Kinetic inductance coupling for circuit QED with spins

Abstract

In contrast to the commonly used qubit resonator transverse coupling via the $σ_{xy}$-degree of freedom, longitudinal coupling through $σ_z$ presents a tantalizing alternative: it does not hybridize the modes, eliminating Purcell decay, and it enables quantum-non-demolishing 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_\text{spin}=$0-15 GHz. For 2 GHz detuning from the readout resonator, we measure a $1/e$ spin relaxation time $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.

Figures (6)

• Figure 1: Implementation of kinetic inductance coupling between a superconducting resonator and molecular spins.a Sketch of the resonator-spin system: high kinetic inductance granular aluminum (grAl) resonator (red strip) coupled to a {Cr$_7$Ni} micro-crystal (green). Left inset: grAl consists of crystalline aluminum grains (red) in an amorphous AlO$_x$ matrix (gray) Deutscher1973Jan. Right inset: the single magnetic molecule $\mathrm{Cr_7NiF_8(O_2CCMe_3)_{16}} \equiv${Cr$_7$Ni} has effective spin $S=1/2$ and $g=1.8$Larsen2003JanArdavan2007Jan. The magnetization of the spin ensemble due to in-plane field $B_\parallel$ (blue arrow) creates a magnetic field $B_\uparrow$ (green lines). This field locally enhances the kinetic inductance of the grAl resonator and provides the mechanism for longitudinal coupling. b Optical microscope image of the magnetic crystal (green) placed on top of the grAl readout resonator. The crystal is attached with Apiezon NApiezonN vacuum grease (transparent gray), which provides thermal conductivity at cryogenic temperatures, low levels of magnetic susceptibility and low microwave losses (cf. \ref{['sec:supp:grAlInField']}). To excite the spins (black arrows) composing the crystal we use inductively coupled niobium lines, visible above and below the grAl resonator and shown as dashed white outlines in the region underneath the crystal. The 150-long inductor sections of the drive resonators parallel to the central grAl strip generate radio-frequency magnetic fields perpendicular to the spin quantization axis (defined by $B_\parallel$). c Zoom-out: The grAl readout resonator (center) is flanked by two low-impedance niobium resonators (top, bottom), necessary to excite the spins. Note that the frequencies $f_{\text{Nb},1}$, $f_{\text{Nb},2}$ of the niobium drive resonators are several detuned from the readout $f_\text{r}$.
• Figure 2: Detuning independent readout of the spin ensemble magnetization.a Resonance frequency shift of the grAl readout resonator coupled to the molecular spin ensemble $\delta f_\mathrm{r}(B_\parallel,M)$ (blue markers) in magnetic field. The parabolic frequency shift of the bare resonator $\delta f_\mathrm{r}(B_\parallel,M=0)$ (black dashed line) is given by the suppression of the grAl superconducting gap $\delta(B_\parallel)$ in magnetic field Borisov2020. The resonator experiences an additional frequency shift $\delta f_\mathrm{M}$ due to the magnetization of the spin ensemble (red shift), yielding the measured total frequency shift $\delta f_\mathrm{r}(B_\parallel,M)$. To disentangle these two contributions, we fit a parabolic frequency shift $\delta f_\mathrm{r}(B_\parallel,M_\mathrm{S})$ to the tail of the data for $B_\parallel > 0.32T$ (black solid) where the magnetization of the crystal is saturated $(M=M_\mathrm{S})$. We can then remove the vertical offset to get $\delta f_\mathrm{r}(B_\parallel,0)$ (dashed black) and $\delta f_\mathrm{M}$ (red arrows). At $B_\parallel=\qty{0.29}{\tesla}$, as expected for $g=1.83$, we measure an avoided level crossing due to the transverse coupling of the spin ensemble to the geometric inductance of the resonator. At $B_\parallel=\qty{0.26}{\tesla}$ we observe an additional feature corresponding to transverse coupling of $g=2$ spurious ESR