Abstract
Arbitrary linear transformations are of crucial importance in a plethora of photonic applications spanning classical signal processing, communication systems, quantum information processing and machine learning. Here, we present a photonic architecture to achieve arbitrary linear transformations by harnessing the synthetic frequency dimension of photons. Our structure consists of dynamically modulated microring resonators that implement tunable couplings between multiple frequency modes carried by a single waveguide. By inverse design of these short and longrange couplings using automatic differentiation, we realize arbitrary scattering matrices in synthetic space between the input and output frequency modes with nearunity fidelity and favorable scaling. We show that the same physical structure can be reconfigured to implement a wide variety of manipulations including singlefrequency conversion, nonreciprocal frequency translations, and unitary as well as nonunitary transformations. Our approach enables compact, scalable and reconfigurable integrated photonic architectures to achieve arbitrary linear transformations in both the classical and quantum domains using current stateoftheart technology.
Introduction
Arbitrary linear transformations in photonics^{1,2,3} are of central importance for optical quantum computing^{4}, classical signal processing and deep learning^{5,6,7,8,9,10}. A variety of architectures are being actively studied to implement linear transformations for quantum computation and photonic neural networks, including those based on Mach–Zender interferometers (MZI)^{4,5}, microring weight banks^{6,7,11}, phasechange materials^{8,9}, and diffractive metasurfaces^{10}. All such approaches use path encoding of photons in real space. By contrast, implementing such linear transformations in the frequency space would open avenues beyond those possible with previously reported architectures, which are inherently timeinvariant. For example, frequencyspace transformations allow spectrotemporal shaping of light and generation of new frequencies, with wideranging applications in frequency metrology, spectroscopy, communication networks, classical signal processing^{12,13,14} and linear optical quantum information processing^{15,16,17,18,19,20,21,22,23,24}. Nonlinear optics has traditionally been the workhorse for such spectrotemporal shaping, but the requirement of highpower fields and the difficulty of implementing arbitrary linear transformations motivates new architectures for manipulating states in the frequency domain. To that end, photonic synthetic dimensions offer an attractive solution to implement linear transformations in a single physical waveguide by harnessing the internal degrees of freedom of a photon^{25,26,27,28,29,30,31,32}. Synthetic frequency dimensions in particular offer a small spatial footprint and inherent reconfigurability since multiple frequency modes can be addressed simultaneously, and the short and longrange coupling^{29,30,33,34} between them can be controlled by applying an appropriate timedomain signal to a modulator.
Previous works have considered implementing photonic linear transformations using different frequency channels in parallel but without frequency conversions among them^{6,7,9,11} by demultiplexing the different frequencies into separate spatial channels. Additionally, optimized fast modulation has been used for tailoring single photon spectra from twolevel quantum emitters^{35}, or for quantum frequency conversion^{15} and linear optical quantum computation^{17,36}, where the modulator is used as a generalized beam splitter in synthetic frequency dimensions. However, the design of an entire scattering matrix that implements an arbitrary N × N linear transformation in synthetic space, which is essential for many applications in quantum information processing and neural networks, has not yet been shown.
Here, we show that arbitrary linear transformations can be performed directly in the synthetic space spanned by the different frequency modes carried by a single physical waveguide. We use gradientbased inverse design to automate the process of designing the linear transformations, and demonstrate that a wide variety of transformations can be realized. As examples, we show singlefrequency conversion, nonreciprocal frequency translations as well as general arbitrary unitary and nonunitary transformations, all achieved with high fidelities in a fully reconfigurable fashion.
Results
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 groupvelocity dispersion, the eigenmodes of the ring occur at frequencies ω_{m} = ω_{0} + mΩ_{R}, where ω_{0} 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 \({e}^{i({m}_{0}+m)\phi }\), where m_{0} denotes the angular momentum of the 0^{th} mode and ϕ is the azimuthal coordinate of the ring. Corresponding to these eigenmodes, we define \({a}_{m}(t){e}^{i{\omega }_{m}t}\) 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 \({s}_{m}^{\pm }(t){e}^{i{\omega }_{m}t}\) 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 electrooptic modulator in the form \(\mathop{\sum }\nolimits_{l = 1}^{{N}_{{\rm{f}}}}\delta {\epsilon }_{l}(\phi )\cos (l{{{\Omega }}}_{{\rm{R}}}t+{\theta }_{l})\), 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 electrooptic modulator to a specific range of ϕ, as shown in Fig. 1. The dynamics of the coupled ringwaveguide system can be described by a coupledmode theory (see Supplementary Note 1) given by:
where
is the modulationinduced coupling between the modes of the ring, with α_{l} describing the radial and zenithangle overlap of the eigenmodes of the ring with the electrooptic modulator (see Supplementary Note 1).
