Synthetic phonons enable nonreciprocal coupling to arbitrary resonator networks

Nonreciprocal coupling to engineered dark states

We consider our representative system (Fig. 1B) consisting of a two-port waveguide having a frequency-dependent propagation constant k and a resonator supporting a single mode at angular frequency ω₀. The resonator is side-coupled to the waveguide at N independent sites that are evenly separated on the waveguide by a constant length ℓ. For simplicity, we assume that each coupling site is located at the same spatial location on the resonator and that the waveguide is lossless and only supports a single mode. We define forward propagation in the waveguide from port 1 toward port 2.

This system can be characterized by analyzing the coupling between the waveguide and the resonator using the framework of temporal coupled-mode theory (32, 33). Since each coupling site is independent, the coupling constants C₁ and C₂ between the waveguide and resonator (see Fig. 1C) are evaluated as a superposition of the contributions from each siteC1=∑n=1Ncne−ikℓ(n−1), C2=∑n=1Ncneikℓ(n−1)(1)where c_n is the coupling constant at the nth site. The exponential term in these definitions accounts for propagation in the waveguide between adjacent coupling sites spaced by ℓ and differs between C₁ and C₂ due to the opposite propagation directions. The coupling constants are also related to the resonator’s decay, which can be described by the decay rate γ = (κ_i + κ_ex)/2. Here, κ_i is the intrinsic decay rate of the resonator, and κ_ex = |C₁|² + |C₂|² is the external decay rate due to coupling with the waveguide (33).

Equation 1 reveals that the contribution from the nth coupling site carries a phase kℓ(n − 1). When summed, these contributions interfere such that the maximum coupling rate occurs only if all N contributions are in-phase (phase-matched coupling). The coupling rate decreases away from this maximum and reaches zero when the contributions perfectly destructively interfere. In the case of a complete phase mismatch, we obtain κ_ex = 0, and the resonator can be classified as a dark state since it cannot be excited by (or decay to) the waveguide. Since phase matching in this system is determined by the product kℓ, it is possible to arrange a dark state from an arbitrary waveguide and resonator by selecting the appropriate ℓ.

We now discuss how a dark state created by a total phase mismatch can be coupled to the accompanying waveguide through a synthetic phonon bias (Fig. 1, A and B). We consider synthetic phonons having angular frequency Ω, momentum q, and amplitude δ_c, which are a modulation of each site’s coupling ratecn=c0+δccos(Ωt−qℓ(n−1))(2)

The product qℓ is equivalent to a phase offset on the modulation applied to adjacent sites; thus, any phonon momentum q can be selected by modulating each site with a phase offset θ_n = qℓ(n − 1). The synthetic phonon bias breaks time-reversal symmetry, and thus induces nonreciprocal coupling, if the momentum q satisfies qℓ ≠ zπ, where z is an integer. This condition is equivalent to having synthetic phonons with a nonzero momentum, since if qℓ = zπ, then a standing wave is formed.

When this spatiotemporally modulated coupling is substituted into Eq. 1, it is instructive to separate the resulting terms into frequency components as followsC1=c0∑n=1Ne−ikℓ(n−1)⏞C10+δc2eiΩt∑n=1Ne−i(k+q)ℓ(n−1)⏞C1++δc2e−iΩt∑n=1Ne−i(k−q)ℓ(n−1)⏞C1−(3)C2=c0∑n=1Neikℓ(n−1)⏞C20+δc2eiΩt∑n=1Nei(k−q)ℓ(n−1)⏞C2++δc2e−iΩt∑n=1Nei(k+q)ℓ(n−1)⏞C2−(4)

For brevity, from here we refer to the terms that make up the coupling constants as Cm=Cm0+Cm++Cm− for m = 1, 2. The first term Cm0 does not depend on the modulation amplitude δ_c and describes coupling that would occur without synthetic phonons. The remaining terms only describe coupling enabled by interactions with synthetic phonons: Cm+ corresponds to coupling where a synthetic phonon is annihilated and the photon shifts up in frequency, and Cm− corresponds to coupling where a synthetic phonon is created and the photon shifts down in frequency. Because of energy and momentum conservation, both terms incorporate a frequency shift (e^±iΩt) and momentum shift (k ± q), as depicted in Fig. 1A. The momentum shift modifies the original phase matching condition and can enable coupling to a resonator which would otherwise be dark.

