1. INTRODUCTION

Light interacting with nano-mechanical motion via the radiation pressure force is studied in the field of optomechanics. The field has seen rapid progress in the last few years (see the recent review [1]). So far, most experimental achievements have been realized in setups comprising one optical mode coupled to one vibrational mode. Obviously, one of the next frontiers will be the combination of many such optomechanical cells into an optomechanical array, enabling the optical in situ investigation of (quantum) many-body dynamics of interacting photons and phonons. Many experimental platforms can be scaled up to arrays. However, optomechanical crystals seem to be the best-suited candidates at the present stage. Optomechanical crystals are formed by the periodic spatial patterning of regular dielectric and elastic materials, resulting in an enhanced coupling between optical and acoustic waves via moving boundary or electrostriction radiation pressure effects. Two-dimensional (2D) optomechanical crystals with both photonic and phononic bandgaps [2] can be fabricated by standard microfabrication techniques through the lithographic patterning, plasma etching, and release of a thin-film material [3]. These 2D crystals for light and sound can be used to create a circuit architecture for the routing and localization of photons and phonons [3–7].

Optomechanical arrays promise to be a versatile platform for exploring optomechanical many-body physics. Several aspects have already been investigated theoretically, e.g., synchronization [8–10], long-range interactions [11,12], reservoir engineering [13], entanglement [14,15], correlated quantum many-body states [10], slow light [16], transport in a one-dimensional (1D) chain [17], and graphene-like Dirac physics [18].

One of the central aims in photonics is to build waveguides that are robust against disorder and do not display backscattering. Recently there have been several proposals [19–23] to engineer nonreciprocal transport for photons. On the lattice, this corresponds to an artificial magnetic field, which would (among other effects) enable chiral edge states that display the desired robustness against disorder. First experiments have shown such edge states [24–26]. These developments in photonics are related to a growing effort across various fields to produce synthetic gauge fields for neutral particles [27–30].

In this paper we will propose two schemes to generate arbitrary artificial magnetic fields for photons on a lattice. These schemes break time-reversal symmetry by incorporating amplitude-modulated laser fields with different modulation phases at different sites. In contrast to any previous proposals or experiments for photonic magnetic fields on a lattice, these would be controlled all-optically, and, crucially, they would be tunable in situ by changing the properties of a laser field (frequency, intensity, and phase pattern). They require no more than a patterned dielectric slab illuminated by two laser beams with suitably engineered optical phase fields. The crucial ingredient is the optomechanical interaction.

2. OPTOMECHANICAL SCHEMES FOR PHOTON MAGNETIC FIELDS

On the classical level, a charged particle subject to a magnetic field experiences a Lorentz force. In the quantum regime, the appearance of Landau levels leads to the integer and fractional quantum Hall effects, where topologically protected chiral edge states are responsible for a quantized Hall conductance. On a closed orbit, a particle with charge $q$ will pick up a phase that is given by the magnetic flux $\mathrm{\Phi }$ through the circumscribed area, where $\mathrm{\Phi }=(q/\hslash )\int \mathbf{B}·\mathrm{d}\mathbf{S}$ in units of the flux quantum, with $\mathbf{B}$ denoting the magnetic field. On a lattice, a charged particle hopping from site $i$ to $j$ acquires a Peierls phase ${\varphi }_{ij}=(q/\hslash ){\int }_{{\mathbf{r}}_i}^{{\mathbf{r}}_j}\mathbf{A}\mathrm{d}\mathbf{r}$ determined by the vector potential $\mathbf{A}$. Conversely, if we can engineer a Hamiltonian for neutral particles containing arbitrary Peierls phases,

A. Optomechanical Interaction as a Tool for Generating Synthetic Magnetic Fields

Every defect in an optomechanical crystal [3–7] supports a localized vibrational mode (annihilation operator $\widehat{b}$, eigenfrequency ${\mathrm{\Omega }}_0$) and an optical mode ($\widehat{a}$, frequency ${\omega }_{\mathrm{cav}}$) that interact via radiation pressure, giving rise to the standard optomechanical interaction [1],