If the δϵ_{l}’s are real, i.e., only the real part of the refractiveindex 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 \({\bf{a}}={\{\ldots {a}_{1},{a}_{0},{a}_{1},\ldots \}}^{t}\), \({{\bf{s}}}^{\pm }={\{\ldots {s}_{1}^{\pm },{s}_{0}^{\pm },{s}_{1}^{\pm },\ldots \}}^{t}\), \({{\Gamma }}={\rm{diag}}(\ldots {\gamma }_{1}^{e},{\gamma }_{0}^{e},{\gamma }_{1}^{e},\ldots )\), Δω is a constant detuning of the equally spaced frequencies of input comb s^{+} from the ring’s resonant frequencies, and \({{\mathcal{K}}}_{mm^{\prime} }\equiv {\kappa }_{mm^{\prime} }\) as defined by Eq. (3). Then, from Eq. (2), we obtain \({{\bf{s}}}^{}={\mathcal{M}}{{\bf{s}}}^{+}\), where
A direct verification of the unitarity of \({\mathcal{M}}\) is included in Supplementary Note 2. In the idealized situation as described above, where the ringwaveguide system is assumed to be singlemoded over a broad bandwidth and is free from group velocity dispersion, the matrix \({\mathcal{M}}\) is infinitedimensional. 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 ringwaveguide 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 \({(2{N}_{{\rm{sb}}}+1)}^{2}\) 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_{sb}Ω_{R} are used. To overcome this problem, notice that products of unitary transformations are also unitary^{37}. 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_{f}N_{r}, exceeds \({(2{N}_{{\rm{sb}}}+1)}^{2}\), 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_{f}N_{r} degrees of freedom to enable physical approximation of arbitrary unitary and certain nonunitary 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 \(\langle U,V\rangle ={\sum }_{ij}{U}_{ij}^{* }{V}_{ij}\) is the elementwise inner product and \(  U { }_{{\rm{F}}}=\sqrt{{\sum }_{ij} {U}_{ij}{ }^{2}}\) 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 \({\mathcal{F}}(U,V)=1\), 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 gradientbased 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 framework^{38}. While explicitly defined adjoint variable methods have been widely used for photonic inverse design^{39}, 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 structures^{40} as well as photonic neural networks^{41}, 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 reversemode differentiation. Using the gradients from automatic differentiation, the Limitedmemory Broyden–Fletcher–Goldfarb–Shanno (LBFGS) algorithm^{42} is used for optimization.
Implementation of linear transformations
For the results in this Section, we assume that the ringwaveguide 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 nonuniform 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 modelocked laser or an electrooptic frequency comb with a tailored amplitude/phase spectrum to implement the input vector. Alternatively, active phase stabilization could be implemented to compensate for slowtimescale 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 ringwaveguide networks to implement highfidelity frequency translation that is useful for frequencydomain beamsplitters or singlequbit 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 inversedesign 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 highfidelity 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 timevarying nature of the dynamically modulated system^{43}. This is in sharp contrast with MZIbased 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 U_{13} = U_{24} = U_{35} = U_{42} = U_{51} = 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 inversedesign 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 inversedesign 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 nonunitary. The architecture presented thus far can also be used to implement nonunitary matrices with singular values less than or equal to one using one of two techniques. First, such nonunitary matrices can provably be embedded in larger unitary matrices^{44} 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 nonunitary 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 elementwise 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 inversedesign 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 nonunitary matrices since the transformation of Eq. (5) is nonunitary 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.
Discussion
We have shown that combining the concepts of synthetic dimensions and inverse design enables the implementation of versatile linear transformations in photonics. A major advantage of using synthetic frequency dimensions for implementing an N × N linear transformation is that only O(N) photonic elements (modulators in our case) need to be electrically controlled. This is in contrast to realspace dimensions using pathencoding, such as MZI meshes or crossbar arrays, where the full O(N^{2}) degrees of freedom need to be electrically controlled. Such control is nontrivial both from a scalability perspective as well as from a practical geometrical perspective of connecting N^{2} tunable elements (e.g. phaseshifters) to their driving electronics offchip. The reduction in the number of individually controlled elements from O(N^{2}) to O(N) in our scheme comes from the fact that the driving signal on each of the N_{r} EOMs can simultaneously address N_{f} frequency modes in the synthetic dimension.