Borisov2020Gunzler2025JanSamkharadze2016AprKroll2019Jun. b Extracted frequency shift $\delta f_\mathrm{M}$ (red markers) due to the magnetic field of the crystal and corresponding magnetization $M\propto \sqrt{\delta f_\mathrm{M}}$ (green markers). Above $B_\parallel = \qty{0.32}{\tesla}$, the crystal magnetization saturates, and $\delta f_\mathrm{M}$ remains constant. In the range $B_\parallel = \qtyrange{0.244}{0.302}{\tesla}$, anti-crossings (see double arrow markers) arising from residual transverse coupling to $g=2$ and $g=1.83$ spins prevent a reliable extraction of the magnetization. Note that the magnetization curve shown here is measured on a sample without Nb drive resonators (cf. \ref{['fig:sample']}) in order to avoid field distortions (cf. \ref{['sec:supp:Nb_magnetization_curve']}).
• Figure 3: Excitation and decay of the spin ensemble $\mathbf{\qty{2}{\giga\hertz}}$ detuned from the readout.a Continuous wave two-tone spectroscopy of the spin ensemble: To excite the spins, we sweep a drive tone $f_\mathrm{d}$ in the vicinity of the niobium resonator frequency $f_\text{Nb1} = \qty{9,84}{\giga\hertz}$ while monitoring the readout resonator at $f_\text{r}= \qty{7,87}{\giga\hertz}$. The relative change of the ensemble magnetization $\delta M/M_\text{S}$ (green colorbar) is calibrated to the measured shift in $f_\text{r}$ utilizing a magnetization curve (cf. \ref{['fig:magnetization']}). The excitation of the spin ensemble is only effective within the bandwith $\kappa = \qty{2}{\mega\hertz}$ of the Nb drive resonator. The dashed line indicates the center frequency of the spin distribution given by $\text{h}f = g\mu_\text{B} B_\parallel$ with $g=1.83\pm0.01$. b, c Time domain characterization of the spin ensemble at $B_\parallel = \qty{0.384}{\tesla}$ (red marker in a): excitation to saturation (b) and decay from saturation (c) for different drive powers, with a $1/e$ time $\tau= \qty{0.38}{\second}$.
• Figure 4: Towards longitudinal coupling between superconducting circuits and single spins.a Implementation of a spin-modulated Josephson junction: scanning electron microscope image of a grAl nanojunction with a single spin sketched on top. The nanojunction consists of a $\approx(\qty{20}{\nano\metre})^3$ grAl volume Gralmonium, which remains coherent in Tesla-scale magnetic fields Gunzler2025Jan. When biased in magnetic field $B_\parallel$, the two spin orientations generate different local magnetic fields, leading to distinct Josephson energies $E_\mathrm{J}^{\uparrow}$ and $E_\mathrm{J}^{\downarrow}$. b We use the spin-state-dependent nanojunction to build a fluxonium quantum circuit with junction capacitance $C_\mathrm{J}$, superinductor $L_\mathrm{q}$ and external flux $\Phi_\mathrm{ext}$. The spin polarization tunes the Josephson energy $E_\mathrm{J}$ and, consequently, the fluxonium transition frequency $f_\mathrm{q}$. c Simulation of the qubit frequency shift $\delta f_\mathrm{q}$ for a spin-flip versus distance $d$ from the nanojunction surface for several $B_\parallel$ bias values. Note that for the calculation we used a spin with magnetic moment $\mu = 10\mu_\mathrm{B}$ (see Ref. Moreno-Pineda2021Sep) on top of a regular gralmonium qubit Gralmonium.
• Figure 5: Resilience of grAl resonators in parallel field. Internal quality factor $Q_\mathrm{i}$ (blue markers) corresponding Fano uncertainty range Rieger2023Jul (blue shaded area) of the grAl resonator in magnetic field $B_\parallel$. The resonator is measured at an average photon number $\Bar{n}\approx10$ with a coupling quality factor $Q_\mathrm{c}=\qty{25e3}{}$. The dips in $Q_\mathrm{i}$, shown as peaks in the decay rate $\Gamma_\mathrm{i}=1/Q_\mathrm{i}$ in (b), correspond to ESR of spurious electronic spins with $g=2$ (left) and the molecular spin ensemble with $g=1.83$ (right). The associated resonance frequency shifts are presented in \ref{['fig:supp:magnetization']}.