Research ArticleAPPLIED PHYSICS

Synthetic phonons enable nonreciprocal coupling to arbitrary resonator networks

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Science Advances  08 Jun 2018:
Vol. 4, no. 6, eaat0232
DOI: 10.1126/sciadv.aat0232
  • Fig. 1 Theoretical description of nonreciprocal coupling.

    (A) Interactions between photons and phonons enable coupling to photonic dark states by modifying the phase matching condition. Phonon-enabled coupling occurs only when the frequency and momentum difference between the waveguide mode and resonator mode are matched by the phonon frequency and momentum. This coupling is inherently nonreciprocal: The forward (right-traveling) guided mode only couples to the dark resonant state through phonon annihilation at the Stokes sideband frequency (ω0 − Ω), and the backward (left-traveling) guided mode only couples to the resonator through phonon creation at the anti-Stokes sideband frequency (ω0 + Ω). (B) A resonator and two-port waveguide are coupled at multiple spatially separated sites. The coupling at these sites can be modulated to create synthetic phonons, which can enable nonreciprocal coupling between the waveguide and resonator. (C) Schematic describing the coupling constants C1 and C2. C1 describes coupling into the resonator from the forward waveguide mode and coupling out of the resonator to the backward waveguide mode and is associated with port 1. C2 describes coupling into the resonator from the backward waveguide mode and coupling out of the resonator to the forward waveguide mode and is associated with port 2.

  • Fig. 2 Experimentally measured nonreciprocal resonant absorption from synthetic phonon–enabled coupling between a microstrip waveguide and resonator.

    (A) Picture of the experimental circuit and measured power transmission without synthetic phonon bias applied. The dark resonance does not interact with the waveguide due to destructive interference from multiple coupling sites, and no dip in transmission is observed. (B) Synthetic phonons with momentum q are applied such that −k0 + Ω) − q = 0, enabling nonreciprocal coupling to the resonator. Absorption occurs at the Stokes sideband for forward propagation (S21 measurement) and the anti-Stokes sideband for reverse propagation (S12 measurement). The resonance is broadened in momentum space since the finite number of coupling sites (N = 3) only completely destructively interfere for k = ±2π/3ℓ. (C) There is no momentum shift when synthetic phonons with zero momentum (q = 0) are applied. However, coupling is observed at the sideband frequencies due to waveguide dispersion. The system is reciprocal because these synthetic phonons do not break time-reversal symmetry.

  • Fig. 3 Experimental demonstration of essential nonreciprocal functions.

    Nonreciprocal coupling is enabled by synthetic phonons with the same momentum q but varying amplitude δc in all plots. All measurements are focused on the anti-Stokes sideband. (A) Nearly zero transmission (≤92 dB) is measured through the waveguide in the direction that is critically coupled (Embedded Image). (B) Measured power isolation contrast for the case shown in (A). (C) For an overcoupled resonator, there is a π nonreciprocal phase shift at the anti-Stokes sideband frequency. (D) When the resonator is undercoupled, there is no phase shift at the anti-Stokes sideband frequency.

  • Fig. 4 Higher-order nonreciprocal transfer functions created through nonreciprocal coupling to resonator networks.

    (A) Nonreciprocal coupling can be engaged to an arbitrary band-limited impedance network. Several of these impedance networks can be simultaneously coupled in either direction to create customizable responses. (B) Photograph and schematic of the circuit use to demonstrate customizable nonreciprocal transfer functions. (C) Measured power transmission showing a flat band over which a constant isolation response is obtained. (D) Experimental demonstration of four distinct nonreciprocal transfer functions obtained by tuning the resonator network.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/4/6/eaat0232/DC1

    section S1. Steady-state transmission coefficients

    section S2. Resonant decay rate

    section S3. Circulator

    fig. S1. Schematic of a waveguide-resonator system with one resonance and two ports.

    fig. S2. Theoretical description of a circulator implemented with nonreciprocal coupling.

    fig. S3. Experimentally measured circulator implemented with nonreciprocal coupling.

  • Supplementary Materials

    This PDF file includes:

    • section S1. Steady-state transmission coefficients
    • section S2. Resonant decay rate
    • section S3. Circulator
    • fig. S1. Schematic of a waveguide-resonator system with one resonance and two ports.
    • fig. S2. Theoretical description of a circulator implemented with nonreciprocal coupling.
    • fig. S3. Experimentally measured circulator implemented with nonreciprocal coupling.

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