Multimode Superconducting Inductor Architecture

• Multimode superconducting inductor architecture is a circuit paradigm that integrates engineered inductance with distributed capacitance to support multiple, well-defined resonant modes.
• It leverages geometric and electrical designs—such as metamaterial left-handed LC lines and planar spiral inductors—to achieve tunable coupling with high mode density.
• This architecture facilitates frequency-multiplexed readout and scalable qubit entanglement, with experimental validations showing high Q factors and precise modal control.

A multimode superconducting inductor architecture is a circuit paradigm in which the engineered structure of an inductor, combined with distributed capacitance, gives rise to a set of well-defined electromagnetic resonant modes within a compact superconducting layout. Such architectures are essential for applications in quantum information science, notably as coupling buses between artificial atoms, for scalable fast readout of spin and charge qubits, and more generally as compact, low-loss, frequency-multiplexed elements in cryogenic RF systems. Multimode operation leverages distributed parameters of the inductor to support several discrete resonances, each characterized by its own effective inductance, capacitance, impedance, and quality factor, thus enabling simultaneous or tunable coupling to diverse quantum and classical devices (Rivard et al., 4 Dec 2025, McBroom-Carroll et al., 2023, Tolpygo et al., 2021).

1. Circuit Topologies Enabling Multimode Inductance

Multimode operation is typically obtained by geometrically or electrically distributing capacitance along a superconducting inductor. Two principal classes are established:

• Metamaterial Left-Handed LC Lines and Rings: A left-handed transmission line is formed by periodic cells each comprising a series capacitor ($C_L$) followed by one or more shunt inductors ($L_L$) to ground. Such lines, folded into a ring of $N$ cells, give rise to a dense set of modes with nontrivial dispersion above a finite infrared cutoff. For example, with $C_L \approx 303$ fF and $L_L \approx 1.04$ nH, a 24-cell ring (cell length $\Delta x \approx 200$ μm) achieves a $\sim2$ GHz band populated by multiple modes, supporting high mode density within a 4.8 mm perimeter (McBroom-Carroll et al., 2023).
• Planar Spiral Inductors with Distributed Inter-turn Capacitance: In planar NbN spirals (e.g., 150 turns, conductor width 1 μm, spacing 1 μm, outer diameter 500 μm), adjacent turns are capacitively coupled ($C_{\text{turn}}$ ~ fF). The result is a discrete ladder of standing-wave resonances up to $2$ GHz, in contrast to a single lumped-element resonance (Rivard et al., 4 Dec 2025).

The mode structure in both cases is governed by the interplay of distributed $L'$ and $C'$, supporting resonant patterns analogous to those in finite transmission lines, but with critical distinctions depending on cell topology (left-handed vs. right-handed) and terminations (ring vs. open or shorted line).

2. Mode Structure, Frequency Dispersion, and Parameter Extraction

The spectrum of a multimode architecture can be derived analytically or numerically. For a left-handed ring resonator, the mode frequencies are given by

$$\omega_n = \frac{1}{2\,\sqrt{L_L\,C_L}\,\sin\left(\frac{\pi n}{N}\right)},\qquad n=1,\ldots,N-1$$

with an infrared cutoff frequency $\omega_\text{IR} = 1/\left(2\sqrt{L_LC_L}\right)$ below which no traveling-wave modes exist (McBroom-Carroll et al., 2023). For the spiral inductor, a similar discretization applies:

$$\omega_n^2 = \frac{2}{L_0C_0} \left[1 - \cos\left(\frac{n\pi}{N+1}\right)\right], \qquad n=1\ldots N$$

or, in the continuum limit, via appropriate boundary conditions (open–short, open–open) and velocity $v=1/\sqrt{L'C'}$ (Rivard et al., 4 Dec 2025).

Effective Modal Inductance and Capacitance: For each mode $n$, the effective $L_{\text{eff},n}$ and $C_{\text{eff},n}$ are:

$$L_{\text{eff},n} = \int_0^\ell L' \left[\frac{I_n(x)}{I_{n0}}\right]^2 dx,\qquad C_{\text{eff},n} = \int_0^\ell C' \left[\frac{V_n(x)}{V_{n0}}\right]^2 dx$$

where $I_n(x)$ and $V_n(x)$ are modal current and voltage patterns, typically with $\sin$ and $\cos$ standing wave dependence.

Quality Factors and Impedance: Each mode is characterized by

$$Z_{\text{char}, n} = \sqrt{L_{\text{eff},n}/ C_{\text{eff}, n}},\qquad Q_n = R_{\parallel, n} \sqrt{C_{\text{eff},n}/L_{\text{eff},n}}$$

with $R_{\parallel, n}$ the mode-specific parallel resistance determined by sample load, inductor dissipation, and matching elements.

