Nonreciprocal Multijunction Model Overview

• The Nonreciprocal Multijunction Model is a framework that enables uni-directional transport through cascading junctions by engineering asymmetry in interconnected modes.
• It applies to diverse implementations such as superconducting circuits, photonic stacks, and multijunction solar cells, all optimized via phase tuning and material design.
• Analytic methods quantify performance metrics like isolation and gain, providing practical guidelines for enhancing devices in communications and energy applications.

The Nonreciprocal Multijunction Model generalizes the principle of nonreciprocal transport—originally associated with time-reversal symmetry breaking in electronics and photonics—to cascaded or networked systems comprising multiple interconnected junctions or modes. This framework appears in diverse contexts, including superconducting circuits, optical and photonic architectures, and the thermodynamic modeling of multijunction solar cells, where unidirectional coupling or the suppression of backward flows enables enhanced performance or new functionality. A central feature of such models is that internal transfer processes (e.g., of photons, excitations, or signals) are engineered to be asymmetric: typically, energy or information preferentially flows in one direction, which leads to nonreciprocal transmission, improved isolation, or higher thermodynamic efficiency. The following sections survey the abstract principles, representative model architectures, metrics of performance, and analytic results across physical implementations.

1. Principles of Nonreciprocity and Multijunction Architectures

Nonreciprocity denotes the breakdown of symmetry between forward and backward transport in a physical network, such that the response matrix elements $S_{ij}$ and $S_{ji}$ (transfer from port $j$ to port $i$ and vice versa) are unequal. In the multijunction context, this asymmetry is engineered into a system of interconnected modes, physical junctions, or structured interfaces, either by explicit symmetry breaking in the Hamiltonian, circuit parameters, or transfer laws.

Major architectures that instantiate nonreciprocal multijunction models include:

• Multimode bosonic networks: The "diamond configuration" introduces four coupled resonators with complex-valued hopping (phase-engineered) and parametric (two-mode squeezing) interactions. The closed topology and phase sums around the network enable strong nonreciprocity and even directional amplification (Khorasani, 2016).
• Optoelectronic stacks: In multijunction solar cells, optical transfer across interfaces is modeled as uni-directional via "transfer coefficients," effectively capturing the one-way radiative exchange, which is justified by energy conservation and the narrow overlap of emission/absorption windows in cascaded bandgaps (Strandberg, 2022).
• Transmission-mode nonreciprocal junctions: In planar multilayer photonics, magneto-optic mid-infrared stacks display polarization-independent nonreciprocal transmission when concatenated, enabling unidirectional energy flow across a multijunction ladder (Picardi et al., 2024).
• Superconducting circuits: Multi-terminal Josephson junctions (MTJJs) acquire nonreciprocal response via network asymmetry (e.g., branch critical currents $I_{c,k}$ unequal), leading to a rectifying S-matrix (Zhang et al., 2023); similar effects arise in voltage-tunable hybrid junctions (Leroux et al., 2022).

In all such systems, the multijunction character is crucial: it enables not merely nonreciprocal behavior at a single interface, but cumulative or even enhanced asymmetry in cascades.

2. Representative Model Realizations

(a) Diamond Configuration for Microwave Transmission

The diamond model (Khorasani, 2016) is a four-mode network with the following topology:

• Nodes 1,3 at resonance $\omega$, 2,4 at $\Omega$ ($\Omega\neq\omega$).
• Edges: strong, phase-encoded "hopping" interactions (1–2:$g$, 2–3:$h$, 3–4:$f$, 4–1:$k$).
• Diagonals: weak, real parametric (two-mode squeezing) terms (1–3, 2–4: $\gamma$).

The total Hamiltonian in the rotating-wave approximation is

$$\mathbb{H}_0 = \hbar\omega(\hat{a}_1^\dagger\hat{a}_1+\hat{a}_3^\dagger\hat{a}_3) + \hbar\Omega(\hat{a}_2^\dagger\hat{a}_2+\hat{a}_4^\dagger\hat{a}_4)$$

$$\mathbb{H}_i = \hbar \bigl[ g\hat{a}_1\hat{a}_2^\dagger + h\hat{a}_2\hat{a}_3^\dagger + f\hat{a}_3\hat{a}_4^\dagger + k\hat{a}_4\hat{a}_1^\dagger + \gamma(\hat{a}_1\hat{a}_3 + \hat{a}_2\hat{a}_4) + \text{h.c.} \bigr]$$

By optimizing the phases of $g,h,f,k$ to yield a total round-trip phase $\theta=\pi$, path-interference allows constructive transmission in one direction and destructive in the reverse.

(b) Radiatively Coupled Multijunction Solar Cells

The nonreciprocal transfer-coefficient model (Strandberg, 2022) describes cascaded subcells with transfer fractions $\tau_{i,i+1}$ for photons emitted by subcell $i$ and absorbed by $i+1$, neglecting backflow ($\tau_{i+1,i}=0$): $\tau_{i,i+1} = \frac{n_i^2}{1+2n_i^2}$ where $n_i$ is the refractive index of cell $i$. Net transfer fractions $T_i$ accumulate recursively. The stack current-voltage relation is

$$V(J) = \frac{kT}{q}\ln\Biggl[ \prod_{i=1}^N \frac{J_{G,i}+T_i\Delta J_{G,i}-J}{(1-T_i)\tilde J_{0,i}} \Biggr]$$

This formalism allows closed-form predictions of $J_{SC}$ and $V_{OC}$ enhancements due to nonreciprocal transfer.

