Observation of Classical-Quantum Crossover of $1/f$ Flux Noise and Its Paramagnetic Temperature Dependence

By analyzing the dissipative dynamics of a tunable gap flux qubit, we extract both sides of its two-sided environmental flux noise spectral density over a range of frequencies around $2k_{B}T/h\approx 1\mathrm{GHz}$, allowing for the observation of a classical-quantum crossover. Below the crossover point, the symmetric noise component follows a $1/f$ power law that matches the magnitude of the $1/f$ noise near 1 Hz. The antisymmetric component displays a $1/T$ dependence below 100 mK, providing dynamical evidence for a paramagnetic environment. Extrapolating the two-sided spectrum predicts the linewidth and reorganization energy of incoherent resonant tunneling between flux qubit wells.

Figure 1

(a) Optical micrograph of Al-sapphire fluxmon qubit and bias lines, inductively coupled to a CPW readout resonator. The main qubit loop and bias utilize a gradiometric CPW current (arrows). (b) Four stages of measurement, involving double-well projection followed by dispersive readout. (c) Qubit spectroscopy vs frequency and ${\mathrm{\Phi }}_t^x$ for two values of $\mathrm{\Delta }({\mathrm{\Phi }}_{\mathrm{ba}}^x)$.

Figure 2

(a) $T_1$ vs $f_{10}$ at zero tilt measured with swap spectroscopy. Inset illustrates the pulse sequence. (b) $p_{\text{stray}}$ vs $f_{10}$ at zero tilt. Inset: $f_{10}$ at zero tilt vs barrier bias. (c) $S_{\mathrm{\Phi }}^{\pm }(f)$ extracted from (a) and (b) and numerical evaluation of $⟨0|\widehat{\mathrm{\Phi }}|1⟩$. (d) Effective temperature $T_{\text{eff}}$ of the data in (b). (e) Low-frequency quasistatic flux noise [25] together with $S_{\mathrm{\Phi }}^{+}(f)$ from (c) on the same axes.

Figure 3

Temperature dependence of $S_{\mathrm{\Phi }}^{\pm }$. $S_{\mathrm{\Phi }}^{-}(f=500\mathrm{MHz})$ shows a $1/(T_{\text{eff}}+T_0)$ dependence, as explicitly plotted in the inset.

Figure 4

(a) Pulse sequence for the MRT experiment. (b) MRT data with $\mathrm{\Delta }/h\approx 0.5\mathrm{MHz}$, showing a small offset in the tilt bias that maximizes tunneling, consistent with the measured $S_{\mathrm{\Phi }}^{-}$.