Non-Reciprocal Inductance in Advanced Circuits

• Non-reciprocal inductance is a finite-frequency antisymmetric reactive response found in circuits with broken input–output symmetry, as seen in multiterminal Josephson and Hall devices.
• It arises from mechanisms like Berry-curvature effects and gyrator-based flux–charge coupling, leading to an antisymmetric admittance matrix and directional signal behavior.
• The phenomenon distinguishes dynamic nonreciprocity from static symmetric inductance, offering practical insights for designing superconducting circulators and microwave isolators.

Searching arXiv for recent and foundational papers on non-reciprocal inductance and closely related circuit realizations. First search: multiterminal Josephson junction linear-response nonreciprocity and reactive admittance. Second search: passive non-reciprocal microwave devices and effective circuit models with chiral impedance. Third search: superconducting-circuit gyrators from flux–charge coupling and reciprocal-vs-nonreciprocal circuit distinctions. Non-reciprocal inductance denotes a reactive circuit response that is not invariant under interchange of source and response channels. In strict linear-response form, it appears as an antisymmetric component of a susceptibility or admittance matrix, rather than as an ordinary symmetric inductance matrix derived from an equilibrium energy functional. In contemporary arXiv literature, the term spans several closely related but not identical regimes: finite-frequency multiterminal Josephson response with $\chi_{ij}(\omega)\neq \chi_{ji}(\omega)$, gyrator-like flux–charge couplings in superconducting circuits, and effective chiral impedances in Hall-based microwave devices. Other works provide only indirect or effective analogues, such as direction-dependent Josephson dynamics or momentum-resolved apparent non-reciprocity in reciprocal RLC networks (Virtanen et al., 2023, Leroux et al., 2022, Bosco, 31 Aug 2025).

1. Reciprocity, inductance matrices, and the scope of the term

In the circuit literature represented here, non-reciprocal means that a system “behaves differently under interchange of input and response,” whereas non-Hermitian means that it exchanges energy with its environment (Hofmann et al., 2019). These notions are distinct. In reciprocal topolectrical circuits, reciprocity is expressed as

$J^{\top}(k_x,k_y)=J(-k_x,-k_y),$

and a system may remain reciprocal even when it is dissipative.

For Josephson systems, the sharpest distinction is between static equilibrium response and finite-frequency response. The linear electromagnetic response is written as

$$J^i(\omega)=\sum_j \chi_{ij}(\omega)\, \frac{V_j(\omega)}{-i\omega},\qquad Y_{ij}(\omega)=\frac{\chi_{ij}(\omega)}{i\omega}.$$

Here, reciprocal response means $\chi_{ij}(\omega)=\chi_{ji}(\omega)$, while nonreciprocal response means $\chi_{ij}(\omega)\neq \chi_{ji}(\omega)$ (Virtanen et al., 2023).

At zero frequency, however, the Josephson response is always reciprocal: $$\chi_{ij}(0)=\chi_{ji}(0)=(L^{-1})_{ij} =\frac{4e^2}{\hbar^2}\frac{\partial^2 F}{\partial \varphi_i \partial \varphi_j}.$$ Because this static response is the Hessian of the free energy, the conventional equilibrium Josephson inductance matrix is symmetric. This is why recent work treats non-reciprocal inductance primarily as a finite-frequency or dynamical phenomenon rather than as a static linear inductance (Virtanen et al., 2023).

A complementary Lagrangian criterion appears in circuit models with explicit nonreciprocal couplings. In that formalism, the necessary and sufficient condition for nonreciprocity is

$\theta \neq \theta^{\mathrm{T}},$

while a reciprocal circuit has $\theta=\theta^{\mathrm{T}}$ (Rouhi et al., 2023).