Future work could leverage synthetic frequency dimensions for complicated quantum information protocols beyond singlequdit unitary transformations, such as realizing probabilistic entangling gates for linear optical quantum computing (LOQC)^{17,36}. In particular, spectral LOQC using EOMs and pulse shapers has been shown to be universal for quantum computation^{17}. However, pulse shapers involve demultiplexing the frequency modes into distinct spatial channels using gratings to apply modebymode phase shifts, and limit the number of modes that can be accommodated within the modulator bandwidth due to a finite spectral resolution, thus reducing the benefit of using synthetic frequency dimensions. Such pulse shapers are also lossy and challenging to integrate on chip. Our architecture obviates the pulse shaper by exclusively using EOMs. The advent of ultralowloss nanophotonic EOMs in lithium niobate^{45,46}, as well as progress in silicon^{47,48} and aluminum nitride^{49} makes our architecture fully compatible with onchip integration, since modulation at frequencies exceeding the ring’s FSR have been demonstrated^{14,47,50}.
For applications in neural networks, the performance of our architecture in terms of the speed, compute density and energy consumption for multiplyandaccumulate (MAC) operations is important^{51}. Assuming we need N modulation tones and N rings with FSR Δf = Ω_{R}/2π to implement a matrix, we can input information encoded in the N frequencies and read out the matrixvector product, which amounts to N^{2} MAC operations. Since we need a frequencyresolved measurement, the fastest readout bandwidth is Δf. We assume that the input data can be prepared at speed comparable to or faster than the readout speed. Then, the computational speed in MACs per second is given by
The maximum number of channels is limited by the FSR and the modulation bandwidth. If we utilize the whole available bandwidth, B = NΔf, then the speed is
For a modulation bandwidth of 100 GHz and an FSR of 100 MHz (such that N = 1000), this yields a speed C = 10^{14} MACs per second or 100 TMAC per second, which is comparable with MZI meshes^{5,51,52}. Although achieving such small FSRs on chip is challenging, recent progress in integrating lowloss delay lines on chip^{53,54} holds promise, since meterscale delays were reported in an 8 mm^{2} footprint using spiral resonators, corresponding to an equivalent FSR of ~350 MHz^{53}. These design techniques can be extended to lithium niobate rings with high modulation bandwidths^{14,46}.
To optimize for computation density, i.e. MACs per second per unit area^{51}, one can use a larger FSR Δf = 1 GHz, in a 1mm^{2} footprint, and combine synthetic frequency dimensions within each 100GHz modulation bandwidth with wavelengthdivision multiplexed channels separated by 100GHzwide stopbands, to parallelize several uncoupled MAC operations across the 5 THz telecommunications band, as has been done for crossbar arrays^{6,7,9,51}. This leads to a compute density of ~10 TMAC s^{−1} mm^{−2}, which is much better than MZI meshes and comparable with standard silicon microring crossbar arrays^{51}, with the added advantage of only O(N) electronically controlled elements. We anticipate that future progress in modulation speed and power using highconfinement integrated photonic platforms will push these current estimates further, leading to experimental implementations of MAC operations using the architecture proposed here with improvements in complexity, speed, power and footprint.
Data availability
The data related to this study is available in the manuscript and the Supplementary Materials. Additional data is available from the authors upon reasonable request. Correspondence and requests for materials should be addressed to S.F.
References
Reck, M., Zeilinger, A., Bernstein, H. J. & Bertani, P. Experimental realization of any discrete unitary operator. Phys. Rev. Lett. 73, 58–61 (1994).
Clements, W. R., Humphreys, P. C., Metcalf, B. J., Kolthammer, W. S. & Walmsley, I. A. Optimal design for universal multiport interferometers. Optica 3, 1460–1465 (2016).
Miller, D. A. Selfconfiguring universal linear optical component. Photonics Res. 1, 1–15 (2013).
Carolan, J. et al. Universal linear optics. Science 349, 711–716 (2015).
Shen, Y. et al. Deep learning with coherent nanophotonic circuits. Nat. Photonics 11, 441–446 (2017).
Tait, A. N., Nahmias, M. A., Shastri, B. J. & Prucnal, P. R. Broadcast and weight: an integrated network for scalable photonic spike processing. J. Lightwave Technol. 32, 4029–4041 (2014).
Tait, A. N. et al. Neuromorphic photonic networks using silicon photonic weight banks. Sci. Rep. 7, 1–10 (2017).
