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

TL;DR

This work introduces lumped-element superconducting resonators (LERs) engineered to maximize magnetic coupling to molecular spin qubits, enabling record single-spin couplings up to $100\,\mathrm{kHz}$ and collective couplings beyond $10\,\mathrm{MHz}$. By using high-inductance designs for large spin ensembles and low-inductance, nano-constricted wires for enhanced per-spin coupling, the authors demonstrate tunable spin–photon interactions with PTMr radicals embedded in a polymer matrix. They report dispersive readout, Purcell-enhanced relaxation that reveals the distribution of individual spin couplings, and coherent spin manipulation using independent pump lines, illustrating a viable path toward integrated molecular-spin quantum processors. The results offer a scalable platform combining initialization, control, and readout on a single chip, with design strategies such as nano-constrictions and mode-structure optimization to approach strong coupling at the level of individual spins.

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 were made 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 microwire. 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 has been used to infer the distribution of single spin photon couplings in a given device. For resonators with a 50 nm wide constriction fabricated at the center of its 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 side 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.

Figures (18)

• Figure 1: (a) Optical microscopy image of a chip hosting ten low-impedance Nb LERs with different resonance frequencies, all coupled to the same readout transmission line and hosting dry deposits of free radical PTMr molecules embedded in polystyrene. (b) Image of a $1.971$ GHz LER with a large meandering inductor, designed to optimally couple to large spin sample volumes. (c) Image of a $1.767$ GHz low-impedance LER with a small-size inductor (a $12$$\mu m$ wide wire near the readout line) tailored to optimally couple to small sample volumes.
• Figure 2: Simulation of the microwave magnetic field amplitude $b_{\rm mw}$ generated by a LER vacuum fluctuations $1~\mu \mathrm{m}$ above the chip surface. Two LER designs are considered, with (a) a meandered inductor $L_{\rm HL}$ and (b) a single wire inductor $L_{\rm LL}$ with a central constriction. Both simulations were performed for the same photon energy ($\omega_{\rm r} \slash 2 \pi = 1.7144$ GHz). The ratio between the maximum microwave currents of each design is approximately $0.22$, in agreement with $(L_{\rm LL} / L_{\rm HL})^{1/2} \approx 0.17$.
• Figure 3: SEM images of the center of the inductor of a low-impedance Nb LER (a) before and (b) after the fabrication of a $50$ nm wide nanoconstriction. The white spots in (b) are nanoscopic drops of PTMr embedded in a polymer matrix. (c) Microwave transmission through this chip measured at $T=12$ mK and at zero magnetic field near the LER resonance frequency before (green) and after (red) the fabrication of the nanoconstriction.
• Figure 4: Finite element simulations of the magnetic field generated by single-wire inductors of different widths (a: $w = 50$ nm, b: $w = 2$$\mu$m, c: $w = 12$$\mu$m), calculated for various heights ($10$ nm, $20$ nm, $100$ nm and $200$ nm) above the chip surface. The magnetic fields are normalized to the current in order to account for differing inductances, while keeping the photon energy constant ($\omega_{\rm r} / 2 \pi = 1.71 \, \mathrm{GHz}$). The inductance values are as follows: $0.429$ nH for the wire with a $50$ nm nanoconstriction, $0.420$ nH for the wire with a $2$$\mu$m constriction, and $0.231$ nH for the $12$$\mu$m wire, with corresponding current values of $39.39$ nA, $36.78$ nA, and $40.96$ nA, respectively.
• Figure 5: (a) Molecular structure of PTMr, showing the free electron at its centre. (b) Topographic Atomic Force Microscopy profiles measured on a PTMr/PS drop deposited onto one of the Nb LERs shown in Fig. \ref{['fig:LERs']}. (c) Scheme of the set-up for transmission and pump-probe experiments. The cryostat, either a $^{3}$He-$^{4}$He dilution or a adiabatic demagnetization refrigerator, has several cryogenic coaxial cables that drive the different input and output microwave signals to and from the chip, respectively. Input lines incorporate either 0 dB or -10 dB attenuators at each constant temperature plate. Output readout lines are amplified at $T = 4.2$ K before reaching the digital readout electronics while excitation lines, which drive higher power pulses, are directly fed into a digital high frequency oscilloscope. (d) Chip hosting multiple low-inductance NbTiN LERs coupled to independent readout and control transmission lines. The latter, inductively coupled to the PTMr molecular deposits, can be used to induce spin excitations through strong microwave pulses whereas the former allows reading out the LER. (e) Zoom of one of the LERs in the same chip.