Platform	Nonreciprocal object	Relation to non-reciprocal inductance
Multiterminal Josephson junctions	Antisymmetric finite-frequency admittance	Direct linear-response realization (Virtanen et al., 2023)
Voltage-tunable superconducting junction circuits	Gyrator term $\propto \dot\Phi_2\Phi_1-\dot\Phi_1\Phi_2$	Direct dynamical route (Leroux et al., 2022)
Hall-material microwave devices	Ideal circulator plus reactive stubs	Effective chiral impedance (Bosco, 31 Aug 2025)
Asymmetric Josephson diode dynamics	Direction-dependent nonlinear impedance/capacitance/inductive-like response	Indirect, effective only (Misaki et al., 2020)
Reciprocal topolectrical skin-effect circuits	Momentum-resolved apparent non-reciprocity	Not true non-reciprocal inductance (Hofmann et al., 2019)
Reciprocal EPD circuits	Same Jordan form as gyrator circuit	Spectral degeneracy is not diagnostic (Rouhi et al., 2023)

2. Static Josephson asymmetry versus genuine non-reciprocal inductive response

An important precursor is the theory of the nonreciprocal Josephson effect. In that setting, the mechanism is an asymmetry in charge accumulation on the two sides of the junction, encoded as a non-symmetric charging energy,

$E_{\rm ch}(Q)\neq E_{\rm ch}(-Q),$

with higher-order terms such as

$J^{\top}(k_x,k_y)=J(-k_x,-k_y),$0

Because the voltage is

$J^{\top}(k_x,k_y)=J(-k_x,-k_y),$1

positive and negative accumulated charge produce different responses, leading to direction-dependent thresholds

$J^{\top}(k_x,k_y)=J(-k_x,-k_y),$2

and to nonreciprocal $J^{\top}(k_x,k_y)=J(-k_x,-k_y),$3-$J^{\top}(k_x,k_y)=J(-k_x,-k_y),$4 curves (Misaki et al., 2020).

In the classical or semiclassical regime, the model is a resistively shunted Josephson junction with finite capacitance: $J^{\top}(k_x,k_y)=J(-k_x,-k_y),$5 with $J^{\top}(k_x,k_y)=J(-k_x,-k_y),$6. In the overdamped limit, by contrast, the response reduces to the usual symmetric Josephson behavior. The source of asymmetry is therefore not a static inductance matrix, but the combination of finite capacitance and asymmetric charging dynamics.

This distinction is central for the topic of non-reciprocal inductance. The work supports a direction-dependent effective reactive response, and it explicitly connects the phenomenon to nonlinear impedance and to the Josephson diode. It does not directly derive a static relation $J^{\top}(k_x,k_y)=J(-k_x,-k_y),$7. A precise reading is therefore that the paper establishes nonreciprocal Josephson diode behavior and a direction-dependent nonlinear circuit response, only indirectly supporting the language of non-reciprocal inductance (Misaki et al., 2020).

3. Finite-frequency Josephson non-reciprocity as a reactive inductive effect

The most explicit linear-response realization appears in multiterminal Josephson junctions. The key statement is that static Josephson diodes involve nonlinear $J^{\top}(k_x,k_y)=J(-k_x,-k_y),$8 asymmetry, whereas finite-frequency driving produces a genuine small-signal nonreciprocity with

$J^{\top}(k_x,k_y)=J(-k_x,-k_y),$9

The static response remains reciprocal, but finite-frequency response can acquire an antisymmetric reactive component (Virtanen et al., 2023).

The bound-state contribution is given by the Kubo formula

$$J^i(\omega)=\sum_j \chi_{ij}(\omega)\, \frac{V_j(\omega)}{-i\omega},\qquad Y_{ij}(\omega)=\frac{\chi_{ij}(\omega)}{i\omega}.$$0

Nonreciprocity requires transitions for which

$$J^i(\omega)=\sum_j \chi_{ij}(\omega)\, \frac{V_j(\omega)}{-i\omega},\qquad Y_{ij}(\omega)=\frac{\chi_{ij}(\omega)}{i\omega}.$$1