B. Modulated-Link Scheme

Recently Fang et al. [22,31] proposed to create a photonic gauge field by electro-optically modulating the photon hopping rate $J_{ij}=J\cos (\mathrm{\Omega }t+{\varphi }_{ij}$) between neighboring cavities. This would require locally wired electrodes for each link of the lattice. Here we propose a potentially more powerful all-optical implementation of that idea. We employ optomechanically driven photon transitions, as first discussed in [32], but extended to a scheme with modulated interface modes, depicted in Fig. 1(a). We now discuss the leftmost three optical modes in the first row, ${\widehat{a}}_A$, ${\widehat{a}}_I$, and ${\widehat{a}}_B$ (from left to right), exemplary of the full grid. Their coherent dynamics are governed by the Hamiltonian

 

Fig. 1. Proposed schemes to create a photonic gauge field in optomechanical arrays by engineering photon hopping phases. (a) Modulated-link scheme. (b) Corresponding optical spectrum of a row with relevant sidebands (dashed). Driven vibrational modes (yellow) optomechanically modulate the frequency of optical link modes. Tunneling photons are thus upconverted to the first sideband and pick up the phase of the modulation. Arrows in (b) indicate the resonant photon transmission process in a row. (c) Wavelength-conversion scheme and (d) corresponding optical spectrum: neighboring modes in a row couple to a vibrational mode (yellow) optomechanically (red lines, denoting the linearized optomechanical interaction). Two lasers, driving the optical modes close to the red sidebands [wiggly arrows in (d)], give rise to resonant photon–phonon–photon conversion with the phase set by the lasers. Different rows are connected via simple photon hopping (blue lines), without a phase, in both schemes (a),(c). The indicated phase configuration corresponds to a constant magnetic field.

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Note that interface sites are not required for the creation of a spatially constant magnetic field. Instead, it is sufficient to modulate the sites themselves, as proposed in [33] for the case of optical lattices (see also Refs. [34,35]). The modulation can be accomplished by driving a suitable vibrational mode on the site in the same way as discussed for the links before. In this situation, a photon is upconverted at site $A$ by the modulation, picks up the corresponding phase ${\varphi }_A$, and subsequently tunnels resonantly to the neighboring site $B$ (which has to be chosen resonant with the sideband, i.e., ${\omega }_B={\omega }_A+\mathrm{\Omega }$). In addition, there is another equally important contribution to the effective tunneling rate: the photon can virtually tunnel into the neighboring site $B$, where it is resonantly upconverted, acquiring the phase ${\varphi }_B$. For a constant magnetic field, the phases in a row are equal, i.e., ${\varphi }_A={\varphi }_B$, and both processes add up equally to the effective hopping rate. However, if one wants to go to arbitrary flux patterns, the phases would have to become unequal, and, as a result, the magnitude of the hopping amplitude, $|J_{\mathrm{eff}}|\sim |\exp (i{\varphi }_A)+\exp (i{\varphi }_B)|$, would start to depend on the values of those phases. This would be an undesirable feature, since it couples the hopping amplitudes to the flux distribution. Therefore, the links remain beneficial in a more general situation with arbitrary spatial flux configurations, allowing for more flexibility. In addition, if one wants to include photon–phonon coupling on the sites (as discussed in Section 4), then the links permit us to spatially separate the externally driven vibrational mode that generates the artificial field from the dynamical vibrational modes on the sites themselves.

C. Wavelength-Conversion Scheme

There is another, alternative way of engineering an optical Peierls phase, and it is related to optomechanical wavelength conversion [36,37]. In wavelength-conversion setups, low-frequency photons in one mode are upconverted to a higher frequency in another mode by exploiting the modes’ mutual optomechanical coupling to a vibrational mode. We propose to scale up this idea into a grid as depicted in Fig. 1(c). The leftmost three modes in the first row depict (in this order) an optical mode (annihilation operator ${\widehat{a}}_A$, frequency ${\omega }_A$), a mechanical mode ($\widehat{b}$, ${\mathrm{\Omega }}_0$), and another optical mode (${\widehat{a}}_B$, ${\omega }_B\ne {\omega }_A$). The mechanical mode couples optomechanically to both optical modes. $A$ and $B$ are driven by a laser with frequency ${\omega }_{L1}$ and ${\omega }_{L2}$, respectively: for mode $A$, we require ${\omega }_A-{\omega }_{L1}={\mathrm{\Omega }}_0+\delta \equiv \mathrm{\Omega }$, where ${\omega }_{L1}$ denotes the driving laser’s frequency and $\delta \ll {\mathrm{\Omega }}_0$ is the detuning from the red sideband. For mode $B$, a similar relation ${\omega }_B-{\omega }_{L2}=\mathrm{\Omega }$ holds, as depicted in the spectrum in Fig. 1(d). After application of the standard linearization and RWA procedure [1], the dynamics in a frame rotating with the drive are governed by the Hamiltonian

