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arXiv:2511.00857

Optimizing magnetic coupling in lumped element superconducting resonators for molecular spin qubits

Marcos Rubín-Osanz, David Rodríguez, Ignacio Gimeno, Wenzel Kersten, Nerea González-Prato, María Carmen Pallarés, Sebastián Roca-Jerat, Marina C. de Ory, Marta Mas‐Torrent, J. Alejandro de Sousa, Lorenzo Tesi, Daniel Granados, Jaume Veciana, David Zueco, Anabel Lostao, Jörg Schmiedmayer, Imma Ratera, Joris van Slageren, Núria Crivillers, A. Gómez, Fernando M. Luis

Published Nov 2, 2025• 1 citation
arXivHTMLPDF

Abstract

We engineer lumped-element superconducting resonators that maximize magnetic coupling to molecular spin qubits, achieving record single-spin couplings up to $100$ kHz and collective couplings exceeding $10$ MHz. The resonators interact with PTMr organic free radicals, model spin systems with $S=1/2$ and a quasi-isotropic $g \simeq 2$, 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 $G_{1}$ 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 $50$ nm wide constriction at the inductor center, single maximum $G_{1}$ values reach $\sim 100$ 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.

DOI
10.48550/arxiv.2511.00857

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•Jan 1, 2015•3 citations
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.
We study the simultaneous interaction of a sample of organic free radicals, which realize the spin-1/2 model with a quasi-isotropic g factor ≈ 2, with three ≈ 5.5 GHz lumped-element superconducting resonators in a chip. Each resonator consists of a finger capacitor in parallel with a single wire inductor, which maximizes the coupling intensity of the spins to the photon magnetic field. Their frequencies have been detuned relative to one another in order to make them individually addressable. Each resonator within the trio is capacitively coupled to its nearest neighbor and inductively coupled to a common readout transmission line, and only the central one hosts a molecular deposit. The device has been characterized by microwave transmission experiments performed as a function of frequency (between 5.3 and 5.7 GHz) and magnetic field (170–220 mT) at very low temperatures (T ≃ 13 mK). The results evidence a strong local interaction of the spins to the central resonator, which reaches the coherent coupling regime. In addition, they show that the same spins also couple to the modes associated with the distant resonators. These data have been analyzed on the basis of a theoretical model that incorporates the direct spin-photon interactions as well as those mediated by the circuit. The results provide a relatively simple, practical method for investigating spin resonance of a material at multiple frequencies. Besides, this method allows designing each resonator differently, to fulfill a specific role, e.g., for optimized control or readout of the spin states, which can be of relevance for the implementation of hybrid quantum computation schemes based on molecular spin qubits.
The ability to establish coherent communication channels is key for scaling up quantum devices. Here, we engineer interactions between distant polaritons, hybrid spin-photon excitations formed at different lumped-element superconducting resonators within a chip. The chip consists of several resonator pairs, slightly detuned in frequency to make them addressable, capacitively coupled within each pair and inductively coupled to a common readout line. They interact locally with samples of PTMr and Tripak−^{-}− organic free radicals, deposited onto their inductors, which provide model S=1/2S = 1/2S=1/2, g≃2g \simeq 2g≃2 spin ensembles. Frequency-dependent microwave transmission experiments, performed at very low temperatures, measure polariton frequencies as a function of magnetic field in different scenarios. When only one resonator within a pair hosts a molecular sample, the results evidence that spins couple remotely to the empty LER as well as to the local cavity mode. If both resonators interact with a spin ensemble, the magnetic field tunes the polariton frequencies relative to each other, on account of the different spin-photon interactions at each LER. When polaritons are brought into mutual resonance, an avoided level crossing emerges that gives direct spectroscopic evidence for a coherent polariton-polariton interaction mediated by the circuit. Pump-probe experiments reveal that the excitation of a polariton within a connected pair is felt, thus it can be read out, by the other one. These observations, backed by model calculations, illustrate the control and detection of distant photon-photon and spin-spin correlations and entanglement in a scalable modular chip.
A promising route toward the realization of a molecular spin quantum processor relies on coupling magnetic molecules to individual photons confined within superconducting resonators. As a simple approximation to such a hybrid scheme, here we explore the conditions that determine the collective coupling of DPPH organic free radicals to lumped-element LC superconducting resonators. In these chips, multiple resonators are coupled to a single readout line. This enables designing the relevant resonator properties, such as resonance frequency, cavity volume, and impedance while keeping a perfect transmission for the device. Here, we exploit these design possibilities to achieve a coherent spin-photon coupling regime. Besides, we study how this coupling depends on the relative orientation of the external dc magnetic field concerning the photon magnetic field and the spins locations concerning the chip surface.
