phase in the frequency space [Supplement 1, Note 1]. Similarly, such nonreciprocal phases also hold up for anti-Stokes processes in the GM. Note that, unlike previous cases of phase-matched scenarios where frequency conversions are only permitted in one phase-matched direction while inhibited in the other [19–21], here frequency conversions are allowed in both directions thanks to such *bidirectional near-phase-matching*; meanwhile, nonreciprocal phases are induced by directional phonon modulations in our GMs.

As mentioned above, the key ingredient in our GM to support gauge potential is the directional nonreciprocal phases. Based on an AB interferometer loop [Fig. 4(a)] interfering with an EOM and a GM, we can effectively exam such nonreciprocal phases during frequency conversions for both directions. Here within an AB interferometer, the GM is configured using the aforementioned method with a pump laser, while the EOM is modulated at the same frequency as the GM’s with a phase control using an external microwave source. A full scan of the phase spectrum at the center modulation frequency can be revealed for both directions using an optical switch. To ensure a constant phonon phase, we have implemented a phase stabilization technique using a secondary modulated seed beam to lock the Stokes beam’s phase; as a result, the probe can experience the acoustic modulation with the same phase (more details in Supplement 1, Note 5). By comparing the interference patterns, both Stokes [Figs. 4(b) and 4(d)] and anti-Stokes [Figs. 4(c) and 4(e)] cases exhibit nonreciprocal phases between forward and backward propagations. Furthermore, the relative differences between the two interferences stay the same level for both the Stokes and anti-Stokes processes [Figs. 4(d) and 4(e)] during the probe’s frequency detuning (${\sim}{{20}}\;{\rm{MHz}}$). This could be understood from the above argument in Eq. (2), which explicitly indicates the origin of phonon-induced nonreciprocal phase depending only on the phonon’s wave vector $G$. As a result, $0.42 \pi$ phase difference is observed for the acoustic phonon at 66.5 MHz (Fig. 4), while in another experiment (Fig. 5), $1.05 \pi$ phase difference is observed for the phonon at 141.7 MHz. In the following, we will explore this nonreciprocal phase in GMs to construct effective gauge potential for unique applications in unidirectional frequency converters and photonic isolators for synthetic frequency space.

#### E. Coherently Controlled Unidirectional Frequency Converter

The aforementioned configuration of the AB interferometer can be seamlessly adapted for an application of a unidirectional frequency converter (Fig. 5), which permits frequency conversion along one way, while prohibits it from the other one, also in a phase-controllable manner. Previously this has been realized based on fixed and phase-matched schemes [19]; instead, the near-phase-matching condition and the current AB interferometer configuration allow flexible coherent control of conversion directions. For input photons at the frequency ${\omega _0}$ (not in-resonance), we first consider the case that the photon is injected at the left side, and the interference signal out of the parallel-connected EOM and GM is transmitted at the right side. In this forward Stokes setup, the phase difference at the right side is ${{\Delta}}{\phi _ +} = {{\Delta}}{k_{\rm{FSS}}}L + {{\varphi}}$, while, for the backward Stokes case, the phase difference at the left side is ${{\Delta}}{\phi _ -} = {{\Delta}}{k_{\rm{BSS}}}L + {{\varphi}}$. Here the reciprocal phase ${{\varphi}}$ is the EOM’s modulation phase, which can be externally controlled through the EOM [Fig. 4(a)]. Therefore, interference between the paths with EOM and GM leads to a flexible way for coherent controlling frequency conversion through active phase control, and the switching speed is supposed to be potentially ultrafast given the nonresonant and interference nature. One can purposely select a regime where the conversion in the forward direction is constructive while the conversion in the backward direction is destructive, resulting in unidirectional frequency conversion in the forward direction, as shown in Fig. 5(c). More than 20 dB signal contrast of unidirectional conversions can be obtained in the experiment, and the extinction ratio can be further enhanced by increasing the pump’s power (details in Supplement 1, Note 5). Meanwhile, with an additional ${{\pi}}$ phase added to the EOM, such conversion can be reversed into the backward direction while inhibiting the forward conversion with a similar signal contrast [Fig. 5(d)].