D. Comparison with Other Schemes

In general, a variety of schemes exist for nonreciprocal photon transport on a lattice, both as proposals and (partially) as proof-of-principle implementations. In comparison to those, the optomechanical schemes provide all-optical control as a crucial novel advantage. The resulting in situ tunability sets them apart from existing schemes based on geometry (e.g., those based on suitably coupled ring resonators [20]). At the same time, everything is controlled by the laser phase pattern, avoiding the challenging fabrication overhead required in other tunable schemes (e.g., those with local electrodes for modulation of photon tunneling between localized optical modes [22]).

Another optomechanical scheme for nonreciprocal photon transport that could potentially be extended to a lattice is based on optical microring resonators [23], but the connection of such rather large rings via waveguides would presumably result in a less compact structure than what can be done with the photonic-crystal-based approaches analyzed here.

E. Limitations

We now discuss the limitations imposed on the achievable effective hopping $J_{\mathrm{eff}}$. The important end result will be that $J_{\mathrm{eff}}$ is limited to about the mechanical frequency ${\mathrm{\Omega }}_0$, even though perturbation theory would seem to imply a far smaller limit (for possible technical limitations connected to the driving strength, see Supplement 1).

We denote as $ϵ\ll 1$ the order of the three small parameters $J/|{\omega }_A-{\omega }_I|$, $J/|{\omega }_B-{\omega }_I|$, and $g_0|\beta |/\mathrm{\Omega }$ in the modulated-link scheme [Figs. 1(a) and 1(b)]. Then the effective coupling strength in the perturbative regime reads $J_{\mathrm{eff}}=\mathcal{O}({ϵ}^3)\mathrm{\Omega }$. Even though the modulation frequency $\mathrm{\Omega }$ need not equal the eigenfrequency ${\mathrm{\Omega }}_0$, they should usually be close to yield a significant mechanical response and avoid other resonances. For the wavelength-conversion scheme, where $|g_{A,B}|/\delta =\mathcal{O}(ϵ)$, we recover $J_{\mathrm{eff}}=\mathcal{O}({ϵ}^2)\delta =\mathcal{O}({ϵ}^3){\mathrm{\Omega }}_0$, since RWA requires $\delta /{\mathrm{\Omega }}_0$ to be small as well. In any experimental realization, photons will decay at the rate $\kappa$. Thus they travel $\sim J_{\mathrm{eff}}/\kappa \sim ({\mathrm{\Omega }}_0/\kappa )\mathcal{O}({ϵ}^3)$ sites. In order for the photons to feel the magnetic field (or to find nontrivial transport at all), this number should be larger than 1. That precludes being in the deep perturbative limit $ϵ\ll 1$, even for a fairly well sideband-resolved system (where typically $\kappa \sim 0.1{\mathrm{\Omega }}_0$). Similar considerations apply for other proposed (nonoptomechanical) schemes based on modulation [22].

