Theory

Consider a ring of radius R formed by a single mode waveguide with a refractive index n. The ring is coupled to an external waveguide of the same refractive index. Assuming sufficiently weak coupling between the ring and the external waveguide and neglecting group-velocity dispersion, the eigenmodes of the ring occur at frequencies ω_m = ω₀ + mΩ_R, where ω₀ is the central frequency, m is an integer and Ω_R = c/nR is the free spectral range (FSR) of the ring in angular frequency units, with c being the speed of light in vacuum. These eigenmodes take the form $$ e^{-i\left(m_0+m\right)\varphi }$$, where m₀ denotes the angular momentum of the 0^th mode and ϕ is the azimuthal coordinate of the ring. Corresponding to these eigenmodes, we define $$ a_m\left(t\right)e^{i{\omega }_{m}t}$$ to be the amplitude of the mode centered at ω_m, normalized such that ∣a_m(t)∣² corresponds to the photon number in the m^th mode. Likewise, we define $$ s_m^{\pm }\left(t\right)e^{i{\omega }_{m}t}$$ to be the amplitudes of the modes of the external waveguide at the input and output ports, respectively, as shown in Fig. 1b. The coupling between the ring modes and waveguide modes at frequency ω_m is described by an external coupling rate $$ {\gamma }_m^e$$, while other losses occurring in the ring, such as absorption or bending loss, are captured by an internal decay rate $$ {\gamma }_m^i$$. Lastly, we assume that the dielectric constant of the ring is modulated using an electro-optic modulator in the form $$ {\sum }_{l=1}^{N_{\mathrm{f}}}\delta {ϵ}_l\left(\varphi \right)\cos \left(l{\mathrm{\Omega }}_{\mathrm{R}}t+{\theta }_l\right)$$, where δϵ_l is the depth of the modulation and θ_l is the phase of the modulation at frequency lΩ_R. The angular dependence δϵ_l(ϕ) occurs due to the physical localization of the electro-optic modulator to a specific range of ϕ, as shown in Fig. 1. The dynamics of the coupled ring-waveguide system can be described by a coupled-mode theory (see Supplementary Note 1) given by:

whereis the modulation-induced coupling between the modes of the ring, with α_l describing the radial and zenith-angle overlap of the eigenmodes of the ring with the electro-optic modulator (see Supplementary Note 1).

If the δϵ_l’s are real, i.e., only the real part of the refractive-index is modulated, then $$ {\kappa }_l^{*}={\kappa }_{-l}$$. Therefore, the modulation conserves the total photon number summed across all frequency channels. Further, if $$ {\gamma }_m^i$$ are negligible, then no photons are lost to absorption or radiation. Under these conditions, the setup of Eqs. (1–2) implements a unitary transformation between the fields $$ s_m^{+}$$ at the input ports and the fields $$ s_m^{-}$$ at the output ports. This unitary transformation can be obtained by first converting Eq. (1) to the frequency domain, resulting in

where $$ \mathbf{a}={\left\{\dots a_{-1},a_0,a_1,\dots \right\}}^t$$, $$ {\mathbf{s}}^{\pm }={\left\{\dots s_{-1}^{\pm },s_0^{\pm },s_1^{\pm },\dots \right\}}^t$$, $$ \mathrm{\Gamma }=\mathrm{diag}\left(\dots {\gamma }_{-1}^e,{\gamma }_0^e,{\gamma }_1^e,\dots \right)$$, Δω is a constant detuning of the equally spaced frequencies of input comb s⁺ from the ring’s resonant frequencies, and $$ {\mathcal{K}}_{m{m}^{\prime }}\equiv {\kappa }_{m-m^{\prime }}$$ as defined by Eq. (3). Then, from Eq. (2), we obtain $$ {\mathbf{s}}^{-}=\mathcal{M}{\mathbf{s}}^{+}$$, whereA direct verification of the unitarity of $$ \mathcal{M}$$ is included in Supplementary Note 2. In the idealized situation as described above, where the ring-waveguide system is assumed to be single-moded over a broad bandwidth and is free from group velocity dispersion, the matrix $$ \mathcal{M}$$ is infinite-dimensional. In practice, the dimensionality of the scattering matrix can be controlled by introducing a “truncation” along the frequency dimension. Such a truncation can be implemented using one or more auxiliary rings coupled to the main ring (see Supplementary Note 3). The auxiliary rings couple to and perturb a few modes immediately outside the (2N_sb + 1) modes around the 0^th mode, dispersively shifting and splitting them. These perturbed modes have frequencies such that the modulation tones of lΩ_R cannot couple these modes to the (2N_sb + 1) modes of interest. Therefore, the total number of modes under consideration in the coupled ring-waveguide system is 2N_sb + 1, and the scattering matrix defined in Eq. (5) is of size (2N_sb + 1) × (2N_sb + 1).