Feldmann, J., Youngblood, N., Wright, C., Bhaskaran, H. & Pernice, W. Alloptical spiking neurosynaptic networks with selflearning capabilities. Nature 569, 208–214 (2019).
Feldmann, J. et al. Parallel convolutional processing using an integrated photonic tensor core. Nature 589, 52–58 (2021).
Lin, X. et al. Alloptical machine learning using diffractive deep neural networks. Science 361, 1004–1008 (2018).
Ohno, S., Toprasertpong, K., Takagi, S. & Takenaka, M. Si microring resonator crossbar arrays for deep learning accelerator. Jpn. J. Appl. Phys. 59, SGGE04 (2020).
Cundiff, S. T. & Weiner, A. M. Optical arbitrary waveform generation. Nat. Photonics 4, 760–766 (2010).
Supradeepa, V. et al. Combbased radiofrequency photonic filters with rapid tunability and high selectivity. Nat. Photonics 6, 186–194 (2012).
Zhang, M. et al. Broadband electrooptic frequency comb generation in a lithium niobate microring resonator. Nature 568, 373–377 (2019).
Lu, H.H. et al. Electrooptic frequency beam splitters and tritters for highfidelity photonic quantum information processing. Phys. Rev. Lett. 120, 030502 (2018).
Menicucci, N. C., Flammia, S. T. & Pfister, O. Oneway quantum computing in the optical frequency comb. Phys. Rev. Lett. 101, 130501 (2008).
Lukens, J. M. & Lougovski, P. Frequencyencoded photonic qubits for scalable quantum information processing. Optica 4, 8–16 (2017).
Lu, H.H., Weiner, A. M., Lougovski, P. & Lukens, J. M. Quantum information processing with frequencycomb qudits. IEEE Photonics Technol. Lett. 31, 1858–1861 (2019).
Joshi, C., Farsi, A., Clemmen, S., Ramelow, S. & Gaeta, A. L. Frequency multiplexing for quasideterministic heralded singlephoton sources. Nat. Commun. 9, 847 (2018).
Roslund, J., de Araújo, R. M., Jiang, S., Fabre, C. & Treps, N. Wavelengthmultiplexed quantum networks with ultrafast frequency combs. Nat. Photonics 8, 109–112 (2014).
Reimer, C. et al. Highdimensional oneway quantum processing implemented on d level cluster states. Nat. Phys. 15, 148–153 (2019).
Joshi, C. et al. Frequencydomain quantum interference with correlated photons from an integrated microresonator. Phys. Rev. Lett. 124, 143601 (2020).
Zhu, X. et al. Graph state engineering by phase modulation of the quantum optical frequency comb. http://arxiv.org/abs/1912.11215 (2019).
Hu, Y. et al. Reconfigurable electrooptic frequency shifter. arXiv preprint http://arxiv.org/abs/2005.09621 (2020).
Yuan, L., Lin, Q., Xiao, M. & Fan, S. Synthetic dimension in photonics. Optica 5, 1396–1405 (2018).
Ozawa, T. & Price, H. M. Topological quantum matter in synthetic dimensions. Nat. Rev. Phys. 1, 349–357 (2019).
Yuan, L., Shi, Y. & Fan, S. Photonic gauge potential in a system with a synthetic frequency dimension. Opt. Lett. 41, 741–744 (2016).
Ozawa, T., Price, H. M., Goldman, N., Zilberberg, O. & Carusotto, I. Synthetic dimensions in integrated photonics: from optical isolation to fourdimensional quantum Hall physics. Phys. Rev. A 93, 043827 (2016).
Bell, B. A. et al. Spectral photonic lattices with complex longrange coupling. Optica 4, 1433–1436 (2017).
Wang, K. et al. Multidimensional synthetic chiraltube lattices via nonlinear frequency conversion. Light. Sci. Appl. 9, 132 (2020).
Dutt, A. et al. A single photonic cavity with two independent physical synthetic dimensions. Science 367, 59–64 (2020).
Qin, C. et al. Spectrum control through discrete frequency diffraction in the presence of photonic gauge potentials. Phys. Rev. Lett. 120, 133901 (2018).
Dutt, A. et al. Experimental band structure spectroscopy along a synthetic dimension. Nat. Commun. 10, 3122 (2019).
Yuan, L., Xiao, M., Lin, Q. & Fan, S. Synthetic space with arbitrary dimensions in a few rings undergoing dynamic modulation. Phys. Rev. B 97, 104105 (2018).