At low frequency and $$J^i(\omega)=\sum_j \chi_{ij}(\omega)\, \frac{V_j(\omega)}{-i\omega},\qquad Y_{ij}(\omega)=\frac{\chi_{ij}(\omega)}{i\omega}.$$2,

$$J^i(\omega)=\sum_j \chi_{ij}(\omega)\, \frac{V_j(\omega)}{-i\omega},\qquad Y_{ij}(\omega)=\frac{\chi_{ij}(\omega)}{i\omega}.$$3

where

$$J^i(\omega)=\sum_j \chi_{ij}(\omega)\, \frac{V_j(\omega)}{-i\omega},\qquad Y_{ij}(\omega)=\frac{\chi_{ij}(\omega)}{i\omega}.$$4

is the Andreev-bound-state Berry curvature. The static term is symmetric, whereas the term linear in $$J^i(\omega)=\sum_j \chi_{ij}(\omega)\, \frac{V_j(\omega)}{-i\omega},\qquad Y_{ij}(\omega)=\frac{\chi_{ij}(\omega)}{i\omega}.$$5 is antisymmetric in $$J^i(\omega)=\sum_j \chi_{ij}(\omega)\, \frac{V_j(\omega)}{-i\omega},\qquad Y_{ij}(\omega)=\frac{\chi_{ij}(\omega)}{i\omega}.$$6. This is the direct link between Berry-phase geometry and non-reciprocal reactive response.

The paper separates the nonreciprocal component as

$$J^i(\omega)=\sum_j \chi_{ij}(\omega)\, \frac{V_j(\omega)}{-i\omega},\qquad Y_{ij}(\omega)=\frac{\chi_{ij}(\omega)}{i\omega}.$$7

Its real part is dissipative and its imaginary part is reactive. The reactive regime is especially important because outside exact Andreev resonances the nonreciprocal response is predominantly reactive, so the microwave transmission is essentially non-dissipative.

An explicit three-terminal example makes the inductive interpretation concrete. At

$$J^i(\omega)=\sum_j \chi_{ij}(\omega)\, \frac{V_j(\omega)}{-i\omega},\qquad Y_{ij}(\omega)=\frac{\chi_{ij}(\omega)}{i\omega}.$$8

the strong nonreciprocal regime yields

$$J^i(\omega)=\sum_j \chi_{ij}(\omega)\, \frac{V_j(\omega)}{-i\omega},\qquad Y_{ij}(\omega)=\frac{\chi_{ij}(\omega)}{i\omega}.$$9

This admittance matrix is purely reactive, explicitly antisymmetric, and scales as $\chi_{ij}(\omega)=\chi_{ji}(\omega)$0. It is one of the clearest formulas in the literature for a non-reciprocal inductance-like response. The same work shows that, with suitable impedance matching, the microwave scattering can approach complete non-reciprocity, enabling an Andreev-bound-state Josephson circulator operated away from dissipative resonances (Virtanen et al., 2023).

The symmetry requirements are equally explicit: time-reversal symmetry and permutation or spatial symmetry between terminals must be broken. Nonzero superconducting phase differences are generically sufficient, and flux biasing is usually required; systems with intrinsic $\chi_{ij}(\omega)=\chi_{ji}(\omega)$1 shift effects can show nonreciprocity even without external flux bias.

4. Gyrators, flux–charge coupling, and superconducting implementations

A second direct route to non-reciprocal inductive behavior is through gyration. In voltage-tunable hybrid superconducting–semiconducting circuits, the Josephson energy depends on both gate voltage and superconducting flux,

$\chi_{ij}(\omega)=\chi_{ji}(\omega)$2

In the weak-transmission regime this becomes

$\chi_{ij}(\omega)=\chi_{ji}(\omega)$3

The decisive step is to let the gate voltage of one junction depend on the time derivative of another mode’s flux,

$\chi_{ij}(\omega)=\chi_{ji}(\omega)$4

Expanding in $\chi_{ij}(\omega)=\chi_{ji}(\omega)$5 and $\chi_{ij}(\omega)=\chi_{ji}(\omega)$6 produces a mixed charge–flux coupling (Leroux et al., 2022).

