Synthetic dimensions in integrated photonics: From optical isolation to four-dimensional quantum Hall physics

Recent technological advances in integrated photonics have spurred on the study of topological phenomena in engineered bosonic systems. Indeed, the controllability of silicon ring-resonator arrays has opened up new perspectives for building lattices for photons with topologically nontrivial bands and integrating them into photonic devices for practical applications. Here, we push these developments even further by exploiting the different modes of a silicon ring resonator as an extra dimension for photons. Tunneling along this synthetic dimension is implemented via an external time-dependent modulation that allows for the generation of engineered gauge fields. We show how this approach can be used to generate a variety of exciting topological phenomena in integrated photonics, ranging from a topologically-robust optical isolator in a spatially one-dimensional (1D) ring-resonator chain to a driven-dissipative analog of the 4D quantum Hall effect in a spatially 3D resonator lattice. Our proposal paves the way towards the use of topological effects in the design of novel photonic lattices supporting many frequency channels and displaying higher connectivities.

Figure 1

(a) An illustration of the synthetic dimension scheme. The dielectric properties of a ring resonator (black) are modulated by external light fields, leading to a linear coupling between the angular momentum eigenmodes of the resonator (colored rings represent different eigenfrequencies). (b) An effective lattice made of a 1D array of resonators with coupling in a synthetic dimension. Here, $J_x$ denotes the tunneling amplitude along the real dimension by evanescent proximity coupling, and $J_w$ denotes the coupling in the synthetic dimension. Spatial variation of the dielectric modulation of each resonator makes the complex hopping amplitude $J_w$ spatially dependent. We can design this modulation so that the phase, $\mathrm{\Phi }$, obtained by a photon hopping around a plaquette in the synthetic 2D lattice (shaded in blue) is uniform across the lattice. The resulting Hamiltonian is analogous to that of the Harper-Hofstadter model [31], of a charged particle hopping on a 2D lattice in the presence of a uniform magnetic flux $\mathrm{\Phi }$. (c) A sketch of an effective 4D lattice composed of a 3D resonator lattice and one synthetic dimension. In the $xy$ plane the resonators are coupled via link resonators with directional couplers (gray rings). Making the position of the link resonators along the $y$ direction depend on $z$ engineers a gauge field and generates 2D quantum Hall physics in the $yz$ planes [8]. Similar to (b) the coupling of different eigenmodes along the synthetic dimension $w$ can be made to depend on $x$, leading to 2D quantum Hall systems in the $xw$ planes. The resulting 4D physics is that of the 4D integer quantum Hall effect on a hypercube [26, 32].

Figure 2

(Left) The energy spectrum $\mathcal{E}\left(k_w\right)$ of the Harper-Hofstadter model with $\mathrm{\Phi }=1/4$. We take the $w$ direction to be periodic, and the $x$ direction to have a finite width $L_x$. There are four bands, with two bands touching at the zero energy. Two states inside each band gap are chiral edge states, one localized at one edge and the other localized at the other edge. (Right) Schematic representation of the edge states labeled $\mathcal{A}$ and $\mathcal{B}$ in the left figure. The edge state $\mathcal{A}$ is localized at the edge $x=0$, and the other edge state $\mathcal{B}$ is localized at the edge $x=L_x$. These edge states in the lower gap would propagate in the counterclockwise direction in a planar geometry.

Figure 3

(a), (b) The steady-state photon distribution ${\beta }_{x,w}$ in (6) for the Harper-Hofstadter model with $\mathrm{\Phi }=1/4$ when the central resonator is driven. The synthetic electric field $\eta e{E}_w$ is zero for (a), and $\eta e{E}_w=0.3J$ for (b). The driving frequency is tuned to $-2.7J$ within the lowest-energy band, and the uniform loss is $\gamma =0.3J$, which is large enough to cover the lowest band. Different colors schematically represent that different sites in the synthetic dimension correspond to different frequencies. More opaque circles represent higher intensity; color bars shown in (a), also applicable to (b), show the value of the steady-state photon distribution calculated at each lattice site. (c) The numerically calculated displacement $\langle x\rangle$ (solid line) and the analytical prediction (dashed line) as a function of the electric field $E_w$. (d) The estimated Chern number, as a function of the nonzero force $|e{E}_w|$. The upper (blue) solid line is the estimate from Eq. (8) without the subleading term $O\left({\gamma }^0\right)$, and the points with error bars are estimated Chern numbers for systems with $\pm 10\%$ random fluctuations on the loss rate ${\gamma }_{\mathbf{r},w}$, averaged over 20 different realizations. The lower (red) solid line is the estimate taking into account the term $O\left({\gamma }^0\right)$. When the force $E_w$ is small, the estimated Chern number is close to the true value of $-1$, which is indicated by a dashed line.

Figure 4

Simulation of the steady-state photon-field distribution for $\mathrm{\Phi }=1/4$. Light is pumped into the system with a driving frequency ${\mathrm{\Omega }}_{\mathrm{drive}}=-2J$, chosen in the bulk band gaps of the 2D Harper-Hofstadter model (see Fig. 2). The uniform loss rate is $\gamma =0.1J$. The pumped modes and positions are indicated by arrows, and the central modes (indicated by black circles) have losses equal to (a) $\gamma$ (i.e., no extra losses), (b) $10\gamma$, (c) $100\gamma$, and (d) $1000\gamma$. The color bars show the value of the steady-state photon distribution calculated at each lattice site. In (a) where there are no extra losses in the central row, the light remains spatially localized at either end of the chain, while changing its frequency depending on the chirality of the edge state driven. When the loss of the central row becomes very large, as in (d), the lossy mode acts like an artificial edge in the synthetic dimension. Depending on the frequency of the pumped mode, light propagates either to the left or the right. The unidirectional propagation of these edge states can be exploited to provide topological optical isolation (see also Fig. 5).