3. Quantum Device Coupling and Mediated Interactions

Multimode superconducting inductor architectures are central as coupling buses in hybrid quantum circuits:

• Artificial Atom–Mode Coupling: Transmon qubits are capacitively coupled to selected unit cells or spiral taps. Analytical formulas show that for parity-defined standing wave modes (even/odd), the coupling strength $g$ oscillates as a cosine or sine function of the mode number and qubit spacing (McBroom-Carroll et al., 2023).
• Entangling Interactions: The multimode bus enables two fundamental forms of interaction between otherwise uncoupled qubits:
  • Transverse Exchange ($J$):

  $$J = \frac{1}{2} \sum_i g_i^A g_i^B \left( \frac{1}{\Delta_i^A} + \frac{1}{\Delta_i^B} \right), \quad \Delta_i^q = \omega_{01}^q - \omega_i$$

  The sign and magnitude of $J$ vary with detuning, showing zero crossings and reversals as the qubit frequencies sweep through the modal band. - Longitudinal $ZZ$ Interaction ($\zeta$):

  $\zeta = E_{00} + E_{11} - E_{01} - E_{10}$

  obtained via full diagonalization of the multi-level Hamiltonian. $\zeta$ exhibits sign changes, smooth zero crossings, and discontinuous jumps (e.g., $|11\rangle$ anticrossing with $|20\rangle$ states).

Small changes in qubit frequency ($<50$ MHz) can traverse multiple $J$ or $\zeta$ zero crossings, allowing fast entangling gate control without large frequency excursions (McBroom-Carroll et al., 2023).

4. Experimental Implementation and Characterization

Practical realization covers:

• Superconductor Materials: Devices employ films such as 100 nm NbN on sapphire for spirals (Rivard et al., 4 Dec 2025) or niobium circuits in multilayered structures (Tolpygo et al., 2021).
• Planarization and Lithography: Dimension control at the $\mu$m scale is critical; designs use widths down to $1\,\mu$m and fine control of inter-turn or inter-cell spacing.

Resonator Spectroscopy: Cryogenic ($\sim4$ K) network analyzer measurements reveal mode frequencies, loaded $Q_n$, and impedance matching:

Mode n	$f_n$ (MHz)	$Q_n$ (loaded)	$|\Delta S_{11}|$ at resonance (dB)
1	$\sim$150	$\sim$200	6
2	222.5	650	12
3	360.2	780	15
4	523.8	840	17

(Rivard et al., 4 Dec 2025)

Measured $Q$ values up to $\sim 840$ are obtained with this geometry. For metamaterial rings, 13 modes are clearly resolved between 4.3 and 6.5 GHz, with $|g_{A,B}|/2\pi$ spanning $13.7$–$66.3$ MHz (McBroom-Carroll et al., 2023).

Device Calibration: Analytical formulae for self- and mutual inductance of microstrips, striplines, and meander geometries provide design recipes with $<2$\% accuracy. Calibration proceeds via SQUID-based extraction of $L$ and $M$ for each geometry (Tolpygo et al., 2021).

5. Applications: Spin Qubit Readout and Quantum Buses

The multimode inductor architecture enables:

• Frequency-Multiplexed Spin Qubit Readout: Each mode offers a distinct resonance for reflectometry, permitting simultaneous or rapid sequential probing of different quantum dot or RF-SET sensors. In an experimental realization, single-shot singlet-triplet spin readout in a double quantum dot achieved $98\%$ fidelity with $8\,\mu$s integration time at $245.1$ MHz ($Q=870$) (Rivard et al., 4 Dec 2025).
• Entangling Gate Implementation: In multimode left-handed ring architectures, the ability to swiftly switch interaction sign and magnitude between qubits by small local frequency tuning underlies scalable, parallelizable entanglement generation (McBroom-Carroll et al., 2023).
• Device Characterization: Each mode’s distinct impedance and bandwidth enables mapping of tunneling rates, identification of charge defects, and fine tuning of the sample environment over a wide frequency window without structural reconfiguration.

6. Design Methodology and Scalability Considerations

Design and deployment of a multimode superconducting inductor array follow an analytical–experimental loop:

1. Target Mode Specification: Choose desired $\omega_k$ and extract or assign appropriate capacitances $C_i$ from device layout.
2. Dimension Selection: Use self-inductance formulas (e.g., $L(w,h,\lambda)$) to solve for trace widths or layer heights. For strong coupling, tune mutual inductance via $M_{ij}(d_{ij},\dots)$, including vertical stacking or ground-plane slits as needed (Tolpygo et al., 2021).
3. Layout Calibration: Lay out test structures, measure $L$ and $M$, fit to extract actual device parameters ($\lambda$, layer heights), and feed into simulation for sub-1\% dimensional agreement.
4. Resonator Extraction: Recompute $\omega_k$ and coupling $\kappa_{ij}$ for the full array, iterate as needed.

These architectures offer compact footprints (e.g., $\sim4$ mm perimeter for a 24-cell ring vs. $\sim15$ mm for a single-mode $\lambda/2$ resonator at 5 GHz) and support direct scalability: more qubits can be located around the ring or spiral, and multiple devices networked in modular fashion (McBroom-Carroll et al., 2023, Rivard et al., 4 Dec 2025).

Advantages:

• High mode density and frequency multiplexing allow rapid, broadband measurements.
• Flexible impedance matching facilitates adaptation to varying device loads.
• Footprint is minimized compared to single-mode designs for equivalent bandwidth.

Limitations:

• High-frequency modes ($>$2 GHz) may experience crowding and $Q$ degradation.
• Each mode requires independent cryogenic calibration.
• Crosstalk can arise without careful layout and filtering.

A plausible implication is that, combined with near-quantum-limited amplification chains, these multimode superconducting inductor architectures can provide scaling paths toward $>99.5\%$ single-shot qubit readout fidelity with sub-microsecond integration in quantum information processors (Rivard et al., 4 Dec 2025).