#### F. Coherently Controlled Photonic Isolator in the Frequency Space

In a similar manner, a phase controllable photonic isolator in the frequency space can be readily demonstrated based on the same AB interferometer platform. This dual-functionality is enabled by the near-phase-matching condition that allows the Stokes/anti-Stokes processes in both directions, unlike the phase-matching case where only one particular conversion is permitted along one direction. Instead of observing forward and backward Stokes processes in the previous unidirectional converter, here in Fig. 6, we can observe a nonreciprocal frequency conversion between a forward Stokes ($\;{\omega _0} \to {\omega _1}$) and its corresponding backward anti-Stokes (${\omega _0} \leftarrow {\omega _1}$) due to the fact ${{\Delta}}{k_{\rm{FSS}}} \ne {{\Delta}}{k_{\rm{BAS}}}$ as discussed above. Moreover, similar to the unidirectional converter case, such a photonic isolator can be coherently controlled through the EOM’s phase, making its isolation direction switchable, as shown in Figs. 6(c) and 6(d). The signal contrast is also more than 20 dB. The photonic isolators ensure the nonreciprocal light propagation in the spatial domain (forward and backward) as well as the frequency one (${\omega _0}$, ${\omega _1}$), opening up a new door for all-optical signal processing in the synthetic dimension.

## 3. DISCUSSION AND CONCLUSIONS

The conversion efficiency in the current device can be improved, for example, through resonant enhancement. As mentioned above, the optical WGM might allow enhancing the probe’s Stokes/anti-Stokes conversions. Meanwhile, resonant enhancement can be also applied to the acoustic phonons to improve the overall conversion efficiency. For example, the mechanical $Q$-factor can be further enlarged with a cooling and vacuum system [48] to reduce the loss of acoustic waves. Also, levitated waveguides or nanowires for guiding acoustic phonons are possible for enhancing acoustic field intensity in a compact and integrated platform [49].

At last, we would like to discuss the potential applications of such phase-controlled unidirectional converters and photonic isolators. As compared with unidirectional phase-matched frequency converters, e.g., AOMs, our proposed devices feature controllable and broadband frequency conversion thanks to the near-phase-matching condition. Like the prominent case of multi-wavelength signal processing in optical fibers, i.e., DWDM, on-chip multi-frequency signal processing promises a bright future for dramatically increasing signal bandwidths and processing capacity [50]. Here optical signals may need to be constantly converted back and forth in the frequency domain using modulators or nonlinear optics; in this case, this unidirectional converter and the photonic isolator can play the same role in such multi-frequency signal processing preventing signals from penetrating unwanted directions or frequency channels, similar to their counterpart of photonic isolators in the spatial domain. Although the current work due to its working frequency at megahertz may not be directly applicable for DWDM with gigahertz spacing, it could be potentially beneficial for optical orthogonal frequency-division multiplexing [51] and microwave photonics [52] with megahertz channel spacing. For the photonic synthetic dimensions, our proposed GM provides an alternative modulator for either the electro-optic modulation or the four-wave-mixing process [53] which has been used to construct the synthetic dimension along the frequency axis of light.

The photonic isolator in the frequency space and its associated anomalous gauge potential is the essential ingredient for constructing an effective magnetic field in the synthetic frequency dimensions. More importantly, they can also be coherently controlled in directions offering more functionality such as signal routers or switches. On the other hand, the key features demonstrated in the current work are enabled by the near-phase-matching condition, due to the small wave vectors of phonons. We expect similar approaches may be also implemented with commercially available AOMs for free-space applications [15].