(c) Planar Transmission-Mode Multijunctions

A three-layer stack (Si/WSM/SiO$_2$) creates transmission-mode nonreciprocity via a magneto-optic Weyl semimetal layer with off-diagonal permittivity: $$\epsilon(\omega) = \begin{pmatrix} \epsilon^d_r + i\epsilon^d_i & 0 & +i\epsilon^a \ 0 & \epsilon^d_r + i\epsilon^d_i & 0 \ -i\epsilon^a & 0 & \epsilon^d_r + i\epsilon^d_i \end{pmatrix}$$ Concatenation of $N$ such junctions, each with a distinct bandgap/absorption window, enables directionally-biased photon flow toward the lowest bandgap cell, approaching Landsberg/Carnot limits for harvesting (Picardi et al., 2024).

3. Analytic Frameworks and Performance Metrics

Analytic evaluation centers on the calculation of the system S-matrix $S(\omega)$, quantification of isolation and gain, and analysis of nonreciprocity ratios. For the diamond network (Khorasani, 2016):

• The S-matrix is computed as $$S(\omega) = I - \sqrt{\Gamma}[i\omega I + M]^{-1}\sqrt{\Gamma}$$.
• Isolation between two ports (e.g., 1 and 3) is

$$R(\omega) = \frac12\left[ \left|\frac{S_{31}}{S_{13}}\right|^2 + \left|\frac{S_{13}}{S_{31}}\right|^2 \right]$$

and can exceed 12 dB intrinsically and 130 dB extrinsically (with auxiliary pumping).

• Directional gain is quantified as $G_\text{fwd} = |S_{31}|^2$ and $G_\text{bwd}=|S_{13}|^2$.
• Cascading systems multiplies isolation in dB, enabling >170 dB total.

In solar cell models (Strandberg, 2022, Lee, 24 Jan 2026), forward-only photon transfer eliminates upward luminescence entropy losses, raising the open-circuit voltage and stack efficiency by $\approx 1$–2 percentage points for $N\geq5$ ladders (Lee, 24 Jan 2026). In planar photonics (Picardi et al., 2024), the transmission contrast $\Delta T(\omega) = T_f - T_b$ serves as the nonreciprocity metric.

4. Physical Mechanisms and Symmetry Breaking

Nonreciprocity in multijunction systems is achieved via one or more of:

• Phase engineering in multimode Hamiltonians: Collective phase sums around closed loops control directionality.
• Time-reversal symmetry breaking elements: Magneto-optic materials (WSMs), external fields, or opportune spatio-temporal modulations in photonic stacks.
• Structural or circuit asymmetries: Disparate branch parameters (Josephson couplings, critical currents) induce mirror-symmetry breaking; e.g., $I_{c,3} \gg I_{c,1,2}$ in three-terminal JJs (Zhang et al., 2023).
• Unidirectional transfer coefficients: Energy conservation and Boltzmann spectral constraints in cascaded bandgap devices permit one-way radiative coupling, formalized as zero backflow in transfer-coefficient models (Strandberg, 2022).

A recurring principle is that by canceling symmetry between forward and backward pathways—either dynamically (phases/pumping) or statically (material or network asymmetry)—the system suppresses dissipation or loss in one direction, ratcheting up forward performance.

5. Practical Optimization and Implementation Guidelines

Optimization focuses on several control "knobs" unique to each platform:

• Hamiltonian and phase parameters: In the diamond model, tuning the hopping phases ($\phi_g, \phi_h, \phi_f, \phi_k$) to achieve loop-phase $\theta=\pm\pi$ is the most sensitive lever for nonreciprocity; parametric gain $\gamma$ requires careful balance to optimize intrinsic isolation (Khorasani, 2016).
• Structural dimensions: In planar stacks, dielectric and magneto-optic layer thicknesses must satisfy Fabry–Perot resonance and impedance-matching criteria; WSM rotation angle $\varphi$ is scanned to equalize nonreciprocal response for all polarizations (Picardi et al., 2024).
• Material bandgap engineering: In solar applications, bandgap ladders are optimized via dynamic programming for photon splitting objectives; the position and refractive indices of interfaces tune the transfer coefficients $T_i$ (Lee, 24 Jan 2026, Strandberg, 2022).
• Cascading and modular design: Chaining multiple nonreciprocal junctions compounds isolation and enhances cumulative directionality or energy flow (Khorasani, 2016, Picardi et al., 2024).

Parametric regimes for maximum performance typically require a careful trade-off between bandwidth, gain, and acceptable levels of loss or noise, all quantifiable via the above analytic expressions.

6. Broader Implications and Applications

The nonreciprocal multijunction paradigm unlocks performance enhancements in microwave and photonic signal processing (directional amplifiers, circulators, isolators), next-generation photovoltaics and thermophotovoltaics (by approaching thermodynamic efficiency bounds via one-way luminescence), and superconducting device engineering (supercurrent rectification, field-free superconducting diodes) (Khorasani, 2016, Picardi et al., 2024, Zhang et al., 2023, Lee, 24 Jan 2026). In all cases, multijunction architectures are leveraged to accumulate the benefits of local nonreciprocity throughout the network or stack, with optimization rules tailored to the particulars of the physical system and the target operational regime.

Analytically tractable transfer-coefficient models reproduce full detailed-balance and numerical simulations to sub-percent accuracy, while allowing for inclusion of nonidealities such as finite external radiative efficiency, nonradiative recombination, and imperfect optical isolation (Strandberg, 2022). This suggests the abstract nonreciprocal multijunction model is a powerful unifying tool for both theoretical and applied device research across multiple platforms.