3. NUMERICAL SIMULATIONS

A. Optical Spectrum of Photons Subject to Synthetic Magnetic Field

We now explore numerically the full dynamics, beyond the perturbative limit. The optical local density of states (LDOS) is experimentally accessible by measuring the reflection when probing an optical defect mode via a tapered fiber, and it reveals the spectrum of the Hamiltonian. It thus provides a reasonable way to assess the validity of the effective Hamiltonian beyond the perturbative limit. Figure 2(a) shows the LDOS in the bulk calculated with the ideal effective Hofstadter model (1) for a spatially constant magnetic field, depicting the famous fractal Hofstadter butterfly structure [38]. For comparison, we plot the LDOS of the modulated-link scheme in Figs. 2(b) and 2(c). It is obtained by numerically calculating the Floquet Green’s function of the full equations of motion (with time-periodic coefficients); see Supplement 1. The results indicate that the scheme works even for $J_{\mathrm{eff}}\sim 0.1\mathrm{\Omega }>\kappa$, although perturbation theory clearly breaks down in this regime. We stress that the butterfly in Figs. 2(b) and 2(c) could even be observed experimentally at room temperature, since the spectrum is insensitive to thermal fluctuations. One would also observe sidebands; see Fig. 2(d). Similar results hold for the wavelength-conversion scheme (not shown here). Note that some states may not show up in the LDOS (or their contribution may be suppressed), depending on their wavefunction and the location. This is particularly true for the contribution of the edge states, since we chose a location in the middle of the sample. Nevertheless the features of the usual Hofstadter butterfly spectrum (which is the eigenvalue spectrum) manifest themselves in the experimentally accessible LDOS, as evidenced by Fig. 2(a).

 

Fig. 2. Comparison between (a) experimentally accessible optical spectrum (LDOS) of the ideal effective Hofstadter model and (b)–(d) actual results from the proposed modulated-link scheme, for different optical decay rates and magnetic fluxes. The simulation results indicate that the scheme works even beyond the perturbative regime. The resulting Hofstadter butterfly could be observed by a local tapered fiber probe. Modulation of links produces higher sidebands (d). The phase configuration corresponds to a constant magnetic flux $\mathrm{\Phi }$ per plaquette; see Fig. 1. [Parameters: grid (a) $10\times 10$, (b)–(d) $12\times 12$; (a)–(d) $J_{\mathrm{eff}}=0.108{\mathrm{\Omega }}_0$; (a),(c),(d) $\kappa =0.01{\mathrm{\Omega }}_0$, (b) $\kappa =0.05{\mathrm{\Omega }}_0$; (b)–(d) $J=0.3{\mathrm{\Omega }}_0$; (b)–(d) $g_0|\beta |=0.3{\mathrm{\Omega }}_0$; optical eigenfrequencies (relative to first mode) in a row (left to right) including interface modes: (b)–(d) $(0,0.5,1,1.5,\dots ){\mathrm{\Omega }}_0$.]

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B. Spatially Resolved Photon Transport

In addition to measuring the optical spectrum, it is also possible to look at photon transport in a spatially resolved manner, by injecting a probe laser locally and then imaging the photons leaving the sample. This provides another way to observe the effects of the artificial gauge field, which gives rise to distinct transport phenomena as depicted in Figs. 3(a) and 3(b). For small magnetic fields, $|\varphi |\ll 2\pi$, the dynamics can be understood in the continuum limit when probing the bulk: one recovers the standard Landau level picture for electrons in a constant magnetic field [38,39], with effective mass $m^{*}=\hslash /2J{a}^2$ and cyclotron frequency ${\omega }_{\mathrm{cyc}}=2\varphi J$, where $a$ is the lattice constant. In Fig. 3(a) the $n=1$ Landau level is selected via the probe’s detuning ${\mathrm{\Delta }}_p$ with respect to the drive. The circles indicate the semiclassical cyclotron orbits with radius $R=a\sqrt{(2n+1)/\varphi }$. In this semiclassical picture, the momentum of a photon injected locally at a site in the bulk is equally distributed over all directions, since the position is well-defined. Thus, the observed response resembles a superposition of semiclassical circular Lorentz trajectories with different initial velocity directions. A probe injected closer to the edge excites chiral integer quantum Hall effect edge states; see Fig. 3(b).