Lukin, D. M. et al. Spectrally reconfigurable quantum emitters enabled by optimized fast modulation. npj Quantum Inf. 6, 1–9 (2020).
Lu, H.H. et al. A controlledNOT gate for frequencybin qubits. npj Quantum Inf. 5, 1–8 (2019).
Huhtanen, M. & Perämäki, A. Factoring matrices into the product of circulant and diagonal matrices. J. Fourier Anal. Appl. 21, 1018–1033 (2015).
Maclaurin, D., Duvenaud, D. & Adams, R. P. Autograd: Effortless gradients in numpy. In ICML 2015 AutoML Workshop, vol. 238 (ICML, 2015).
Molesky, S. et al. Outlook for inverse design in nanophotonics. Nat. Photonics 12, 659–670 (2018).
Minkov, M. et al. Inverse design of photonic crystals through automatic differentiation. ACS Photonics 7, 1729–1741 (2020).
Hughes, T. W., Williamson, I. A. D., Minkov, M. & Fan, S. Wave physics as an analog recurrent neural network. Sci. Adv. 5, eaay6946 (2019).
Byrd, R. H., Lu, P., Nocedal, J. & Zhu, C. A limited memory algorithm for bound constrained optimization. SIAM J. Sci. Comput. 16, 1190–1208 (1995).
Fang, K., Yu, Z. & Fan, S. Photonic aharonovbohm effect based on dynamic modulation. Phys. Rev. Lett. 108, 153901 (2012).
Tischler, N., Rockstuhl, C. & Słowik, K. Quantum optical realization of arbitrary linear transformations allowing for loss and gain. Phys. Rev. X 8, 021017 (2018).
Wang, C., Zhang, M., Stern, B., Lipson, M. & Lončar, M. Nanophotonic lithium niobate electrooptic modulators. Opt. Express 26, 1547–1555 (2018).
Wang, C. et al. Integrated lithium niobate electrooptic modulators operating at CMOScompatible voltages. Nature 562, 101–104 (2018).
Tzuang, L. D., Soltani, M., Lee, Y. H. D. & Lipson, M. High RF carrier frequency modulation in silicon resonators by coupling adjacent freespectralrange modes. Opt. Lett. 39, 1799–1802 (2014).
Van Laer, R., Patel, R. N., McKenna, T. P., Witmer, J. D. & SafaviNaeini, A. H. Electrical driving of Xband mechanical waves in a silicon photonic circuit. APL Photonics 3, 086102 (2018).
Tian, H. et al. Hybrid integrated photonics using bulk acoustic resonators. Nat. Commun. 11, 3073 (2020).
Hu, Y. et al. Realization of highdimensional frequency crystals in electrooptic microcombs. Optica 7, 1189–1194 (2020).
Nahmias, M. A. et al. Photonic multiplyaccumulate operations for neural networks. IEEE J. Sel. Top. Quantum Electron. 26, 1–18 (2019).
Williamson, I. A. D. et al. Reprogrammable electrooptic nonlinear activation functions for optical neural networks. IEEE J. Sel. Top. Quantum Electron. 26, 1–12 (2020).
Ji, X. et al. Onchip tunable photonic delay line. APL Photonics 4, 090803 (2019).
Ji, X. et al. Ultralowloss onchip resonators with submilliwatt parametric oscillation threshold. Optica 4, 619–624 (2017).
Acknowledgements
This work is supported by the U.S. Air Force Office of Scientific Research (FA95501710002, FA95501810379). S.B. acknowledges the support of a Stanford Graduate Fellowship.
Author information
Authors and Affiliations
Contributions
S.B., M.M., and I.A.D.W. conceived the project. S.B., A.D., and M.M. performed the research. S.F. supervised the research. All authors analyzed the results and contributed to discussions. S.B., A.D. and S.F. wrote the manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Peer review information Nature Communications thanks the anonymous reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Buddhiraju, S., Dutt, A., Minkov, M. et al. Arbitrary linear transformations for photons in the frequency synthetic dimension. Nat Commun 12, 2401 (2021). https://doi.org/10.1038/s41467021226707
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s41467021226707
This article is cited by

Terasamplepersecond arbitrary waveform generation in a synthetic dimension
Communications Physics (2023)

Metaprogrammable analog differentiator
Nature Communications (2022)

Enabling scalable optical computing in synthetic frequency dimension using integrated cavity acoustooptics
Nature Communications (2022)

Creating boundaries along a synthetic frequency dimension
Nature Communications (2022)

Topological complexenergy braiding of nonHermitian bands
Nature (2021)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.