Coupling to the resonator, including coupling enabled by the action of synthetic phonons, has a significant impact on wave transmission through the waveguide due to resonant absorption or reflection. In our proposed system, the steady-state forward transmission coefficient (S₂₁) as a function of frequency ω is evaluated (see the Supplementary Materials for a complete derivation) to beS21=e−ikℓ(N−1)−C20C10i(ω−ω0)+γ−C2−C1+i(ω+Ω−ω0)+γ−C2+C1−i(ω−Ω−ω0)+γ(5)

Here, S₂₁ is a linear transfer function, and terms corresponding to transmission with a frequency shift have been dropped. The steady-state backward transmission coefficient (S₁₂) is similarlyS12=e−ikℓ(N−1)−C10C20i(ω−ω0)+γ−C1−C2+i(ω+Ω−ω0)+γ−C1+C2−i(ω−Ω−ω0)+γ(6)

From the above equations, we find that the synthetic phonon–enabled coupling results in a distinct transmission spectrum where resonant absorption can occur at the original resonance frequency ω₀ and at the shifted frequencies ω₀ ± Ω. We will hereafter refer to absorption at ω₀ − Ω as the Stokes sideband and absorption at ω₀ + Ω as the anti-Stokes sideband (see Fig. 1A). Since the sideband coupling constants are not required to be equal, that is, C1+≠C2+, transmission at these sidebands can be strongly nonreciprocal. In addition, we note that absorption at the sidebands is in general asymmetric due to the frequency dependence of k.

To experimentally validate our theory, we implemented a waveguide-resonator system with three coupling sites (N = 3) using a microstrip waveguide and stub resonator (Fig. 2A, bottom). The fabricated resonator has a loaded resonant frequency ω₀/2π ≈ 1.4 GHz. The waveguide and resonator are coupled through variable capacitors (varactor diodes) which enable dynamic control over the coupling constants c_n (see Methods). We design the coupling site separation such that kℓ = 2π/3 at ω = ω₀, resulting in a complete phase mismatch and creating a dark state. Without a synthetic phonon bias, the measured response for this circuit (Fig. 2A, top) does not indicate any dips in transmission corresponding to resonant absorption, confirming that interactions between the resonator and waveguide are suppressed by the phase mismatch. The broadband background transmission losses are caused by reflection from and losses in the capacitive coupling network.

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We next apply synthetic phonons with frequency Ω/2π = 104 MHz and momentum q = −k at ω = ω₀ + Ω, implemented through a phase offset θn=5π3(n−1), to the system, as described by Eq. 2. This phonon momentum was empirically tuned to maximize the coupling rate C1+C2− between the resonator and backward waveguide mode at the anti-Stokes sideband. From Eq. 5, we see that, neglecting any frequency dependence, C1+C2− also describes coupling for the forward waveguide mode at the Stokes sideband. Thus, resonant absorption should occur nonreciprocally: at the anti-Stokes sideband for backward transmission and at the Stokes sideband for forward transmission. The coupling rate C2+C1− is simultaneously minimized by this choice of phonon momentum, so no absorption is expected at these frequencies for the opposite directions (anti-Stokes for forward transmission and Stokes for backward transmission).

The measured forward (S₂₁) and backward (S₁₂) transmission coefficients for this system are shown in Fig. 2B. As predicted, resonant absorption occurs at ≈1.3 GHz only in the forward direction and at ≈1.5 GHz only in the backward direction. The frequency dependence of k creates a slight phase mismatch at the Stokes sideband, resulting in reduced absorption. The measured absorption is highly nonreciprocal, no resonant absorption is observed at ≈1.3 GHz in the backward direction or at ≈1.5 GHz in the forward direction, validating that C1−=C2+≈0.

In this experiment, we have shown that synthetic phonons can facilitate coupling to dark states by modifying the original phase matching condition. Such phonon-assisted coupling results in nonreciprocal transmission if this modified phase matching condition is not satisfied for both directions simultaneously. However, synthetic phonons that do not modify the phase matching condition, that is, phonons with zero momentum (q = 0), can also enable coupling to a dark resonator due to the frequency dependence of k. This case, which we show experimentally in Fig. 2C, demonstrates that both waveguide directions can couple to a dark state simultaneously, resulting in reciprocal transmission. We note that because a partial phase mismatch remains, the coupling is weaker at the sidebands and absorption is reduced compared to Fig. 2B.

Demonstration of elementary nonreciprocal devices

Nonreciprocal devices such as isolators and circulators are important tools for controlling wave propagation and have a wide range of uses, from protecting lasers against reflections (34) to facilitating full duplex communications (35). The gyrator, a fundamental nonreciprocal building block that introduces a unidirectional π phase shift, can be used to produce a variety of nonreciprocal circuits including isolators and circulators. Below, we experimentally show how both isolators and gyrators can be directly created through synthetic phonon–enabled coupling to a single dark state. In addition, in the Supplementary Materials, we provide preliminary experimental evidence of a four-port circulator implemented using synthetic phonon–enabled coupling between a dark state and two waveguides.