Figure 5

Demonstration of how the synthetic two-dimensional lattice can be used as an optical isolator. (a), (b) Steady-state photon configurations when (a) the rightmost and (b) the leftmost sites are driven (as indicated by the arrows). The loss rate of all modes is taken to be uniform and equal to $\gamma =0.1J$, except for the mode directly above the driven one [indicated as dashed circles in (a) and (b)] whose loss is set to be 1000 times larger. The color bars show the value of the steady-state photon distribution calculated at each lattice site. (c) Plot of the steady-state intensity at one end of the chain, when the other end is driven, as a function of the length of the chain $N_x$. The three topmost curves (blue, yellow, and purple stars) correspond to a drive located on the rightmost resonator. These three curves differ by the value of the extra losses of the mode defining the topological edge, which is equal, from top to bottom, to $1000\gamma$, $100\gamma$, and $10\gamma$. The bottom curve (red circles) corresponds to a drive located on the leftmost resonator with an extra loss rate of $1000\gamma$: the almost vanishing value of the intensity on the rightmost exit resonator shows that the system indeed acts as a good optical isolator. The ratio between the stars and circles quantitatively characterizes the efficiency of the optical isolation. The log-scale zoom of the curves for the extra loss rate of $1000\gamma$ in the region $N_x=6$ to 15 shown in the inset further illustrates that there is always a large difference in transmission in opposite directions.

Figure 6

(a)–(d) Simulation of photon propagation in the 4D effective lattice sketched in Fig. 1, composed of a 3D resonator lattice and one synthetic dimension. The 4D effective lattice has ${13}^4$ sites, with strong engineered gauge fields in the $xw$ and $yz$ planes of strengths $\left(\eta a\right)e{B}_{xw}=2\pi /6$ and $a^{2}e{B}_{yz}=2\pi /7$. The lowest band of this effective 4D topological lattice has a nontrivial second Chern number (11), ${\nu }_2=1$. The central site of the lattice is pumped with frequency $\mathrm{\Omega }=-6.3J$, resonant with the lowest band. We take the uniform loss $\gamma =0.05J$. The steady-state light intensity distribution is projected onto $xy$ plane for different values of the weak perturbative fields $E_x$ and $\delta B_{zw}$. (a) $ae{E}_x=0$ and $\left(\eta a\right)e\delta B_{zw}=0$, (b) $ae{E}_x=J/10$ and $\left(\eta a\right)e\delta B_{zw}=0$, (c) $ae{E}_x=0$ and $\left(\eta a\right)e\delta B_{zw}=2\pi /10$, (d) $ae{E}_x=J/10$ and $\left(\eta a\right)e\delta B_{zw}=2\pi /10$. The scales show the value of the projected steady-state photon distribution. As can be seen, only in (d) in the presence of both fields is there a clear displacement along $y$: this is the optical analog of the 4D quantum Hall effect. (e) The displacement $\langle y\rangle$ as a function of $E_x$ for a fixed value of $\left(\eta a\right)e\delta B_{zw}=0$ (red horizontal line) and $\left(\eta a\right)e\delta B_{zw}=2\pi /30$ (blue curve). (f) The displacement $\langle y\rangle$ as a function of $\delta B_{zw}$ for a fixed value of $ae{E}_x=0$ (red horizontal line) and $ae{E}_x=J/30$ (blue curve). We take the loss in (e) and (f) to be $\gamma =0.15J$. The dashed lines in (e) and (f) are the theoretical prediction from Eq. (12) without the term $O\left({\gamma }^0\right)$ for a band with ${\nu }_2=1$. (g) and (h) The extracted second Chern number ${\nu }_2^{\mathrm{expt}}$ for the configuration of (e) and (f), respectively. The lower (blue) curve at small fields is the estimate using Eq. (12) with one value of loss $\gamma =0.15J$, ignoring the term $O\left({\gamma }^0\right)$. The points with error bars are estimated second Chern numbers for systems with $\pm 10\%$ random fluctuations on the loss rate ${\gamma }_{\mathbf{r},w}$, averaged over 20 different realizations. The upper (red) curve at small fields is the estimate taking into account the term $O\left({\gamma }^0\right)$ by using two different values of loss $\gamma =0.15J$ and $0.16J$ and calculating the difference in the displacement. The true value of the second Chern number ${\nu }_1=1$ is indicated by dashed lines.

Figure 7

The solid lines show the spatial profile of the light intensity ${\left|\psi \left(x\right)\right|}^2$, in units of ${(F_0\sigma /v_{\mathrm{vg}})}^2$, in a unidirectionally propagating mode of a cavity. The driving field $F\left(x\right)$ is monochromatic, satisfying the resonance condition ${\mathrm{\Omega }}_{\mathrm{drive}}={\omega }_0$ with an exponentially shaped spot. The dashed line plots the intensity of the driving field scaled as $4|F\left(x\right)/F_0{|}^2$. The different solid curves correspond to different values of the cavity decay as indicated in the legend.