## Funding

National Natural Science Foundation of China (92050113, 11674228, 11974245, 12122407); National Key Research and Development Program of China (2016YFA0302500, 2017YFA0303700); Shanghai MEC Scientific Innovation Program (E00075); National Science Foundation (CBET-1641069).

## Acknowledgment

S. F. acknowledges support from the U.S. National Science Foundation.

## Disclosures

The authors declare that they have no conflicts of interest.

## Data availability

The data that support the findings of this study are available from the authors on reasonable request.

## Supplemental document

See Supplement 1 for supporting content.

## REFERENCES

**1. **Y. Aharonov and D. Bohm, “Significance of electromagnetic potentials in the quantum theory,” Phys. Rev. **115**, 485–491 (1959). [CrossRef]

**2. **L. D. Tzuang, K. Fang, P. Nussenzveig, S. Fan, and M. Lipson, “Non-reciprocal phase shift induced by an effective magnetic flux for light,” Nat. Photonics **8**, 701–705 (2014). [CrossRef]

**3. **K. Fang, Z. Yu, and S. Fan, “Photonic Aharonov-Bohm effect based on dynamic modulation,” Phys. Rev. Lett. **108**, 153901 (2012). [CrossRef]

**4. **K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nat. Photonics **6**, 782–787 (2012). [CrossRef]

**5. **R. O. Umucalılar and I. Carusotto, “Artificial gauge field for photons in coupled cavity arrays,” Phys. Rev. A **84**, 043804 (2011). [CrossRef]

**6. **F. Liu and J. Li, “Gauge field optics with anisotropic media,” Phys. Rev. Lett. **114**, 103902 (2015). [CrossRef]

**7. **M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics **7**, 1001–1005 (2013). [CrossRef]

**8. **L. Yuan and S. Fan, “Three-dimensional dynamic localization of light from a time-dependent effective gauge field for photons,” Phys. Rev. Lett. **114**, 243901 (2015). [CrossRef]

**9. **L. Yuan, Y. Shi, and S. Fan, “Photonic gauge potential in a system with a synthetic frequency dimension,” Opt. Lett. **41**, 741–744 (2016). [CrossRef]

**10. **T. Ozawa, H. M. Price, N. Goldman, O. Zilberberg, and I. Carusotto, “Synthetic dimensions in integrated photonics: from optical isolation to four-dimensional quantum Hall physics,” Phys. Rev. A **93**, 043827 (2016). [CrossRef]

**11. **A. Celi, P. Massignan, J. Ruseckas, N. Goldman, I. B. Spielman, G. Juzeliūnas, and M. Lewenstein, “Synthetic gauge fields in synthetic dimensions,” Phys. Rev. Lett. **112**, 043001 (2014). [CrossRef]

**12. **L. Yuan, Q. Lin, A. Zhang, M. Xiao, X. Chen, and S. Fan, “Photonic gauge potential in one cavity with synthetic frequency and orbital angular momentum dimensions,” Phys. Rev. Lett. **122**, 083903 (2019). [CrossRef]

**13. **A. Dutt, Q. Lin, L. Yuan, M. Minkov, M. Xiao, and S. Fan, “A single photonic cavity with two independent physical synthetic dimensions,” Science **367**, 59–64 (2020). [CrossRef]

**14. **E. Lustig, S. Weimann, Y. Plotnik, Y. Lumer, M. A. Bandres, A. Szameit, and M. Segev, “Photonic topological insulator in synthetic dimensions,” Nature **567**, 356–360 (2019). [CrossRef]

**15. **E. Li, B. J. Eggleton, K. Fang, and S. Fan, “Photonic Aharonov–Bohm effect in photon–phonon interactions,” Nat. Commun. **5**, 3225 (2014). [CrossRef]

**16. **T. Ozawa and I. Carusotto, “Synthetic dimensions with magnetic fields and local interactions in photonic lattices,” Phys. Rev. Lett. **118**, 013601 (2017). [CrossRef]

