Applying weak in-plane magnetic fields (B≤100B\le 100 G, Zeeman energy below the device temperature) to a superconducting flux qubit transforms its 1/f1/f magnetic flux noise: below ∼10\sim 10 Hz the spectrum crosses over from 1/f1/f to approximately Lorentzian, above ∼1\sim 1 MHz the noise is suppressed and its exponent grows, and several features are non-monotonic in field (Rower et al., Phys. Rev. Lett. 130, 220602 (2023)). The open problem is to determine which microscopic mechanism(s) of the surface spin bath — growth of interacting spin clusters, inhomogeneous broadening, partial spin polarization, independent spin relaxation, or a combination — produce these trends, to build a quantitative microscopic model that accounts for all of them simultaneously (including the non-monotonic behavior and the observed Ramsey beating), and thereby to constrain the electronic and chemical identity of the magnetic defects responsible for flux noise.

Background

Superconducting quantum circuits such as flux qubits and SQUIDs (superconducting quantum interference devices — superconducting loops interrupted by Josephson junctions, whose interference response makes them exquisitely sensitive to magnetic flux) are limited by an excess magnetic-flux noise with a power spectral density (PSD) approximately of the 1/f1/f form, i.e. SΦ(f)∝1/fS_\Phi(f)\propto 1/f, that has persisted for decades despite extensive study. A broad consensus attributes this noise to a bath of weakly interacting microscopic magnetic moments (spins) residing on the device surfaces and interfaces, at areal densities of order 1017 m−210^{17}\ \mathrm{m}^{-2}; however, neither the chemical/electronic identity of these defects nor the dynamics by which they generate 1/f1/f noise is established. Phenomenology that any microscopic account must reproduce includes: noise amplitude scaling with the total length of superconducting wiring rather than loop area; a paramagnetic temperature dependence; spin diffusion (transport of spin magnetization by mutual spin flip-flops) proposed as the source of the high-frequency part of the spectrum; and suggestions that interacting surface spins sit near a magnetic phase transition, forming clusters (groups of mutually interacting spins that can behave as effective macrospins with slow, telegraphic dynamics — random switching between discrete states whose PSD is Lorentzian, with an ensemble of widely distributed switching rates generating 1/f1/f noise).

The qubit itself acts as a noise spectrometer. In a Ramsey experiment (free precession followed by a measurement, sensitive to slow frequency fluctuations) the pure-dephasing rate measures low-frequency noise; a spin-echo sequence (Ramsey with a refocusing pulse in the middle, filtering slower fluctuations) measures higher-frequency noise; and dynamical-decoupling sequences (repeated refocusing pulses at set intervals) and spin-locking (continuously driving the qubit so that its precession is locked to a rotating frame, where it senses noise near the drive frequency) extend spectroscopy to the MHz range. In 2023, Rower et al. performed the first systematic study of flux-noise-limited dephasing and flux noise as a function of applied in-plane field, up to B=100B=100 G, on a capacitively shunted flux qubit (a superconducting loop whose two lowest states form an artificial atom whose transition frequency is tunable by flux). Because the electron Zeeman energy at 100100 G (about 280280 MHz) is below the thermal scale kBTeff/hk_BT_\mathrm{eff}/h (at least about 800800 MHz at the ≈40\approx 40 mK mixing-chamber temperature), the spins should be largely unpolarized — so any field dependence reflects changes in bath dynamics rather than simple spin freezing. They observed: (i) at low frequencies (≲10\lesssim 10 Hz), increasing noise and a 1/f→1/f\to approximately Lorentzian transition for B≳20B\gtrsim 20 G, with the Lorentzian cutoff frequency peaking near B≈80B\approx 80 G and the noise apparently decreasing from 8080 G to 100100 G; (ii) at high frequencies (≳1\gtrsim 1 MHz), noise suppression with an increasing exponent α\alpha in 1/fα1/f^\alpha; (iii) beating in Ramsey decays for 50 G≲B≲9050\ \mathrm{G}\lesssim B\lesssim 90 G, suggestive of discrete telegraph fluctuators. The authors showed the trends are qualitatively consistent with growing spin clusters and with inhomogeneous broadening (a spread of spin resonance frequencies, e.g. from spatially nonuniform Meissner screening of the applied field, that inhibits resonant flip-flops and hence slows spin diffusion), and argued thermal polarization alone cannot explain them; they placed only a rough bound on the spin-spin interaction strength, J/h≲150J/h\lesssim 150 MHz. They explicitly left the microscopic mechanism, the non-monotonic features, the relation between the Ramsey beating and the Lorentzian spectra, the material and field-angle dependence, and the identity of the defects as open questions. They also list follow-up measurements that could adjudicate these mechanisms and help constrain the identity of the defects — probing higher fields and field-resilient metallizations (Nb, granular Al, thinner Al), mapping the field-angle dependence, and searching for field-dependent resonant peaks at ≳10\gtrsim 10 MHz; spin polarization has so far been established as the dominant effect only in the high-field regime, where the Zeeman energy exceeds kBTk_BT.

Problem Statement

Determine the microscopic mechanism(s) by which weak applied magnetic fields (in-plane fields up to ∼100\sim 100 G, where the electron Zeeman energy lies below the thermal energy kBTeffk_BT_\mathrm{eff}) reshape the 1/f1/f flux noise of superconducting qubits, and establish a quantitative microscopic model of the surface spin bath that simultaneously accounts for the observed trends: the low-frequency (≲10\lesssim 10 Hz) crossover from 1/f1/f to approximately Lorentzian spectra for B≳20B\gtrsim 20 G; the suppression of noise above ∼1\sim 1 MHz and the growth of the exponent α\alpha in 1/fα1/f^\alpha; the non-monotonic field dependence (Lorentzian cutoff frequency and amplitude peaking near B≈80B\approx 80 G and the apparent noise decrease from B=80B=80 G to B=100B=100 G); and the beating observed in Ramsey decays for 50 G≲B≲9050\ \mathrm{G}\lesssim B\lesssim 90 G and whether it arises from the same discrete telegraph fluctuators as the Lorentzian spectra. The common objective is a validated microscopic attribution: the answer must determine which microscopic mechanism(s) — among those proposed, including growth of interacting spin clusters, inhomogeneous broadening of the spin bath, partial spin polarization, and independent-spin-flip relaxation with direct (phonon-emission) and cross (two-level-system-mediated) channels (Ref. 2), or any other mechanism consistent with the data — reproduce the observed trends, and derive from the data quantitative bath parameters appropriate to the prevailing mechanism(s) (e.g. spin-spin interaction strengths, cluster-size distributions, spin diffusion coefficients) that constrain the electronic and chemical identity of the magnetic defects. A valid solution must name the prevailing mechanism(s) and support the attribution with a quantitative microscopic model that reproduces all of the trends listed above simultaneously — within tolerances the solution states explicitly — and must settle whether the Ramsey beating shares the discrete telegraph fluctuators responsible for the Lorentzian spectra; matching any one of these trends alone is at most a partial result.

References

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