The main objective of our paper is to show that an arbitrary scattering matrix of size (2N_sb + 1) × (2N_sb + 1) can be created. To that end, we first note that the number of real degrees of freedom in the scattering matrix (Eq. (5)) of a single ring under modulation is equal to twice the number of distinct modulation tones, 2N_f, provided the modulation amplitudes δϵ_l and phases θ_l are independently controllable. Since the system is truncated to have 2N_sb + 1 frequencies, the largest harmonic of Ω_R that will result in nonzero coupling between any two modes is 2N_sb, i.e., N_f ≤ 2N_sb. Since an arbitrary unitary matrix of size (2N_sb + 1) × (2N_sb + 1) has $$ {\left(2N_{\mathrm{sb}}+1\right)}^2$$ real degrees of freedom whereas N_f≤2N_sb, we conclude that a single modulated ring is insufficient to approximate an arbitrary unitary matrix to a high degree of accuracy, even if all modulation tones up to 2N_sbΩ_R are used. To overcome this problem, notice that products of unitary transformations are also unitary³⁷. Therefore, as shown in Fig. 1a, instead of a single ring, we consider a sequence of N_r number of rings with each ring providing N_f complex degrees of freedom. Thus, if the total degrees of freedom in series of rings coupled to the waveguide, given by 2N_fN_r, exceeds $$ {\left(2N_{\mathrm{sb}}+1\right)}^2$$, then the setup of Fig. 1a should be able to approximate an arbitrary unitary transformation to a high degree of accuracy.

Below, we optimize these 2N_fN_r degrees of freedom to enable physical approximation of arbitrary unitary and certain non-unitary transformations. For unitary transformations or parts thereof, we use as the objective function the fidelity, which measures the accuracy of an approximation V to a unitary transformation U:

where $$ ⟨U,V⟩={\sum }_{ij}{U}_{ij}^{*}{V}_{ij}$$ is the element-wise inner product and $$ \mid \mid U\mid {\mid }_{\mathrm{F}}=\sqrt{{\sum }_{ij}\mid U_{ij}{\mid }^2}$$ is the Frobenius norm. The use of an absolute value in Eq. (6) allows for the tolerance of a single global phase, i.e., if $$ \mathcal{F}\left(U,V\right)=1$$, then the transformation V achieved by the architecture is equal to Ue^iΦ for some phase Φ. To achieve a high fidelity for a given target matrix we use gradient-based inverse design to optimize the parameters of the modulated system. To enable such optimization, we implemented a numerical model of the unitary transformations defined by Eq. (5) in an automatic differentiation framework³⁸. While explicitly defined adjoint variable methods have been widely used for photonic inverse design³⁹, automatic differentiation is the generalization of the adjoint variable methods to arbitrary computational graphs. Automatic differentiation has recently been successfully applied to the inverse design of photonic band structures⁴⁰ as well as photonic neural networks⁴¹, where explicit adjoint methods are challenging to implement. Here, automatic differentiation enables the efficient computation of the gradients of a scalar objective function with respect to complex control parameters, which in this case are the coupling constants κ_±l as defined in Eq. (3). The advantage of using automatic differentiation is that one needs only to implement the computational model as described above, while the automatic differentiation framework manages the gradient computation through an efficient reverse-mode differentiation. Using the gradients from automatic differentiation, the Limited-memory Broyden–Fletcher–Goldfarb–Shanno (LBFGS) algorithm⁴² is used for optimization.

Implementation of linear transformations

For the results in this Section, we assume that the ring-waveguide system under consideration operates with N_sb = 2, i.e., 5 equally spaced lines followed by at least 4 perturbed lines on each side. The five relevant modes are indexed {−2, −1, 0, 1, 2}. For simplicity, we assume that all five ring modes couple to the waveguide with equal strength, i.e., $$ {\gamma }_m^e\equiv \gamma $$ and $$ {\gamma }_m^i=0\forall m$$. We also assume that the source frequencies in the waveguide are on resonance with the ring, i.e., Δω = 0 in Eq. (5). Examples of finite intrinsic loss ($$ {\gamma }_m^i\ne 0$$) and non-uniform detuning (Δω ≠ 0) are considered in Supplementary Notes 4 and 5. Note that the different source frequencies’ phases should not drift with respect to each other during the timescale of the transformation. To ensure such phase coherence between the different input frequency modes, the source could be a mode-locked laser or an electro-optic frequency comb with a tailored amplitude/phase spectrum to implement the input vector. Alternatively, active phase stabilization could be implemented to compensate for slow-timescale phase drifts. Under the assumptions made in this Section, the transformation in Eq. (5) is completely determined by the ratios κ_l/γ, where κ_l is controlled by the index perturbation amplitude δϵ_l and phase θ_l, as described by Eq. (3). Therefore, we optimize the amplitude and phase of κ_l (in units of γ) for N_r rings and N_f modulation tones per ring to implement a variety of transformations. Note that since we only optimize for the ratios κ_l/γ, our approach is robust to variations in γ during fabrication.

