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Abstract Random matrices are fundamental in photonic computing because of their ability to model and enhance complex light interactions and signal processing capabilities. In manipulating classical light, random operations are utilized for random projections and dimensionality reduction, which are important for analog signal processing, computing, and imaging. In quantum information processing, random unitary operations are essential to boson sampling algorithms for multiphoton states in linear photonic circuits. Random operations are typically realized in photonic circuits through fixed disordered structures or through large meshes of interferometers with reconfigurable phase shifters, requiring a large number of active components. In this article, we introduce a compact photonic circuit for generating random matrices by utilizing programmable phase modulation layers interlaced with a fixed mixing operator. We show that using only two random phase layers is sufficient for producing output optical signals with a white-noise profile, even for highly sparse input optical signals. We experimentally demonstrate these results using a silicon-based photonic circuit with tunable thermal phase shifters and waveguide lattices as mixing layers. The proposed circuit offers a practical method for generating random matrices for photonic information processing and for applications in data encryption. 

Abstract The quantum conductance and its classical wave analogue, the transmittance, are given by the sum of the eigenvalues of the transmission matrix. However, neither measurements nor theoretical analysis of the transmission eigenchannels have been carried out to explain the dips in conductance found in simulations as new channels are introduced. Here, we measure the microwave transmission matrices of random waveguides and find the spectra of all transmission eigenvalues, even at dips in the lowest transmission eigenchannel that are orders of magnitude below the noise in the transmission matrix. Transmission vanishes both at topological transmission zeros, where the energy density at the sample output vanishes, and at crossovers to new channels, where the longitudinal velocity vanishes. Zeros of transmission pull down all the transmission eigenvalues and thereby produce dips in the transmittance. These dips and the ability to probe the characteristics of even the lowest transmission eigenchannel are due to correlation among the eigenvalues. The precise tracking of dips in the conductance by peaks in the density of states points to a further correlation between zeros and poles of the transmission matrix. The conductance approaches Ohm’s law as the sample width increases in accord with the correspondence principle. 

Abstract The diffusion model is used to calculate both the time-averaged flow of particles in stochastic media and the propagation of waves averaged over ensembles of disordered static configurations. For classical waves exciting static disordered samples, such as a layer of paint or a tissue sample, the flux transmitted through the sample may be dramatically enhanced or suppressed relative to predictions of diffusion theory when the sample is excited by a waveform corresponding to a transmission eigenchannel. Even so, it is widely assumed that the velocity of waves is irretrievably randomized in scattering media. Here we demonstrate in microwave measurements and numerical simulations that the statistics of velocity of different transmission eigenchannels are distinct and remains so on all length scales and are identical on the incident and output surfaces. The interplay between eigenchannel velocities and transmission eigenvalues determines the energy density within the medium, the diffusion coefficient, and the dynamics of propagation. The diffusion coefficient and all scattering parameters, including the scattering mean free path, oscillate with the width of the sample as the number and shape of the propagating channels in the medium change. 

The statistics of transmission through random 1D media are generally presumed to be universal and to depend only upon a single dimensionless parameter—the ratio of the sample length and the mean free path, s = L/ℓ. Here we show in numerical simulations and optical measurements of random binary systems, and most prominently in systems for which s < 1, that the statistics of the logarithm of transmission, lnT, are universal for transmission near the upper cutoff of unity and depend distinctively upon the reflectivity of the layer interfaces and their number near a lower cutoff. The universal segment of the probability distribution function of the logarithm of transmission, P(lnT) is manifest with as few as three binary layers. For a given value of s, P(lnT) evolves towards a universal distribution as the number of layers increases. Optical measurements in stacks of 5 and 20 glass coverslips exhibit statistics at low and moderate values of transmission that are close to those found in simulations for 1D layered media, while differences appear at higher transmission where the dwell time in the medium is longer and the wave explores the transverse nonuniformity of the sample.