 

Fig. 3. Microscopic simulation of the wavelength-conversion scheme, Eq. (4), indicating its feasibility: spatial distribution of light intensity upon local injection of a probe laser (a) in the bulk and (b) at the edge, for a constant artificial magnetic field. Bulk transport (a) is governed by Landau levels and can be understood as a superposition of classical cyclotron orbits (yellow circles) for different momentum directions. (b) At the edges robust edge channels exist. (c) Optical Aharonov–Bohm effect in minimal symmetric setup. (d) The interference pattern (normalized probe laser transmission intensity) is shifted by the magnetic flux through the ring. [Parameters: (a),(b) $22\times 22$ grid; (a),(b) $\delta =0.3{\mathrm{\Omega }}_0$, (d) $\delta =0.1{\mathrm{\Omega }}_0$; (a),(b) $g=0.2{\mathrm{\Omega }}_0$, (d) $g=0.01{\mathrm{\Omega }}_0$; (a),(b) $\kappa =0.01{\mathrm{\Omega }}_0$; $\mathrm{\Gamma }=\kappa /10$; (a),(b) $\mathrm{\Phi }=2\pi /8$; (a),(b) $J=0.13{\mathrm{\Omega }}_0$, (d) $J=0.001{\mathrm{\Omega }}_0$, (a) ${\mathrm{\Delta }}_p=1.278{\mathrm{\Omega }}_0$, (b) ${\mathrm{\Delta }}_p=1.260{\mathrm{\Omega }}_0$, (d) ${\mathrm{\Delta }}_p=1.103{\mathrm{\Omega }}_0$.]

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The Aharonov–Bohm effect [40] is one of the most intriguing features of quantum mechanics. In an interferometer, electrons can acquire a phase difference determined by the magnetic flux enclosed by the interfering pathways, even though they never feel any force due to the magnetic field. Figure 3(c) depicts a setup that is based on the wavelength-conversion scheme and realizes an optical analog of the Aharonov–Bohm effect: a local probe is transmitted via two pathways, leading to an interference pattern in the transmission. The pattern is shifted according to the flux through the “ring” [see Fig. 3(d)], confirming the effect.

All the effects displayed in Fig. 3 have been simulated numerically for the wavelength-conversion scheme (see Supplement 1), but similar results hold for the modulated-link scheme.

4. GAUGE FIELDS IN SYNTHETIC DIMENSIONS

So far we have analyzed schemes to engineer hopping phases for photons. We now ask about situations in which the phonons are not only employed as auxiliary virtual excitations, but rather occur as real excitations, which can be interconverted with the photons. This means, in addition to the modes making up the lattices described above (in either of the two schemes), we now consider on-site vibrational modes ${\widehat{b}}_j$ coupled optomechanically to the corresponding optical modes ${\widehat{a}}_j$. Using the standard approach [1], we arrive at a linearized optomechanical interaction of the form $-g{\widehat{a}}_j^{†}{\widehat{b}}_j+\text{h.c}$. Moreover, to be general (and generate nontrivial features connected to the gauge field structure), we will assume that the neighboring phonon modes may also be coupled, as described by a tight-binding Hamiltonian of the form $-K\sum _{〈ij〉}{\widehat{b}}_j^{†}{\widehat{b}}_i+\text{h.c}$.

When discussing the effects of gauge fields in such a setting, the system is best understood within the concept of ‘synthetic’ dimensions [41–44]. The optomechanical interaction can be viewed in terms of an extension of the 1D or 2D lattice into such an additional synthetic dimension. In our case, this dimension only has two discrete locations, corresponding to photons versus phonons. In that picture, the optomechanical interaction, converting photons to phonons, corresponds to a simple hopping between sites along the additional direction. Figure 4(a) sketches this for an optomechanical ring: photons and phonons represent two layers separated along the synthetic dimension. Applying any of our two previously discussed schemes, a photon hopping from site $i$ to $j$ will acquire a phase ${\varphi }_{ij}={\int }_{{\mathbf{r}}_i}^{{\mathbf{r}}_j}\mathrm{d}\mathbf{r}\mathbf{A}$. The gauge field $\mathbf{A}$ must now be viewed as a vector field in this new three-dimensional (3D) space, where one of the dimensions is synthetic. A finite hopping phase $\varphi$ at one of the optical links creates a magnetic flux through the optical plaquette as desired; see Fig. 4(a). However, and this is the important point, since the magnetic field $\mathbf{B}$ is divergence-free, the field must penetrate at least one additional plaquette, causing the opposite magnetic flux in the synthetic dimension (assuming $g\in \mathbb{R}$). In general, realizing that there is this kind of behavior is crucial to avoid puzzles about seeming violations of gauge symmetry in situations with photon magnetic fields in optomechanical arrays. It is necessary to keep track of the full vector potential in the space that includes the synthetic dimension.