The leading interaction is

$\chi_{ij}(\omega)=\chi_{ji}(\omega)$7

and at the gyrator operating point the effective interaction is

$\chi_{ij}(\omega)=\chi_{ji}(\omega)$8

This is the Lagrangian of a gyrator with conductance $\chi_{ij}(\omega)=\chi_{ji}(\omega)$9. Ordinary inductance is encoded in symmetric flux–flux couplings, but the gyrator introduces an antisymmetric flux–charge coupling. In the paper’s own circuit language, the dynamical impedance is

$\chi_{ij}(\omega)\neq \chi_{ji}(\omega)$0

The term $\chi_{ij}(\omega)\neq \chi_{ji}(\omega)$1 is the antisymmetric response responsible for nonreciprocal transport.

The mechanism requires static external magnetic flux biases. In the weak-transmission approximation, the coupling is proportional to $\chi_{ij}(\omega)\neq \chi_{ji}(\omega)$2, and it is largest at

$\chi_{ij}(\omega)\neq \chi_{ji}(\omega)$3

Biasing the two halves at opposite fluxes makes the antisymmetric coupling survive while unwanted reciprocal terms cancel.

The associated ideal two-port scattering matrix is

$\chi_{ij}(\omega)\neq \chi_{ji}(\omega)$4

which is the canonical signature of a gyrator. The same architecture is presented as compact and passive, with no parametric drive required. The paper also states that the effective conductance is power dependent,

$\chi_{ij}(\omega)\neq \chi_{ji}(\omega)$5

so stronger intracircuit fields compress the nonreciprocal response, and it gives explicit disorder and parasitic-coupling conditions under which the perturbations remain small (Leroux et al., 2022).

5. Hall materials, kinetic inductance, and effective chiral microwave impedance

Passive non-reciprocal microwave devices based on Hall materials provide a third framework. Here the microscopic starting point is a semiclassical Hall conductor with

$\chi_{ij}(\omega)\neq \chi_{ji}(\omega)$6

and conductivity tensor

$\chi_{ij}(\omega)\neq \chi_{ji}(\omega)$7

In the local-capacitance approximation, the Hall response converts the boundary dynamics into a chiral edge-plasmon problem, while capacitive coupling to external electrodes converts that internal dynamics into a terminal microwave response (Bosco, 31 Aug 2025).

The exact terminal admittance $\chi_{ij}(\omega)\neq \chi_{ji}(\omega)$8 is derived as a Fourier sum. In the quantum Hall limit $\chi_{ij}(\omega)\neq \chi_{ji}(\omega)$9, the result simplifies to a universal circuit interpretation: the device behaves like an ideal circulator with characteristic impedance

$$\chi_{ij}(0)=\chi_{ji}(0)=(L^{-1})_{ij} =\frac{4e^2}{\hbar^2}\frac{\partial^2 F}{\partial \varphi_i \partial \varphi_j}.$$0

scattering matrix

$$\chi_{ij}(0)=\chi_{ji}(0)=(L^{-1})_{ij} =\frac{4e^2}{\hbar^2}\frac{\partial^2 F}{\partial \varphi_i \partial \varphi_j}.$$1

and terminal-dependent stubs

$$\chi_{ij}(0)=\chi_{ji}(0)=(L^{-1})_{ij} =\frac{4e^2}{\hbar^2}\frac{\partial^2 F}{\partial \varphi_i \partial \varphi_j}.$$2

This mapping is not a literal lumped inductor with different forward and backward inductance values. Rather, the Hall material functions as an ideal chiral current conveyor or circulator dressed by reactive delay-line stubs.

Dissipation is incorporated by allowing the effective parameters to become complex: $$\chi_{ij}(0)=\chi_{ji}(0)=(L^{-1})_{ij} =\frac{4e^2}{\hbar^2}\frac{\partial^2 F}{\partial \varphi_i \partial \varphi_j}.$$3 which produces a lossy characteristic impedance and lossy stubs. The paper states that this effective circuit model is accurate in the experimentally relevant regime of low dissipation and low frequencies, and that dissipation mainly broadens resonances and reduces transmission amplitude.

