Resolving Abrikosov vortex entry in superconducting nano-string resonators via displacement-noise spectroscopy in cavity-optomechanics

TL;DR

The paper tackles how vortices in type-II superconductors affect current flow and coherence in superconducting quantum devices. It introduces a chip-scale, cavity-optomechanical platform where a superconducting Al nanostring is embedded in a SQUID-terminated microwave cavity, enabling displacement-noise spectroscopy under magnetic fields. A key finding is the observation of discrete, attonewton-scale force steps corresponding to single-vortex entry, alongside a smooth Campbell-regime background with a scaling $\Omega_0^2 \propto B_{ip}^{k}$ where $k \approx 1.81$, yielding a Labusch parameter $\alpha_L \sim 10^{14}\ \mathrm{N\,m^{-4}}$ and single-vortex pinning energies in the 0.1–0.4 eV range. This method provides a sensitive, in-situ probe of vortex dynamics and decoherence pathways in superconducting circuits, with potential to guide the design of devices approaching single-photon strong coupling in cavity optomechanics.

Abstract

Abrikosov vortices in type-II superconductors critically influence current flow and coherence, thereby imposing fundamental limits on superconducting quantum technologies. Quantum circuits employ superconducting elements at micro- and mesoscopic scales, where individual vortices can significantly impact device performance, necessitating investigation of vortex entry, motion, and pinning in these constrained geometries. Cavity-optomechanical platforms combining flux-tunable microwave resonators with superconducting nanomechanical elements offer a promising route to the single-photon strong-coupling regime and enable highly sensitive probing of the mechanical degree of freedom under elevated magnetic fields. Here, we exploit this platform to investigate vortex entry processes at the single-event level. We observe discrete jumps of the mechanical resonance frequency attributable to individual vortex entry, corresponding to attonewton-scale forces and allowing quantitative extraction of single-vortex pinning energies. These signatures are superimposed on a smooth power-law background characteristic of the collective Campbell-regime of vortex elasticity. Our results establish optomechanics-inspired sensing as a powerful method for exploring fundamental superconducting properties and identifying decoherence pathways in quantum circuits. Beyond advancing vortex physics, this work opens new opportunities for integrating mechanical sensing into superconducting device architectures, bridging condensed matter physics and quantum information science.

Figures (4)

• Figure 1: Nanomechanical detection of magnetic flux lines. (a) Superconducting $\lambda/4$ coplanar waveguide resonator capacitively coupled to a microwave feedline and terminated to ground by a dc SQUID. $\kappa_\mathrm{ext}$ and $\kappa_\mathrm{in}$ denote the external and internal coupling rates of the resonator, respectively. The magnified view shows the SQUID loop incorporating a suspended Al nanostring $(l,w,d)=(30,\,0.2,\,0.11)\,µm)$ and two Josephson junctions (JJs), with labels indicating the nanostring displacement $\delta x$, circulating SQUID current $J$, and applied magnetic control fields $B_\mathrm{ip}$ (in-plane) and $B_\mathrm{oop}$ (out-of-plane). (b) Lumped-element circuit representation of the distributed $\lambda/4$ CPW resonator with distributed inductance $\tilde{L}$ and capacitance $\tilde{C}$ per unit length, external coupling capacitor $C_\mathrm{ex}$ and the feedline ports ($S_\mathrm{in}$ and $S_\mathrm{out}$). The total resonator inductance and capacitance are denoted as $L$ and $C$, respectively. The SQUID provides flux-tunable termination through geometric loop inductance ($L_\mathrm{l}$, distributed as $L_1/2$ on each branch) and Josephson inductances ($L_{J1}(\Phi)$ and $L_{J2}(\Phi)$). (c) Conceptual depiction of magnetic flux penetration into thin-film Al nanostring, where vortices become trapped at pinning centers (blue spheres).
• Figure 2: Measurement scheme for determining the resonance frequency of the string-resonator. (a) Sketch of all relevant modes of the cavity optomechanical system. The mechanical mode at $\Omega_\mathrm{m}$ interacts with the probe tone $\omega_\mathrm{p}$ resulting in two mechanical sidebands at $\omega_\mathrm{p} \pm \Omega_\mathrm{m}$. For the determination of the thermal displacement noise, used to infer $\Omega_\mathrm{m}$, we record the anti-Stokes field for $\omega_\mathrm{p}=\omega_\mathrm{c}$. (b) Representative voltage power spectral density of the anti-Stokes field downconverted to $\omega_\mathrm{p}$ showing mechanical resonance, from which we determine the mechanical resonance frequency $\Omega_\mathrm{m}$ and linewidth $\Gamma_\mathrm{m}/2\pi=20Hz$. (c) Evolution of the mechanical frequency $\Omega_\mathrm{m}(\Phi_\mathrm{b}/\Phi_0)$ as a function of $B_\mathrm{oop}$ measured for different $B_\mathrm{ip}$. (d) Extracted intrinsic mechanical frequency $\Omega_0(B_\mathrm{ip})$ versus in-plane magnetic field, obtained by fitting the flux-dependent data to the theoretical model (Eq. \ref{['fre']}). The solid line is a power-law fit $A_0+C\,B_{\mathrm{ip}}^{\,k}$ with exponent $k=1.81\pm0.05$.
• Figure 3: Stochastic vortex entry signatures across independent cooldowns. Squared mechanical frequency shift versus in-plane magnetic field at 85 mK for (a) two successive datasets within the same cooldown (blue and red circles), referenced to $\Omega_0(20mT)$ and (b) an independent dataset taken after warming to room temperature and re-cooling, referenced to $\Omega_0(35mT)$. The dash-dot line shows the power-law fit $A_0 + CB_{\mathrm{ip}}^k$ with $k=1.81\pm0.05$. Arrows mark abrupt frequency steps attributed to vortex entry events.
• Figure 4: Collective vortex stiffening and extracted Labusch parameter. Squared resonance frequency shift (relative to 35 mT) versus in-plane field for high-resolution data (blue circles with error bars) and previous measurements luschmann2022mechanical (black squares). The black line represents the fitting function with $k=1.81\pm 0.05$. Inset shows extracted Labusch parameter $\alpha_L(B_{\mathrm{ip}})$ as a function of $B_{\mathrm{ip}}$.