 

Fig. 4. Optomechanical gauge fields within the concept of synthetic dimensions. (a) The optomechanical coupling, $g$, can be viewed as connecting sites along a synthetic dimension (photons versus phonons). A phase for the photon hopping, engineered using the schemes from above, creates a flux in the optical plaquette (blue, top) and in the adjacent synthetic plaquette (gray). Hence, the magnetic field (black lines) in the full space is divergence-free. (b) Engineering exclusively the phases of $g$ allows us to create magnetic fields/fluxes, but only perpendicular to the synthetic dimension. Shining a single tilted laser on a 1D chain yields a synthetic optomechanical ladder system with constant synthetic magnetic flux. (c) 2D array, with the field inside the (physical) plane generated by an arbitrary laser phase pattern.

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We now take a step back, getting rid of the previously discussed engineered schemes that required two lasers and some arrangement of ‘link’ modes. Rather we will consider simple optomechanical arrays, i.e., lattices of optical and vibrational modes, with photon and phonon tunnel coupling between modes and with the optomechanical interaction. We ask: What is the effect of an arbitrary, spatially varying optical phase field in the driving laser that sets the strength of the optomechanical coupling? It turns out that the resulting spatially varying phase of the optomechanical coupling, $g_j=|g_j|e^{i{\phi }_j}$, can be chosen to create arbitrary magnetic fields perpendicular to the synthetic dimension. A particularly simple example is a simple linear chain of optomechanical cells. Shining a tilted laser (i.e., with a phase gradient, ${\phi }_j=j·\delta \phi$) onto such a 1D optomechanical array creates a constant magnetic flux through the plaquettes of the “optomechanical synthetic ladder” that can be drawn to understand the situation; cf. Fig. 4(b). The quantum mechanics of excitations tunneling between the two ‘rails’ of the ladder (corresponding to photon–phonon conversion) are directly analogous to experiments on electron tunneling between parallel wires in a magnetic field [45]. The magnetic field shifts the momenta of the tunneling particles, giving rise to resonance phenomena when the shifted dispersion curves $\omega (k)$ of the excitations match. Via phase front engineering one could create arbitrary synthetic magnetic fields also in 2D grids; see Fig. 4(c). We note, though, that this method is constrained since it cannot directly create magnetic fluxes through optical or mechanical plaquettes, and in general only the schemes discussed above provide full flexibility. On the other hand, if either the photon or the phonon modes are occupied only virtually, then effective fluxes can still be generated for the remaining real excitations, even with a single laser, and this works best for phonons (see [46,47]).

5. POSSIBLE EXPERIMENTAL REALIZATIONS

We now discuss the most salient aspects of the experimental realization. Both the ‘butterfly’ optical spectrum and spatially resolved transport can be probed using homodyne techniques, which are insensitive to noise. Real-space optical imaging is feasible, as the defects are a few micrometers apart. The optical phase pattern can be engineered using spatial light modulators. No time-dependent changes of the pattern are needed here, since the time dependence is generated via the beat-note between the two laser beams.

For the modulated-link scheme, the mechanical oscillation amplitude $\beta$ used for the modulation should overwhelm any thermal fluctuations. In the example of Fig. 2, we assumed $g_0\beta =0.3{\mathrm{\Omega }}_0$. At recently achieved parameters [7] $g_0/2\pi =220\mathrm{kHz}$ and ${\mathrm{\Omega }}_0/2\pi =9\mathrm{GHz}$, this would imply $\beta \sim {10}^4$, i.e., a phonon number of ${10}^8$ reached by driving, certainly larger than the thermal population. If we drive the mechanical vibration using a radiation pressure force oscillating at resonance (assuming the quoted $g_0$ also for the optical mode used in that driving), then we have $\beta =2g_0{n}_c/\mathrm{\Gamma }$, where $n_c$ is the circulating photon number and $\mathrm{\Gamma }$ is the mechanical damping rate. Given a mechanical quality factor of ${\mathrm{\Omega }}_0/\mathrm{\Gamma }=2·{10}^5$, this requires $n_c\sim {10}^3$ photons for Fig. 2, a realistic number. We note that thermal fluctuations of the mechanical amplitude give rise to a fractional deviation of $\sqrt{{\overline{n}}_{\mathrm{th}}}/\beta$ in $J_{\mathrm{eff}}$, with a slow drift on the time scale ${\mathrm{\Gamma }}^{-1}$. At typical temperatures used in experiments, we have ${\overline{n}}_{\mathrm{th}}\sim 100-1000$, and so the fractional change is on the order of a percent, which will not noticeably impact transport.

