Fig. 1. Temporally indistinguishable photons within the temporal resolution δt propagating through different delay lines can be temporally distinguishable given the delay provided by the ring resonator waveguides is sensitive to disorder, i.e., random red or blue shifts of the individual resonators. Insets below illustrate various possible effects of disorder: (a) phase shift via the difference in phase velocities, (b) difference in arrival times due to variation of the group velocities, and (c) wavepacket distortion due to higher-order dispersion and wavelength-dependent reflection.

Fig. 2. Schematic of the helical coupled-resonator optical waveguide (H-CROW). Pseudospin-momentum locking is achieved after a certain propagation distance, where each sublattice exhibits definite momentum for designated circulation mode, thereby facilitating a disorder-resistant transport. As opposite circulations exhibit opposite helicity, two co-propagating channels can be realized.

Fig. 3. Classical wave transport through H-CROWs and CROWs in the presence of moderate disorder U=0.8J and intrinsic losses κin=0.1J. (a), (b) Disorder-averaged field intensity profiles at ω=0 in the first 10 rings of an L=20 (a) H-CROW and (b) CROW. (c) Frequency-dependent transmission spectra. Solid lines indicate the disorder average; shaded regions represent 65% confidence interval. Maximum of average is −15.8 and −22.8  dB at ω=0 for the H-CROW and CROW, respectively. (d) Dependence of the transmission at ω=0 on the disorder strength U. (e) Wave packet delay time as a function of the input frequency. (f) Distribution of delay times at ω=0, where τ¯ is the root mean square delay.

Fig. 4. (a), (b) Schematics of coincidence measurement using tight-binding models of an H-CROW and a pair of regular CROWs. We consider measurements for two photons exhibiting opposite helicity with controlled delay time τc before a 50:50 beam splitter (BS) and resulting coincidence probability of two photons to produce simultaneous “clicks” of single-photon detectors. (c) Coincidence versus controlled delay time for 20-site long CROW structures. (d) Minimum coincidence values with respect to the number of sites. Blue solid line and dots represent the average for H-CROWs and red for CROWs. Error bars indicate 65% confidence interval for 500 disorder realizations.

Fig. 5. Disorder-robust transmission of N=2 N00N states using the H-CROW. (a), (b) Statistics of the output coincidence probability for the (a) H-CROW and (b) regular CROWs. (c) Output state purity versus controlled delay time for the H-CROW (blue) and regular CROW (red), with error bars indicating 65% confidence interval. (d) Exponentiated entanglement entropy of the upper output port, exp(Sa), as a function of the photon number N. exp(Sa) distinguishes maximally entangled states exp(Sa)=2 from separable states exp(Sa)=1. We use an ensemble of 500 disorder realizations and disorder strength U=0.8J.