The use of circuit quantum electrodynamics with single-atom electron spins could lead to scalable, highly coherent qubit schemes for quantum information processing. Reaching the regime of strong magnetic coupling of a photon to a single spin is challenging, though, because of the relatively small spin magnetic moment, and the relatively weak magnetic field of a typical superconducting circuit resonator. The authors show that using a lumped-element resonator with a nanoscale spiral inductor can greatly enhance the photon's magnetic field in the vicinity of the spin trapped in a donor potential. This enables strong magnetic coupling of the resonator to the spin.
High Resolution Image Download MS PowerPoint Slide A strategy toward the realization of a quantum spin processor involves the coupling of spin qubits and qudits to photons within superconducting resonators. To enable the realization of such hybrid architecture, here we first explore the design of a chip with multiple lumped-element LC superconducting resonators optimized for their coupling to distinct transitions of a vanadyl porphyrin electronuclear qudit. The controlled integration of the vanadyl qudit onto the superconducting device, both in terms of number and orientation, is then attained using the in situ formation of nanosheets of a 2D framework built on the vanadyl qudit as a node. Low-temperature transmission experiments demonstrate the coupling of photons in resonators with different frequencies to the targeted electronuclear transitions of the vanadyl qudit, also confirming the control over the vanadyl qudit node orientation. The derived collective spin-photon couplings in the 0.3–1.6 MHz range then allow to estimate enhanced, optimal, single spin photon couplings up to 4 Hz.
I discuss the design, fabrication and measurement at millikelvin-temperatures of Al/AlOx_xx​/Al Josephson junction-based transmon qubits coupled to superconducting thin-film lumped element microwave resonators made of aluminum on sapphire. The resonators had a center frequency of around 6 6\,6GHz, and a total quality factor ranging from 15,000 to 70,000 for the various devices. The area of the transmon junctions was about 150 nm×150 nm150\, \mathrm{nm} \times 150\, \mathrm{nm}150nm×150nm and with Josephson energy EJE_JEJ​ such that 10 GHz≤EJ/h≤30 10\,\text{GHz} \leq E_J / h \leq 30\,10GHz≤EJ​/h≤30GHz. The charging energy of the transmons arising mostly from the large interdigital shunt capacitance, was Ec/h≈300 E_c / h \approx 300\,Ec​/h≈300MHz. I present microwave spectroscopy of the devices in the strongly dispersive regime of circuit quantum electrodynamics. In this limit the ac Stark shift due to a single photon in the resonator is greater than the linewidth of the qubit transition. When the resonator is driven coherently using a coupler tone, the transmon spectrum reveals individual ``photon number'' peaks, each corresponding to a single additional photon in the resonator. Using a weighted average of the peak heights in the qubit spectrum, I calculated the average number of photons nˉ\bar{n}nˉ in the resonator. I also observed a nonlinear variation of nˉ\bar{n}nˉ with the applied power of the coupler tone PrfP_{rf}Prf​. I studied this nonlinearity using numerical simulations and found good qualitative agreement with data. In the absence of a coherent drive on the resonator, a thermal population of 5.474 5.474\,5.474GHz photons in the resonator, at an effective temperature of 120 120\,120mK resulted in a weak n=1n=1n=1 thermal photon peak in the qubit spectrum. In the presence of independent coupler and probe tones, the n=1n=1n=1 thermal photon peak revealed an Autler-Townes splitting. The observed effect was explained accurately using the four lowest levels of the dispersively dressed Jaynes-Cummings transmon-resonator system, and numerical simulations of the steady-state master equation for the coupled system. I also present time-domain measurements on transmons coupled to lumped-element resonators. From T1T_1T1​ and Rabi oscillation measurements, I found that my early transmon devices (called design LEv5) had lifetimes (T1∼1 μT_1 \sim 1\,\muT1​∼1μs) limited by strong coupling to the 50 Ω50\,\Omega50Ω transmission line. This coupling was characterized by the the rate of change of the Rabi oscillation frequency with the change in the drive voltage (dfRabi /dV\mathrm{d}f_{Rabi}\, / \mathrm{d}VdfRabi​/dV) -- also termed the Rabi coupling to the drive. I studied the design of the transmon-resonator system using circuit analysis and microwave simulations with the aim being to reduce the Rabi coupling to the drive. By increasing the resonance frequency of the resonator ωr/2π\omega_r/2\piωr​/2π from 5.4 \,GHz to 7.2 \,GHz, lowering the coupling of the resonator to the transmission line and thereby increasing the external quality factor QeQ_eQe​ from 20,000 to 70,000, and reducing the transmon-resonator coupling g/2πg/2\pig/2π from 70 \,MHz to 40 \,MHz, I reduced the Rabi coupling to the drive by an order of magnitude (∼\sim∼ factor of 20). The T1∼4 μT_1 \sim 4\,\muT1​∼4μs of devices in the new design (LEv6) was longer than that of the early devices, but still much shorter than the lifetimes predicted from Rabi coupling, suggesting the presence of alternative sources of noise causing qubit relaxation. Microwave simulations and circuit analysis in the presence of a dielectric loss tangent tan⁡δ≃5×10−6\tan \delta \simeq 5\times10^{-6}tanδ≃5×10−6 agree reasonably well with the measured T1T_1T1​ values, suggesting that surface dielectric loss may be causing relaxation of transmons in the new designs.