Qubit Noise Spectroscopy of Superconducting Dynamics in a Magnetic Field

Abstract

An applied magnetic field affects a superconductor in two ways -- by promoting pairing fluctuations, and by inducing topological defects called vortices that carry quantized magnetic flux. A quantitative characterization of the resultant field-induced superconducting dynamics with spatio-temporal resolution remains challenging, particularly in two-dimensional materials. In this work, we analyze magnetic noise measured by the depolarization rate of a proximate single spin-qubit as a non-invasive probe of such dynamical fluctuations. We demonstrate that the temperature dependence of the magnetic noise spectrum near $T_c$ deviates from predictions based on quasiparticle excitations due to critical superconducting fluctuations, which in turn are enhanced by a weak applied field. By analyzing the magnetic noise due to vortex dynamics, we further show that noise spectroscopy is not only able to distinguish between different vortex phases, but also extract key physical quantities of interest, such as oscillation frequencies of pinned vortices, phonon dispersion of vortex lattices and vortex diffusivity in a vortex liquid. Complementing recent work on noise magnetometry of quasiparticle excitations and Berezinskii-Kosterlitz-Thouless transitions in two-dimensional superconductors without an applied field, our work highlights the ability of noise spectroscopy to reveal a wealth of superconducting dynamical phenomena in an applied field.

Figures (8)

• Figure 1: Schematic of the set-up: a qubit sensor with spin-quantization axis $\hat{\bm{n}}_q$ is placed at a distance $z_0$ for a two dimensional superconducting sample in a perpendicular magnetic field $\bm{H} = H \hat{z}$. Close to the critical temperature $T_c$, the magnetic noise is dominated by critical Cooper-pair fluctuations. Below $T_c$, qualitatively new sources of noise arise in the form of vortex fluctuations: these vortices may be mobile in a liquid phase or freeze into a vortex lattice (or a glass) at lower temperatures.
• Figure 2: (a)/(c) Rescaled transverse noise on the metallic/superconducting side $\mathcal{N}_{\rm m/sc}/\mathcal{N}_{\rm ref}$ at $H=0$ as a function of $t=T-T_c$; the dashed line indicates the scaling $\sim 1/t$. (b)/(d) Rescaled transverse noise as a function of probe distance $z_0$ for different $t$; the dashed line indicates the scaling $\sim 1/z_0^{2}$. In each panel, the reference noise is $\mathcal{N}_{\rm ref}=\mathcal{N}_{\rm m/sc}(1\,\text{K},\,50\,\text{nm})$. The coherence length at zero temperature is chosen as $\xi_0 = 1.6$ nm, $\xi_{\rm sc}(t)=\xi_0\sqrt{T_c/|t|}$, $\xi_{\rm m} = \sqrt{2}\xi_{\rm sc}$, and transition temperature $T_c = 92.75~$K, which are typical values for high $T_c$ cuprates.
• Figure 3: Left inset (schematic): As a magnetic field $H$ is turned on, the peak of the noise $\mathcal{N}_H(T)$ moves to a lower temperature and increases in magnitude in the weak-field regime. Main figure: The relative change in noise $\mathcal{N}_H(\tilde{z}, s) / \mathcal{N}_{\rm m}(\tilde{z}) - 1$ as a function of $t = T - T_c(H)$, with $(\tilde{z},s)= (z_0/\xi_H,\ell/\xi_H)$ and $T_c(H) = T_c(0)\sqrt{1 - H/H_{c2}(0)}$. The zero-field reference noise $\mathcal{N}_{\rm m}(\tilde{z} = z_0/\xi_{\rm m})$ is evaluated using $\xi_{\rm m} = \xi_0 \sqrt{2 T_c/t}$ with $\xi_0 = 1.6~\rm{nm}$ and $z_0 = 200~\rm{nm}$, while the field-dependent noise $\mathcal{N}_H$ is computed from Eq. \ref{['eq:NB_int']}, using the correlation length $\xi_H$ and relaxation time $\tau_H$ given by Eq. \ref{['eq:tHxiH']}. As $T$ approaches $T_c(H)$, the relative noise initially exhibits a $1/t$ scaling, but saturates as $t \to 0$. Right inset: at weak fields, the relative increase in noise scales as $1/s^2 \propto H$, i.e., linearly with the applied field.
• Figure 4: Schematics of magnetic noise due to thermal fluctuations of a single pinned vortex in various regimes. (a) Langevin dynamics of a single line vortex, showing the different forces acting on it BlatterRMP. (b) Transverse noise response $\mathcal{N}_{xx}(\Omega)$ for a given $k_z$ mode. In the weak damping limit, the response peaks at $\Omega_c \approx \overline{K}(k_z) / (n_sh)$ for a massless vortex, and at $\Omega_l=(\overline{K}(k_z) / \mu_v)^{\frac{1}{2}}$ for a massive vortex. The linewidth of the peak decreases for heavier vortex mass. Inset shows the resonant peaks in $\mathcal{N}_{xx}$ at ($\Omega_{l,1},\Omega_{l,2},\ldots,\Omega_{l,n}$), expected when multiple $k_z$ modes are thermally populated. (c) Longitudinal noise response $\mathcal{N}_{zz}(\Omega)$: the response peaks at $2\Omega_c$ for a massless vortex and peaks at $\Omega = 0$ and $2\Omega_l$ for a massive vortex.
• Figure 5: Transverse (longitudinal) noise $\mathcal{N}_{xx}$ ($\mathcal{N}_{zz}$) as a function of qubit-sample distance $z_0$ for a single fluctuating vortex. (a), (b) Characteristic noise from a Pearl (point) vortex, with asymptotes following Eq. \ref{['eq:N2DpinnedNFLim']} and Eq. \ref{['eq:N2DpinnedFFLim']}. (c), (d) Characteristic noise from an Abrikosov (line) vortex, with asymptotes following Eq. \ref{['eq:N3DpinnedNFLim']} and Eq. \ref{['eq:N3DpinnedFFLim']}. In each panel, the reference noise is chosen as $\mathcal{N}_{\rm ref}=\mathcal{N}(z_0 = \lambda_{\rm scr})$.