Flux noise in disordered spin systems

Impurity spins randomly distributed at the surfaces and interfaces of superconducting wires are known to cause flux noise in superconducting quantum interference devices (SQUIDs), providing a dominant mechanism for decoherence in flux-tunable superconducting qubits. While flux noise is well characterized experimentally, the microscopic model underlying spin dynamics remains a great puzzle. The main problem is that first-principles theories based on an integration of the quantum Heisenberg equations of motion for interacting spins are too computationally expensive to capture spin diffusion over large length scales, hindering comparisons between microscopic models and experimental data. In contrast, third-principles approaches lump spin dynamics into a single phenomenological spin-diffusion operator $D{\ensuremath{\nabla}}^{2}$ that is not able to describe the quantum noise regime and connect to microscopic models and different disorder scenarios such as spin clusters. Here we propose an intermediate ``second-principles'' method to describe general spin dissipation and flux noise in the quantum regime. It leads to the interpretation that flux noise arises from the density of paramagnon excitations at the edge of the superconducting wire, with paramagnon-paramagnon interactions leading to spin diffusion, and interactions between paramagnons and other degrees of freedom such as phonons, electrons, and two-level systems leading to spin energy relaxation. At high frequency $\ensuremath{\omega}$, we obtain an upper bound for flux noise, showing that the (super)Ohmic noise observed in experiments is not originating from interacting spin impurities. We apply the method to Heisenberg models in two-dimensional square lattices with a random distribution of vacancies, with nearest-neighbor spins coupled by a constant exchange. Explicit numerical calculations of flux noise show that it follows the observed power law $A/{\ensuremath{\omega}}^{\ensuremath{\alpha}}$, with amplitude $A$ and exponent $\ensuremath{\alpha}$ depending on temperature and inhomogeneities such as spatial confinement and disorder. These results are compared to experiments in niobium and aluminum devices. The method establishes a connection between flux noise experiments and microscopic Hamiltonians with the goal of identifying relevant microscopic mechanisms and guiding strategies for reducing flux noise.

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