We first consider the case of an isolator with high transmission amplitude in the forward direction and zero transmission in the backward direction, operating at the anti-Stokes sideband frequency ω₀ + Ω. Examining Eqs. 5 and 6, we find that this case occurs when C1+C2−=γ [the critical coupling condition (36)] and C2+C1−=0. We experimentally investigated this case using the circuit shown in Fig. 2A. The resonance frequency was tuned to ω₀/2π ≈ 1.42 GHz, and synthetic phonons were again applied with frequency Ω/2π = 104 MHz and q = −k(ω₀ + Ω). The synthetic phonon amplitude δ_c was increased until the critical coupling condition C1+C2−=γ was reached. The measured forward (S₂₁) and backward (S₁₂) transmission coefficients for synthetic phonons with this critical amplitude are presented in Fig. 3A. We observe a large Lorentzian dip in the measured backward transmission, which drops to below −89 dB at 1.52 GHz. No resonant absorption is visible in the forward direction. The measured isolation contrast (Fig. 3B) exceeds 82 dB with a 10-dB bandwidth of approximately 12 MHz.

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We next analyze the case of a gyrator, where high transmission amplitude occurs in both directions, but the backward transmission is phase-shifted by π in comparison to forward transmission. Considering the same system as above, it is evident from Eqs. 5 and 6 that this case occurs if the phonon amplitude is increased such that C1+C2−≈2γ (strong overcoupling) while the opposite direction remains uncoupled. We experimentally realize nonreciprocal overcoupling by further increasing the synthetic phonon amplitude such that C1+C2−>γ and observe the anticipated nonreciprocal π phase shift at the anti-Stokes sideband frequency ≈1.52 GHz (Fig. 3B). Unfortunately, we were unable to realize the required synthetic phonon amplitude to achieve C1+C2−≈2γ due to limitations caused by nonlinearity in the varactor diodes. For comparison, we also show measured transmission amplitude and phase for the undercoupled case, where C1+C2−<γ and there is no nonreciprocal π phase shift at the anti-Stokes sideband.

Higher-order nonreciprocal transfer functions

While high-order filters are often necessary for signal processing applications (37, 38), a platform for integrating such functionality into nonmagnetic nonreciprocal systems has not yet been shown. Frequency-selective nonreciprocal devices in the literature have been mainly limited to Lorentzian-shaped transfer functions (14, 15, 17–19, 23, 35). Synthetic phonon–enabled coupling is a technique uniquely suited to address this challenge because it can permit unidirectional access to arbitrary band-limited load impedances (Fig. 4A), producing arbitrary nonreciprocal responses. In addition, different frequency responses could be simultaneously achieved in opposite directions by coupling each direction to an appropriate resonator network (39).

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Implementing this idea, we experimentally demonstrate non-Lorentzian nonreciprocal transfer functions using the circuit shown in Fig. 4B, which is a modified version of that in Fig. 2A. Here, two additional microstrip stub resonators with tunable resonance frequencies are coupled to the original stub resonator used in previous experiments (Fig. 4B), providing six additional degrees of freedom: the additional resonance frequencies ω₁, ω₂, interresonator coupling rates κ₁, κ₂, and linewidths γ₁, γ₂.

A maximally flat nonreciprocal filter with constant isolation over an appreciable bandwidth is arguably one of the most important functionalities that cannot be implemented using a single resonant response. Such a flat response can be approximated in the three-resonator network using γ₁ = γ₂ = γ, κ₁ = κ₂ = 9γ/14, ω₁ = ω₀ + 3γ/7, and ω₂ = ω₀ − 3γ/7, where the loss rate γ and resonance frequency ω₀ are associated with the original resonator. We empirically tuned both the resonance frequencies (ω₁, ω₂) and coupling rates (κ₁, κ₂) of the additional resonators in our circuit (Fig. 4B) near these values until the desired transfer function was achieved. Since the resonators are fabricated on the same substrate and conductor, their linewidths are intrinsically equal. The experimentally measured transmission through the waveguide (Fig. 4C) exhibits nearly constant isolation of 14 dB over a 10-MHz bandwidth. In Fig. 4D, we present four additional examples of arbitrary nonreciprocal transfer functions obtained by varying the interresonator coupling strength and frequency separation of the three resonators. In these experiments, we observe consistently flat forward transmission (S₂₁) even though the reverse transmission (S₁₂) varies, demonstrating that propagation in the uncoupled direction is largely unaffected by changes to the impedance network.