Evolution of $1/f$ Flux Noise in Superconducting Qubits with Weak Magnetic Fields

The microscopic description of $1/f$ magnetic flux noise in superconducting circuits has remained an open question for several decades despite extensive experimental and theoretical investigation. Recent progress in superconducting devices for quantum information has highlighted the need to mitigate sources of qubit decoherence, driving a renewed interest in understanding the underlying noise mechanism(s). Though a consensus has emerged attributing flux noise to surface spins, their identity and interaction mechanisms remain unclear, prompting further study. Here, we apply weak in-plane magnetic fields to a capacitively shunted flux qubit (where the Zeeman splitting of surface spins lies below the device temperature) and study the flux-noise-limited qubit dephasing, revealing previously unexplored trends that may shed light on the dynamics behind the emergent $1/f$ noise. Notably, we observe an enhancement (suppression) of the spin-echo (Ramsey) pure-dephasing time in fields up to $B=100\mathrm{G}$. With direct noise spectroscopy, we further observe a transition from a $1/f$ to approximately Lorentzian frequency dependence below 10 Hz and a reduction of the noise above 1 MHz with increasing magnetic field. We suggest that these trends are qualitatively consistent with an increase of spin cluster sizes with magnetic field. These results should help to inform a complete microscopic theory of $1/f$ flux noise in superconducting circuits.

Figure 1

Flux qubit in a magnetic field. A simplified schematic of the experimental setup. The Josephson-junction (JJ) loop (gray) of a flux qubit is placed in a uniform magnetic field provided by Helmholtz coils (blue). The field is oriented in the plane of the device, and the device is tilted at a 45° in-plane angle relative to the field. Surface spins (red) in proximity to the loop generate flux noise that dephases the qubit. The inset shows an example flux qubit spectrum, with frequency $f_{01}$ as a function of an independent flux bias $\mathrm{\Phi }$. The blue diamond indicates the point of first-order flux insensitivity (the so-called “sweet spot”). The red circle highlights an example operating point where the qubit displays flux-noise-limited dephasing.

Figure 2

Evolution of qubit coherence with an in-plane magnetic field. Data taken at the sweet spot ($\partial f_{01}/\partial \mathrm{\Phi }=0$, blue diamonds) and off the sweet spot ($|\partial f_{01}/\partial \mathrm{\Phi }|=26.0\mathrm{GHz}/{\mathrm{\Phi }}_0$, red circles). (a) Energy relaxation rate ${\mathrm{\Gamma }}_1$. (b) Ramsey pure-dephasing rate ${\mathrm{\Gamma }}_{\varphi }^R$. (c) Spin-echo pure dephasing rate ${\mathrm{\Gamma }}_{\varphi }^E$. Insets in (b),(c) show dephasing rates at the sweet spot. Data was taken during nine field sweeps, with ${\mathrm{\Gamma }}_1$, ${\mathrm{\Gamma }}_{\varphi }^R$, and ${\mathrm{\Gamma }}_{\varphi }^E$ measured consecutively at each bias point and field. Individual rate measurements are presented as partially transparent small markers with error bars given by the fit uncertainty. Average rates at each field are presented as large opaque markers. The outlier dephasing at $B=40\mathrm{G}$ is likely dominated by noise in the applied field (see Supplemental Material [40] for details).

Figure 3

Evolution of flux noise with an in-plane magnetic field. (a) Low-frequency noise spectroscopy taken with single-shot Ramsey measurements. Data for $B<100\mathrm{G}$ were taken in one upward sweep with $|\partial f_{01}/\partial \mathrm{\Phi }|=22.0\mathrm{GHz}/{\mathrm{\Phi }}_0$, and data at $B=100\mathrm{G}$ was taken in a separate upwards sweep with $|\partial f_{01}/\partial \mathrm{\Phi }|=21.0\mathrm{GHz}/{\mathrm{\Phi }}_0$. Gray dash-dotted lines serve as guides to the eye displaying power laws $1/f^{0.88}$ (bottom) and $1/f^2$ (top, characteristic of a Lorentzian roll-off). The $B=0\mathrm{G}$ data is fit to a $1/f$ + white noise model (purple, dashed line), and data at each nonzero field is fit to a $\mathrm{Lorentzian}+\text{white}$ noise model (solid line, color of corresponding data). We attribute the white noise floor to readout infidelity. For further details, see Supplemental Material [40]. (b) Spin-locking noise spectroscopy. Data was taken in four separate field sweeps with $|\partial f_{01}/\partial \mathrm{\Phi }|=30.0\mathrm{GHz}/{\mathrm{\Phi }}_0$ for $B\lesssim 10\mathrm{G}$ and $|\partial f_{01}/\partial \mathrm{\Phi }|=31.0\mathrm{GHz}/{\mathrm{\Phi }}_0$ for $B\gtrsim 20\mathrm{G}$. Individual measurements are presented as partially transparent small markers with error bars given by the spin-locking decay fit uncertainty. Averages at each field are presented with opaque markers. Gray dash-dotted lines serve as guides to the eye displaying the power laws $1/f^{\alpha }$ with $\alpha =0.88\pm 0.02$ (top, $\alpha$ from fit to $B=0\mathrm{G}$ data), $\alpha =1.07\pm 0.02$ (middle, from fit to $B=20\mathrm{G}$ data), and $\alpha =1.5$ (bottom, characteristic of the asymptotic behavior of spin diffusion noise). At higher fields, we note a suppression of the measured flux noise, denoted by an annotated black arrow.