Interacting defects generate stochastic fluctuations in superconducting qubits

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Interacting defects generate stochastic fluctuations in superconducting qubits

J. H. Béjanin, C. T. Earnest, A. S. Sharafeldin, and M. Mariantoni

Phys. Rev. B 104, 094106 – Published 21 September 2021

Abstract

Amorphous dielectric materials have been known to host two-level systems (TLSs) for more than four decades. Recent developments on superconducting resonators and qubits enable detailed studies on the physics of TLSs. In particular, measuring the loss of a device over long time periods (a few days) allows us to investigate stochastic fluctuations due to the interaction between TLSs. We measure the energy relaxation time of a frequency-tunable planar superconducting qubit over time and frequency. The experiments show a variety of stochastic patterns that we are able to explain by means of extensive simulations. The model used in our simulations assumes a qubit interacting with high-frequency TLSs, which, in turn, interact with thermally activated low-frequency TLSs. Our simulations match the experiments and suggest the density of low-frequency TLSs is about three orders of magnitude larger than that of high-frequency ones.

Received 29 June 2021
Revised 18 August 2021
Accepted 1 September 2021

DOI:https://doi.org/10.1103/PhysRevB.104.094106

Physics Subject Headings (PhySH)

Crystal defects Impurities Quantum computation

Amorphous materials Josephson junctions Superconducting qubits

Monte Carlo methods

Condensed Matter, Materials & Applied PhysicsQuantum Information, Science & Technology

Authors & Affiliations

J. H. Béjanin , C. T. Earnest, A. S. Sharafeldin^*, and M. Mariantoni ^†

Institute for Quantum Computing, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada N2L 3G1 and Department of Physics and Astronomy, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada N2L 3G1

^*Present address: Department of Engineering Physics, McMaster University, Hamilton, Ontario, Canada L8S 4L7.
^†Corresponding author: matteo.mariantoni@uwaterloo.ca

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Vol. 104, Iss. 9 — 1 September 2021

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Images

Figure 1
Experimental [(a), (b), and (c); data sets $1, 2$ and 3, respectively, in Table 1] and simulated [(d), (e), and (f)] spectrotemporal charts of $T_{1}$ vs $f_{q}$ and $t$ , where the panels in each column display an experiment and the corresponding simulation. Spectral-diffusion patterns in the experiments are highlighted with boxes. Band-limited diffusive: dashed purple boxes. Fast narrow-band telegraphic: solid orange boxes. Slow wide-band telegraphic: dashed-dotted red boxes. In the simulations, we add a background time series of Gaussian white noise with a standard deviation of $2 k Hz$ , which is comparable to the fitting error of our $T_{1}$ experiments.
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Figure 2
Three spectral-diffusion patterns. (a) Q-TLS frequency $f_{Q-TLS}$ vs $t$ for Q-TLS 1 (left purple line), 2 (middle orange line), and 3 (right red line). (b), (c) Simulated spectrotemporal charts of $T_{1}$ vs $f_{q}$ and $t$ for $g = 50$ and $100 k Hz$ , respectively. The color map for $T_{1}$ is the same as in Fig. 1.
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Figure 3
Comparison between the statistical analyses of two simulated time series. (a) Simulated time series of $T_{1}$ vs $t$ . The series for $M_{T-TLS} = 1$ is vertically offset by 40 $μ s$ for clarity. (b) Estimated overlapping AD $σ$ vs $τ$ and associated fitting curves from Eq. (16). We find $A_{0} = 5.84 (5)$ and 4.98(6) $μ s$ , $h_{0} = 749 (111)$ and 583(119) $μ s^{2} {Hz}^{- 1}$ , and $1 / τ_{0} = 195 (7)$ and 195(10) $μ Hz$ for $M_{T-TLS} = 1$ and 4, respectively. Note that we are fitting $σ^{2}$ with the Levenberg-Marquardt algorithm but plotting $σ$ . The overlapping AD is computed at logarithmically spaced points. (c) Estimated PSD $S$ vs $f$ . We use the fitting parameters from (b) to overlay the model of Eq. (17) to the data. The PSD is estimated using the Welch's method with $25 h$ overlapping segments (rectangular window). The value of $τ_{0}$ fitted for $M_{T-TLS} = 1$ matches (within the confidence interval) that chosen in the simulations and reported in Table 3; the fitted $τ_{0}$ for $M_{T-TLS} = 4$ , instead, does not match any of the values in Table 3.
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Figure 4
Scatter plot of $T_{1}$ vs $f_{q}$ and $t$ for data set 2 [same data set as in Fig. 1] at the actual measurement time; the color map for $T_{1}$ is the same as in Fig. 1. Note that the vertical axis is truncated at $t = 5 h$ to display the relative measurement times more clearly.
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Figure 5
Time series of $T_{1}$ vs $t$ showing the relative time between measurements for data set 1 (blue dots) and 3 (yellow triangles) [same data sets as in Figs. 1 and 1] and an additional data set (green squares); all data sets are for $f_{q} = 4.529 G Hz$ .
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Figure 6
Qubit electric field $| | {\vec{E}}_{q} | |$ for $ϕ_{0} = 1 V$ vs width $x$ at one value of the height $z = 1.5 n m$ . The origin of the graph is at $x = 0$ , corresponding to the middle point of the strip. Due to the symmetry of the CPW segment with respect to its longitudinal axis (i.e., the $y$ axis; not shown), we display $| | {\vec{E}}_{q} (x) | |$ only for half of the CPW segment, for $x \geq 0$ . The extent of the conducting sections of the CPW is indicated by the thick blue lines. The dashed black vertical lines are placed at the edge of each conductor; the left line corresponds to the edge of the strip and the right line to the edge of the ground plane.
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