In the alternative wavelength-conversion scheme, one should strive for a large photon-enhanced optomechanical coupling rate $g=g_0\alpha$. A general estimate implies that we always need the photon number to be larger than ${(\kappa /g_0)}^2$ in order to see the butterfly spectrum and the transport effects. This condition (compatible with Fig. 3) would require a circulating photon number of around $3·{10}^5$ for the parameters demonstrated in a recent successful wavelength-conversion setup based on optomechanical crystals [36]. It is also important to estimate the unwanted influx of thermal excitations from the phonon subsystem into the photon subsystem, at least if the setup is to be applied in the quantum regime, for observing the transport of single photons in the presence of a magnetic field. In the wavelength-conversion scheme, there is a detuning $\delta \gg \kappa$ between the red sideband of the laser and the phonon mode, such that photon–phonon conversion is suppressed. Nevertheless, it still happens at a rate $\gamma \overline{n}$, where $\gamma =g^2/\delta$ is the “cooling rate” (for the detuned case applicable here) and $\overline{n}$ is the number of phonons in the mode. Fortunately, this phonon number is also reduced by the very same off-resonant cooling process. Balancing the inflow and outflow of excitations, we find that there will be a remaining unwanted photon occupation of ${\overline{n}}_{\mathrm{phot}}^{\mathrm{th}}={\overline{n}}_{\mathrm{th}}\mathrm{\Gamma }/\kappa$ due to the conversion of thermal phonons into photons, where ${\overline{n}}_{\mathrm{th}}$ is the bulk thermal phonon occupation. The factor $\mathrm{\Gamma }/\kappa$ suppresses this number strongly, and it should be possible to reach the regime ${\overline{n}}_{\mathrm{phot}}^{\mathrm{th}}\ll 1$ in low-temperature setups.

Reducing fabrication-induced disorder will be crucial for any future applications of photonic crystals, including the one envisaged here (as well as other photonic magnetic field schemes). In first experimental attempts, the optical and mechanical disorder is on the percent level, which makes especially the fluctuations of the optical resonance frequencies significant. Nevertheless, strong reductions of the disorder will be possible by post-fabrication methods [48–50], such as local laser-induced oxidation. These are expected to reduce the fluctuations down to the level of ${10}^{-5}$ relative optical frequency fluctuations. This is enough to suppress the optical disorder to some fraction of the photon hopping rate $J_{\mathrm{eff}}\sim \mathrm{\Omega }$, which will enable near-ideal photon transport (e.g., Anderson localization lengths would be at least hundreds of sites, larger than the typical arrays). Disorder in the mechanical frequencies can be reduced by similar techniques, but is much less problematic, due to the difference in absolute frequency scales between optics and mechanics.

6. OUTLOOK

Optomechanical crystals represent an interesting system for the realization of artificial photonic magnetic fields. The fact that the magnetic field is not generated by the geometry of device fabrication but rather controlled all-optically offers the significant advantage of in situ tunability. At the same time, the robustness of the resulting edge states against backscattering by disorder is better than in topological-insulator inspired photonic schemes, since the latter are still susceptible to certain kinds of disorder.

The rich nonlinear (quantum) dynamics [10] of optomechanical arrays could be explored in the presence of such an artificial magnetic field. In general, the very flexible optical control could be used to create and explore novel features, e.g., varying the optomechanical coupling strength spatially and/or temporally, both adiabatically and with sudden quenches. Moreover, a second strong control laser could be used to create a spatially and temporarily varying optical on-site potential landscape, such that the combined effect of that potential and the magnetic field on the photon transport can be explored.