Low-temperature $1/f$ noise in microwave dielectric constant of amorphous dielectrics in Josephson qubits

The analytical solution for the low-temperature $1/f$ noise in the microwave dielectric constant of amorphous films at frequency ${\nu }_0\sim 5\mathrm{GHz}$ due to tunneling two-level systems (TLSs) is derived within the standard tunneling model including the weak dipolar or elastic TLS-TLS interactions. The $1/f$ frequency dependence is caused by TLS spectral diffusion characterized by the width growing logarithmically with time. Temperature and field dependencies are predicted for the noise spectral density in typical glasses with universal TLSs. The satisfactory interpretation of the recent experiment by J. Burnett et al. [Nat. Commun. 5, 4119 (2014)] in Pt-capped Nb superconducting resonators is attained by assuming a smaller density of TLSs compared to ordinary glasses, which is consistent with the very high internal quality factor in those samples.

Figure 1

Tunneling two-level system (TLS) characterized by well asymmetry $\mathrm{\Delta }$ and tunneling amplitude ${\mathrm{\Delta }}_0$.

Figure 2

Frequency dependence of the noise parameter ${\alpha }_{\mathrm{TLS}}$ for typical glasses with $\chi =5\times {10}^{-4}$ (red) and for glasses with a smaller density of TLSs, with $\chi ={10}^{-6}$ (blue). The curves with diamond markers describe the numerical evaluation of the integral in Eq. (18) with $T=0.1\mathrm{K},\omega =2\pi \times 5\mathrm{GHz},p=5\mathrm{D}$, and $A={10}^8{\mathrm{s}}^{-1}{\mathrm{K}}^{-3}$. Dashed lines are fit to ${\alpha }_{\mathrm{TLS}}\propto f^{\beta }$.

Figure 3

Temperature dependence of the noise parameter ${\alpha }_{\mathrm{TLS}}$ in typical glasses $\left(\chi =5\times {10}^{-4}\right)$ at frequency $f=0.1\mathrm{Hz}$ for various electric fields. Other parameters are the same as in Fig. 2. Dashed line is a fit of the zero-field curve to ${\alpha }_{\mathrm{TLS}}\propto T^{-\eta }$. The values of the field $F_{\mathrm{ac}}$ are in units of $\mathrm{V}/\mathrm{m}$.

Figure 4

Temperature dependence of the noise power spectral density $S_y$ in typical glasses $\left(\chi =5\times {10}^{-4}\right)$ at frequency $f=0.1\mathrm{Hz}$ for various electric fields.

Figure 5

Temperature dependence of $S_y/{tanh}^2(\hslash {\omega }_0/2k_{\mathrm{B}}T)$ for the noise power spectral density $S_y$ obtained in a Pt-capped Nb resonator at resonance frequency ${\nu }_0=5.55\mathrm{GHz}$ and at the smallest measuring field [16].

Figure 6

Fit of the experimental data by Burnett et al. [16] at resonance frequency ${\nu }_0=5.55\mathrm{GHz}$ (markers) to Eq. (29) (solid lines).

Figure 7

Fit of the experimental data by Burnett et al. [16] at resonance frequency ${\nu }_0=6.68\mathrm{GHz}$ (markers) to Eq. (29) (solid lines).