First, we consider the application of such ring-waveguide networks to implement high-fidelity frequency translation that is useful for frequency-domain beam-splitters or single-qubit gates. As an example, we show a design where an input signal in mode 0, after forward propagation through the network, results in a complete conversion to mode +2. Using our inverse-design framework, such a frequency translation corresponds to designing only one column of a unitary transformation and can be achieved with a fidelity exceeding 1 − 10⁻⁵ using just two rings and two modulation tones per ring, as shown in Fig. 2a, b. In Fig. 2c, we present the error function versus the number of iterations. The error function is defined as 1 − F₊₂, where F₊₂ is the normalized output photon flux in the mode +2. After a few iterations, almost all the photon flux is converted to frequency ω₊₂ at the output.

In addition to such high-fidelity frequency conversion implemented in forward propagation through the network, the transformations achieved in this architecture can be different in forward and reverse propagation due to the relative phase shift between the modulation tones across the different rings and the explicit time-varying nature of the dynamically modulated system⁴³. This is in sharp contrast with MZI-based architectures, which are inherently reciprocal. As an example, we show in Fig. 3 that we can simultaneously realize with a fidelity exceeding 1 − 10⁻⁵ a frequency shift, say, 0 → 2, in forward propagation (Fig. 3a) and a different shift, say, 2 → 1, in reverse propagation (Fig. 3b) with three modulated rings.

Achieving frequency shifts using modulated rings, as shown in Figs. 2 and 3, requires designing only one and two columns of the 5 × 5 unitary matrix, respectively. On the other hand, if the number of modulation tones N_f and/or the number of rings N_r are increased, an arbitrary unitary transformation can be achieved with a high fidelity. As an example, we depict in Fig. 4a a 5 × 5 permutation matrix U, defined by U₁₃ = U₂₄ = U₃₅ = U₄₂ = U₅₁ = 1, and zero otherwise. In Fig. 4b, we present the amplitudes of the matrix achieved using one ring and four modulation tones, resulting in a fidelity of 1 − 5.9 × 10⁻³. With four rings and four modulation tones, the fidelity is boosted to over 1 − 3.8 × 10⁻⁶, as shown by the amplitudes in Fig. 4c. In Fig. 4d, we tabulate as a function of N_r and N_f one minus the maximum fidelities obtained in approximating the 5 × 5 permutation matrix, showing that very high fidelities can be achieved using a wide variety of N_r and N_f combinations.

In Fig. 4, we considered only the accuracy of the amplitudes achieved by our inverse-design approach. We now show that our architecture can also capture the phase of an arbitrary unitary transformation with a high fidelity. To demonstrate this, we consider a normalized 5 × 5 Vandermonde matrix, which is used to implement the discrete Fourier transform. This unitary transformation, defined by $$ U_{mn}=e^{-2\pi imn/5}/\sqrt{5}$$, has a constant amplitude across its matrix elements but significantly varying phase, as shown in Fig. 5a. With the use of one ring and four modulation tones, the inverse-design algorithm is able to achieve a fidelity of 0.8, with the corresponding phase profile shown in Fig. 5b up to a global phase of 0.0099π. As depicted in Fig. 5c, a significantly better performance is possible with the use of four rings and four modulation tones per ring, achieving a fidelity of 1 − 7.25 × 10⁻⁷ with a global phase of 0.596π. A map of one minus the maximum fidelities achieved by our inverse design approach as a function of the number of rings and modulation tones is shown in Fig. 5d.

While unitary transformations are usually required for quantum information processing, matrices used in classical signal processing and in neural networks are in general non-unitary. The architecture presented thus far can also be used to implement non-unitary matrices with singular values less than or equal to one using one of two techniques. First, such non-unitary matrices can provably be embedded in larger unitary matrices⁴⁴ using their singular value decomposition. Subsequently, the larger unitaries can be implemented using refractive index modulation as discussed thus far. As an example, we consider the following 3 × 3 non-unitary matrix that was randomly generated subject to the constraint that its largest singular value is equal to one:

The singular values of M are 1, 0.3755 and 0.1421, respectively. Since there are two singular values less than 1, M can be extended into a unitary matrix by adding two dimensions. The element-wise amplitude and phase corresponding to the extended 5 × 5 unitary matrix are shown in Fig. 6a, c, respectively. Using four rings (N_r = 4) and four modulation tones per ring (N_f = 4), our inverse-design algorithm achieves the extended unitary matrix with a fidelity exceeding 1 − 10⁻⁵, as shown in Fig. 6b, d. Notice that the phase of element (5, 4) is significantly different between Fig. 6c, d, but this is because the target amplitude for this element is zero. As an alternative approach, amplitude modulation, where the imaginary part of the refractive index is also modulated, can also be used to directly implement non-unitary matrices since the transformation of Eq. (5) is non-unitary under modulation of the imaginary part of the refractive index. Lastly, in order to implement matrices with singular values greater than 1, a gain element is necessary. For such matrices, a scaled version such that the singular values are below 1 can first be implemented using the methods outlined above, after which a uniform amplification for all frequency channels can rescale the matrix to its intended form.