A distinctive contribution of this work is its treatment of the full AC response of the Hall material. In the AC Drude model, $$\chi_{ij}(0)=\chi_{ji}(0)=(L^{-1})_{ij} =\frac{4e^2}{\hbar^2}\frac{\partial^2 F}{\partial \varphi_i \partial \varphi_j}.$$4 and $$\chi_{ij}(0)=\chi_{ji}(0)=(L^{-1})_{ij} =\frac{4e^2}{\hbar^2}\frac{\partial^2 F}{\partial \varphi_i \partial \varphi_j}.$$5 become complex and frequency dependent, which lets the theory capture the intrinsic reactive response, identified in the paper as kinetic inductance. This generates additional resonance branches: the usual chiral plasmon resonances scale as

$$\chi_{ij}(0)=\chi_{ji}(0)=(L^{-1})_{ij} =\frac{4e^2}{\hbar^2}\frac{\partial^2 F}{\partial \varphi_i \partial \varphi_j}.$$6

whereas the new counter-circulating resonances scale as

$$\chi_{ij}(0)=\chi_{ji}(0)=(L^{-1})_{ij} =\frac{4e^2}{\hbar^2}\frac{\partial^2 F}{\partial \varphi_i \partial \varphi_j}.$$7

The paper argues that these counterpropagating features can be used to selectively control the direction of signal propagation, so the material’s reactive response is not a perturbative correction but a structural ingredient of the device’s nonreciprocity (Bosco, 31 Aug 2025).

6. Reciprocal look-alikes, effective analogues, and diagnostic cautions

Two additional lines of work clarify what should not be identified as non-reciprocal inductance.

First, the reciprocal skin effect in topolectrical circuits shows that a passive RLC network can display momentum-resolved apparent non-reciprocity while remaining reciprocal as a whole. The inductors in that circuit are used in a standard reciprocal way: one inductor per plaquette implements the $$\chi_{ij}(0)=\chi_{ji}(0)=(L^{-1})_{ij} =\frac{4e^2}{\hbar^2}\frac{\partial^2 F}{\partial \varphi_i \partial \varphi_j}.$$8-flux sign structure, and a grounding inductor cancels unwanted diagonal terms. The paper explicitly states that there is no active gyrator, amplifier, or time-modulated element, and that the circuit realizes the effect “without active non-reciprocal elements such as operational amplifiers.” The apparent asymmetry arises only in fixed-$$\chi_{ij}(0)=\chi_{ji}(0)=(L^{-1})_{ij} =\frac{4e^2}{\hbar^2}\frac{\partial^2 F}{\partial \varphi_i \partial \varphi_j}.$$9 slices, while the full 2D system satisfies the reciprocity condition $\theta \neq \theta^{\mathrm{T}},$0 (Hofmann et al., 2019).

Second, exceptional-point circuits can mimic some spectral signatures of nonreciprocal systems while remaining perfectly reciprocal. A reciprocal 4-element, two-loop, lossless circuit with a shared capacitor and parallel inductance can realize a second-order exceptional point of degeneracy and even share the same Jordan canonical form as an earlier gyrator-based nonreciprocal circuit. In the reciprocal circuit,

$\theta \neq \theta^{\mathrm{T}},$1

whereas the nonreciprocal gyrator circuit has

$\theta \neq \theta^{\mathrm{T}},$2

with gyrator contribution

$\theta \neq \theta^{\mathrm{T}},$3

The paper’s conclusion is that nonreciprocity is manifested in the Lagrangian and in the breakdown of eigenvector symmetries, but “cannot be captured in the Jordan canonical form and consequently in the eigenvalues of the circuit matrix” (Rouhi et al., 2023).

These cautions sharpen the encyclopedia meaning of the topic. Non-reciprocal inductance should not be inferred from non-Hermiticity alone, from exceptional-point spectra alone, or from direction-dependent nonlinear transport alone. In the strongest sense represented by current arXiv work, it is a dynamical antisymmetric reactive response: in multiterminal Josephson junctions through Berry-curvature and Andreev-bound-state physics, in superconducting gyrators through antisymmetric flux–charge coupling, and in Hall-based microwave devices through chiral admittance and kinetic-inductance-dressed edge dynamics (Virtanen et al., 2023, Leroux et al., 2022, Bosco, 31 Aug 2025).