**17. **Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics **3**, 91–94 (2009). [CrossRef]

**18. **H. Lira, Z. Yu, S. Fan, and M. Lipson, “Electrically driven nonreciprocity induced by interband photonic transition on a silicon chip,” Phys. Rev. Lett. **109**, 033901 (2012). [CrossRef]

**19. **E. Kittlaus, N. Otterstrom, P. Kharel, S. Gertler, and P. Rakich, “Non-reciprocal interband Brillouin modulation,” Nat. Photonics **12**, 613–619 (2018). [CrossRef]

**20. **E. Kittlaus, N. Otterstrom, and P. Rakich, “On-chip inter-modal Brillouin scattering,” Nat. Commun. **8**, 15819 (2017). [CrossRef]

**21. **D. B. Sohn, S. Kim, and G. Bahl, “Time-reversal symmetry breaking with acoustic pumping of nanophotonic circuits,” Nat. Photonics **12**, 91–97 (2018). [CrossRef]

**22. **J. Kim, M. Kuzyk, K. Han, H. Wang, and G. Bahl, “Non-reciprocal Brillouin scattering induced transparency,” Nat. Phys. **11**, 275–280 (2015). [CrossRef]

**23. **L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science **335**, 447–450 (2012). [CrossRef]

**24. **S. Manipatruni, J. T. Robinson, and M. Lipson, “Optical nonreciprocity in optomechanical structures,” Phys. Rev. Lett. **102**, 213903 (2009). [CrossRef]

**25. **Y. Shi, Z. Yu, and S. Fan, “Limitations of nonlinear optical isolators due to dynamic reciprocity,” Nat. Photonics **9**, 388–392 (2015). [CrossRef]

**26. **L. Yuan, Q. Lin, M. Xiao, and S. Fan, “Synthetic dimension in photonics,” Optica **5**, 1396–1405 (2018). [CrossRef]

**27. **T. Ozawa and H. M. Price, “Topological quantum matter in synthetic dimensions,” Nat. Rev. Phys. **1**, 349–357 (2019). [CrossRef]

**28. **B. A. Bell, K. Wang, A. S. Solntsev, D. N. Neshev, A. A. Sukhorukov, and B. J. Eggleton, “Spectral photonic lattices with complex long-range coupling,” Optica **4**, 1433–1436 (2017). [CrossRef]

**29. **C. Qin, F. Zhou, Y. Peng, D. Sounas, X. Zhu, B. Wang, J. Dong, X. Zhang, A. Alù, and P. Lu, “Spectrum control through discrete frequency diffraction in the presence of photonic gauge potentials,” Phys. Rev. Lett. **120**, 133901 (2018). [CrossRef]

**30. **Q. Lin, M. Xiao, L. Yuan, and S. Fan, “Photonic Weyl point in a two-dimensional resonator lattice with a synthetic frequency dimension,” Nat. Commun. **7**, 13731 (2016). [CrossRef]

**31. **L. Yuan, M. Xiao, Q. Lin, and S. Fan, “Synthetic space with arbitrary dimensions in a few rings undergoing dynamic modulation,” Phys. Rev. B **97**, 104105 (2018). [CrossRef]

**32. **A. Dutt, M. Minkov, Q. Lin, L. Yuan, D. A. Miller, and S. Fan, “Experimental band structure spectroscopy along a synthetic dimension,” Nat. Commun. **10**, 3122 (2019). [CrossRef]

**33. **X. W. Luo, X. Zhou, C. F. Li, J. S. Xu, G. C. Guo, and Z. W. Zhou, “Quantum simulation of 2D topological physics in a 1D array of optical cavities,” Nat. Commun. **6**, 7704 (2015). [CrossRef]

**34. **X. W. Luo, X. Zhou, J. S. Xu, C. F. Li, G. C. Guo, C. Zhang, and Z. W. Zhou, “Synthetic-lattice enabled all-optical devices based on orbital angular momentum of light,” Nat. Commun. **8**, 16097 (2017). [CrossRef]

