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arXiv:2502.07605 · Physical review. B./Physical review. B

Kinetic inductance coupling for circuit QED with spins

Simon Günzler, D. Rieger, Martin Spiecker, Thomas Koch, Grigore A. Timco, Richard E. P. Winpenny, Ioan M. Pop, Wolfgang Wernsdorfer

Published Aug 14, 2025• 3 citations
arXivHTMLPDF

Abstract

While transverse coupling ( σ x ) is commonly used in qubit-resonator systems, longitudinal coupling via σ z presents a tantalizing alternative: it does not hybridize the modes, eliminating Purcell decay, and it enables quantum-nondemolition qubit readout independent of the qubit-resonator frequency detuning. Here, we demonstrate longitudinal coupling between a { Cr 7 Ni } molecular spin qubit ensemble and the kinetic inductance of a granular aluminum superconducting microwave resonator. The inherent frequency independence of this coupling allows for the utilization of a 7.8-GHz readout resonator to measure the full { Cr 7 Ni } magnetization curve spanning 0–600 mT, corresponding to a spin frequency range of f spin = 0 –15 GHz. For 2-GHz detuning from the readout resonator, we measure a 1 / e spin relaxation time of τ = 0.38 s, limited by phonon decay to the substrate. Based on these results, we propose a path towards longitudinal coupling of single spins to a superconducting fluxonium qubit.

DOI
10.1103/jz24-qpqv

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While transverse coupling ( σ x ) is commonly used in qubit-resonator systems, longitudinal coupling via σ z presents a tantalizing alternative: it does not hybridize the modes, eliminating Purcell decay, and it enables quantum-nondemolition qubit readout independent of the qubit-resonator frequency detuning. Here, we demonstrate longitudinal coupling between a { Cr 7 Ni } molecular spin qubit ensemble and the kinetic inductance of a granular aluminum superconducting microwave resonator. The inherent frequency independence of this coupling allows for the utilization of a 7.8-GHz readout resonator to measure the full { Cr 7 Ni } magnetization curve spanning 0–600 mT, corresponding to a spin frequency range of f spin = 0 –15 GHz. For 2-GHz detuning from the readout resonator, we measure a 1 / e spin relaxation time of τ = 0.38 s, limited by phonon decay to the substrate. Based on these results, we propose a path towards longitudinal coupling of single spins to a superconducting fluxonium qubit.
We engineer lumped-element superconducting resonators that maximize magnetic coupling to molecular spin qubits, achieving record single-spin couplings up to 100100100 kHz and collective couplings exceeding 101010 MHz. The resonators interact with PTMr organic free radicals, model spin systems with S=1/2S=1/2S=1/2 and a quasi-isotropic g≃2g \simeq 2g≃2, dispersed in polymer matrices. The highest collective spin-photon coupling strengths are attained with resonators having large inductors, which therefore interact with most spins in the molecular ensemble. By contrast, the coupling of each individual spin G1G_{1}G1​ is maximized in resonators having a minimum size inductor, made of a single wire. The same platform has been used to study spin relaxation and spin coherent dynamics in the dispersive regime, when spins are energetically detuned from the resonator. We find evidences for the Purcell effect, i.e. the photon induced relaxation of those spins that are most strongly coupled to the circuit. The rate of this process gives access to the distribution of single spin photon couplings in a given device. For resonators with a 505050 nm wide constriction at the inductor center, single maximum G1G_{1}G1​ values reach ∼100\sim 100∼100 kHz. Pumping the spins with strong pulses fed through an independent transmission line induces coherent Rabi oscillations. The spin excitation then proceeds via either direct resonant processes induced by the main pulse frequency or, in the case of square-shaped pulses, via the excitation of the cavity by sideband frequency components. The latter process measures the cavity mode hybridization with the spins and can be eliminated by using Gaussian shaped pulses. These results establish a scalable route toward integrated molecular-spin quantum processors.
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We present a circuit construction for a fixed-frequency superconducting qubit and show how it can be scaled up to a grid with strictly local interactions. The circuit QED realization we propose implements ${\ensuremath{\sigma}}_{z}$ type coupling between a superconducting qubit and any number of LCLCLC resonators. The resulting longitudinal coupling is inherently different from the usual ${\ensuremath{\sigma}}_{x}$ type transverse coupling, which is the one that has been most commonly used for superconducting qubits. In a grid of fixed-frequency qubits and resonators with a particular pattern of always-on interactions, coupling is strictly confined to nearest and next-nearest neighbor resonators; there is never any direct qubit-qubit coupling. We note that just a single unique qubit frequency suffices for the scalability of this scheme. The same is true for the resonators, if the resonator-resonator coupling constants are varied instead. A controlled phase gate between two neighboring qubits can be realized with microwave drives on the qubits, without affecting the other qubits. This fact is a significant advantage for the scalability of this scheme.
The measurement of a superconducting qubit is implemented by coupling it to a resonator. The common choice is transverse coupling, which, in the dispersive approximation, introduces an interaction term that enables the measurement. This cross-Kerr term provides a qubit-state dependent dispersive shift in the resonator frequency with the device parameters chosen carefully to get sufficient signal while minimizing Purcell decay of the qubit. We introduce a two-mode circuit, nicknamed quantromon, with two orthogonal modes implementing a qubit and a resonator. Unlike before, where the coupling term emerges as a perturbative expansion, the quantromon has intrinsic cross-Kerr coupling by design. Our experiments implemented in a hybrid 2D–3D circuit QED architecture demonstrate some unique features of the quantromon like weak dependence of the dispersive shift on the qubit-resonator detuning and intrinsic Purcell protection. In a tunable qubit-frequency device, we show that the dispersive shift (2χ/2π) changes by only 0.8 MHz, while the qubit-resonator detuning (Δ/2π) is varied between 0.398 and 3.288 GHz. We also demonstrate Purcell protection in a second device where we tune the orthogonality between the two modes. Finally, we demonstrate a single-shot readout fidelity of 98.3%, which is comparable to the state-of-the-art measurements without the use of a parametric amplifier and suggests a potential simplification of the measurement circuitry for scaling up quantum processors.