Abstract We introduce procedures for decomposing N × N unitary matrices into smaller M × M unitary matrices. Our procedures enable designing modular and optimal architectures for implementing arbitrary discrete unitary transformations on light. Such architectures rely on systematically combining the M -mode linear optical interferometers together to implement a given N -mode transformation. Thus this work enables the implementation of large linear optical transformations using smaller modules that act on the spatial or the internal degrees of freedom of light such as polarization, time or orbital angular momentum. The architectures lead to a rectangular gate structure, which is optimal in the sense that realizing arbitrary transformations on these architectures needs a minimal number of optical elements and minimal circuit depth. Moreover, the rectangular structure ensures that each of the different optical modes incurs balanced optical losses, so the architectures promise substantially enhanced process fidelities as compared to existing schemes.
We present two hybrid linear-optical architectures that simultaneously exploit spatial and temporal degrees of freedom of light to effect arbitrary discrete unitary transformations. Our architectures combine the benefits of spatial implementations of linear optics, namely, low loss and parallel operation, with those of temporal implementations, namely, modest resource requirements and access to transformations of potentially unbounded size. We arrive at our architectures by devising and employing decompositions of large discrete unitary transformations into smaller ones, decompositions we expect to have broad utility beyond spatiotemporal linear optics. We show that hybrid architectures promise important advantages over both spatial-only and temporal-only architectures.
Programmable linear optical interferometers are a core primitive in optical signal processing, quantum information processing, and photonic computing. Existing photonic-integrated implementations realize arbitrary
M-mode unitaries using Mach--Zehnder-interferometer meshes whose footprint and accumulated loss scale with
O(M2) optical components. Here we analyze and experimentally demonstrate a programmable architecture for implementing linear optical transformations directly on spatially tiled free-space {\em pixel modes} using multi-plane light conversion (MPLC). In this architecture,
M spatial modes arranged on a transverse lattice undergo a unitary transformation and are mapped to
M output modes of identical geometry through a sequence of programmable phase masks separated by free-space propagation segments. Numerical simulations show that arbitrary
M-mode unitaries can be compiled to a desired high fidelity using a number of phase planes that scales approximately linearly with
M. Using a spatial-light-modulator-based MPLC, we experimentally demonstrate programmable interferometers acting on up to
16 spatial pixel modes, including tunable beamsplitters, Hadamard unitaries, spatial permutations, and partial unitaries on select subsets of modes. These results establish MPLC-based pixel-mode interferometers as a promising architecture for programmable linear optics with applications in classical and quantum optical interconnects, photonic switching, and quantum information processing.
Any lossless transformation on
ns spatial and
np internal modes of light can be described by an ${n}_{s}{n}_{p}\ifmmode\times\else\texttimes\fi{}{n}_{s}{n}_{p}$ unitary matrix, but there is no known procedure to effect an arbitrary ${n}_{s}{n}_{p}\ifmmode\times\else\texttimes\fi{}{n}_{s}{n}_{p}$ unitary matrix on light in
ns spatial and
np internal modes. We devise an algorithm to realize an arbitrary discrete unitary transformation on the combined spatial and internal degrees of freedom of light. Our realization uses beam splitters and operations on internal modes to effect arbitrary linear transformations. The number of beam splitters required to realize a unitary transformation is reduced as compared to existing realization by a factor
np2/2 at the cost of increasing the number of internal optical elements by a factor of 2. Our algorithm thus enables the optical implementation of higher dimensional unitary transformations.
We propose and demonstrate a modular architecture for reconfigurable on-chip linear-optical circuits. Each module contains 10 independent phase-controlled Mach-Zehnder interferometers; several such modules can be connected to each other to build large reconfigurable interferometers. With this architecture, large interferometers are easier to build and characterize than with traditional, bespoke, monolithic designs. We demonstrate our approach by fabricating three modules in the form of UV-written silica-on-silicon chips. We characterize these chips, connect them to each other, and implement a wide range of linear optical transformations. We envisage that this architecture will enable many future experiments in quantum optics.
We introduce a novel parameterization of complex unitary matrices, which allows for the efficient photonic implementation of arbitrary linear discrete unitary operators. The proposed architecture is built on factorizing an
N×N unitary matrix into interlaced discrete fractional Fourier transforms and
N-parameter diagonal phase shifts. We show that such a configuration can represent arbitrary unitary operators with
N+1 phase layers. We discuss a gradient-based algorithm for finding the optimal phase parameters for implementing a given unitary matrix. By increasing the number of phase layers beyond the critical value of
N+1, the optimization consistently converges faster as the system becomes over-determined. We propose an integrated photonic circuit realization of this architecture with coupled waveguide arrays and reconfigurable phase modulators. The proposed architecture can pave the way for developing novel families of programmable photonic circuits for optical classical and quantum information processing.