**35. **A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity–time synthetic photonic lattices,” Nature **488**, 167–171 (2012). [CrossRef]

**36. **C. A. Brackett, “Dense wavelength division multiplexing networks: principles and applications,” IEEE J. Sel. Areas Commun. **8**, 948–964 (1990). [CrossRef]

**37. **D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is—and what is not—an optical isolator,” Nat. Photonics **7**, 579–582 (2013). [CrossRef]

**38. **C. R. Doerr, N. Dupuis, and L. Zhang, “Optical isolator using two tandem phase modulators,” Opt. Lett. **36**, 4293–4295 (2011). [CrossRef]

**39. **C. R. Doerr, L. Chen, and D. Vermeulen, “Silicon photonics broadband modulation-based isolator,” Opt. Express **22**, 4493–4498 (2014). [CrossRef]

**40. **C. H. Dong, Z. Shen, C. L. Zou, Y. L. Zhang, W. Fu, and G. C. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. **6**, 6193 (2015). [CrossRef]

**41. **J. Yang, T. Qin, F. Zhang, X. Chen, X. Jiang, and W. Wan, “Multiphysical sensing of light, sound and microwave in a microcavity Brillouin laser,” Nanophotonics **9**, 2915–2925 (2020). [CrossRef]

**42. **T. Carmon, L. Yang, and K. J. Vahala, “Dynamical thermal behavior and thermal self-stability of microcavities,” Opt. Express **12**, 4742–4750 (2004). [CrossRef]

**43. **A. M. Scott and K. D. Ridley, “A review of Brillouin-enhanced four-wave mixing,” IEEE J. Quantum Electron. **25**, 438–459 (1989). [CrossRef]

**44. **Y. Feng, F. Zhang, Y. Zheng, L. Chen, D. Shen, W. Liu, and W. Wan, “Coherent control of acoustic phonons by seeded Brillouin scattering in polarization-maintaining fibers,” Opt. Lett. **44**, 2270–2273 (2019). [CrossRef]

**45. **F. Zhang, Y. Feng, X. Chen, L. Ge, and W. Wan, “Synthetic anti-PT symmetry in a single microcavity,” Phys. Rev. Lett. **124**, 053901 (2020). [CrossRef]

**46. **E. P. Ippen and R. H. Stolen, “Stimulated Brillouin scattering in optical fibers,” Appl. Phys. Lett. **21**, 539–541 (1972). [CrossRef]

**47. **G. Bahl, J. Zehnpfennig, M. Tomes, and T. Carmon, “Stimulated optomechanical excitation of surface acoustic waves in a microdevice,” Nat. Commun. **2**, 403 (2011). [CrossRef]

**48. **A. Safavi-Naeini, S. Gröblacher, J. Hill, J. Chan, M. Aspelmeyer, and O. Painter, “Squeezed light from a silicon micromechanical resonator,” Nature **500**, 185–189 (2013). [CrossRef]

**49. **R. Van Laer, B. Kuyken, D. Van Thourhout, and R. Baets, “Interaction between light and highly confined hypersound in a silicon photonic nanowire,” Nat. Photonics **9**, 199–203 (2015). [CrossRef]

**50. **P. Dong, “Silicon photonic integrated circuits for wavelength-division multiplexing applications,” IEEE J. Sel. Top. Quantum Electron. **22**, 370–378 (2016). [CrossRef]

**51. **J. Armstrong, “OFDM for optical communications,” J. Lightwave Technol. **27**, 189–204 (2009). [CrossRef]

**52. **J. Yao, “Microwave photonics,” J. Lightwave Technol. **27**, 314–335 (2009). [CrossRef]

**53. **Y. Zheng, J. Yang, Z. Shen, J. Cao, X. Chen, X. Liang, and W. Wan, “Optically induced transparency in a micro-cavity,” Light Sci. Appl. **5**, e16072 (2016). [CrossRef]